# Enhancing Performance on Seen and Unseen Dialogue Scenarios using Retrieval-
Augmented End-to-End Task-Oriented System

Jianguo Zhang1111This work was partially conducted during Jianguo’s internship
and Stephen’s full-time employment at Meta AI Research (FAIR). Stephen Roller2
Kun Qian3 Zhiwei Liu1 Rui Meng1
Shelby Heinecke1 Huan Wang1 Silvio Savarese1 Caiming Xiong1
1Salesforce AI 2Character.AI 3Columbia University
<EMAIL_ADDRESS>
###### Abstract

End-to-end task-oriented dialogue (TOD) systems have achieved promising
performance by leveraging sophisticated natural language understanding and
natural language generation capabilities of pre-trained models. This work
enables the TOD systems with more flexibility through a simple cache. The
cache provides the flexibility to dynamically update the TOD systems and
handle both existing and unseen dialogue scenarios. Towards this end, we first
fine-tune a retrieval module to effectively retrieve the most relevant
information entries from the cache. We then train end-to-end TOD models that
can refer to and ground on both dialogue history and retrieved information
during TOD generation. The cache is straightforward to construct, and the
backbone models of TOD systems are compatible with existing pre-trained
generative models. Extensive experiments demonstrate the superior performance
of our framework, with a notable improvement in non-empty joint goal accuracy
by $6.7\%$ compared to strong baselines.

##  1 Introduction

Task-oriented dialogue (TOD) systems play an important role in various
applications, such as restaurant booking, alarm setting, and recommendations
Gao et al. (2018); Xie et al. (2022). These systems can be broadly categorized
into two groups: pipeline-based dialogue systems and end-to-end dialogue
systems. Pipeline-based dialogue systems consist of four separate modules: a
natural language understanding (NLU) module to detect user intents, a dialogue
state tracking (DST) module to track user belief states across dialogue turns,
a dialogue management (DM) module to decide system actions based on dialogue
states, and a natural language generation (NLG) module to generate natural-
language responses. However, the pipeline-based approach is annotation-
intensive, prone to error propagation, and challenging to scale Hosseini-Asl
et al. (2020); Zhang et al. (2020); Feng et al. (2023).

Figure 1: An example of the auto-regressive TOD with retrieved slot
information from cache. The APICALL generation process is shown, with $N$ set
to 3 for the retrieval module.

Recently, various approaches have been proposed to utilize sequence-to-
sequence models for generating dialogue states and responses in an end-to-end
manner Ham et al. (2020); Lin et al. (2020); Yang et al. (2021); Gao et al.
(2021); Chen et al. (2021); Peng et al. (2021); Liu et al. (2021); He et al.
(2022b); Feng et al. (2023). Compared with the pipeline-based systems, these
approaches demonstrate effectiveness on public datasets requiring fewer direct
annotations, such as user intents and dialogue acts. Additionally, they
leverage the capabilities of large-scale pre-trained language models, such as
GPT-2 Radford et al. (2019), T5 Raffel et al. (2019), and BART Lewis et al.
(2020a), for improved performance in NLU and NLG tasks. However, these
approaches are limited in their ability to dynamically handle both existing
and emerging intents and slots, particularly in the context of unseen dialogue
scenarios such as new domains and services Hosseini-Asl et al. (2020); Peng et
al. (2021); Rastogi et al. (2020a).

Simultaneously, research on open-domain question answering and dialogue
systems has explored the use of retrieval-augmented models. These models
retrieve relevant information from a passage, database, APIs, etc., and
incorporate it into the generation process, improving answer quality or
dialogue responses Karpukhin et al. (2020); Izacard and Grave (2021); Dinan et
al. (2018); Lewis et al. (2020b); Shuster et al. (2021). Inspired by these
ideas, we combine both worlds and propose an end-to-end TOD framework with a
retrieval system that addresses the challenge of handling both existing and
zero-shot unseen dialogue scenarios.

Our approach involves training the end-to-end TOD models with a cache that
contains accessible domains, intents, slots, and APIs. The cache can be
constructed based on the schema or database, or by extracting information from
accessible dialogues when the schema or database is not fully accessible. The
cache serves as a reference point, allowing the models to ground their
responses in the retrieved information. By incorporating a retrieval module
and leveraging this cache of knowledge, our system enhances the flexibility
and adaptability to handle both existing and unseen intents and slots. It also
ensures robust performance in novel dialogue domains and services where the
model hasn’t been explicitly trained. Figure 1 shows an illustrative example
of our approach, demonstrating how the RETRIEVAL module retrieves relevant
information, such as slots in this case, from the cache to enrich the system’s
understanding and generate more accurate responses. The APICALL represents the
dialogue states from the system side, and APIRESP returns information from
external API interactions between the system and system databases.

To build an accurate end-to-end TOD system with the benefits of a simple
cache, we fine-tune a retrieval module to effectively retrieve the most
relevant and informative information from the cache, using a Top-$N$ retrieval
strategy. Then we integrate the retrieval module into the generative model to
facilitate end-to-end TOD generation. We evaluate our approach on the publicly
available Google Schema-Guided Dialogue dataset (SGD) Rastogi et al. (2020b),
which includes a significant number of unseen dialogue domains and services in
the development and test sets.

The contributions of this paper are as follows: (1) We design a simple yet
effective end-to-end TOD framework with a cache that enables dynamic handling
of intents and slots. The framework is compatible with existing pre-trained
generative models, and enhances the system’s robustness. (2) We provide
experimental results that demonstrate the superior performance of our approach
compared to strong baselines. It achieves $6.7\%$ improvement in non-empty
joint goal accuracy, demonstrating the effectiveness in handling various
dialogue scenarios, including the challenging zero-shot unseen dialogues. (3)
We conduct comprehensive ablation studies and analyses to provide insights
into the impact of different components and design choices within our
framework.

##  2 Related Work

#### End-to-End TOD Systems

End-to-end TOD models have shown promising performance on public dataset Ham
et al. (2020); Lin et al. (2020); Yang et al. (2021); Gao et al. (2021); Chen
et al. (2021); Peng et al. (2021); Liu et al. (2021); He et al. (2022a, b);
Feng et al. (2023); Bang et al. (2023). These approaches typically follow
common patterns: (1) Rely on powerful pre-trained seq2seq models. (2) Use
language modeling objectives to generate NLU and NLG outputs, sometimes
augmented with auxiliary multi-task goals like DST loss. (3) Either fine-tune
models directly on the target dataset or conduct pre-training on multiple TOD
dialogue datasets. (4) Employ data augmentation techniques such as back-
translation and entity replacement due to the challenges in collecting large-
scale TOD corpora. For example, Hosseini-Asl et al. (2020) fine-tunes
DistilGPT2 for TOD. The model generates user belief states and system
responses in an auto-regressive way. Peng et al. (2021) introduces two
auxiliary tasks for belief state prediction and grounded response generation
and pre-train language models first on multitple TOD dataset. Gao et al.
(2021) enables the belief state to interact with both structured and
unstructured knowledge. Feng et al. (2023) designs a reward-function learning
objective to guide the model’s generation. While these methods have
demonstrated effectiveness on public datasets, they have limitations in
handling unseen dialogue scenarios such as unseen domains and services.

#### Retrieval-Augmented Models

Retrieval augmented approaches have been widely used in open-domain question
answering. For instance, Karpukhin et al. (2020) proposes a BERT-based Devlin
et al. (2019) dual-encoder framework to retrieve passages from Wikipedia,
which is further incorporated into open-domain conversations to reduce
hallucination and enrich engagement with users Shuster et al. (2021); Komeili
et al. (2021). These models retrieve information related to the query from a
knowledge base of sentences and ground the generation response on this
information Dinan et al. (2018); Lewis et al. (2020b). Inspired by these
works, we explore the integration of retrieval modules into end-to-end TOD
systems, leveraging the retrieval-augmented approach to enhance the system’s
performance in handling both existing and novel dialogue scenarios.

##  3 TOD Systems with a Simple Cache

We present an end-to-end transformer-based framework with a simple cache that
is compatible with multiple generative models, including BART, T5, GPT2, etc.
Our framework enables dynamic handling of intents, slots, and APIs while
maintaining flexibility in choosing the backbone model.

Generally, our framework consists of two parts: a retrieval model for
retrieving the most relevant and informative information from the cache, and
an end-to-end TOD model that generates APICALLs and system responses based on
the dialogue history and the retrieved information. The retrieval model
functions by retrieving intents, slots, APIs, and other relevant information
from the cache.

Figure 2 illustrates one simple variant of our framework, which is an encoder-
decoder architecture. In this variant, the retrieved information such as slots
are stacked together. We also introduce another variant in Sec. 4.2, where
each retrieved information is concatenated with the dialogue history and then
all the information are concatenated together before being sent to the
decoder.

Figure 2: Illustration of the end-to-end framework with a simple cache. The
left figure illustrates the generation of an APICALL. In this process, the
retrieval module extracts relevant information, like slots, from the cache.
The decoder then uses the retrieved information, combined with the dialogue
history, to generate the APICALL. The right figure depicts the continuation of
the dialogue, generating the system response. The system retrieves additional
information from the cache and incorporates all previous information to
generate a system response. By decoupling the generation of APICALLs from
system responses, we aim to clearly represent the framework’s components and
their interactions in an end-to-end setting.

Cache Templates | Examples
---|---
INTENT: intent name, SLOT: slot name | INTENT: findrestaurants, SLOT: city
| intent name, slot name, service description,
---
intent description, slot description
| findrestaurants, city, a leading provider for restaurant search and
reservations,
---
find a restaurant of a particular cuisine in a city, city in which the
restaurant is located
API-information |  | api_name = FindRestaurants; optArg = has_live_music, price_range,
---
serves_alcohol; reqArg = city, cuisine

Table 1: Several typical templates of the simple cache construction, where
each template represents one type of cache. Some other templates can be found
in Table 4.

###  3.1 Construction of Cache

In this section, we describe the construction of a simple cache that provides
necessary information for the model’s referencing and grounding procedure. The
cache consists of intents, slots, and APIs extracted from the schema and
database. In cases where the schema or database is not fully accessible, we
extract information from accessible dialogues. During training, it is
important to note that the cache exclusively contains information relevant to
the training dialogues and does not incorporate any unseen information of
dialogues in the test set.

Since there are different ways to construct a cache, we design various
templates to formalize the retrieved information. Table 1 presents several
templates that we utilize. One example is the “API-information” template,
where an API includes all the intents and relevant slots mentioned throughout
the whole dialogue. Although this template may contain redundant information
as some intents and slots may not be mentioned initially, it allows us to
evaluate the model’s ability to disregard irrelevant details.

In addition to the listed templates, we explore several other templates with
special tokens such as “[INTENT] intent name [SLOT] slot name”, as well as
different orderings of intents and slots, such as “intent name, intent
description, slot name, slot description” and “intent name, slot name, intent
description, slot description”. We conduct an in-depth analysis of the effects
of different cache templates in the experimental section.

###  3.2 Retrieval Module

After constructing the cache, we fine-tune a retrieval model to effectively
retrieve the most relevant and informative information for the dialogue
context. Given a dialogue history $c$, the TOD system utilizes a retrieval
module to retrieve Top-$N$ most relevant information $s_{1},\ldots,s_{N}$ from
the cache. Firstly, based on the dialogue history, the system triggers the
retrieval module to generate an APICALL, which includes relevant mentioned
intents, slots and values. Subsequently, the system continues to use the
retrieval module to generate a system response based on all previous
information.

