Properties

Label 756.2.x.a
Level 756
Weight 2
Character orbit 756.x
Analytic conductor 6.037
Analytic rank 0
Dimension 16
CM no
Inner twists 4

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Newspace parameters

Level: \( N \) = \( 756 = 2^{2} \cdot 3^{3} \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 756.x (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(6.03669039281\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 3^{6} \)
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.

\(f(q)\) \(=\) \( q + ( \beta_{1} - \beta_{15} ) q^{5} + ( -\beta_{6} + \beta_{13} ) q^{7} +O(q^{10})\) \( q + ( \beta_{1} - \beta_{15} ) q^{5} + ( -\beta_{6} + \beta_{13} ) q^{7} -\beta_{5} q^{11} + ( -\beta_{1} + \beta_{6} - \beta_{7} - \beta_{10} + \beta_{12} - \beta_{15} ) q^{13} + ( 2 \beta_{6} - \beta_{8} - 2 \beta_{10} + \beta_{11} + \beta_{12} - \beta_{13} - 2 \beta_{15} ) q^{17} + ( -2 \beta_{1} + \beta_{3} - \beta_{7} + \beta_{8} + \beta_{13} + \beta_{15} ) q^{19} + ( \beta_{9} - \beta_{13} ) q^{23} + ( -\beta_{8} + 2 \beta_{9} - \beta_{10} - \beta_{13} - \beta_{14} ) q^{25} + ( 2 + \beta_{2} - \beta_{8} + \beta_{10} + \beta_{13} + \beta_{14} ) q^{29} + ( -\beta_{1} - \beta_{15} ) q^{31} + ( 1 + 2 \beta_{2} - \beta_{3} + \beta_{6} - \beta_{7} - \beta_{8} + \beta_{9} - 2 \beta_{10} - \beta_{13} - \beta_{14} - \beta_{15} ) q^{35} + ( 1 - \beta_{4} - 2 \beta_{5} ) q^{37} + ( 2 \beta_{1} - 2 \beta_{3} + \beta_{7} + 2 \beta_{11} - \beta_{12} - 2 \beta_{15} ) q^{41} + ( -\beta_{2} - 2 \beta_{4} - \beta_{5} + \beta_{6} + \beta_{8} + \beta_{10} - \beta_{13} ) q^{43} + ( -\beta_{1} + 2 \beta_{3} - \beta_{7} + \beta_{11} + \beta_{12} ) q^{47} + ( \beta_{4} - \beta_{5} - \beta_{7} - \beta_{10} + \beta_{12} ) q^{49} + ( 2 + 4 \beta_{2} + 2 \beta_{4} - 3 \beta_{6} - \beta_{8} - \beta_{9} - 2 \beta_{10} + 2 \beta_{13} + \beta_{14} ) q^{53} + ( -\beta_{11} + \beta_{12} ) q^{55} + ( -\beta_{1} + \beta_{3} - \beta_{6} - \beta_{7} + 2 \beta_{8} + \beta_{10} - \beta_{11} + 2 \beta_{13} + \beta_{15} ) q^{59} + ( 2 \beta_{1} + \beta_{6} - \beta_{7} - \beta_{8} - \beta_{10} - \beta_{11} - \beta_{12} - \beta_{13} - 4 \beta_{15} ) q^{61} + ( 2 + \beta_{2} + \beta_{5} + \beta_{6} - \beta_{8} + \beta_{10} + \beta_{13} ) q^{65} + ( -\beta_{4} + \beta_{5} + \beta_{6} + \beta_{9} - \beta_{10} - \beta_{13} - 2 \beta_{14} ) q^{67} + ( -3 - 6 \beta_{2} - \beta_{6} - \beta_{9} + \beta_{13} + \beta_{14} ) q^{71} + ( 2 \beta_{1} + \beta_{3} - \beta_{7} + \beta_{8} + \beta_{13} - \beta_{15} ) q^{73} + ( 1 - \beta_{2} - \beta_{3} + \beta_{7} + 2 \beta_{8} + \beta_{9} + \beta_{10} + \beta_{11} - \beta_{13} ) q^{77} + ( -\beta_{2} + 2 \beta_{4} + \beta_{5} - \beta_{6} - 2 \beta_{9} + 2 \beta_{13} + \beta_{14} ) q^{79} + ( -2 \beta_{1} - \beta_{3} - \beta_{7} - 2 \beta_{11} + \beta_{12} ) q^{83} + ( 1 + \beta_{2} + \beta_{6} + \beta_{10} ) q^{85} + ( 3 \beta_{3} + 2 \beta_{6} + 3 \beta_{7} - \beta_{8} - 2 \beta_{10} + \beta_{11} + \beta_{12} - \beta_{13} ) q^{89} + ( -1 - 2 \beta_{1} - \beta_{3} + \beta_{7} - \beta_{8} - \beta_{11} + \beta_{12} + \beta_{15} ) q^{91} + ( 2 - 2 \beta_{2} - \beta_{4} - \beta_{5} + \beta_{6} + 2 \beta_{8} - 2 \beta_{9} + \beta_{10} ) q^{95} + ( -\beta_{1} - \beta_{3} - \beta_{6} + \beta_{7} + \beta_{8} + \beta_{10} + 2 \beta_{11} + \beta_{12} + \beta_{13} + 2 \beta_{15} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q - q^{7} + O(q^{10}) \) \( 16q - q^{7} - 6q^{11} - 6q^{23} - 8q^{25} + 12q^{29} + 4q^{37} + 4q^{43} - 5q^{49} + 24q^{65} + 14q^{67} + 21q^{77} + 20q^{79} + 6q^{85} - 18q^{91} + 60q^{95} + O(q^{100}) \)

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

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{15} + 15 \nu^{13} + 72 \nu^{11} + 153 \nu^{9} - 423 \nu^{7} - 891 \nu^{5} + 1944 \nu^{3} + 17496 \nu \)\()/15309\)
\(\beta_{2}\)\(=\)\((\)\( -5 \nu^{14} - 12 \nu^{12} + 18 \nu^{10} + 369 \nu^{8} - 153 \nu^{6} - 1782 \nu^{4} - 4617 \nu^{2} + 9477 \)\()/5103\)
\(\beta_{3}\)\(=\)\((\)\( -2 \nu^{15} - 9 \nu^{13} - 18 \nu^{11} + 72 \nu^{9} + 657 \nu^{7} - 297 \nu^{5} - 3888 \nu^{3} - 9477 \nu \)\()/5103\)
\(\beta_{4}\)\(=\)\((\)\( -\nu^{14} + 3 \nu^{12} + 9 \nu^{10} + 9 \nu^{8} - 225 \nu^{6} + 81 \nu^{4} + 2187 \)\()/729\)
\(\beta_{5}\)\(=\)\((\)\( 2 \nu^{14} + 3 \nu^{12} - 45 \nu^{10} - 99 \nu^{8} + 369 \nu^{6} + 1134 \nu^{4} - 729 \nu^{2} - 6561 \)\()/729\)
