Properties

Label 189.2.o.a
Level 189
Weight 2
Character orbit 189.o
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.o (of order \(6\), degree \(2\), not 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: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 3^{2} \)
Twist minimal: no (minimal twist has level 63)
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.

\(f(q)\) \(=\) \( q + ( \beta_{1} - \beta_{3} - \beta_{5} ) q^{2} + ( -\beta_{3} - \beta_{5} - \beta_{8} ) q^{4} + \beta_{10} q^{5} + ( \beta_{3} - \beta_{4} - \beta_{8} + \beta_{9} ) q^{7} + ( 1 - \beta_{1} + \beta_{3} - \beta_{4} - \beta_{5} - 2 \beta_{8} ) q^{8} +O(q^{10})\) \( q + ( \beta_{1} - \beta_{3} - \beta_{5} ) q^{2} + ( -\beta_{3} - \beta_{5} - \beta_{8} ) q^{4} + \beta_{10} q^{5} + ( \beta_{3} - \beta_{4} - \beta_{8} + \beta_{9} ) q^{7} + ( 1 - \beta_{1} + \beta_{3} - \beta_{4} - \beta_{5} - 2 \beta_{8} ) q^{8} + ( \beta_{2} + \beta_{6} - \beta_{9} - \beta_{11} ) q^{10} + ( \beta_{3} - \beta_{4} + \beta_{5} + \beta_{8} ) q^{11} + ( -\beta_{2} - 2 \beta_{7} - \beta_{10} ) q^{13} + ( 1 - \beta_{1} - \beta_{2} + \beta_{5} + 2 \beta_{9} - \beta_{10} ) q^{14} + ( -\beta_{3} + \beta_{4} + \beta_{8} ) q^{16} + ( \beta_{6} + \beta_{11} ) q^{17} + ( \beta_{2} - \beta_{6} + \beta_{7} - \beta_{9} + 2 \beta_{10} + \beta_{11} ) q^{19} + ( 2 \beta_{2} - \beta_{9} ) q^{20} + ( -1 + 2 \beta_{3} + 3 \beta_{5} + 2 \beta_{8} ) q^{22} + ( -1 - \beta_{1} - \beta_{5} ) q^{23} + ( \beta_{1} + \beta_{3} ) q^{25} + ( -2 \beta_{2} - \beta_{6} - \beta_{7} - 2 \beta_{9} - \beta_{11} ) q^{26} + ( -\beta_{1} - 2 \beta_{2} + \beta_{4} - \beta_{6} - \beta_{7} + 2 \beta_{9} - 2 \beta_{10} + \beta_{11} ) q^{28} + ( -4 + \beta_{1} - \beta_{3} + \beta_{5} ) q^{29} + ( 2 \beta_{6} + 2 \beta_{7} + \beta_{10} ) q^{31} + ( 1 + \beta_{1} - 2 \beta_{3} + 4 \beta_{4} - \beta_{5} + 2 \beta_{8} ) q^{32} + ( \beta_{7} - \beta_{10} ) q^{34} + ( 1 + \beta_{1} + \beta_{3} + \beta_{4} - \beta_{5} - \beta_{6} + \beta_{7} + 2 \beta_{8} - \beta_{11} ) q^{35} + ( -1 + \beta_{1} - \beta_{4} ) q^{37} + ( 4 \beta_{2} + 2 \beta_{6} - 2 \beta_{9} - \beta_{11} ) q^{38} + ( \beta_{2} - 2 \beta_{6} + 2 \beta_{7} + \beta_{10} ) q^{40} + ( -\beta_{2} + 2 \beta_{9} - 2 \beta_{10} ) q^{41} + ( -2 \beta_{1} - 3 \beta_{3} + \beta_{4} - 2 \beta_{5} + \beta_{8} ) q^{43} + ( -2 + 3 \beta_{3} + 7 \beta_{5} ) q^{44} + ( -2 - 3 \beta_{1} + 2 \beta_{3} - \beta_{4} + 2 \beta_{5} ) q^{46} + ( -2 \beta_{2} - 2 \beta_{6} + \beta_{7} + \beta_{9} + \beta_{10} + \beta_{11} ) q^{47} + ( -1 + 2 \beta_{1} - 4 \beta_{3} - 3 \beta_{5} - 2 \beta_{6} ) q^{49} + ( 2 + \beta_{1} - \beta_{3} + 2 \beta_{4} + \beta_{5} + \beta_{8} ) q^{50} + ( -\beta_{7} - 5 \beta_{9} + \beta_{10} - \beta_{11} ) q^{52} + ( -1 - 2 \beta_{1} - \beta_{3} - 2 \beta_{4} + \beta_{5} - 4 \beta_{8} ) q^{53} + ( -2 \beta_{2} + \beta_{6} + 2 \beta_{9} - \beta_{11} ) q^{55} + ( -6 + 2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} + \beta_{5} - 2 \beta_{6} - \beta_{7} + \beta_{9} - \beta_{10} + \beta_{11} ) q^{56} + ( 2 - 2 \beta_{1} + 3 \beta_{3} + \beta_{5} - \beta_{8} ) q^{58} + ( 2 \beta_{2} - \beta_{6} - 4 \beta_{9} + \beta_{10} + 2 \beta_{11} ) q^{59} + ( \beta_{9} + \beta_{11} ) q^{61} + ( \beta_{2} + \beta_{6} + 2 \beta_{7} + \beta_{9} + \beta_{11} ) q^{62} + ( -2 + 7 \beta_{1} - 4 \beta_{3} + \beta_{4} - 4 \beta_{5} ) q^{64} + ( 10 - 2 \beta_{1} + \beta_{3} + \beta_{4} - 4 \beta_{5} - \beta_{8} ) q^{65} + ( 1 + 2 \beta_{1} - \beta_{3} - 2 \beta_{5} + 3 \beta_{8} ) q^{67} + ( -\beta_{2} + \beta_{6} + 2 \beta_{9} - 2 \beta_{11} ) q^{68} + ( 2 \beta_{1} - \beta_{3} + 3 \beta_{4} + 2 \beta_{5} - \beta_{7} + 3 \beta_{8} + \beta_{9} + \beta_{10} + \beta_{11} ) q^{70} + ( 2 + 2 \beta_{1} - 3 \beta_{3} + 2 \beta_{4} - 7 \beta_{5} + 4 \beta_{8} ) q^{71} + ( 3 \beta_{2} + 3 \beta_{6} - \beta_{7} - 3 \beta_{9} - 2 \beta_{10} - 3 \beta_{11} ) q^{73} + ( 2 - \beta_{1} + \beta_{3} ) q^{74} + ( 4 \beta_{2} + 2 \beta_{6} + 2 \beta_{7} + \beta_{10} ) q^{76} + ( 1 + \beta_{2} + \beta_{3} - 2 \beta_{4} + 2 \beta_{5} + \beta_{6} - \beta_{8} - 2 \beta_{9} + 2 \beta_{10} - 2 \beta_{11} ) q^{77} + ( -\beta_{1} + 2 \beta_{3} - 3 \beta_{4} - 3 \beta_{8} ) q^{79} + ( \beta_{6} - \beta_{7} + \beta_{11} ) q^{80} + ( -5 \beta_{2} - 2 \beta_{6} - \beta_{7} + 5 \beta_{9} - 2 \beta_{10} + 2 \beta_{11} ) q^{82} + ( -4 \beta_{2} + 2 \beta_{6} - 2 \beta_{7} + 2 \beta_{9} - 2 \beta_{10} - \beta_{11} ) q^{83} + ( 3 - 6 \beta_{1} + 6 \beta_{3} + 3 \beta_{5} - 6 \beta_{8} ) q^{85} + ( -5 - 3 \beta_{1} + 2 \beta_{3} - 4 \beta_{4} - 3 \beta_{5} - 2 \beta_{8} ) q^{86} + ( \beta_{1} + 2 \beta_{3} - \beta_{4} + 6 \beta_{5} - \beta_{8} ) q^{88} + ( 3 \beta_{2} + 3 \beta_{9} ) q^{89} + ( -5 \beta_{1} + 4 \beta_{3} - 3 \beta_{4} + 4 \beta_{5} - 2 \beta_{6} + \beta_{7} + 2 \beta_{10} + 2 \beta_{11} ) q^{91} + ( -2 - 2 \beta_{1} + 4 \beta_{3} - 2 \beta_{4} + 5 \beta_{5} + 2 \beta_{8} ) q^{92} + ( -2 \beta_{2} + \beta_{6} - 4 \beta_{7} - 2 \beta_{10} ) q^{94} + ( -6 + 3 \beta_{1} + 3 \beta_{3} - 6 \beta_{4} - 3 \beta_{5} - 3 \beta_{8} ) q^{95} + ( -\beta_{7} + \beta_{10} - \beta_{11} ) q^{97} + ( 4 - 2 \beta_{1} - \beta_{3} - 2 \beta_{4} - 9 \beta_{5} - 2 \beta_{7} - 4 \beta_{8} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12q + 6q^{2} + 2q^{4} - 2q^{7} + O(q^{10}) \) \( 12q + 6q^{2} + 2q^{4} - 2q^{7} + 12q^{14} + 2q^{16} - 10q^{22} - 24q^{23} - 8q^{28} - 30q^{29} + 12q^{32} - 4q^{37} - 10q^{43} - 40q^{46} + 6q^{49} + 36q^{50} - 42q^{56} + 2q^{58} + 16q^{64} + 78q^{65} + 12q^{67} + 18q^{70} + 12q^{74} + 24q^{77} - 6q^{79} - 6q^{85} - 96q^{86} + 34q^{88} - 24q^{91} - 30q^{92} - 72q^{95} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{12} - 7 x^{10} + 37 x^{8} - 78 x^{6} + 123 x^{4} - 36 x^{2} + 9\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( -28 \nu^{10} + 148 \nu^{8} + 446 \nu^{6} - 3807 \nu^{4} + 17052 \nu^{2} + 4446 \)\()/12897\)
\(\beta_{2}\)\(=\)\((\)\( -49 \nu^{11} + 259 \nu^{9} - 1369 \nu^{7} + 861 \nu^{5} - 252 \nu^{3} - 15864 \nu \)\()/4299\)
\(\beta_{3}\)\(=\)\((\)\( -164 \nu^{10} + 1481 \nu^{8} - 7214 \nu^{6} + 17007 \nu^{4} - 16197 \nu^{2} - 3438 \)\()/12897\)
\(\beta_{4}\)\(=\)\((\)\( -175 \nu^{10} + 925 \nu^{8} - 3661 \nu^{6} - 1224 \nu^{4} + 16296 \nu^{2} - 30249 \)\()/12897\)
\(\beta_{5}\)\(=\)\((\)\( 148 \nu^{10} - 987 \nu^{8} + 5217 \nu^{6} - 10175 \nu^{4} + 17343 \nu^{2} - 777 \)\()/4299\)
\(\beta_{6}\)\(=\)\((\)\( 70 \nu^{11} - 370 \nu^{9} + 1751 \nu^{7} - 1230 \nu^{5} + 360 \nu^{3} + 8947 \nu \)\()/1433\)
\(\beta_{7}\)\(=\)\((\)\( -730 \nu^{11} + 5701 \nu^{9} - 30748 \nu^{7} + 76698 \nu^{5} - 122898 \nu^{3} + 66168 \nu \)\()/12897\)
\(\beta_{8}\)\(=\)\((\)\( -877 \nu^{10} + 6478 \nu^{8} - 34855 \nu^{6} + 79281 \nu^{4} - 123654 \nu^{2} + 31473 \)\()/12897\)
\(\beta_{9}\)\(=\)\((\)\( 543 \nu^{11} - 3689 \nu^{9} + 19499 \nu^{7} - 39839 \nu^{5} + 64821 \nu^{3} - 18972 \nu \)\()/4299\)
\(\beta_{10}\)\(=\)\((\)\( 2237 \nu^{11} - 15509 \nu^{9} + 81362 \nu^{7} - 167049 \nu^{5} + 249792 \nu^{3} - 42912 \nu \)\()/12897\)
\(\beta_{11}\)\(=\)\((\)\( -890 \nu^{11} + 6342 \nu^{9} - 33522 \nu^{7} + 71935 \nu^{5} - 111438 \nu^{3} + 32616 \nu \)\()/4299\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-\beta_{10} - 2 \beta_{7} + \beta_{6} - \beta_{2}\)\()/3\)
