Properties

Label 324.3.d.g
Level 324
Weight 3
Character orbit 324.d
Analytic conductor 8.828
Analytic rank 0
Dimension 8
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) \(=\) \( 324 = 2^{2} \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 324.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(8.82836056527\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.1919698923024.1
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 36)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q -\beta_{1} q^{2} + ( \beta_{1} + \beta_{2} + \beta_{5} - \beta_{6} ) q^{4} + ( 1 + \beta_{4} + \beta_{7} ) q^{5} + ( \beta_{3} + \beta_{4} - \beta_{7} ) q^{7} + ( 4 - 2 \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} ) q^{8} +O(q^{10})\) \( q -\beta_{1} q^{2} + ( \beta_{1} + \beta_{2} + \beta_{5} - \beta_{6} ) q^{4} + ( 1 + \beta_{4} + \beta_{7} ) q^{5} + ( \beta_{3} + \beta_{4} - \beta_{7} ) q^{7} + ( 4 - 2 \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} ) q^{8} + ( 1 - \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} - 2 \beta_{7} ) q^{10} + ( 1 - 2 \beta_{1} - \beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} - \beta_{7} ) q^{11} + ( 6 - \beta_{2} - \beta_{5} + 2 \beta_{6} ) q^{13} + ( -2 - \beta_{2} + 2 \beta_{3} + \beta_{5} - 3 \beta_{6} - 2 \beta_{7} ) q^{14} + ( 3 - 2 \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} + \beta_{5} + 2 \beta_{6} + 2 \beta_{7} ) q^{16} + ( -2 + 2 \beta_{1} + 3 \beta_{2} + 2 \beta_{5} - 6 \beta_{6} ) q^{17} + ( 1 - 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} - \beta_{4} - \beta_{5} + \beta_{7} ) q^{19} + ( 5 - \beta_{1} + 3 \beta_{3} - \beta_{4} - 3 \beta_{6} + 2 \beta_{7} ) q^{20} + ( -6 + 2 \beta_{1} + 5 \beta_{3} - \beta_{4} + 3 \beta_{5} - \beta_{6} - 2 \beta_{7} ) q^{22} + ( 3 - 6 \beta_{1} - \beta_{2} - \beta_{3} - 2 \beta_{4} - 3 \beta_{5} + 2 \beta_{7} ) q^{23} + ( 1 + 6 \beta_{1} + 2 \beta_{2} - 3 \beta_{4} - \beta_{5} - 4 \beta_{6} - 3 \beta_{7} ) q^{25} + ( -2 - 5 \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} - 2 \beta_{5} ) q^{26} + ( \beta_{1} - 6 \beta_{2} + 3 \beta_{3} + 3 \beta_{4} + 3 \beta_{5} + \beta_{6} ) q^{28} + ( 4 + 3 \beta_{2} - 2 \beta_{4} + 3 \beta_{5} - 6 \beta_{6} - 2 \beta_{7} ) q^{29} + ( 5 - 10 \beta_{1} + \beta_{2} - 3 \beta_{3} - 5 \beta_{5} ) q^{31} + ( 11 - 2 \beta_{1} + 7 \beta_{2} - 5 \beta_{3} - \beta_{4} + \beta_{5} - 6 \beta_{6} - 2 \beta_{7} ) q^{32} + ( -2 - 2 \beta_{1} - 5 \beta_{2} - 3 \beta_{3} - 3 \beta_{4} + 4 \beta_{5} + 2 \beta_{6} ) q^{34} + ( 3 - 6 \beta_{1} - 4 \beta_{2} + 5 \beta_{3} + \beta_{4} - 3 \beta_{5} - \beta_{7} ) q^{35} + ( 1 + 12 \beta_{1} + \beta_{2} + 3 \beta_{4} - 5 \beta_{5} - 2 \beta_{6} + 3 \beta_{7} ) q^{37} + ( -13 + 2 \beta_{1} + 5 \beta_{2} - 7 \beta_{3} + \beta_{4} + \beta_{5} - 6 \beta_{6} + 2 \beta_{7} ) q^{38} + ( -7 - 4 \beta_{1} - 8 \beta_{3} + 4 \beta_{4} + 3 \beta_{5} + 2 \beta_{6} + 2 \beta_{7} ) q^{40} + ( 10 - 4 \beta_{1} + 3 \beta_{2} - 3 \beta_{4} + 5 \beta_{5} - 6 \beta_{6} - 3 \beta_{7} ) q^{41} + ( 6 - 12 \beta_{1} - 5 \beta_{3} - 2 \beta_{4} - 6 \beta_{5} + 2 \beta_{7} ) q^{43} + ( 15 + \beta_{1} - 5 \beta_{2} - 2 \beta_{3} + 6 \beta_{4} - 2 \beta_{5} - 3 \beta_{6} + 2 \beta_{7} ) q^{44} + ( -19 + 6 \beta_{1} + 7 \beta_{2} - 4 \beta_{3} + 2 \beta_{4} + 4 \beta_{5} + \beta_{6} + 4 \beta_{7} ) q^{46} + ( 2 - 4 \beta_{1} - 6 \beta_{2} + 3 \beta_{3} - 3 \beta_{4} - 2 \beta_{5} + 3 \beta_{7} ) q^{47} + ( -12 + 18 \beta_{1} + \beta_{2} - 8 \beta_{5} - 2 \beta_{6} ) q^{49} + ( -23 - 6 \beta_{1} - 11 \beta_{2} + \beta_{3} + \beta_{4} - 5 \beta_{5} + 6 \beta_{6} + 6 \beta_{7} ) q^{50} + ( -19 + 9 \beta_{1} + 6 \beta_{2} - \beta_{3} - \beta_{4} + 6 \beta_{5} - 9 \beta_{6} - 2 \beta_{7} ) q^{52} + ( 7 - 8 \beta_{1} - 3 \beta_{2} - \beta_{4} + \beta_{5} + 6 \beta_{6} - \beta_{7} ) q^{53} + ( 9 - 18 \beta_{1} - 9 \beta_{2} - 7 \beta_{3} + 2 \beta_{4} - 9 \beta_{5} - 2 \beta_{7} ) q^{55} + ( 35 - 4 \beta_{1} - 6 \beta_{2} + 6 \beta_{3} + 6 \beta_{4} + \beta_{5} + 6 \beta_{6} - 6 \beta_{7} ) q^{56} + ( 4 - 7 \beta_{1} - 5 \beta_{2} - \beta_{3} - \beta_{4} + 4 \beta_{5} + 4 \beta_{7} ) q^{58} + ( -2 + 4 \beta_{1} - 6 \beta_{2} + 3 \beta_{3} + 6 \beta_{4} + 2 \beta_{5} - 6 \beta_{7} ) q^{59} + ( 6 + 12 \beta_{1} - 7 \beta_{2} + 12 \beta_{4} - 13 \beta_{5} + 14 \beta_{6} + 12 \beta_{7} ) q^{61} + ( -39 + 10 \beta_{1} + 11 \beta_{2} + 2 \beta_{3} - 4 \beta_{4} + 10 \beta_{5} - 9 \beta_{6} ) q^{62} + ( -15 - 10 \beta_{1} + \beta_{2} + \beta_{3} - 11 \beta_{4} + 7 \beta_{5} - 6 \beta_{6} + 2 \beta_{7} ) q^{64} + ( -1 - 2 \beta_{1} - 3 \beta_{2} + 6 \beta_{4} - 2 \beta_{5} + 6 \beta_{6} + 6 \beta_{7} ) q^{65} + ( 3 - 6 \beta_{1} - 9 \beta_{2} + 11 \beta_{3} - \beta_{4} - 3 \beta_{5} + \beta_{7} ) q^{67} + ( 49 - \beta_{2} + 5 \beta_{3} + 5 \beta_{4} - 3 \beta_{5} + 12 \beta_{6} + 6 \beta_{7} ) q^{68} + ( -22 + 6 \beta_{1} + \beta_{2} + 2 \beta_{3} + 8 \beta_{4} + 7 \beta_{5} - 5 \beta_{6} - 2 \beta_{7} ) q^{70} + ( -1 + 2 \beta_{1} + 5 \beta_{2} - 22 \beta_{3} - 5 \beta_{4} + \beta_{5} + 5 \beta_{7} ) q^{71} + ( 4 + 6 \beta_{1} - 5 \beta_{2} - 12 \beta_{4} - 8 \beta_{5} + 10 \beta_{6} - 12 \beta_{7} ) q^{73} + ( -43 - 8 \beta_{1} - 10 \beta_{2} - 4 \beta_{3} - 4 \beta_{4} - 7 \beta_{5} + 12 \beta_{6} - 6 \beta_{7} ) q^{74} + ( -5 + 10 \beta_{1} - \beta_{2} - \beta_{3} - 13 \beta_{4} + 5 \beta_{5} + 6 \beta_{6} - 2 \beta_{7} ) q^{76} + ( -45 + 16 \beta_{1} + \beta_{4} - 8 \beta_{5} + \beta_{7} ) q^{77} + ( 11 - 22 \beta_{1} - 5 \beta_{2} + 11 \beta_{3} - 4 \beta_{4} - 11 \beta_{5} + 4 \beta_{7} ) q^{79} + ( 38 + 8 \beta_{1} + 8 \beta_{2} + 8 \beta_{3} - 12 \beta_{4} + 6 \beta_{5} - 8 \beta_{7} ) q^{80} + ( 19 - 11 \beta_{1} - 2 \beta_{2} + 7 \beta_{5} - 4 \beta_{6} + 6 \beta_{7} ) q^{82} + ( -5 + 10 \beta_{1} + 26 \beta_{2} - \beta_{3} + \beta_{4} + 5 \beta_{5} - \beta_{7} ) q^{83} + ( 13 + 12 \beta_{1} + 9 \beta_{2} + 3 \beta_{4} + 3 \beta_{5} - 18 \beta_{6} + 3 \beta_{7} ) q^{85} + ( -41 + 12 \beta_{1} + 14 \beta_{2} - \beta_{3} - 3 \beta_{4} + 10 \beta_{5} - 3 \beta_{6} + 4 \beta_{7} ) q^{86} + ( -8 - 14 \beta_{1} - 7 \beta_{2} + 9 \beta_{3} - 3 \beta_{4} + 8 \beta_{5} + 12 \beta_{6} - 12 \beta_{7} ) q^{88} + ( 25 + 8 \beta_{1} - 9 \beta_{2} + 3 \beta_{4} - 13 \beta_{5} + 18 \beta_{6} + 3 \beta_{7} ) q^{89} + ( 4 - 8 \beta_{1} + 11 \beta_{2} + \beta_{3} + 4 \beta_{4} - 4 \beta_{5} - 4 \beta_{7} ) q^{91} + ( 28 + 9 \beta_{1} + 6 \beta_{2} - 9 \beta_{3} - 13 \beta_{4} - 5 \beta_{5} - 3 \beta_{6} - 4 \beta_{7} ) q^{92} + ( -4 + 4 \beta_{1} + \beta_{2} - 6 \beta_{3} + 12 \beta_{4} + \beta_{5} + 11 \beta_{6} + 6 \beta_{7} ) q^{94} + ( -14 + 28 \beta_{1} + 22 \beta_{2} + 4 \beta_{3} + 2 \beta_{4} + 14 \beta_{5} - 2 \beta_{7} ) q^{95} + ( 16 + \beta_{2} - 9 \beta_{4} + \beta_{5} - 2 \beta_{6} - 9 \beta_{7} ) q^{97} + ( -70 + 2 \beta_{1} - 19 \beta_{2} - \beta_{3} - \beta_{4} - 16 \beta_{5} + 18 \beta_{6} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q - 3q^{2} + 5q^{4} + 6q^{5} + 27q^{8} + O(q^{10}) \) \( 8q - 3q^{2} + 5q^{4} + 6q^{5} + 27q^{8} + 10q^{10} + 46q^{13} - 12q^{14} + 17q^{16} - 6q^{17} + 36q^{20} - 33q^{22} + 30q^{25} - 36q^{26} + 6q^{28} + 42q^{29} + 87q^{32} - 11q^{34} + 28q^{37} - 99q^{38} - 68q^{40} + 84q^{41} + 111q^{44} - 132q^{46} - 58q^{49} - 219q^{50} - 110q^{52} + 36q^{53} + 270q^{56} + 