\(\beta_{6}\)\(=\)\((\)\( -32 \nu^{15} + 69 \nu^{14} + 87 \nu^{13} - 99 \nu^{12} + 342 \nu^{11} - 702 \nu^{10} + 774 \nu^{9} - 3051 \nu^{8} - 5175 \nu^{7} + 11637 \nu^{6} - 5508 \nu^{5} + 25272 \nu^{4} + 12636 \nu^{3} + 6561 \nu^{2} + 67797 \nu - 247131 \)\()/30618\)
\(\beta_{7}\)\(=\)\((\)\( 5 \nu^{15} + 12 \nu^{13} - 18 \nu^{11} - 369 \nu^{9} + 153 \nu^{7} + 1782 \nu^{5} + 4617 \nu^{3} - 19683 \nu \)\()/5103\)
\(\beta_{8}\)\(=\)\((\)\( 5 \nu^{15} - 18 \nu^{14} + 12 \nu^{13} + 27 \nu^{12} - 72 \nu^{11} + 162 \nu^{10} - 207 \nu^{9} + 648 \nu^{8} + 396 \nu^{7} - 2349 \nu^{6} + 2268 \nu^{5} - 3888 \nu^{4} - 243 \nu^{3} - 2916 \nu^{2} - 8748 \nu + 41553 \)\()/4374\)
\(\beta_{9}\)\(=\)\((\)\( 35 \nu^{15} + 48 \nu^{14} + 84 \nu^{13} - 225 \nu^{12} - 504 \nu^{11} - 1080 \nu^{10} - 1449 \nu^{9} - 2862 \nu^{8} + 2772 \nu^{7} + 18819 \nu^{6} + 15876 \nu^{5} + 25272 \nu^{4} - 1701 \nu^{3} - 49572 \nu^{2} - 61236 \nu - 369603 \)\()/30618\)
\(\beta_{10}\)\(=\)\((\)\( 32 \nu^{15} + 69 \nu^{14} - 87 \nu^{13} - 99 \nu^{12} - 342 \nu^{11} - 702 \nu^{10} - 774 \nu^{9} - 3051 \nu^{8} + 5175 \nu^{7} + 11637 \nu^{6} + 5508 \nu^{5} + 25272 \nu^{4} - 12636 \nu^{3} + 6561 \nu^{2} - 67797 \nu - 247131 \)\()/30618\)
\(\beta_{11}\)\(=\)\((\)\( -10 \nu^{15} - 24 \nu^{13} + 36 \nu^{11} + 738 \nu^{9} - 306 \nu^{7} - 3564 \nu^{5} - 9234 \nu^{3} + 24057 \nu \)\()/5103\)
\(\beta_{12}\)\(=\)\((\)\( 11 \nu^{15} - 3 \nu^{13} - 27 \nu^{11} - 207 \nu^{9} + 261 \nu^{7} + 27 \nu^{5} + 2673 \nu^{3} + 3645 \nu \)\()/5103\)
\(\beta_{13}\)\(=\)\((\)\( 5 \nu^{15} + 18 \nu^{14} + 12 \nu^{13} - 27 \nu^{12} - 72 \nu^{11} - 162 \nu^{10} - 207 \nu^{9} - 648 \nu^{8} + 396 \nu^{7} + 2349 \nu^{6} + 2268 \nu^{5} + 3888 \nu^{4} - 243 \nu^{3} + 2916 \nu^{2} - 8748 \nu - 41553 \)\()/4374\)
\(\beta_{14}\)\(=\)\((\)\( -32 \nu^{15} + 273 \nu^{14} + 87 \nu^{13} - 63 \nu^{12} + 342 \nu^{11} - 3024 \nu^{10} + 774 \nu^{9} - 9261 \nu^{8} - 5175 \nu^{7} + 34209 \nu^{6} - 5508 \nu^{5} + 71442 \nu^{4} + 12636 \nu^{3} - 45927 \nu^{2} + 67797 \nu - 535815 \)\()/30618\)
\(\beta_{15}\)\(=\)\((\)\( -61 \nu^{15} + 30 \nu^{13} + 711 \nu^{11} + 2574 \nu^{9} - 8217 \nu^{7} - 18792 \nu^{5} - 1215 \nu^{3} + 157464 \nu \)\()/15309\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-\beta_{11} - 2 \beta_{7}\)\()/3\)
