Properties

Label 1764.2.bm.b
Level $1764$
Weight $2$
Character orbit 1764.bm
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.bm (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.

\(f(q)\) \(=\) \( q + \beta_{1} q^{3} + \beta_{15} q^{5} + \beta_{2} q^{9} +O(q^{10})\) \( q + \beta_{1} q^{3} + \beta_{15} q^{5} + \beta_{2} q^{9} + ( -\beta_{2} - \beta_{8} ) q^{11} + ( \beta_{1} + \beta_{5} + \beta_{7} + \beta_{11} + \beta_{15} ) q^{13} + ( -\beta_{4} - \beta_{6} + \beta_{8} + \beta_{13} + \beta_{14} ) q^{15} + ( \beta_{7} - \beta_{9} + 2 \beta_{11} + \beta_{15} ) q^{17} + ( -\beta_{1} + \beta_{3} + \beta_{5} + \beta_{7} - \beta_{9} + 2 \beta_{11} + 2 \beta_{15} ) q^{19} + ( \beta_{12} - \beta_{14} ) q^{23} + ( \beta_{2} + \beta_{4} - 2 \beta_{8} - \beta_{13} - \beta_{14} ) q^{25} + ( \beta_{7} + \beta_{9} + \beta_{11} ) q^{27} + ( -2 + \beta_{4} + \beta_{6} - \beta_{8} + \beta_{12} - 2 \beta_{13} - \beta_{14} ) q^{29} + ( -\beta_{3} - 2 \beta_{15} ) q^{31} + ( -\beta_{3} - \beta_{5} - 2 \beta_{11} - \beta_{15} ) q^{33} + ( -2 \beta_{2} - \beta_{6} - \beta_{8} + \beta_{12} + \beta_{14} ) q^{37} + ( \beta_{2} - \beta_{4} + 2 \beta_{6} - \beta_{8} - \beta_{12} + \beta_{13} ) q^{39} + ( -\beta_{3} + \beta_{5} - \beta_{7} - \beta_{9} - 3 \beta_{10} + \beta_{11} - \beta_{15} ) q^{41} + ( \beta_{2} + \beta_{4} + \beta_{6} + \beta_{8} - \beta_{14} ) q^{43} + ( \beta_{1} + 2 \beta_{10} + 3 \beta_{15} ) q^{45} + ( 3 \beta_{1} - 2 \beta_{3} - \beta_{5} - 2 \beta_{7} + \beta_{9} - \beta_{11} - 2 \beta_{15} ) q^{47} + ( -1 - 2 \beta_{2} + \beta_{4} + \beta_{6} + 3 \beta_{12} + \beta_{14} ) q^{51} + ( -2 + \beta_{2} - 2 \beta_{4} - 2 \beta_{6} - \beta_{8} - \beta_{12} + \beta_{13} + 2 \beta_{14} ) q^{53} + ( -\beta_{1} + \beta_{3} + \beta_{5} + \beta_{7} - \beta_{9} - 2 \beta_{10} + \beta_{11} + \beta_{15} ) q^{55} + ( -4 - 2 \beta_{2} + 3 \beta_{6} + \beta_{12} + \beta_{13} + \beta_{14} ) q^{57} + ( \beta_{3} + \beta_{5} + \beta_{9} ) q^{59} + ( 2 \beta_{1} + 3 \beta_{3} + \beta_{5} - 2 \beta_{9} + 3 \beta_{10} + \beta_{11} ) q^{61} + ( 1 + \beta_{2} - 2 \beta_{4} + \beta_{6} - \beta_{12} + \beta_{13} + \beta_{14} ) q^{65} + ( -2 \beta_{2} + \beta_{4} - 2 \beta_{8} + 2 \beta_{14} ) q^{67} + ( -3 \beta_{3} - 3 \beta_{5} - 2 \beta_{7} + \beta_{9} - 2 \beta_{11} - 3 \beta_{15} ) q^{69} + ( 3 - 6 \beta_{6} + \beta_{12} - \beta_{14} ) q^{71} + ( -2 \beta_{3} - \beta_{5} - \beta_{10} - \beta_{11} + \beta_{15} ) q^{73} + ( -\beta_{1} - \beta_{3} - \beta_{5} + 2 \beta_{7} + 2 \beta_{9} - \beta_{10} - 4 \beta_{15} ) q^{75} + ( 1 + 2 \beta_{2} - \beta_{6} + \beta_{8} + 2 \beta_{12} - \beta_{14} ) q^{79} + ( 4 - \beta_{2} + \beta_{4} - 2 \beta_{6} - 2 \beta_{8} + \beta_{13} ) q^{81} + ( \beta_{3} - \beta_{5} - 2 \beta_{7} + \beta_{9} + 3 \beta_{10} - \beta_{11} - \beta_{15} ) q^{83} + ( 1 - \beta_{6} + \beta_{13} ) q^{85} + ( -3 \beta_{1} - 3 \beta_{3} - \beta_{7} + 2 \beta_{9} - 3 \beta_{10} - \beta_{11} - 6 \beta_{15} ) q^{87} + ( -3 \beta_{1} - \beta_{3} - 2 \beta_{5} - 3 \beta_{7} + 4 \beta_{9} + 3 \beta_{10} - 3 \beta_{11} - 3 \beta_{15} ) q^{89} + ( 1 + \beta_{4} + \beta_{6} - 2 \beta_{8} - 2 \beta_{13} - 2 \beta_{14} ) q^{93} + ( 4 + \beta_{2} + \beta_{4} - 2 \beta_{6} - 3 \beta_{8} - 2 \beta_{13} - 3 \beta_{14} ) q^{95} + ( \beta_{3} + \beta_{5} + 2 \beta_{7} - \beta_{9} + 4 \beta_{10} ) q^{97} + ( 5 + \beta_{2} - \beta_{4} + 2 \beta_{6} + 2 \beta_{8} - \beta_{13} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q + 6q^{9} + O(q^{10}) \) \( 16q + 6q^{9} - 12q^{15} + 16q^{25} - 12q^{29} - 2q^{37} + 18q^{39} + 4q^{43} + 6q^{51} - 36q^{53} - 42q^{57} + 24q^{65} + 14q^{67} + 20q^{79} + 54q^{81} + 6q^{85} + 30q^{93} + 60q^{95} + 90q^{99} + O(q^{100}) \)

