Properties

Label 147.3.h.e
Level $147$
Weight $3$
Character orbit 147.h
Analytic conductor $4.005$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 147 = 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 147.h (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(4.00545988610\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: 8.0.39033114624.8
Defining polynomial: \(x^{8} - 2 x^{7} + 6 x^{6} - 30 x^{5} + 34 x^{4} - 102 x^{3} + 486 x^{2} - 730 x + 373\)
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 3 \)
Twist minimal: no (minimal twist has level 21)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{5} q^{2} + ( -\beta_{2} + \beta_{6} ) q^{3} + ( 2 \beta_{2} - 3 \beta_{4} ) q^{4} + ( -1 - \beta_{2} - 2 \beta_{3} - \beta_{4} - 2 \beta_{5} + \beta_{7} ) q^{5} + ( 4 + \beta_{3} + \beta_{4} + \beta_{6} + 2 \beta_{7} ) q^{6} + ( -2 - \beta_{1} - 4 \beta_{3} - 5 \beta_{4} + \beta_{5} - 5 \beta_{6} + 2 \beta_{7} ) q^{8} + ( 4 - 2 \beta_{2} - 2 \beta_{3} + 4 \beta_{4} - 3 \beta_{5} + 2 \beta_{7} ) q^{9} +O(q^{10})\) \( q + \beta_{5} q^{2} + ( -\beta_{2} + \beta_{6} ) q^{3} + ( 2 \beta_{2} - 3 \beta_{4} ) q^{4} + ( -1 - \beta_{2} - 2 \beta_{3} - \beta_{4} - 2 \beta_{5} + \beta_{7} ) q^{5} + ( 4 + \beta_{3} + \beta_{4} + \beta_{6} + 2 \beta_{7} ) q^{6} + ( -2 - \beta_{1} - 4 \beta_{3} - 5 \beta_{4} + \beta_{5} - 5 \beta_{6} + 2 \beta_{7} ) q^{8} + ( 4 - 2 \beta_{2} - 2 \beta_{3} + 4 \beta_{4} - 3 \beta_{5} + 2 \beta_{7} ) q^{9} + ( \beta_{2} + 7 \beta_{4} ) q^{10} + ( -2 \beta_{1} - 2 \beta_{4} - 2 \beta_{6} ) q^{11} + ( 5 - \beta_{2} - \beta_{3} + 5 \beta_{4} + 6 \beta_{5} + \beta_{7} ) q^{12} + ( -9 - \beta_{7} ) q^{13} + ( 5 + 3 \beta_{1} - 4 \beta_{3} - \beta_{4} - 3 \beta_{5} - \beta_{6} - 2 \beta_{7} ) q^{15} + ( -9 + 4 \beta_{2} - 9 \beta_{4} - 4 \beta_{7} ) q^{16} + ( -2 \beta_{1} - 2 \beta_{2} + 2 \beta_{6} ) q^{17} + ( 6 \beta_{1} - 2 \beta_{2} + 21 \beta_{4} + 8 \beta_{6} ) q^{18} + ( -3 + 5 \beta_{2} - 3 \beta_{4} - 5 \beta_{7} ) q^{19} + ( 3 - 2 \beta_{1} + 6 \beta_{3} + 4 \beta_{4} + 2 \beta_{5} + 4 \beta_{6} - 3 \beta_{7} ) q^{20} + ( -14 - 4 \beta_{7} ) q^{22} + ( -2 - 2 \beta_{2} - 4 \beta_{3} - 2 \beta_{4} + 4 \beta_{5} + 2 \beta_{7} ) q^{23} + ( 6 \beta_{1} + 6 \beta_{2} - 27 \beta_{4} + 3 \beta_{6} ) q^{24} + ( -10 \beta_{2} - 3 \beta_{4} ) q^{25} + ( 1 + \beta_{2} + 2 \beta_{3} + \beta_{4} - 10 \beta_{5} - \beta_{7} ) q^{26} + ( 2 + 6 \beta_{1} - \beta_{3} + 5 \beta_{4} - 6 \beta_{5} + 