To ensure accurate retrieval from the cache, we fine-tune a dense passage
retriever (DPR) model (Karpukhin et al., 2020), which is a BERT-based dual-
encoder framework optimized via contrastive learning. Specifically, we obtain
the hidden representation $\mathbf{h}_{c}$ for the dialogue history using an
encoder model, e.g., $\mathbf{h}_{c}=\text{BERT}_{c}(c)$. Similarly, we use
another BERT encoder to obtain the feature representation $\mathbf{h}_{s}$ for
each retrieved information entry from the cache, i.e.,
$\mathbf{h}_{s}=\text{BERT}_{s}(s)$. The similarity between the dialogue
history and the retrieved information entry is:
$sim(c,s)=\mathbf{h}_{c}^{T}\odot\mathbf{h}_{s}$.

For each dialogue history, there are $n$ relevant (positive) entries and $m$
irrelevant (negative) entries, where $n$ and $m$ may vary as each dialogue
history would contain different active intents and slots. Our objective is to
learn a function that minimizes the distance between pairs of relevant
dialogue histories and information entries than the irrelevant pairs. The
corresponding loss function for a specific pair is as follows:

$\mathcal{L}_{\text{api}}(c,s_{1}^{+},s_{1}^{-},\ldots,s_{m}^{-})=-\log\frac{\exp(\text{sim}(\mathbf{h}_{c},\mathbf{h}_{s_{1}^{+}}))}{\sum_{j=1}^{m}\exp(\text{sim}(\mathbf{h}_{c},\mathbf{h}_{s_{j}^{-}}))}\,.$
(1)

Once the retrieval module is fine-tuned, it is incorporated into the end-to-
end sequence-to-sequence task-oriented dialogue generative model. The
parameters of the retrieval module remain fixed during training of the
generative model.

#### Negative Sampling

In the training process, we employ negative sampling to include retrieved
information entries that are irrelevant to the dialogue history. We utilize
both natural and hard negative pairs to enhance the robustness and performance
of the retrieval module.

For natural negative pairs, we consider pairs such as “irrelevant intent,
irrelevant slots” as counterparts to the positive pairs of “relevant intent,
relevant slots”. Additionally, we construct hard negative pairs that pose a
more challenge to the retrieval module. These hard negative pairs include
combinations such as “relevant intent, irrelevant slots from the same relevant
intent” and “irrelevant intents that are semantically similar to the relevant
intent, along with relevant slots from the relevant intent”. By incorporating
these hard negative pairs, we encourage the retrieval module to learn to
differentiate between relevant and irrelevant information effectively.

###  3.3 End-to-End TOD Systems

Our end-to-end TOD framework generates the APICALL and system response in an
auto-regressive manner. Figure 1 provides an example of this process. The
APICALL represents the dialogue states from the system side, and same with
previous work (Hosseini-Asl et al., 2020; Peng et al., 2021), it is an
intermediate step of the system response generation, and they share the same
model framework to generate tokens autoregressively.

For each dialogue turn, the TOD framework triggers the retrieval module twice.
The system first retrieves the Top-$N$ information entries from the
constructed cache, i.e.,

$\text{Top-$N$ info}=\text{Retrieval}(c)\,.$ (2)

Then it generates an APICALL using the retrieved information, i.e.,

$\text{APICALL}=\text{TOD}(c,\text{Top-$N$ info)}\,.$ (3)

After that the TOD framework retrieves another set of Top-$N$ information
entries from the cache, considering the generated APICALL, i.e.,

$\text{Top-$N$ info}=\text{Retrieval}(c,\text{APICALL},\text{APIRESP})\,,$ (4)

where APIRESP is automatically obtained from corresponding API, without the
need for prediction.

Finally, the system generates a system response using the following inputs:

$\text{Response}=\text{TOD($c$, APICALL, APIRESP, Top-$N$ slots)}\,.$ (5)

##  4 Experimental Settings

###  4.1 Dataset

A substantial number of end-to-end TOD works (Hosseini-Asl et al., 2020; Peng
et al., 2021; Lin et al., 2020; Yang et al., 2021; Su et al., 2021; He et al.,
2022b; Feng et al., 2023) commonly employ the MultiWOZ datasets (Budzianowski
et al., 2018; Zang et al., 2020). However, these studies primarily focus on
full-shot and few-shot learning, and the scope for zero-shot evaluation
appears somewhat constrained given that MultiWOZ only has five domains and
approximately $35$ slots, all of them are presented in the training set. In
contrast, our work aims to assess the system across large-scale multi-domain
dialogue scenarios. We utilize the Google Schema-Guided Dialogue (SGD) dataset
Rastogi et al. (2020c). 111SGD processed dataset. SGD provides a more
expansive dialogue landscape with over 16,000 multi-domain conversations that
span more than 16 domains, 26 services, and 200 slots. Notably, the SGD
includes a significant number of unseen domains/APIs in both the development
and test sets, thereby increasing the complexity and diversity of the dataset.
A substantial 45% of the dialogue turns in the development set, and a
significant 77% of the dialogue turns in the test set, incorporate at least
one service not included in the training set. Table 2 summarizes the
statistics of SGD.

| Dialogues | Services | ZS Services | ZS Turns
---|---|---|---|---
Train | 16142 | 26 | - | -
Dev. | 2482 | 17 | 8 | 45%
Test | 4201 | 21 | 11 | 77%

Table 2: Data Statistics of SGD. ZS: Zero-Shot.

| PPL | Overall JGA | Non-Empty JGA | Token EM | BLEU-4
---|---|---|---|---|---
MinT (BART-Large) (Chen et al., 2021) | 2.385 | 0.812 | 0.364 | 0.497 | 0.179
T5DST (Lee et al., 2022) | 2.419 | 0.810 | 0.361 | 0.491 | 0.170
FiD-TOD | 2.133 | 0.829 | 0.431 | 0.501 | 0.179

Table 3: Testing results on the SGD dataset.

Cache Templates | Top-1 | Top-2 | Top-3 | Top-4 | Top-5
---|---|---|---|---|---
INTENT: intent name, SLOT: slot name | 0.833 | 0.882 | 0.914 | 0.945 | 0.960
| INTENT: intent name, service description,
---
intent description, SLOT: slot name, slot description
0.887 | 0.922 | 0.952 | 0.976 | 0.980
intent name, slot name, intent description, slot description | 0.835 | 0.906 | 0.928 | 0.946 | 0.955
| intent name, slot name, service description,
---
intent description, slot description
0.913 | 0.943 | 0.965 | 0.977 | 0.981
API-information | 0.844 | 0.927 | 0.956 | 0.962 | 0.967

Table 4: Top-5 retrieval accuracy on the test set of SGD.

###  4.2 Models

In term of baselines, we adopt Lin et al. (2020); Chen et al. (2021) and
implement their model MinTL (BART-Large). We also implement T5DST from Lee et
al. (2022), which achieves strong performance on MultiWOZ 2.2 (Zang et al.,
2020). Since our end-to-end TOD framework is compatible with existing pre-
trained generative models, we experiment with BART, GPT2 and T5.
Interestingly, we found that BART-Large (406M) perform comparably with
T5-Large (770M), despite having fewer parameters. Moreover, it outperformed
many models developed by teams in DSTC8 (Rastogi et al., 2020a), where the
majority of models are BERT-based classification models. Thus, we select BART-
Large as our primary backbone model.

Inspired by previous model designs in open-domain question answering (Lewis et
al., 2020b; Izacard and Grave, 2021), we design two variants for end-to-end
TOD systems. The first, named Fusion-in-Decoder TOD (FiD-TOD), is illustrated
in Figure 2, In this model, the retrieved information such as slots, are
stacked together. Notably, when the retrieval model is not incorporated, FiD-
TOD becomes identical to MinTL (BART-Large). The second variant FiD-TOD-
NoStack, is depicted in Figure 3 and is used as ablation study. In this model,
the retrieved information is not directly stacked, and instead, the dialogue
history is concatenated with each retrieved information entry and then sent to
the shared encoder.

Regarding the generative model, we truncate the tokens of dialogue history to
256, and retrieve Top-5 most relevant information entries from the cache,
unless otherwise specified. For DPR fine-tuning, we align one hard negative
pair to each positive pair. We employ the preset hyperparameters from the
ParLAI code, 222ParLAI platform. such as setting the learning rate to 5e-5,
batch size to 32, etc. Initially, we conducted experiments with slight
alterations in hyperparameters and observed no statistically significant
difference on performance. We selected the best model based on its performance
on the development set.

The retrieve model is fine-tuned up to 3 epochs based on open-sourced DPR
(Karpukhin et al., 2020), and the generative model is fine-tuned up to 4
epochs with an overall batch size of $64$ on $8$ Nvidia Tesla V100 GPUs. All
experiments are based on public code from the ParLAI platform.

Figure 3: Illustration of FiD-TOD-NoStack framework.

###  4.3 Evaluation Metrics

We evaluate the end-en-end TOD framework using the following ParLAI metrics:
(1) Top-$N$ accuracy: It evaluates the retrieval module through checking
whether the ground-truth slot appears in the Top-$N$ predicted candidates
Karpukhin et al. (2020). (2) Joint Goal Accuracy (Overall JGA): It evaluates
whether the predicted APICALL on both seen and unseen services is correct or
not, specifically. JGA is $1$ if the model correctly predicts all intent,
slots and corresponding values in the APICALL. Otherwise, JGA is $0$. (3) Non-
Empty JGA: It evaluates whether overall JGA is correct if the model calls the
API on both seen and unseen scenarios. In SGD, most dialogue turns would not
trigger an API retrieval, resulting in empty APICALLs, and identifying Empty
JGA is relatively quite easy (Chen et al., 2021). Moreover, most services,
intents and slots in the test set are unseen. Therefore, we focus on Non-Empty
APICALL turns and treat it as the most crucial metric for evaluating the
model’s performance on both seen scenarios and its zero-shot generalization
ability on unseen scenarios. (4) Token EM: It evaluates the utterance-level
token accuracy. Roughly corresponds to perfection under greedy search
(generative only). (5) Perplexity (PPL): It measures the generative model’s
ability to predict individual tokens. (6) BLEU-4: It measures the BLEU score
Papineni et al. (2002) between the predicted system response and the reference
response.

| PPL | Overall JGA | Non-Empty JGA | Token EM | BLEU-4
---|---|---|---|---|---
MinT (BART-Large) (Chen et al., 2021) | 1.700 | 0.876 | 0.586 | 0.538 | 0.221
INTENT: intent name, SLOT: slot name | 1.688 | 0.889 | 0.633 | 0.538 | 0.212
intent name, slot name, intent description, slot description | 1.679 | 0.895 | 0.661 | 0.541 | 0.215
| intent name, slot name, service description,
---
intent description, slot description
1.679 | 0.894 | 0.649 | 0.541 | 0.212
| INTENT: intent name, service description,
---
intent description, SLOT: slot name, slot description
1.676 | 0.897 | 0.660 | 0.545 | 0.217

Table 5: Performance of FiD-TOD on the development set with variations of
cache templates.

| PPL | Overall JGA | Non-Empty | Token EM | BLEU-4
---|---|---|---|---|---
MinT (BART-Large) (Chen et al., 2021) | 1.700 | 0.876 | 0.586 | 0.538 | 0.221
FiD-TOD w/ API-information (N=1) | 1.653 | 0.896 | 0.658 | 0.543 | 0.218
FiD-TOD w/ API-information (N=5) | 1.655 | 0.897 | 0.663 | 0.544 | 0.219
FiD-TOD-NoStack | 1.683 | 0.895 | 0.653 | 0.543 | 0.215
FiD-TOD | 1.676 | 0.897 | 0.660 | 0.545 | 0.217

Table 6: Results on development set. By default, retrieval module retrieves
Top-5 information entries from cache.

