Properties

Label 462.2.c.b
Level $462$
Weight $2$
Character orbit 462.c
Analytic conductor $3.689$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 462 = 2 \cdot 3 \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 462.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(3.68908857338\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
Defining polynomial: \(x^{12} + 16 x^{10} + 94 x^{8} + 246 x^{6} + 277 x^{4} + 114 x^{2} + 9\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{2} + \beta_{6} q^{3} + q^{4} -\beta_{11} q^{5} + \beta_{6} q^{6} -\beta_{2} q^{7} + q^{8} + ( \beta_{1} + \beta_{4} ) q^{9} +O(q^{10})\) \( q + q^{2} + \beta_{6} q^{3} + q^{4} -\beta_{11} q^{5} + \beta_{6} q^{6} -\beta_{2} q^{7} + q^{8} + ( \beta_{1} + \beta_{4} ) q^{9} -\beta_{11} q^{10} + ( 1 - \beta_{2} - \beta_{5} - \beta_{9} - \beta_{10} ) q^{11} + \beta_{6} q^{12} + ( -2 \beta_{2} + \beta_{5} - \beta_{6} ) q^{13} -\beta_{2} q^{14} + ( \beta_{1} + \beta_{2} - \beta_{4} - \beta_{5} - \beta_{8} - \beta_{11} ) q^{15} + q^{16} + ( \beta_{3} - \beta_{4} - \beta_{5} - \beta_{6} + \beta_{7} ) q^{17} + ( \beta_{1} + \beta_{4} ) q^{18} + ( 2 \beta_{8} - 2 \beta_{9} + 2 \beta_{10} + \beta_{11} ) q^{19} -\beta_{11} q^{20} -\beta_{8} q^{21} + ( 1 - \beta_{2} - \beta_{5} - \beta_{9} - \beta_{10} ) q^{22} + ( -\beta_{3} - \beta_{4} - \beta_{5} + \beta_{6} - \beta_{8} + \beta_{9} ) q^{23} + \beta_{6} q^{24} + ( -1 - \beta_{3} + \beta_{4} - 2 \beta_{7} + 2 \beta_{8} + 2 \beta_{9} ) q^{25} + ( -2 \beta_{2} + \beta_{5} - \beta_{6} ) q^{26} + ( -1 + \beta_{1} - \beta_{2} + \beta_{5} - \beta_{7} - 2 \beta_{9} + \beta_{10} + \beta_{11} ) q^{27} -\beta_{2} q^{28} + ( -2 \beta_{1} + \beta_{5} + \beta_{6} - 2 \beta_{7} + \beta_{8} + \beta_{9} ) q^{29} + ( \beta_{1} + \beta_{2} - \beta_{4} - \beta_{5} - \beta_{8} - \beta_{11} ) q^{30} + ( 2 - \beta_{3} + \beta_{4} - \beta_{5} - \beta_{6} + \beta_{7} ) q^{31} + q^{32} + ( -2 + 3 \beta_{2} + \beta_{6} + \beta_{7} - \beta_{8} - \beta_{9} - \beta_{10} ) q^{33} + ( \beta_{3} - \beta_{4} - \beta_{5} - \beta_{6} + \beta_{7} ) q^{34} + \beta_{7} q^{35} + ( \beta_{1} + \beta_{4} ) q^{36} + ( -\beta_{5} - \beta_{6} + 2 \beta_{7} + \beta_{8} + \beta_{9} ) q^{37} + ( 2 \beta_{8} - 2 \beta_{9} + 2 \beta_{10} + \beta_{11} ) q^{38} + ( 3 - \beta_{1} - \beta_{4} - 2 \beta_{8} ) q^{39} -\beta_{11} q^{40} + ( -2 - 2 \beta_{1} + \beta_{3} - \beta_{4} + \beta_{5} + \beta_{6} - \beta_{7} - 2 \beta_{8} - 2 \beta_{9} ) q^{41} -\beta_{8} q^{42} + ( \beta_{3} + \beta_{4} + \beta_{5} - \beta_{6} + \beta_{8} - \beta_{9} - 2 \beta_{10} + 2 \beta_{11} ) q^{43} + ( 1 - \beta_{2} - \beta_{5} - \beta_{9} - \beta_{10} ) q^{44} + ( -2 + 2 \beta_{1} - \beta_{3} - \beta_{4} + \beta_{7} + 2 \beta_{9} ) q^{45} + ( -\beta_{3} - \beta_{4} - \beta_{5} + \beta_{6} - \beta_{8} + \beta_{9} ) q^{46} + ( 2 \beta_{2} + 2 \beta_{3} + 2 \beta_{4} + \beta_{5} - \beta_{6} + \beta_{11} ) q^{47} + \beta_{6} q^{48} - q^{49} + ( -1 - \beta_{3} + \beta_{4} - 2 \beta_{7} + 2 \beta_{8} + 2 \beta_{9} ) q^{50} + ( -1 - \beta_{2} + \beta_{3} - \beta_{4} - 2 \beta_{5} - \beta_{6} + 2 \beta_{7} + \beta_{9} + \beta_{11} ) q^{51} + ( -2 \beta_{2} + \beta_{5} - \beta_{6} ) q^{52} + ( 4 \beta_{2} + \beta_{3} + \beta_{4} - 2 \beta_{10} ) q^{53} + ( -1 + \beta_{1} - \beta_{2} + \beta_{5} - \beta_{7} - 2 \beta_{9} + \beta_{10} + \beta_{11} ) q^{54} + ( 1 - \beta_{1} - \beta_{2} + 2 \beta_{4} + 2 \beta_{5} - \beta_{7} - 2 \beta_{9} + \beta_{10} ) q^{55} -\beta_{2} q^{56} + ( -2 - \beta_{1} + 5 \beta_{2} + 2 \beta_{3} + \beta_{4} + \beta_{5} - 2 \beta_{7} + \beta_{8} + 2 \beta_{9} + \beta_{11} ) q^{57} + ( -2 \beta_{1} + \beta_{5} + \beta_{6} - 2 \beta_{7} + \beta_{8} + \beta_{9} ) q^{58} + ( -\beta_{5} + \beta_{6} - 2 \beta_{8} + 2 \beta_{9} ) q^{59} + ( \beta_{1} + \beta_{2} - \beta_{4} - \beta_{5} - \beta_{8} - \beta_{11} ) q^{60} + ( -2 \beta_{3} - 2 \beta_{4} + \beta_{5} - \beta_{6} + 2 \beta_{10} ) q^{61} + ( 2 - \beta_{3} + \beta_{4} - \beta_{5} - \beta_{6} + \beta_{7} ) q^{62} + ( -\beta_{3} + \beta_{10} ) q^{63} + q^{64} + ( -2 \beta_{1} - \beta_{3} + \beta_{4} + \beta_{5} + \beta_{6} + 2 \beta_{7} + \beta_{8} + \beta_{9} ) q^{65} + ( -2 + 3 \beta_{2} + \beta_{6} + \beta_{7} - \beta_{8} - \beta_{9} - \beta_{10} ) q^{66} + ( 2 - 2 \beta_{1} + 2 \beta_{3} - 2 \beta_{4} - \beta_{5} - \beta_{6} + \beta_{8} + \beta_{9} ) q^{67} + ( \beta_{3} - \beta_{4} - \beta_{5} - \beta_{6} + \beta_{7} ) q^{68} + ( -2 - 2 \beta_{2} - \beta_{3} + \beta_{4} + 2 \beta_{5} + \beta_{7} + 2 \beta_{9} - \beta_{11} ) q^{69} + \beta_{7} q^{70} + ( 2 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} + 2 \beta_{8} - 2 \beta_{9} + 4 \beta_{10} ) q^{71} + ( \beta_{1} + \beta_{4} ) q^{72} + ( 4 \beta_{2} + \beta_{5} - \beta_{6} - 2 \beta_{8} + 2 \beta_{9} + \beta_{11} ) q^{73} + ( -\beta_{5} - \beta_{6} + 2 \beta_{7} + \beta_{8} + \beta_{9} ) q^{74} + ( -3 - \beta_{1} - 5 \beta_{2} + 2 \beta_{5} + \beta_{6} - 3 \beta_{7} - 3 \beta_{10} - \beta_{11} ) q^{75} + ( 2 \beta_{8} - 2 \beta_{9} + 2 \beta_{10} + \beta_{11} ) q^{76} + ( -1 + \beta_{1} - \beta_{2} + \beta_{5} - \beta_{9} ) q^{77} + ( 3 - \beta_{1} - \beta_{4} - 2 \beta_{8} ) q^{78} + ( -2 \beta_{2} - 2 \beta_{8} + 2 \beta_{9} - 2 \beta_{11} ) q^{79} -\beta_{11} q^{80} + ( 1 + 4 \beta_{2} - \beta_{3} + \beta_{4} + 2 \beta_{5} - 2 \beta_{7} + 2 \beta_{9} + 2 \beta_{11} ) q^{81} + ( -2 - 2 \beta_{1} + \beta_{3} - \beta_{4} + \beta_{5} + \beta_{6} - \beta_{7} - 2 \beta_{8} - 2 \beta_{9} ) q^{82} + ( -4 + 2 \beta_{1} - \beta_{3} + \beta_{4} + \beta_{7} ) q^{83} -\beta_{8} q^{84} + ( \beta_{5} - \beta_{6} + \beta_{8} - \beta_{9} - 4 \beta_{10} - 2 \beta_{11} ) q^{85} + ( \beta_{3} + \beta_{4} + \beta_{5} - \beta_{6} + \beta_{8} - \beta_{9} - 2 \beta_{10} + 2 \beta_{11} ) q^{86} + ( 1 - \beta_{1} - \beta_{2} - \beta_{3} + \beta_{4} - 2 \beta_{5} + 2 \beta_{6} - 2 \beta_{7} + 2 \beta_{9} - 3 \beta_{10} - 2 \beta_{11} ) q^{87} + ( 1 - \beta_{2} - \beta_{5} - \beta_{9} - \beta_{10} ) q^{88} + ( 2 \beta_{2} + 2 \beta_{5} - 2 \beta_{6} - 2 \beta_{8} + 2 \beta_{9} ) q^{89} + ( -2 + 2 \beta_{1} - \beta_{3} - \beta_{4} + \beta_{7} + 2 \beta_{9} ) q^{90} + ( -2 + \beta_{8} + \beta_{9} ) q^{91} + ( -\beta_{3} - \beta_{4} - \beta_{5} + \beta_{6} - \beta_{8} + \beta_{9} ) q^{92} + ( -3 - 2 \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} + 2 \beta_{5} + \beta_{6} - 3 \beta_{9} + 2 \beta_{10} - \beta_{11} ) q^{93} + ( 2 \beta_{2} + 2 \beta_{3} + 2 \beta_{4} + \beta_{5} - \beta_{6} + \beta_{11} ) q^{94} + ( -2 + 4 \beta_{1} + \beta_{3} - \beta_{4} + 2 \beta_{8} + 2 \beta_{9} ) q^{95} + \beta_{6} q^{96} + ( -4 - 2 \beta_{1} + 2 \beta_{7} ) q^{97} - q^{98} + ( 2 + \beta_{1} + 3 \beta_{2} + \beta_{4} - 3 \beta_{6} + 2 \beta_{7} + 3 \beta_{8} - 2 \beta_{9} + \beta_{10} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12q + 12q^{2} + 4q^{3} + 12q^{4} + 4q^{6} + 12q^{8} + O(q^{10}) \) \( 12q + 12q^{2} + 4q^{3} + 12q^{4} + 4q^{6} + 12q^{8} + 8q^{11} + 4q^{12} + 4q^{15} + 12q^{16} + 4q^{17} + 8q^{22} + 4q^{24} - 28q^{25} - 8q^{27} - 8q^{29} + 4q^{30} + 12q^{31} + 12q^{32} - 16q^{33} + 4q^{34} + 4q^{35} + 36q^{39} - 20q^{41} + 8q^{44} - 12q^{45} + 4q^{48} - 12q^{49} - 28q^{50} - 8q^{51} - 8q^{54} + 4q^{55} - 28q^{57} - 8q^{58} + 4q^{60} + 12q^{62} - 4q^{63} + 12q^{64} - 16q^{66} + 24q^{67} + 4q^{68} - 20q^{69} + 4q^{70} - 40q^{75} - 4q^{77} + 36q^{78} + 4q^{81} - 20q^{82} - 44q^{83} - 8q^{87} + 8q^{88} - 12q^{90} - 24q^{91} - 24q^{93} + 4q^{96} - 48q^{97} - 12q^{98} + 20q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{12} + 16 x^{10} + 94 x^{8} + 246 x^{6} + 277 x^{4} + 114 x^{2} + 9\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{6} + 8 \nu^{4} + 17 \nu^{2} + 9 \)\()/2\)
\(\beta_{2}\)\(=\)\((\)\( \nu^{11} + 13 \nu^{9} + 55 \nu^{7} + 72 \nu^{5} - 29 \nu^{3} - 60 \nu \)\()/18\)
\(\beta_{3}\)\(=\)\((\)\( \nu^{7} + 10 \nu^{5} + 29 \nu^{3} + 2 \nu^{2} + 21 \nu + 6 \)\()/2\)
\(\beta_{4}\)\(=\)\((\)\( \nu^{7} + 10 \nu^{5} + 29 \nu^{3} - 2 \nu^{2} + 21 \nu - 6 \)\()/2\)