5 \beta_{6} - 8 \beta_{7} ) q^{27} + ( 2 - 6 \beta_{1} + 4 \beta_{3} - 2 \beta_{4} + 6 \beta_{5} - 2 \beta_{6} - 2 \beta_{7} ) q^{29} + ( 11 + 8 \beta_{2} + 8 \beta_{3} + 11 \beta_{4} + 3 \beta_{5} - 8 \beta_{7} ) q^{30} + ( 2 \beta_{2} + 34 \beta_{4} ) q^{31} + ( -9 \beta_{1} - 4 \beta_{2} - 5 \beta_{4} - \beta_{6} ) q^{32} + ( -8 + 4 \beta_{2} - 2 \beta_{3} - 8 \beta_{4} - 4 \beta_{7} ) q^{33} + 6 \beta_{7} q^{34} + ( 34 + 3 \beta_{1} + 10 \beta_{3} + 13 \beta_{4} - 3 \beta_{5} + 13 \beta_{6} + 8 \beta_{7} ) q^{36} + ( -4 - 14 \beta_{2} - 4 \beta_{4} + 14 \beta_{7} ) q^{37} + ( -8 \beta_{1} + 5 \beta_{2} - 13 \beta_{4} - 18 \beta_{6} ) q^{38} + ( -3 \beta_{1} + 8 \beta_{2} - 6 \beta_{4} - 11 \beta_{6} ) q^{39} + ( -21 - 15 \beta_{2} - 21 \beta_{4} + 15 \beta_{7} ) q^{40} + ( -8 - 6 \beta_{1} - 16 \beta_{3} - 22 \beta_{4} + 6 \beta_{5} - 22 \beta_{6} + 8 \beta_{7} ) q^{41} + ( -40 + 6 \beta_{7} ) q^{43} + ( 4 + 4 \beta_{2} + 8 \beta_{3} + 4 \beta_{4} - 10 \beta_{5} - 4 \beta_{7} ) q^{44} + ( -6 \beta_{1} - 13 \beta_{2} - 39 \beta_{4} - 2 \beta_{6} ) q^{45} + ( 18 \beta_{2} - 42 \beta_{4} ) q^{46} + ( 4 + 4 \beta_{2} + 8 \beta_{3} + 4 \beta_{4} + 16 \beta_{5} - 4 \beta_{7} ) q^{47} + ( 7 - 12 \beta_{1} - 5 \beta_{3} - 17 \beta_{4} + 12 \beta_{5} - 17 \beta_{6} + 5 \beta_{7} ) q^{48} + ( 10 + 7 \beta_{1} + 20 \beta_{3} + 27 \beta_{4} - 7 \beta_{5} + 27 \beta_{6} - 10 \beta_{7} ) q^{50} + ( 18 - 6 \beta_{3} + 18 \beta_{4} - 6 \beta_{5} ) q^{51} + ( -21 \beta_{2} + 41 \beta_{4} ) q^{52} + ( 22 \beta_{1} + 16 \beta_{2} + 6 \beta_{4} - 10 \beta_{6} ) q^{53} + ( 47 - \beta_{2} + 17 \beta_{3} + 47 \beta_{4} - 6 \beta_{5} + \beta_{7} ) q^{54} + ( 14 - 2 \beta_{7} ) q^{55} + ( 17 - 15 \beta_{1} + 2 \beta_{3} - 13 \beta_{4} + 15 \beta_{5} - 13 \beta_{6} - 2 \beta_{7} ) q^{57} + ( -28 + 2 \beta_{2} - 28 \beta_{4} - 2 \beta_{7} ) q^{58} + ( 28 \beta_{1} + \beta_{2} + 27 \beta_{4} + 26 \beta_{6} ) q^{59} + ( -9 \beta_{1} - 2 \beta_{2} + 30 \beta_{4} - \beta_{6} ) q^{60} + ( 39 - 7 \beta_{2} + 39 \beta_{4} + 7 \beta_{7} ) q^{61} + ( -2 + 32 \beta_{1} - 4 \beta_{3} + 28 \beta_{4} - 32 \beta_{5} + 28 \beta_{6} + 2 \beta_{7} ) q^{62} + ( 1 + 18 \beta_{7} ) q^{64} + ( 6 + 6 \beta_{2} + 12 \beta_{3} + 6 \beta_{4} + 14 \beta_{5} - 6 \beta_{7} ) q^{65} + ( -12 \beta_{1} + 10 \beta_{2} - 24 \beta_{4} - 22 \beta_{6} ) q^{66} + ( -8 \beta_{2} - 6 \beta_{4} ) q^{67} + ( 2 + 2 \beta_{2} + 4 \beta_{3} + 2 \beta_{4} + 14 \beta_{5} - 2 \beta_{7} ) q^{68} + ( 42 + 6 \beta_{1} + 6 \beta_{4} - 6 \beta_{5} + 6 \beta_{6} + 12 \beta_{7} ) q^{69} + ( -6 + 30 \beta_{1} - 12 \beta_{3} + 18 \beta_{4} - 30 \beta_{5} + 18 \beta_{6} + 6 \beta_{7} ) q^{71} + ( -9 - 36 \beta_{2} - 18 \beta_{3} - 9 \beta_{4} + 18 \beta_{5} + 36 \beta_{7} ) q^{72} + ( 26 \beta_{2} - 8 \beta_{4} ) q^{73} + ( 10 \beta_{1} - 14 \beta_{2} + 24 \beta_{4} + 38 \beta_{6} ) q^{74} + ( -43 - 13 \beta_{2} - 13 \beta_{3} - 43 \beta_{4} - 30 \beta_{5} + 13 \beta_{7} ) q^{75} + ( -79 - 21 \beta_{7} ) q^{76} + ( -53 - 3 \beta_{1} - 8 \beta_{3} - 11 \beta_{4} + 3 \beta_{5} - 11 \beta_{6} - 22 \beta_{7} ) q^{78} + ( -32 + 36 \beta_{2} - 32 \beta_{4} - 36 \beta_{7} ) q^{79} + ( 2 \beta_{1} - 3 \beta_{2} + 5 \beta_{4} + 8 \beta_{6} ) q^{80} + ( -24 \beta_{1} - 22 \beta_{2} - 39 \beta_{4} - 20 \beta_{6} ) q^{81} + ( -98 + 52 \beta_{2} - 98 \beta_{4} - 52 \beta_{7} ) q^{82} + ( 9 - 36 \beta_{1} + 18 \beta_{3} - 18 \beta_{4} + 36 \beta_{5} - 18 \beta_{6} - 9 \beta_{7} ) q^{83} + ( 42 - 18 \beta_{7} ) q^{85} + ( -6 - 6 \beta_{2} - 12 \beta_{3} - 6 \beta_{4} - 34 \beta_{5} + 6 \beta_{7} ) q^{86} + ( -6 \beta_{1} + 8 \beta_{2} + 6 \beta_{4} + 4 \beta_{6} ) q^{87} + ( -24 \beta_{2} + 42 \beta_{4} ) q^{88} + ( -16 - 16 \beta_{2} - 32 \beta_{3} - 16 \beta_{4} + 10 \beta_{5} + 16 \beta_{7} ) q^{89} + ( -17 - 24 \beta_{1} + 22 \beta_{3} - 2 \beta_{4} + 24 \beta_{5} - 2 \beta_{6} - 13 \beta_{7} ) q^{90} + ( -10 - 44 \beta_{1} - 20 \beta_{3} - 64 \beta_{4} + 44 \beta_{5} - 64 \beta_{6} + 10 \beta_{7} ) q^{92} + ( 42 + 36 \beta_{2} + 36 \beta_{3} + 42 \beta_{4} + 6 \beta_{5} - 36 \beta_{7} ) q^{93} + ( 12 \beta_{2} - 84 \beta_{4} ) q^{94} + ( -14 \beta_{1} - 12 \beta_{2} - 2 \beta_{4} + 10 \beta_{6} ) q^{95} + ( 16 + 10 \beta_{2} - 17 \beta_{3} + 16 \beta_{4} - 12 \beta_{5} - 10 \beta_{7} ) q^{96} + ( -2 - 8 \beta_{7} ) q^{97} + ( 26 - 12 \beta_{1} - 4 \beta_{3} - 16 \beta_{4} + 12 \beta_{5} - 16 \beta_{6} + 4 \beta_{7} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + 2q^{3} + 12q^{4} + 28q^{6} + 20q^{9} + O(q^{10}) \) \( 8q + 2q^{3} + 12q^{4} + 28q^{6} + 20q^{9} - 28q^{10} + 22q^{12} - 72q^{13} + 56q^{15} - 36q^{16} - 56q^{18} - 12q^{19} - 112q^{22} + 126q^{24} + 12q^{25} + 20q^{27} + 28q^{30} - 136q^{31} - 28q^{33} + 232q^{36} - 16q^{37} - 4q^{39} - 84q^{40} - 320q^{43} + 140q^{45} + 168q^{46} + 76q^{48} + 84q^{51} - 164q^{52} + 154q^{54} + 112q^{55} + 128q^{57} - 112q^{58} - 140q^{60} + 156q^{61} + 8q^{64} + 28q^{66} + 