##  5 Experimental Results

###  5.1 End-to-End TOD Performance

Table 3 shows the overall performance on the test set. FiD-TOD outperforms
baselines across most metrics. Specifically, it improves the essential NLU
metric, i.e., Non-Empty JGA, by $6.7\%$. This demonstrates the model’s
enhanced capability in handling both seen dialogue scenarios and, notably, its
capacity for zero-shot handling of unseen scenarios. As MinT (BART-Large)
corresponds to the FiD-TOD without the retrieval model from the cache, this
comparison highlights the significant benefits that our design brings to the
handling of unseen dialogs. Additionally, the other metrics related to NLG are
also slightly improved.

###  5.2 Retrieval Performance

We hope the model can generalize well as there could be many new intents and
slots in real world.

#### General

As shown in Table 4, our model shows effective Top-5 retrieval accuracy on the
test set, keeping in mind that more than half of the services and slots are
unseen in this set. The model shows good Top-1 accuracy and above $96\%$ Top-5
accuracy, demonstrating strong abilities for handling both seen and unseen
intents and slots. Compared to only using names, adding related service and
intent descriptions improves the Top-1 accuracy by more than $5\%$. This
suggests that incorporating descriptions can enhance the model’s ability to
generalize to unseen dialogue scenarios.

#### API-information

When evaluating the “API-information”, where a single API entry in the cache
encompasses all intents and slots information for the whole dialogue. We see
that the model has high Top-1 accuracy and Top-5 accuracy. This suggests that
the model has a high potential to retrieve all the related intents and slots
information with a single retrieval attempt.

#### Orders and Special Tokens

We test with different templates, such as switching orders of intents and
slots, and find no significant differences. We also find that adding the
special tokens “INTENT” and “SLOT” slightly decreases the Top-1 accuracy.

#### Negative Sampling

Experiments with both normal and hard negative pairs, including varying
numbers of hard negative pairs, showed no significant impact on retrieval
performance. This could be attributed to the fact that, unlike longer passages
in question answering, dialogue intents, slots, and APIs are generally easier
to distinguish when they are referenced in the dialogue context.

###  5.3 Performance of Variants of Cache on End-to-End TOD

As our design involves several templates for the cache, we aim to assess the
impact of various cache templates on the performance of the end-to-end TOD
system. Table 5 shows that FiD-TOD using only names already outperforms MinT
(BART-Large), and adding descriptions further improves the performance. For
instance, FiD-TOD with cache template “INTENT: intent name, service
description, intent description, SLOT: slot name, slot description” surpassess
both MinT (BART-Large) and FiD-TOD with cache template “INTENT: intent name,
SLOT: slot name” by $7.4\%$ and $2.7\%$ in terms of Non-Empty JGA,
respectively.

… | …
---|---
SYSTEM: | Do you want to make a reservation for 2 people in the restaurant?
USER: | Yes, thanks. What’s their phone number?
|  | INTENT: ReserveRestaurant , a popular restaurant search and reservation service , make
---
a table reservation at a restaurant , SLOT:  number_of_seats , number of seats
to reserve at the restaurant
2-2 |  | INTENT: ReserveRestaurant , a popular restaurant search and reservation service , make
---
a table reservation at a restaurant , SLOT: time , tentative time of
restaurant reservation
2-2 |  | INTENT: ReserveRestaurant , a popular restaurant search and reservation service , make
---
a table reservation at a restaurant , SLOT: date , tentative date of
restaurant reservation
2-2 |  | INTENT: ReserveRestaurant , a popular restaurant search and reservation service , make
---
a table reservation at a restaurant , SLOT: restaurant_name, name of the
restaurant
2-2  RETRIEVAL: (Predicted Top-5) |  | INTENT: ReserveRestaurant, a popular restaurant search and reservation service , make
---
a table reservation at a restaurant , SLOT:  location , city where the
restaurant is located
APICALL: (Gold) |  | api_name = ReserveRestaurant ; date = 2019-03-01 ;  location = San Jose ;  number_of_seats = 2 ;
---
restaurant_name = Sino ; time = 11:30
APICALL: (Predicted) |  | api_name = ReserveRestaurant ; date = 2019-03-01 ;  city = San Jose ;  party_size = 2 ;
---
restaurant_name = Sino ; time = 11:30
APIRESP: |  | city = San Jose ; cuisine = Asian ; has_live_music = False ; phone_number = 408-247-8880 ;
---
price_range = moderate ; restaurant_name: Sino; serves_alcohol = False ;
street_address = 377 Santana Row
SYSTEM: | The phone number is 408-247-8880.

Table 7: A predicted example on the development set. Red colors indicate
incorrect predictions and light blue colors indicate correct slots.

#### Influence of Irrelevant Information on the End-to-End TOD

Given the potential emergence of unseen intents and slots in real-world
scenarios, it is challenging to expect a perfect retrieval module.

In this section, we first investigate the ability of the TOD to ignore
irrelevant retrieved information. In this section, we first investigate “the
TOD’s ability in learning to ignore irrelevant retrieved information”. Table 6
shows the corresponding results. As described in Sec 3.1, API-information
includes all intents and slots for the whole dialogue. The retrieval module
exhibits an $84.4\%$ Top-1 accuracy in retrieving all slot information in a
single attempt, as shown in Table 4. However, when setting $N$ to 5, the
retriever returns similar yet irrelevant information despite a near $100\%$
Top-5 recall accuracy. This results in the inclusion of lots of irrelevant
intents and slots into the generative model. Interestingly, “API-information
(N=1)” performs similar to “API-information (N=5)”, suggesting that the TOD is
capable of learning to ignore irrelevant retrieved information.

Second, we investigate “if the TOD generator relies more on the retriever when
all retrieved information entries are stacked together”. In pursuit of this
objective, we compare FiD-TOD and FiD-TOD-NoStack, with the difference being
whether the retrieved information entries are handled collectively or
separately. As shown in Row 3 of Table 6, FiD-TOD-NoStack performs slightly
worse when not stacking all retrieved information directly with a single
dialogue context. This could be attributed to the design of FiD-TOD-NoStack,
which results in repeated dialogue context during each retrieval attempt and
may hinder the retrieved information.

#### Error Analysis

Despite the retrieval module demonstrating relatively high Top-5 accuracy,
there is still room for improvement in the Joint Goal Accuracy (JGA).
Therefore, we examine potential reasons for this discrepancy. Table 7 shows
one most frequently appeared error type, where the retrieval module
successfully retrieve Top-5 information entries from the cache. In terms of
APICALL prediction, the TOD accurately generates the intent and associated
values. Among the generated slots,“city” and “party_size” are semantically
similar to “location” and “number_of_seats”, respectively. However, the two
generated slots are incorrect as they belongs to different services. Upon
further inspection, we find these terms are from the training cache. This
suggests that the TOD generator does not completely rely on the retriever, and
it tends to memorize the training slot information entries from the training
cache, pointing towards the need for better generalized abilities.
Furthermore, approximately $20\%$ dialogue turns on the developments set shows
this issue, suggesting a huge space to improve the performance. We hypothesize
that data augmentation, such as entity replacements in dialogue history, could
be one possible way to mitigate this problem. We leave further exploration of
this issue to future work.

##  6 Conclusion

This paper aims to enhance the performance of end-to-end TOD systems by
incorporating a simple cache. We begin by constructing a simple cache
containing intents and slots. Subsequently, we fine-tune a retrieval module to
extract the most relevant information entries. Next, we train the end-to-end
TOD model, enabling it to reference and ground both the dialogue history and
the retrieved information during TOD generation. Experimental results, based
on a large-scale SGD dataset, demonstrate that our approach outperforms strong
baselines.

##  7 Acknowledgements

We extend our gratitude to Moya Chen, Paul A. Crook, and Hu Xu for their
insightful discussions. Additionally, we appreciate the valuable feedback
provided by our anonymous reviewers.

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# Root number bias for newforms

Kimball Martin<EMAIL_ADDRESS>Department of Mathematics, University of
Oklahoma, Norman, OK 73019 USA

###### Abstract.

Previously we observed that newforms obey a strict bias towards root number
$+1$ in squarefree levels: at least half of the newforms in
$S_{k}(\Gamma_{0}(N))$ with root number $+1$ for $N$ squarefree, and it is
strictly more than half outside of a few special cases. Subsequently, other
authors treated levels which are cubes of squarefree numbers. Here we treat
arbitrary levels, and find that if the level is not the square of a squarefree
number, this strict bias still holds for any weight. In fact the number of
such exceptional levels is finite for fixed weight, and 0 if $k<12$. We also
investigate some variants of this question to better understand the
exceptional levels.

##  1\. Introduction

Throughout, $N$ and $k$ are positive integers with $k$ even. Let
$S_{k}(N)=S_{k}(\Gamma_{0}(N))$ be the space of weight $k$, level $N$ cusp
forms, and $S_{k}^{\mathrm{new}}(N)$ be the new subspace. The root number of a
newform $f\in S_{k}^{\mathrm{new}}(N)$ is the sign in the functional equation
for $L(s,f)$, and equals $(-1)^{k/2}$ times the eigenvalue of the Fricke
involution $W_{N}=\prod_{p|N}W_{p}$. It is known that 50% of newforms have
root number $+1$ and 50% of newforms have root number $-1$. We examine the
distribution of root numbers more precisely.

Let $S_{k}^{\mathrm{new}}(N)^{\pm}$ denote the subspace generated by newforms
with root number $\pm 1$. Set

$\Delta(N,k)=\dim S_{k}^{\mathrm{new}}(N)^{+}-\dim
S_{k}^{\mathrm{new}}(N)^{-}.$

In [me:refdim], we observed that the trace formula for $W_{N}$ implies that
newforms obey a strict bias towards root number $+1$ for squarefree levels
$N$, in the sense that $\Delta(N,k)\geq 0$ for all $k$ and all squarefree
$N$.111The “strict” in strict bias here refers to the fact that $\Delta(N,k)$
is never negative (for squarefree $N$), not that it is always strictly
positive, though it typically is. However, the results can also be interpreted
as strict positivity on finite unions $\bigcup S_{k}^{\mathrm{new}}(N)$
varying over $k\leq C$ or $N\leq C$ for any $C\geq 11$. Moreover,
$\Delta(N,k)$ is an elementary expression in the class number of
$\mathbb{Q}(\sqrt{-N})$, and is strictly positive unless (i) $N=2,3$ and $k$
satisfies a certain congruence, or (ii) $k=2$ and either $\dim
S_{2}^{\mathrm{new}}(N)=0$ or $N\in\\{37,58\\}$.

Subsquently, these results were extended to the case of cubes of squarefree
levels by [PQ] ($k>2$) and [LPW] ($k\geq 2$, which also considers Hilbert
modular forms). The methods in those papers are much more involved, and use
Petersson and Jacquet–Zagier trace formulas, respectively.

Here we explicitly compute the trace of $W_{N}$ on $S_{k}^{\mathrm{new}}(N)$
for arbitrary $(N,k)$, which yields the following.

######  Theorem 1.1.

  1. (1)

If $N$ is not the square of a squarefree number, or if $k\leq 10$ or $k=14$,
then there is a strict bias of newforms toward root number $+1$, i.e.,
$\Delta(N,k)\geq 0$.

  2. (2)

Suppose $N$ is the square of a squarefree number, i.e., a cubefree square. For
sufficiently large $k$, $\Delta(N,k)<0$ if and only if
$(-1)^{k/2}=-\mu(\sqrt{N})=-\prod_{p|N}(-1)$.

