11institutetext: Main Astronomical Observatory, National Academy of Sciences
of Ukraine, Kiev, 03143, Ukraine 22institutetext: Special Astrophysical
Observatory of the Russian Academy of Sciences, N.Arkhyz, KChR, 369167, Russia

# The M 101 galaxy group as a node in the nearby cosmic filament

Valentina E. Karachentseva The M 101 galaxy group as a node in the nearby
cosmic filamentThe M 101 galaxy group as a node in the nearby cosmic filament
Igor D. Karachentsev The M 101 galaxy group as a node in the nearby cosmic
filamentThe M 101 galaxy group as a node in the nearby cosmic filament Elena
I. Kaisina contact e-mail<EMAIL_ADDRESS>M 101 galaxy group as a
node in the nearby cosmic filamentThe M 101 galaxy group as a node in the
nearby cosmic filament Serafim S. Kaisin The M 101 galaxy group as a node in
the nearby cosmic filamentThe M 101 galaxy group as a node in the nearby
cosmic filament

We performed a search for faint low surface brightness dwarf galaxies around
the major spiral galaxy M 101 and in the large rectangular area within SGL =
[30 – 80]∘, SGB =[10 – 37]∘ spanning a chain of galaxies: M 63, M 51, M 101,
and NGC 6503, based on the data from DESI Legacy Imaging Surveys. Six new
supposed dwarf members of the complex were discovered. We present a list of 25
prospective members of the M 101 group and estimate the total mass and the
total-mass-to-$K$-band luminosity ratio of the group as $(1.02\pm 0.42)\times
10^{12}\leavevmode\nobreak\ M_{\odot}$ and $(16.0\pm 6.5)\leavevmode\nobreak\
M_{\odot}/L_{\odot}$, respectively. We notice that the average dark mass-to-
luminosity ratio in the groups around M 63, M 51, and M 101 is $(12\pm
4)M_{\odot}/L_{\odot}$ that almost an order of magnitude lower than the global
cosmic ratio, $(102\pm 5)M_{\odot}/L_{\odot}$.

###### Key Words.:

galaxies: dwarf — galaxies: groups: individual: M 101 — cosmology: large-scale
structure of Universe

##  1 Introduction

The filaments as the extending elongated structures, which consist of many
galaxies, are one of main elements of the large-scale structure of the
Universe, together with walls (sheets, pancakes), rich clusters, and cosmic
voids (Joeveer et al., 1978; Bond et al., 1995; Cautun et al., 2014). The
reality of the cosmic filaments is confirmed obviously by many N-body
simulations made in the standard $\Lambda$CDM cosmological model (Aragon-Calvo
et al., 2008; Chen et al., 2015; Libeskind et al., 2018, 2020). Characteristic
linear dimensions, population, and a degree of “straightness” of the filaments
remain not yet quite fixed, because they depend strongly on the criteria, by
which the cosmic filaments have been highlighted. The most rigorous definition
of a filament as one-dimensional sequence (a chain) of galaxies was proposed
by Tempel et al. (2014). They did publish a catalog of filaments based on the
Sloan Digital Sky Survey (SDSS) Data Release 8 (DR8) (York et al., 2000;
Aihara et al., 2011). Linear dimensions of these filaments are equal to
$\sim(3-60)$ Mpc. Here is still not clear understanding of whether the space
between the members of the filament is filled with neutral or hot gas and what
is the proportion of dark-to-light matter in these structures.

The answer to these important questions can be obtained by studying the
closest objects of this type. Within the Local supercluster, the best-known
filament is Virgo Southern Extension (VSE) (Tully & Shaya, 1984), whose length
reaches approximately 6 Mpc. Using the data on distances and radial velocities
of galaxies in VSE, Kashibadze et al. (2020) showed that galaxies of this
filament are moving to the Virgo cluster center, replenishing its population.
Kim et al. (2016) and Castignani et al. (2022) marked around the Virgo cluster
a dozen of other filament structures of a similar length. However, the
population and kinematics of these filaments are still almost not
investigated.

In the nearest Local volume limited by a distance of $D\sim 10$ Mpc around the
Milky Way, several candidates to linear structures have been marked. Jerjen et
al. (1998), Karachentsev et al. (2003), and Pruzhinskaya et al. (2020)
suggested that nearby galaxies in Sculptor constellation, i.e. NGC 55, NGC
300, NGC 247, NGC 253, NGC 24, NGC 45 and their satellites form a diffuse
chain which is elongated in the radial direction in the distance interval from
2 to 7 Mpc. According to Sandage & Tammann (1974), the northern sky bright
galaxies: IC 342, NGC 2403, M 81, NGC 4236 with their satellites can be
members of a chain located approximately in the picture plane at a distance of
3 – 4 Mpc from us. Müller et al. (2017) and Karachentsev et al. (2020)
supposed that the galaxy groups around massive spiral galaxies M 63 (NGC
5055), M 51 (NGC 5194) and M 101 (NGC 5457) are part of a filament that
extends towards the Local Void, including possibly the NGC 6503 galaxy too.

Karachentsev et al. (2020) and Karachentsev & Kashibadze (2021) reviewed
recently the structure and kinematics of the galaxy groups around M 63 and M
51. In this work, we discuss new data about the M 101 group population in the
context of its membership in the M 63/M 51/M 101 filament. The proximity of
this chain of groups allows make a clearer portrait of the cosmic filament,
while it is easier to measure distances to nearby galaxies and to exclude
projected foreground and background objects.

The paper is outlined as follows. We begin in Section 2 by describing the
searches for supposed new low surface brightness dwarf galaxies and their
photometry. Using the radial velocities and projected separations for eight M
101 group galaxies, we estimate the total (orbital) mass of the group. In
Section 3, we discuss the structure and population of a chain of galaxy
groups, which includes galaxies M 101, M 51, M 63, and possibly NGC 6503,
together with their satellites. We calculate the dark-to-luminous mass matter
in the filament groups, which is almost an order of magnitude lower than the
global cosmic ratio. In Section 4, we give our primary results, focusing on
the general size of the galaxy filament and briefly summarize our results. It
is also noticed that the environment of the filament is extremely poor, partly
belonging to the Local Void.

In Appendix LABEL:appendix, we present a list of 62 supposed members of the
filament as well as 10 galaxies in the immediate environment of the filament.
In all calculations we use the Hubble parameter, $H_{0}=73$ km s-1 Mpc-1.

##  2 Search for new dwarf galaxies in the M 101 group

For a long time, it was believed that only a few late-type dwarf galaxies,
i.e. NGC 5474, NGC 5477, NGC 5585, and Holmberg IV are associated with massive
spiral galaxy M 101. In the last decade, searches have been undertaken for
fainter satellites of M 101. Merritt et al. (2014) and Karachentsev et al.
(2015) performed a survey of the surroundings of M 101 using long exposures on
small telescopes with a large field of view. As a result, 10 low surface
brightness candidates to the satellites of M 101 were discovered. Bennet et
al. (2017) did repeat the survey of the virial zone around M 101, obtaining
deeper images of this region at the 3.6-m CFHT telescope. On these images,
made with high resolution, they detected 38 dwarf spheroidal galaxies with
apparent magnitudes ranging between 19 and 21 mag in the B-band, and angular
diameters of $\sim(10-20)\arcsec$. However, it later turned out that the most
part of them are associated with distant galaxy groups NGC 5422 and NGC 5485
located behind M 101 at a distance of $D\simeq 29$ Mpc (Karachentsev &
Makarova, 2019). Müller et al. (2017) undertook a search for dwarf galaxies
outside the virial zone of M 101 using the data of SDSS DR 11-12 (Alam et al.,
2015). As a result, they found six new candidates to the M 101 satellites.

At the Hubble Space Telescope (HST) with the ACS camera, there were obtained
images in filters F606W and F814W for two dozen supposed satellites of M 101
(the programs GO 13682, PI van Dokkum and GO 14796, PI Crnojevich). Based on
these images, distances to a number of dwarfs resolved into stars were
determined by the luminosity of the tip of the red giant branch ($=$TRGB)
(Merritt et al., 2016; Danieli et al., 2017; Karachentsev & Makarova, 2019).
Using the images obtained with CFHT, Carlsten et al. (2022) performed a
surface photometry of many dwarf galaxies in the M 101 virial zone and
determined their distances by the surface brightness fluctuation (sbf) method.

The galaxy group around M 101 is located in the area of DESI Legacy Imaging
Surveys (Dey et al., 2019). This survey is deeper than the SDSS sky survey by
about 1.5 mag. We use the DESI Legacy Imaging Surveys data to search for new
dwarf satellites of M 101. Our search region covered an area with a radius of
6∘ around M 101, which is almost 4 times the virial radius of the group.
Emphasis was placed on finding objects of low and very low surface brightness.
As a result, we found 5 candidates for M 101 peripheral satellites. The data
of them are presented in Table 1. Its columns indicate: the galaxy name;
equatorial coordinates in degrees; supergalactic coordinates in degrees; an
angular diameter of the galaxy in arc minutes, maximum visible in the DESI
Legacy Imaging Surveys; the apparent axis ratio; morphological type; the total
$g$ and $r$ band mag and the total $B$ mag estimated via the relation
$B=g+0.313(g-r)+0.227$, recommened by
Lupton111https://www.sdss3.org/dr10/algorithms/
sdssUBVRITransform.php#Lupton2005. The last raw of Table 1 contains the data
on a new dwarf galaxy found by us slightly away from the M 63 and M 51 groups.

Table 1: New candidates for M 101 satellites

Name | RA (2000.0) DEC | SGL | SGB | $a$ | $b/a$ | Type | $g$ | $r$ | $B$
---|---|---|---|---|---|---|---|---|---
| degree | degree | degree | arcmin |  |  | mag | mag | mag
KK 207 aa$a$Re-discovered by us. See note to this objects. | 203.357+56.500 | 61.43 | 18.32 | 0.97 | 0.70 | dIrr | 18.41 | 18.07 | 18.75
Dw1348+60 | 207.024+60.067 | 57.61 | 20.18 | 0.83 | 0.75 | dTr | 19.45 | 18.67 | 19.92
Dw1351+50 | 207.996+50.248 | 68.07 | 21.08 | 0.50 | 0.82 | dIrr | 20.10 | 19.56 | 20.50
Dw1358+52 | 209.531+52.918 | 65.18 | 21.95 | 0.72 | 0.92 | dSph | 19.86 | 19.03 | 20.35
Dw1409+51 | 212.304+51.225 | 66.89 | 23.76 | 1.70 | 0.58 | dIrr | 18.60 | 18.15 | 18.97
Dw1341+42 | 205.495+42.069 | 76.72 | 19.08 | 0.42 | 0.77 | dIrr | 19.80 | 19.75 | 20.04

222

The apparent $g$ and $r$ magnitudes were obtained from surface photometry of
galaxies in the DESI Legacy Imaging Surveys DR 9. We measured fluxes of these
galaxies in $g$\- and $r$-bands using the standard methods of processing in
the ESO-MIDAS software package. Foreground stars were removed from the frames
by fitting a second-degree surface in circular pixel area. The sky background
on the image was approximated by a tilted plane, created from a 2-dimensional
polynomial using the least-squares method (fit/flat sky). The mean uncertainty
introduced by the inaccuracy of the sky background determination is $\leq
0.12$ mag, being primarily caused by the background variations across the
frame. A part of our sample is comprised of galaxies of extremely low surface
brightness. The objects are characterized by an irregular, clumpy structure.
For this reason, we did not approximate the galaxies by ellipses, but used
circular apertures. The center of each galaxy was determined interactively. To
measure fluxes in $g$\- and $r$-bands the integrated photometry was performed
in increasing circular apertures from a pre chosen center to the faint
outskirts of the galaxies. The total flux was then estimated as the asymptotic
value of the obtained radial growth curve. The uncertainties of the total flux
determination were $0.06$ mag in $g$\- and $0.08$ mag in $r$-bands. We
obtained the apparent magnitudes according to the relation
$m=22.5-2.5\log{\rm(flux)}$333https://www.legacysurvey.org/dr10/description/.

The mosaic of images of these galaxies, taken from the DESI Legacy Imaging
Surveys, is given in Fig.1. One side of an image corresponds to 2$\arcmin$.
North is in the top, east is in the left.

Figure 1: Images of 6 new dwarf galaxies from the DESI Legacy Imaging Surveys
found in the M 63/M 51/M 101/NGC 6503 filament. Each image size is
$2^{\prime}\times 2^{\prime}$. North is to the top, East is to the left.

Below, we note some features of the discovered galaxies.

