g) + b e + c d + d e + d g + e f]}, \\
\delta\omega_p &= \omega_{\PP,+} - \omega_{\PP,-}, \label{eq:deltaomega}
\end{align}

which gives $\mathbb{R}$ through $\bm{W}_j = \mathbb{R} \bm{Q}_j$ as

\begin{align}
L_k^{(1)} &= Q_{0,j}^{(1)} + Q_{+,j}^{(1)} + Q_{-,j}^{(1)}, \\
S_{1,k}^{(1)} &= \frac{be + bg + cg}{de + dg + ef} Q_{0,j}^{(1)} +
\frac{2(g-b)}{b+c - (d-e)-(f+g) + \delta \omega_p}  Q_{+,j}^{(1)}
+\frac{2(g-b)}{b+c- (d-e)-(f+g)  - \delta \omega_p} Q_{-,j}^{(1)}, \\
S_{2,k}^{(1)} &= \frac{cd + bf + cf}{de + dg + ef} Q_{0,j}^{(1)} +
\frac{2(f-c)}{b+c + (d-e)-(f+g) + \delta \omega_p}  Q_{+,j}^{(1)}
+\frac{2(f-c)}{b+c + (d-e)-(f+g)  - \delta \omega_p} Q_{-,j}^{(1)}.
\end{align}

The amplitudes $B_{i,j}^{(1,0)}$ with $i= +$ or $-$, and $j = 1$ or $2$ can be
computed using Eqs. (<ref>-<ref>) at $t=0$, with
$Q_{i,j}^{(1)} \to
B_{i,j}^{(1,0)}$ and $L_k^{(1)} \to - S_{1,k} - S_{2,k}$. Using
$S_{j,k}(t = 0) = m_j^2 \chi_j \hat{S}_{j,k,0}$, we get

\begin{align}
B_{+,1}^{(1,0)} &= T_{+,1} \hat{S}_{1,x,0} + T_{+,2} \hat{S}_{2,x,0}, \\
B_{+,2}^{(1,0)} &= T_{+,1} \hat{S}_{1,y,0} + T_{+,2} \hat{S}_{2,y,0}, \\
B_{-,1}^{(1,0)} &= T_{-,1} \hat{S}_{1,x,0} + T_{-,2} \hat{S}_{2,x,0}, \\
B_{-,2}^{(1,0)} &= T_{-,1} \hat{S}_{1,y,0} + T_{-,2} \hat{S}_{2,y,0}, \\
T_{+,1} &= - \frac{(T_4 + T_2) T_5}{T_3}, \\
T_{+,2} &= - \frac{(T_4 + T_2) (T_6 + m_2 T_2)}{T_3}, \\
T_{-,1} &= \frac{(T_4 - T_2) T_5}{T_3}, \\
T_{-,2} &= \frac{(T_4 - T_2) (T_6 - m_2 T_2)}{T_3},
\end{align}

\begin{align}
T_1 &= m_1 m_2 (\chi_1 - \chi_2) \xi_0 +
m_2^2 (1 - 2 \chi_2 \xi_0) - m_1^2 (1 - 2 \chi_1 \xi_0), \\
T_2 &= \left[ 4 m_1 m_2\chi_1 \chi_2 (M -
m_1 \chi_1 \xi_0) (M - m_2 \chi_2 \xi_0) \xi_0^2  + T_1^2
\right]^{1/2}, \\
T_3 &= 4 (M - m_1 \chi_1 \xi_0) T_2  \xi_0, \\
T_4 &= - M (m_1 - m_2) +  \left[ 2 m_1^2 \chi_1 + m_1 m_2 (\chi_1 +
\chi_2 ) \right] \xi_0 - 2 m_1 m_2 \chi_1 \chi_2 \xi_0^2, \\
T_5 &= 2 m_1^2 \chi_1 (M -
m_1 \chi_1 \xi_0) \xi_0, \\
T_6 &= m_2 \left[m_1^2 (1 - 2 \chi_1 \xi_0) - m_1 m_2 (\chi_1 - \chi_2) \xi_0
- m_2^2 (1 - 2 \chi_2 \xi_0)\right].
\end{align}

where $\xi_0 = \xi(t=0)$.

To describe the time dependence of the amplitudes $A_{i,j}^{(1,0)}$, we solve
Eq. (<ref>).
When $\delta m = (m_1-m_2)/M \gg \mathcal{O}(c^{-1})$, we get

\begin{align}
A_{+,j}^{(1,0)}(t) &= B_{+,j}^{(1,0)} \left[ 1 + \left(\frac{m_1
\chi_1}{m_1 - m_2} - \frac{m_2 \chi_2}{m_1} \right) \xi(t) \right] +
\mathcal{O}(c^{-2}) , \\
A_{-,j}^{(1,0)}(t) &= B_{-,j}^{(1,0)} \left[ 1 - \left(\frac{m_1
\chi_1}{m_2} - \frac{m_2 \chi_2}{m_1 - m_2} \right) \xi(t) \right] +
\mathcal{O}(c^{-2}) ;
\end{align}

when $\delta m = (m_1-m_2)/M \sim \mathcal{O}(c^{-1})$, we get

\begin{align}
A_{+,j}^{(1,0)}(t) &= B_{+,j}^{(1,0)} \left[ 1 + \frac{
\chi_1}{2 \delta m} \xi(t) \right] +
\mathcal{O}(c^{-2}) , \\
A_{-,j}^{(1,0)}(t) &= B_{-,j}^{(1,0)} \left[ 1 + \frac{
\chi_1 - 3 \chi_2}{4\delta m} \xi(t)
\right] +
\mathcal{O}(c^{-2}) ;
\end{align}

when $\delta m = (m_1-m_2)/M \ll \mathcal{O}(c^{-1})$, we get

\begin{align}
A_{+,j}^{(1,0)}(t) &= B_{+,j}^{(1,0)} \left\{1 - \frac{1}{18} \left[3 T + 9
\chi_1 + 5 \chi_2 + \frac{42 \chi_1 \chi_2}{T} - \frac{(44\chi_1 - 36
\chi_2)\chi_2^2}{T^2} \right]\xi(t) \right\} +
\mathcal{O}(c^{-2}) , \\
A_{-,j}^{(1,0)}(t) &= B_{-,j}^{(1,0)} \left\{1 + \frac{1}{18} \left[3 T - 9
\chi_1 - 5 \chi_2 + \frac{42 \chi_1 \chi_2}{T} + \frac{(44\chi_1 - 36
\chi_2)\chi_2^2}{T^2} \right]\xi(t) \right\} +
\mathcal{O}(c^{-2}) , \\
T &= \sqrt{9 \chi_1^2 - 2 \chi_1 \chi_2 + 9 \chi_2^2}.
\end{align}

§ PRECESSION PHASES

In this section, we give the exact expressions for the precession phases that
we used in our implementation. $\phi_{\PP,\pm}$ is calculated from
$\phi_{\PP,\pm} = \phi_{\PP,m} \pm \delta \phi_\PP$.
We give them in terms of
the $a_i$ given in appendix <ref>, and $\delta m = (m_1 -