\(\nu^{2}\)\(=\)\(\beta_{8} + 2 \beta_{5} + \beta_{4} - \beta_{3}\)
\(\nu^{3}\)\(=\)\((\)\(-4 \beta_{11} - 8 \beta_{10} + \beta_{9} - 4 \beta_{7} + 4 \beta_{6} - \beta_{2}\)\()/3\)
\(\nu^{4}\)\(=\)\(4 \beta_{8} + 6 \beta_{5} - 6 \beta_{3} + 5 \beta_{1} - 12\)
\(\nu^{5}\)\(=\)\((\)\(-19 \beta_{11} - 16 \beta_{10} - 2 \beta_{9} + 16 \beta_{7}\)\()/3\)
\(\nu^{6}\)\(=\)\(-7 \beta_{5} - 16 \beta_{4} - 7 \beta_{3} + 30 \beta_{1} - 51\)
\(\nu^{7}\)\(=\)\((\)\(67 \beta_{10} + 134 \beta_{7} - 88 \beta_{6} - 23 \beta_{2}\)\()/3\)
\(\nu^{8}\)\(=\)\(-67 \beta_{8} - 118 \beta_{5} - 67 \beta_{4} + 104 \beta_{3} + 37 \beta_{1}\)
\(\nu^{9}\)\(=\)\((\)\(400 \beta_{11} + 578 \beta_{10} + 134 \beta_{9} + 289 \beta_{7} - 400 \beta_{6} - 134 \beta_{2}\)\()/3\)
\(\nu^{10}\)\(=\)\(-289 \beta_{8} - 333 \beta_{5} + 645 \beta_{3} - 467 \beta_{1} + 978\)
\(\nu^{11}\)\(=\)\((\)\(1801 \beta_{11} + 1267 \beta_{10} + 668 \beta_{9} - 1267 \beta_{7}\)\()/3\)

Character values

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

\(n\) \(29\) \(136\)
\(\chi(n)\) \(1 - \beta_{5}\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
62.1
−1.82904 + 1.05600i
1.82904 1.05600i
0.474636 0.274031i
−0.474636 + 0.274031i
−1.29589 + 0.748185i
1.29589 0.748185i
−1.82904 1.05600i
1.82904 + 1.05600i
0.474636 + 0.274031i
−0.474636 0.274031i
−1.29589 0.748185i
1.29589 + 0.748185i
−1.02704 0.592963i 0 −0.296790 0.514055i −1.41899 2.45776i 0 −2.07253 + 1.64457i 3.07579i 0 3.36562i
62.2 −1.02704 0.592963i 0 −0.296790 0.514055i 1.41899 + 2.45776i 0 −0.387972 + 2.61715i 3.07579i 0 3.36562i
62.3 0.555632 + 0.320794i 0 −0.794182 1.37556i −1.10552 1.91482i 0 −0.906161 2.48573i 2.30225i 0 1.41858i
62.4 0.555632 + 0.320794i 0 −0.794182 1.37556i 1.10552 + 1.91482i 0 2.60579 0.458109i 2.30225i 0 1.41858i
62.5 1.97141 + 1.13819i 0 1.59097 + 2.75564i −0.717144 1.24213i 0 2.16235 + 1.52455i 2.69056i 0 3.26499i
62.6 1.97141 + 1.13819i 0 1.59097 + 2.75564i 0.717144 + 1.24213i 0 −2.40147 1.11037i 2.69056i 0 3.26499i
125.1 −1.02704 + 0.592963i 0 −0.296790 + 0.514055i −1.41899 + 2.45776i 0 −2.07253 1.64457i 3.07579i 0 3.36562i
125.2 −1.02704 + 0.592963i 0 −0.296790 + 0.514055i 1.41899 2.45776i 0 −0.387972 2.61715i 3.07579i 0 3.36562i
125.3 0.555632 0.320794i 0 −0.794182 + 1.37556i −1.10552 + 1.91482i 0 −0.906161 + 2.48573i 2.30225i 0 1.41858i