KK 207. This dwarf galaxy of irregular type was found earlier (Karachentseva &
Karachentsev, 1998) on the Second Palomar Observatory Sky Survey, but not
assigned to the M 101 group. Its structure seems as granulated one on the DESI
Legacy Imaging Surveys. The galaxy is also detected in the GALEX survey
(Martin et al., 2005) and has the apparent magnitude of $m({\rm FUV})=20.68$
mag.

Dw 1348+60. The dwarf galaxy of spheroidal (dSph) or transition (Tr) type of
very low surface brightness. Probably it is a companion of a distant galaxy
NGC 5322 with $V_{\rm LG}=1932$ km s-1 and $D=31.9$ Mpc (sbf), which has a
number other small dSph satellites. However, at a distance of 31.9 Mpc, the
galaxy Dw 1348+60 would have a linear diameter of 8 kpc, which is quite
unusual for a dwarf galaxy with such a low surface brightness. On the other
hand, Dw 1348+60 is located on far outskirts of the M 101 group, which raises
the question of the cause of suppressed star-formation in this galaxy.

Dw 1351+50. The dwarf galaxy of irregular- or transition-type of very low
surface brightness, which is barely visible in the SDSS survey.

Dw 1358+52. This dwarf spheroidal galaxy of very low surface brightness is
placed in the survey zone of Carlsten et al. (2022) but is not noticed by
them.

Dw 1409+51. This is a dwarf irregular galaxy of low surface brightness with a
blue central knot having in GALEX $m({\rm NUV})=20.09$ mag. A distant Sd-
galaxy UGC 9050 is lying at $8\arcmin$ to West from it. This neighboring
galaxy has a heliocentric velocity of $V_{h}=2001$ km s-1 and a distance
estimate of 32 Mpc. Both galaxies can form a physical pair. But in this case
the linear diameter of Dw 1409+51 could reach about 16 kpc, which is atypical
for such a dwarf object.444In a recent paper, C. Fielder et al.
(arXiv:2306.06164) studied Dw1409+51 = UGC 9050-Dw1 with Hubble Space
Telescope and Very Large Array. They found the object to be a background
ultra-diffuse galaxy with a heliocentric velocity of $1952$ km s-1.

Dw 1341+42. Isolated dwarf irregular galaxy with slightly granulated structure
and GALEX mag of $m({\rm NUV})=22.14$ mag.

Table 2: Galaxies associated with M 101

Name | RA (2000.0) DEC | $D$ | meth | $V_{\rm LG}$ | $\log L_{K}$ | Type | $R_{p}$ | Ref | $M_{\rm orb}$
---|---|---|---|---|---|---|---|---|---
(1) | (2) | (3) | (4) | (5) | (6) | (7) | (8) | (9) | (10)
KK 207 | 133325.7+563000 | 6.95 | mem | - | 6.47 | dIrr | 575 | (1) | -
dw1343+58 | 134307.0+581340 | 5.59 | TRGB | 338 | 7.46 | BCD | 582 | (2) | 1.10
Dw1348+60 | 134805.8+600401 | 6.95 | mem | - | 6.75 | dTr | 737 | (1) | -
dw1350+5441 | 135058.3+544120 | 6.27 | sbf | - | 7.54 | dSph | 220 | (3) | -
Dw1351+50 | 135159.0+501453 | 6.95 | mem | - | 5.75 | dTr | 545 | (1) | -
Holmberg IV | 135445.1+535417 | 7.24 | TRGB | 272 | 8.62 | Sm | 163 | (2) | 2.16
GBT1355+54 | 135450.6+543850 | 6.95 | mem | 345 | 7.10 | HIcld | 152 | (4) | 0.21
dw1355+51 | 135511.0+515429 | 6.95 | mem | - | 7.07 | dSph | 334 | (5) | -
M101Dw9 | 135544.6+550845 | 7.73 | TRGB | - | 6.26 | dSph | 166 | (6) | -
UGC 8882 | 135714.6+540603 | 6.95 | sbf | 482 | 7.51 | dEn | 110 | (3) | 1.40
Dw1358+52 | 135807.4+525505 | 6.95 | mem | - | 6.52 | dSph | 199 | (1) | -
M101-df3 | 140305.7+533656 | 6.52 | TRGB | - | 7.27 | dSph | 97 | (7) | -
M 101 | 140312.8+542102 | 6.95 | TRGB | 378 | 10.79 | Scd | 0 | (2) | -
M101-df1 | 140345.0+535640 | 6.37 | TRGB | - | 6.17 | dTr | 53 | (7) | -
NGC 5474 | 140502.1+533947 | 6.98 | TRGB | 424 | 9.21 | Sm | 96 | (2) | 0.24
NGC 5477 | 140533.1+542739 | 6.76 | TRGB | 451 | 8.26 | Im | 44 | (2) | 0.28
M101-DwA | 140650.2+534432 | 6.65 | TRGB | - | 6.68 | dSph | 102 | (6) | -
M101Dw7 | 140721.0+550351 | 6.95 | mem | - | 6.14 | dSph | 115 | (8) | -
M101-df2 | 140837.5+541931 | 6.87 | TRGB | - | 6.74 | dSph | 96 | (7) | -
Dw1409+51 | 140913.0+511330 | 6.95 | mem | - | 6.38 | dIrr | 403 | (1) | -
dw1412+56 | 141211.0+560831 | 6.95 | mem | - | 6.79 | dSph | 272 | (5) | -
dw1416+57 | 141659.0+575439 | 6.95 | mem | - | 6.95 | dSph | 499 | (5) | -
NGC 5585 | 141948.3+564349 | 7.00 | TRGB | 457 | 9.21 | Sdm | 410 | (2) | 3.02
DDO 194 | 143524.6+571524 | 5.81 | TRGB | 381 | 8.09 | Sm | 661 | (2) | 0.01
dw1446+58 | 144600.0+583404 | 6.95 | mem | - | 6.49 | dIrr | 911 | (5) | -

(1) this work; (2) Anand et al. (2021); (3) Carlsten et al. (2022); (4) Mihos
et al. (2012); (5) Müller et al. (2017); (6) Karachentsev & Makarova (2019);
(7) Danieli et al. (2017); (8) Bennet et al. (2017).

At present, a list of members of the M 101 group includes 25 candidates. A
summary of them is presented in Table 2. Its columns contain: (1) — galaxy
name; (2) — equatorial coordinates; (3) — distance to galaxy in Mpc; (4) —
method, by which the galaxy distance estimation is made: “TRGB” — by the
luminosity of the tip of red giants branch, “sbf” — by the surface brightness
fluctuations, “mem” — by supposed group membership; (5) — radial velocity of
galaxy in km s-1 relative to the centroid of the Local Group; (6) — logarithm
of the integral luminosity of galaxy in the $K$-band , in units of the
luminosity of the Sun, taken from Updated Nearby Galaxy Catalog ($=$UNGC)
(Karachentsev et al., 2013); (7) — morphological type taken from UNGC; (8) —
projected separation of the galaxy from M 101 in kpc assuming that all objects
are at the same distance as M 101; (9) — link to source of the distance data;
(10) — orbital estimate of the total mass of the group by projected
separation, $R_{p}$, and radial velocity difference, $\Delta V$, of each
satellite relative to M 101: $M_{\rm orb}=(16/\pi\times G)R_{p}\Delta V^{2}$
(Karachentsev & Kashibadze, 2021), where $G$ is the gravitational constant and
the mass is expressed in units of $M_{\odot}$. In the list of group members we
included also the intergalactic hydrogen cloud GBT 1355+54 found by Mihos et
al. (2012).

According to these data, the average value of the orbital estimate of the
total mass of the M 101 group is $\langle M_{\rm orb}\rangle=(1.05\pm
0.42)\times 10^{12}\leavevmode\nobreak\ M_{\odot}$. With the integral
luminosity of the group $\Sigma L_{K}=6.56\times 10^{10}\leavevmode\nobreak\
L_{\odot}$ the total mass-to-luminosity ratio, $\langle M_{\rm
orb}\rangle/\Sigma L_{K}$, is equal to $(16.0\pm 6.5)\leavevmode\nobreak\
M_{\odot}/L_{\odot}$. This quantity is typical for the groups where the late-
type spiral galaxy dominates.

It should be noted that the membership of some peripheral galaxies in the M
101 group is questionable. So, the dwarf galaxies of very low surface
brightness: Dw 1348+60 and Dw 1409+51 may be satellites of distant galaxies
NGC 5322 and UGC 9050 at a distance of about 32 Mpc. Besides, dwarf galaxies
dw 1446+58 and dw 1416+57, found by Müller et al. (2017), look in the DESI
Legacy Imaging Surveys as background galaxies. Being without measured radial
velocities, these 4 galaxies do not participate in the orbital mass
estimation.

##  3 The chain of groups M 63/M 51/M 101/NGC 6503

The basic parameters of these four groups are presented in Table 3, whose
columns contain: (1) — the main galaxy name; (2) — its supergalactic
coordinates; (3, 4) — distance (Mpc) and radial velocity (km s-1) of the host
galaxy; (5) — number of the group member candidates; (6) — number of
satellites with a measured radial velocity; (7) — the total (orbital) mass in
$M_{\odot}$ units; (8) — virial radius of the group (kpc) estimated from Tully
(2015) relation ($R_{\rm vir}/215$ kpc) $=(M_{T}/10^{12}\leavevmode\nobreak\
M_{\odot})^{1/3}$ ; (9) — the total mass-to-luminosity ratio. For the groups
of galaxies M 63 and M 51 their mass estimates are taken from Karachentsev &
Kashibadze (2021) , and for NGC 6503 the mass estimate is made by only one
satellite (Karachentsev et al., 2022).

Table 3:  Galaxy groups in the M 101 filament

Name | SGL SGB | $D$ | $V_{\rm LG}$ | $n$ | $n_{v}$ | $\log(M_{T})$ | $R_{\rm vir}$ | $M_{T}/L_{K}$
---|---|---|---|---|---|---|---|---
(1) | (2) | (3) | (4) | (5) | (6) | (7) | (8) | (9)
M 63 | 76.20+14.25 | 9.04 | 562 | 21 | 7 | 11.71 | 172 | 5$\pm$2
M 51 | 71.12+17.33 | 8.40 | 538 | 16 | 5 | 12.13 | 237 | 10$\pm$8
M 101 | 63.58+22.61 | 6.95 | 378 | 25 | 8 | 12.02 | 220 | 16$\pm$6
N 6503 | 33.14+34.63 | 6.25 | 309 | 2 | 1 | 11.5 | 126 | 40

Figure 2: Relation between radial velocities and distances of the principal
galaxies in the M 101 filament. The horizontal bars correspond to 5-percent
errors in the galaxy distances. The oblique line indicates the linear
regression with a slope of $H=(95\pm 7)$ km s-1Mpc-1.

As can be seen from these data, the galaxy NGC 6503 with its satellite KK 242
is quite far from other members of the filament. Nevertheless, its distance
and radial velocity fit well with the general relation between $V_{LG}$ and
$D$ (see Fig. 2). The linear regression in the Fig. 2 passes through each a 5%
— distance error interval for all the galaxies. The slope of the regression
line in the Fig. 2 corresponds formally to the Hubble parameter $H=(95\pm 7)$
km s-1Mpc-1. The large value of the Hubble parameter is explained in the
Numerical Action Methods model (Shaya et al., 2017), which describes the local
velocity field taking into account the expansion of the Local Void and the
Virgo-centric flow. According to a ”Distance - Velocity” diagram in the
Extragalactic Distance Database555http://edd.ifa.hawaii.edu/NAMcalculator
(Kourkchi et al., 2020), the galaxy M 63 at the far end of the filament has a
peculiar radial velocity of $-50$ km s-1 relative to the unperturbed Hubble
flow with $H=73$ km s-1Mpc-1, whereas, the galaxy NGC 6503 at the close end of
the chain has a peculiar velocity of $-115$ km s-1. This difference in
peculiar velocities results in a steeper observed slope, $H=96$ km s-1Mpc-1
instead of $73$ km s-1Mpc-1.

Figure 3: Survey area of about 1200 square degrees in the M 63/M 51/M 101/NGC
6503 filament region. The filled circles identify supposed satellites of the
principal galaxies in the chain of groups, which are shown as squares. Crosses
identify other galaxies with distances from 5 to 10 Mpc, not assosiated with
the filament. The dwarf galaxies discovered by us are shown by asteriscs. The
red symbols correspond to early-type galaxies ($T<0$), while the blue symbols
represent late-type ones. Small and large circles around the principal
galaxies indicate their virial radius, $R_{\rm vir}$, and the zero-velocity
radius, $R_{0}$.