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

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} \)
\(\beta_{3}\)\(=\)\((\)\( \nu^{15} + 15 \nu^{13} + 72 \nu^{11} + 153 \nu^{9} - 423 \nu^{7} - 891 \nu^{5} + 1944 \nu^{3} + 17496 \nu \)\()/15309\)
\(\beta_{4}\)\(=\)\((\)\( \nu^{14} + 15 \nu^{12} + 72 \nu^{10} + 153 \nu^{8} - 423 \nu^{6} - 891 \nu^{4} + 1944 \nu^{2} + 12393 \)\()/5103\)
\(\beta_{5}\)\(=\)\((\)\( -\nu^{15} + 48 \nu^{13} - 72 \nu^{11} + 414 \nu^{9} - 1845 \nu^{7} - 1944 \nu^{5} - 10449 \nu^{3} + 59049 \nu \)\()/15309\)
\(\beta_{6}\)\(=\)\((\)\( 5 \nu^{14} + 12 \nu^{12} - 18 \nu^{10} - 369 \nu^{8} + 153 \nu^{6} + 1782 \nu^{4} + 4617 \nu^{2} - 9477 \)\()/5103\)
\(\beta_{7}\)\(=\)\((\)\( 2 \nu^{15} + 9 \nu^{13} + 18 \nu^{11} - 72 \nu^{9} - 657 \nu^{7} + 297 \nu^{5} + 3888 \nu^{3} + 9477 \nu \)\()/5103\)
\(\beta_{8}\)\(=\)\((\)\( \nu^{14} - 3 \nu^{12} - 9 \nu^{10} - 9 \nu^{8} + 225 \nu^{6} - 81 \nu^{4} - 729 \nu^{2} - 2187 \)\()/729\)
\(\beta_{9}\)\(=\)\((\)\( 2 \nu^{15} + 3 \nu^{13} - 45 \nu^{11} - 99 \nu^{9} + 369 \nu^{7} + 1134 \nu^{5} - 6561 \nu \)\()/2187\)
\(\beta_{10}\)\(=\)\((\)\( -5 \nu^{15} - 12 \nu^{13} + 18 \nu^{11} + 369 \nu^{9} - 153 \nu^{7} - 1782 \nu^{5} - 4617 \nu^{3} + 9477 \nu \)\()/5103\)
\(\beta_{11}\)\(=\)\((\)\( -20 \nu^{15} - 48 \nu^{13} + 261 \nu^{11} + 909 \nu^{9} - 612 \nu^{7} - 8829 \nu^{5} + 3645 \nu^{3} + 17496 \nu \)\()/15309\)
\(\beta_{12}\)\(=\)\((\)\( -20 \nu^{14} + 15 \nu^{12} + 72 \nu^{10} + 342 \nu^{8} - 1179 \nu^{6} + 243 \nu^{4} - 1458 \nu^{2} + 2187 \)\()/5103\)
\(\beta_{13}\)\(=\)\((\)\( 23 \nu^{14} - 33 \nu^{12} - 234 \nu^{10} - 1017 \nu^{8} + 3879 \nu^{6} + 8424 \nu^{4} + 2187 \nu^{2} - 82377 \)\()/5103\)
\(\beta_{14}\)\(=\)\((\)\( \nu^{14} + \nu^{12} - 12 \nu^{10} - 36 \nu^{8} + 81 \nu^{6} + 306 \nu^{4} + 54 \nu^{2} - 1215 \)\()/189\)