125.4 0.555632 0.320794i 0 −0.794182 + 1.37556i 1.10552 1.91482i 0 2.60579 + 0.458109i 2.30225i 0 1.41858i
125.5 1.97141 1.13819i 0 1.59097 2.75564i −0.717144 + 1.24213i 0 2.16235 1.52455i 2.69056i 0 3.26499i
125.6 1.97141 1.13819i 0 1.59097 2.75564i 0.717144 1.24213i 0 −2.40147 + 1.11037i 2.69056i 0 3.26499i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 125.6
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 189.2.o.a 12
3.b odd 2 1 63.2.o.a 12
4.b odd 2 1 3024.2.cc.a 12
7.b odd 2 1 inner 189.2.o.a 12
7.c even 3 1 1323.2.i.c 12
7.c even 3 1 1323.2.s.c 12
7.d odd 6 1 1323.2.i.c 12
7.d odd 6 1 1323.2.s.c 12
9.c even 3 1 63.2.o.a 12
9.c even 3 1 567.2.c.c 12
9.d odd 6 1 inner 189.2.o.a 12
9.d odd 6 1 567.2.c.c 12
12.b even 2 1 1008.2.cc.a 12
21.c even 2 1 63.2.o.a 12
21.g even 6 1 441.2.i.c 12
21.g even 6 1 441.2.s.c 12
21.h odd 6 1 441.2.i.c 12
21.h odd 6 1 441.2.s.c 12
28.d even 2 1 3024.2.cc.a 12
36.f odd 6 1 1008.2.cc.a 12
36.h even 6 1 3024.2.cc.a 12
63.g even 3 1 441.2.i.c 12
63.h even 3 1 441.2.s.c 12
63.i even 6 1 1323.2.s.c 12
63.j odd 6 1 1323.2.s.c 12
63.k odd 6 1 441.2.i.c 12
63.l odd 6 1 63.2.o.a 12
63.l odd 6 1 567.2.c.c 12
63.n odd 6 1 1323.2.i.c 12
63.o even 6 1 inner 189.2.o.a 12
63.o even 6 1 567.2.c.c 12
63.s even 6 1 1323.2.i.c 12
63.t odd 6 1 441.2.s.c 12
84.h odd 2 1 1008.2.cc.a 12
252.s odd 6 1 3024.2.cc.a 12
252.bi even 6 1 1008.2.cc.a 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
63.2.o.a 12 3.b odd 2 1
63.2.o.a 12 9.c even 3 1
63.2.o.a 12 21.c even 2 1
63.2.o.a 12 63.l odd 6 1
189.2.o.a 12 1.a even 1 1 trivial
189.2.o.a 12 7.b odd 2 1 inner
189.2.o.a 12 9.d odd 6 1 inner
189.2.o.a 12 63.o even 6 1 inner
441.2.i.c 12 21.g even 6 1
441.2.i.c 12 21.h odd 6 1
441.2.i.c 12 63.g even 3 1
441.2.i.c 12 63.k odd 6 1
441.2.s.c 12 21.g even 6 1
441.2.s.c 12 21.h odd 6 1
441.2.s.c 12 63.h even 3 1
441.2.s.c 12 63.t odd 6 1
567.2.c.c 12 9.c even 3 1
567.2.c.c 12 9.d odd 6 1
567.2.c.c 12 63.l odd 6 1
567.2.c.c 12 63.o even 6 1
1008.2.cc.a 12 12.b even 2 1
1008.2.cc.a 12 36.f odd 6 1
1008.2.cc.a 12 84.h odd 2 1
1008.2.cc.a 12 252.bi even 6 1
1323.2.i.c 12 7.c even 3 1
1323.2.i.c 12 7.d odd 6 1
1323.2.i.c 12 63.n odd 6 1
1323.2.i.c 12 63.s even 6 1
1323.2.s.c 12 7.c even 3 1
1323.2.s.c 12 7.d odd 6 1
1323.2.s.c 12 63.i even 6 1
1323.2.s.c 12 63.j odd 6 1
3024.2.cc.a 12 4.b odd 2 1
3024.2.cc.a 12 28.d even 2 1
3024.2.cc.a 12 36.h even 6 1
