Properties

Label 1764.2.w.a
Level $1764$
Weight $2$
Character orbit 1764.w
Analytic conductor $14.086$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 1764 = 2^{2} \cdot 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1764.w (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(14.0856109166\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Defining polynomial: \(x^{16} - 3 x^{14} - 9 x^{12} - 9 x^{10} + 225 x^{8} - 81 x^{6} - 729 x^{4} - 2187 x^{2} + 6561\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{4} \)
Twist minimal: no (minimal twist has level 252)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{8} q^{3} + \beta_{15} q^{5} -\beta_{11} q^{9} +O(q^{10})\) \( q + \beta_{8} q^{3} + \beta_{15} q^{5} -\beta_{11} q^{9} + ( -\beta_{7} - \beta_{11} - \beta_{13} ) q^{11} + ( -\beta_{2} - \beta_{5} + \beta_{9} + \beta_{10} - \beta_{14} ) q^{13} + ( -1 - \beta_{3} + \beta_{4} + \beta_{7} + \beta_{12} + \beta_{13} ) q^{15} + ( -\beta_{2} + \beta_{6} + \beta_{9} + \beta_{10} ) q^{17} + ( -\beta_{5} - \beta_{6} + \beta_{8} - \beta_{9} + \beta_{15} ) q^{19} + ( -\beta_{1} + \beta_{7} + \beta_{11} ) q^{23} + ( -\beta_{3} + \beta_{7} + \beta_{11} - \beta_{13} ) q^{25} + ( -\beta_{2} + \beta_{6} - \beta_{14} + \beta_{15} ) q^{27} + ( -1 - \beta_{1} + 2 \beta_{3} - \beta_{4} - \beta_{7} + \beta_{11} - \beta_{12} ) q^{29} + ( \beta_{2} + \beta_{15} ) q^{31} + ( \beta_{2} + 2 \beta_{6} + \beta_{14} - \beta_{15} ) q^{33} + ( -2 \beta_{1} - \beta_{4} - \beta_{7} + \beta_{11} + \beta_{13} ) q^{37} + ( -2 - \beta_{1} + \beta_{3} + \beta_{11} - \beta_{13} ) q^{39} + ( -\beta_{2} - 3 \beta_{5} + \beta_{6} + 3 \beta_{8} + \beta_{9} - \beta_{10} + \beta_{14} - \beta_{15} ) q^{41} + ( 1 - \beta_{1} - \beta_{4} - 2 \beta_{7} - \beta_{11} - \beta_{12} - \beta_{13} ) q^{43} + ( -3 \beta_{2} - 2 \beta_{5} + \beta_{8} ) q^{45} + ( 3 \beta_{5} - \beta_{6} - 3 \beta_{8} + 2 \beta_{9} + \beta_{10} - \beta_{14} ) q^{47} + ( 1 - 3 \beta_{1} + \beta_{3} - \beta_{7} + \beta_{11} ) q^{51} + ( 2 - \beta_{1} + 2 \beta_{3} + 2 \beta_{4} + \beta_{7} + \beta_{11} - \beta_{12} - 2 \beta_{13} ) q^{53} + ( -\beta_{2} - \beta_{5} - \beta_{6} + 2 \beta_{8} + \beta_{9} + \beta_{10} - \beta_{14} + \beta_{15} ) q^{55} + ( -1 + \beta_{1} - \beta_{3} - 3 \beta_{4} + 2 \beta_{7} + \beta_{13} ) q^{57} + ( 2 \beta_{2} - \beta_{6} - \beta_{9} + \beta_{14} - \beta_{15} ) q^{59} + ( -3 \beta_{2} + 2 \beta_{5} - \beta_{6} + \beta_{8} + 2 \beta_{10} - \beta_{14} ) q^{61} + ( -1 - \beta_{3} + 2 \beta_{4} - \beta_{7} - \beta_{11} - \beta_{12} ) q^{65} + ( -\beta_{3} + \beta_{7} - 2 \beta_{11} + \beta_{12} - \beta_{13} ) q^{67} + ( 2 \beta_{2} + \beta_{6} - 3 \beta_{9} - 3 \beta_{10} + 2 \beta_{14} - 2 \beta_{15} ) q^{69} + ( -3 + 6 \beta_{4} - \beta_{11} + \beta_{13} ) q^{71} + ( -4 \beta_{2} + \beta_{5} + \beta_{9} - \beta_{14} + 2 \beta_{15} ) q^{73} + ( 3 \beta_{2} - \beta_{5} + 2 \beta_{6} - 2 \beta_{9} - 4 \beta_{10} + 3 \beta_{14} - 6 \beta_{15} ) q^{75} + ( -1 - \beta_{1} + \beta_{7} - \beta_{11} - \beta_{13} ) q^{79} + ( 2 - \beta_{3} - 4 \beta_{4} - \beta_{11} + 2 \beta_{12} - \beta_{13} ) q^{81} + ( 2 \beta_{2} + 3 \beta_{5} + 2 \beta_{6} - 3 \beta_{8} - \beta_{9} - 2 \beta_{10} + 2 \beta_{14} ) q^{83} + ( \beta_{1} - \beta_{3} + \beta_{4} - \beta_{11} ) q^{85} + ( 3 \beta_{2} + 3 \beta_{5} - 2 \beta_{6} - 3 \beta_{8} - \beta_{9} + \beta_{10} + 3 \beta_{15} ) q^{87} + ( -2 \beta_{2} - 3 \beta_{5} + 6 \beta_{8} + 3 \beta_{9} - \beta_{10} - \beta_{14} ) q^{89} + ( -1 - \beta_{3} + 2 \beta_{4} + 2 \beta_{7} + 2 \beta_{12} + 2 \beta_{13} ) q^{93} + ( 2 - \beta_{3} - 4 \beta_{4} - \beta_{7} + \beta_{11} - \beta_{12} - 2 \beta_{13} ) q^{95} + ( -3 \beta_{2} + 4 \beta_{5} - 3 \beta_{6} - 4 \beta_{8} + \beta_{9} + 3 \beta_{10} - 2 \beta_{14} + 3 \beta_{15} ) q^{97} + ( 7 - 3 \beta_{1} + \beta_{3} - 2 \beta_{4} + \beta_{11} + \beta_{12} + \beta_{13} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q - 6q^{9} + O(q^{10}) \) \( 16q - 6q^{9} - 6q^{11} - 12q^{15} - 6q^{23} - 8q^{25} - 12q^{29} - 2q^{37} - 36q^{39} + 4q^{43} + 12q^{51} + 36q^{53} - 42q^{57} - 28q^{67} - 40q^{79} - 18q^{81} + 6q^{85} - 6q^{93} + 90q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{16} - 3 x^{14} - 9 x^{12} - 9 x^{10} + 225 x^{8} - 81 x^{6} - 729 x^{4} - 2187 x^{2} + 6561\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu^{2} \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{15} + 15 \nu^{13} + 72 \nu^{11} + 153 \nu^{9} - 423 \nu^{7} - 891 \nu^{5} + 1944 \nu^{3} + 17496 \nu \)\()/15309\)
\(\beta_{3}\)\(=\)\((\)\( \nu^{14} + 15 \nu^{12} + 72 \nu^{10} + 153 \nu^{8} - 423 \nu^{6} - 891 \nu^{4} + 1944 \nu^{2} + 12393 \)\()/5103\)
\(\beta_{4}\)\(=\)\((\)\( 5 \nu^{14} + 12 \nu^{12} - 18 \nu^{10} - 369 \nu^{8} + 153 \nu^{6} + 1782 \nu^{4} + 4617 \nu^{2} - 9477 \)\()/5103\)
\(\beta_{5}\)\(=\)\((\)\( -\nu^{15} + 3 \nu^{13} + 9 \nu^{11} + 9 \nu^{9} - 225 \nu^{7} + 81 \nu^{5} + 729 \nu^{3} + 2187 \nu \)\()/2187\)