\(\beta_{5}\)\(=\)\((\)\( -\nu^{11} - 16 \nu^{9} + 3 \nu^{8} - 94 \nu^{7} + 33 \nu^{6} - 243 \nu^{5} + 111 \nu^{4} - 253 \nu^{3} + 114 \nu^{2} - 63 \nu + 27 \)\()/12\)
\(\beta_{6}\)\(=\)\((\)\( \nu^{11} + 16 \nu^{9} + 3 \nu^{8} + 94 \nu^{7} + 33 \nu^{6} + 243 \nu^{5} + 111 \nu^{4} + 253 \nu^{3} + 114 \nu^{2} + 63 \nu + 27 \)\()/12\)
\(\beta_{7}\)\(=\)\((\)\( \nu^{10} + 16 \nu^{8} + 91 \nu^{6} + 219 \nu^{4} + 202 \nu^{2} + 42 \)\()/6\)
\(\beta_{8}\)\(=\)\((\)\( -\nu^{11} + 2 \nu^{10} - 16 \nu^{9} + 29 \nu^{8} - 94 \nu^{7} + 149 \nu^{6} - 243 \nu^{5} + 315 \nu^{4} - 253 \nu^{3} + 230 \nu^{2} - 75 \nu + 33 \)\()/12\)
\(\beta_{9}\)\(=\)\((\)\( \nu^{11} + 2 \nu^{10} + 16 \nu^{9} + 29 \nu^{8} + 94 \nu^{7} + 149 \nu^{6} + 243 \nu^{5} + 315 \nu^{4} + 253 \nu^{3} + 230 \nu^{2} + 75 \nu + 33 \)\()/12\)
\(\beta_{10}\)\(=\)\((\)\( \nu^{11} + 16 \nu^{9} + 94 \nu^{7} + 243 \nu^{5} + 259 \nu^{3} + 93 \nu \)\()/6\)
\(\beta_{11}\)\(=\)\((\)\( 7 \nu^{11} + 109 \nu^{9} + 610 \nu^{7} + 1458 \nu^{5} + 1354 \nu^{3} + 345 \nu \)\()/18\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{9} - \beta_{8} - \beta_{6} + \beta_{5}\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(-\beta_{4} + \beta_{3} - 6\)\()/2\)
\(\nu^{3}\)\(=\)\((\)\(2 \beta_{10} - 5 \beta_{9} + 5 \beta_{8} + 3 \beta_{6} - 3 \beta_{5}\)\()/2\)
\(\nu^{4}\)\(=\)\((\)\(-\beta_{9} - \beta_{8} + 2 \beta_{7} - \beta_{6} - \beta_{5} + 5 \beta_{4} - 5 \beta_{3} + 26\)\()/2\)
\(\nu^{5}\)\(=\)\((\)\(2 \beta_{11} - 14 \beta_{10} + 23 \beta_{9} - 23 \beta_{8} - 13 \beta_{6} + 13 \beta_{5} + \beta_{4} + \beta_{3} - 2 \beta_{2}\)\()/2\)
\(\nu^{6}\)\(=\)\((\)\(8 \beta_{9} + 8 \beta_{8} - 16 \beta_{7} + 8 \beta_{6} + 8 \beta_{5} - 23 \beta_{4} + 23 \beta_{3} + 4 \beta_{1} - 124\)\()/2\)
\(\nu^{7}\)\(=\)\(-10 \beta_{11} + 41 \beta_{10} - 53 \beta_{9} + 53 \beta_{8} + 32 \beta_{6} - 32 \beta_{5} - 4 \beta_{4} - 4 \beta_{3} + 10 \beta_{2}\)
\(\nu^{8}\)\(=\)\((\)\(-51 \beta_{9} - 51 \beta_{8} + 102 \beta_{7} - 47 \beta_{6} - 47 \beta_{5} + 106 \beta_{4} - 106 \beta_{3} - 44 \beta_{1} + 612\)\()/2\)
\(\nu^{9}\)\(=\)\((\)\(146 \beta_{11} - 456 \beta_{10} + 496 \beta_{9} - 496 \beta_{8} - 328 \beta_{6} + 328 \beta_{5} + 47 \beta_{4} + 47 \beta_{3} - 158 \beta_{2}\)\()/2\)
\(\nu^{10}\)\(=\)\((\)\(307 \beta_{9} + 307 \beta_{8} - 602 \beta_{7} + 243 \beta_{6} + 243 \beta_{5} - 496 \beta_{4} + 496 \beta_{3} + 340 \beta_{1} - 3074\)\()/2\)
\(\nu^{11}\)\(=\)\((\)\(-942 \beta_{11} + 2484 \beta_{10} - 2359 \beta_{9} + 2359 \beta_{8} + 1707 \beta_{6} - 1707 \beta_{5} - 243 \beta_{4} - 243 \beta_{3} + 1134 \beta_{2}\)\()/2\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/462\mathbb{Z}\right)^\times\).