The mean precession phase we used in our implementation is

\begin{align}
\phi_{\PP,m} &= \frac{1}{2} \int \left( \omega_{\PP,+} + \omega_{\PP,-} \right)
\frac{dt}{d\xi}
d\xi = \frac{1}{2} \int \frac{ b+c+d+e+f+g}{a\xi} d\xi \nonumber\\
&= \phi_{\PP,m,0} + \phi_{\PP,m}^{(-3)} \xi^{-3} + \phi_{\PP,m}^{(-2)} \xi^{-2}
\phi_{\PP,m}^{(-1)} \xi^{-1} + \phi_{\PP,m}^{(0)} \log\xi + \phi_{\PP,m}^{(1)}
\xi +
\phi_{\PP,m}^{(2)} \xi^{2} + \phi_{\PP,m}^{(3)} \xi^{3}, \\
\phi_{\PP,m}^{(-3)} &= - \frac{7 - \delta m^2}{8a_0}, \\
\phi_{\PP,m}^{(-2)} &= -\frac{3}{32a_0} [(5 + 4 \delta m - \delta m^2) \chi_1
+ (5 - 4 \delta m - \delta m^2) \chi_2 ], \\
\phi_{\PP,m}^{(-1)} &= \frac{3}{8 a_0}   [ a_2 (7 - \delta m^2) + 3 (1 -
\delta m^2) \chi_1 \chi_2 ] , \\
\phi_{\PP,m}^{(0)} &= - \frac{3}{16 a_0} \{ 2 a_3 (7 - \delta m^2) + a_2 [(5 + 4
\delta m - \delta m^2) \chi_1
+ (5 - 4 \delta m - \delta m^2) \chi_2 ] \} , \\
\phi_{\PP,m}^{(1)} &= \frac{3}{16 a_0} \{ 2 (a_2^2 - a_4) (7 - \delta m^2) - a_3
[(5 + 4 \delta m - \delta m^2) \chi_1
+ (5 - 4 \delta m - \delta m^2) \chi_2 ] +
6 a_2 (1 - \delta m^2) \chi_1 \chi_2 \} , \\
\phi_{\PP,m}^{(2)} &= \frac{3}{32a_0} \{ (4 a_2 a_3 - 2 a_5) (7 - \delta m^2) +
(a_2^2 - a_4) [(5 + 4
\delta m - \delta m^2) \chi_1
+ (5 - 4 \delta m - \delta m^2) \chi_2 ] + 6 a_3 (1 -
\delta m^2) \chi_1 \chi_2 \} , \\
\phi_{\PP,m}^{(3)} &= -\frac{1}{16 a_0} \{ (2 a_2^3 -
2 a_3^2 - 4 a_2 a_4 ) (7 - \delta m^2) \nonumber\\
&- (2 a_2 a_3 - a_5) [(5 +
4 \delta m - \delta m^2) \chi_1
+ (5 - 4 \delta m - \delta m^2) \chi_2 ] + 6 (a_2^2 - a_4) (1 -
\delta m^2) \chi_1 \chi_2 \} .
\end{align}

$\delta\phi_\PP$ for $\delta m \gg \mathcal{O}(c^{-1})$ is given by

\begin{align}
\delta \phi_{\PP,1} &= \frac{1}{2} \int \left( \omega_{\PP,+} - \omega_{\PP,-}
\right)
\frac{dt}{d\xi}
d\xi = \frac{1}{2} \int \frac{\sqrt{(b-c+d-e+f-g)^2 + 4(c-f)(b-g)}}{a\xi} d\xi
\nonumber\\
&= \delta \phi_{\PP,1,0} + \delta\phi_{\PP,1}^{(-3)} \xi^{-3} +
\delta\phi_{\PP,1}^{(-2)} \xi^{-2} + \delta\phi_{\PP,1}^{(-1)}
\xi^{-1} + \delta\phi_{\PP,1}^{(0)} \log\xi + \delta\phi_{\PP,1}^{(1)} \xi +
\delta\phi_{\PP,1}^{(2)}
\xi^2 + \delta\phi_{\PP,1}^{(3)}
\xi^3 , \\
\delta \phi_{\PP,1}^{(-3)} &= - \frac{3 \delta m}{4a_0} , \\
\delta \phi_{\PP,1}^{(-2)} &= \frac{9}{32 a_0} [(3 + 4 \delta m +
\delta m^2) \chi_1 - (3 - 4 \delta m +
\delta m^2) \chi_2]
, \\
\delta \phi_{\PP,1}^{(-1)} &= \frac{9}{8 a_0 \delta m} [2 a_2 \delta m^2
- (1 -
\delta m^2) \chi_1 \chi_2 ], \\
\delta \phi_{\PP,1}^{(0)} &= -\frac{9}{32 a_0 \delta m^2} \{ 8 a_3 \delta m^3
- 2 a_2 \delta m^2 [(3 + 4 \delta m +
\delta m^2) \chi_1 - (3 - 4 \delta m +
\delta m^2) \chi_2] \nonumber\\
&- (1 -
\delta m^2) [(3 + 2 \delta m +
\delta m^2) \chi_1 - (3 - 2 \delta m +
\delta m^2) \chi_2] \chi_1 \chi_2 \} , \\
\delta \phi_{\PP,1}^{(1)} &= \frac{9}{128 a_0\delta m^3} \{ 8 \delta
m^5 a_3 ( \chi_1 - \chi_2) + 32 \delta
m^4 [ a_2^2 - a_4 + a_3 (\chi_1 + \chi_2)] + 24 \delta m^3 a_3(
\chi_1 - \chi_2) \nonumber\\
& - [ 16 \delta m^2 a_2 - (1 + \delta m)^2 ( 9 - \delta m^2) \chi_1^2
+ 2 (\delta m^4 - 12 \delta m^2 + 11) \chi_1 \chi_2
- (1 -
\delta m)^2 ( 9 - \delta m^2) \chi_2^2 ] (1 - \delta m^2) \chi_1 \chi_2
\} , \\
\delta \phi_{\PP,1}^{(2)} &= -\frac{9}{1024 a_0 \delta m^4} \bm{[} 32
\delta m^6 (a_2^2 - a_4 ) ( \chi_1 - \chi_2) + 128 \delta m^5 [ a_5 - 2 a_2 a_3
+ (a_2^2 - a_4) (\chi_1 + \chi_2) ] \nonumber \\
&+ 96 \delta m ^4 ( a_2^2 - a_4 ) ( \chi_1
- \chi_2 ) + \bm{(} \delta m^6 ( \chi_1^2 - \chi_2^2) ( \chi_1 + \chi_2) + 2
\delta m^5 (3 \chi_1^2 + 2\chi_1 \chi_2 + 3\chi_2^2 ) (\chi_1 + \chi_2 )
\nonumber\\
& - \delta m^4 [ 16 a_2 - 3 ( \chi_1 + \chi_2 )^2 ] ( \chi_1 - \chi_2 ) + 4
\delta m^3 \{ 16 a_3 + [8 a_2 - 11 ( \chi_1 + \chi_2 )^2 ] ( \chi_1 + \chi_2 )\}
\nonumber\\
&+ \delta m^2 (48 a_2 - 105 \chi_1^2 - 226 \chi_1 \chi_2 - 105
\chi_2^2) ( \chi_1 - \chi_2) - 2 \delta m (45 \chi_1^2 - 106 \chi_1 \chi_2 + 45
\chi_2^2 ) (\chi_1 + \chi_2 ) \nonumber\\
&- 9 ( 3 \chi_1^2 - 10 \chi_1 \chi_2 + 3
\chi_2^2) ( \chi_1 - \chi_2)
\bm{)} (1 - \delta m^2 ) \chi_1 \chi_2
\bm{]}, \\
\delta \phi_{\PP,1}^{(3)} &= \frac{3}{128 a_0 \delta m^3} \bm{(} 8 \delta m^3
- 2 a_2 a_3) [(3 + 4 \delta m +
\delta m^2) \chi_1 - (3 - 4 \delta m +
\delta m^2) \chi_2] + \{ \delta m^4 a_2 ( \chi_1 + \chi_2 )^2 \nonumber\\
&+ 2 \delta m^3 [2 a_3 + a_2(\chi_1 + \chi_2 )] ( \chi_1 - \chi_2)
- 8 \delta m^2 [ 2(a_4 - a_2^2 ) + a_3 (\chi_1 + \chi_2 ) + a_2 ( \chi_1^2 + 3
\chi_1 \chi_2 + \chi_2^2 )] \nonumber \\
& - 6 \delta m [ 2 a_3 + 3 a_2 ( \chi_1 + \chi_2 ) ] ( \chi_1 - \chi_2 )
- a_2 ( 9 \chi_1^2 - 22 \chi_1 \chi_2 + 9 \chi_2^2 )
\} (1 - \delta m^2 ) \chi_1 \chi_2
\bm{)}.
\end{align}

