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: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
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: \( 3^{6} \)
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})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - q^{7} + O(q^{10}) \) \( 16 q - q^{7} - 6 q^{11} - 6 q^{23} - 8 q^{25} + 12 q^{29} + 4 q^{37} + 4 q^{43} - 5 q^{49} + 24 q^{65} + 14 q^{67} + 21 q^{77} + 20 q^{79} + 6 q^{85} - 18 q^{91} + 60 q^{95} + O(q^{100}) \)

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

$p$ $F_p(T)$
$2$ \( T^{16} \)
$3$ \( T^{16} \)
$5$ \( 81 + 567 T^{2} + 2916 T^{4} + 6939 T^{6} + 12168 T^{8} + 2682 T^{10} + 459 T^{12} + 24 T^{14} + T^{16} \)
$7$ \( 5764801 + 823543 T + 352947 T^{2} + 336140 T^{3} + 127253 T^{4} + 7203 T^{5} - 3626 T^{6} + 2485 T^{7} - 1314 T^{8} + 355 T^{9} - 74 T^{10} + 21 T^{11} + 53 T^{12} + 20 T^{13} + 3 T^{14} + T^{15} + T^{16} \)
$11$ \( ( 3969 - 2268 T - 891 T^{2} + 756 T^{3} + 342 T^{4} - 63 T^{5} - 18 T^{6} + 3 T^{7} + T^{8} )^{2} \)
$13$ \( 81 - 4536 T^{2} + 248994 T^{4} - 280368 T^{6} + 287163 T^{8} - 25776 T^{10} + 1746 T^{12} - 48 T^{14} + T^{16} \)
$17$ \( ( 900 - 6741 T^{2} + 1467 T^{4} - 78 T^{6} + T^{8} )^{2} \)
$19$ \( ( 900 + 5787 T^{2} + 1476 T^{4} + 75 T^{6} + T^{8} )^{2} \)
$23$ \( ( 50625 - 20250 T - 6075 T^{2} + 3510 T^{3} + 1206 T^{4} - 117 T^{5} - 36 T^{6} + 3 T^{7} + T^{8} )^{2} \)
$29$ \( ( 245025 + 236115 T + 38718 T^{2} - 35775 T^{3} + 4176 T^{4} + 450 T^{5} - 63 T^{6} - 6 T^{7} + T^{8} )^{2} \)
$31$ \( 531441 - 1240029 T^{2} + 2125764 T^{4} - 1686177 T^{6} + 985608 T^{8} - 72414 T^{10} + 4131 T^{12} - 72 T^{14} + T^{16} \)
$37$ \( ( 610 + 23 T - 66 T^{2} - T^{3} + T^{4} )^{4} \)
$41$ \( 331869318561 + 96021181080 T^{2} + 22187899809 T^{4} + 1414696806 T^{6} + 64225080 T^{8} + 1385487 T^{10} + 21618 T^{12} + 177 T^{14} + T^{16} \)
$43$ \( ( 461041 + 131047 T + 88174 T^{2} - 11759 T^{3} + 5332 T^{4} - 236 T^{5} + 79 T^{6} - 2 T^{7} + T^{8} )^{2} \)
$47$ \( 24685405970481 + 2494659194541 T^{2} + 168710132016 T^{4} + 6221777481 T^{6} + 165301362 T^{8} + 2722068 T^{10} + 32499 T^{12} + 222 T^{14} + T^{16} \)
$53$ \( ( 41990400 + 2848203 T^{2} + 56133 T^{4} + 414 T^{6} + T^{8} )^{2} \)
$59$ \( 194481 + 8037225 T^{2} + 331075026 T^{4} + 44366103 T^{6} + 4198680 T^{8} + 197694 T^{10} + 6777 T^{12} + 96 T^{14} + T^{16} \)
$61$ \( 1247449349924001 - 74160462871782 T^{2} + 2899235658015 T^{4} - 64949934240 T^{6} + 1054472814 T^{8} - 10802655 T^{10} + 80460 T^{12} - 351 T^{14} + T^{16} \)
$67$ \( ( 3940225 - 1992940 T + 787681 T^{2} - 139234 T^{3} + 21334 T^{4} - 1231 T^{5} + 160 T^{6} - 7 T^{7} + T^{8} )^{2} \)
$71$ \( ( 15876 + 97443 T^{2} + 11250 T^{4} + 207 T^{6} + T^{8} )^{2} \)
$73$ \( ( 76176 + 527499 T^{2} + 19620 T^{4} + 243 T^{6} + T^{8} )^{2} \)
$79$ \( ( 319225 + 470645 T + 746434 T^{2} - 66169 T^{3} + 16414 T^{4} - 736 T^{5} + 193 T^{6} - 10 T^{7} + T^{8} )^{2} \)
$83$ \( 30237384321 + 13715668764 T^{2} + 4535917299 T^{4} + 671688342 T^{6} + 72720468 T^{8} + 2430279 T^{10} + 61596 T^{12} + 267 T^{14} + T^{16} \)
$89$ \( ( 211004676 - 9432045 T^{2} + 133731 T^{4} - 648 T^{6} + T^{8} )^{2} \)
$97$ \( 83955602727441 - 9642663582291 T^{2} + 789930535230 T^{4} - 29657333385 T^{6} + 800598564 T^{8} - 10788390 T^{10} + 103725 T^{12} - 372 T^{14} + T^{16} \)
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