  3. (3)

For a fixed $k$, there are only finitely many $N$ such that $\Delta(N,k)<0$.

Moreover, in (1), $\Delta(N,k)$ is typically strictly positive. Precise
conditions are given in 3.3. We will discuss possible reasons for the curious
difference in behaviour for cubefree square levels in Section 1.2.

It is not too hard to make statements (2) and (3) effective. We do not
explicate this, but refer the interested reader to the proof of 3.3(2).

As in the squarefree case, we in fact get an elementary expression for the
exact size of the bias, and it is essentially the class number of
$\mathbb{Q}(\sqrt{-N})$. The precise formula for $\Delta(N,k)$ breaks up into
several cases according to whether $N$ is 1, 2, 3 or 4 times the square of a
squarefree number, or none of these (the generic case). Here we just describe
the formula in the generic case, and refer to 3.2 for all cases.

Write $N=N_{1}N_{2}^{2}$, where $N_{1}$ is squarefree. Define
$\beta(N)\in\\{1,2,3,4\\}$ by

(1.1) $\beta(N)=\begin{cases}1&N_{1}\equiv 1,2\bmod 4,\text{ or }2\parallel
N_{2};\\\ 2&N_{1}\equiv 3\bmod 4\quad\text{and }4\mid N_{2};\\\
3-{-N_{1}\overwithdelims( )2}&N_{1}\equiv 3\bmod 4\quad\text{and }2\nmid
N_{2}.\end{cases}$

For a fundamental discriminant $-D<0$, let $h^{\prime}(-D)$ be the “unit group
weighted” class number. Precisely, set $h^{\prime}(-4)=\frac{1}{2}$ and
$h^{\prime}(-3)=\frac{1}{3}$, and take
$h^{\prime}(-D)=h_{\mathbb{Q}(\sqrt{-D})}$ to be usual class number for $D>4$.

######  Theorem 1.2.

Suppose $N$ is not $1$, $2$, $3$ or $4$ times the square of a squarefree
number. Let $-D\in\\{-N_{1},-4N_{1}\\}$ be the discriminant of
$\mathbb{Q}(\sqrt{-N})$. Then

$\Delta(N,k)=\frac{1}{2}\beta(N)\prod_{p|N_{2}}\left(\phi(p^{v_{p}(N_{2})})-\phi(p^{v_{p}(N_{2})-1)}){-D\overwithdelims(
)p}\right)h^{\prime}(-D)-\delta,$

where $\phi$ is the Euler phi function, $\delta=1$ if $(N,k)=(1,2)$ and
$\delta=0$ otherwise.

In particular, we always have $\Delta(N,k)\geq 0$. Moreover, $\Delta(N,k)=0$
if and only if (i) $k=2$ and either $\dim S_{k}^{\mathrm{new}}(N)=0$ or
$N\in\\{37,58\\}$; or (ii) $2\parallel N_{2}$ and $N_{1}\equiv 7\bmod 8$.

Note that if $v_{p}(N)$ is odd for all $p\mid N$, then ${-D\overwithdelims(
)p}=0$ for all $p\mid N_{2}$, and thus the main term in the above formula is
just $\frac{1}{2}\beta(N)\phi(N_{2})h^{\prime}(-D)$. In particular, this
recovers the formulas in [me:refdim] and [PQ].

###  1.1. Remarks on proof

The proof has in essence two steps. The quantity $\Delta(N,k)$ equals the
trace of $(-1)^{k/2}W_{N}$ on $S_{k}^{\mathrm{new}}(N)$. First, we use a trace
formula due to Yamauchi [yamauchi] and Skoruppa–Zagier [SZ] which expresses
the trace of $W_{N}$ on $S_{k}(N)$ as an alternating sum of class numbers.
Using class number relations and elementary but careful analysis, we rewrite
this trace in terms of a single weighted class number $h^{\prime}(-D)$ (see
3.1). Second, one can express the trace of $W_{N}$ on the new subspace as an
alternating sum over $M^{2}\mid N$ of traces on $S_{k}(N/M^{2})$. Once again,
we use class number relations and elementary analysis to obtain our formula
for $\Delta(N,k)$ (see 3.2).

We remark that our approach is much more straightforward and simple than the
approaches taken in [PQ] and [LPW]. In [me:refdim], we restricted to
squarefree levels because our main focus in that paper was dimension formulas
with prescribed local signs at multiple ramified places, and that problem
seems considerably more complicated for non-squarefree levels. The result
therein about root number bias for squarefree levels was merely a curious
observation we made along the way, which was already immediate from the
Yamauchi/Skoruppa–Zagier trace formulas (though, to our knowledge, it had not
been noticed before). For squarefree levels, both steps in our present proof
are trivial because (i) the trace formula for $W_{N}$ on $S_{k}(N)$ is very
simple for squarefree $N$ and (ii) there is no oldform contribution to this
trace that we need to subtract off.

###  1.2. Further questions

1.1 naturally prompts 2 questions: (i) what is the reason for this bias, and
(ii) what is the reason for these exceptions.

We briefly speculated on (i) in [me:refdim], and mentioned two ideas that at
least support the existence of a bias towards root number $+1$. First, the
“$L$-functions from nothing” perspective suggests that $L$-functions that
barely exist must have root number $+1$. Second, when $k=2$, at least for
prime levels, $\Delta(p,2)\geq 0$ is forced upon us by the Jacquet–Langlands
correspondence together with the fact that the type number is at least
$\frac{1}{2}$ times the class number for maximal orders in quaternion algebras
of prime discriminant.

Anna Medvedovsky has since suggested to me that one can also reinterpret the
$k=2$ case for squarefree $N$ by comparing the genus of $X_{0}(N)$ with its
quotient by the Fricke involution. One can note that the formulas for
$\Delta(N,k)$ are essentially independent of $k$, and thus are controlled by
the $k=2$ case. One heuristic reason for this latter fact is that the main
term in dimension formulas of spaces of modular forms is a product of local
factors, including one at infinity for the weight $k$.

Thus there are at least some hints for the above root number bias that do not
only rely on the trace formula, but I do not have a compelling existential
explanation for this bias.

Moving on to (ii), let me first explain how the exceptions to the strict bias
arise in the proof. First, the trace of $(-1)^{k/2}W_{N}$ on $S_{k}(N)$ is
always non-negative (see 3.1). The only question now is whether this still
holds when we subtract off the oldform contribution, i.e., if
$(-1)^{k/2}\sum_{M^{2}|N}\mu(M)\operatorname{tr}W_{N/M^{2}}\geq 0$. For a
fixed $N$, the the terms for each $M^{2}\mid N$ are all similar looking
expressions in $h^{\prime}(-D)$ that can be collected together, except in the
special case that $N$ is a perfect square. In that case, the term with
$M=\sqrt{N}$ is $\mu(M)$ times the trace of $W_{N/M^{2}}=W_{1}=\mathrm{Id}$ on
$S_{k}(1)$, i.e., $\mu(M)\dim S_{k}(1)$. Since $\dim
S_{k}(1)\simeq\frac{k}{12}$, we find that
$\Delta(N,k)\approx\frac{1}{4}\phi(M)+(-1)^{k/2}\mu(M)\frac{k}{12}$. This
yields parts (2) and (3) of 1.1.

One thing that is special about levels which are perfect squares is that these
are precisely the levels where the newspaces are not generated by theta series
attached to definite quaternion algebras (see [HPS] and [me:basis]). In
particular if the level $N=M^{2}$, where $M$ is squarefree, then
$S_{k}^{\mathrm{new}}(N)$ is generated by theta series together with twists of
forms of level 1 and level $M$ (now including forms with nebentypus).

The discussion of the proof above, combined with this consideration about
theta series, suggests that the reason the strict bias towards root number
$+1$ does not persist in (certain) square levels may be due to newforms which
are twists from full level.

In Section 4, we describe what happens if one removes the contribution from
twists of level 1 forms. Specficially, denote by
$S_{k}^{\mathrm{new}}(N)^{\prime}$ be the subspace of
$S_{k}^{\mathrm{new}}(N)$ generated by newforms which are not twists from
level $1$. Let $\Delta(N,k)^{\prime}$ be the number of newforms in
$S_{k}^{\mathrm{new}}(N)^{\prime}$ with root number $+1$ minus the number of
newforms with root number $-1$.

######  Proposition 1.3.

Let $M>1$ be squarefree. The spaces $S_{k}^{\mathrm{new}}(M^{2})^{\prime}$
have a strict bias towards root number $+1$, i.e.,
$\Delta(M^{2},k)^{\prime}\geq 0$, for all $k$ if and only if $M$ is odd and
$M$ has an odd number of prime factors which are $3\bmod 4$. Otherwise, for
sufficiently large $k$, the sign of $\Delta(M^{2},k)^{\prime}$ is
$(-1)^{k/2}\mu(M)$.

Thus even though the proof of our theorems suggests the difference in root
number bias behavior for cubefree square levels comes from level 1 forms, it
can sometimes but not always be explained by twists of level 1 forms. In fact,
when $M$ is even and squarefree, there are no twists from level 1, so
$\Delta(M^{2},k)^{\prime}=\Delta(M^{2},k)$.

From this perspective, it is natural to ask what happens if one excludes all
newforms which are twists from smaller levels, i.e., if one restricts to
minimal newforms. Since every newform in $S_{k}^{\mathrm{new}}(4)$ is minimal,
and $\Delta(4,k)<0$ whenever $k\equiv 0\bmod 4$ and $k\geq 12$, the sum of
root numbers over minimal newforms is not always positive. We also exhibit
examples of $p\equiv 3\bmod 4$ such that $S_{k}^{\mathrm{new}}(p^{2})$ has
more minimal newforms with root number $-1$ than $+1$. On the other hand, we
give a simple local condition which forces root numbers to be perfectly
equidistributed for minimal forms in certain (not necessarily square) levels.

######  Proposition 1.4.

Let $N\geq 1$, and let $S_{k}^{\min}(N)$ be the subspace of
$S_{k}^{\mathrm{new}}(N)$ generated by newforms which are not twists from
smaller levels. Suppose there exists $p^{2}\parallel N$ such that
${-N/p^{2}\overwithdelims( )p}=1$. Then $\dim S_{k}^{\min}(N)^{+}=\dim
S_{k}^{\min}(N)^{-}$, i.e., root numbers are perfectly balanced for minimal
newforms.

The congruence conditions on primes dividing $N$ in these propositions arise
from the way that local root numbers behave under twisting by quadratic
characters.

###  1.3. Further remarks

As in [me:refdim], our formula for $\Delta(N,k)$ implies an exact formula for
the dimensions

$\dim S^{\mathrm{new}}_{k}(N)^{\pm}=\frac{1}{2}\left(\dim
S^{\mathrm{new}}_{k}(N)\pm\Delta(N,k)\right).$

We do not write down these dimension formulas explicitly, but the interested
reader can work them out by comparing with the explicit formula for $\dim
S_{k}^{\mathrm{new}}(N)$ in [martin].

Consequently, our formula for $\Delta(N,k)$ implies that root numbers of are
$+1$ (or $-1$) for 50% of newforms as $k+N\to\infty$. We are not aware of an
explicit proof of this fact in the literature for non-squarefree $N$, though
proofs were surely known to experts. In any case, our formulas together with
standard class number bounds yield equidistribution of root numbers with a
very good error estimate.