\(\beta_{6}\)\(=\)\((\)\( -\nu^{15} + 3 \nu^{13} - 18 \nu^{11} + 90 \nu^{9} + 18 \nu^{7} + 324 \nu^{5} - 3159 \nu^{3} + 2187 \nu \)\()/2187\)
\(\beta_{7}\)\(=\)\((\)\( \nu^{14} - 3 \nu^{12} - 9 \nu^{10} - 9 \nu^{8} + 225 \nu^{6} - 81 \nu^{4} - 729 \nu^{2} - 2187 \)\()/729\)
\(\beta_{8}\)\(=\)\((\)\( 13 \nu^{15} + 6 \nu^{13} - 9 \nu^{11} - 279 \nu^{9} - 396 \nu^{7} + 324 \nu^{5} + 6561 \nu^{3} + 13122 \nu \)\()/15309\)
\(\beta_{9}\)\(=\)\((\)\( 5 \nu^{15} + 12 \nu^{13} - 18 \nu^{11} - 369 \nu^{9} + 153 \nu^{7} + 1782 \nu^{5} + 4617 \nu^{3} - 4374 \nu \)\()/5103\)
\(\beta_{10}\)\(=\)\((\)\( -20 \nu^{15} + 78 \nu^{13} + 72 \nu^{11} + 909 \nu^{9} - 3447 \nu^{7} - 2592 \nu^{5} - 9963 \nu^{3} + 63423 \nu \)\()/15309\)
\(\beta_{11}\)\(=\)\((\)\( -20 \nu^{14} + 15 \nu^{12} + 72 \nu^{10} + 342 \nu^{8} - 1179 \nu^{6} + 243 \nu^{4} - 1458 \nu^{2} + 2187 \)\()/5103\)
\(\beta_{12}\)\(=\)\((\)\( 23 \nu^{14} - 33 \nu^{12} - 234 \nu^{10} - 1017 \nu^{8} + 3879 \nu^{6} + 8424 \nu^{4} + 2187 \nu^{2} - 82377 \)\()/5103\)
\(\beta_{13}\)\(=\)\((\)\( \nu^{14} + \nu^{12} - 12 \nu^{10} - 36 \nu^{8} + 81 \nu^{6} + 306 \nu^{4} + 54 \nu^{2} - 1215 \)\()/189\)
\(\beta_{14}\)\(=\)\((\)\( 11 \nu^{15} - 3 \nu^{13} - 153 \nu^{11} - 396 \nu^{9} + 1395 \nu^{7} + 2862 \nu^{5} - 162 \nu^{3} - 16767 \nu \)\()/5103\)
\(\beta_{15}\)\(=\)\((\)\( 62 \nu^{15} - 15 \nu^{13} - 639 \nu^{11} - 2421 \nu^{9} + 7794 \nu^{7} + 17901 \nu^{5} + 3159 \nu^{3} - 139968 \nu \)\()/15309\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-\beta_{15} + \beta_{14} - \beta_{10} + \beta_{9} + \beta_{6} + \beta_{2}\)\()/3\)
\(\nu^{2}\)\(=\)\(\beta_{1}\)
\(\nu^{3}\)\(=\)\(\beta_{14} - \beta_{9} - \beta_{8} - \beta_{6} + 2 \beta_{5} + 2 \beta_{2}\)
\(\nu^{4}\)\(=\)\(\beta_{12} - 2 \beta_{7} - 2 \beta_{4} + \beta_{3} - \beta_{1} + 4\)
\(\nu^{5}\)\(=\)\(-3 \beta_{10} + 3 \beta_{6} + 6 \beta_{5} + 3 \beta_{2}\)
\(\nu^{6}\)\(=\)\(3 \beta_{11} + 6 \beta_{7} + 3 \beta_{4} + 3 \beta_{3} + 3 \beta_{1} + 15\)
\(\nu^{7}\)\(=\)\(-3 \beta_{15} + 6 \beta_{14} - 3 \beta_{10} + 3 \beta_{9} - 12 \beta_{8} - 3 \beta_{5} + 18 \beta_{2}\)
\(\nu^{8}\)\(=\)\(3 \beta_{13} + 3 \beta_{12} - 6 \beta_{7} - 24 \beta_{4} + 12 \beta_{3} + 9 \beta_{1} - 24\)
\(\nu^{9}\)\(=\)\(12 \beta_{15} + 6 \beta_{14} + 3 \beta_{10} - 39 \beta_{9} - 9 \beta_{8} - 12 \beta_{6} + 45 \beta_{5} + 51 \beta_{2}\)
\(\nu^{10}\)\(=\)\(-18 \beta_{13} + 18 \beta_{12} - 9 \beta_{7} + 18 \beta_{4} + 45 \beta_{3} - 45 \beta_{1} + 72\)