When $\delta m \sim \mathcal{O} ( c^{-1} )$, $\delta \phi_\PP$ is given by

\begin{align}
\delta \phi_{\PP,2} &= \frac{1}{2} \int \left( \omega_{\PP,+} - \omega_{\PP,-}
\right)
\frac{dt}{d\xi}
d\xi = \frac{1}{2} \int \frac{\sqrt{(b-c+d-e+f-g)^2 + 4(c-f)(b-g)}}{a\xi} d\xi
\nonumber\\
&= \delta \phi_{\PP,2,0} + \frac{1}{T \xi^3} \left( \delta \phi_{\PP,2}^{(-3)} +
\delta \phi_{\PP,2}^{(-2)} \xi + \delta \phi_{\PP,2}^{(-1)} \xi^2 + \delta
\phi_{\PP,2}^{(0)} \xi^3 + \delta \phi_{\PP,2}^{(1)} \xi^4 + \delta
\phi_{\PP,2}^{(2)} \xi^5 \right) \nonumber\\
& + \delta \phi_{\PP,2}^{(l,1)} [ \log\xi - \log | 4 \delta m - 3 ( \chi_1 -
\chi_2) \xi + T | ] \nonumber\\
&+ \delta
\phi_{\PP,2}^{(l,2)} \log | 12 \delta m ( \chi_1 - \chi_2 ) -
( 9 \chi_1^2 - 2
\chi_1 \chi_2 + 9 \chi_2^2 ) \xi - (9 \chi_1^2 - 2
\chi_1 \chi_2 + 9 \chi_2^2)^{1/2} T |, \\
T &= [ 16 \delta m^2 - 24 \delta m ( \chi_1 - \chi_2 ) \xi + ( 9 \chi_1^2 - 2
\chi_1 \chi_2 + 9 \chi_2^2 ) \xi^2 ]^{1/2}, \\
\delta \phi_{\PP,2}^{(-3)} &= - \frac{3}{16 a_0} , \\
\delta \phi_{\PP,2}^{(-2)} &= \frac{9}{128 a_0 \delta m} [ (1 + 4 \delta m +
\delta m^2) \chi_1 - ( 1 - 4 \delta m^2 + \delta m^2 ) \chi_2 ] , \\
\delta \phi_{\PP,2}^{(-1)} &= \frac{3}{512 a_0 \delta m^2} [ 96 a_2 \delta m^2
(1 + 4 \delta m + \delta
m^2) \chi_1^2 - 10 ( 5 - 3 \delta m^2) \chi_1 \chi_2 + 9
(1 - 4 \delta m + \delta
m^2) \chi_2^2]  , \\
\delta \phi_{\PP,2}^{(0)} &= \frac{3}{16 a_0 ( 9 \chi_1^2 - 2 \chi_1 \chi_2 + 9
\chi_2^2)^4 } \{ 3456 \delta m^4 a_3 (81 \chi_1^6 - 567 \chi_1^5 \chi_2 +
1123 \chi_1^4 \chi_2^2 -
1530 \chi_1^3 \chi_2^3 + 1123 \chi_1^2 \chi_2^4 -
567 \chi_1 \chi_2^5 \nonumber \\
&+ 81 \chi_2^6) (\chi_1 + \chi_2 )^2 + 864 \delta m^3 a_2 (27 \chi_1^4 - 63
\chi_1^3 \chi_2 + 88 \chi_1^2 \chi_2^2 -
63 \chi_1 \chi_2^3 + 27 \chi_2^4 )  (9 \chi_1^2 -
2 \chi_1 \chi_2 \nonumber \\
&+ 9 \chi_2^2) (\chi_1 +
\chi_2)^2  (\chi_1 - \chi_2) - \delta m^2 a_3 (6561 \chi_1^8 + 17496
\chi_1^7
\chi_2 +
89676 \chi_1^6 \chi_2^2 - 229976 \chi_1^5 \chi_2^3 +
355366 \chi_1^4 \chi_2^4 \nonumber\\
&- 229976 \chi_1^3 \chi_2^5 +
89676 \chi_1^2 \chi_2^6 + 17496 \chi_1 \chi_2^7 +
6561 \chi_2^8) - 6 \delta m [ 6 a_3 (9 \chi_1^2 -
8 \chi_1 \chi_2 + 9 \chi_2^2)(9 \chi_1^2 -
2 \chi_1 \chi_2 \nonumber \\
&+ 9 \chi_2^2)(\chi_1 + \chi_2) + 2 a_2
(27 \chi_1^4 + 225 \chi_1^3 \chi_2 - 184 \chi_1^2 \chi_2^2 +
225 \chi_1 \chi_2^3 + 27 \chi_2^4) \chi_1 \chi_2
+ 9 (9 \chi_1^2 -
2 \chi_1 \chi_2 + 9 \chi_2^2)^2 ( 3
\chi_1^2 \nonumber \\
& - \chi_1 \chi_2 + 3 \chi_2^2)  (\chi_1 + \chi_2)^2
] (9 \chi_1^2 -
2 \chi_1 \chi_2 + 9 \chi_2^2) ( \chi_1 - \chi_2 )
- 3 [ a_3 (9 \chi_1^2 -
2 \chi_1 \chi_2 + 9 \chi_2^2) \nonumber \\
&- 4 a_2 ( \chi_1 + \chi_2 ) \chi_1
\chi_2 ](9 \chi_1^2 -
2 \chi_1 \chi_2 + 9 \chi_2^2)^3
\} , \\
\delta \phi_{\PP,2}^{(1)} &= -\frac{9}{32 a_0 ( 9 \chi_1^2 - 2 \chi_1 \chi_2 + 9
\chi_2^2)^4 } \{
576 \delta m^3 a_3 (243 \chi_1^6 - 2781 \chi_1^5 \chi_2 + 5577
\chi_1^4 \chi_2^2 -
7870 \chi_1^3 \chi_2^3 + 5577 \chi_1^2 \chi_2^4 \nonumber \\
&-  2781 \chi_1 \chi_2^5 + 243 \chi_2^6) ( \chi_1 + \chi_2 )^2 (\chi_1 -
\chi_2 ) + 144 \delta m^2 a_2 (81 \chi_1^6 - 567 \chi_1^5 \chi_2 + 1123
\chi_1^4 \chi_2^2 -
1530 \chi_1^3 \chi_2^3 + 1123 \chi_1^2 \chi_2^4 \nonumber \\
&- 567 \chi_1 \chi_2^5 + 81 \chi_2^6)(9 \chi_1^2 - 2 \chi_1 \chi_2 + 9
\chi_2^2) (\chi_1 + \chi_2)^2 + 4 \delta m a_3  (27 \chi_1^4 + 230 \chi_1^3
\chi_2 - 194 \chi_1^2 \chi_2^2 +
230 \chi_1 \chi_2^3 \nonumber \\
&+ 27 \chi_2^4) (9 \chi_1^2 - 2 \chi_1 \chi_2 + 9
\chi_2^2) (\chi_1 - \chi_2) \chi_1 \chi_2 - 4  a_3 (9 \chi_1^2 - 2 \chi_1 \chi_2
+ 9 \chi_2^2)^3 (\chi_1 + \chi_2 ) \chi_1 \chi_2 + 10 a_2 ( 9
\chi_1^2 - 2 \chi_1 \chi_2 \nonumber \\
&+ 9 \chi_2^2)^2 (\chi_1 - \chi_2 )^2 \chi_1^2
\chi_2^2 - 9 (9 \chi_1^4 - 13 \chi_1^3 \chi_2 + 24 \chi_1^2 \chi_2^2 -
13 \chi_1 \chi_2^3 + 9 \chi_2^4) ( 9 \chi_1^2 - 2 \chi_1
\chi_2 + 9 \chi_2^2)^3 (\chi_1 + \chi_2 )^2
\} , \\
\delta \phi_{\PP,2}^{(2)} &= -\frac{15 a_3 ( \chi_1 - \chi_2 )^2 \chi_1^2
\chi_2^2}{8 a_0 ( 9 \chi_1^2 - 2 \chi_1 \chi_2 + 9 \chi_2^2)^2 } , \\
\delta \phi_{\PP,2}^{(l,1)} &= -\frac{9}{32 a_0 \delta m^2} \{ 8 \delta m^3 a_3
-2 \delta m^2 a_2 [ ( 3 + 4 \delta m + \delta m^2 ) \chi_1 - ( 3 - 4 \delta m +
\delta m^2 ) \chi_2 ] \nonumber \\
&- [ (3 + 2 \delta m - 4 \delta m^2) \chi_1 -
(3 - 2 \delta m - 4 \delta m^2 ) \chi_2 ]
\}, \\
\delta \phi_{\PP,2}^{(l,2)} &= - \frac{9}{16 a_0 ( 9 \chi_1^2 - 2 \chi_1
\chi_2 + 9 \chi_2^2)^{9/2}} \{
32 \delta m^3 a_3 (12393 \chi_1^6 + 5238 \chi_1^5 \chi_2 -
25065 \chi_1^4 \chi_2^2 + 53780 \chi_1^3 \chi_2^3 -
25065 \chi_1^2 \chi_2^4 \nonumber\\
&+ 5238 \chi_1 \chi_2^5 +
12393 \chi_2^6) (\chi_1 - \chi_2 ) \chi_1 \chi_2 - \delta m^2 [ 64 a_3 (45
\chi_1^2 - 34 \chi_1 \chi_2 + 45 \chi_2^2) (9 \chi_1^2 - 2 \chi_1 \chi_2
\nonumber\\
&+ 9 \chi_2^2)^2 (\chi_1 + \chi_2) \chi_1 \chi_2 - a_2 (2187 \chi_1^8 + 5184
\chi_1^7 \chi_2 +   25668 \chi_1^6 \chi_2^2 - 61184 \chi_1^5 \chi_2^3 +
97250 \chi_1^4 \chi_2^4 - 61184 \chi_1^3 \chi_2^5 \nonumber\\
&+  25668 \chi_1^2 \chi_2^6 + 5184 \chi_1 \chi_2^7 + 2187 \chi_2^8) (9
\chi_1^2 - 2 \chi_1 \chi_2 + 9 \chi_2^2) ]
- 12 \delta m [a_3 (9 \chi_1^2 - 2 \chi_1 \chi_2 + 9 \chi_2^2) \nonumber\\
&- 3 a_2  (3
\chi_1^2 - 2 \chi_1 \chi_2 + 3 \chi_2^2)  (\chi_1 + \chi_2) ]  (9
\chi_1^2 - 2 \chi_1 \chi_2 + 9 \chi_2^2)^3 ( \chi_1 - \chi_2 ) + [ a_2 (9
\chi_1^2 - 2 \chi_1 \chi_2 + 9 \chi_2^2)^2 \nonumber\\
&- 10 ( \chi_1^2 - \chi_2^2
) \chi_1^2 \chi_2^2 ] (9 \chi_1^2 - 2 \chi_1 \chi_2 + 9 \chi_2^2)^3
\}.
\end{align}