\(n\) \(155\) \(199\) \(211\)
\(\chi(n)\) \(-1\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
197.1
2.18803i
2.18803i
0.822410i
0.822410i
1.18147i
1.18147i
2.33939i
2.33939i
0.319443i
0.319443i
1.88826i
1.88826i
1.00000 −1.64459 0.543441i 1.00000 0.118610i −1.64459 0.543441i 1.00000i 1.00000 2.40934 + 1.78747i 0.118610i
197.2 1.00000 −1.64459 + 0.543441i 1.00000 0.118610i −1.64459 + 0.543441i 1.00000i 1.00000 2.40934 1.78747i 0.118610i
197.3 1.00000 −0.742446 1.56486i 1.00000 3.23163i −0.742446 1.56486i 1.00000i 1.00000 −1.89755 + 2.32364i 3.23163i
197.4 1.00000 −0.742446 + 1.56486i 1.00000 3.23163i −0.742446 + 1.56486i 1.00000i 1.00000 −1.89755 2.32364i 3.23163i
197.5 1.00000 0.482127 1.66360i 1.00000 0.885172i 0.482127 1.66360i 1.00000i 1.00000 −2.53511 1.60413i 0.885172i
197.6 1.00000 0.482127 + 1.66360i 1.00000 0.885172i 0.482127 + 1.66360i 1.00000i 1.00000 −2.53511 + 1.60413i 0.885172i
197.7 1.00000 0.806630 1.53276i 1.00000 3.86104i 0.806630 1.53276i 1.00000i 1.00000 −1.69870 2.47274i 3.86104i
197.8 1.00000 0.806630 + 1.53276i 1.00000 3.86104i 0.806630 + 1.53276i 1.00000i 1.00000 −1.69870 + 2.47274i 3.86104i
197.9 1.00000 1.37401 1.05456i 1.00000 3.92876i 1.37401 1.05456i 1.00000i 1.00000 0.775791 2.89796i 3.92876i
197.10 1.00000 1.37401 + 1.05456i 1.00000 3.92876i 1.37401 + 1.05456i 1.00000i 1.00000 0.775791 + 2.89796i 3.92876i
197.11 1.00000 1.72427 0.163987i 1.00000 1.55439i 1.72427 0.163987i 1.00000i 1.00000 2.94622 0.565516i 1.55439i
197.12 1.00000 1.72427 + 0.163987i 1.00000 1.55439i 1.72427 + 0.163987i 1.00000i 1.00000 2.94622 + 0.565516i 1.55439i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 197.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
33.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 462.2.c.b yes 12
3.b odd 2 1 462.2.c.a 12
11.b odd 2 1 462.2.c.a 12
33.d even 2 1 inner 462.2.c.b yes 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
462.2.c.a 12 3.b odd 2 1
462.2.c.a 12 11.b odd 2 1
462.2.c.b yes 12 1.a even 1 1 trivial
462.2.c.b yes 12 33.d even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{17}^{6} - 2 T_{17}^{5} - 50 T_{17}^{4} + 64 T_{17}^{3} + 348 T_{17}^{2} - 328 T_{17} - 184 \) acting on \(S_{2}^{\mathrm{new}}(462, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( -1 + T )^{12} \)