A general shape of the chain of considered groups in supergalactic coordinates
is shown in Fig. 3. The early-type and late-type galaxies depicted by red and
blue symbols, respectively. The host galaxies in groups are indicated by
squares. Small and large circles around them correspond to the virial radius
of group, $R_{\rm vir}$, and the radius of zero-velocity sphere, $R_{0}\simeq
3.5\leavevmode\nobreak\ R_{\rm vir}$, which separates the volume of the group
from the general expanding cosmic space. Six new dwarf galaxies found by us
are indicated by asterisks. They can be identified by their supergalactic
coordinates presented in Table 1.

In addition to the alleged members of specified groups, we show by crosses in
Fig. 3 other galaxies from the UNGC catalog with their distance estimate in
the interval (5 – 10) Mpc. Almost all of them (9 out of 10) lie near the Local
supergalactic plane (SGB $<15^{\circ}$).

General summary of data on 72 galaxies in the region SGL =
$[30^{\circ}-80^{\circ}]$, SGB = $[10^{\circ}-37^{\circ}]$ with the distance
estimates $D=(5-10)$ Mpc is presented in Appendix LABEL:appendix. The columns
of this table contain: (1) — the galaxy name; (2,3) — supergalactic
coordinates in degrees; (4) — $K$-band integral luminosity in the Sun
luminosity units, taken from UNGC; (5) — morphological type from UNGC in de
Vaucouleurs digital scale; (6) — radial velocity relative to the Local group
centroid, in km s-1; (7) — distance in Mpc; (8) method, by which the distance
estimation is made; here “SN” denotes the distance estimated from the
luminosity of supernovae, “TF” is the estimate by Tully-Fisher relation
between the galaxy rotation amplitude and luminosity, “NAM” is a kinematic
estimate taking into account the local velocity field (Shaya et al., 2017),
and “txt” is the rough estimate by a galaxy texture. The galaxies in each
group are ranked according to their SGL. In Appendix LABEL:appendix and in
Table 2, we distinguish between the dwarf galaxies that we discovered and
known from the literature as ”Dw” and ”dw”, respectively.

We examined in DESI Legacy Imaging Surveys all sky region outlined in Fig. 3
of a total area of about 1200 square degrees and found only one dwarf galaxy
with supposed distance of $D<10$ Mpc. The galaxy is shown in last row of Table
1. This fact tell us that the main body of filament M 63/M 51/M 101/ NGC 6503
is indeed surrounded by almost empty space, being part of the Local Void,
which extends towards the northern supergalactic pole.

The M 63 group as the farthest side of filament from us is the closest one to
the Local Sheet plane. Wherein, we did not find any massive group in the Local
Sheet itself which could serve as a southward continuation of the filament
under consideration.

As seen from Fig. 3, the zones of gravitational influence of the massive
spiral galaxies M 63, M 51, and M 101, outlined by the radius $R_{0}$,
intersect with each other. This means that in the future these groups will
merge into a single system. The galaxy NGC 6503 with its companion stand apart
from the main body of the filament.

The dwarf spheroidal galaxies without obvious sings of star formation are
located inside the $R_{0}$ zones, demonstrating the well-known segregation
effect of the early type galaxies ($T<0$) against the late type ones in
dependence on the density of environment.

From the data of Table 3 follows that the average weighted ratio of the total
mass-to-$K$-band luminosity for the M 63, M 51, and M 101 groups is $(12\pm
4)\leavevmode\nobreak\ M_{\odot}/L_{\odot}$. According to Driver et al.
(2012), the average global density of the K- luminosity is equal to $(4.3\pm
0.2)\times 10^{8}\leavevmode\nobreak\ L_{\odot}$/Mpc3 at $H_{0}=73$ km
s-1Mpc-1. At the value of the critical cosmic density $1.46\times
10^{11}M_{\odot}$ Mpc-3 and the parameter $\Omega_{m}=0.3$, the ratio of the
global density of dark matter to the global density of $K$-luminosity is
$(102\pm 5)\leavevmode\nobreak\ M_{\odot}/L_{\odot}$. Therefore, the ratio of
dark-to-luminous matter in the M 63/ M 51/ M 101 filament groups turns out to
be 8.5 times less than the global cosmic ratio. In order for this chain of
groups not to look like a “dent” in the general cosmological field, it is
necessary to assume that the bulk of dark matter of the filament is
distributed beyond the virial radii of these three groups. In such a case, the
estimates of virial radius from the Tully’s relation would be rather
uncertain. The reason for the low dark-to-stellar mass ratio in the groups
under consideration remains unclear to us. Possibly, the dark mass deficit of
the filament groups can be caused by their proximity to the Local Void. A
special deep survey of this filament and its vicinity in the 21 cm line could
reveal new small dwarfs and HI- clouds in the system, thereby increasing the
statistical reliability of estimates of the total mass of the groups.

##  4 Concluding remarks on the filament

The extended galaxy chains, i.e. filaments, are a common configuration in the
large-scale structure of Universe. According to Tempel et al. (2014), about 35
– 40% of all galaxies enter into cosmic filaments. We have considered one of
their nearest representatives, which is inside the Local Volume. Main elements
of this filament are the groups around the massive spiral galaxies M 63, M 51,
and M 101, which are in contact with each other. The spiral galaxy NGC 6503
with its satellite KK 242 may serve as a continuation of this chain. The M 63,
M 51, M 101, and NGC 6503 radial velocities as well as their distances from
the observer smoothly change along the body of the filament. The far end of
this chain at the distance of 9 Mpc is adjacent to the Local Sheet plane,
while the near end at the distance of 6 Mpc is embedded in the Local Void
volume. A total filament length, taking into account NGC 6503, is $\sim
60^{\circ}$ or 8 Mpc.

Using the data of DESI Legacy Imaging Surveys we have undertaken the searches
for new dwarf galaxies in a wide region of $50^{\circ}\times 27^{\circ}$
designated in Fig. 3. In the M 101 group itself, 5 new candidates for
satellites of the host galaxy were discovered, and only one dwarf galaxy was
found outside the filament body. The list of 72 galaxies in this region with
the distance estimates from 5 Mpc to 10 Mpc is presented in Appendix
LABEL:appendix. Of these, only 10 galaxies are located outside the filament
body, and most of them, 9 out of 10, have a low supergalactic latitude,
tending towards the Local Sheet.

The proximity of this cosmic filament made it possible for the first time to
estimate the ratio of dark to light matter in it, $M_{T}/L_{K}$. The average
weighted ratio $\langle M_{T}/L_{K}\rangle\simeq(12\pm 4)\leavevmode\nobreak\
M_{\odot}/L_{\odot}$ for the filament groups turned out to be almost an order
of magnitude less than the global ratio $(102\pm 5)\leavevmode\nobreak\
M_{\odot}/L_{\odot}$ in the Standard Model $\Lambda$CDM. The reason for this
paradox deserves careful analysis.

Despite great efforts made in the last quarter of the century to measure
accurate distances with the Hubble Space Telescope, more than 60% galaxies in
the zone of the nearest filament (see Appendix LABEL:appendix) do not yet have
reliable distance estimates. Obviously, this gap needs to be filled in order
to better understand the structure of the cosmic filament closest to us.

###### Acknowledgements.

We are grateful to the anonymous referee for a prompt report that helped us
improve the manuscript. This work has made use of the DESI Legacy Imaging
Surveys666https://www.legacysurvey.org/, the Sloan Digital Sky Survey
(SDSS)777https://www.sdss3.org/, the NASA/IPAC Extragalactic
Database888http://www.ned.ipac.caltech.edu (NED), The Galaxy Evolution
Explorer (GALEX)999http://www.galex.caltech.edu/index.html,
HyperLeda101010http://leda.univ-lyon1.fr/, and the revised version of the
Local Volume galaxy database111111http://www.sao.ru/lv/lvgdb. The Local Volume
galaxies database has been updated within the framework of grant
075–15–2022–262 (13.MNPMU.21.0003) of the Ministry of Science and Higher
Education of the Russian Federation.

## References

  * Aihara et al. (2011) Aihara, H., Allende, P.C., An, D., et al., 2011, ApJS, 193, 29
  * Alam et al. (2015) Alam, S., Albareti, F.D., Allende, P.C., et al., 2015, ApJS, 219, 12
  * Anand et al. (2021) Anand, G.S., Rizzi, L., Tully, R.B., et al., 2021, AJ, 162, 80
  * Aragon-Calvo et al. (2008) Aragon-Calvo, M.A., Platen, E., van de Weygaert, R., & Szalay, A.S., 2008, ApJ, 723, 364
  * Bennet et al. (2017) Bennet, P., Sand, D.J., Crnojevich, D., et al., 2017, ApJ, 805, 109
  * Bond et al. (1995) Bond, J.R., Kofman, L., & Pogosyan, D., 1995, Nature, 380, 603
  * Carlsten et al. (2022) Carlsten, S.G., Greene, J.E., & Beaton, R.L., 2022, ApJ, 933, id.47
  * Castignani et al. (2022) Castignani, G., Vulcani, B., Finn, R.A., et al., 2022, ApJS, 259,43
  * Cautun et al. (2014) Cautun, M., van de Weygaert, R., Jones, B.J.T., & Frenk, C.S., 2014, MNRAS, 441, 2923
  * Chen et al. (2015) Chen, Y.C., Ho, S., Tenneti, A., et al., 2015, MNRAS, 454, 3341
  * Danieli et al. (2017) Danieli, S., van Dokkum, P., Merritt, A., et al., 2017, ApJ, 837, 136
  * Dey et al. (2019) Dey, A., Schlegel, D.J., Lang, D., et al., 2019, AJ, 157, 168
  * Driver et al. (2012) Driver, S.P., Robotham, A.S.G., Kelvin, L., et al., 2012, MNRAS, 427, 3244
  * Jerjen et al. (1998) Jerjen, H., Freeman, K.C., & Binggeli, B., 1998, AJ, 116, 2873
  * Joeveer et al. (1978) Joeveer, M., Einasto, J., & Tago, E., 1978, MNRAS, 185, 357
  * Karachentseva & Karachentsev (1998) Karachentseva, V.E., & Karachentsev, I.D., 1998, A&AS, 127, 409
  * Karachentsev et al. (2022) Karachentsev, I.D., Cannon, J.M., Fuson, J., et al., 2022, AJ, 163, 51
  * Karachentsev & Kashibadze (2021) Karachentsev, I., & Kashibadze, O., 2021, Astr. Nachr., 342, 999
  * Karachentsev et al. (2020) Karachentsev, I.D., Neyer, F., Spani, R., & Zilch, T., 2020, Astr. Nachr., 341, 1037
  * Karachentsev & Makarova (2019) Karachentsev, I.D., & Makarova, L.N., 2019, Ap, 62, 293
  * Karachentsev et al. (2015) Karachentsev, I.D., Riepe, P., Zilch, T., et al., 2015, AstBull, 70, 379
  * Karachentsev et al. (2013) Karachentsev, I.D., Makarov, D.I., & Kaisina, E.I., 2013, AJ, 145, 101
  * Karachentsev et al. (2003) Karachentsev, I.D., Grebel, E.K., Sharina, M.E., et al., 2003, A & A, 404, 93
  * Kashibadze et al. (2020) Kashibadze, O.G., Karachentsev, I.D., & Karachentseva, V.E., 2020, A & A, 635A, 135
  * Kim et al. (2016) Kim, S., Rey, S.-C., Bureau, M., et al., 2016, ApJ, 833, 207
  * Kourkchi et al. (2020) Kourkchi, E., Courtois, H.M., Graziani, R., et al., 2020, AJ, 159, 67
  * Libeskind et al. (2020) Libeskind, N.I., Carlesi, E., Grand, R.J., et al., 2020, MNRAS, 498, 2968
  * Libeskind et al. (2018) Libeskind, N.I., van de Weygaert, R., Cautun, M., et al., 2018, MNRAS, 473, 1195
  * Martin et al. (2005) Martin, D.C., Fanson, J., Schiminovich, D., et al., 2005, ApJ, 619, L1
  * Merritt et al. (2016) Merritt, A., van Dokkum, P., Danieli, S., et al., 2016, ApJ, 833., 168
  * Merritt et al. (2014) Merritt, A., van Dokkum, P., & Abraham, R., 2014, ApJ, 787, L37
  * Mihos et al. (2012) Mihos, J.C., Keating, K., Holley-Bockelmann, K., et al, 2012, ApJ, 761, 186
  * Müller et al. (2017) Müller, O., Scalera, R., Binggeli, B., & Jerjen, H., 2017, A & A, 602, A119
  * Pruzhinskaya et al. (2020) Pruzhinskaya, M.V., Chernin, A.D., & Karachentsev, I.D., 2020, Ap&SS, 365, 120
  * Sandage & Tammann (1974) Sandage, A., & Tammann, G.A., 1974, ApJ, 191, 603
  * Shaya et al. (2017) Shaya, E.L., Tully, R.B., Hoffman, Y., & Pomarede, D., 2017, ApJ, 850, 207
  * Tempel et al. (2014) Tempel, E., Stoica, R.S., Martinez, V.J., et al., 2014, MNRAS, 438, 3465
  * Tully (2015) Tully, R.B., 2015, AJ, 149, 54
  * Tully & Shaya (1984) Tully, R.B., & Shaya, E.J., 1984, ApJ, 281, 31
  * York et al. (2000) York, D.G., Adelman, J., Anderson, J.E.Jr., et al., 2000, AJ, 120, 1579