When $\delta m \ll \mathcal{O} ( c^{-1} )$, $\delta \phi_\PP$ is given by

\begin{align}
\delta \phi_{\PP,3} &= \frac{1}{2} \int \left( \omega_{\PP,+} - \omega_{\PP,-}
\right)
\frac{dt}{d\xi}
d\xi = \frac{1}{2} \int \frac{\sqrt{(b-c+d-e+f-g)^2 + 4(c-f)(b-g)}}{a\xi} d\xi
\nonumber\\
&= \delta \phi_{\PP,3,0} + \delta
\phi_{\PP,3}^{(0)} + \frac{T}{\xi^3} \left( \delta \phi_{\PP,3}^{(-3)} +
\delta \phi_{\PP,3}^{(-2)} \xi + \delta \phi_{\PP,3}^{(-1)} \xi^2  + \delta
\phi_{\PP,3}^{(1)} \xi^4 \right) \nonumber\\
&+ \delta
\phi_{\PP,3}^{(l,1)}  \log \left| \frac{2 (\chi_1 + \chi_2 ) - 2 \chi_1 \chi_2
\xi
- T}{2 (\chi_1 + \chi_2 ) - (9 \chi_1^2 - 2 \chi_1
\chi_2 + 9 \chi_2^2)^{1/2}} \right| \nonumber\\
\delta
\phi_{\PP,3}^{(l,2)}  [ \log \xi - \log | 9 \chi_1^2 - 2 \chi_1 \chi_2 + 9
\chi_2^2 - 4 ( \chi_1 + \chi_2 ) \chi_1 \chi_2 \xi + (9 \chi_1^2 - 2 \chi_1
\chi_2 + 9 \chi_2^2)^{1/2} T | ], \\
T &= [ (9 \chi_1^2 - 2 \chi_1 \chi_2 + 9
\chi_2^2) - 8 (
\chi_1 + \chi_2 ) \chi_1 \chi_2 \xi + 4 \chi_1^2 \chi_2^2 \xi^2]^{1/2}, \\
\delta \phi_{\PP,3}^{(-3)} &= \frac{9 \delta m (\chi_1 - \chi_2)}{4 a_0 (9
\chi_1^2 - 2 \chi_1
\chi_2 + 9 \chi_2^2)} , \\
\delta \phi_{\PP,3}^{(-2)} &= - \frac{9}{32 a_0} - \frac{9 \delta m (27
\chi_1^2
26 \chi_1 \chi_2 + 27 \chi_2^2) ( \chi_1^2 - \chi_2^2) }{8 a_0 (9 \chi_1^2
- 2 \chi_1
\chi_2 + 9 \chi_2^2)^2} , \\
\delta \phi_{\PP,3}^{(-1)} &= \frac{9 ( \chi_1 + \chi_2 ) \chi_1 \chi_2}{8 a_0
\chi_1^2
- 2 \chi_1
\chi_2 + 9 \chi_2^2)} - \frac{27 a_2 \delta m (\chi_1 - \chi_2)}{4 a_0 (9
\chi_1^2
- 2 \chi_1
\chi_2 + 9 \chi_2^2)} \nonumber\\
&- \frac{9 \delta m (81 \chi_1^4 + 276 \chi_1^3
\chi_2 - 170 \chi_1^2 \chi_2^2 +
276 \chi_1 \chi_2^3 + 81 \chi_2^4) ( \chi_1 - \chi_2) \chi_1 \chi_2 }{4
a_0 (9
\chi_1^2
- 2 \chi_1
\chi_2 + 9 \chi_2^2)^3}
\delta \phi_{\PP,3}^{(0)} &= - \frac{9 a_2 T}{16 a_0} + \frac{9 a_3 [ \chi_1 +
\chi_2 + 2 \delta m (\chi_1 - \chi_2 )]}{32 a_0 \chi_1 \chi_2} \left( T -
\sqrt{9
\chi_1^2
- 2 \chi_1
\chi_2 + 9 \chi_2^2} \right) ,
\\
\delta \phi_{\PP,3}^{(1)} &= - \frac{9 a_3}{32 a_0} , \\
\delta \phi_{\PP,3}^{(l,1)} &= \frac{9 \{ a_3 [(5 + 16 \delta m) \chi_1 - (5 -
16\delta m) \chi_2 ] ( \chi_1 - \chi_2 ) - 8 a_2 [ \chi_1 + \chi_2 + \delta m (
\chi_1 - \chi_2 )] \chi_1 \chi_2 \} }{64 a_0 \chi_1 \chi_2} , \\
\delta \phi_{\PP,3}^{(l,2)} &= - \frac{9}{16 a_0 (9 \chi_1^2 - 2 \chi_1
\chi_2 + 9 \chi_2^2)^{7/2}} \{(9 \chi_1^2 - 2 \chi_1
\chi_2 + 9 \chi_2^2)^2 [
a_2 (9 \chi_1^2 - 2 \chi_1
\chi_2 + 9 \chi_2^2)^2 - 10 ( \chi_1 - \chi_2)^2 \chi_1^2 \chi_2^2 ] \nonumber\\
- \delta m
[ 12 a_3  (9 \chi_1^2 - 2 \chi_1
\chi_2 + 9 \chi_2^2)^3 ( \chi_1 - \chi_2)
-36 a_2 (9 \chi_1^2 - 2 \chi_1
\chi_2 + 9 \chi_2^2)^2 (3 \chi_1^2 - 2 \chi_1
\chi_2 + 3 \chi_2^2) ( \chi_1^2 - \chi_2^2 ) \nonumber \\
&-8 (81 \chi_1^4 - 840 \chi_1^3 \chi_2 + 878 \chi_1^2 \chi_2^2 -
840 \chi_1 \chi_2^3 + 81 \chi_2^4)  ( \chi_1^2 - \chi_2^2 ) \chi_1^2
\chi_2^2
\end{align}

Note that despite appearances, $\delta \phi_{\PP,3}$ is regular in the
limits $\chi_1 \to 0$ and $\chi_2 \to 0$.