\(\nu^{2}\)\(=\)\((\)\(-\beta_{14} + \beta_{13} - \beta_{9} + \beta_{6} + \beta_{5} - \beta_{4}\)\()/3\)
\(\nu^{3}\)\(=\)\(-2 \beta_{15} - \beta_{13} + \beta_{11} - 2 \beta_{10} - \beta_{8} + 2 \beta_{6} - \beta_{3} + \beta_{1}\)
\(\nu^{4}\)\(=\)\(2 \beta_{10} - \beta_{9} + \beta_{8} + 2 \beta_{6} + \beta_{4} + 2 \beta_{2} + 4\)
\(\nu^{5}\)\(=\)\(-4 \beta_{12} + 3 \beta_{10} - 3 \beta_{6} - 5 \beta_{3} + 6 \beta_{1}\)
\(\nu^{6}\)\(=\)\(-3 \beta_{13} + 3 \beta_{10} + 3 \beta_{8} + 3 \beta_{6} - 6 \beta_{4} - 3 \beta_{2} + 15\)
\(\nu^{7}\)\(=\)\(-6 \beta_{15} - 3 \beta_{13} - 3 \beta_{11} - 6 \beta_{10} - 3 \beta_{8} - 9 \beta_{7} + 6 \beta_{6} + 6 \beta_{3} + 12 \beta_{1}\)
\(\nu^{8}\)\(=\)\(3 \beta_{13} + 15 \beta_{10} - 15 \beta_{9} + 12 \beta_{8} + 15 \beta_{6} + 3 \beta_{5} - 6 \beta_{4} + 24 \beta_{2} - 24\)
\(\nu^{9}\)\(=\)\(-27 \beta_{15} - 9 \beta_{13} - 9 \beta_{12} + 39 \beta_{11} - 18 \beta_{10} - 9 \beta_{8} + 33 \beta_{7} + 18 \beta_{6} - 27 \beta_{3} + 54 \beta_{1}\)
\(\nu^{10}\)\(=\)\(21 \beta_{14} - 39 \beta_{13} + 63 \beta_{10} - 6 \beta_{9} + 45 \beta_{8} + 42 \beta_{6} - 39 \beta_{5} - 15 \beta_{4} - 18 \beta_{2} + 72\)
\(\nu^{11}\)\(=\)\(72 \beta_{15} + 9 \beta_{13} - 63 \beta_{12} - 54 \beta_{11} + 126 \beta_{10} + 9 \beta_{8} + 18 \beta_{7} - 126 \beta_{6} - 9 \beta_{3} + 180 \beta_{1}\)
\(\nu^{12}\)\(=\)\(36 \beta_{14} - 90 \beta_{13} + 63 \beta_{10} - 36 \beta_{9} + 126 \beta_{8} + 27 \beta_{6} + 18 \beta_{5} - 162 \beta_{4} - 153 \beta_{2} - 198\)
\(\nu^{13}\)\(=\)\(81 \beta_{13} + 144 \beta_{12} - 189 \beta_{10} + 81 \beta_{8} + 27 \beta_{7} + 189 \beta_{6} + 342 \beta_{3} + 108 \beta_{1}\)
\(\nu^{14}\)\(=\)\(297 \beta_{14} + 81 \beta_{13} + 378 \beta_{10} - 378 \beta_{9} + 297 \beta_{8} + 81 \beta_{6} - 270 \beta_{5} + 27 \beta_{4} + 432 \beta_{2} - 1026\)
\(\nu^{15}\)\(=\)\(297 \beta_{15} + 189 \beta_{13} + 189 \beta_{12} + 540 \beta_{11} + 540 \beta_{10} + 189 \beta_{8} + 1107 \beta_{7} - 540 \beta_{6} - 324 \beta_{3} + 945 \beta_{1}\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/756\mathbb{Z}\right)^\times\).