\(\nu^{11}\)\(=\)\(9 \beta_{15} - 72 \beta_{14} - 45 \beta_{10} + 54 \beta_{9} + 9 \beta_{8} + 54 \beta_{6} - 45 \beta_{5} + 54 \beta_{2}\)
\(\nu^{12}\)\(=\)\(54 \beta_{13} - 63 \beta_{12} + 108 \beta_{11} + 180 \beta_{7} + 153 \beta_{4} + 126 \beta_{3} + 36 \beta_{1} - 198\)
\(\nu^{13}\)\(=\)\(108 \beta_{15} - 27 \beta_{14} + 297 \beta_{10} - 243 \beta_{8} - 216 \beta_{6} - 216 \beta_{5} + 270 \beta_{2}\)
\(\nu^{14}\)\(=\)\(27 \beta_{13} + 81 \beta_{12} - 351 \beta_{11} - 378 \beta_{7} - 432 \beta_{4} + 297 \beta_{3} - 243 \beta_{1} - 1026\)
\(\nu^{15}\)\(=\)\(459 \beta_{15} - 567 \beta_{14} + 216 \beta_{10} - 540 \beta_{9} + 1242 \beta_{8} - 27 \beta_{6} - 216 \beta_{5} + 135 \beta_{2}\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1764\mathbb{Z}\right)^\times\).

\(n\) \(785\) \(883\) \(1081\)
\(\chi(n)\) \(\beta_{4}\) \(1\) \(\beta_{4}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
509.1
−0.604587 + 1.62311i
−1.71965 + 0.206851i
0.744857 + 1.56371i
−1.69483 0.357142i
1.69483 + 0.357142i
−0.744857 1.56371i
1.71965 0.206851i
0.604587 1.62311i
−0.604587 1.62311i
−1.71965 0.206851i
0.744857 1.56371i
−1.69483 + 0.357142i
1.69483 0.357142i
−0.744857 + 1.56371i
1.71965 + 0.206851i
0.604587 + 1.62311i
0 −1.70794 0.287965i 0 0.266780 + 0.462077i 0 0 0 2.83415 + 0.983658i 0
509.2 0 −1.03897 + 1.38584i 0 2.09336 + 3.62580i 0 0 0 −0.841101 2.87968i 0
509.3 0 −0.981784 1.42692i 0 0.276914 + 0.479629i 0 0 0 −1.07220 + 2.80185i 0
509.4 0 −0.538121 + 1.64634i 0 −1.21244 2.10001i 0 0 0 −2.42085 1.77186i 0
509.5 0 0.538121 1.64634i 0 1.21244 + 2.10001i 0 0 0 −2.42085 1.77186i 0
509.6 0 0.981784 + 1.42692i 0 −0.276914 0.479629i 0 0 0 −1.07220 + 2.80185i 0
509.7 0 1.03897 1.38584i 0 −2.09336 3.62580i 0 0 0 −0.841101 2.87968i 0
509.8 0 1.70794 + 0.287965i 0 −0.266780 0.462077i 0 0 0 2.83415 + 0.983658i 0
1109.1 0 −1.70794 + 0.287965i 0 0.266780 0.462077i 0 0 0 2.83415 0.983658i 0
1109.2 0 −1.03897 1.38584i 0 2.09336 3.62580i 0 0 0 −0.841101 + 2.87968i 0
1109.3 0 −0.981784 + 1.42692i 0 0.276914 0.479629i 0 0 0 −1.07220 2.80185i 0
1109.4 0 −0.538121 1.64634i 0 −1.21244 + 2.10001i 0 0 0 −2.42085 + 1.77186i 0
1109.5 0 0.538121 + 1.64634i 0 1.21244 2.10001i 0 0 0 −2.42085 + 1.77186i 0
1109.6 0 0.981784 1.42692i 0 −0.276914 + 0.479629i 0 0 0 −1.07220 2.80185i 0
1109.7 0 1.03897 + 1.38584i 0 −2.09336 + 3.62580i 0 0 0 −0.841101 + 2.87968i 0
1109.8 0 1.70794 0.287965i 0 −0.266780 + 0.462077i 0 0 0 2.83415 0.983658i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1109.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