From $\omega_{\PP,m}(\xi)$, $\delta \omega_{\PP}(\xi)$ and $\dot{\xi}$, we
can compute $\ddot{\phi}_{\PP,\pm} = \dot{\omega}_{\PP,\pm} =
\dot{\omega}_{\PP,m} \pm \delta \dot{\omega}_{\PP}$. We get

\begin{align}
\dot{\omega}_{\PP,m} &= \dot{\omega} \left\{ \frac{5}{24} \left( 7 - \delta
m^2\right) \xi^2 + \frac{1}{8} \left[ \left( 5 + 4 \delta m - \delta m^2\right)
\chi_1 + \left(5 - 4 \delta m - \delta m^2 \right) \chi_2 \right] \xi^3 -
\frac{7}{8} \left( 1 - \delta m^2 \right) \chi_1 \chi_2 \xi^4 \right\}, \\
\delta \dot{\omega}_{\PP} &= \frac{\dot{\omega}}{\delta \omega_\PP} \bigg\{
\frac{15 \delta m^2}{16} \xi^7 + \frac{33 \delta m}{64} \left[ \left( 3 + 4
\delta m  + \delta m^2 \right) \chi_1 - \left( 3 - 4 \delta m + \delta
m^2\right) \chi_2 \right] \xi^8 \nonumber\\
&+ \frac{9}{128} \left[ \left( 3 + 4 \delta m  +
\delta m^2 \right)^2 \chi_1^2 - 2 \left(1 - \delta m^2\right)^2 \chi_1 \chi_2
+ \left( 3 - 4 \delta m + \delta m^2\right)^2 \chi_2^2 \right] \xi^9 \nonumber\\
& + \frac{39}{64} \left(1 - \delta m^2\right) \left[\left(1+ \delta m\right)
\chi_1 + \left(1 - \delta m\right)\chi_2\right] \chi_1 \chi_2 \xi^{10} +
\frac{21}{64} \left( 1 + \delta m\right)^2 \left(1 - \delta m\right)^2 \chi_1^2
\chi_2^2 \xi^{11}
\bigg\}.
\end{align}

§ AMPLITUDE AND PHASE MODULATIONS

Using $\bm{V}_1 = \uvec{z} \times \uvec{N}$ and $\bm{V}_2 = \bm{z} - \uvec{N}
(\uvec{z} \cdot \uvec{N})$,
the amplitudes and phases of Eq. (<ref>) are given by

\begin{align}
A_{0,n,k,m} &= k \arccos \hat{N}_z + m \arctan \frac{V_{1,z}}{V_{2,z}} \\
A_{\delta \phi,\pm} &=  \mbox{sign}\left( \hat{N}_x
B_{\pm,2}^{(1,0)} - \hat{N}_y
B_{\pm,1}^{(1,0)} \right) \left[
\frac{\left.B_{\pm,1}^{(1,0)}\right.^2 +
\left.B_{\pm,2}^{(1,0)}\right.^2}{\hat{N}_x^2 + \hat{N}_y^2} \right]^{1/2}
\hat{N}_z \frac{\xi}{\nu}, \\
\phi_{\delta\phi,\pm}^{(0)} &= - \arctan \left( \frac{\hat{N}_x
B_{\pm,1}^{(1,0)} + \hat{N}_y
B_{\pm,2}^{(1,0)} - \hat{N}_y
B_{\pm,1}^{(1,0)}} \right), \\
A_{i_L,\pm} &= -\mbox{sign}\left( \hat{N}_x
B_{\pm,1}^{(1,0)} + \hat{N}_y
B_{\pm,2}^{(1,0)} \right) \left[
\left.B_{\pm,1}^{(1,0)}\right.^2 +
\left.B_{\pm,2}^{(1,0)}\right.^2 \right]^{1/2} \frac{\xi}{\nu}, \\
\phi_{i_L,\pm}^{(0)} &= \arctan \left( \frac{\hat{N}_x
B_{\pm,2}^{(1,0)} - \hat{N}_y
B_{\pm,1}^{(1,0)} + \hat{N}_y
B_{\pm,2}^{(1,0)}} \right), \\
A_{\psi_N,\pm} &= \mbox{sign} \Big[ V_{1,z} \Big( B_{\pm,1}^{(1,0)}
V_{2,x} +
B_{\pm,2}^{(1,0)} V_{2,y} \Big) - V_{2,z} \Big(
B_{\pm,1}^{(1,0)} V_{1,x} +
B_{\pm,2}^{(1,0)} V_{1,y} \Big) \Big] \nonumber\\
\bigg\{ \Big[ V_{1,z} \Big( B_{\pm,1}^{(1,0)}
V_{2,x} +
B_{\pm,2}^{(1,0)} V_{2,y} \Big) - V_{2,z} \Big(
B_{\pm,1}^{(1,0)} V_{1,x} +
B_{\pm,2}^{(1,0)} V_{1,y} \Big) \Big]^2 \nonumber\\
&+ \Big[ V_{1,z} \Big(
B_{\pm,1}^{(1,0)} V_{2,y} -
B_{\pm,2}^{(1,0)} V_{2,x} \Big) - V_{2,z} \Big( B_{\pm,1}^{(1,0)} V_{1,y} -
B_{\pm,2}^{(1,0)} V_{1,x}\Big) \Big]^2 \bigg\}^{1/2}  \left( V_{1,z}^2
+ V_{2,z}^2   \right)^{-1} \frac{\xi}{\nu} , \\
\phi_{\psi_N,\pm}^{(0)} &= - \arctan \bigg\{ \Big[ V_{1,z} \Big(
B_{\pm,1}^{(1,0)} V_{2,y} -
B_{\pm,2}^{(1,0)} V_{2,x} \Big) - V_{2,z} \Big( B_{\pm,1}^{(1,0)} V_{1,y} -
B_{\pm,2}^{(1,0)} V_{1,x}\Big) \Big] \nonumber\\
&\times\Big[ V_{1,z} \Big( B_{\pm,1}^{(1,0)}
V_{2,x} +
B_{\pm,2}^{(1,0)} V_{2,y} \Big) - V_{2,z} \Big(
B_{\pm,1}^{(1,0)} V_{1,x} +
B_{\pm,2}^{(1,0)} V_{1,y} \Big) \Big]^{-1} \bigg\}.
\end{align}

Using this, we can express

\begin{align}
A_{\pm,n,k,m} &= \mbox{sign}(A_{c,\pm}) \sqrt{A_{c,\pm}^2 + A_{s,\pm}^2}, \\
\phi_{\pm,n,k,m} &= \arctan \left( \frac{A_{s,\pm}}{A_{c,\pm}} \right), \\
A_{c,\pm} &= n A_{\delta\phi,\pm} \cos (\phi_{\delta\phi,\pm}^{(0)}) + k
A_{i_L,\pm} \cos (\phi_{i_L,\pm}^{(0)})
+ m
A_{\psi_N,\pm} \cos ( \phi_{\psi_N,\pm}^{(0)}), \\
A_{s,\pm} &= n A_{\delta\phi,\pm} \sin
(\phi_{\delta\phi,\pm}^{(0)}) + k
A_{i_L,\pm} \sin (\phi_{i_L,\pm}^{(0)})
+ m
A_{\psi_N,\pm} \sin ( \phi_{\psi_N,\pm}^{(0)}) ,
\end{align}

§ MODE-DECOMPOSED, TIME-DOMAIN AMPLITUDES

The waveforms are expressed as

\begin{align}
h(t) = \frac{G\mu \xi^2}{D_L c^2} \sum_{n \geq 0} \ \sum_{k \in \mathbb{Z}} \
\sum_{m = -2,2}
\mathcal{A}_{n,k,m}(\theta_N, \phi_N) e^{i [ n\psi_{\C} + n\delta\phi + k
i_L + m \psi_N]} + \rm{c.c.} \, ,
\end{align}

where $\psi_\C = \phi_\C - (6 - 3 \nu \xi^2) \xi^3 \log\xi$.