\(n\) \(29\) \(325\) \(379\)
\(\chi(n)\) \(1 + \beta_{2}\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
125.1
1.71965 + 0.206851i
−1.69483 + 0.357142i
−0.744857 + 1.56371i
0.604587 + 1.62311i
−0.604587 1.62311i
0.744857 1.56371i
1.69483 0.357142i
−1.71965 0.206851i
1.71965 0.206851i
−1.69483 0.357142i
−0.744857 1.56371i
0.604587 1.62311i
−0.604587 + 1.62311i
0.744857 + 1.56371i
1.69483 + 0.357142i
−1.71965 + 0.206851i
0 0 0 −2.09336 + 3.62580i 0 −2.64522 + 0.0532130i 0 0 0
125.2 0 0 0 −1.21244 + 2.10001i 0 1.05649 + 2.42566i 0 0 0
125.3 0 0 0 −0.276914 + 0.479629i 0 −1.98718 1.74675i 0 0 0
125.4 0 0 0 −0.266780 + 0.462077i 0 2.54716 0.715531i 0 0 0
125.5 0 0 0 0.266780 0.462077i 0 −1.89325 + 1.84814i 0 0 0
125.6 0 0 0 0.276914 0.479629i 0 −0.519138 2.59432i 0 0 0
125.7 0 0 0 1.21244 2.10001i 0 1.57244 + 2.12778i 0 0 0
125.8 0 0 0 2.09336 3.62580i 0 1.36869 2.26422i 0 0 0
629.1 0 0 0 −2.09336 3.62580i 0 −2.64522 0.0532130i 0 0 0
629.2 0 0 0 −1.21244 2.10001i 0 1.05649 2.42566i 0 0 0
629.3 0 0 0 −0.276914 0.479629i 0 −1.98718 + 1.74675i 0 0 0
629.4 0 0 0 −0.266780 0.462077i 0 2.54716 + 0.715531i 0 0 0
629.5 0 0 0 0.266780 + 0.462077i 0 −1.89325 1.84814i 0 0 0
629.6 0 0 0 0.276914 + 0.479629i 0 −0.519138 + 2.59432i 0 0 0
629.7 0 0 0 1.21244 + 2.10001i 0 1.57244 2.12778i 0 0 0
629.8 0 0 0 2.09336 + 3.62580i 0 1.36869 + 2.26422i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 629.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
9.d odd 6 1 inner
63.o even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 756.2.x.a 16
3.b odd 2 1 252.2.x.a 16
4.b odd 2 1 3024.2.cc.c 16
7.b odd 2 1 inner 756.2.x.a 16
7.c even 3 1 5292.2.w.a 16
7.c even 3 1 5292.2.bm.b 16
7.d odd 6 1 5292.2.w.a 16
7.d odd 6 1 5292.2.bm.b 16
9.c even 3 1 252.2.x.a 16
9.c even 3 1 2268.2.f.b 16
9.d odd 6 1 inner 756.2.x.a 16
9.d odd 6 1 2268.2.f.b 16
12.b even 2 1 1008.2.cc.c 16
21.c even 2 1 252.2.x.a 16
21.g even 6 1 1764.2.w.a 16
21.g even 6 1 1764.2.bm.b 16
21.h odd 6 1 1764.2.w.a 16
21.h odd 6 1 1764.2.bm.b 16
28.d even 2 1 3024.2.cc.c 16
36.f odd 6 1 1008.2.cc.c 16
36.h even 6 1 3024.2.cc.c 16
63.g even 3 1 1764.2.w.a 16
63.h even 3 1 1764.2.bm.b 16
63.i even 6 1 5292.2.bm.b 16
63.j odd 6 1 5292.2.bm.b 16
63.k odd 6 1 1764.2.w.a 16
63.l odd 6 1 252.2.x.a 16
63.l odd 6 1 2268.2.f.b 16
63.n odd 6 1 5292.2.w.a 16
63.o even 6 1 inner 756.2.x.a 16
63.o even 6 1 2268.2.f.b 16
63.s even 6 1 5292.2.w.a 16
63.t odd 6 1 1764.2.bm.b 16
84.h odd 2 1 1008.2.cc.c 16
252.s odd 6 1 3024.2.cc.c 16
252.bi even 6 1 1008.2.cc.c 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