24q^{67} + 336q^{69} + 32q^{73} - 146q^{75} - 632q^{76} - 392q^{78} - 128q^{79} + 68q^{81} - 392q^{82} + 336q^{85} - 28q^{87} - 168q^{88} - 224q^{90} + 96q^{93} + 336q^{94} + 98q^{96} - 16q^{97} + 224q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - 2 x^{7} + 6 x^{6} - 30 x^{5} + 34 x^{4} - 102 x^{3} + 486 x^{2} - 730 x + 373\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( 50 \nu^{7} + 15 \nu^{6} - 1344 \nu^{5} - 2577 \nu^{4} - 16648 \nu^{3} + 10993 \nu^{2} + 13664 \nu + 174191 \)\()/46998\)
\(\beta_{2}\)\(=\)\((\)\( 62 \nu^{7} + 1809 \nu^{6} + 3078 \nu^{5} + 13500 \nu^{4} - 11602 \nu^{3} - 37037 \nu^{2} - 174898 \nu + 164120 \)\()/46998\)
\(\beta_{3}\)\(=\)\((\)\( 140 \nu^{7} - 704 \nu^{6} - 257 \nu^{5} - 7514 \nu^{4} + 3666 \nu^{3} - 6072 \nu^{2} + 120543 \nu - 105932 \)\()/15666\)
\(\beta_{4}\)\(=\)\((\)\( -163 \nu^{7} + 63 \nu^{6} - 945 \nu^{5} + 3276 \nu^{4} - 469 \nu^{3} + 15883 \nu^{2} - 51751 \nu + 36554 \)\()/6714\)
\(\beta_{5}\)\(=\)\((\)\( 1223 \nu^{7} + 255 \nu^{6} + 7365 \nu^{5} - 20310 \nu^{4} - 655 \nu^{3} - 114503 \nu^{2} + 346799 \nu - 202912 \)\()/46998\)
\(\beta_{6}\)\(=\)\((\)\( -616 \nu^{7} + 39 \nu^{6} - 2823 \nu^{5} + 12099 \nu^{4} + 7667 \nu^{3} + 62674 \nu^{2} - 160597 \nu + 51101 \)\()/23499\)
\(\beta_{7}\)\(=\)\((\)\( -152 \nu^{7} + 29 \nu^{6} - 808 \nu^{5} + 3373 \nu^{4} + 986 \nu^{3} + 16429 \nu^{2} - 45746 \nu + 19023 \)\()/5222\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-\beta_{7} + 2 \beta_{6} + \beta_{5} + \beta_{4} + 2 \beta_{3} + 2 \beta_{2} + \beta_{1} + 1\)\()/3\)
\(\nu^{2}\)\(=\)\((\)\(2 \beta_{7} + 2 \beta_{6} + \beta_{5} - 5 \beta_{4} - 4 \beta_{3} - 4 \beta_{2} + 4 \beta_{1} - 8\)\()/3\)
\(\nu^{3}\)\(=\)\((\)\(4 \beta_{7} - 11 \beta_{6} + 11 \beta_{5} + 17 \beta_{4} - 2 \beta_{3} - 8 \beta_{2} - 16 \beta_{1} + 38\)\()/3\)
\(\nu^{4}\)\(=\)\(8 \beta_{7} - 2 \beta_{6} - 2 \beta_{5} - 6 \beta_{4} + 8 \beta_{3} + 10 \beta_{2} + 2 \beta_{1} + 11\)
\(\nu^{5}\)\(=\)\((\)\(-47 \beta_{7} + 136 \beta_{6} - 91 \beta_{5} - 199 \beta_{4} - 44 \beta_{3} + 10 \beta_{2} + 107 \beta_{1} - 163\)\()/3\)
\(\nu^{6}\)\(=\)\((\)\(-170 \beta_{7} + 22 \beta_{6} + 377 \beta_{5} + 575 \beta_{4} + 4 \beta_{3} - 86 \beta_{2} - 136 \beta_{1} - 100\)\()/3\)
\(\nu^{7}\)\(=\)\((\)\(1190 \beta_{7} - 1309 \beta_{6} + 301 \beta_{5} + 37 \beta_{4} - 280 \beta_{3} - 490 \beta_{2} - 434 \beta_{1} + 1036\)\()/3\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/147\mathbb{Z}\right)^\times\).