63.i even 6 1 inner
63.j odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1764.2.w.a 16
3.b odd 2 1 5292.2.w.a 16
7.b odd 2 1 inner 1764.2.w.a 16
7.c even 3 1 252.2.x.a 16
7.c even 3 1 1764.2.bm.b 16
7.d odd 6 1 252.2.x.a 16
7.d odd 6 1 1764.2.bm.b 16
9.c even 3 1 5292.2.bm.b 16
9.d odd 6 1 1764.2.bm.b 16
21.c even 2 1 5292.2.w.a 16
21.g even 6 1 756.2.x.a 16
21.g even 6 1 5292.2.bm.b 16
21.h odd 6 1 756.2.x.a 16
21.h odd 6 1 5292.2.bm.b 16
28.f even 6 1 1008.2.cc.c 16
28.g odd 6 1 1008.2.cc.c 16
63.g even 3 1 756.2.x.a 16
63.h even 3 1 2268.2.f.b 16
63.h even 3 1 5292.2.w.a 16
63.i even 6 1 inner 1764.2.w.a 16
63.i even 6 1 2268.2.f.b 16
63.j odd 6 1 inner 1764.2.w.a 16
63.j odd 6 1 2268.2.f.b 16
63.k odd 6 1 756.2.x.a 16
63.l odd 6 1 5292.2.bm.b 16
63.n odd 6 1 252.2.x.a 16
63.o even 6 1 1764.2.bm.b 16
63.s even 6 1 252.2.x.a 16
63.t odd 6 1 2268.2.f.b 16
63.t odd 6 1 5292.2.w.a 16
84.j odd 6 1 3024.2.cc.c 16
84.n even 6 1 3024.2.cc.c 16
252.n even 6 1 3024.2.cc.c 16
252.o even 6 1 1008.2.cc.c 16
252.bl odd 6 1 3024.2.cc.c 16
252.bn odd 6 1 1008.2.cc.c 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
252.2.x.a 16 7.c even 3 1
252.2.x.a 16 7.d odd 6 1
252.2.x.a 16 63.n odd 6 1
252.2.x.a 16 63.s even 6 1
756.2.x.a 16 21.g even 6 1
756.2.x.a 16 21.h odd 6 1
756.2.x.a 16 63.g even 3 1
756.2.x.a 16 63.k odd 6 1
1008.2.cc.c 16 28.f even 6 1
1008.2.cc.c 16 28.g odd 6 1
1008.2.cc.c 16 252.o even 6 1
1008.2.cc.c 16 252.bn odd 6 1
1764.2.w.a 16 1.a even 1 1 trivial
1764.2.w.a 16 7.b odd 2 1 inner
1764.2.w.a 16 63.i even 6 1 inner
1764.2.w.a 16 63.j odd 6 1 inner
1764.2.bm.b 16 7.c even 3 1
1764.2.bm.b 16 7.d odd 6 1
1764.2.bm.b 16 9.d odd 6 1
1764.2.bm.b 16 63.o even 6 1
2268.2.f.b 16 63.h even 3 1
2268.2.f.b 16 63.i even 6 1
2268.2.f.b 16 63.j odd 6 1
2268.2.f.b 16 63.t odd 6 1
3024.2.cc.c 16 84.j odd 6 1
3024.2.cc.c 16 84.n even 6 1
3024.2.cc.c 16 252.n even 6 1
3024.2.cc.c 16 252.bl odd 6 1
5292.2.w.a 16 3.b odd 2 1
5292.2.w.a 16 21.c even 2 1
5292.2.w.a 16 63.h even 3 1
5292.2.w.a 16 63.t odd 6 1
5292.2.bm.b 16 9.c even 3 1
5292.2.bm.b 16 21.g even 6 1
5292.2.bm.b 16 21.h odd 6 1
5292.2.bm.b 16 63.l odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{5}^{16} + \cdots\) acting on \(S_{2}^{\mathrm{new}}(1764, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \)
$3$ \( 6561 + 2187 T^{2} + 729 T^{4} - 243 T^{6} - 99 T^{8} - 27 T^{10} + 9 T^{12} + 3 T^{14} + T^{16} \)
$5$ \( 81 + 567 T^{2} + 2916 T^{4} + 6939 T^{6} + 12168 T^{8} + 2682 T^{10} + 459 T^{12} + 24 T^{14} + T^{16} \)