##  Appendix A M101 filament of galaxies

Table 1: The list of 72 galaxies in the inspected region with distance estimates from 5 Mpc to 10 Mpc Name | SGL | SGB | $\log L_{K}$ | T | $V_{\rm LG}$ | $D$ | meth
---|---|---|---|---|---|---|---
(1) | (2) | (3) | (4) | (5) | (6) | (7) | (8)
M 63 | 76.20 | 14.25 | 11.00 | 4 | 562 | 9.04 | TRGB
KK 194 | 73.82 | 14.85 | 7.15 | 10 | - | 9.04 | mem
Dw1311+4317 | 74.80 | 13.70 | 7.21 | -2 | - | 9.04 | mem
Dw1315+4304 | 75.10 | 14.25 | 7.13 | -2 | - | 9.04 | mem
dw1303+42 | 75.46 | 12.00 | 6.88 | 10 | - | 9.04 | mem
dw1321+4226 | 75.94 | 15.24 | 6.50 | -2 | - | 7.30 | sbf
UGC 8313 | 75.96 | 13.92 | 8.44 | 8 | 683 | 8.77 | sbf
dw1305+41 | 76.02 | 12.33 | 7.28 | 10 | - | 9.04 | mem
KK 191 | 76.12 | 13.85 | 6.83 | 10 | - | 8.28 | TRGB
TBGdw1 | 76.15 | 13.59 | 6.20 | 10 | - | 9.26 | sbf
dw1310+4153 | 76.19 | 13.28 | 6.90 | -1 |  | 7.38 | sbf
KKH 82 | 76.36 | 13.69 | 7.77 | 10 | 588 | 7.58 | sbf
TBGdw5 | 76.63 | 13.91 | 6.52 | -2 | - | 9.04 | mem
KK 193 | 76.73 | 14.11 | 7.03 | 10 | - | 9.65 | sbf
dw1315+4123 | 76.83 | 14.05 | 6.43 | -2 | - | 8.80 | sbf
dw1308+40 | 77.13 | 12.76 | 7.54 | -1 | - | 9.04 | mem
Dw1311+4051 | 77.26 | 13.30 | 6.55 | 10 | - | 9.04 | mem
CGCG 217-018 | 77.62 | 13.47 | 8.19 | 9 | 608 | 8.99 | TRGB
DDO 182 | 79.27 | 18.99 | 7.89 | 10 | 730 | 8.90 | TRGB
dw1305+38 | 79.85 | 11.73 | 7.03 | 10 | - | 9.04 | mem
M 51 | 71.12 | 17.33 | 10.97 | 5 | 538 | 8.40 | SN
dw1327+51 | 66.44 | 17.19 | 7.00 | -1 | - | 8.40 | mem
dw1338+50 | 68.29 | 18.96 | 7.08 | -1 | - | 8.40 | mem
UGCA 361 | 68.44 | 17.96 | 8.13 | -1 | - | 8.40 | mem
LV J1328+4937 | 68.61 | 17.28 | 7.12 | 10 | 497 | 7.73 | TRGB
KK 206 | 69.21 | 18.04 | 8.25 | 9 | 690 | 9.31 | TF
LV J1342+4840 | 69.72 | 19.49 | 7.60 | 9 | 543 | 8.40 | mem
NGC 5229 | 70.46 | 18.08 | 8.56 | 7 | 456 | 8.95 | TRGB
NGC 5195 | 71.09 | 17.35 | 10.59 | 0 | 548 | 7.66 | sbf
dw1328+4703 | 71.28 | 17.06 | 6.51 | -2 | - | 8.35 | sbf
dw1313+46 | 71.45 | 14.40 | 6.98 | 10 | - | 8.40 | mem
dw1340+45 | 72.87 | 19.07 | 6.74 | 10 | - | 8.40 | mem
MCG+08-25-028 | 73.98 | 18.31 | 7.78 | 10 | 565 | 8.47 | TRGB
PGC 2229803 | 74.65 | 16.66 | 7.42 | 9 | 529 | 6.58 | NAM
Dw 1341+42 | 76.72 | 19.08 | 5.93 | 10 | - | 8.40 | mem
M 101 | 63.58 | 22.61 | 10.79 | 6 | 378 | 6.95 | TRGB
Dw1348+60 | 57.61 | 20.18 | 6.75 | 10 | - | 6.95 | mem
dw1446+58 | 57.83 | 27.82 | 6.49 | 10 | - | 6.95 | mem
dw1416+57 | 59.49 | 24.10 | 6.95 | -2 | - | 6.95 | mem
dw1343+58 | 59.59 | 19.62 | 7.46 | 9 | 338 | 5.59 | TRGB
DDO 194 | 59.68 | 26.65 | 8.09 | 8 | 381 | 5.81 | TRGB
NGC 5585 | 60.70 | 24.67 | 9.21 | 7 | 457 | 7.00 | TRGB
KK 207 | 61.43 | 18.32 | 6.47 | 10 | - | 6.95 | mem
dw1412+56 | 61.50 | 23.70 | 6.79 | -2 | - | 6.95 | mem
M101Dw7 | 62.75 | 23.15 | 6.14 | -2 | - | 6.95 | mem
M101Dw9 | 62.80 | 21.48 | 6.26 | -2 | - | 7.73 | TRGB
dw1350+5441 | 63.32 | 20.82 | 7.54 | -2 | - | 6.27 | sbf
GBT1355+54 | 63.34 | 21.38 | 7.10 | 11 | 345 | 6.95 | mem
NGC 5477 | 63.43 | 22.94 | 8.26 | 9 | 451 | 6.76 | TRGB
M101-df2 | 63.53 | 23.40 | 6.74 | -2 | - | 6.87 | TRGB
UGC 8882 | 63.91 | 21.76 | 7.51 | -1 | 482 | 6.95 | sbf
M101-df1 | 64.01 | 22.72 | 6.17 | 10 | - | 6.37 | TRGB
NGC 5474 | 64.30 | 22.93 | 9.21 | 8 | 424 | 6.98 | TRGB
M101-df3 | 64.38 | 22.65 | 7.27 | -2 | - | 6.52 | TRGB
M101-DwA | 64.19 | 23.19 | 6.68 | -2 | - | 6.65 | TRGB
Holm IV | 64.14 | 21.40 | 8.62 | 8 | 272 | 7.24 | TRGB
Dw1358+52 | 65.18 | 21.95 | 6.52 | -2 | - | 6.95 | mem
dw1355+51 | 66.28 | 21.54 | 7.07 | -2 | - | 6.95 | mem
Dw1409+51 | 66.89 | 23.76 | 6.38 | 10 | - | 6.95 | mem
Dw1351+50 | 68.07 | 21.08 | 5.75 | 10 | - | 6.95 | mem
NGC 6503 | 33.14 | 34.63 | 10.00 | 6 | 309 | 6.25 | TRGB
KK 242 | 32.81 | 34.73 | 6.47 | 10 | 179 | 6.46 | TRGB
UGC 7748 | 48.59 | 12.71 | 7.68 | 9 | 590 | 8.70 | TF
Dw1245+6158 | 55.23 | 12.76 | 6.41 | 10 | - | 5.60 | mem
NGC 4605 | 55.47 | 12.02 | 9.70 | 8 | 284 | 5.55 | TRGB
KDG 162 | 58.56 | 10.76 | 6.87 | 10 | - | 10.0 | txt
KKH 87 | 60.42 | 23.98 | 7.73 | 10 | 473 | 8.87 | TRGB
KDG 192 | 63.23 | 11.02 | 9.66 | 10 | 544 | 9.66 | TRGB
UGC 7950 | 65.67 | 11.02 | 8.36 | 9 | 599 | 8.90 | TF
NGC 4707 | 66.16 | 11.15 | 8.25 | 10 | 588 | 6.52 | TRGB
BTS 157 | 70.34 | 10.80 | 7.22 | 10 | 644 | 9.04 | mem
NGC 5023 | 74.05 | 13.89 | 6.05 | 7 | 476 | 6.05 | TRGB
Table 1: continued.