The original motivation for the dimension formulas for newforms with
prescribed Atkin–Lehner signs in [me:refdim] was to use them to obtain mod 2
congruences in [me:cong2]. Namely, we showed that perfect equidistribution of
Atkin–Lehner sign patterns in squarefree levels essentially means that every
newform is mod 2 congruent to one with any desired Atkin–Lehner signs at those
places. It may be interesting to see whether the cases of perfect
equidistribution of root numbers in non-squarefree levels we give here are
similarly related to mod 2 congruences.

As a check on our formulas, we compared them numerically with newform and root
number calculations in Sage [sage] for various ranges of values of $(N,k)$.

### Acknowledgements

I was inspired to revisit the question of root number biases after a series of
individual discussions with Andrew Knightly, Anna Medvedovsky, and Andrew
Sutherland. I also thank them for subsequent comments. This research was
partially supported by grants from the Simons Foundation (Collaboration Grant
512927) and the Japan Society for the Promotion of Science (Invitational
Fellowship L22540). This work was carried out during visits to MIT and Osaka
Metropolitan University, and I thank them for their hospitality.

##  2\. Preliminaries

###  2.1. Notation

Throughout, $M$ and $N$ denote positive integers, and $k\geq 2$ is even. Let
$v_{p}(N)=\max\\{r\in\mathbb{Z}:p^{r}\mid N\\}$. For $N\in\mathbb{N}$, denote
by $N^{\mathrm{odd}}$ the odd part of $N$, i.e.,
$N=2^{v_{2}(N)}N^{\mathrm{odd}}$. Let $\square\subset\mathbb{N}$ be the
multiplicative submonoid of squares. Let $\mu$ be the Möbius function.

Denote by $\operatorname{tr}W_{N}$ the trace of $W_{N}$ on the full cusp space
$S_{k}(N)$, and by $\operatorname{tr}W_{N}^{\mathrm{new}}$ the trace on the
new subspace $S_{k}^{\mathrm{new}}(N)$. These traces also depend on the weight
$k$, but we suppress it in our notation.

For a statement $*$, we use $\delta_{*}$ to mean 1 if the statement $*$ is
satisfied, and $0$ otherwise. E.g., $\delta_{k=2}$ represents the Kronecker
delta function $\delta_{k,2}$.

The following function arises in the trace formula for $W_{N}$. For
$s\in\mathbb{Z}_{\geq 0}$, set
$p_{k}(s)=\frac{\rho^{k-1}-\bar{\rho}^{k-1}}{\rho-\bar{\rho}}$ where
$\rho,\bar{\rho}$ are the roots of $X^{2}-sX+1$ if $s^{2}\neq 4$, and set
$p_{k}(2)=k-1$.

###  2.2. Class number relations

Let $-D<0$ be a discriminant, and $h(-D)$ be the class number of the quadratic
order $\mathcal{O}_{D}$ of discriminant $-D$. Let $h^{\prime}(-D)$ be the
class number weighted by
$[\mathcal{O}_{D}^{\times}:\mathbb{Z}^{\times}]^{-1}$, i.e.,
$h^{\prime}(-D)=h(-D)$ unless $D<5$, and then $h^{\prime}(-4)=\frac{1}{2}$ and
$h^{\prime}(-3)=\frac{1}{3}$. Define the Hurwitz class number $H(-D)$ to be
the number of positive definite binary quadratic forms of discrimimant
$\Delta$ up to equivalence, where we weight forms of discriminant $-4$ and
$-3$ by $\frac{1}{2}$ and $\frac{1}{3}$, respectively. (The class numbers
$h(-D)$ and $h^{\prime}(-D)$ can be interpreted in terms of counting primitive
forms.) We also set $H(0)=-\frac{1}{12}$.

Now suppose $-D$ is a fundamental discriminant, and $\lambda>0$. Then

(2.1)
$h^{\prime}(-\lambda^{2}D)=\lambda\prod_{p|\lambda}(1-{-D\overwithdelims(
)p}\frac{1}{p})\,h^{\prime}(-D)=\lambda\sum_{t|\lambda}\mu(t){-D\overwithdelims(
)t}\frac{1}{t}\,h^{\prime}(-D).$

We can write the Hurwitz class number in terms of $h^{\prime}$ by

(2.2) $H(-\lambda^{2}D)=\sum_{t|\lambda}h^{\prime}(-Dt^{2}).$

Then

(2.3) $H(-\lambda^{2}D)=\sum_{m|\lambda}\sum_{t|m}\mu(t){-D\overwithdelims(
)t}\frac{m}{t}h^{\prime}(-D)=\sum_{t|\lambda}\mu(t){-D\overwithdelims(
)t}{\sigma(\lambda/t)}h^{\prime}(-D).$

##  3\. Traces on newspace

Since, $\Delta(N,k)=(-1)^{k/2}\operatorname{tr}W_{N}^{\mathrm{new}}$ we want
to compute $\operatorname{tr}W_{N}^{\mathrm{new}}$. A formula for the trace of
a product $T_{n}W_{M}$ of Hecke and Atkin–Lehner operators on $S_{k}(N)$,
together with a relation for how the trace on the newspace is related to
traces on full cusp spaces was given by Yamauchi [yamauchi]. Unfortunately,
Yamauchi’s paper contains some clerical errors. A corrected formulation of
these traces was later given by Skoruppa and Zagier [SZ], and we will use
their formulation.

We just need the case that $n=1$ and $M=N$, but some effort is still required
to get from the Skoruppa–Zagier formula to a relatively simple expression for
$\Delta(N,k)$. We will first find a simple expression for
$\operatorname{tr}W_{N}$, and then use it get our desired formula for
$\operatorname{tr}W_{N}^{\mathrm{new}}$.

As explained in [SZ, pp. 132–133], a newform $f\in S_{k}^{\mathrm{new}}(M)$
with root number $(-1)^{k/2}w_{f}$ contributes $w_{f}$ to
$\operatorname{tr}W_{N}$ if $N/M\in\square$ and $0$ otherwise. Thus

$\operatorname{tr}W_{N}=\sum_{M\mid
N,\,N/M\in\square}\operatorname{tr}W_{M}^{\mathrm{new}},$

whence

(3.1) $\operatorname{tr}W_{N}^{\mathrm{new}}=\sum_{Q^{2}\mid
N}\mu(Q)\operatorname{tr}W_{N/Q^{2}}.$

Now a special case of [SZ, (2.7)] yields

(3.2)
$\operatorname{tr}W_{N}=-\frac{1}{2}\sum_{M|N,N/M\in\square}\mu(\sqrt{N/M})\sum_{s}I_{s}(k,M)-\frac{1}{2}\delta_{N\in\\{1,4\\}}+\delta_{k=2},$

where

$I_{s}(k,M)=p_{k}(s/\sqrt{M})H(s^{2}-4M),$

and $0\leq s\leq 2\sqrt{M}$ such that $\sqrt{MN}\mid s$. The only way that
$s>0$ is possible then is if $N\leq 4$. Note that $p_{k}(0)=(-1)^{(k-2)/2}$,
so when $s=0$ we get

$I_{0}(k,M)=(-1)^{(k-2)/2}H(-4M).$

###  3.1. Small levels

We first deal with levels $N\leq 4$.

The Fricke involution is trivial if $N=1$. In particular we have

$\operatorname{tr}W_{1}=\dim
S_{k}(1)=\begin{cases}\lfloor\frac{k}{12}\rfloor&k\not\equiv 2\bmod 12\\\
\lfloor\frac{k}{12}\rfloor-1+\delta_{k=2}&k\equiv 2\bmod 12.\\\ \end{cases}$

The cases $N=2,3$ are included in the case of $N$ squarefree (e.g., see
[me:refdim, Thm 2.2]). Explicitly, we have

$\operatorname{tr}W_{2}=\operatorname{tr}W_{2}^{\mathrm{new}}=\begin{cases}(-1)^{k/2}(1-\delta_{k=2})&k\equiv
0,2\bmod 8\\\ 0&k\equiv 4,6\bmod 8,\end{cases}$

and

$\operatorname{tr}W_{3}=\operatorname{tr}W_{3}^{\mathrm{new}}=\begin{cases}(-1)^{k/2}(1-\delta_{k=2})&k\equiv
0,2,6,8\bmod 12\\\ 0&k\equiv 4,10\bmod 12.\end{cases}$

If $N=4$, we have

$\operatorname{tr}W_{4}=\frac{1}{2}\left(I_{0}(k,1)+I_{2}(k,1)-I_{0}(k,4)-I_{4}(k,4)\right)-\frac{1}{2}+\delta_{k=2}$

Note that $I_{0}(k,1)-I_{0}(k,4)=(-1)^{(k-2)/2}(H(-4)-H(-16))=(-1)^{k/2}$, and
$I_{2}(k,1)=p_{k}(2)H(0)=I_{4}(k,4)$. Thus

$\operatorname{tr}W_{4}=\begin{cases}0&k\equiv 0\bmod 4\\\
-1+\delta_{k=2}&k\equiv 2\bmod 4.\end{cases}$

Then

$\operatorname{tr}W_{4}^{\mathrm{new}}=\operatorname{tr}W_{4}-\operatorname{tr}W_{1}=\begin{cases}-\lfloor\frac{k}{12}\rfloor&k\equiv
0,2,4,8\bmod 12\\\ -\lfloor\frac{k}{12}\rfloor-1&k\equiv 6,10\bmod
12.\end{cases}$

###  3.2. Generic levels: trace on full cusp space

Now suppose $N>4$. Then the only $s$-terms in (3.2) are $s=0$, and we have

$\operatorname{tr}W_{N}=(-1)^{k/2}\frac{1}{2}\sum_{M|N,N/M\in\square}\mu(\sqrt{N/M})H(-4M)+\delta_{k=2}.$

We can uniquely write $N=N_{1}N_{2}^{2}$ where $N_{1}$ is squarefree. If $N/M$
is a square, then we can write $M=N_{1}M_{2}^{2}$ where $M_{2}\mid N_{2}$.
Hence

$\operatorname{tr}W_{N}=(-1)^{k/2}\frac{1}{2}\sum_{M_{2}|N_{2}}\mu(N_{2}/M_{2})H(-4N_{1}M_{2}^{2})+\delta_{k=2}.$

######  Proposition 3.1.

Suppose $N>4$, and write $N=N_{1}N_{2}^{2}$ with $N_{1}$ squarefree as above.
Then

$(-1)^{k/2}\operatorname{tr}W_{N}-\delta_{k=2}=\begin{cases}\frac{1}{2}h^{\prime}(-4N)&N_{1}\equiv
1,2\bmod 4,\\\ \frac{1}{2}\left(3-{-N_{1}\overwithdelims(
)2}\right)h^{\prime}(-N)&N_{1}\equiv 0,3\bmod 4,\quad N_{2}\text{ odd},\\\
h^{\prime}(-N)&N_{1}\equiv 0,3\bmod 4,\quad N_{2}\text{ even}.\\\ \end{cases}$

###### Proof.

First suppose $-4N_{1}$ is a fundamental discriminant, i.e., $-N_{1}\equiv
2,3\bmod 4$. We compute that

(3.3) $\displaystyle(-1)^{k/2}\operatorname{tr}W_{N}-\delta_{k=2}$
$\displaystyle=\frac{1}{2}\sum_{M_{2}|N_{2}}\sum_{t|M_{2}}\mu(N_{2}/M_{2})\sigma(M_{2}/t)\mu(t){-4N_{1}\overwithdelims(
)t}h^{\prime}(-4N_{1})$ (3.4)
$\displaystyle=\frac{1}{2}\sum_{t|N_{2}}\frac{N_{2}}{t}\mu(t){-4N_{1}\overwithdelims(
)t}h^{\prime}(-4N_{1})$ $\displaystyle=\frac{1}{2}h^{\prime}(-4N).$

Here we got from (3.3) to (3.4) by interchanging the order of summation and
observing that for fixed $t\mid N_{2}$,

$\sum_{t|M_{2}|N_{2}}\mu(N_{2}/M_{2})\sigma(M_{2}/t)=\sum_{M_{2}^{\prime}|(N_{2}/t)}\mu((N_{2}/t)/M_{2}^{\prime})\sigma(M_{2}^{\prime}),$

where we have written $M_{2}=tM_{2}^{\prime}$. Then observe that this
expression is just the Dirichlet convolution
$(\mu*\sigma)(N_{2}/t)=\mathrm{Id}(N_{2}/t)=N_{2}/t$. We will use this latter
fact again in the remaining cases.