$3$ \( 729 - 972 T + 648 T^{2} - 216 T^{3} - 9 T^{4} + 36 T^{5} - 24 T^{6} + 12 T^{7} - T^{8} - 8 T^{9} + 8 T^{10} - 4 T^{11} + T^{12} \)
$5$ \( 64 + 4672 T^{2} + 8784 T^{4} + 4240 T^{6} + 680 T^{8} + 44 T^{10} + T^{12} \)
$7$ \( ( 1 + T^{2} )^{6} \)
$11$ \( 1771561 - 1288408 T + 790614 T^{2} - 393976 T^{3} + 167343 T^{4} - 61952 T^{5} + 19316 T^{6} - 5632 T^{7} + 1383 T^{8} - 296 T^{9} + 54 T^{10} - 8 T^{11} + T^{12} \)
$13$ \( 64 + 5504 T^{2} + 10976 T^{4} + 6240 T^{6} + 1108 T^{8} + 60 T^{10} + T^{12} \)
$17$ \( ( -184 - 328 T + 348 T^{2} + 64 T^{3} - 50 T^{4} - 2 T^{5} + T^{6} )^{2} \)
$19$ \( 12222016 + 37304256 T^{2} + 6560080 T^{4} + 433584 T^{6} + 13256 T^{8} + 188 T^{10} + T^{12} \)
$23$ \( 36864 + 98304 T^{2} + 88576 T^{4} + 30592 T^{6} + 3280 T^{8} + 120 T^{10} + T^{12} \)
$29$ \( ( -320 + 64 T + 480 T^{2} - 144 T^{3} - 84 T^{4} + 4 T^{5} + T^{6} )^{2} \)
$31$ \( ( 5288 + 6600 T + 2492 T^{2} + 128 T^{3} - 82 T^{4} - 6 T^{5} + T^{6} )^{2} \)
$37$ \( ( 3008 - 6400 T + 2192 T^{2} + 128 T^{3} - 96 T^{4} + T^{6} )^{2} \)
$41$ \( ( 21832 + 20232 T + 2844 T^{2} - 1152 T^{3} - 130 T^{4} + 10 T^{5} + T^{6} )^{2} \)
$43$ \( 1478656 + 2799616 T^{2} + 1655040 T^{4} + 370048 T^{6} + 27872 T^{8} + 320 T^{10} + T^{12} \)
$47$ \( 22353984 + 538788480 T^{2} + 59152240 T^{4} + 2376512 T^{6} + 42844 T^{8} + 344 T^{10} + T^{12} \)
$53$ \( 4096 + 112640 T^{2} + 369920 T^{4} + 254848 T^{6} + 13344 T^{8} + 208 T^{10} + T^{12} \)
$59$ \( 2005056 + 77623296 T^{2} + 10588768 T^{4} + 569664 T^{6} + 15076 T^{8} + 196 T^{10} + T^{12} \)
$61$ \( 9424526400 + 1431044736 T^{2} + 85912288 T^{4} + 2602080 T^{6} + 41716 T^{8} + 332 T^{10} + T^{12} \)
$67$ \( ( -46784 - 22208 T + 3280 T^{2} + 1216 T^{3} - 112 T^{4} - 12 T^{5} + T^{6} )^{2} \)
$71$ \( 40771686400 + 8622024704 T^{2} + 551165696 T^{4} + 13524736 T^{6} + 138096 T^{8} + 616 T^{10} + T^{12} \)
$73$ \( 318551104 + 168251264 T^{2} + 22384880 T^{4} + 1145344 T^{6} + 26748 T^{8} + 280 T^{10} + T^{12} \)
$79$ \( 2166784 + 17197056 T^{2} + 4771584 T^{4} + 418560 T^{6} + 15344 T^{8} + 232 T^{10} + T^{12} \)
$83$ \( ( -136 + 624 T - 660 T^{2} - 32 T^{3} + 124 T^{4} + 22 T^{5} + T^{6} )^{2} \)
$89$ \( 7573504 + 19691520 T^{2} + 10305280 T^{4} + 758016 T^{6} + 21104 T^{8} + 248 T^{10} + T^{12} \)
$97$ \( ( -121152 - 98688 T - 27920 T^{2} - 2816 T^{3} + 52 T^{4} + 24 T^{5} + T^{6} )^{2} \)
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