252.2.x.a 16 3.b odd 2 1
252.2.x.a 16 9.c even 3 1
252.2.x.a 16 21.c even 2 1
252.2.x.a 16 63.l odd 6 1
756.2.x.a 16 1.a even 1 1 trivial
756.2.x.a 16 7.b odd 2 1 inner
756.2.x.a 16 9.d odd 6 1 inner
756.2.x.a 16 63.o even 6 1 inner
1008.2.cc.c 16 12.b even 2 1
1008.2.cc.c 16 36.f odd 6 1
1008.2.cc.c 16 84.h odd 2 1
1008.2.cc.c 16 252.bi even 6 1
1764.2.w.a 16 21.g even 6 1
1764.2.w.a 16 21.h odd 6 1
1764.2.w.a 16 63.g even 3 1
1764.2.w.a 16 63.k odd 6 1
1764.2.bm.b 16 21.g even 6 1
1764.2.bm.b 16 21.h odd 6 1
1764.2.bm.b 16 63.h even 3 1
1764.2.bm.b 16 63.t odd 6 1
2268.2.f.b 16 9.c even 3 1
2268.2.f.b 16 9.d odd 6 1
2268.2.f.b 16 63.l odd 6 1
2268.2.f.b 16 63.o even 6 1
3024.2.cc.c 16 4.b odd 2 1
3024.2.cc.c 16 28.d even 2 1
3024.2.cc.c 16 36.h even 6 1
3024.2.cc.c 16 252.s odd 6 1
5292.2.w.a 16 7.c even 3 1
5292.2.w.a 16 7.d odd 6 1
5292.2.w.a 16 63.n odd 6 1
5292.2.w.a 16 63.s even 6 1
5292.2.bm.b 16 7.c even 3 1
5292.2.bm.b 16 7.d odd 6 1
5292.2.bm.b 16 63.i even 6 1
5292.2.bm.b 16 63.j odd 6 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(756, [\chi])\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ \( 1 - 16 T^{2} + 159 T^{4} - 998 T^{6} + 4298 T^{8} - 12066 T^{10} + 21316 T^{12} - 64633 T^{14} + 271566 T^{16} - 1615825 T^{18} + 13322500 T^{20} - 188531250 T^{22} + 1678906250 T^{24} - 9746093750 T^{26} + 38818359375 T^{28} - 97656250000 T^{30} + 152587890625 T^{32} \)
$7$ \( 1 + T + 3 T^{2} + 20 T^{3} + 53 T^{4} + 21 T^{5} - 74 T^{6} + 355 T^{7} - 1314 T^{8} + 2485 T^{9} - 3626 T^{10} + 7203 T^{11} + 127253 T^{12} + 336140 T^{13} + 352947 T^{14} + 823543 T^{15} + 5764801 T^{16} \)
$11$ \( ( 1 + 3 T + 26 T^{2} + 69 T^{3} + 265 T^{4} + 756 T^{5} + 2585 T^{6} + 8259 T^{7} + 34846 T^{8} + 90849 T^{9} + 312785 T^{10} + 1006236 T^{11} + 3879865 T^{12} + 11112519 T^{13} + 46060586 T^{14} + 58461513 T^{15} + 214358881 T^{16} )^{2} \)
$13$ \( 1 + 56 T^{2} + 1590 T^{4} + 26848 T^{6} + 274865 T^{8} + 1615656 T^{10} + 12863830 T^{12} + 363405056 T^{14} + 6601006116 T^{16} + 61415454464 T^{18} + 367403848630 T^{20} + 7798462921704 T^{22} + 224215824627665 T^{24} + 3701224789161952 T^{26} + 37043955344744790 T^{28} + 220493077599160184 T^{30} + 665416609183179841 T^{32} \)
$17$ \( ( 1 + 58 T^{2} + 1603 T^{4} + 30013 T^{6} + 497674 T^{8} + 8673757 T^{10} + 133884163 T^{12} + 1399979002 T^{14} + 6975757441 T^{16} )^{2} \)
$19$ \( ( 1 - 77 T^{2} + 3034 T^{4} - 84368 T^{6} + 1811980 T^{8} - 30456848 T^{10} + 395393914 T^{12} - 3622532837 T^{14} + 16983563041 T^{16} )^{2} \)
