Properties

Label 600.2.b.f
Level 600
Weight 2
Character orbit 600.b
Analytic conductor 4.791
Analytic rank 0
Dimension 8
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 600 = 2^{3} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 600.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(4.79102412128\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.1649659456.5
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 120)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{1} q^{2} + \beta_{6} q^{3} + ( \beta_{1} + \beta_{3} + \beta_{4} ) q^{4} + ( -1 + \beta_{2} ) q^{6} + ( -\beta_{1} + \beta_{2} - \beta_{3} + \beta_{5} ) q^{7} + ( 1 - \beta_{3} + \beta_{5} + \beta_{6} + \beta_{7} ) q^{8} + ( -\beta_{1} + \beta_{5} - \beta_{7} ) q^{9} +O(q^{10})\) \( q + \beta_{1} q^{2} + \beta_{6} q^{3} + ( \beta_{1} + \beta_{3} + \beta_{4} ) q^{4} + ( -1 + \beta_{2} ) q^{6} + ( -\beta_{1} + \beta_{2} - \beta_{3} + \beta_{5} ) q^{7} + ( 1 - \beta_{3} + \beta_{5} + \beta_{6} + \beta_{7} ) q^{8} + ( -\beta_{1} + \beta_{5} - \beta_{7} ) q^{9} -2 \beta_{3} q^{11} + ( 2 - \beta_{1} + \beta_{3} - \beta_{4} ) q^{12} + ( \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} + \beta_{6} ) q^{13} + ( 1 - \beta_{2} + \beta_{3} - 2 \beta_{4} + \beta_{7} ) q^{14} + ( -1 + 2 \beta_{1} + \beta_{3} - \beta_{5} + \beta_{6} + \beta_{7} ) q^{16} + ( \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} - \beta_{6} ) q^{17} + ( -1 - \beta_{1} - \beta_{2} - \beta_{3} - 2 \beta_{4} + 2 \beta_{5} + \beta_{7} ) q^{18} + ( -\beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} + \beta_{6} + 2 \beta_{7} ) q^{19} + ( 1 - 2 \beta_{1} - \beta_{2} - 3 \beta_{3} - 3 \beta_{4} + 2 \beta_{5} + \beta_{6} + \beta_{7} ) q^{21} + ( -2 + 2 \beta_{1} + 2 \beta_{3} + 2 \beta_{4} - 2 \beta_{5} - 2 \beta_{6} ) q^{22} + ( -\beta_{1} - \beta_{2} - \beta_{3} - 2 \beta_{4} + \beta_{5} + 2 \beta_{7} ) q^{23} + ( 1 - \beta_{3} - 2 \beta_{4} + \beta_{5} + \beta_{6} - \beta_{7} ) q^{24} + ( -2 + 2 \beta_{5} + 2 \beta_{6} ) q^{26} + ( 2 - 4 \beta_{1} - 2 \beta_{3} - \beta_{4} ) q^{27} + ( 2 \beta_{6} - 2 \beta_{7} ) q^{28} + ( 2 \beta_{1} + 2 \beta_{4} + 2 \beta_{6} + 2 \beta_{7} ) q^{29} + ( \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} + 3 \beta_{5} + \beta_{6} ) q^{31} + ( 1 + 2 \beta_{2} + \beta_{3} + 2 \beta_{4} - \beta_{5} + \beta_{6} - \beta_{7} ) q^{32} + ( 2 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} + 2 \beta_{4} ) q^{33} + ( -2 \beta_{3} + 2 \beta_{4} ) q^{34} + ( -2 - \beta_{1} - 2 \beta_{2} + \beta_{3} - \beta_{4} + 2 \beta_{6} ) q^{36} + ( \beta_{1} - \beta_{2} - 3 \beta_{3} + \beta_{4} + \beta_{5} - \beta_{6} ) q^{37} + ( -2 - 2 \beta_{5} + 2 \beta_{6} ) q^{38} + ( -2 - 3 \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} - \beta_{6} - 2 \beta_{7} ) q^{39} -2 \beta_{5} q^{41} + ( -5 + 2 \beta_{1} - \beta_{2} + 3 \beta_{3} - 2 \beta_{5} - \beta_{7} ) q^{42} + ( -\beta_{4} - \beta_{6} ) q^{43} + ( 4 - 2 \beta_{1} - 2 \beta_{3} + 2 \beta_{4} ) q^{44} + ( -1 - \beta_{2} + \beta_{3} - 2 \beta_{5} + 2 \beta_{6} - \beta_{7} ) q^{46} + ( 4 - 5 \beta_{1} - \beta_{2} - \beta_{3} - 2 \beta_{4} + \beta_{5} - 2 \beta_{7} ) q^{47} + ( -3 + 2 \beta_{1} + 3 \beta_{3} + \beta_{5} - \beta_{6} - \beta_{7} ) q^{48} + ( -3 + 4 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} + 2 \beta_{4} - 2 \beta_{5} - 2 \beta_{6} - 2 \beta_{7} ) q^{49} + ( 2 + 3 \beta_{1} + \beta_{2} + 3 \beta_{3} + 3 \beta_{4} - 3 \beta_{5} - \beta_{6} ) q^{51} + ( -2 - 2 \beta_{1} - 2 \beta_{4} + 2 \beta_{5} + 2 \beta_{6} + 2 \beta_{7} ) q^{52} + ( 2 + 2 \beta_{4} + 2 \beta_{6} ) q^{53} + ( -2 - \beta_{3} - 2 \beta_{4} - 2 \beta_{5} - 2 \beta_{6} - \beta_{7} ) q^{54} + ( -4 + 2 \beta_{2} + 2 \beta_{5} - 2 \beta_{6} ) q^{56} + ( 3 \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} - \beta_{6} - 2 \beta_{7} ) q^{57} + ( 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{4} - 2 \beta_{5} + 2 \beta_{6} + 2 \beta_{7} ) q^{58} + ( -2 \beta_{1} + 2 \beta_{2} - 2 \beta_{5} ) q^{59} + ( 2 \beta_{4} + 2 \beta_{5} - 2 \beta_{6} ) q^{61} + ( -4 + 2 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} + 2 \beta_{5} + 2 \beta_{6} + 2 \beta_{7} ) q^{62} + ( -4 + \beta_{1} + \beta_{2} - 3 \beta_{3} - \beta_{5} - 2 \beta_{7} ) q^{63} + ( 3 + 2 \beta_{2} - \beta_{3} - 2 \beta_{4} + \beta_{5} - \beta_{6} + \beta_{7} ) q^{64} + ( 2 + 4 \beta_{1} + 4 \beta_{3} + 2 \beta_{4} - 2 \beta_{5} - 2 \beta_{6} + 2 \beta_{7} ) q^{66} + ( 4 + 4 \beta_{1} + \beta_{4} + \beta_{6} + 4 \beta_{7} ) q^{67} + ( -2 + 2 \beta_{1} + 2 \beta_{4} - 2 \beta_{5} - 2 \beta_{6} + 2 \beta_{7} ) q^{68} + ( -3 + 4 \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} - 2 \beta_{5} - \beta_{6} - \beta_{7} ) q^{69} + ( 4 - 6 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} - 4 \beta_{4} + 2 \beta_{5} ) q^{71} + ( -5 - 4 \beta_{1} + 2 \beta_{2} - 3 \beta_{3} + \beta_{5} + \beta_{6} - \beta_{7} ) q^{72} + ( 2 - 6 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} - 4 \beta_{4} + 2 \beta_{5} ) q^{73} + ( -4 + 4 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} + 4 \beta_{4} - 2 \beta_{5} - 2 \beta_{6} + 2 \beta_{7} ) q^{74} + ( -2 - 2 \beta_{1} + 4 \beta_{2} + 2 \beta_{4} - 2 \beta_{5} - 2 \beta_{6} - 2 \beta_{7} ) q^{76} + ( 4 \beta_{1} - 2 \beta_{4} - 2 \beta_{6} + 4 \beta_{7} ) q^{77} + ( 2 - 6 \beta_{1} - 2 \beta_{3} - 4 \beta_{4} + 2 \beta_{5} - 2 \beta_{6} - 2 \beta_{7} ) q^{78} + ( -\beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} + \beta_{6} ) q^{79} + ( 1 + 2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} + 2 \beta_{4} + 2 \beta_{6} ) q^{81} + ( 2 \beta_{2} + 2 \beta_{4} - 2 \beta_{5} - 2 \beta_{6} - 2 \beta_{7} ) q^{82} + ( 2 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} + \beta_{4} - 2 \beta_{5} - \beta_{6} ) q^{83} + ( -6 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} - 2 \beta_{7} ) q^{84} + ( 1 - \beta_{2} + \beta_{3} - \beta_{7} ) q^{86} + ( 4 + 2 \beta_{1} + 2 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} - 2 \beta_{7} ) q^{87} + ( -2 + 4 \beta_{1} - 2 \beta_{3} - 2 \beta_{5} - 2 \beta_{6} + 2 \beta_{7} ) q^{88} + ( 4 \beta_{1} - 4 \beta_{2} + 4 \beta_{3} ) q^{89} + ( 4 - 6 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} - 4 \beta_{4} + 2 \beta_{5} ) q^{91} + ( -4 - 2 \beta_{1} + 4 \beta_{2} - 2 \beta_{3} + 2 \beta_{4} - 2 \beta_{6} - 2 \beta_{7} ) q^{92} + ( -5 \beta_{1} + 3 \beta_{2} - 3 \beta_{3} - 3 \beta_{4} - \beta_{5} - \beta_{6} - 2 \beta_{7} ) q^{93} + ( -5 - \beta_{2} - 3 \beta_{3} - 4 \beta_{4} + 2 \beta_{5} - 2 \beta_{6} - \beta_{7} ) q^{94} + ( 3 - 4 \beta_{1} - 2 \beta_{2} - \beta_{3} - 2 \beta_{4} + 5 \beta_{5} + 3 \beta_{6} + \beta_{7} ) q^{96} + ( -2 + 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} + 4 \beta_{4} - 2 \beta_{5} - 4 \beta_{7} ) q^{97} + ( 6 - \beta_{1} + 2 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} - 2 \beta_{6} ) q^{98} + ( 2 + 2 \beta_{2} - 4 \beta_{3} - 2 \beta_{4} + 4 \beta_{5} + 2 \beta_{6} + 2 \beta_{7} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + q^{2} + q^{4} - 7q^{6} + 7q^{8} + O(q^{10}) \) \( 8q + q^{2} + q^{4} - 7q^{6} + 7q^{8} + 15q^{12} + 6q^{14} - 7q^{16} - 11q^{18} - 4q^{19} + 4q^{21} - 14q^{22} - 4q^{23} + 9q^{24} - 16q^{26} + 12q^{27} + 2q^{28} + 11q^{32} + 4q^{33} - 19q^{36} - 16q^{38} - 16q^{39} - 38q^{42} + 30q^{44} - 8q^{46} + 28q^{47} - 21q^{48} - 16q^{49} + 20q^{51} - 20q^{52} + 16q^{53} - 15q^{54} - 30q^{56} + 4q^{57} + 2q^{58} - 34q^{62} - 28q^{63} + 25q^{64} + 18q^{66} + 32q^{67} - 16q^{68} - 20q^{69} + 24q^{71} - 41q^{72} + 8q^{73} - 32q^{74} - 12q^{76} + 12q^{78} + 8q^{81} + 4q^{82} - 2q^{84} + 8q^{86} + 36q^{87} - 14q^{88} + 24q^{91} - 28q^{92} - 40q^{94} + 17q^{96} - 8q^{97} + 47q^{98} + 16q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - x^{7} - 2 x^{5} + 4 x^{4} - 4 x^{3} - 8 x + 16\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{6} + \nu^{5} - 4 \)\()/4\)