\begin{align}
\mathcal{A}_F &= \frac{1}{2} \left( 1 + \cos^2 \theta_N
\right) \cos 2\phi_N, \\
\mathcal{B}_F &= \cos \theta_N \sin 2\phi_N,
\end{align}

and using $\delta m = (m_1 - m_2)/M$ the dimensionless mass difference,
we can express the $\mathcal{A}_{n,k,m}$ at 2PN for $n \neq 0$ as
(see [Arun et al., 2004])

\begin{align}
\mathcal{A}_{1,1,2} &= - \mathcal{A}_{1,-1,2} = - \delta m ( \mathcal{B}_F - i
\mathcal{A}_F ) \left\{
\frac{21}{128}\xi - \left( \frac{575}{6144} - \frac{367 \nu}{3072}\right) \xi^3
+ \left[ \frac{21\pi}{128} + i \left(\frac{21 \log 2}{64} + \frac{51}{640}
\right) \right] \xi^4 \right\}, \\
\mathcal{A}_{1,2,2} &= - \mathcal{A}_{1,-2,2} = \delta m ( \mathcal{B}_F - i
\mathcal{A}_F ) \left\{
\frac{3}{32}\xi - \left( \frac{121}{1536} - \frac{41 \nu}{768}\right) \xi^3
+ \left[ \frac{3\pi}{32} + i \left(\frac{3 \log 2}{16} + \frac{9}{160}
\right) \right] \xi^4 \right\}, \\
\mathcal{A}_{1,3,2} &= - \mathcal{A}_{1,-3,2} = - \delta m ( \mathcal{B}_F - i
\mathcal{A}_F ) \left\{
\frac{1}{128}\xi - \left( \frac{79}{4096} + \frac{17 \nu}{2048}\right) \xi^3
+ \left[ \frac{\pi}{128} + i \left(\frac{\log 2}{64} + \frac{7}{640}
\right) \right] \xi^4 \right\}, \\
\mathcal{A}_{1,4,2} &= - \mathcal{A}_{1,-4,2} = \delta m ( \mathcal{B}_F - i
\mathcal{A}_F ) \left( \frac{5}{3072}
- \frac{5 \nu}{1536}\right) \xi^3, \\
\mathcal{A}_{1,5,2} &= - \mathcal{A}_{1,-5,2} = - \delta m ( \mathcal{B}_F - i
\mathcal{A}_F ) \left( \frac{1}{12288}
- \frac{\nu}{6144}\right) \xi^3, \\
\mathcal{A}_{1,1,-2} &= - \mathcal{A}_{1,-1,-2} = \delta m ( \mathcal{B}_F + i
\mathcal{A}_F ) \left\{
\frac{21}{128}\xi - \left( \frac{575}{6144} - \frac{367 \nu}{3072}\right) \xi^3
+ \left[ \frac{21\pi}{128} + i \left(\frac{21 \log 2}{64} + \frac{51}{640}
\right) \right] \xi^4 \right\}, \\
\mathcal{A}_{1,2,-2} &= - \mathcal{A}_{1,-2,-2} = \delta m ( \mathcal{B}_F + i
\mathcal{A}_F ) \left\{
\frac{3}{32}\xi - \left( \frac{121}{1536} - \frac{41 \nu}{768}\right) \xi^3
+ \left[ \frac{3\pi}{32} + i \left(\frac{3 \log 2}{16} + \frac{9}{160}
\right) \right] \xi^4 \right\}, \\
\mathcal{A}_{1,3,-2} &= - \mathcal{A}_{1,-3,-2} = \delta m ( \mathcal{B}_F + i
\mathcal{A}_F ) \left\{
\frac{1}{128}\xi - \left( \frac{79}{4096} + \frac{17 \nu}{2048}\right) \xi^3
+ \left[ \frac{\pi}{128} + i \left(\frac{\log 2}{64} + \frac{7}{640}
\right) \right] \xi^4 \right\}, \\
\mathcal{A}_{1,4,-2} &= - \mathcal{A}_{1,-4,-2} = \delta m ( \mathcal{B}_F + i
\mathcal{A}_F ) \left( \frac{5}{3072}
- \frac{5 \nu}{1536}\right) \xi^3, \\
\mathcal{A}_{1,5,-2} &= - \mathcal{A}_{1,-5,-2} = \delta m ( \mathcal{B}_F + i
\mathcal{A}_F ) \left( \frac{1}{12288}
- \frac{\nu}{6144}\right) \xi^3, \\
\mathcal{A}_{2,0,2} &= - (\mathcal{A}_F + i \mathcal{B}_F) \left[ \frac{3}{4}
\left( \frac{91}{48} -
\frac{15\nu}{16} \right) \xi^2 + \frac{3\pi}{2} \xi^3 - \left( \frac{291}{256}
+ \frac{8137\nu}{2304} - \frac{257\nu^2}{256} \right) \xi^4 \right], \\
\mathcal{A}_{2,1,2} &= \mathcal{A}_{2,-1,2} = (\mathcal{A}_F + i
\mathcal{B}_F)
\left[ \frac{1}{2} - \left(
\frac{7}{6} -
\frac{\nu}{3} \right) \xi^2 + \pi \xi^3 - \left( \frac{365}{384}
+ \frac{1861\nu}{1152} - \frac{103\nu^2}{384} \right) \xi^4 \right], \\
\mathcal{A}_{2,2,2} &= \mathcal{A}_{2,-2,2} = -(\mathcal{A}_F + i
\mathcal{B}_F) \left[ \frac{1}{8} - \left(
\frac{7}{48} +
\frac{17\nu}{48} \right) \xi^2 + \frac{\pi}{4} \xi^3 - \left( \frac{8581}{15360}
+ \frac{7247\nu}{9216} - \frac{1541\nu^2}{3072} \right) \xi^4 \right], \\
\mathcal{A}_{2,3,2} &= \mathcal{A}_{2,-3,2} = (\mathcal{A}_F + i
\mathcal{B}_F)
\left[ \left(
\frac{1}{12} -
\frac{\nu}{4} \right) \xi^2 - \left( \frac{829}{3840}
- \frac{1735\nu}{2304} + \frac{205\nu^2}{768} \right) \xi^4 \right], \\
\mathcal{A}_{2,4,2} &= \mathcal{A}_{2,-4,2} = -(\mathcal{A}_F + i
\mathcal{B}_F) \left[ \left(
\frac{1}{96} -
\frac{\nu}{32} \right) \xi^2 - \left( \frac{55}{1536}
- \frac{457\nu}{4608} - \frac{29\nu^2}{1536} \right) \xi^4 \right], \\
\mathcal{A}_{2,5,2} &= \mathcal{A}_{2,-5,2} = (\mathcal{A}_F + i
\mathcal{B}_F)
\left( \frac{1}{256}
- \frac{5\nu}{256} + \frac{5\nu^2}{256} \right) \xi^4, \\
\mathcal{A}_{2,6,2} &= \mathcal{A}_{2,-6,2} = -(\mathcal{A}_F + i
\mathcal{B}_F)  \left( \frac{1}{3072}
- \frac{5\nu}{3072} + \frac{5\nu^2}{3072} \right) \xi^4, \\
\mathcal{A}_{2,0,-2} &= - (\mathcal{A}_F - i \mathcal{B}_F) \left[ \frac{3}{4}
- \left( \frac{91}{48} -
\frac{15\nu}{16} \right) \xi^2 + \frac{3\pi}{2} \xi^3 - \left( \frac{291}{256}
+ \frac{8137\nu}{2304} - \frac{257\nu^2}{256} \right) \xi^4 \right], \\
\mathcal{A}_{2,1,-2} &= \mathcal{A}_{2,-1,-2} = -(\mathcal{A}_F - i
\mathcal{B}_F) \left[ \frac{1}{2} - \left(
\frac{7}{6} -
\frac{\nu}{3} \right) \xi^2 + \pi \xi^3 - \left( \frac{365}{384}
+ \frac{1861\nu}{1152} - \frac{103\nu^2}{384} \right) \xi^4 \right], \\
\mathcal{A}_{2,2,-2} &= \mathcal{A}_{2,-2,-2} = -(\mathcal{A}_F - i
\mathcal{B}_F) \left[ \frac{1}{8} - \left(
\frac{7}{48} +
\frac{17\nu}{48} \right) \xi^2 + \frac{\pi}{4} \xi^3 - \left( \frac{8581}{15360}
+ \frac{7247\nu}{9216} - \frac{1541\nu^2}{3072} \right) \xi^4 \right], \\