11 | $d_{R}^{c2}$ | $\stackrel{{\scriptstyle 03}}{{(+i)}}\,\stackrel{{\scriptstyle 12}}{{[+]}}|\stackrel{{\scriptstyle 56}}{{(-)}}\,\stackrel{{\scriptstyle 78}}{{[-]}}||\stackrel{{\scriptstyle 9\;10}}{{[-]}}\;\;\stackrel{{\scriptstyle 11\;12}}{{(+)}}\;\;\stackrel{{\scriptstyle 13\;14}}{{[-]}}$ | 1 | $\frac{1}{2}$ | 0 | $-\frac{1}{2}$ | $-\frac{1}{2}$ | $\frac{1}{2\,\sqrt{3}}$ | $\frac{1}{6}$ | $-\frac{1}{3}$ | $-\frac{1}{3}$
12 | $d_{R}^{c2}$ | $\stackrel{{\scriptstyle 03}}{{[-i]}}\,\stackrel{{\scriptstyle 12}}{{(-)}}|\stackrel{{\scriptstyle 56}}{{(-)}}\,\stackrel{{\scriptstyle 78}}{{[-]}}||\stackrel{{\scriptstyle 9\;10}}{{[-]}}\;\;\stackrel{{\scriptstyle 11\;12}}{{(+)}}\;\;\stackrel{{\scriptstyle 13\;14}}{{[-]}}$ | 1 | $-\frac{1}{2}$ | 0 | $-\frac{1}{2}$ | $-\frac{1}{2}$ | $\frac{1}{2\,\sqrt{3}}$ | $\frac{1}{6}$ | $-\frac{1}{3}$ | $-\frac{1}{3}$
13 | $d_{L}^{c2}$ | $\stackrel{{\scriptstyle 03}}{{[-i]}}\,\stackrel{{\scriptstyle 12}}{{[+]}}|\stackrel{{\scriptstyle 56}}{{(-)}}\,\stackrel{{\scriptstyle 78}}{{(+)}}||\stackrel{{\scriptstyle 9\;10}}{{[-]}}\;\;\stackrel{{\scriptstyle 11\;12}}{{(+)}}\;\;\stackrel{{\scriptstyle 13\;14}}{{[-]}}$ | -1 | $\frac{1}{2}$ | $-\frac{1}{2}$ | 0 | $-\frac{1}{2}$ | $\frac{1}{2\,\sqrt{3}}$ | $\frac{1}{6}$ | $\frac{1}{6}$ | $-\frac{1}{3}$
14 | $d_{L}^{c2}$ | $-\stackrel{{\scriptstyle 03}}{{(+i)}}\,\stackrel{{\scriptstyle 12}}{{(-)}}|\stackrel{{\scriptstyle 56}}{{(-)}}\,\stackrel{{\scriptstyle 78}}{{(+)}}||\stackrel{{\scriptstyle 9\;10}}{{[-]}}\;\;\stackrel{{\scriptstyle 11\;12}}{{(+)}}\;\;\stackrel{{\scriptstyle 13\;14}}{{[-]}}$ | -1 | $-\frac{1}{2}$ | $-\frac{1}{2}$ | 0 | $-\frac{1}{2}$ | $\frac{1}{2\,\sqrt{3}}$ | $\frac{1}{6}$ | $\frac{1}{6}$ | $-\frac{1}{3}$
15 | $u_{L}^{c2}$ | $-\stackrel{{\scriptstyle 03}}{{[-i]}}\,\stackrel{{\scriptstyle 12}}{{[+]}}|\stackrel{{\scriptstyle 56}}{{[+]}}\,\stackrel{{\scriptstyle 78}}{{[-]}}||\stackrel{{\scriptstyle 9\;10}}{{[-]}}\;\;\stackrel{{\scriptstyle 11\;12}}{{(+)}}\;\;\stackrel{{\scriptstyle 13\;14}}{{[-]}}$ | -1 | $\frac{1}{2}$ | $\frac{1}{2}$ | 0 | $-\frac{1}{2}$ | $\frac{1}{2\,\sqrt{3}}$ | $\frac{1}{6}$ | $\frac{1}{6}$ | $\frac{2}{3}$
16 | $u_{L}^{c2}$ | $\stackrel{{\scriptstyle 03}}{{(+i)}}\,\stackrel{{\scriptstyle 12}}{{(-)}}|\stackrel{{\scriptstyle 56}}{{[+]}}\,\stackrel{{\scriptstyle 78}}{{[-]}}||\stackrel{{\scriptstyle 9\;10}}{{[-]}}\;\;\stackrel{{\scriptstyle 11\;12}}{{(+)}}\;\;\stackrel{{\scriptstyle 13\;14}}{{[-]}}$ | -1 | $-\frac{1}{2}$ | $\frac{1}{2}$ | 0 | $-\frac{1}{2}$ | $\frac{1}{2\,\sqrt{3}}$ | $\frac{1}{6}$ | $\frac{1}{6}$ | $\frac{2}{3}$
17 | $u_{R}^{c3}$ | $\stackrel{{\scriptstyle 03}}{{(+i)}}\,\stackrel{{\scriptstyle 12}}{{[+]}}|\stackrel{{\scriptstyle 56}}{{[+]}}\,\stackrel{{\scriptstyle 78}}{{(+)}}||\stackrel{{\scriptstyle 9\;10}}{{[-]}}\;\;\stackrel{{\scriptstyle 11\;12}}{{[-]}}\;\;\stackrel{{\scriptstyle 13\;14}}{{(+)}}$ | 1 | $\frac{1}{2}$ | 0 | $\frac{1}{2}$ | $0$ | $-\frac{1}{\sqrt{3}}$ | $\frac{1}{6}$ | $\frac{2}{3}$ | $\frac{2}{3}$
18 | $u_{R}^{c3}$ | $\stackrel{{\scriptstyle 03}}{{[-i]}}\,\stackrel{{\scriptstyle 12}}{{(-)}}|\stackrel{{\scriptstyle 56}}{{[+]}}\,\stackrel{{\scriptstyle 78}}{{(+)}}||\stackrel{{\scriptstyle 9\;10}}{{[-]}}\;\;\stackrel{{\scriptstyle 11\;12}}{{[-]}}\;\;\stackrel{{\scriptstyle 13\;14}}{{(+)}}$ | 1 | $-\frac{1}{2}$ | 0 | $\frac{1}{2}$ | $0$ | $-\frac{1}{\sqrt{3}}$ | $\frac{1}{6}$ | $\frac{2}{3}$ | $\frac{2}{3}$
19 | $d_{R}^{c3}$ | $\stackrel{{\scriptstyle 03}}{{(+i)}}\,\stackrel{{\scriptstyle 12}}{{[+]}}|\stackrel{{\scriptstyle 56}}{{(-)}}\,\stackrel{{\scriptstyle 78}}{{[-]}}||\stackrel{{\scriptstyle 9\;10}}{{[-]}}\;\;\stackrel{{\scriptstyle 11\;12}}{{[-]}}\;\;\stackrel{{\scriptstyle 13\;14}}{{(+)}}$ | 1 | $\frac{1}{2}$ | 0 | $-\frac{1}{2}$ | $0$ | $-\frac{1}{\sqrt{3}}$ | $\frac{1}{6}$ | $-\frac{1}{3}$ | $-\frac{1}{3}$
20 | $d_{R}^{c3}$ | $\stackrel{{\scriptstyle 03}}{{[-i]}}\,\stackrel{{\scriptstyle 12}}{{(-)}}|\stackrel{{\scriptstyle 56}}{{(-)}}\,\stackrel{{\scriptstyle 78}}{{[-]}}||\stackrel{{\scriptstyle 9\;10}}{{[-]}}\;\;\stackrel{{\scriptstyle 11\;12}}{{[-]}}\;\;\stackrel{{\scriptstyle 13\;14}}{{(+)}}$ | 1 | $-\frac{1}{2}$ | 0 | $-\frac{1}{2}$ | $0$ | $-\frac{1}{\sqrt{3}}$ | $\frac{1}{6}$ | $-\frac{1}{3}$ | $-\frac{1}{3}$
21 | $d_{L}^{c3}$ | $\stackrel{{\scriptstyle 03}}{{[-i]}}\,\stackrel{{\scriptstyle 12}}{{[+]}}|\stackrel{{\scriptstyle 56}}{{(-)}}\,\stackrel{{\scriptstyle 78}}{{(+)}}||\stackrel{{\scriptstyle 9\;10}}{{[-]}}\;\;\stackrel{{\scriptstyle 11\;12}}{{[-]}}\;\;\stackrel{{\scriptstyle 13\;14}}{{(+)}}$ | -1 | $\frac{1}{2}$ | $-\frac{1}{2}$ | 0 | $0$ | $-\frac{1}{\sqrt{3}}$ | $\frac{1}{6}$ | $\frac{1}{6}$ | $-\frac{1}{3}$
22 | $d_{L}^{c3}$ | $-\stackrel{{\scriptstyle 03}}{{(+i)}}\,\stackrel{{\scriptstyle 12}}{{(-)}}|\stackrel{{\scriptstyle 56}}{{(-)}}\,\stackrel{{\scriptstyle 78}}{{(+)}}||\stackrel{{\scriptstyle 9\;10}}{{[-]}}\;\;\stackrel{{\scriptstyle 11\;12}}{{[-]}}\;\;\stackrel{{\scriptstyle 13\;14}}{{(+)}}$ | -1 | $-\frac{1}{2}$ | $-\frac{1}{2}$ | 0 | $0$ | $-\frac{1}{\sqrt{3}}$ | $\frac{1}{6}$ | $\frac{1}{6}$ | $-\frac{1}{3}$
23 | $u_{L}^{c3}$ | $-\stackrel{{\scriptstyle 03}}{{[-i]}}\,\stackrel{{\scriptstyle 12}}{{[+]}}|\stackrel{{\scriptstyle 56}}{{[+]}}\,\stackrel{{\scriptstyle 78}}{{[-]}}||\stackrel{{\scriptstyle 9\;10}}{{[-]}}\;\;\stackrel{{\scriptstyle 11\;12}}{{[-]}}\;\;\stackrel{{\scriptstyle 13\;14}}{{(+)}}$ | -1 | $\frac{1}{2}$ | $\frac{1}{2}$ | 0 | $0$ | $-\frac{1}{\sqrt{3}}$ | $\frac{1}{6}$ | $\frac{1}{6}$ | $\frac{2}{3}$
24 | $u_{L}^{c3}$ | $\stackrel{{\scriptstyle 03}}{{(+i)}}\,\stackrel{{\scriptstyle 12}}{{(-)}}|\stackrel{{\scriptstyle 56}}{{[+]}}\,\stackrel{{\scriptstyle 78}}{{[-]}}||\stackrel{{\scriptstyle 9\;10}}{{[-]}}\;\;\stackrel{{\scriptstyle 11\;12}}{{[-]}}\;\;\stackrel{{\scriptstyle 13\;14}}{{(+)}}$ | -1 | $-\frac{1}{2}$ | $\frac{1}{2}$ | 0 | $0$ | $-\frac{1}{\sqrt{3}}$ | $\frac{1}{6}$ | $\frac{1}{6}$ | $\frac{2}{3}$
25 | $\nu_{R}$ | $\stackrel{{\scriptstyle 03}}{{(+i)}}\,\stackrel{{\scriptstyle 12}}{{[+]}}|\stackrel{{\scriptstyle 56}}{{[+]}}\,\stackrel{{\scriptstyle 78}}{{(+)}}||\stackrel{{\scriptstyle 9\;10}}{{(+)}}\;\;\stackrel{{\scriptstyle 11\;12}}{{(+)}}\;\;\stackrel{{\scriptstyle 13\;14}}{{(+)}}$ | 1 | $\frac{1}{2}$ | 0 | $\frac{1}{2}$ | $0$ | $0$ | $-\frac{1}{2}$ | $0$ | $0$
26 | $\nu_{R}$ | $\stackrel{{\scriptstyle 03}}{{[-i]}}\,\stackrel{{\scriptstyle 12}}{{(-)}}|\stackrel{{\scriptstyle 56}}{{[+]}}\,\stackrel{{\scriptstyle 78}}{{(+)}}||\stackrel{{\scriptstyle 9\;10}}{{(+)}}\;\;\stackrel{{\scriptstyle 11\;12}}{{(+)}}\;\;\stackrel{{\scriptstyle 13\;14}}{{(+)}}$ | 1 | $-\frac{1}{2}$ | 0 | $\frac{1}{2}$ | $0$ | $0$ | $-\frac{1}{2}$ | $0$ | $0$
27 | $e_{R}$ | $\stackrel{{\scriptstyle 03}}{{(+i)}}\,\stackrel{{\scriptstyle 12}}{{[+]}}|\stackrel{{\scriptstyle 56}}{{(-)}}\,\stackrel{{\scriptstyle 78}}{{[-]}}||\stackrel{{\scriptstyle 9\;10}}{{(+)}}\;\;\stackrel{{\scriptstyle 11\;12}}{{(+)}}\;\;\stackrel{{\scriptstyle 13\;14}}{{(+)}}$ | 1 | $\frac{1}{2}$ | 0 | $-\frac{1}{2}$ | $0$ | $0$ | $-\frac{1}{2}$ | $-1$ | $-1$
28 | $e_{R}$ | $\stackrel{{\scriptstyle 03}}{{[-i]}}\,\stackrel{{\scriptstyle 12}}{{(-)}}|\stackrel{{\scriptstyle 56}}{{(-)}}\,\stackrel{{\scriptstyle 78}}{{[-]}}||\stackrel{{\scriptstyle 9\;10}}{{(+)}}\;\;\stackrel{{\scriptstyle 11\;12}}{{(+)}}\;\;\stackrel{{\scriptstyle 13\;14}}{{(+)}}$ | 1 | $-\frac{1}{2}$ | 0 | $-\frac{1}{2}$ | $0$ | $0$ | $-\frac{1}{2}$ | $-1$ | $-1$
29 | $e_{L}$ | $\stackrel{{\scriptstyle 03}}{{[-i]}}\,\stackrel{{\scriptstyle 12}}{{[+]}}|\stackrel{{\scriptstyle 56}}{{(-)}}\,\stackrel{{\scriptstyle 78}}{{(+)}}||\stackrel{{\scriptstyle 9\;10}}{{(+)}}\;\;\stackrel{{\scriptstyle 11\;12}}{{(+)}}\;\;\stackrel{{\scriptstyle 13\;14}}{{(+)}}$ | -1 | $\frac{1}{2}$ | $-\frac{1}{2}$ | 0 | $0$ | $0$ | $-\frac{1}{2}$ | $-\frac{1}{2}$ | $-1$
30 | $e_{L}$ | $-\stackrel{{\scriptstyle 03}}{{(+i)}}\,\stackrel{{\scriptstyle 12}}{{(-)}}|\stackrel{{\scriptstyle 56}}{{(-)}}\,\stackrel{{\scriptstyle 78}}{{(+)}}||\stackrel{{\scriptstyle 9\;10}}{{(+)}}\;\;\stackrel{{\scriptstyle 11\;12}}{{(+)}}\;\;\stackrel{{\scriptstyle 13\;14}}{{(+)}}$ | -1 | $-\frac{1}{2}$ | $-\frac{1}{2}$ | 0 | $0$ | $0$ | $-\frac{1}{2}$ | $-\frac{1}{2}$ | $-1$