\(\beta_{15}\)\(=\)\((\)\( 61 \nu^{15} - 30 \nu^{13} - 711 \nu^{11} - 2574 \nu^{9} + 8217 \nu^{7} + 18792 \nu^{5} + 1215 \nu^{3} - 157464 \nu \)\()/15309\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2}\)
\(\nu^{3}\)\(=\)\(\beta_{11} + \beta_{9} + \beta_{7}\)
\(\nu^{4}\)\(=\)\(\beta_{13} - 2 \beta_{8} - 2 \beta_{6} + \beta_{4} - \beta_{2} + 4\)
\(\nu^{5}\)\(=\)\(-\beta_{15} - 4 \beta_{11} + 3 \beta_{10} + \beta_{9} + \beta_{7} - 4 \beta_{5} + 2 \beta_{3} + 3 \beta_{1}\)
\(\nu^{6}\)\(=\)\(3 \beta_{12} + 6 \beta_{8} + 3 \beta_{6} + 3 \beta_{4} + 3 \beta_{2} + 15\)
\(\nu^{7}\)\(=\)\(3 \beta_{11} - 3 \beta_{10} + 3 \beta_{9} - 6 \beta_{7} + 9 \beta_{3} + 12 \beta_{1}\)
\(\nu^{8}\)\(=\)\(3 \beta_{14} + 3 \beta_{13} - 6 \beta_{8} - 24 \beta_{6} + 12 \beta_{4} + 9 \beta_{2} - 24\)
\(\nu^{9}\)\(=\)\(27 \beta_{10} + 18 \beta_{9} + 18 \beta_{7} - 9 \beta_{5} + 36 \beta_{3} - 36 \beta_{1}\)
\(\nu^{10}\)\(=\)\(-18 \beta_{14} + 18 \beta_{13} - 9 \beta_{8} + 18 \beta_{6} + 45 \beta_{4} - 45 \beta_{2} + 72\)
\(\nu^{11}\)\(=\)\(-9 \beta_{15} - 72 \beta_{11} - 54 \beta_{9} - 54 \beta_{7} - 63 \beta_{5} + 126 \beta_{3} + 27 \beta_{1}\)
\(\nu^{12}\)\(=\)\(54 \beta_{14} - 63 \beta_{13} + 108 \beta_{12} + 180 \beta_{8} + 153 \beta_{6} + 126 \beta_{4} + 36 \beta_{2} - 198\)
\(\nu^{13}\)\(=\)\(-45 \beta_{15} + 63 \beta_{11} - 216 \beta_{10} + 126 \beta_{9} - 198 \beta_{7} + 144 \beta_{5} + 333 \beta_{3} - 324 \beta_{1}\)
\(\nu^{14}\)\(=\)\(27 \beta_{14} + 81 \beta_{13} - 351 \beta_{12} - 378 \beta_{8} - 432 \beta_{6} + 297 \beta_{4} - 243 \beta_{2} - 1026\)
\(\nu^{15}\)\(=\)\(432 \beta_{15} + 513 \beta_{10} - 540 \beta_{9} + 513 \beta_{7} + 189 \beta_{5} + 1323 \beta_{3} - 1323 \beta_{1}\)