\(n\) \(50\) \(52\)
\(\chi(n)\) \(-1\) \(-1 - \beta_{4}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
116.1
−0.279898 3.02113i
−1.85391 + 1.90397i
1.03103 0.478705i
2.10277 0.136187i
−0.279898 + 3.02113i
−1.85391 1.90397i
1.03103 + 0.478705i
2.10277 + 0.136187i
−3.03622 + 1.75296i −2.90987 0.729839i 4.14575 7.18065i 1.07558 0.620984i 10.1144 2.88494i 0 15.0457i 7.93467 + 4.24747i −2.17712 + 3.77089i
116.2 −1.13198 + 0.653548i 2.97489 0.387321i −1.14575 + 1.98450i 6.39086 3.68977i −3.11438 + 2.38267i 0 8.22359i 8.69997 2.30448i −4.82288 + 8.35347i
116.3 1.13198 0.653548i −1.15202 2.76999i −1.14575 + 1.98450i −6.39086 + 3.68977i −3.11438 2.38267i 0 8.22359i −6.34572 + 6.38215i −4.82288 + 8.35347i
116.4 3.03622 1.75296i 2.08699 + 2.15510i 4.14575 7.18065i −1.07558 + 0.620984i 10.1144 + 2.88494i 0 15.0457i −0.288920 + 8.99536i −2.17712 + 3.77089i
128.1 −3.03622 1.75296i −2.90987 + 0.729839i 4.14575 + 7.18065i 1.07558 + 0.620984i 10.1144 + 2.88494i 0 15.0457i 7.93467 4.24747i −2.17712 3.77089i
128.2 −1.13198 0.653548i 2.97489 + 0.387321i −1.14575 1.98450i 6.39086 + 3.68977i −3.11438 2.38267i 0 8.22359i 8.69997 + 2.30448i −4.82288 8.35347i
128.3 1.13198 + 0.653548i −1.15202 + 2.76999i −1.14575 1.98450i −6.39086 3.68977i −3.11438 + 2.38267i 0 8.22359i −6.34572 6.38215i −4.82288 8.35347i
128.4 3.03622 + 1.75296i 2.08699 2.15510i 4.14575 + 7.18065i −1.07558 0.620984i 10.1144 2.88494i 0 15.0457i −0.288920 8.99536i −2.17712 3.77089i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 128.4
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 147.3.h.e 8
3.b odd 2 1 inner 147.3.h.e 8
7.b odd 2 1 147.3.h.c 8
7.c even 3 1 21.3.b.a 4
7.c even 3 1 inner 147.3.h.e 8
7.d odd 6 1 147.3.b.f 4
7.d odd 6 1 147.3.h.c 8
21.c even 2 1 147.3.h.c 8
21.g even 6 1 147.3.b.f 4
21.g even 6 1 147.3.h.c 8
21.h odd 6 1 21.3.b.a 4
21.h odd 6 1 inner 147.3.h.e 8
28.g odd 6 1 336.3.d.c 4
35.j even 6 1 525.3.c.a 4
35.l odd 12 2 525.3.f.a 8
56.k odd 6 1 1344.3.d.b 4
56.p even 6 1 1344.3.d.f 4
63.g even 3 1 567.3.r.c 8
63.h even 3 1 567.3.r.c 8
63.j odd 6 1 567.3.r.c 8
63.n odd 6 1 567.3.r.c 8
84.n even 6 1 336.3.d.c 4
105.o odd 6 1 525.3.c.a 4
105.x even 12 2 525.3.f.a 8
168.s odd 6 1 1344.3.d.f 4