\(\beta_{3}\)\(=\)\((\)\( \nu^{7} + \nu^{6} + 4 \nu^{2} - 8 \nu - 8 \)\()/8\)
\(\beta_{4}\)\(=\)\((\)\( -\nu^{7} - \nu^{6} + 4 \nu^{2} + 8 \)\()/8\)
\(\beta_{5}\)\(=\)\((\)\( \nu^{7} + \nu^{6} - 4 \nu^{4} + 4 \nu^{3} + 4 \nu^{2} - 16 \)\()/8\)
\(\beta_{6}\)\(=\)\((\)\( \nu^{7} - \nu^{6} + 2 \nu^{5} + 4 \nu^{3} - 4 \nu^{2} - 8 \)\()/8\)
\(\beta_{7}\)\(=\)\((\)\( -\nu^{7} + \nu^{6} - 2 \nu^{5} + 4 \nu^{4} + 4 \nu^{2} - 8 \nu + 8 \)\()/8\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{4} + \beta_{3} + \beta_{1}\)
\(\nu^{3}\)\(=\)\(\beta_{7} + \beta_{6} + \beta_{5} - \beta_{3} + 1\)
\(\nu^{4}\)\(=\)\(\beta_{7} + \beta_{6} - \beta_{5} + \beta_{3} + 2 \beta_{1} - 1\)
\(\nu^{5}\)\(=\)\(-\beta_{7} + \beta_{6} - \beta_{5} + 2 \beta_{4} + \beta_{3} + 2 \beta_{2} + 1\)
\(\nu^{6}\)\(=\)\(\beta_{7} - \beta_{6} + \beta_{5} - 2 \beta_{4} - \beta_{3} + 2 \beta_{2} + 3\)
\(\nu^{7}\)\(=\)\(-\beta_{7} + \beta_{6} - \beta_{5} - 2 \beta_{4} + 5 \beta_{3} - 2 \beta_{2} + 4 \beta_{1} + 5\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/600\mathbb{Z}\right)^\times\).

\(n\) \(151\) \(301\) \(401\) \(577\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\) \(1\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
251.1
−1.13622 0.842022i
−1.13622 + 0.842022i
−0.578647 1.29041i
−0.578647 + 1.29041i
0.814732 1.15595i
0.814732 + 1.15595i
1.40014 0.199044i
1.40014 + 0.199044i
−1.13622 0.842022i 1.71822 0.218455i 0.581998 + 1.91345i 0 −2.13622 1.19857i 3.64426i 0.949886 2.66415i 2.90455 0.750707i 0
251.2 −1.13622 + 0.842022i 1.71822 + 0.218455i 0.581998 1.91345i 0 −2.13622 + 1.19857i 3.64426i 0.949886 + 2.66415i 2.90455 + 0.750707i 0
251.3 −0.578647 1.29041i −0.751690 1.56044i −1.33034 + 1.49339i 0 −1.57865 + 1.87293i 4.28591i 2.69688 + 0.852541i −1.86993 + 2.34593i 0
251.4 −0.578647 + 1.29041i −0.751690 + 1.56044i −1.33034 1.49339i 0 −1.57865 1.87293i 4.28591i 2.69688 0.852541i −1.86993 2.34593i 0
251.5 0.814732 1.15595i −1.48716 + 0.887900i −0.672424 1.88357i 0 −0.185268 + 2.44247i 0.797253i −2.72515 0.757320i 1.42327 2.64089i 0
251.6 0.814732 + 1.15595i −1.48716 0.887900i −0.672424 + 1.88357i 0 −0.185268 2.44247i 0.797253i −2.72515 + 0.757320i 1.42327 + 2.64089i 0
251.7 1.40014 0.199044i 0.520627 1.65195i 1.92076 0.557378i 0 0.400136 2.41659i 1.92736i 2.57839 1.16272i −2.45790 1.72010i 0