For the rest of the proof, assume $-N_{1}$ is a fundamental discriminant. Then

(3.5)
$(-1)^{k/2}\operatorname{tr}W_{N}-\delta_{k=2}=\frac{1}{2}\sum_{M_{2}|N_{2}}\sum_{t|2M_{2}}\mu(N_{2}/M_{2})\sigma(2M_{2}/t)\mu(t){-N_{1}\overwithdelims(
)t}h^{\prime}(-N_{1}).$

First consider the case that $N_{2}$ is odd. Note that for $t$ odd and
$M_{2}\mid N_{2}$, $\sigma(2M_{2}/t)=3\sigma(M_{2}/t)$. Consequently,

$\displaystyle\sum_{M_{2}|N_{2}}\sum_{t|2M_{2}}\mu(N_{2}/M_{2})\sigma(2M_{2}/t)\mu(t){-N_{1}\overwithdelims(
)t}$ $\displaystyle=$
$\displaystyle\sum_{t|M_{2}}\sum_{M_{2}|N_{2}}\left(\sigma(2M_{2}/t)\mu(t){-N_{1}\overwithdelims(
)t}+\sigma(M_{2}/t)\mu(2t){-N_{1}\overwithdelims( )2t}\right)$
$\displaystyle=$
$\displaystyle\sum_{t|N_{2}}\sum_{t|M_{2}|N_{2}}\sigma(M_{2}/t)\mu(t){-N_{1}\overwithdelims(
)t}\left(3-{-N_{1}\overwithdelims( )2t}\right)$ $\displaystyle=$
$\displaystyle(\mu*\sigma)(N_{2}/t){-N_{1}\overwithdelims(
)t}\left(3-{-N_{1}\overwithdelims( )2t}\right)$
$\displaystyle=\frac{N_{2}}{t}{-N_{1}\overwithdelims(
)t}\left(3-{-N_{1}\overwithdelims( )2t}\right)$

Now plugging this into (3.5) yields the second case of the proposition.

Finally suppose that $N_{2}$ is even, so we can write
$N_{2}=2^{e}N_{2}^{\mathrm{odd}}$ where $e\geq 1$. For an $M_{2}\mid N_{2}$
appearing in the double sum in (3.5), we must have
$M_{2}=2^{f}M_{2}^{\mathrm{odd}}$ where $f\in\\{e-1,e\\}$ and
$M_{2}^{\mathrm{odd}}\mid N_{2}^{\mathrm{odd}}$ due to the factor
$\mu(N_{2}/M_{2})$. Furthermore the $\mu(t)$ factor means that we can restrict
to $t\mid 2M_{2}^{\mathrm{odd}}$ in the same double sum. Hence (3.5) is equal
to $\frac{1}{2}h^{\prime}(-N_{1})$ times

$\sum_{M_{2}^{\mathrm{odd}}|N_{2}^{\mathrm{odd}}}\sum_{t|2M_{2}^{\mathrm{odd}}}\left(\mu(2N_{2}^{\mathrm{odd}}/M_{2}^{\mathrm{odd}})\sigma(2^{e}M_{2}^{\mathrm{odd}}/t)\mu(t){-N_{1}\overwithdelims(
)t}+\right.\\\
\left.\mu(N_{2}^{\mathrm{odd}}/M_{2}^{\mathrm{odd}})\sigma(2^{e+1}M_{2}^{\mathrm{odd}}/t)\mu(t){-N_{1}\overwithdelims(
)t}\right)=\\\
\sum_{M_{2}^{\mathrm{odd}}|N_{2}^{\mathrm{odd}}}\sum_{t|2M_{2}^{\mathrm{odd}}}\mu(N_{2}^{\mathrm{odd}}/M_{2}^{\mathrm{odd}})\left(\sigma(2^{e+1}M_{2}^{\mathrm{odd}}/t)-\sigma(2^{e}M_{2}^{\mathrm{odd}}/t)\right)\mu(t){-N_{1}\overwithdelims(
)t}.$

Note that $\sigma(2^{j}m)=(2^{j+1}-1)\sigma(m)$ for $m>0$ odd and $j\geq 0$.
Now considering the $t$ odd and $t$ even terms separately, we rewrite the
above as

$\displaystyle
2^{e}\sum_{M_{2}^{\mathrm{odd}}|N_{2}^{\mathrm{odd}}}\sum_{t|M_{2}^{\mathrm{odd}}}\mu(N_{2}^{\mathrm{odd}}/M_{2}^{\mathrm{odd}})\sigma(M_{2}^{\mathrm{odd}}/t)\mu(t){-N_{1}\overwithdelims(
)t}\left(2-{-N_{1}\overwithdelims( )2}\right)$ $\displaystyle=$ $\displaystyle
2^{e}\left(2-{-N_{1}\overwithdelims(
)2}\right)\sum_{t|N_{2}^{\mathrm{odd}}}(\mu*\sigma)(N_{2}^{\mathrm{odd}}/t)\mu(t){-N_{1}\overwithdelims(
)t}$ $\displaystyle=$ $\displaystyle 2^{e}\left(2-{-N_{1}\overwithdelims(
)2}\right)\sum_{t|N_{2}^{\mathrm{odd}}}\frac{N_{2}^{\mathrm{odd}}}{t}\mu(t){-N_{1}\overwithdelims(
)t}.$

Multiplying this expression by $\frac{1}{2}h^{\prime}(-N_{1})$ and applying
both equalities in (2.1) yields

$2^{e-1}\left(2-{-N_{1}\overwithdelims(
)2}\right)h^{\prime}(-N_{1}(N_{2}^{\mathrm{odd}})^{2})=h^{\prime}(-N).$

This completes the last case of the proposition. ∎

###  3.3. Alternating class number sums

Let $N\geq 1$ and write $N=N_{1}N_{2}^{2}>4$ with $N_{1}$ squarefree. Then by
(3.1), we have

(3.6)
$\operatorname{tr}W_{N}^{\mathrm{new}}=\sum_{Q|N_{2}}\mu(N_{2}/Q)\operatorname{tr}W_{N_{1}Q^{2}}.$

Comparing this with 3.1, to get a simplified formula for
$\operatorname{tr}W_{N}^{\mathrm{new}}$, we essentially just need to evaluate

(3.7) $\sum_{Q|N_{2}}\mu(N_{2}/Q)h^{\prime}(-DQ^{2}),$

where $-D=-4N_{1}$ or $-D=-N_{1}$, whichever is a fundamental discriminant. We
will also need to include “correction terms” when $k=2$ or when
$N_{1}Q^{2}\leq 4$, but let us first evaluate (3.7).

We can rewrite the sum in (3.7) as

$\sum_{Q|N_{2}}\sum_{t|Q}\mu(N_{2}/Q)\frac{Q}{t}\mu(t){-D\overwithdelims(
)t}h^{\prime}(-D)=\sum_{t|N_{2}}\left(\sum_{Q^{\prime}|N_{2}/t}\mu((N_{2}/t)/Q^{\prime})Q^{\prime}\right)\mu(t){-D\overwithdelims(
)t}h^{\prime}(-D).$

The inner sum is the Dirichlet convolution
$(\mu*\mathrm{Id})(N_{2}/t)=\phi(N_{2}/t)$. Hence

(3.8)
$\sum_{Q|N_{2}}\mu(N_{2}/Q)h^{\prime}(-DQ^{2})=c(-D,N_{2})h^{\prime}(-D),$

where

$\displaystyle c(-D,n)$
$\displaystyle=\sum_{t|n}\phi(n/t)\mu(t){-D\overwithdelims( )t}$ (3.9)
$\displaystyle=\prod_{p||n}\left(p-1-{-D\overwithdelims(
)p}\right)\prod_{p^{2}|n}p^{v_{p}(n)-2}(p-1)\left(p-{-D\overwithdelims(
)p}\right).$

The second equality follows from observing $c(-D,n)$ is a Dirichlet
convolution of multiplicative functions in $n$, and thus multiplicative in
$n$, and then noting that
$c(-D,p^{r})=\phi(p^{r})-\phi(p^{r-1}){-D\overwithdelims( )p}$.

We make two remarks. First, $c(-D,N_{2})\geq 0$ for all $N$, and
$c(-D,N_{2})=0$ if and only if $2\parallel N_{2}$ and ${-D\overwithdelims(
)2}=1$. Second, if $N_{2}\mid D$ (which, when $-D$ is odd, is equivalent to
$v_{p}(N)\neq 2$ for all $p$), then we simply have that
$c(-D,N_{2})=\phi(N_{2})$.

###  3.4. Correction terms

As above, write $N=N_{1}N_{2}^{2}$ ($N\geq 1$) with $N_{1}$ squarefree, and
let $-D$ be the discriminant of $\mathbb{Q}(\sqrt{N})$. We will write

$\operatorname{tr}W_{N}^{\mathrm{new}}=(-1)^{k/2}\sum_{Q|N_{2}}\mu(N_{2}/Q)b(N_{1}Q^{2})h^{\prime}(-DQ^{2})+\xi(N,k),$

where

$b(N)=\begin{cases}\frac{1}{2}&-N_{1}\equiv 2,3\bmod 4,\\\
\frac{1}{2}\left(3-{-N_{1}\overwithdelims( )2}\right)&-N_{1}\equiv 1\bmod
4,\quad N_{2}\text{ odd},\\\ 1&-N_{1}\equiv 1\bmod 4,\quad N_{2}\text{
even}.\\\ \end{cases}$

and $\xi(N,k)$ consists of some correction terms that we describe below.

Note that $b(N_{1}Q^{2})=b(N)$ for all $Q\mid N_{2}$ unless $-N_{1}$ is a
fundamental discrimimant, $N_{2}$ is even, $Q$ is odd and
${-N_{1}\overwithdelims( )2}\neq 1$. In the latter instance, $b(N)=1$ and for
$Q\mid N_{2}$, $b(N_{1}Q^{2})$ is 1 or 2 when $Q$ is even or odd,
respectively. Bearing this in mind, using (3.8) we can rewrite the above
equation as

(3.10)
$\operatorname{tr}W_{N}^{\mathrm{new}}=(-1)^{k/2}b(N)c^{\prime}(-D,N_{2})h^{\prime}(-D)+\xi(N,k),$

where

$c^{\prime}(-D,N_{2})=\begin{cases}\frac{1}{2}c(-D,N_{2})&N_{1}\equiv 3\bmod
8,\,2\parallel N_{2}\\\ c(-D,N_{2})&\text{ else}.\end{cases}$

The $\frac{1}{2}c(-D,N_{2})$ case arises as
$c(-D,N_{2})-c(-D,N_{2}^{\mathrm{odd}})$, and then observing that
$c(-D,N_{2})=2c(-D,N_{2}^{\mathrm{odd}})$ when $N_{1}\equiv 3\bmod 8$ and
$2\parallel N_{2}$. Consequently, the function $c^{\prime}(-D,N_{2})$ also has
image in $\mathbb{Z}_{\geq 0}$ and is 0 precisely when $c(-D,N_{2})$ is. Note
that in general we can write

$b(N)c^{\prime}(-D,N_{2})=\frac{1}{2}\beta(N)c(-D,N_{2})$

where $\beta(N)$ is as in (1.1).

