Properties

Label 189.2.p.d
Level 189
Weight 2
Character orbit 189.p
Analytic conductor 1.509
Analytic rank 0
Dimension 12
CM no
Inner twists 4

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Newspace parameters

Level: \( N \) = \( 189 = 3^{3} \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 189.p (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.50917259820\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 3^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(f(q)\) \(=\) \( q -\beta_{1} q^{2} + ( 1 - \beta_{3} - \beta_{7} + \beta_{9} ) q^{4} + ( \beta_{5} + \beta_{6} ) q^{5} + ( -1 + \beta_{4} - \beta_{9} ) q^{7} + ( \beta_{8} - \beta_{10} + \beta_{11} ) q^{8} +O(q^{10})\) \( q -\beta_{1} q^{2} + ( 1 - \beta_{3} - \beta_{7} + \beta_{9} ) q^{4} + ( \beta_{5} + \beta_{6} ) q^{5} + ( -1 + \beta_{4} - \beta_{9} ) q^{7} + ( \beta_{8} - \beta_{10} + \beta_{11} ) q^{8} + ( -2 \beta_{2} - 2 \beta_{4} + \beta_{7} + \beta_{9} ) q^{10} + ( \beta_{1} - \beta_{10} + \beta_{11} ) q^{11} + ( -1 + \beta_{2} + 2 \beta_{3} - \beta_{9} ) q^{13} + ( \beta_{6} - \beta_{8} + \beta_{10} - \beta_{11} ) q^{14} + ( 2 \beta_{2} - \beta_{3} + \beta_{4} - \beta_{7} ) q^{16} + ( -\beta_{1} + \beta_{11} ) q^{17} + ( 2 \beta_{7} - \beta_{9} ) q^{19} + ( \beta_{8} + \beta_{10} ) q^{20} + ( -3 + \beta_{2} + 2 \beta_{4} + 2 \beta_{7} - \beta_{9} ) q^{22} + ( -\beta_{1} + \beta_{5} - \beta_{6} - \beta_{8} ) q^{23} + ( -4 + \beta_{2} + 4 \beta_{3} - \beta_{4} - 2 \beta_{7} + 2 \beta_{9} ) q^{25} + ( -2 \beta_{1} - \beta_{5} - \beta_{6} - 4 \beta_{11} ) q^{26} + ( 2 - 2 \beta_{2} + \beta_{3} - \beta_{4} + 2 \beta_{7} - 2 \beta_{9} ) q^{28} + ( \beta_{8} - \beta_{10} - \beta_{11} ) q^{29} + ( -1 - \beta_{2} - \beta_{3} - \beta_{4} - \beta_{7} - \beta_{9} ) q^{31} + ( -2 \beta_{5} - \beta_{6} ) q^{32} + ( 3 - 6 \beta_{3} - \beta_{7} + 2 \beta_{9} ) q^{34} + ( 3 \beta_{1} - \beta_{6} - \beta_{8} + 2 \beta_{10} + 3 \beta_{11} ) q^{35} + ( -2 \beta_{2} + 2 \beta_{3} - \beta_{4} + \beta_{7} + \beta_{9} ) q^{37} + ( 2 \beta_{1} - 2 \beta_{8} + \beta_{10} - 2 \beta_{11} ) q^{38} + ( \beta_{4} - 3 \beta_{7} + \beta_{9} ) q^{40} + ( 2 \beta_{1} - \beta_{5} + \beta_{8} + \beta_{10} + \beta_{11} ) q^{41} + ( 1 + \beta_{2} + 2 \beta_{4} - \beta_{7} - \beta_{9} ) q^{43} + ( 6 \beta_{1} - \beta_{5} + \beta_{6} - 2 \beta_{8} ) q^{44} + ( 3 + \beta_{2} - 3 \beta_{3} - \beta_{4} ) q^{46} + ( 2 \beta_{1} - \beta_{5} - \beta_{6} + 4 \beta_{11} ) q^{47} + ( 4 - \beta_{2} - 3 \beta_{3} + \beta_{7} + 2 \beta_{9} ) q^{49} + ( -\beta_{5} - 2 \beta_{6} + 3 \beta_{8} - 3 \beta_{10} - \beta_{11} ) q^{50} + ( 4 + 4 \beta_{3} - \beta_{7} - \beta_{9} ) q^{52} + ( -2 \beta_{1} + 2 \beta_{5} + \beta_{6} - \beta_{10} - 2 \beta_{11} ) q^{53} + ( -\beta_{2} - \beta_{7} + 3 \beta_{9} ) q^{55} + ( -4 \beta_{1} - \beta_{6} - \beta_{8} + 2 \beta_{10} - 4 \beta_{11} ) q^{56} + ( 2 \beta_{2} + 3 \beta_{3} + \beta_{4} - \beta_{7} ) q^{58} + ( \beta_{1} - \beta_{6} - \beta_{11} ) q^{59} + ( 2 - \beta_{3} + 2 \beta_{4} + 2 \beta_{7} - 2 \beta_{9} ) q^{61} + ( -4 \beta_{1} + \beta_{5} + 2 \beta_{8} + 2 \beta_{10} - 2 \beta_{11} ) q^{62} + ( 2 - \beta_{7} ) q^{64} + ( -3 \beta_{1} - \beta_{5} + \beta_{6} - 3 \beta_{8} ) q^{65} + ( 7 + 2 \beta_{2} - 7 \beta_{3} - 2 \beta_{4} - \beta_{7} + \beta_{9} ) q^{67} + ( 3 \beta_{1} + \beta_{8} - 2 \beta_{10} + 6 \beta_{11} ) q^{68} + ( -9 - \beta_{2} - 3 \beta_{4} + 2 \beta_{7} ) q^{70} + ( \beta_{5} + 2 \beta_{6} - \beta_{11} ) q^{71} + ( -3 + 3 \beta_{2} - 3 \beta_{3} + 3 \beta_{4} - 2 \beta_{7} - 2 \beta_{9} ) q^{73} + ( -\beta_{1} + 2 \beta_{5} + \beta_{6} + \beta_{10} - \beta_{11} ) q^{74} + ( -6 - 3 \beta_{2} + 12 \beta_{3} + 3 \beta_{7} - 3 \beta_{9} ) q^{76} + ( -4 \beta_{1} - \beta_{5} - \beta_{6} + \beta_{8} + \beta_{10} ) q^{77} + ( -4 \beta_{2} - 5 \beta_{3} - 2 \beta_{4} + 2 \beta_{7} + \beta_{9} ) q^{79} + ( -3 \beta_{1} + \beta_{6} - 2 \beta_{8} + \beta_{10} + 3 \beta_{11} ) q^{80} + ( -6 + 3 \beta_{3} - \beta_{4} - \beta_{7} + \beta_{9} ) q^{82} + ( 4 \beta_{1} + \beta_{5} + 2 \beta_{11} ) q^{83} + ( -2 \beta_{2} - 4 \beta_{4} + \beta_{7} + 2 \beta_{9} ) q^{85} + ( -2 \beta_{1} - \beta_{5} + \beta_{6} - \beta_{8} ) q^{86} + ( -12 - 2 \beta_{2} + 12 \beta_{3} + 2 \beta_{4} + 5 \beta_{7} - 5 \beta_{9} ) q^{88} + ( -2 \beta_{1} + \beta_{8} - 2 \beta_{10} - 4 \beta_{11} ) q^{89} + ( -2 + 4 \beta_{2} + 4 \beta_{3} + 3 \beta_{4} - 2 \beta_{7} ) q^{91} + ( \beta_{5} + 2 \beta_{6} + 3 \beta_{8} - 3 \beta_{10} ) q^{92} + ( -6 + 2 \beta_{2} - 6 \beta_{3} + 2 \beta_{4} + \beta_{7} + \beta_{9} ) q^{94} + ( 2 \beta_{5} + \beta_{6} - 3 \beta_{10} ) q^{95} + ( 4 + 2 \beta_{2} - 8 \beta_{3} - 2 \beta_{9} ) q^{97} + ( 4 \beta_{1} + \beta_{5} + \beta_{6} - \beta_{8} - \beta_{10} + 7 \beta_{11} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12q + 8q^{4} - 8q^{7} + O(q^{10}) \) \( 12q + 8q^{4} - 8q^{7} - 6q^{10} - 4q^{16} - 6q^{19} - 40q^{22} - 24q^{25} + 28q^{28} - 12q^{31} + 8q^{37} + 12q^{40} + 20q^{43} + 14q^{46} + 24q^{49} + 78q^{52} + 20q^{58} + 18q^{61} + 28q^{64} + 36q^{67} - 120q^{70} - 42q^{73} - 36q^{79} - 54q^{82} - 12q^{85} - 74q^{88} + 6q^{91} - 114q^{94} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{12} - 9 x^{10} + 59 x^{8} - 180 x^{6} + 403 x^{4} - 198 x^{2} + 81\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( 81 \nu^{11} - 531 \nu^{9} + 3481 \nu^{7} - 3627 \nu^{5} + 1782 \nu^{3} + 76298 \nu \)\()/21995\)