$23$ \( ( 1 + 3 T + 56 T^{2} + 159 T^{3} + 1321 T^{4} + 3510 T^{5} + 30863 T^{6} + 67173 T^{7} + 842170 T^{8} + 1544979 T^{9} + 16326527 T^{10} + 42706170 T^{11} + 369669961 T^{12} + 1023378537 T^{13} + 8290009784 T^{14} + 10214476341 T^{15} + 78310985281 T^{16} )^{2} \)
$29$ \( ( 1 - 6 T + 53 T^{2} - 246 T^{3} + 580 T^{4} - 540 T^{5} - 22066 T^{6} + 299625 T^{7} - 1268282 T^{8} + 8689125 T^{9} - 18557506 T^{10} - 13170060 T^{11} + 410222980 T^{12} - 5045742654 T^{13} + 31525636013 T^{14} - 103499257854 T^{15} + 500246412961 T^{16} )^{2} \)
$31$ \( 1 + 176 T^{2} + 16407 T^{4} + 1045570 T^{6} + 50906954 T^{8} + 2036295270 T^{10} + 71233973380 T^{12} + 2306080860179 T^{14} + 72156636517806 T^{16} + 2216143706632019 T^{18} + 65786070329870980 T^{20} + 1807219547727888870 T^{22} + 43418084810021264714 T^{24} + \)\(85\!\cdots\!70\)\( T^{26} + \)\(12\!\cdots\!27\)\( T^{28} + \)\(13\!\cdots\!96\)\( T^{30} + \)\(72\!\cdots\!81\)\( T^{32} \)
$37$ \( ( 1 - T + 82 T^{2} - 88 T^{3} + 3940 T^{4} - 3256 T^{5} + 112258 T^{6} - 50653 T^{7} + 1874161 T^{8} )^{4} \)
$41$ \( 1 - 151 T^{2} + 9564 T^{4} - 353897 T^{6} + 12352577 T^{8} - 588962616 T^{10} + 30503470855 T^{12} - 1573469011633 T^{14} + 72313421958936 T^{16} - 2645001408555073 T^{18} + 86195518306695655 T^{20} - 2797633820052054456 T^{22} + 98634403731959794817 T^{24} - \)\(47\!\cdots\!97\)\( T^{26} + \)\(21\!\cdots\!84\)\( T^{28} - \)\(57\!\cdots\!11\)\( T^{30} + \)\(63\!\cdots\!41\)\( T^{32} \)
$43$ \( ( 1 - 2 T - 93 T^{2} - 64 T^{3} + 3956 T^{4} + 12192 T^{5} - 158990 T^{6} - 321485 T^{7} + 7972668 T^{8} - 13823855 T^{9} - 293972510 T^{10} + 969349344 T^{11} + 13524776756 T^{12} - 9408540352 T^{13} - 587886763557 T^{14} - 543637222214 T^{15} + 11688200277601 T^{16} )^{2} \)
$47$ \( 1 - 154 T^{2} + 7683 T^{4} - 245888 T^{6} + 25028264 T^{8} - 1538143278 T^{10} + 38475997042 T^{12} - 2602469435029 T^{14} + 205422570296046 T^{16} - 5748854981979061 T^{18} + 187750591721903602 T^{20} - 16579977600415908462 T^{22} + \)\(59\!\cdots\!04\)\( T^{24} - \)\(12\!\cdots\!12\)\( T^{26} + \)\(89\!\cdots\!03\)\( T^{28} - \)\(39\!\cdots\!26\)\( T^{30} + \)\(56\!\cdots\!21\)\( T^{32} \)
$53$ \( ( 1 - 10 T^{2} + 3133 T^{4} + 54785 T^{6} + 5778574 T^{8} + 153891065 T^{10} + 24720876973 T^{12} - 221643611290 T^{14} + 62259690411361 T^{16} )^{2} \)
$59$ \( 1 - 376 T^{2} + 75453 T^{4} - 10695122 T^{6} + 1193881406 T^{8} - 111003832932 T^{10} + 8882998756942 T^{12} - 624280670070511 T^{14} + 38988958528434078 T^{16} - 2173121012515448791 T^{18} + \)\(10\!\cdots\!62\)\( T^{20} - \)\(46\!\cdots\!12\)\( T^{22} + \)\(17\!\cdots\!26\)\( T^{24} - \)\(54\!\cdots\!22\)\( T^{26} + \)\(13\!\cdots\!93\)\( T^{28} - \)\(23\!\cdots\!36\)\( T^{30} + \)\(21\!\cdots\!41\)\( T^{32} \)