31 | $\nu_{L}$ | $-\stackrel{{\scriptstyle 03}}{{[-i]}}\,\stackrel{{\scriptstyle 12}}{{[+]}}|\stackrel{{\scriptstyle 56}}{{[+]}}\,\stackrel{{\scriptstyle 78}}{{[-]}}||\stackrel{{\scriptstyle 9\;10}}{{(+)}}\;\;\stackrel{{\scriptstyle 11\;12}}{{(+)}}\;\;\stackrel{{\scriptstyle 13\;14}}{{(+)}}$ | -1 | $\frac{1}{2}$ | $\frac{1}{2}$ | 0 | $0$ | $0$ | $-\frac{1}{2}$ | $-\frac{1}{2}$ | $0$
32 | $\nu_{L}$ | $\stackrel{{\scriptstyle 03}}{{(+i)}}\,\stackrel{{\scriptstyle 12}}{{(-)}}|\stackrel{{\scriptstyle 56}}{{[+]}}\,\stackrel{{\scriptstyle 78}}{{[-]}}||\stackrel{{\scriptstyle 9\;10}}{{(+)}}\;\;\stackrel{{\scriptstyle 11\;12}}{{(+)}}\;\;\stackrel{{\scriptstyle 13\;14}}{{(+)}}$ | -1 | $-\frac{1}{2}$ | $\frac{1}{2}$ | 0 | $0$ | $0$ | $-\frac{1}{2}$ | $-\frac{1}{2}$ | $0$
33 | $\bar{d}_{L}^{\bar{c1}}$ | $\stackrel{{\scriptstyle 03}}{{[-i]}}\,\stackrel{{\scriptstyle 12}}{{[+]}}|\stackrel{{\scriptstyle 56}}{{[+]}}\,\stackrel{{\scriptstyle 78}}{{(+)}}||\stackrel{{\scriptstyle 9\;10}}{{[-]}}\;\;\stackrel{{\scriptstyle 11\;12}}{{(+)}}\;\;\stackrel{{\scriptstyle 13\;14}}{{(+)}}$ | -1 | $\frac{1}{2}$ | 0 | $\frac{1}{2}$ | $-\frac{1}{2}$ | $-\frac{1}{2\,\sqrt{3}}$ | $-\frac{1}{6}$ | $\frac{1}{3}$ | $\frac{1}{3}$
34 | $\bar{d}_{L}^{\bar{c1}}$ | $\stackrel{{\scriptstyle 03}}{{(+i)}}\,\stackrel{{\scriptstyle 12}}{{(-)}}|\stackrel{{\scriptstyle 56}}{{[+]}}\,\stackrel{{\scriptstyle 78}}{{(+)}}||\stackrel{{\scriptstyle 9\;10}}{{[-]}}\;\;\stackrel{{\scriptstyle 11\;12}}{{(+)}}\;\;\stackrel{{\scriptstyle 13\;14}}{{(+)}}$ | -1 | $-\frac{1}{2}$ | 0 | $\frac{1}{2}$ | $-\frac{1}{2}$ | $-\frac{1}{2\,\sqrt{3}}$ | $-\frac{1}{6}$ | $\frac{1}{3}$ | $\frac{1}{3}$
35 | $\bar{u}_{L}^{\bar{c1}}$ | $-\stackrel{{\scriptstyle 03}}{{[-i]}}\,\stackrel{{\scriptstyle 12}}{{[+]}}|\stackrel{{\scriptstyle 56}}{{(-)}}\,\stackrel{{\scriptstyle 78}}{{[-]}}||\stackrel{{\scriptstyle 9\;10}}{{[-]}}\;\;\stackrel{{\scriptstyle 11\;12}}{{(+)}}\;\;\stackrel{{\scriptstyle 13\;14}}{{(+)}}$ | -1 | $\frac{1}{2}$ | 0 | $-\frac{1}{2}$ | $-\frac{1}{2}$ | $-\frac{1}{2\,\sqrt{3}}$ | $-\frac{1}{6}$ | $-\frac{2}{3}$ | $-\frac{2}{3}$
36 | $\bar{u}_{L}^{\bar{c1}}$ | $-\stackrel{{\scriptstyle 03}}{{(+i)}}\,\stackrel{{\scriptstyle 12}}{{(-)}}|\stackrel{{\scriptstyle 56}}{{(-)}}\,\stackrel{{\scriptstyle 78}}{{[-]}}||\stackrel{{\scriptstyle 9\;10}}{{[-]}}\;\;\stackrel{{\scriptstyle 11\;12}}{{(+)}}\;\;\stackrel{{\scriptstyle 13\;14}}{{(+)}}$ | -1 | $-\frac{1}{2}$ | 0 | $-\frac{1}{2}$ | $-\frac{1}{2}$ | $-\frac{1}{2\,\sqrt{3}}$ | $-\frac{1}{6}$ | $-\frac{2}{3}$ | $-\frac{2}{3}$
37 | $\bar{d}_{R}^{\bar{c1}}$ | $\stackrel{{\scriptstyle 03}}{{(+i)}}\,\stackrel{{\scriptstyle 12}}{{[+]}}|\stackrel{{\scriptstyle 56}}{{[+]}}\,\stackrel{{\scriptstyle 78}}{{[-]}}||\stackrel{{\scriptstyle 9\;10}}{{[-]}}\;\;\stackrel{{\scriptstyle 11\;12}}{{(+)}}\;\;\stackrel{{\scriptstyle 13\;14}}{{(+)}}$ | 1 | $\frac{1}{2}$ | $\frac{1}{2}$ | 0 | $-\frac{1}{2}$ | $-\frac{1}{2\,\sqrt{3}}$ | $-\frac{1}{6}$ | $-\frac{1}{6}$ | $\frac{1}{3}$
38 | $\bar{d}_{R}^{\bar{c1}}$ | $-\stackrel{{\scriptstyle 03}}{{[-i]}}\,\stackrel{{\scriptstyle 12}}{{(-)}}|\stackrel{{\scriptstyle 56}}{{[+]}}\,\stackrel{{\scriptstyle 78}}{{[-]}}||\stackrel{{\scriptstyle 9\;10}}{{[-]}}\;\;\stackrel{{\scriptstyle 11\;12}}{{(+)}}\;\;\stackrel{{\scriptstyle 13\;14}}{{(+)}}$ | 1 | $-\frac{1}{2}$ | $\frac{1}{2}$ | 0 | $-\frac{1}{2}$ | $-\frac{1}{2\,\sqrt{3}}$ | $-\frac{1}{6}$ | $-\frac{1}{6}$ | $\frac{1}{3}$
39 | $\bar{u}_{R}^{\bar{c1}}$ | $\stackrel{{\scriptstyle 03}}{{(+i)}}\,\stackrel{{\scriptstyle 12}}{{[+]}}|\stackrel{{\scriptstyle 56}}{{(-)}}\,\stackrel{{\scriptstyle 78}}{{(+)}}||\stackrel{{\scriptstyle 9\;10}}{{[-]}}\;\;\stackrel{{\scriptstyle 11\;12}}{{(+)}}\;\;\stackrel{{\scriptstyle 13\;14}}{{(+)}}$ | 1 | $\frac{1}{2}$ | $-\frac{1}{2}$ | 0 | $-\frac{1}{2}$ | $-\frac{1}{2\,\sqrt{3}}$ | $-\frac{1}{6}$ | $-\frac{1}{6}$ | $-\frac{2}{3}$
40 | $\bar{u}_{R}^{\bar{c1}}$ | $\stackrel{{\scriptstyle 03}}{{[-i]}}\,\stackrel{{\scriptstyle 12}}{{(-)}}|\stackrel{{\scriptstyle 56}}{{(-)}}\,\stackrel{{\scriptstyle 78}}{{(+)}}||\stackrel{{\scriptstyle 9\;10}}{{[-]}}\;\;\stackrel{{\scriptstyle 11\;12}}{{(+)}}\;\;\stackrel{{\scriptstyle 13\;14}}{{(+)}}$ | 1 | $-\frac{1}{2}$ | $-\frac{1}{2}$ | 0 | $-\frac{1}{2}$ | $-\frac{1}{2\,\sqrt{3}}$ | $-\frac{1}{6}$ | $-\frac{1}{6}$ | $-\frac{2}{3}$
41 | $\bar{d}_{L}^{\bar{c2}}$ | $\stackrel{{\scriptstyle 03}}{{[-i]}}\,\stackrel{{\scriptstyle 12}}{{[+]}}|\stackrel{{\scriptstyle 56}}{{[+]}}\,\stackrel{{\scriptstyle 78}}{{(+)}}||\stackrel{{\scriptstyle 9\;10}}{{(+)}}\;\;\stackrel{{\scriptstyle 11\;12}}{{[-]}}\;\;\stackrel{{\scriptstyle 13\;14}}{{(+)}}$ | -1 | $\frac{1}{2}$ | 0 | $\frac{1}{2}$ | $\frac{1}{2}$ | $-\frac{1}{2\,\sqrt{3}}$ | $-\frac{1}{6}$ | $\frac{1}{3}$ | $\frac{1}{3}$
42 | $\bar{d}_{L}^{\bar{c2}}$ | $\stackrel{{\scriptstyle 03}}{{(+i)}}\,\stackrel{{\scriptstyle 12}}{{(-)}}|\stackrel{{\scriptstyle 56}}{{[+]}}\,\stackrel{{\scriptstyle 78}}{{(+)}}||\stackrel{{\scriptstyle 9\;10}}{{(+)}}\;\;\stackrel{{\scriptstyle 11\;12}}{{[-]}}\;\;\stackrel{{\scriptstyle 13\;14}}{{(+)}}$ | -1 | $-\frac{1}{2}$ | 0 | $\frac{1}{2}$ | $\frac{1}{2}$ | $-\frac{1}{2\,\sqrt{3}}$ | $-\frac{1}{6}$ | $\frac{1}{3}$ | $\frac{1}{3}$
43 | $\bar{u}_{L}^{\bar{c2}}$ | $-\stackrel{{\scriptstyle 03}}{{[-i]}}\,\stackrel{{\scriptstyle 12}}{{[+]}}|\stackrel{{\scriptstyle 56}}{{(-)}}\,\stackrel{{\scriptstyle 78}}{{[-]}}||\stackrel{{\scriptstyle 9\;10}}{{(+)}}\;\;\stackrel{{\scriptstyle 11\;12}}{{[-]}}\;\;\stackrel{{\scriptstyle 13\;14}}{{(+)}}$ | -1 | $\frac{1}{2}$ | 0 | $-\frac{1}{2}$ | $\frac{1}{2}$ | $-\frac{1}{2\,\sqrt{3}}$ | $-\frac{1}{6}$ | $-\frac{2}{3}$ | $-\frac{2}{3}$
44 | $\bar{u}_{L}^{\bar{c2}}$ | $-\stackrel{{\scriptstyle 03}}{{(+i)}}\,\stackrel{{\scriptstyle 12}}{{(-)}}|\stackrel{{\scriptstyle 56}}{{(-)}}\,\stackrel{{\scriptstyle 78}}{{[-]}}||\stackrel{{\scriptstyle 9\;10}}{{(+)}}\;\;\stackrel{{\scriptstyle 11\;12}}{{[-]}}\;\;\stackrel{{\scriptstyle 13\;14}}{{(+)}}$ | -1 | $-\frac{1}{2}$ | 0 | $-\frac{1}{2}$ | $\frac{1}{2}$ | $-\frac{1}{2\,\sqrt{3}}$ | $-\frac{1}{6}$ | $-\frac{2}{3}$ | $-\frac{2}{3}$
45 | $\bar{d}_{R}^{\bar{c2}}$ | $\stackrel{{\scriptstyle 03}}{{(+i)}}\,\stackrel{{\scriptstyle 12}}{{[+]}}|\stackrel{{\scriptstyle 56}}{{[+]}}\,\stackrel{{\scriptstyle 78}}{{[-]}}||\stackrel{{\scriptstyle 9\;10}}{{(+)}}\;\;\stackrel{{\scriptstyle 11\;12}}{{[-]}}\;\;\stackrel{{\scriptstyle 13\;14}}{{(+)}}$ | 1 | $\frac{1}{2}$ | $\frac{1}{2}$ | 0 | $\frac{1}{2}$ | $-\frac{1}{2\,\sqrt{3}}$ | $-\frac{1}{6}$ | $-\frac{1}{6}$ | $\frac{1}{3}$
46 | $\bar{d}_{R}^{\bar{c2}}$ | $-\stackrel{{\scriptstyle 03}}{{[-i]}}\,\stackrel{{\scriptstyle 12}}{{(-)}}|\stackrel{{\scriptstyle 56}}{{[+]}}\,\stackrel{{\scriptstyle 78}}{{[-]}}||\stackrel{{\scriptstyle 9\;10}}{{(+)}}\;\;\stackrel{{\scriptstyle 11\;12}}{{[-]}}\;\;\stackrel{{\scriptstyle 13\;14}}{{(+)}}$ | 1 | $-\frac{1}{2}$ | $\frac{1}{2}$ | 0 | $\frac{1}{2}$ | $-\frac{1}{2\,\sqrt{3}}$ | $-\frac{1}{6}$ | $-\frac{1}{6}$ | $\frac{1}{3}$
47 | $\bar{u}_{R}^{\bar{c2}}$ | $\stackrel{{\scriptstyle 03}}{{(+i)}}\,\stackrel{{\scriptstyle 12}}{{[+]}}|\stackrel{{\scriptstyle 56}}{{(-)}}\,\stackrel{{\scriptstyle 78}}{{(+)}}||\stackrel{{\scriptstyle 9\;10}}{{(+)}}\;\;\stackrel{{\scriptstyle 11\;12}}{{[-]}}\;\;\stackrel{{\scriptstyle 13\;14}}{{(+)}}$ | 1 | $\frac{1}{2}$ | $-\frac{1}{2}$ | 0 | $\frac{1}{2}$ | $-\frac{1}{2\,\sqrt{3}}$ | $-\frac{1}{6}$ | $-\frac{1}{6}$ | $-\frac{2}{3}$
48 | $\bar{u}_{R}^{\bar{c2}}$ | $\stackrel{{\scriptstyle 03}}{{[-i]}}\,\stackrel{{\scriptstyle 12}}{{(-)}}|\stackrel{{\scriptstyle 56}}{{(-)}}\,\stackrel{{\scriptstyle 78}}{{(+)}}||\stackrel{{\scriptstyle 9\;10}}{{(+)}}\;\;\stackrel{{\scriptstyle 11\;12}}{{[-]}}\;\;\stackrel{{\scriptstyle 13\;14}}{{(+)}}$ | 1 | $-\frac{1}{2}$ | $-\frac{1}{2}$ | 0 | $\frac{1}{2}$ | $-\frac{1}{2\,\sqrt{3}}$ | $-\frac{1}{6}$ | $-\frac{1}{6}$ | $-\frac{2}{3}$