\mathcal{A}_{2,3,-2} &= \mathcal{A}_{2,-3,-2} = -(\mathcal{A}_F - i
\mathcal{B}_F) \left[ \left(
\frac{1}{12} -
\frac{\nu}{4} \right) \xi^2 - \left( \frac{829}{3840}
- \frac{1735\nu}{2304} + \frac{205\nu^2}{768} \right) \xi^4 \right], \\
\mathcal{A}_{2,4,-2} &= \mathcal{A}_{2,-4,-2} = -(\mathcal{A}_F - i
\mathcal{B}_F) \left[ \left(
\frac{1}{96} -
\frac{\nu}{32} \right) \xi^2 - \left( \frac{55}{1536}
- \frac{457\nu}{4608} - \frac{29\nu^2}{1536} \right) \xi^4 \right], \\
\mathcal{A}_{2,5,-2} &= \mathcal{A}_{2,-5,-2} = -(\mathcal{A}_F - i
\mathcal{B}_F)  \left( \frac{1}{256}
- \frac{5\nu}{256} + \frac{5\nu^2}{256} \right) \xi^4, \\
\mathcal{A}_{2,6,-2} &= \mathcal{A}_{2,-6,-2} = -(\mathcal{A}_F - i
\mathcal{B}_F)  \left( \frac{1}{3072}
- \frac{5\nu}{3072} + \frac{5\nu^2}{3072} \right) \xi^4, \\
\mathcal{A}_{3,1,2} &= - \mathcal{A}_{3,-1,2} = \delta m ( \mathcal{B}_F - i
\mathcal{A}_F ) \left\{
\frac{45}{128}\xi - \left( \frac{5895}{4096} - \frac{1575 \nu}{2048}\right)
\xi^3
+ \left[ \frac{135\pi}{128} - i \left(\frac{135 \log(3/2)}{64} -
\frac{189}{128}
\right) \right] \xi^4 \right\}, \\
\mathcal{A}_{3,2,2} &= - \mathcal{A}_{3,-2,2} = -\delta m ( \mathcal{B}_F - i
\mathcal{A}_F ) \left\{
\frac{9}{32}\xi - \left( \frac{1071}{1024} - \frac{207 \nu}{512}\right)
\xi^3
+ \left[ \frac{27\pi}{32} + i \left(\frac{27 \log (3/2)}{16} -
\frac{189}{160}
\right) \right] \xi^4 \right\}, \\
\mathcal{A}_{3,3,2} &= - \mathcal{A}_{3,-3,2} = \delta m ( \mathcal{B}_F - i
\mathcal{A}_F ) \left\{
\frac{9}{128}\xi - \left( \frac{1197}{8192} + \frac{531 \nu}{4096}\right)
\xi^3
+ \left[ \frac{27\pi}{128} - i \left(\frac{27 \log(3/2)}{64} -
\frac{189}{640}
\right) \right] \xi^4 \right\}, \\
\mathcal{A}_{3,4,2} &= - \mathcal{A}_{3,-4,2} = -\delta m ( \mathcal{B}_F - i
\mathcal{A}_F ) \left( \frac{135}{2048} -
\frac{135 \nu}{1024}\right)
\xi^3, \\
\mathcal{A}_{3,5,2} &= - \mathcal{A}_{3,-5,2} = \delta m ( \mathcal{B}_F - i
\mathcal{A}_F ) \left( \frac{81}{8192} -
\frac{81 \nu}{4096}\right)
\xi^3, \\
\mathcal{A}_{3,1,-2} &= - \mathcal{A}_{3,-1,-2} = - \delta m ( \mathcal{B}_F +
i \mathcal{A}_F ) \left\{
\frac{45}{128}\xi - \left( \frac{5895}{4096} - \frac{1575 \nu}{2048}\right)
\xi^3
+ \left[ \frac{135\pi}{128} - i \left(\frac{135 \log(3/2)}{64} -
\frac{189}{128}
\right) \right] \xi^4 \right\}, \\
\mathcal{A}_{3,2,-2} &= - \mathcal{A}_{3,-2,-2} = -\delta m ( \mathcal{B}_F +
i \mathcal{A}_F ) \left\{
\frac{9}{32}\xi - \left( \frac{1071}{1024} - \frac{207 \nu}{512}\right)
\xi^3
+ \left[ \frac{27\pi}{32} - i \left(\frac{27 \log (3/2)}{16} -
\frac{189}{160}
\right) \right] \xi^4 \right\}, \\
\mathcal{A}_{3,3,-2} &= - \mathcal{A}_{3,-3,-2} = -\delta m ( \mathcal{B}_F +
i \mathcal{A}_F ) \left\{
\frac{9}{128}\xi - \left( \frac{1197}{8192} + \frac{531 \nu}{4096}\right)
\xi^3
+ \left[ \frac{27\pi}{128} - i \left(\frac{27 \log(3/2)}{64} -
\frac{189}{640}
\right) \right] \xi^4 \right\}, \\
\mathcal{A}_{3,4,-2} &= - \mathcal{A}_{3,-4,-2} = -\delta m ( \mathcal{B}_F +
i \mathcal{A}_F ) \left( \frac{135}{2048}
\frac{135 \nu}{1024}\right)
\xi^3, \\
\mathcal{A}_{3,5,-2} &= - \mathcal{A}_{3,-5,-2} = -\delta m ( \mathcal{B}_F +
i \mathcal{A}_F ) \left( \frac{81}{8192}
\frac{81 \nu}{4096}\right)
\xi^3, \\
\mathcal{A}_{4,0,2} &= - (\mathcal{A}_F + i \mathcal{B}_F) \left[ \left(
\frac{5}{12} -
\frac{5 \nu}{4} \right) \xi^2  - \left( \frac{89}{40}
- \frac{571 \nu}{72} + \frac{77\nu^2}{24} \right) \xi^4 \right], \\
\mathcal{A}_{4,1,2} &= \mathcal{A}_{4,-1,2} = (\mathcal{A}_F + i
\mathcal{B}_F)
\left[ \left(
\frac{1}{6} -
\frac{\nu}{2} \right) \xi^2 - \left( \frac{49}{60}
- \frac{101\nu}{36} + \frac{11 \nu^2}{12} \right) \xi^4 \right], \\
\mathcal{A}_{4,2,2} &= \mathcal{A}_{4,-2,2} = (\mathcal{A}_F + i
\mathcal{B}_F)
\left[ \left(
\frac{1}{6} -
\frac{\nu}{2} \right) \xi^2 - \left( \frac{119}{120}
- \frac{265 \nu}{72} + \frac{43 \nu^2}{24} \right) \xi^4 \right], \\
\mathcal{A}_{4,3,2} &= \mathcal{A}_{4,-3,2} = - (\mathcal{A}_F + i
\mathcal{B}_F) \left[ \left(
\frac{1}{6} -
\frac{\nu}{2} \right) \xi^2 - \left( \frac{13}{15}
- \frac{55 \nu}{18} + \frac{7 \nu^2}{6} \right) \xi^4 \right], \\
\mathcal{A}_{4,4,2} &= \mathcal{A}_{4,-4,2} = (\mathcal{A}_F + i
\mathcal{B}_F)
\left[ \left(
\frac{1}{24} -
\frac{\nu}{8} \right) \xi^2 - \left( \frac{31}{240}
- \frac{47 \nu}{144} - \frac{7 \nu^2}{48} \right) \xi^4 \right], \\
\mathcal{A}_{4,5,2} &= \mathcal{A}_{4,-5,2} = -(\mathcal{A}_F + i
\mathcal{B}_F)  \left( \frac{1}{20}
- \frac{\nu}{4} + \frac{\nu^2}{4} \right) \xi^4, \\
\mathcal{A}_{4,6,2} &= \mathcal{A}_{4,-6,2} = (\mathcal{A}_F + i
\mathcal{B}_F)
\left( \frac{1}{120}
- \frac{\nu}{24} + \frac{\nu^2}{24} \right) \xi^4, \\
\mathcal{A}_{4,0,-2} &= - (\mathcal{A}_F - i \mathcal{B}_F) \left[ \left(
\frac{5}{12} -
\frac{5 \nu}{4} \right) \xi^2  - \left( \frac{89}{40}
- \frac{571 \nu}{72} + \frac{77\nu^2}{24} \right) \xi^4 \right], \\
\mathcal{A}_{4,1,-2} &= \mathcal{A}_{4,-1,-2} = -(\mathcal{A}_F - i
\mathcal{B}_F) \left[ \left(
\frac{1}{6} -
\frac{\nu}{2} \right) \xi^2 - \left( \frac{49}{60}