3024.2.cc.a 12 252.s odd 6 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(189, [\chi])\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - 3 T + 7 T^{2} - 12 T^{3} + 17 T^{4} - 24 T^{5} + 31 T^{6} - 48 T^{7} + 68 T^{8} - 96 T^{9} + 112 T^{10} - 96 T^{11} + 64 T^{12} )^{2} \)
$3$ 1
$5$ \( 1 - 15 T^{2} + 84 T^{4} - 457 T^{6} + 3921 T^{8} - 20514 T^{10} + 80841 T^{12} - 512850 T^{14} + 2450625 T^{16} - 7140625 T^{18} + 32812500 T^{20} - 146484375 T^{22} + 244140625 T^{24} \)
$7$ \( 1 + 2 T - T^{2} - 30 T^{3} - 50 T^{4} + 58 T^{5} + 715 T^{6} + 406 T^{7} - 2450 T^{8} - 10290 T^{9} - 2401 T^{10} + 33614 T^{11} + 117649 T^{12} \)
$11$ \( ( 1 + 22 T^{2} + 242 T^{4} - 66 T^{5} + 2665 T^{6} - 726 T^{7} + 29282 T^{8} + 322102 T^{10} + 1771561 T^{12} )^{2} \)
$13$ \( 1 + 33 T^{2} + 351 T^{4} + 2144 T^{6} + 53457 T^{8} + 906351 T^{10} + 9826878 T^{12} + 153173319 T^{14} + 1526785377 T^{16} + 10348678496 T^{18} + 286321483071 T^{20} + 4549330231017 T^{22} + 23298085122481 T^{24} \)
$17$ \( ( 1 + 48 T^{2} + 1338 T^{4} + 26845 T^{6} + 386682 T^{8} + 4009008 T^{10} + 24137569 T^{12} )^{2} \)
$19$ \( ( 1 - 51 T^{2} + 1599 T^{4} - 35471 T^{6} + 577239 T^{8} - 6646371 T^{10} + 47045881 T^{12} )^{2} \)
$23$ \( ( 1 + 12 T + 130 T^{2} + 984 T^{3} + 7082 T^{4} + 39654 T^{5} + 209833 T^{6} + 912042 T^{7} + 3746378 T^{8} + 11972328 T^{9} + 36379330 T^{10} + 77236116 T^{11} + 148035889 T^{12} )^{2} \)
$29$ \( ( 1 + 15 T + 184 T^{2} + 1635 T^{3} + 13205 T^{4} + 85158 T^{5} + 500899 T^{6} + 2469582 T^{7} + 11105405 T^{8} + 39876015 T^{9} + 130139704 T^{10} + 307667235 T^{11} + 594823321 T^{12} )^{2} \)
$31$ \( 1 + 57 T^{2} + 3216 T^{4} + 97787 T^{6} + 2702697 T^{8} + 47442966 T^{10} + 1072931313 T^{12} + 45592690326 T^{14} + 2495997436137 T^{16} + 86786322453947 T^{18} + 2742897576410256 T^{20} + 46718812357905657 T^{22} + 787662783788549761 T^{24} \)
$37$ \( ( 1 + T + 107 T^{2} + 73 T^{3} + 3959 T^{4} + 1369 T^{5} + 50653 T^{6} )^{4} \)
$41$ \( 1 - 174 T^{2} + 15930 T^{4} - 1039210 T^{6} + 54933612 T^{8} - 2550462528 T^{10} + 108692576859 T^{12} - 4287327509568 T^{14} + 155229258378732 T^{16} - 4936355828289610 T^{18} + 127199858899897530 T^{20} - 2335542719966517774 T^{22} + 22563490300366186081 T^{24} \)
$43$ \( ( 1 + 5 T - 58 T^{2} - 51 T^{3} + 2155 T^{4} - 7106 T^{5} - 129149 T^{6} - 305558 T^{7} + 3984595 T^{8} - 4054857 T^{9} - 198290458 T^{10} + 735042215 T^{11} + 6321363049 T^{12} )^{2} \)