\(\beta_{2}\)\(=\)\((\)\( 117 \nu^{10} - 767 \nu^{8} + 7472 \nu^{6} - 27234 \nu^{4} + 90554 \nu^{2} - 60864 \)\()/21995\)
\(\beta_{3}\)\(=\)\((\)\( -1298 \nu^{10} + 10953 \nu^{8} - 71803 \nu^{6} + 202311 \nu^{4} - 490451 \nu^{2} + 240966 \)\()/197955\)
\(\beta_{4}\)\(=\)\((\)\( 461 \nu^{10} - 5466 \nu^{8} + 28501 \nu^{6} - 98847 \nu^{4} + 142112 \nu^{2} - 186237 \)\()/65985\)
\(\beta_{5}\)\(=\)\((\)\( 1298 \nu^{11} - 10953 \nu^{9} + 71803 \nu^{7} - 202311 \nu^{5} + 490451 \nu^{3} - 438921 \nu \)\()/197955\)
\(\beta_{6}\)\(=\)\((\)\( 1298 \nu^{11} - 10953 \nu^{9} + 71803 \nu^{7} - 202311 \nu^{5} + 490451 \nu^{3} + 154944 \nu \)\()/197955\)
\(\beta_{7}\)\(=\)\((\)\( -288 \nu^{10} + 1888 \nu^{8} - 9933 \nu^{6} + 12896 \nu^{4} - 6336 \nu^{2} - 68439 \)\()/21995\)
\(\beta_{8}\)\(=\)\((\)\( -288 \nu^{11} + 1888 \nu^{9} - 9933 \nu^{7} + 12896 \nu^{5} - 6336 \nu^{3} - 90434 \nu \)\()/21995\)
\(\beta_{9}\)\(=\)\((\)\( 1138 \nu^{10} - 12348 \nu^{8} + 80948 \nu^{6} - 273351 \nu^{4} + 552916 \nu^{2} - 271656 \)\()/65985\)
\(\beta_{10}\)\(=\)\((\)\( 4712 \nu^{11} - 47997 \nu^{9} + 314647 \nu^{7} - 1022364 \nu^{5} + 2149199 \nu^{3} - 1055934 \nu \)\()/197955\)
\(\beta_{11}\)\(=\)\((\)\( -5192 \nu^{11} + 43812 \nu^{9} - 287212 \nu^{7} + 809244 \nu^{5} - 1763849 \nu^{3} + 172044 \nu \)\()/197955\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{6} - \beta_{5}\)\()/3\)
\(\nu^{2}\)\(=\)\((\)\(-2 \beta_{9} + 2 \beta_{7} + \beta_{4} - 9 \beta_{3} - \beta_{2} + 9\)\()/3\)
\(\nu^{3}\)\(=\)\((\)\(3 \beta_{11} + 8 \beta_{6} + 4 \beta_{5}\)\()/3\)
\(\nu^{4}\)\(=\)\(-\beta_{9} + 2 \beta_{7} - 2 \beta_{4} - 12 \beta_{3} - 4 \beta_{2}\)
\(\nu^{5}\)\(=\)\((\)\(18 \beta_{11} + 3 \beta_{10} + 17 \beta_{6} + 34 \beta_{5} + 18 \beta_{1}\)\()/3\)
\(\nu^{6}\)\(=\)\((\)\(32 \beta_{9} - 5 \beta_{7} - 64 \beta_{4} - 32 \beta_{2} - 153\)\()/3\)
\(\nu^{7}\)\(=\)\((\)\(27 \beta_{8} - 74 \beta_{6} + 74 \beta_{5} + 96 \beta_{1}\)\()/3\)
\(\nu^{8}\)\(=\)\(51 \beta_{9} - 51 \beta_{7} - 55 \beta_{4} + 222 \beta_{3} + 55 \beta_{2} - 222\)
\(\nu^{9}\)\(=\)\((\)\(-495 \beta_{11} - 177 \beta_{10} + 177 \beta_{8} - 656 \beta_{6} - 328 \beta_{5}\)\()/3\)
\(\nu^{10}\)\(=\)\((\)\(-191 \beta_{9} - 835 \beta_{7} + 835 \beta_{4} + 2952 \beta_{3} + 1670 \beta_{2}\)\()/3\)
\(\nu^{11}\)\(=\)\((\)\(-2505 \beta_{11} - 1026 \beta_{10} - 1477 \beta_{6} - 2954 \beta_{5} - 2505 \beta_{1}\)\()/3\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/189\mathbb{Z}\right)^\times\).