- \frac{101\nu}{36} + \frac{11 \nu^2}{12} \right) \xi^4 \right], \\
\mathcal{A}_{4,2,-2} &= \mathcal{A}_{4,-2,-2} = (\mathcal{A}_F - i
\mathcal{B}_F) \left[ \left(
\frac{1}{6} -
\frac{\nu}{2} \right) \xi^2 - \left( \frac{119}{120}
- \frac{265 \nu}{72} + \frac{43 \nu^2}{24} \right) \xi^4 \right], \\
\mathcal{A}_{4,3,-2} &= \mathcal{A}_{4,-3,-2} = (\mathcal{A}_F - i
\mathcal{B}_F) \left[ \left(
\frac{1}{6} -
\frac{\nu}{2} \right) \xi^2 - \left( \frac{13}{15}
- \frac{55 \nu}{18} + \frac{7 \nu^2}{6} \right) \xi^4 \right], \\
\mathcal{A}_{4,4,-2} &= \mathcal{A}_{4,-4,-2} = (\mathcal{A}_F - i
\mathcal{B}_F) \left[ \left(
\frac{1}{24} -
\frac{\nu}{8} \right) \xi^2 - \left( \frac{31}{240}
- \frac{47 \nu}{144} - \frac{7 \nu^2}{48} \right) \xi^4 \right], \\
\mathcal{A}_{4,5,-2} &= \mathcal{A}_{4,-5,-2} = (\mathcal{A}_F - i
\mathcal{B}_F)  \left( \frac{1}{20}
- \frac{\nu}{4} + \frac{\nu^2}{4} \right) \xi^4, \\
\mathcal{A}_{4,6,-2} &= \mathcal{A}_{4,-6,-2} = (\mathcal{A}_F - i
\mathcal{B}_F)  \left( \frac{1}{120}
- \frac{\nu}{24} + \frac{\nu^2}{24} \right) \xi^4, \\
\mathcal{A}_{5,1,2} &= - \mathcal{A}_{5,-1,2} = \delta m ( \mathcal{B}_F - i
\mathcal{A}_F ) \left( \frac{4375}{12288}
- \frac{4375 \nu}{6144}\right)
\xi^3, \\
\mathcal{A}_{5,2,2} &= - \mathcal{A}_{5,-2,2} = - \delta m ( \mathcal{B}_F - i
\mathcal{A}_F ) \left( \frac{625}{3072} -
\frac{625 \nu}{1536}\right)
\xi^3, \\
\mathcal{A}_{5,3,2} &= - \mathcal{A}_{5,-3,2} = - \delta m ( \mathcal{B}_F - i
\mathcal{A}_F ) \left( \frac{625}{8192}
- \frac{625 \nu}{4096}\right)
\xi^3, \\
\mathcal{A}_{5,4,2} &= - \mathcal{A}_{5,-4,2} = \delta m ( \mathcal{B}_F - i
\mathcal{A}_F ) \left( \frac{625}{6144} -
\frac{625 \nu}{3072}\right)
\xi^3, \\
\mathcal{A}_{5,5,2} &= - \mathcal{A}_{5,-5,2} = -\delta m ( \mathcal{B}_F - i
\mathcal{A}_F ) \left( \frac{625}{24576}
\frac{625 \nu}{12288}\right)
\xi^3,  \\
\mathcal{A}_{5,1,-2} &= - \mathcal{A}_{5,-1,-2} = -\delta m ( \mathcal{B}_F +
i \mathcal{A}_F ) \left(
\frac{4375}{12288}
- \frac{4375 \nu}{6144}\right)
\xi^3, \\
\mathcal{A}_{5,2,-2} &= - \mathcal{A}_{5,-2,-2} = -\delta m ( \mathcal{B}_F +
i \mathcal{A}_F ) \left( \frac{625}{3072}
\frac{625 \nu}{1536}\right)
\xi^3, \\
\mathcal{A}_{5,3,-2} &= - \mathcal{A}_{5,-3,-2} = \delta m ( \mathcal{B}_F + i
\mathcal{A}_F ) \left(
\frac{625}{8192}
- \frac{625 \nu}{4096}\right)
\xi^3, \\
\mathcal{A}_{5,4,-2} &= - \mathcal{A}_{5,-4,-2} = \delta m ( \mathcal{B}_F + i
\mathcal{A}_F ) \left( \frac{625}{6144}
\frac{625 \nu}{3072}\right)
\xi^3, \\
\mathcal{A}_{5,5,-2} &= - \mathcal{A}_{5,-5,-2} = \delta m ( \mathcal{B}_F + i
\mathcal{A}_F ) \left(
\frac{625}{24576}
\frac{625 \nu}{12288}\right)
\xi^3,  \\
\mathcal{A}_{6,0,2} &= - (\mathcal{A}_F + i \mathcal{B}_F) \left(
\frac{567}{1280}
- \frac{567 \nu}{256} + \frac{567\nu^2}{256} \right) \xi^4 , \\
\mathcal{A}_{6,1,2} &= \mathcal{A}_{6,-1,2} = (\mathcal{A}_F + i
\mathcal{B}_F)
\left( \frac{81}{640}
- \frac{81\nu}{128} + \frac{81 \nu^2}{128} \right) \xi^4 , \\
\mathcal{A}_{6,2,2} &= \mathcal{A}_{6,-2,2} = (\mathcal{A}_F + i
\mathcal{B}_F)
\left( \frac{1377}{5120}
- \frac{1377 \nu}{1024} + \frac{1377 \nu^2}{1024} \right) \xi^4 , \\
\mathcal{A}_{6,3,2} &= \mathcal{A}_{6,-3,2} = - (\mathcal{A}_F + i
\mathcal{B}_F) \left( \frac{243}{1280}
- \frac{243 \nu}{256} + \frac{243 \nu^2}{256} \right) \xi^4 , \\
\mathcal{A}_{6,4,2} &= \mathcal{A}_{6,-4,2} = -(\mathcal{A}_F + i
\mathcal{B}_F) \left( \frac{81}{2560}
- \frac{81 \nu}{512} + \frac{81 \nu^2}{512} \right) \xi^4 , \\
\mathcal{A}_{6,5,2} &= \mathcal{A}_{6,-5,2} = (\mathcal{A}_F + i
\mathcal{B}_F)
\left( \frac{81}{1280}
- \frac{81\nu}{256} + \frac{81\nu^2}{256} \right) \xi^4, \\
\mathcal{A}_{6,6,2} &= \mathcal{A}_{6,-6,2} = -(\mathcal{A}_F + i
\mathcal{B}_F)  \left( \frac{81}{5120}
- \frac{81\nu}{1024} + \frac{81\nu^2}{1024} \right) \xi^4, \\
\mathcal{A}_{6,0,-2} &= - (\mathcal{A}_F - i \mathcal{B}_F) \left(
\frac{567}{1280}
- \frac{567 \nu}{256} + \frac{567\nu^2}{256} \right) \xi^4 , \\
\mathcal{A}_{6,1,-2} &= \mathcal{A}_{6,-1,-2} = -(\mathcal{A}_F - i
\mathcal{B}_F) \left( \frac{81}{640}
- \frac{81\nu}{128} + \frac{81 \nu^2}{128} \right) \xi^4 , \\
\mathcal{A}_{6,2,-2} &= \mathcal{A}_{6,-2,-2} = (\mathcal{A}_F - i
\mathcal{B}_F) \left( \frac{1377}{5120}
- \frac{1377 \nu}{1024} + \frac{1377 \nu^2}{1024} \right) \xi^4 , \\
\mathcal{A}_{6,3,-2} &= \mathcal{A}_{6,-3,-2} = (\mathcal{A}_F - i
\mathcal{B}_F) \left( \frac{243}{1280}
- \frac{243 \nu}{256} + \frac{243 \nu^2}{256} \right) \xi^4 , \\
\mathcal{A}_{6,4,-2} &= \mathcal{A}_{6,-4,-2} = -(\mathcal{A}_F - i
\mathcal{B}_F) \left( \frac{81}{2560}
- \frac{81 \nu}{512} + \frac{81 \nu^2}{512} \right) \xi^4 , \\
\mathcal{A}_{6,5,-2} &= \mathcal{A}_{6,-5,-2} = -(\mathcal{A}_F - i
\mathcal{B}_F)  \left( \frac{81}{1280}
- \frac{81\nu}{256} + \frac{81\nu^2}{256} \right) \xi^4, \\
\mathcal{A}_{6,6,-2} &= \mathcal{A}_{6,-6,-2} = -(\mathcal{A}_F - i
\mathcal{B}_F)  \left( \frac{81}{5120}
- \frac{81\nu}{1024} + \frac{81\nu^2}{1024} \right) \xi^4,
\end{align}

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