Character values

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

\(n\) \(785\) \(883\) \(1081\)
\(\chi(n)\) \(1 - \beta_{6}\) \(1\) \(\beta_{6}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1685.1
−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 −1.71965 + 0.206851i 0 4.18671 0 0 0 2.91443 0.711425i 0
1685.2 0 −1.69483 0.357142i 0 −2.42488 0 0 0 2.74490 + 1.21059i 0
1685.3 0 −0.744857 1.56371i 0 −0.553827 0 0 0 −1.89038 + 2.32948i 0
1685.4 0 −0.604587 + 1.62311i 0 0.533560 0 0 0 −2.26895 1.96262i 0
1685.5 0 0.604587 1.62311i 0 −0.533560 0 0 0 −2.26895 1.96262i 0
1685.6 0 0.744857 + 1.56371i 0 0.553827 0 0 0 −1.89038 + 2.32948i 0
1685.7 0 1.69483 + 0.357142i 0 2.42488 0 0 0 2.74490 + 1.21059i 0
1685.8 0 1.71965 0.206851i 0 −4.18671 0 0 0 2.91443 0.711425i 0
1697.1 0 −1.71965 0.206851i 0 4.18671 0 0 0 2.91443 + 0.711425i 0
1697.2 0 −1.69483 + 0.357142i 0 −2.42488 0 0 0 2.74490 1.21059i 0
1697.3 0 −0.744857 + 1.56371i 0 −0.553827 0 0 0 −1.89038 2.32948i 0
1697.4 0 −0.604587 1.62311i 0 0.533560 0 0 0 −2.26895 + 1.96262i 0
1697.5 0 0.604587 + 1.62311i 0 −0.533560 0 0 0 −2.26895 + 1.96262i 0
1697.6 0 0.744857 1.56371i 0 0.553827 0 0 0 −1.89038 2.32948i 0
1697.7 0 1.69483 0.357142i 0 2.42488 0 0 0 2.74490 1.21059i 0
1697.8 0 1.71965 + 0.206851i 0 −4.18671 0 0 0 2.91443 + 0.711425i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1697.8
Significant digits:
Format:

Inner twists

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

Twists

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{8} - 24 T_{5}^{6} + 117 T_{5}^{4} - 63 T_{5}^{2} + 9 \) acting on \(S_{2}^{\mathrm{new}}(1764, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \)
$3$ \( 6561 - 2187 T^{2} - 729 T^{4} - 81 T^{6} + 225 T^{8} - 9 T^{10} - 9 T^{12} - 3 T^{14} + T^{16} \)
$5$ \( ( 9 - 63 T^{2} + 117 T^{4} - 24 T^{6} + T^{8} )^{2} \)
$7$ \( T^{16} \)
$11$ \( ( 3969 + 3078 T^{2} + 639 T^{4} + 45 T^{6} + T^{8} )^{2} \)
$13$ \( 81 - 4536 T^{2} + 248994 T^{4} - 280368 T^{6} + 287163 T^{8} - 25776 T^{10} + 1746 T^{12} - 48 T^{14} + T^{16} \)
$17$ \( 810000 + 6066900 T^{2} + 44120781 T^{4} + 9748647 T^{6} + 1625391 T^{8} + 100944 T^{10} + 4617 T^{12} + 78 T^{14} + T^{16} \)
$19$ \( 810000 - 5208300 T^{2} + 32160969 T^{4} - 8406612 T^{6} + 1743651 T^{8} - 99126 T^{10} + 4149 T^{12} - 75 T^{14} + T^{16} \)
$23$ \( ( 50625 + 20250 T^{2} + 2151 T^{4} + 81 T^{6} + T^{8} )^{2} \)
$29$ \( ( 245025 - 236115 T + 38718 T^{2} + 35775 T^{3} + 4176 T^{4} - 450 T^{5} - 63 T^{6} + 6 T^{7} + T^{8} )^{2} \)
$31$ \( 531441 - 1240029 T^{2} + 2125764 T^{4} - 1686177 T^{6} + 985608 T^{8} - 72414 T^{10} + 4131 T^{12} - 72 T^{14} + T^{16} \)
$37$ \( ( 372100 - 14030 T + 40789 T^{2} + 298 T^{3} + 3769 T^{4} - 20 T^{5} + 67 T^{6} + T^{7} + T^{8} )^{2} \)
$41$ \( 331869318561 + 96021181080 T^{2} + 22187899809 T^{4} + 1414696806 T^{6} + 64225080 T^{8} + 1385487 T^{10} + 21618 T^{12} + 177 T^{14} + T^{16} \)
$43$ \( ( 461041 + 131047 T + 88174 T^{2} - 11759 T^{3} + 5332 T^{4} - 236 T^{5} + 79 T^{6} - 2 T^{7} + T^{8} )^{2} \)
$47$ \( 24685405970481 + 2494659194541 T^{2} + 168710132016 T^{4} + 6221777481 T^{6} + 165301362 T^{8} + 2722068 T^{10} + 32499 T^{12} + 222 T^{14} + T^{16} \)
$53$ \( ( 41990400 + 24669360 T + 5822523 T^{2} + 582471 T^{3} + 7047 T^{4} - 2754 T^{5} - 45 T^{6} + 18 T^{7} + T^{8} )^{2} \)
$59$ \( 194481 + 8037225 T^{2} + 331075026 T^{4} + 44366103 T^{6} + 4198680 T^{8} + 197694 T^{10} + 6777 T^{12} + 96 T^{14} + T^{16} \)
$61$ \( 1247449349924001 - 74160462871782 T^{2} + 2899235658015 T^{4} - 64949934240 T^{6} + 1054472814 T^{8} - 10802655 T^{10} + 80460 T^{12} - 351 T^{14} + T^{16} \)
$67$ \( ( 3940225 - 1992940 T + 787681 T^{2} - 139234 T^{3} + 21334 T^{4} - 1231 T^{5} + 160 T^{6} - 7 T^{7} + T^{8} )^{2} \)
$71$ \( ( 15876 + 97443 T^{2} + 11250 T^{4} + 207 T^{6} + T^{8} )^{2} \)
$73$ \( 5802782976 - 40182763824 T^{2} + 276760621881 T^{4} - 10312508844 T^{6} + 256685967 T^{8} - 3712662 T^{10} + 39429 T^{12} - 243 T^{14} + T^{16} \)
$79$ \( ( 319225 + 470645 T + 746434 T^{2} - 66169 T^{3} + 16414 T^{4} - 736 T^{5} + 193 T^{6} - 10 T^{7} + T^{8} )^{2} \)
$83$ \( 30237384321 + 13715668764 T^{2} + 4535917299 T^{4} + 671688342 T^{6} + 72720468 T^{8} + 2430279 T^{10} + 61596 T^{12} + 267 T^{14} + T^{16} \)
$89$ \( 44522973293864976 + 1990205599242420 T^{2} + 60745606555869 T^{4} + 987894749799 T^{6} + 11561010525 T^{8} + 67793598 T^{10} + 286173 T^{12} + 648 T^{14} + T^{16} \)
$97$ \( 83955602727441 - 9642663582291 T^{2} + 789930535230 T^{4} - 29657333385 T^{6} + 800598564 T^{8} - 10788390 T^{10} + 103725 T^{12} - 372 T^{14} + T^{16} \)
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