\(n\) \(29\) \(136\)
\(\chi(n)\) \(-1\) \(\beta_{3}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
26.1
0.617942 + 0.356769i
−1.90412 1.09935i
1.65604 + 0.956115i
−1.65604 0.956115i
1.90412 + 1.09935i
−0.617942 0.356769i
0.617942 0.356769i
−1.90412 + 1.09935i
1.65604 0.956115i
−1.65604 + 0.956115i
1.90412 1.09935i
−0.617942 + 0.356769i
−2.15715 1.24543i 0 2.10220 + 3.64112i −0.617942 + 1.07031i 0 −1.53189 2.15715i 5.49086i 0 2.66599 1.53921i
26.2 −1.58850 0.917122i 0 0.682224 + 1.18165i 1.90412 3.29804i 0 2.11581 + 1.58850i 1.16576i 0 −6.04940 + 3.49262i
26.3 −0.568650 0.328310i 0 −0.784425 1.35866i −1.65604 + 2.86834i 0 −2.58392 + 0.568650i 2.34338i 0 1.88341 1.08739i
26.4 0.568650 + 0.328310i 0 −0.784425 1.35866i 1.65604 2.86834i 0 −2.58392 + 0.568650i 2.34338i 0 1.88341 1.08739i
26.5 1.58850 + 0.917122i 0 0.682224 + 1.18165i −1.90412 + 3.29804i 0 2.11581 + 1.58850i 1.16576i 0 −6.04940 + 3.49262i
26.6 2.15715 + 1.24543i 0 2.10220 + 3.64112i 0.617942 1.07031i 0 −1.53189 2.15715i 5.49086i 0 2.66599 1.53921i
80.1 −2.15715 + 1.24543i 0 2.10220 3.64112i −0.617942 1.07031i 0 −1.53189 + 2.15715i 5.49086i 0 2.66599 + 1.53921i
80.2 −1.58850 + 0.917122i 0 0.682224 1.18165i 1.90412 + 3.29804i 0 2.11581 1.58850i 1.16576i 0 −6.04940 3.49262i
80.3 −0.568650 + 0.328310i 0 −0.784425 + 1.35866i −1.65604 2.86834i 0 −2.58392 0.568650i 2.34338i 0 1.88341 + 1.08739i
80.4 0.568650 0.328310i 0 −0.784425 + 1.35866i 1.65604 + 2.86834i 0 −2.58392 0.568650i 2.34338i 0 1.88341 + 1.08739i
80.5 1.58850 0.917122i 0 0.682224 1.18165i −1.90412 3.29804i 0 2.11581 1.58850i 1.16576i 0 −6.04940 3.49262i
80.6 2.15715 1.24543i 0 2.10220 3.64112i 0.617942 + 1.07031i 0 −1.53189 + 2.15715i 5.49086i 0 2.66599 + 1.53921i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 80.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
7.d odd 6 1 inner
21.g even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 189.2.p.d 12
3.b odd 2 1 inner 189.2.p.d 12
7.c even 3 1 1323.2.c.d 12
7.d odd 6 1 inner 189.2.p.d 12
7.d odd 6 1 1323.2.c.d 12
9.c even 3 1 567.2.i.f 12
9.c even 3 1 567.2.s.f 12
9.d odd 6 1 567.2.i.f 12
9.d odd 6 1 567.2.s.f 12
21.g even 6 1 inner 189.2.p.d 12
21.g even 6 1 1323.2.c.d 12
21.h odd 6 1 1323.2.c.d 12
63.i even 6 1 567.2.s.f 12
63.k odd 6 1 567.2.i.f 12
63.s even 6 1 567.2.i.f 12
63.t odd 6 1 567.2.s.f 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
189.2.p.d 12 1.a even 1 1 trivial
189.2.p.d 12 3.b odd 2 1 inner
189.2.p.d 12 7.d odd 6 1 inner
189.2.p.d 12 21.g even 6 1 inner
567.2.i.f 12 9.c even 3 1