$7$ \( T^{16} \)
$11$ \( ( 3969 - 2268 T - 891 T^{2} + 756 T^{3} + 342 T^{4} - 63 T^{5} - 18 T^{6} + 3 T^{7} + T^{8} )^{2} \)
$13$ \( 81 - 4536 T^{2} + 248994 T^{4} - 280368 T^{6} + 287163 T^{8} - 25776 T^{10} + 1746 T^{12} - 48 T^{14} + T^{16} \)
$17$ \( 810000 + 6066900 T^{2} + 44120781 T^{4} + 9748647 T^{6} + 1625391 T^{8} + 100944 T^{10} + 4617 T^{12} + 78 T^{14} + T^{16} \)
$19$ \( 810000 - 5208300 T^{2} + 32160969 T^{4} - 8406612 T^{6} + 1743651 T^{8} - 99126 T^{10} + 4149 T^{12} - 75 T^{14} + T^{16} \)
$23$ \( ( 50625 - 20250 T - 6075 T^{2} + 3510 T^{3} + 1206 T^{4} - 117 T^{5} - 36 T^{6} + 3 T^{7} + T^{8} )^{2} \)
$29$ \( ( 245025 - 236115 T + 38718 T^{2} + 35775 T^{3} + 4176 T^{4} - 450 T^{5} - 63 T^{6} + 6 T^{7} + T^{8} )^{2} \)
$31$ \( ( 729 + 1701 T^{2} + 1053 T^{4} + 72 T^{6} + T^{8} )^{2} \)
$37$ \( ( 372100 - 14030 T + 40789 T^{2} + 298 T^{3} + 3769 T^{4} - 20 T^{5} + 67 T^{6} + T^{7} + T^{8} )^{2} \)
$41$ \( 331869318561 + 96021181080 T^{2} + 22187899809 T^{4} + 1414696806 T^{6} + 64225080 T^{8} + 1385487 T^{10} + 21618 T^{12} + 177 T^{14} + T^{16} \)
$43$ \( ( 461041 + 131047 T + 88174 T^{2} - 11759 T^{3} + 5332 T^{4} - 236 T^{5} + 79 T^{6} - 2 T^{7} + T^{8} )^{2} \)
$47$ \( ( 4968441 - 502101 T^{2} + 16785 T^{4} - 222 T^{6} + T^{8} )^{2} \)
$53$ \( ( 41990400 - 24669360 T + 5822523 T^{2} - 582471 T^{3} + 7047 T^{4} + 2754 T^{5} - 45 T^{6} - 18 T^{7} + T^{8} )^{2} \)
$59$ \( ( 441 - 18225 T^{2} + 2439 T^{4} - 96 T^{6} + T^{8} )^{2} \)
$61$ \( ( 35319249 + 2099718 T^{2} + 42741 T^{4} + 351 T^{6} + T^{8} )^{2} \)
$67$ \( ( -1985 - 1004 T - 111 T^{2} + 7 T^{3} + T^{4} )^{4} \)
$71$ \( ( 15876 + 97443 T^{2} + 11250 T^{4} + 207 T^{6} + T^{8} )^{2} \)
$73$ \( 5802782976 - 40182763824 T^{2} + 276760621881 T^{4} - 10312508844 T^{6} + 256685967 T^{8} - 3712662 T^{10} + 39429 T^{12} - 243 T^{14} + T^{16} \)
$79$ \( ( 565 - 833 T - 93 T^{2} + 10 T^{3} + T^{4} )^{4} \)
$83$ \( 30237384321 + 13715668764 T^{2} + 4535917299 T^{4} + 671688342 T^{6} + 72720468 T^{8} + 2430279 T^{10} + 61596 T^{12} + 267 T^{14} + T^{16} \)
$89$ \( 44522973293864976 + 1990205599242420 T^{2} + 60745606555869 T^{4} + 987894749799 T^{6} + 11561010525 T^{8} + 67793598 T^{10} + 286173 T^{12} + 648 T^{14} + T^{16} \)
$97$ \( 83955602727441 - 9642663582291 T^{2} + 789930535230 T^{4} - 29657333385 T^{6} + 800598564 T^{8} - 10788390 T^{10} + 103725 T^{12} - 372 T^{14} + T^{16} \)
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