$61$ \( 1 + 137 T^{2} + 306 T^{4} - 368483 T^{6} + 33183791 T^{8} + 2225199582 T^{10} - 143899132121 T^{12} + 1205057298539 T^{14} + 1133086682080440 T^{16} + 4484018207863619 T^{18} - 1992404503385358761 T^{20} + \)\(11\!\cdots\!02\)\( T^{22} + \)\(63\!\cdots\!71\)\( T^{24} - \)\(26\!\cdots\!83\)\( T^{26} + \)\(81\!\cdots\!26\)\( T^{28} + \)\(13\!\cdots\!17\)\( T^{30} + \)\(36\!\cdots\!61\)\( T^{32} \)
$67$ \( ( 1 - 7 T - 108 T^{2} - 293 T^{3} + 11753 T^{4} + 50778 T^{5} - 198827 T^{6} - 2620663 T^{7} - 10701486 T^{8} - 175584421 T^{9} - 892534403 T^{10} + 15272143614 T^{11} + 236836125113 T^{12} - 395586656351 T^{13} - 9769505274252 T^{14} - 42424981237261 T^{15} + 406067677556641 T^{16} )^{2} \)
$71$ \( ( 1 - 361 T^{2} + 64216 T^{4} - 7488268 T^{6} + 623512600 T^{8} - 37748358988 T^{10} + 1631836507096 T^{12} - 46244202495481 T^{14} + 645753531245761 T^{16} )^{2} \)
$73$ \( ( 1 - 341 T^{2} + 62398 T^{4} - 7562288 T^{6} + 647645452 T^{8} - 40299432752 T^{10} + 1771993441918 T^{12} - 51604971164549 T^{14} + 806460091894081 T^{16} )^{2} \)
$79$ \( ( 1 - 10 T - 123 T^{2} + 844 T^{3} + 11042 T^{4} + 2166 T^{5} - 1268066 T^{6} - 531391 T^{7} + 106354368 T^{8} - 41979889 T^{9} - 7913999906 T^{10} + 1067922474 T^{11} + 430086794402 T^{12} + 2597035600756 T^{13} - 29899757029083 T^{14} - 192039089861590 T^{15} + 1517108809906561 T^{16} )^{2} \)
$83$ \( 1 - 397 T^{2} + 87990 T^{4} - 12501089 T^{6} + 1185510017 T^{8} - 57837981066 T^{10} - 2507933243939 T^{12} + 849901181197043 T^{14} - 92913215010718536 T^{16} + 5854969237266429227 T^{18} - \)\(11\!\cdots\!19\)\( T^{20} - \)\(18\!\cdots\!54\)\( T^{22} + \)\(26\!\cdots\!97\)\( T^{24} - \)\(19\!\cdots\!61\)\( T^{26} + \)\(94\!\cdots\!90\)\( T^{28} - \)\(29\!\cdots\!13\)\( T^{30} + \)\(50\!\cdots\!81\)\( T^{32} \)
$89$ \( ( 1 + 64 T^{2} + 9487 T^{4} + 662335 T^{6} + 143358802 T^{8} + 5246355535 T^{10} + 595235640367 T^{12} + 31806802621504 T^{14} + 3936588805702081 T^{16} )^{2} \)
$97$ \( 1 + 404 T^{2} + 81609 T^{4} + 10963666 T^{6} + 1046175986 T^{8} + 57359540784 T^{10} - 1663618077194 T^{12} - 785812116013063 T^{14} - 99406691908304130 T^{16} - 7393706199566909767 T^{18} - \)\(14\!\cdots\!14\)\( T^{20} + \)\(47\!\cdots\!36\)\( T^{22} + \)\(81\!\cdots\!46\)\( T^{24} + \)\(80\!\cdots\!34\)\( T^{26} + \)\(56\!\cdots\!69\)\( T^{28} + \)\(26\!\cdots\!76\)\( T^{30} + \)\(61\!\cdots\!21\)\( T^{32} \)
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