567.2.i.f 12 9.d odd 6 1
567.2.i.f 12 63.k odd 6 1
567.2.i.f 12 63.s even 6 1
567.2.s.f 12 9.c even 3 1
567.2.s.f 12 9.d odd 6 1
567.2.s.f 12 63.i even 6 1
567.2.s.f 12 63.t odd 6 1
1323.2.c.d 12 7.c even 3 1
1323.2.c.d 12 7.d odd 6 1
1323.2.c.d 12 21.g even 6 1
1323.2.c.d 12 21.h odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{12} - 10 T_{2}^{10} + 75 T_{2}^{8} - 232 T_{2}^{6} + 535 T_{2}^{4} - 225 T_{2}^{2} + 81 \) acting on \(S_{2}^{\mathrm{new}}(189, [\chi])\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 + 2 T^{2} - T^{4} - 12 T^{6} - 17 T^{8} + 7 T^{10} + 85 T^{12} + 28 T^{14} - 272 T^{16} - 768 T^{18} - 256 T^{20} + 2048 T^{22} + 4096 T^{24} \)
$3$ \( \)
$5$ \( 1 - 3 T^{2} - 24 T^{4} + 275 T^{6} - 297 T^{8} - 3096 T^{10} + 33369 T^{12} - 77400 T^{14} - 185625 T^{16} + 4296875 T^{18} - 9375000 T^{20} - 29296875 T^{22} + 244140625 T^{24} \)
$7$ \( ( 1 + 4 T + 2 T^{2} - 11 T^{3} + 14 T^{4} + 196 T^{5} + 343 T^{6} )^{2} \)
$11$ \( 1 + 29 T^{2} + 440 T^{4} + 3147 T^{6} - 10709 T^{8} - 688988 T^{10} - 10240799 T^{12} - 83367548 T^{14} - 156790469 T^{16} + 5575102467 T^{18} + 94317907640 T^{20} + 752185313429 T^{22} + 3138428376721 T^{24} \)
$13$ \( ( 1 - 36 T^{2} + 720 T^{4} - 10271 T^{6} + 121680 T^{8} - 1028196 T^{10} + 4826809 T^{12} )^{2} \)
$17$ \( 1 - 72 T^{2} + 2664 T^{4} - 74146 T^{6} + 1759536 T^{8} - 36331632 T^{10} + 658037283 T^{12} - 10499841648 T^{14} + 146958206256 T^{16} - 1789704191074 T^{18} + 18583417822824 T^{20} - 145151560832328 T^{22} + 582622237229761 T^{24} \)
$19$ \( ( 1 + 3 T + 36 T^{2} + 99 T^{3} + 549 T^{4} + 1788 T^{5} + 8413 T^{6} + 33972 T^{7} + 198189 T^{8} + 679041 T^{9} + 4691556 T^{10} + 7428297 T^{11} + 47045881 T^{12} )^{2} \)
$23$ \( 1 + 44 T^{2} + 2132 T^{4} + 55302 T^{6} + 1548472 T^{8} + 27701944 T^{10} + 755610907 T^{12} + 14654328376 T^{14} + 433325952952 T^{16} + 8186680733478 T^{18} + 166959020619092 T^{20} + 1822766493400556 T^{22} + 21914624432020321 T^{24} \)
$29$ \( ( 1 - 137 T^{2} + 8645 T^{4} - 319673 T^{6} + 7270445 T^{8} - 96897497 T^{10} + 594823321 T^{12} )^{2} \)
$31$ \( ( 1 + 6 T + 42 T^{2} + 180 T^{3} - 168 T^{4} - 7836 T^{5} - 41627 T^{6} - 242916 T^{7} - 161448 T^{8} + 5362380 T^{9} + 38787882 T^{10} + 171774906 T^{11} + 887503681 T^{12} )^{2} \)
$37$ \( ( 1 - 4 T - 76 T^{2} + 90 T^{3} + 4144 T^{4} + 688 T^{5} - 181325 T^{6} + 25456 T^{7} + 5673136 T^{8} + 4558770 T^{9} - 142436236 T^{10} - 277375828 T^{11} + 2565726409 T^{12} )^{2} \)
$41$ \( ( 1 + 132 T^{2} + 8472 T^{4} + 388519 T^{6} + 14241432 T^{8} + 373000452 T^{10} + 4750104241 T^{12} )^{2} \)