251.8 1.40014 + 0.199044i 0.520627 + 1.65195i 1.92076 + 0.557378i 0 0.400136 + 2.41659i 1.92736i 2.57839 + 1.16272i −2.45790 + 1.72010i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 251.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
24.f even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 600.2.b.f 8
3.b odd 2 1 600.2.b.e 8
4.b odd 2 1 2400.2.b.f 8
5.b even 2 1 120.2.b.a 8
5.c odd 4 2 600.2.m.c 16
8.b even 2 1 2400.2.b.e 8
8.d odd 2 1 600.2.b.e 8
12.b even 2 1 2400.2.b.e 8
15.d odd 2 1 120.2.b.b yes 8
15.e even 4 2 600.2.m.d 16
20.d odd 2 1 480.2.b.b 8
20.e even 4 2 2400.2.m.d 16
24.f even 2 1 inner 600.2.b.f 8
24.h odd 2 1 2400.2.b.f 8
40.e odd 2 1 120.2.b.b yes 8
40.f even 2 1 480.2.b.a 8
40.i odd 4 2 2400.2.m.c 16
40.k even 4 2 600.2.m.d 16
60.h even 2 1 480.2.b.a 8
60.l odd 4 2 2400.2.m.c 16
120.i odd 2 1 480.2.b.b 8
120.m even 2 1 120.2.b.a 8
120.q odd 4 2 600.2.m.c 16
120.w even 4 2 2400.2.m.d 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
120.2.b.a 8 5.b even 2 1
120.2.b.a 8 120.m even 2 1
120.2.b.b yes 8 15.d odd 2 1
120.2.b.b yes 8 40.e odd 2 1
480.2.b.a 8 40.f even 2 1
480.2.b.a 8 60.h even 2 1
480.2.b.b 8 20.d odd 2 1
480.2.b.b 8 120.i odd 2 1
600.2.b.e 8 3.b odd 2 1
600.2.b.e 8 8.d odd 2 1
600.2.b.f 8 1.a even 1 1 trivial
600.2.b.f 8 24.f even 2 1 inner
600.2.m.c 16 5.c odd 4 2
600.2.m.c 16 120.q odd 4 2
600.2.m.d 16 15.e even 4 2
600.2.m.d 16 40.k even 4 2
2400.2.b.e 8 8.b even 2 1
2400.2.b.e 8 12.b even 2 1
2400.2.b.f 8 4.b odd 2 1
2400.2.b.f 8 24.h odd 2 1
2400.2.m.c 16 40.i odd 4 2
2400.2.m.c 16 60.l odd 4 2
2400.2.m.d 16 20.e even 4 2
2400.2.m.d 16 120.w even 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(600, [\chi])\):

\( T_{7}^{8} + 36 T_{7}^{6} + 384 T_{7}^{4} + 1136 T_{7}^{2} + 576 \)
\( T_{11}^{8} + 48 T_{11}^{6} + 672 T_{11}^{4} + 2560 T_{11}^{2} + 256 \)
\( T_{23}^{4} + 2 T_{23}^{3} - 44 T_{23}^{2} - 188 T_{23} - 192 \)
\( T_{43}^{4} - 12 T_{43}^{2} + 4 T_{43} + 16 \)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 - T - 2 T^{3} + 4 T^{4} - 4 T^{5} - 8 T^{7} + 16 T^{8} \)
$3$ \( 1 - 4 T^{3} - 2 T^{4} - 12 T^{5} + 81 T^{8} \)
$5$ 1
$7$ \( 1 - 20 T^{2} + 244 T^{4} - 2364 T^{6} + 18678 T^{8} - 115836 T^{10} + 585844 T^{12} - 2352980 T^{14} + 5764801 T^{16} \)
$11$ \( 1 - 40 T^{2} + 892 T^{4} - 14424 T^{6} + 178918 T^{8} - 1745304 T^{10} + 13059772 T^{12} - 70862440 T^{14} + 214358881 T^{16} \)
$13$ \( 1 - 52 T^{2} + 1556 T^{4} - 31660 T^{6} + 477814 T^{8} - 5350540 T^{10} + 44440916 T^{12} - 250994068 T^{14} + 815730721 T^{16} \)
$17$ \( 1 - 84 T^{2} + 3604 T^{4} - 101164 T^{6} + 2018902 T^{8} - 29236396 T^{10} + 301009684 T^{12} - 2027555796 T^{14} + 6975757441 T^{16} \)