168.v even 6 1 1344.3.d.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.3.b.a 4 7.c even 3 1
21.3.b.a 4 21.h odd 6 1
147.3.b.f 4 7.d odd 6 1
147.3.b.f 4 21.g even 6 1
147.3.h.c 8 7.b odd 2 1
147.3.h.c 8 7.d odd 6 1
147.3.h.c 8 21.c even 2 1
147.3.h.c 8 21.g even 6 1
147.3.h.e 8 1.a even 1 1 trivial
147.3.h.e 8 3.b odd 2 1 inner
147.3.h.e 8 7.c even 3 1 inner
147.3.h.e 8 21.h odd 6 1 inner
336.3.d.c 4 28.g odd 6 1
336.3.d.c 4 84.n even 6 1
525.3.c.a 4 35.j even 6 1
525.3.c.a 4 105.o odd 6 1
525.3.f.a 8 35.l odd 12 2
525.3.f.a 8 105.x even 12 2
567.3.r.c 8 63.g even 3 1
567.3.r.c 8 63.h even 3 1
567.3.r.c 8 63.j odd 6 1
567.3.r.c 8 63.n odd 6 1
1344.3.d.b 4 56.k odd 6 1
1344.3.d.b 4 168.v even 6 1
1344.3.d.f 4 56.p even 6 1
1344.3.d.f 4 168.s odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(147, [\chi])\):

\( T_{2}^{8} - 14 T_{2}^{6} + 175 T_{2}^{4} - 294 T_{2}^{2} + 441 \)
\( T_{5}^{8} - 56 T_{5}^{6} + 3052 T_{5}^{4} - 4704 T_{5}^{2} + 7056 \)
\( T_{13}^{2} + 18 T_{13} + 74 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + 2 T^{2} - T^{4} - 54 T^{6} - 295 T^{8} - 864 T^{10} - 256 T^{12} + 8192 T^{14} + 65536 T^{16} \)
$3$ \( 1 - 2 T - 8 T^{2} + 12 T^{3} + 27 T^{4} + 108 T^{5} - 648 T^{6} - 1458 T^{7} + 6561 T^{8} \)
$5$ \( 1 + 44 T^{2} + 902 T^{4} - 9504 T^{6} - 531469 T^{8} - 5940000 T^{10} + 352343750 T^{12} + 10742187500 T^{14} + 152587890625 T^{16} \)
$7$ 1
$11$ \( 1 + 428 T^{2} + 108554 T^{4} + 19408944 T^{6} + 2673281075 T^{8} + 284166349104 T^{10} + 23269513968074 T^{12} + 1343247345236588 T^{14} + 45949729863572161 T^{16} \)
$13$ \( ( 1 + 18 T + 412 T^{2} + 3042 T^{3} + 28561 T^{4} )^{4} \)
$17$ \( 1 + 988 T^{2} + 569098 T^{4} + 237123952 T^{6} + 77182165651 T^{8} + 19804829594992 T^{10} + 3969889608158218 T^{12} + 575630770383003868 T^{14} + 48661191875666868481 T^{16} \)
$19$ \( ( 1 + 6 T - 520 T^{2} - 996 T^{3} + 165819 T^{4} - 359556 T^{5} - 67766920 T^{6} + 282275286 T^{7} + 16983563041 T^{8} )^{2} \)
$23$ \( 1 + 1444 T^{2} + 1104970 T^{4} + 607178896 T^{6} + 298907858899 T^{8} + 169913549435536 T^{10} + 86531289405946570 T^{12} + 31644717679837343524 T^{14} + \)\(61\!\cdots\!61\)\( T^{16} \)
$29$ \( ( 1 - 2972 T^{2} + 3611558 T^{4} - 2102039132 T^{6} + 500246412961 T^{8} )^{2} \)