49 | $\bar{d}_{L}^{\bar{c3}}$ | $\stackrel{{\scriptstyle 03}}{{[-i]}}\,\stackrel{{\scriptstyle 12}}{{[+]}}|\stackrel{{\scriptstyle 56}}{{[+]}}\,\stackrel{{\scriptstyle 78}}{{(+)}}||\stackrel{{\scriptstyle 9\;10}}{{(+)}}\;\;\stackrel{{\scriptstyle 11\;12}}{{(+)}}\;\;\stackrel{{\scriptstyle 13\;14}}{{[-]}}$ | -1 | $\frac{1}{2}$ | 0 | $\frac{1}{2}$ | $0$ | $\frac{1}{\sqrt{3}}$ | $-\frac{1}{6}$ | $\frac{1}{3}$ | $\frac{1}{3}$
50 | $\bar{d}_{L}^{\bar{c3}}$ | $\stackrel{{\scriptstyle 03}}{{(+i)}}\,\stackrel{{\scriptstyle 12}}{{(-)}}|\stackrel{{\scriptstyle 56}}{{[+]}}\,\stackrel{{\scriptstyle 78}}{{(+)}}||\stackrel{{\scriptstyle 9\;10}}{{(+)}}\;\;\stackrel{{\scriptstyle 11\;12}}{{(+)}}\;\;\stackrel{{\scriptstyle 13\;14}}{{[-]}}$ | -1 | $-\frac{1}{2}$ | 0 | $\frac{1}{2}$ | $0$ | $\frac{1}{\sqrt{3}}$ | $-\frac{1}{6}$ | $\frac{1}{3}$ | $\frac{1}{3}$
51 | $\bar{u}_{L}^{\bar{c3}}$ | $-\stackrel{{\scriptstyle 03}}{{[-i]}}\,\stackrel{{\scriptstyle 12}}{{[+]}}|\stackrel{{\scriptstyle 56}}{{(-)}}\,\stackrel{{\scriptstyle 78}}{{[-]}}||\stackrel{{\scriptstyle 9\;10}}{{(+)}}\;\;\stackrel{{\scriptstyle 11\;12}}{{(+)}}\;\;\stackrel{{\scriptstyle 13\;14}}{{[-]}}$ | -1 | $\frac{1}{2}$ | 0 | $-\frac{1}{2}$ | $0$ | $\frac{1}{\sqrt{3}}$ | $-\frac{1}{6}$ | $-\frac{2}{3}$ | $-\frac{2}{3}$
52 | $\bar{u}_{L}^{\bar{c3}}$ | $-\stackrel{{\scriptstyle 03}}{{(+i)}}\,\stackrel{{\scriptstyle 12}}{{(-)}}|\stackrel{{\scriptstyle 56}}{{(-)}}\,\stackrel{{\scriptstyle 78}}{{[-]}}||\stackrel{{\scriptstyle 9\;10}}{{(+)}}\;\;\stackrel{{\scriptstyle 11\;12}}{{(+)}}\;\;\stackrel{{\scriptstyle 13\;14}}{{[-]}}$ | -1 | $-\frac{1}{2}$ | 0 | $-\frac{1}{2}$ | $0$ | $\frac{1}{\sqrt{3}}$ | $-\frac{1}{6}$ | $-\frac{2}{3}$ | $-\frac{2}{3}$
53 | $\bar{d}_{R}^{\bar{c3}}$ | $\stackrel{{\scriptstyle 03}}{{(+i)}}\,\stackrel{{\scriptstyle 12}}{{[+]}}|\stackrel{{\scriptstyle 56}}{{[+]}}\,\stackrel{{\scriptstyle 78}}{{[-]}}||\stackrel{{\scriptstyle 9\;10}}{{(+)}}\;\;\stackrel{{\scriptstyle 11\;12}}{{(+)}}\;\;\stackrel{{\scriptstyle 13\;14}}{{[-]}}$ | 1 | $\frac{1}{2}$ | $\frac{1}{2}$ | 0 | $0$ | $\frac{1}{\sqrt{3}}$ | $-\frac{1}{6}$ | $-\frac{1}{6}$ | $\frac{1}{3}$
54 | $\bar{d}_{R}^{\bar{c3}}$ | $-\stackrel{{\scriptstyle 03}}{{[-i]}}\,\stackrel{{\scriptstyle 12}}{{(-)}}|\stackrel{{\scriptstyle 56}}{{[+]}}\,\stackrel{{\scriptstyle 78}}{{[-]}}||\stackrel{{\scriptstyle 9\;10}}{{(+)}}\;\;\stackrel{{\scriptstyle 11\;12}}{{(+)}}\;\;\stackrel{{\scriptstyle 13\;14}}{{[-]}}$ | 1 | $-\frac{1}{2}$ | $\frac{1}{2}$ | 0 | $0$ | $\frac{1}{\sqrt{3}}$ | $-\frac{1}{6}$ | $-\frac{1}{6}$ | $\frac{1}{3}$
55 | $\bar{u}_{R}^{\bar{c3}}$ | $\stackrel{{\scriptstyle 03}}{{(+i)}}\,\stackrel{{\scriptstyle 12}}{{[+]}}|\stackrel{{\scriptstyle 56}}{{(-)}}\,\stackrel{{\scriptstyle 78}}{{(+)}}||\stackrel{{\scriptstyle 9\;10}}{{(+)}}\;\;\stackrel{{\scriptstyle 11\;12}}{{(+)}}\;\;\stackrel{{\scriptstyle 13\;14}}{{[-]}}$ | 1 | $\frac{1}{2}$ | $-\frac{1}{2}$ | 0 | $0$ | $\frac{1}{\sqrt{3}}$ | $-\frac{1}{6}$ | $-\frac{1}{6}$ | $-\frac{2}{3}$
56 | $\bar{u}_{R}^{\bar{c3}}$ | $\stackrel{{\scriptstyle 03}}{{[-i]}}\,\stackrel{{\scriptstyle 12}}{{(-)}}|\stackrel{{\scriptstyle 56}}{{(-)}}\,\stackrel{{\scriptstyle 78}}{{(+)}}||\stackrel{{\scriptstyle 9\;10}}{{(+)}}\;\;\stackrel{{\scriptstyle 11\;12}}{{(+)}}\;\;\stackrel{{\scriptstyle 13\;14}}{{[-]}}$ | 1 | $-\frac{1}{2}$ | $-\frac{1}{2}$ | 0 | $0$ | $\frac{1}{\sqrt{3}}$ | $-\frac{1}{6}$ | $-\frac{1}{6}$ | $-\frac{2}{3}$
57 | $\bar{e}_{L}$ | $\stackrel{{\scriptstyle 03}}{{[-i]}}\,\stackrel{{\scriptstyle 12}}{{[+]}}|\stackrel{{\scriptstyle 56}}{{[+]}}\,\stackrel{{\scriptstyle 78}}{{(+)}}||\stackrel{{\scriptstyle 9\;10}}{{[-]}}\;\;\stackrel{{\scriptstyle 11\;12}}{{[-]}}\;\;\stackrel{{\scriptstyle 13\;14}}{{[-]}}$ | -1 | $\frac{1}{2}$ | 0 | $\frac{1}{2}$ | $0$ | $0$ | $\frac{1}{2}$ | $1$ | $1$
58 | $\bar{e}_{L}$ | $\stackrel{{\scriptstyle 03}}{{(+i)}}\,\stackrel{{\scriptstyle 12}}{{(-)}}|\stackrel{{\scriptstyle 56}}{{[+]}}\,\stackrel{{\scriptstyle 78}}{{(+)}}||\stackrel{{\scriptstyle 9\;10}}{{[-]}}\;\;\stackrel{{\scriptstyle 11\;12}}{{[-]}}\;\;\stackrel{{\scriptstyle 13\;14}}{{[-]}}$ | -1 | $-\frac{1}{2}$ | 0 | $\frac{1}{2}$ | $0$ | $0$ | $\frac{1}{2}$ | $1$ | $1$
59 | $\bar{\nu}_{L}$ | $-\stackrel{{\scriptstyle 03}}{{[-i]}}\,\stackrel{{\scriptstyle 12}}{{[+]}}|\stackrel{{\scriptstyle 56}}{{(-)}}\,\stackrel{{\scriptstyle 78}}{{[-]}}||\stackrel{{\scriptstyle 9\;10}}{{[-]}}\;\;\stackrel{{\scriptstyle 11\;12}}{{[-]}}\;\;\stackrel{{\scriptstyle 13\;14}}{{[-]}}$ | -1 | $\frac{1}{2}$ | 0 | $-\frac{1}{2}$ | $0$ | $0$ | $\frac{1}{2}$ | $0$ | $0$
60 | $\bar{\nu}_{L}$ | $-\stackrel{{\scriptstyle 03}}{{(+i)}}\,\stackrel{{\scriptstyle 12}}{{(-)}}|\stackrel{{\scriptstyle 56}}{{(-)}}\,\stackrel{{\scriptstyle 78}}{{[-]}}||\stackrel{{\scriptstyle 9\;10}}{{[-]}}\;\;\stackrel{{\scriptstyle 11\;12}}{{[-]}}\;\;\stackrel{{\scriptstyle 13\;14}}{{[-]}}$ | -1 | $-\frac{1}{2}$ | 0 | $-\frac{1}{2}$ | $0$ | $0$ | $\frac{1}{2}$ | $0$ | $0$
61 | $\bar{\nu}_{R}$ | $\stackrel{{\scriptstyle 03}}{{(+i)}}\,\stackrel{{\scriptstyle 12}}{{[+]}}|\stackrel{{\scriptstyle 56}}{{(-)}}\,\stackrel{{\scriptstyle 78}}{{(+)}}||\stackrel{{\scriptstyle 9\;10}}{{[-]}}\;\;\stackrel{{\scriptstyle 11\;12}}{{[-]}}\;\;\stackrel{{\scriptstyle 13\;14}}{{[-]}}$ | 1 | $\frac{1}{2}$ | $-\frac{1}{2}$ | 0 | $0$ | $0$ | $\frac{1}{2}$ | $\frac{1}{2}$ | $0$
62 | $\bar{\nu}_{R}$ | $-\stackrel{{\scriptstyle 03}}{{[-i]}}\,\stackrel{{\scriptstyle 12}}{{(-)}}|\stackrel{{\scriptstyle 56}}{{(-)}}\,\stackrel{{\scriptstyle 78}}{{(+)}}||\stackrel{{\scriptstyle 9\;10}}{{[-]}}\;\;\stackrel{{\scriptstyle 11\;12}}{{[-]}}\;\;\stackrel{{\scriptstyle 13\;14}}{{[-]}}$ | 1 | $-\frac{1}{2}$ | $-\frac{1}{2}$ | 0 | $0$ | $0$ | $\frac{1}{2}$ | $\frac{1}{2}$ | $0$
63 | $\bar{e}_{R}$ | $\stackrel{{\scriptstyle 03}}{{(+i)}}\,\stackrel{{\scriptstyle 12}}{{[+]}}|\stackrel{{\scriptstyle 56}}{{[+]}}\,\stackrel{{\scriptstyle 78}}{{[-]}}||\stackrel{{\scriptstyle 9\;10}}{{[-]}}\;\;\stackrel{{\scriptstyle 11\;12}}{{[-]}}\;\;\stackrel{{\scriptstyle 13\;14}}{{[-]}}$ | 1 | $\frac{1}{2}$ | $\frac{1}{2}$ | 0 | $0$ | $0$ | $\frac{1}{2}$ | $\frac{1}{2}$ | $1$
64 | $\bar{e}_{R}$ | $\stackrel{{\scriptstyle 03}}{{[-i]}}\,\stackrel{{\scriptstyle 12}}{{(-)}}|\stackrel{{\scriptstyle 56}}{{[+]}}\,\stackrel{{\scriptstyle 78}}{{[-]}}||\stackrel{{\scriptstyle 9\;10}}{{[-]}}\;\;\stackrel{{\scriptstyle 11\;12}}{{[-]}}\;\;\stackrel{{\scriptstyle 13\;14}}{{[-]}}$ | 1 | $-\frac{1}{2}$ | $\frac{1}{2}$ | 0 | $0$ | $0$ | $\frac{1}{2}$ | $\frac{1}{2}$ | $1$
Table 6:  The left-handed ($\Gamma^{(13,1)}=-1$, Eq. (83)) irreducible
representation of one family of spinors — the product of the odd number of
nilpotents and of projectors, which are eigenvectors of the Cartan subalgebra
of the $SO(13,1)$ group [14, 5], manifesting the subgroup $SO(7,1)$ of the
colour charged quarks and antiquarks and the colourless leptons and
antileptons — is presented. It contains the left-handed ($\Gamma^{(3,1)}=-1$)
weak ($SU(2)_{I}$) charged ($\tau^{13}=\pm\frac{1}{2}$), and $SU(2)_{II}$
chargeless ($\tau^{23}=0$) quarks and leptons, and the right-handed
($\Gamma^{(3,1)}=1$) weak ($SU(2)_{I}$) chargeless and $SU(2)_{II}$ charged
($\tau^{23}=\pm\frac{1}{2}$) quarks and leptons, both with the spin $S^{12}$
up and down ($\pm\frac{1}{2}$, respectively). Quarks distinguish from leptons
only in the $SU(3)\times U(1)$ part: Quarks are triplets of three colours
($c^{i}$ $=(\tau^{33},\tau^{38})$
$=[(\frac{1}{2},\frac{1}{2\sqrt{3}}),(-\frac{1}{2},\frac{1}{2\sqrt{3}}),(0,-\frac{1}{\sqrt{3}})$,
carrying the ”fermion charge” ($\tau^{4}=\frac{1}{6}$). The colourless leptons
carry the ”fermion charge” ($\tau^{4}=-\frac{1}{2}$). The same multiplet
contains also the left handed weak ($SU(2)_{I}$) chargeless and $SU(2)_{II}$
charged antiquarks and antileptons and the right handed weak ($SU(2)_{I}$)
charged and $SU(2)_{II}$ chargeless antiquarks and antileptons. Antiquarks
distinguish from antileptons again only in the $SU(3)\times U(1)$ part:
Antiquarks are anti-triplets carrying the ”fermion charge”
($\tau^{4}=-\frac{1}{6}$). The anti-colourless antileptons carry the ”fermion
charge” ($\tau^{4}=\frac{1}{2}$). $Y=(\tau^{23}+\tau^{4})$ is the hyper
charge, the electromagnetic charge is $Q=(\tau^{13}+Y$).