$19$ \( ( 1 + 2 T + 40 T^{2} + 42 T^{3} + 766 T^{4} + 798 T^{5} + 14440 T^{6} + 13718 T^{7} + 130321 T^{8} )^{2} \)
$23$ \( ( 1 + 2 T + 48 T^{2} - 50 T^{3} + 958 T^{4} - 1150 T^{5} + 25392 T^{6} + 24334 T^{7} + 279841 T^{8} )^{2} \)
$29$ \( ( 1 + 52 T^{2} + 112 T^{3} + 1286 T^{4} + 3248 T^{5} + 43732 T^{6} + 707281 T^{8} )^{2} \)
$31$ \( 1 - 108 T^{2} + 5412 T^{4} - 170900 T^{6} + 4790966 T^{8} - 164234900 T^{10} + 4998095652 T^{12} - 95850397548 T^{14} + 852891037441 T^{16} \)
$37$ \( 1 - 68 T^{2} + 4788 T^{4} - 250460 T^{6} + 9311478 T^{8} - 342879740 T^{10} + 8973482868 T^{12} - 174469395812 T^{14} + 3512479453921 T^{16} \)
$41$ \( 1 - 264 T^{2} + 32668 T^{4} - 2456248 T^{6} + 122337670 T^{8} - 4128952888 T^{10} + 92311960348 T^{12} - 1254027519624 T^{14} + 7984925229121 T^{16} \)
$43$ \( ( 1 + 160 T^{2} + 4 T^{3} + 10078 T^{4} + 172 T^{5} + 295840 T^{6} + 3418801 T^{8} )^{2} \)
$47$ \( ( 1 - 14 T + 168 T^{2} - 1186 T^{3} + 9166 T^{4} - 55742 T^{5} + 371112 T^{6} - 1453522 T^{7} + 4879681 T^{8} )^{2} \)
$53$ \( ( 1 - 8 T + 188 T^{2} - 1144 T^{3} + 14454 T^{4} - 60632 T^{5} + 528092 T^{6} - 1191016 T^{7} + 7890481 T^{8} )^{2} \)
$59$ \( 1 - 312 T^{2} + 41500 T^{4} - 3304648 T^{6} + 204947494 T^{8} - 11503479688 T^{10} + 502870481500 T^{12} - 13160326495992 T^{14} + 146830437604321 T^{16} \)
$61$ \( 1 - 280 T^{2} + 39452 T^{4} - 3783016 T^{6} + 267380710 T^{8} - 14076602536 T^{10} + 546246119132 T^{12} - 14425704821080 T^{14} + 191707312997281 T^{16} \)
$67$ \( ( 1 - 16 T + 216 T^{2} - 2388 T^{3} + 22862 T^{4} - 159996 T^{5} + 969624 T^{6} - 4812208 T^{7} + 20151121 T^{8} )^{2} \)
$71$ \( ( 1 - 12 T + 220 T^{2} - 1564 T^{3} + 18854 T^{4} - 111044 T^{5} + 1109020 T^{6} - 4294932 T^{7} + 25411681 T^{8} )^{2} \)
$73$ \( ( 1 - 4 T + 180 T^{2} - 252 T^{3} + 14966 T^{4} - 18396 T^{5} + 959220 T^{6} - 1556068 T^{7} + 28398241 T^{8} )^{2} \)
$79$ \( 1 - 524 T^{2} + 125604 T^{4} - 18135668 T^{6} + 1736460342 T^{8} - 113184703988 T^{10} + 4892285973924 T^{12} - 127377826693004 T^{14} + 1517108809906561 T^{16} \)
$83$ \( 1 - 512 T^{2} + 125148 T^{4} - 18790160 T^{6} + 1886502918 T^{8} - 129445412240 T^{10} + 5939313956508 T^{12} - 167393471164928 T^{14} + 2252292232139041 T^{16} \)
$89$ \( 1 - 328 T^{2} + 61788 T^{4} - 8336760 T^{6} + 843121542 T^{8} - 66035475960 T^{10} + 3876717586908 T^{12} - 163009863435208 T^{14} + 3936588805702081 T^{16} \)
$97$ \( ( 1 + 4 T + 212 T^{2} + 1948 T^{3} + 21526 T^{4} + 188956 T^{5} + 1994708 T^{6} + 3650692 T^{7} + 88529281 T^{8} )^{2} \)
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