Now we describe the correction terms in $\xi(N,k)$. The first correction terms
are for when the sum in (3.6) includes $\operatorname{tr}W_{N_{1}Q^{2}}$ where
$N_{1}Q^{2}\leq 4$. Specifically, forgetting for the moment the $\delta_{k=2}$
term in $\operatorname{tr}W_{N}$, by 3.1 the difference between
$\operatorname{tr}W_{N}^{\mathrm{new}}$ and
$(-1)^{k/2}\sum_{Q|N_{2}}\mu(N_{2}/Q)b(N_{1}Q^{2})h^{\prime}(-DQ^{2})$ is

$\xi_{0}(N,k):=\sum_{Q|N_{2},\,N_{1}Q^{2}\leq
4}\mu(N_{2}/Q)\left(\operatorname{tr}W_{N_{1}Q^{2}}-(-1)^{k/2}b(N_{1}Q^{2})h^{\prime}(-DQ^{2})\right).$

Terms with $N_{1}Q^{2}\leq 4$ occur if and only if (i) $Q=1$, $N_{2}$ is
squarefree and $N_{1}\leq 3$; or (ii) $Q=2$, $N_{2}$ is twice a squarefree
number and $N_{1}=1$. Hence,

$\xi_{0}(N,k)=\begin{cases}\mu(N_{2})\left(\operatorname{tr}W_{1}-(-1)^{k/2}\cdot\frac{1}{4}\right)&N_{1}=1,\,N_{2}\text{
odd squarefree}\\\
\mu(N_{2})\left(\operatorname{tr}W_{1}-\operatorname{tr}W_{4}+(-1)^{k/2}\cdot\frac{1}{4}\right)&N_{1}=1,\,N_{2}\text{
even squarefree}\\\
\mu(N_{2}/2)\left(\operatorname{tr}W_{4}-(-1)^{k/2}\cdot\frac{1}{2}\right)&N_{1}=1,\,N_{2}/4\text{
odd squarefree}\\\
\mu(N_{2})\left(\operatorname{tr}W_{2}-(-1)^{k/2}\cdot\frac{1}{2}\right)&N_{1}=2,\,N_{2}\text{
squarefree}\\\
\mu(N_{2})\left(\operatorname{tr}W_{3}-(-1)^{k/2}\cdot\frac{2}{3}\right)&N_{1}=3,\,N_{2}\text{
squarefree},\\\ \end{cases}$

and $\xi_{0}(N,k)=0$ otherwise.

When $k=2$, the $\delta_{k=2}$ terms contribute

$\sum_{Q|N_{2}}\mu(N_{2}/Q)\cdot 1=\sum_{Q|N_{2}}\mu(Q)=\delta_{N_{2}=1}$

to the formula for $\operatorname{tr}W_{N}^{\mathrm{new}}$ in (3.6). However,
the correction term $\xi_{0}(N,k)$ also contains a $\delta_{k=2}$ for each
$\operatorname{tr}W_{M}$ appearing where $M\leq 4$. By examining the above
formula for $\xi_{0}(N,k)$, we find that (3.10) holds where

$\xi(N,k)=\xi_{0}(N,k)+\delta_{k=2}\left(\delta_{N_{2}=1}+\varepsilon(N)\right),$

and

$\varepsilon(N)=\begin{cases}-\mu(N_{2})&N_{1}=1,\,N_{2}\text{ odd
squarefree}\\\ -\mu(N_{2}/2)&N_{1}=1,\,N_{2}/4\text{ odd squarefree}\\\
-\mu(N_{2})&N_{1}=2,3,\,N_{2}\text{ squarefree}\\\ 0&\text{else}.\end{cases}$

###  3.5. Formulas for traces on newspaces

Writing out the correction terms $\xi(N,k)$ in (3.10) more explicitly gives
the following.

######  Proposition 3.2.

Let $N\geq 1$, and let $k\geq 2$ be even. Write $N=N_{1}N_{2}^{2}$ where
$N_{1}$ is squarefree, and let $-D\in\\{-N_{1},-4N_{1}\\}$ be the discriminant
of $\mathbb{Q}(\sqrt{-N})$.

  1. (1)

If $N$ is not $1$, $2$, $3$ or $4$ times the square of a squarefree number,
then

$\operatorname{tr}W_{N}^{\mathrm{new}}=\frac{1}{2}(-1)^{k/2}\beta(N)c(-D,N_{2})h^{\prime}(-D)+\delta_{k=2}\delta_{N_{2}=1}.$

  2. (2)

Suppose $N=N_{2}^{2}$, where $N_{2}$ is squarefree. If $N_{2}$ is odd, then

$\operatorname{tr}W_{N}^{\mathrm{new}}=\frac{1}{4}(-1)^{k/2}\left(c(-4,N_{2})-\mu(N_{2})\right)+\mu(N_{2})\left(\left\lfloor\frac{k}{12}\right\rfloor-\kappa_{1}^{\mathrm{odd}}\right)+\delta_{k=2}\delta_{N_{2}=1},$

where $\kappa_{1}^{\mathrm{odd}}=1$ if $k\equiv 2\bmod 12$ and
$\kappa_{1}^{\mathrm{odd}}=0$ otherwise. If $N_{2}$ is even, then

$\operatorname{tr}W_{N}^{\mathrm{new}}=\frac{1}{4}(-1)^{k/2}\left(c(-4,N_{2})+\mu(N_{2})\right)+\mu(N_{2})\left(\left\lfloor\frac{k}{12}\right\rfloor+\kappa_{1}^{\mathrm{even}}\right),$

where $\kappa_{1}^{\mathrm{even}}=1$ if $k\equiv 6,10\bmod 12$ and
$\kappa_{1}^{\mathrm{even}}=0$ otherwise.

  3. (3)

If $N=N_{2}^{2}$, where $N_{2}$ is twice an even squarefree number, then

$\operatorname{tr}W_{N}^{\mathrm{new}}=\frac{1}{4}(-1)^{k/2}c(-4,N_{2})-\frac{1}{2}\mu(N_{2}/2).$

  4. (4)

If $N=2N_{2}^{2}$, where $N_{2}$ is squarefree, then

$\operatorname{tr}W_{N}^{\mathrm{new}}=\frac{1}{2}(-1)^{k/2}\left(c(-8,N_{2})+\kappa_{2}\mu(N_{2})\right)+\delta_{k=2}\delta_{N_{2}=1},$

where $\kappa_{2}=1$ if $k\equiv 0,2\bmod 8$ and $\kappa_{2}=-1$ if $k\equiv
4,6\bmod 8$.

  5. (5)

If $N=3N_{2}^{2}$, where $N_{2}$ is squarefree, then

$\operatorname{tr}W_{N}^{\mathrm{new}}=\frac{1}{3}(-1)^{k/2}\left(\frac{1}{2}\beta(N)c(-3,N_{2})+\kappa_{3}\mu(N_{2})\right)+\delta_{k=2}\delta_{N_{2}=1},$

where $\kappa_{3}=-2$ if $k\equiv 4,10\bmod 12$ and $\kappa_{3}=1$ otherwise.

######  Corollary 3.3.

Keep the notation of 3.2.

  1. (1)

Suppose $N$ is not $1$, $2$, $3$ or $4$ times the square of a squarefree
number. Then $\Delta(N,k)\geq 0$. Furthermore, $\Delta(N,k)=0$ exactly when
(i) $k=2$ and either $\dim S_{k}^{\mathrm{new}}(N)=0$ or $N\in\\{37,58\\}$; or
(ii) $2\parallel N_{2}$ and $N_{1}\equiv 7\bmod 8$.

  2. (2)

Suppose $N=N_{2}^{2}$ where $N_{2}$ is squarefree. Then $\Delta(N,k)\geq 0$
for all $N$ if $k\leq 10$ or $k=14$. For any fixed $k$, $\Delta(N,k)\geq 0$
for $N$ sufficiently large. For fixed $N$ and sufficiently large $k$,
$|\Delta(N,k)|>0$ and the sign of $\Delta(N,k)$ is $(-1)^{k/2}\mu(N_{2})$.

  3. (3)

Suppose $N=N_{2}^{2}$ where $N_{2}$ is twice an even squarefree number. Then
$\Delta(N,k)\geq 0$. Further, $\Delta(N,k)=0$ if and only if $N=16$ and
$k\equiv 2\bmod 4$.

  4. (4)

Suppose $N=2N_{2}^{2}$, where $N_{2}$ is squarefree. Then $\Delta(N,k)\geq 0$.
Further $\Delta(N,k)=0$ if and only if (i) $N\in\\{8,18\\}$ and $k\equiv
0,2\bmod 8$; (ii) $N\in\\{2,72\\}$ and $k\equiv 4,6\bmod 8$; or (iii)
$(N,k)=(2,2)$.

  5. (5)

Suppose $N=3N_{2}^{2}$, where $N_{2}$ is squarefree. Then $\Delta(N,k)\geq 0$.
Further $\Delta(N,k)=0$ if and only if (i) $N=\\{3,108\\}$ and $k\equiv
4,10\bmod 12$; (iii) $N=12$ and $k\not\equiv 4,10\bmod 12$; or (iii)
$(N,k)=(3,2)$.

###### Proof.

Recall that we always have $b(N)\geq\frac{1}{2}$,
$c(-D,N_{2})\in\mathbb{Z}_{\geq 0}$ and $h^{\prime}(-D)\geq\frac{1}{3}$.
Moreover $c(-D,N_{2})=0$ if and only if either $2\parallel N_{2}$ is even and
${-D\overwithdelims( )2}=1$.

Hence the statement that $\Delta(N,k)\geq 0$ in (1) is obvious when $k\neq 2$.
When $k=2$, the only possible negative term in our formula for
$(-1)^{k/2}\operatorname{tr}W_{N}^{\mathrm{new}}$ is $-\delta_{N_{2}=1}$.
However this only occurs when $N$ is squarefree, in which case we already know
$\Delta(N_{1},2)\geq 0$ with equality if and only if $\dim
S_{2}^{\mathrm{new}}(N)=0$ or $N\in\\{37,58\\}$ by [me:refdim]. When $N$ is
not squarefree, $\Delta(N,k)=0$ if and only if $c(-D,N_{2})=0$. This finishes
(1).

Suppose we are in case (2) now. Note
$c(-4,N_{2})=\prod_{p|N_{2}^{\mathrm{odd}}}(p-1-{-4\overwithdelims( )p})\geq
3^{\omega(N_{2}^{\mathrm{odd}})}$. Here $\omega(n)$ denotes the number of
prime divisors of $n$. For fixed $k$, $\Delta(N,k)$ is thus dominated by
$c(-4,N_{2})$ as $N_{2}\to\infty$. For fixed $N$, $\Delta(N,k)$ is dominated
by $(-1)^{k/2}\mu(N_{2})\lfloor k/12\rfloor$ as $k\to\infty$. If $k<12$ and
$k=14$, the only possible negative terms in our formulas for
$(-1)^{k/2}\operatorname{tr}W_{N}^{\mathrm{new}}$ will be a $-1$ (depending on
$\mu(N_{2})$ and $k$). However, in these cases is also a strictly positive
term, and since the trace is an integer,
$(-1)^{k/2}\operatorname{tr}W_{N}^{\mathrm{new}}\geq 0$.