$43$ \( ( 1 - 5 T + 113 T^{2} - 431 T^{3} + 4859 T^{4} - 9245 T^{5} + 79507 T^{6} )^{4} \)
$47$ \( 1 - 147 T^{2} + 9984 T^{4} - 446341 T^{6} + 16664751 T^{8} - 505982232 T^{10} + 16085788593 T^{12} - 1117714750488 T^{14} + 81318668824431 T^{16} - 4811205749161189 T^{18} + 237731886031021824 T^{20} - 7732072438667017203 T^{22} + \)\(11\!\cdots\!41\)\( T^{24} \)
$53$ \( 1 + 200 T^{2} + 21164 T^{4} + 1425678 T^{6} + 67732372 T^{8} + 2466420796 T^{10} + 102771938515 T^{12} + 6928176015964 T^{14} + 534440994350932 T^{16} + 31599242045670462 T^{18} + 1317664087866044204 T^{20} + 34977494073102609800 T^{22} + \)\(49\!\cdots\!41\)\( T^{24} \)
$59$ \( 1 - 291 T^{2} + 46596 T^{4} - 5329417 T^{6} + 480785571 T^{8} - 35925708996 T^{10} + 2284341412497 T^{12} - 125057393015076 T^{14} + 5825852327398131 T^{16} - 224797653055417297 T^{18} + 6841711070610941316 T^{20} - \)\(14\!\cdots\!91\)\( T^{22} + \)\(17\!\cdots\!81\)\( T^{24} \)
$61$ \( ( 1 - 9 T + 159 T^{2} - 1188 T^{3} + 11505 T^{4} - 60363 T^{5} + 676942 T^{6} - 3682143 T^{7} + 42810105 T^{8} - 269653428 T^{9} + 2201488719 T^{10} - 7601366709 T^{11} + 51520374361 T^{12} )^{2} \)
$67$ \( ( 1 - 18 T + 108 T^{2} - 418 T^{3} + 954 T^{4} + 65430 T^{5} - 968637 T^{6} + 4383810 T^{7} + 4282506 T^{8} - 125718934 T^{9} + 2176321068 T^{10} - 24302251926 T^{11} + 90458382169 T^{12} )^{2} \)
$71$ \( ( 1 - 341 T^{2} + 53765 T^{4} - 4892609 T^{6} + 271029365 T^{8} - 8665383221 T^{10} + 128100283921 T^{12} )^{2} \)
$73$ \( ( 1 + 21 T + 342 T^{2} + 4095 T^{3} + 40149 T^{4} + 343806 T^{5} + 3034951 T^{6} + 25097838 T^{7} + 213954021 T^{8} + 1593024615 T^{9} + 9712198422 T^{10} + 43534503453 T^{11} + 151334226289 T^{12} )^{2} \)
$79$ \( ( 1 + 18 T + 72 T^{2} - 1138 T^{3} - 7470 T^{4} + 110574 T^{5} + 1995747 T^{6} + 8735346 T^{7} - 46620270 T^{8} - 561078382 T^{9} + 2804405832 T^{10} + 55387015182 T^{11} + 243087455521 T^{12} )^{2} \)
$83$ \( ( 1 + 339 T^{2} + 56361 T^{4} + 5822683 T^{6} + 388270929 T^{8} + 16088370819 T^{10} + 326940373369 T^{12} )^{2} \)
$89$ \( 1 - 333 T^{2} + 53955 T^{4} - 6596860 T^{6} + 732206169 T^{8} - 72458005383 T^{10} + 6566534648358 T^{12} - 573939860638743 T^{14} + 45940255917084729 T^{16} - 3278515999088982460 T^{18} + \)\(21\!\cdots\!55\)\( T^{20} - \)\(10\!\cdots\!33\)\( T^{22} + \)\(24\!\cdots\!21\)\( T^{24} \)
$97$ \( ( 1 - 378 T^{2} + 65727 T^{4} - 7461452 T^{6} + 618425343 T^{8} - 33464068218 T^{10} + 832972004929 T^{12} )^{2} \)
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