$47$ \( 1 - 189 T^{2} + 18726 T^{4} - 1271869 T^{6} + 67940493 T^{8} - 3218131254 T^{10} + 150306249081 T^{12} - 7108851940086 T^{14} + 331527932822733 T^{16} - 13709749821279901 T^{18} + 445890154028136486 T^{20} - 9941235992571879261 T^{22} + \)\(11\!\cdots\!41\)\( T^{24} \)
$53$ \( ( 1 - 200 T^{2} + 20924 T^{4} - 1371431 T^{6} + 58775516 T^{8} - 1578096200 T^{10} + 22164361129 T^{12} )^{2} \)
$59$ \( 1 - 135 T^{2} + 4578 T^{4} + 47801 T^{6} + 11150547 T^{8} - 1706849076 T^{10} + 108381578457 T^{12} - 5941541633556 T^{14} + 135115203346467 T^{16} + 2016271688573441 T^{18} + 672189743352581538 T^{20} - 69000761695586589135 T^{22} + \)\(17\!\cdots\!81\)\( T^{24} \)
$61$ \( 1 + 342 T^{2} + 66840 T^{4} + 9095042 T^{6} + 949804386 T^{8} + 78844563978 T^{10} + 5324903072391 T^{12} + 293380622562138 T^{14} + 13150840509658626 T^{16} + 468579968669018162 T^{18} + 12813716800738262040 T^{20} + \)\(24\!\cdots\!42\)\( T^{22} + \)\(26\!\cdots\!21\)\( T^{24} \)
$67$ \( ( 1 - 6 T - 150 T^{2} + 506 T^{3} + 17268 T^{4} - 28236 T^{5} - 1220289 T^{6} - 1891812 T^{7} + 77516052 T^{8} + 152186078 T^{9} - 3022668150 T^{10} - 8100750642 T^{11} + 90458382169 T^{12} )^{2} \)
$71$ \( ( 1 - 263 T^{2} + 31517 T^{4} - 2539307 T^{6} + 158877197 T^{8} - 6683272103 T^{10} + 128100283921 T^{12} )^{2} \)
$73$ \( ( 1 - 159 T^{2} + 10563 T^{4} - 623423 T^{6} + 56290227 T^{8} - 4515320319 T^{10} + 151334226289 T^{12} )^{2} \)
$79$ \( ( 1 + 3 T - 204 T^{2} - 151 T^{3} + 27357 T^{4} + 1896 T^{5} - 2498217 T^{6} + 149784 T^{7} + 170735037 T^{8} - 74448889 T^{9} - 7945816524 T^{10} + 9231169197 T^{11} + 243087455521 T^{12} )^{2} \)
$83$ \( 1 - 264 T^{2} + 27306 T^{4} - 2832334 T^{6} + 401548758 T^{8} - 35501027934 T^{10} + 2433955301391 T^{12} - 244566581437326 T^{14} + 19056829854315318 T^{16} - 926004335465713246 T^{18} + 61501091690788653546 T^{20} - \)\(40\!\cdots\!36\)\( T^{22} + \)\(10\!\cdots\!61\)\( T^{24} \)
$89$ \( ( 1 + 210 T^{2} + 31902 T^{4} + 3230233 T^{6} + 252695742 T^{8} + 13175870610 T^{10} + 496981290961 T^{12} )^{2} \)
$97$ \( 1 + 513 T^{2} + 147894 T^{4} + 30072485 T^{6} + 4748892315 T^{8} + 607745530428 T^{10} + 64510991774769 T^{12} + 5718277695797052 T^{14} + 420416022193375515 T^{16} + 25049538123647278565 T^{18} + \)\(11\!\cdots\!34\)\( T^{20} + \)\(37\!\cdots\!37\)\( T^{22} + \)\(69\!\cdots\!41\)\( T^{24} \)
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