##  Appendix E Acknowledgement

The author N.S.M.B. thanks Department of Physics, FMF, University of
Ljubljana, Society of Mathematicians, Physicists and Astronomers of Slovenia
for supporting the research on the spin-charge-family theory, and Matjaž
Breskvar of Beyond Semiconductor for donations, in particular for sponsoring
the annual workshops entitled ”What comes beyond the standard models” at Bled,
in which the ideas and realizations, presented in this paper, were discussed.
The author thanks Holger Beck Nielsen and Milutin Blagojević for fruitful
discussions.

## References

  * [1] N. Mankoč Borštnik, ”Spin connection as a superpartner of a vielbein”, Phys. Lett. B 292 (1992) 25-29.
  * [2] N. Mankoč Borštnik, ”Spinor and vector representations in four dimensional Grassmann space”, J. of Math. Phys. 34 (1993) 3731-3745.
  * [3] N. Mankoč Borštnik, ”Unification of spin and charges in Grassmann space?”, hep-th 9408002, IJS.TP.94/22, Mod. Phys. Lett.A (10) No.7 (1995) 587-595.
  * [4] A. Borštnik Bračič, N. S. Mankoč Borštnik, ”On the origin of families of fermions and their mass matrices”, hep-ph/0512062, Phys. Rev. D 74 073013-28 (2006).
  * [5] N.S. Mankoč Borštnik, H.B.F. Nielsen, J. of Math. Phys. 43, 5782 (2002) [arXiv:hep-th/0111257]. “How to generate families of spinors”, J. of Math. Phys. 44 4817 (2003) [arXiv:hep-th/0303224].
  * [6] N.S. Mankoč Borštnik, D. Lukman, ”Vector and scalar gauge fields with respect to $d=(3+1)$ in Kaluza-Klein theories and in the spin-charge-family theory”, Eur. Phys. J. C 77 (2017) 231.
  * [7] N.S. Mankoč Borštnik, H.B.F. Nielsen, ”Understanding the second quantization of fermions in Clifford and in Grassmann space”, New way of second quantization of fermions — Part I and Part II, [arXiv:2007.03517, arXiv:2007.03516].
  * [8] N. S. Mankoč Borštnik, H. B. Nielsen, ”How does Clifford algebra show the way to the second quantized fermions with unified spins, charges and families, and with vector and scalar gauge fields beyond the standard model”, Progress in Particle and Nuclear Physics, http://doi.org/10.1016.j.ppnp.2021.103890 .
  * [9] N. S. Mankoč Borštnik, ”Clifford odd and even objects offer description of internal space of fermions and bosons, respectively, opening new insight into the second quantization of fields”, The $13^{th}$ Bienal Conference on Classical and Quantum Relativistic Dynamics of Particles and Fields IARD 2022, Prague, $6-9$ June [http://arxiv.org/abs/2210.07004], Journal of Physics: Conference Series, Volume 2482/2023, Proceedings https://iopscience.iop.org/issue/1742-6596/2482/012012.
  * [10] N. S. Mankoč Borštnik, ”How Clifford algebra helps understand second quantized quarks and leptons and corresponding vector and scalar boson fields, opening a new step beyond the standard model”, Reference: NUPHB 994 (2023) 116326 , [arXiv: 2210.06256, physics.gen-ph V2].

  * [11] N. S. Mankoč Borštnik, ”Clifford odd and even objects in even and odd dimensional spaces”, Symmetry 2023,15,818-12-V2 94818, https:doi.org/10.3390/sym15040818, [arxiv.org/abs/2301.04466] , https://www.mdpi.com/2073-8994/15/4/818 Manuscript ID: symmetry-2179313.
  * [12] G. Bregar, M. Breskvar, D. Lukman, N.S. Mankoč Borštnik, ”Predictions for four families by the Approach unifying spins and charges” New J. of Phys. 10 (2008) 093002, G. Bregar, M. Breskvar, D. Lukman, N.S. Mankoč Borštnik, ”Families of Quarks and Leptons and Their Mass Matrices”, Proceedings to the $10^{th}$ international workshop ”What Comes Beyond the Standard Model”, 17 -27 of July, 2007, Ed. Norma Mankoč Borštnik, Holger Bech Nielsen, Colin Froggatt, Dragan Lukman, DMFA Založništvo, Ljubljana December 2007, p.53-70,[hep-ph/0711.4681, hep-ph/0606159, hep/ph-07082846].
  * [13] G. Bregar, N.S. Mankoč Borštnik, ”Does dark matter consist of baryons of new stable family quarks?”, Phys. Rev. D  80, 083534 (2009), 1-16. of new stable family quarks?”, Phys. Rev. D  80, 083534 (2009), 1-16.
  * [14] N.S. Mankoč Borštnik, ”Matter-antimatter asymmetry in the spin-charge-family theory”, Phys. Rev. D 91 (2015) 065004 [arXiv:1409.7791].
  * [15] N.S. Mankoč Borštnik N S, ”The spin-charge-family theory is explaining the origin of families, of the Higgs and the Yukawa couplings”, J. of Modern Phys. 4 (2013) 823 [arXiv:1312.1542].
  * [16] G. Bregar, N.S. Mankoč Borštnik, ”The new experimental data for the quarks mixing matrix are in better agreement with the spin-charge-family theory predictions”, Proceedings to the $17^{t}h$ Workshop ”What comes beyond the standard models”, Bled, 20-28 of July, 2014, Ed. N.S. Mankoč Borštnik, H.B. Nielsen, D. Lukman, DMFA Založništvo, Ljubljana December 2014, p.20-45 [ arXiv:1502.06786v1] [arxiv:1412.5866].
  * [17] N.S. Mankoč Borštnik, M. Rosina, ”Are superheavy stable quark clusters viable candidates for the dark matter?”, International Journal of Modern Physics D (IJMPD) 24 (No. 13) (2015) 1545003.
  * [18] N.S. Mankoč Borštnik, H.B. Nielsen, ”Can the “basis vectors”, describing the internal space of point fermion and boson fields with the Clifford odd (for fermions) and Clifford even (for bosons) objects, be meaningfully extended to strings?”, Proceedings to the $26^{rd}$ Workshop ”What comes beyond the standard models”, 10 - 19 July, 2023, Ed. N.S. Mankoč Borštnik, H.B. Nielsen, M.Yu. Khlopov, A. Kleppe, Založba Univerze v Ljubljani, DOI: 10.51746/9789612972097, December 23, [arxiv:2312.07548].
  * [19] N. S. Mankoč Borštnik, ”How Clifford algebra can help understand second quantization of fermion and boson fields”, [arXiv: 2210.06256. physics.gen-ph V1].
  * [20] N.S. Mankoč Borštnik, H.B. Nielsen, ”Discrete symmetries in the Kaluza-Klein-like theories”, doi:10.1007/ Jour. of High Energy Phys. 04 (2014)165, [http://arxiv.org/abs/1212.2362v3].
  * [21] N.S. Mankoč Borštnik, ”How far has so far the Spin-Charge-Family theory succeeded to offer the explanation for the observed phenomena in elementary particle physics and cosmology”, Proceedings to the $26^{rd}$ Workshop ”What comes beyond the standard models”, 10 - 19 July, 2023, Ed. N.S. Mankoč Borštnik, H.B. Nielsen, M.Yu. Khlopov, A. Kleppe, Založba Univerze v Ljubljani, DOI: 10.51746/9789612972097, December 23, [http://arxiv.org/abs/2312.14946].
  * [22] N.S. Mankoč Borštnik, ”New way of second quantized theory of fermions with either Clifford or Grassmann coordinates and spin-charge-family theory ” [arXiv: 2210.06256. physics.gen-ph V1, arXiv:1802.05554v4, arXiv:1902.10628],
  * [23] N.S. Mankoč Borštnik, ”How do Clifford algebras show the way to the second quantized fermions with unified spins, charges and families, and to the corresponding second quantized vector and scalar gauge field ”, Proceedings to the $24^{rd}$ Workshop ”What comes beyond the standard models”, 5 - 11 of July, 2021, Ed. N.S. Mankoč Borštnik, H.B. Nielsen, D. Lukman, A. Kleppe, DMFA Založništvo, Ljubljana, December 2021, [arXiv:2112.04378].
  * [24] G. Bregar, N.S. Mankoč Borštnik, ”Can we predict the fourth family masses for quarks and leptons?”, Proceedings (arxiv:1403.4441) to the 16 th Workshop ”What comes beyond the standard models”, Bled, 14-21 of July, 2013, Ed. N.S. Mankoč Borštnik, H.B. Nielsen, D. Lukman, DMFA Založništvo, Ljubljana December 2013, p. 31-51, [http://arxiv.org/abs/1212.4055].
  * [25] Ralph Blumenhagen, Dieter L’́ ust, Stephan Theisen, “Basic Concepts of String Theory”, Springer.
  * [26] M. Blagojević, Gravitation and gauge symmetries, IoP Publishing, Bristol 2002.
  * [27] Kewin Wrey, An introduction to String theory, Preprint typeset in JHEP style - PAPER VERSION