$31$ \( ( 1 + 68 T + 1574 T^{2} + 76704 T^{3} + 3935315 T^{4} + 73712544 T^{5} + 1453622054 T^{6} + 60350250308 T^{7} + 852891037441 T^{8} )^{2} \)
$37$ \( ( 1 + 8 T - 1318 T^{2} - 10848 T^{3} - 51853 T^{4} - 14850912 T^{5} - 2470144198 T^{6} + 20525811272 T^{7} + 3512479453921 T^{8} )^{2} \)
$41$ \( ( 1 - 1292 T^{2} + 2832038 T^{4} - 3650883212 T^{6} + 7984925229121 T^{8} )^{2} \)
$43$ \( ( 1 + 80 T + 5046 T^{2} + 147920 T^{3} + 3418801 T^{4} )^{4} \)
$47$ \( 1 + 6148 T^{2} + 18653578 T^{4} + 57698758672 T^{6} + 158252217343891 T^{8} + 281551536415343632 T^{10} + \)\(44\!\cdots\!58\)\( T^{12} + \)\(71\!\cdots\!68\)\( T^{14} + \)\(56\!\cdots\!21\)\( T^{16} \)
$53$ \( 1 - 20 T^{2} + 13350538 T^{4} + 582622000 T^{6} + 115963338015283 T^{8} + 4597167821182000 T^{10} + \)\(83\!\cdots\!18\)\( T^{12} - \)\(98\!\cdots\!20\)\( T^{14} + \)\(38\!\cdots\!21\)\( T^{16} \)
$59$ \( 1 + 3676 T^{2} - 2451290 T^{4} - 30402196256 T^{6} - 55714257061901 T^{8} - 368394387226800416 T^{10} - \)\(35\!\cdots\!90\)\( T^{12} + \)\(65\!\cdots\!56\)\( T^{14} + \)\(21\!\cdots\!41\)\( T^{16} \)
$61$ \( ( 1 - 78 T - 2536 T^{2} - 91884 T^{3} + 37819995 T^{4} - 341900364 T^{5} - 35113052776 T^{6} - 4018589200158 T^{7} + 191707312997281 T^{8} )^{2} \)
$67$ \( ( 1 - 12 T - 8422 T^{2} + 4944 T^{3} + 52578819 T^{4} + 22193616 T^{5} - 169712741062 T^{6} - 1085500586028 T^{7} + 406067677556641 T^{8} )^{2} \)
$71$ \( ( 1 - 10588 T^{2} + 78813510 T^{4} - 269058878428 T^{6} + 645753531245761 T^{8} )^{2} \)
$73$ \( ( 1 - 16 T - 5734 T^{2} + 74688 T^{3} + 6117635 T^{4} + 398012352 T^{5} - 162835513894 T^{6} - 2421347620624 T^{7} + 806460091894081 T^{8} )^{2} \)
$79$ \( ( 1 + 64 T - 338 T^{2} - 515072 T^{3} - 44852861 T^{4} - 3214564352 T^{5} - 13165127378 T^{6} + 15557597153344 T^{7} + 1517108809906561 T^{8} )^{2} \)
$83$ \( ( 1 - 13948 T^{2} + 141899946 T^{4} - 661948661308 T^{6} + 2252292232139041 T^{8} )^{2} \)
$89$ \( 1 + 11468 T^{2} + 10569194 T^{4} - 52049261136 T^{6} + 2439757657758035 T^{8} - 3265687286066845776 T^{10} + \)\(41\!\cdots\!14\)\( T^{12} + \)\(28\!\cdots\!28\)\( T^{14} + \)\(15\!\cdots\!61\)\( T^{16} \)
$97$ \( ( 1 + 4 T + 18374 T^{2} + 37636 T^{3} + 88529281 T^{4} )^{4} \)
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