In case (3), we have
$c(-4,N_{2})=2\prod_{p|N_{2}^{\mathrm{odd}}}(p-1-{-1\overwithdelims( )p})\geq
2$, with equality if and only if $N=16$. This easily gives the assertions in
(3).

Consider case (4). When $N_{2}=1$, by [me:refdim] $\Delta(2,k)\geq 0$ with
equality if and only if $k\equiv 4,6\bmod 8$. Suppose $N_{2}>1$. Then
$\Delta(N,k)\geq 0$ if and only if $c(-8,N_{2})\geq-\kappa_{2}\mu(N_{2})$.
Note $c(-8,N_{2})=\prod_{p|N_{2}}(p-1-{-8\overwithdelims( )p})\geq 1$ and
equals $1$ if and only if $N_{2}|6$. Then the assertions readily follow.

Finally consider case (5). Again, the squarefree case $N=3$ is treated in
[me:refdim], so suppose $N_{2}>1$. Now $\beta(N)c(-3,N_{2})$ equals
$\prod_{p|N_{2}}(p-1-{-3\overwithdelims( )p})\geq 2^{\omega(N_{2})}$ if
$N_{2}$ is odd and
$\frac{1}{2}\prod_{p|N_{2}^{\mathrm{odd}}}(p-1-{-3\overwithdelims( )p})\geq
2^{\omega(N_{2}^{\mathrm{odd}}-1)}$ if $N_{2}$ is even. Comparing with the
proposition now finishes the proof. ∎

This proves the theorems stated in the introduction.

##  4\. Excluding twists for certain levels

Motivated by the different behavior in root number bias for cubefree square
levels, we briefly investigate what happens when we exclude certain twists
from smaller levels. Our main goal is to prove the propositions in Section
1.2.

First we recall some facts about twists of modular forms of cube-free levels.
We will explain things from the point of view of associated local
representations, as this perspective makes things more transparent to us.
However, many of the facts that we recall are also well known from the more
classical “global” approach (see [AL], [atkin-li], [HPS:twists]).

Given a newform $f\in S_{k}^{\mathrm{new}}(N,\psi)$ with nebentypus $\psi$,
there is an associated cuspidal automorphic representation $\pi=\pi_{f}$ of
$\operatorname{GL}_{2}(\mathbb{A}_{\mathbb{Q}})$. We have a decomposition
$\pi=\bigotimes_{p}\pi_{p}\otimes\pi_{\infty}$, where each $\pi_{p}$ is an
infinite-dimensional irreducible admissible complex representation of
$\operatorname{GL}_{2}(\mathbb{Q}_{p})$. We say $\pi_{p}$ is the local
representation at $p$ associated to $f$. The local conductor $c(\pi_{p})$ is
the exponent $v_{p}(N)$ in the level of $f$.

Assume now that $\psi$ is trivial, i.e., $f\in S_{k}^{\mathrm{new}}(N)$. Then
$\pi$, and each $\pi_{p}$, has trivial central character. Moreover, $\pi_{p}$
is an unramified principal series representation (i.e., $c(\pi_{p})=0$) if and
only if $p\nmid N$. Also, $\pi_{p}$ is an unramified quadratic twist (possibly
trivial twist) of the Steinberg representation $\mathrm{St}_{p}$ of
$\operatorname{GL}_{2}(\mathbb{Q}_{p})$ (i.e., $c(\pi_{p})=1$) if and only if
$p\parallel N$.

Now there are 3 possibilities when $c(\pi_{p})=2$: $\pi_{p}$ can be a ramified
principal series, a ramified quadratic twist of Steinberg, or supercuspidal.
However, the former two possibilities do not happen for $p=2$ (e.g., see
[pacetti, Corollary 4.1]). In fact there are 2 possibilities for $\pi_{p}$
being a ramified principal series: either it is a ramified quadratic twist of
an unramified principal series, or a minimal twist (a twist minimizing the
local conductor) of $\pi_{p}$ is a ramified principal series of conductor 1
(necessarily the central character is nontrivial and has conductor 1). In any
event, only the supercuspidal representations are minimal.

Suppose now that $N=M^{2}$ where $M>1$ is squarefree. Then the above
description of local representations means exactly one of the following is
true for a newform $f\in S_{k}^{\mathrm{new}}(N)$:

  1. (1)

$f$ is minimal, i.e., no twist (possibly with nebentypus) has smaller level.
Here $\pi_{p}$ is supercuspidal for each $p\mid N$.

  2. (2)

$f$ is a quadratic twist of a level 1 newform. Here $\pi_{p}$ is a ramified
quadratic twist of an unramified principal series for each $p\mid N$.
Necessarily $N$ is odd.

  3. (3)

A minimal twist $f^{\prime}$ of $f$ (possibly with nebentypus) has level
$N^{\prime}$ strictly between $1$ and $N$. Necessarily
$v_{2}(N^{\prime})=v_{2}(N)$.

The utility of the local representation theory perspective for us comes both
from a straightforward description of the level of twists and the fact that
root numbers can be read off of the local representations. Namely, the root
number of $f$ is the product over $p$ of the local representation root numbers
$w_{p}(\pi_{p})$ times $(-1)^{k/2}$ (which is the local root number of
$\pi_{\infty}$).

Recall $S_{k}^{\mathrm{new}}(N)^{\prime}$ (resp. $S_{k}^{\min}(N)$) is defined
to be the subspace of $S_{k}^{\mathrm{new}}(N)$ generated by newforms of types
(1) and (3) above (resp. of type (1) above). Define
$\operatorname{tr}W_{N}^{\prime}$ and $\operatorname{tr}W_{N}^{\min}$ to be
the traces of $W_{N}$ restricted to the spaces
$S_{k}^{\mathrm{new}}(N)^{\prime}$ and $S_{k}^{\min}(N)$, respectively. First
we describe $\operatorname{tr}W_{N}^{\prime}$.

######  Lemma 4.1.

We have

$\operatorname{tr}W_{N}^{\prime}=\begin{cases}\operatorname{tr}W_{N}^{\mathrm{new}}-(-1)^{k/2}\prod_{p|N}{-1\overwithdelims(
)p}\cdot\dim S_{k}(1)&N\text{ odd}\\\
\operatorname{tr}W_{N}^{\mathrm{new}}&N\text{ even}.\end{cases}$

###### Proof.

Since $S_{k}^{\mathrm{new}}(N)^{\prime}=S_{k}^{\mathrm{new}}(N)$ if $N$ is
even, assume $N$ is odd. Then we simply need to subtract off from
$\operatorname{tr}W_{N}^{\mathrm{new}}$ the trace of $W_{N}$ restricted to the
subspace of $S_{k}^{\mathrm{new}}(N)$ generated by forms of type (2). Now we
simply use the fact that if $\pi_{p}$ is a ramified quadratic twist of an
unramified principal series, then $w_{p}(\pi_{p})={-1\overwithdelims( )p}$
(see [pacetti, Theorem 3.2] in terms of local representations, or [AL, Theorem
6] in terms of modular forms). ∎

Let $\Delta(N,k)^{\prime}=(-1)^{k/2}\operatorname{tr}W_{N}^{\prime}$, which is
the difference between the number of newforms in
$S_{k}^{\mathrm{new}}(N)^{\prime}$ with root number $+1$ and $-1$.

###### Proof of 1.3.

When $N$ is even, we simply have $\Delta(N,k)^{\prime}=\Delta(N,k)$, so assume
$N$ is odd. Then 3.2 combined with the above lemma shows
$\Delta(N,k)^{\prime}$ is

$\frac{1}{4}(c(-4,M)-\mu(M))-\left(\prod_{p|M}{-1\overwithdelims(
)p}-(-1)^{k/2}\mu(M)\right)\dim S_{k}(1)+\mu(M)\delta_{k=2}.$

The sum of the first and the last term is always non-negative. Note the
coefficient of $\dim S_{k}(1)$ is negative if and only if $\mu(M)=(-1)^{k/2}$
and $\prod_{p|M}{-1\overwithdelims( )p}=1$. In this case,
$\Delta(N,k)^{\prime}<0$ if $k$ is sufficiently large. ∎

In general to describe $\operatorname{tr}W_{N}^{\min}$, one needs to isolate
the forms which are supercuspidal at each ramified place. One can presumably
do this in a similar way as 4.1, by subtracting off the contributions from
twists from smaller level. However, we will restrict ourselves to giving a
couple of examples to show that $(-1)^{k/2}\operatorname{tr}W_{N}^{\min}$ can
be negative, and then prove that often there is a quadratic twist which forces
$\operatorname{tr}W_{N}^{\min}=0$ to get 1.4.

The example of $N=4$ was already given in Section 1.2. The following examples
are taken from the LMFDB [LMFDB].

######  Example 4.2.

The space $S^{\mathrm{new}}_{10}(9)$ has 3 newforms, all rational. Two are
twists from $S^{\mathrm{new}}_{10}(3)$ by the quadratic character
${-3\overwithdelims( )\cdot}$, and they have root number $+1$ (though the
corresponding forms on $S^{\mathrm{new}}_{10}(3)$ have opposite root numbers).
The other newform in $S^{\mathrm{new}}_{10}(9)$, which is CM and minimal, has
root number $-1$. Hence the sum of root numbers of minimal newforms in
$S_{10}(9)$ is $-1$, though $\Delta(10,9)=+1$.

######  Example 4.3.

The space $S_{14}^{\min}(49)$ has 3 Galois orbits of newforms: one CM orbit of
size 1 and root number $+1$, one Galois orbit of size 6 and root number $+1$,
and one Galois orbit of size 12 and root number $-1$. Hence the sum of root
numbers for minimal forms is $1\cdot 1+1\cdot 6-1\cdot 12=-5$. On the other
hand, $\Delta(49,14)=2$. We remark that except for the first form, all
newforms are non-CM, and all of the newforms have nontrivial inner twists.

###### Proof of 1.4.

Here we allow any $N>1$, i.e., $N$ need not be a square or cubefree. Suppose
$p^{2}\parallel N^{\mathrm{odd}}$, and let $\chi$ be the quadratic character
of conductor $\pm p$. Then twisting any minimal newform $f\in S_{k}^{\min}(N)$
gives another newform of level $N$. The local argument for this is that
twisting a supercuspidal representation $\pi_{p}$ with $c(\pi_{p})=2$ by
$\chi_{p}$ yields a supercuspidal representation $\pi_{p}\otimes\chi_{p}$ also
of local conductor 2.

Hence twisting by $\chi$ is an involution on the set of minimal newforms. Now
any supercuspidal representation $\pi_{p}$ with $c(\pi_{p})=2$ is necessarily
minimal, and thus by [tunnell, Proposition 3.5], is induced from the
unramified quadratic extension of $\mathbb{Q}_{p}$. Hence by [pacetti, Theorem
3.2], the local root number of $\pi_{p}\otimes\chi_{p}$ is
$-{-1\overwithdelims( )p}$ times the local root number of $\pi_{p}$. Moreover,
for $q\mid p^{-2}N$, twisting by $\chi_{p}$ multiplies the root number of
$\pi_{q}$ by ${q\overwithdelims( )p}^{v_{q}(N)}$. In particular, if
${-N/p^{2}\overwithdelims( )p}=1$, then twisting by $\chi$ flips the sign of
the root number of minimal newforms, which implies
$\operatorname{tr}W_{N}^{\min}=0$. ∎

## References