16q^{58} + 34q^{61} - 258q^{62} - 127q^{64} - 30q^{65} + 375q^{68} - 150q^{70} + 58q^{73} - 372q^{74} + 15q^{76} - 330q^{77} + 360q^{80} + 127q^{82} + 140q^{85} - 273q^{86} - 75q^{88} + 192q^{89} + 258q^{92} - 36q^{94} + 148q^{97} - 585q^{98} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - 3 x^{7} + 2 x^{6} + 12 x^{5} - 36 x^{4} + 48 x^{3} + 32 x^{2} - 192 x + 256\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{7} + \nu^{6} - 10 \nu^{5} + 20 \nu^{4} + 12 \nu^{3} - 32 \nu^{2} + 96 \nu - 64 \)\()/64\)
\(\beta_{3}\)\(=\)\((\)\( \nu^{6} + \nu^{5} - 2 \nu^{4} + 12 \nu^{3} - 4 \nu^{2} - 16 \nu + 80 \)\()/16\)
\(\beta_{4}\)\(=\)\((\)\( -\nu^{7} - 5 \nu^{6} + 6 \nu^{5} - 12 \nu^{4} + 4 \nu^{3} + 48 \nu^{2} - 160 \nu \)\()/64\)
\(\beta_{5}\)\(=\)\((\)\( \nu^{7} - 3 \nu^{6} + 2 \nu^{5} + 12 \nu^{4} - 36 \nu^{3} + 48 \nu^{2} + 32 \nu - 160 \)\()/32\)
\(\beta_{6}\)\(=\)\((\)\( 3 \nu^{7} - 5 \nu^{6} - 6 \nu^{5} + 44 \nu^{4} - 60 \nu^{3} + 224 \nu - 384 \)\()/64\)
\(\beta_{7}\)\(=\)\((\)\( -\nu^{7} + 2 \nu^{6} + \nu^{5} - 6 \nu^{4} + 16 \nu^{3} - 12 \nu^{2} - 32 \nu + 80 \)\()/16\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(-\beta_{6} + \beta_{5} + \beta_{2} + \beta_{1}\)
\(\nu^{3}\)\(=\)\(\beta_{4} + \beta_{3} + \beta_{2} + 2 \beta_{1} - 4\)
\(\nu^{4}\)\(=\)\(2 \beta_{7} + 2 \beta_{6} + \beta_{5} + \beta_{4} + \beta_{3} + \beta_{2} - 2 \beta_{1} + 3\)
\(\nu^{5}\)\(=\)\(2 \beta_{7} + 6 \beta_{6} - \beta_{5} + \beta_{4} + 5 \beta_{3} - 7 \beta_{2} + 2 \beta_{1} - 11\)
\(\nu^{6}\)\(=\)\(2 \beta_{7} - 6 \beta_{6} + 7 \beta_{5} - 11 \beta_{4} + \beta_{3} + \beta_{2} - 10 \beta_{1} - 15\)
\(\nu^{7}\)\(=\)\(-22 \beta_{7} - 6 \beta_{6} - 5 \beta_{5} - 11 \beta_{4} + 17 \beta_{3} - 7 \beta_{2} - 18 \beta_{1} - 43\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/324\mathbb{Z}\right)^\times\).

\(n\) \(163\) \(245\)
\(\chi(n)\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
163.1
1.81766 + 0.834343i
1.81766 0.834343i
1.40960 + 1.41881i
1.40960 1.41881i
0.247102 + 1.98468i
0.247102 1.98468i
−1.97436 + 0.319229i
−1.97436 0.319229i
−1.81766 0.834343i 0 2.60775 + 3.03310i 6.14806 0 0.590679i −2.20934 7.68888i 0 −11.1751 5.12959i
163.2 −1.81766 + 0.834343i 0 2.60775 3.03310i 6.14806 0 0.590679i −2.20934 + 7.68888i 0 −11.1751 + 5.12959i
163.3 −1.40960 1.41881i 0 −0.0260491 + 3.99992i −8.06209 0 4.50627i 5.71184 5.60133i 0 11.3643 + 11.4386i
163.4 −1.40960 + 1.41881i 0 −0.0260491 3.99992i −8.06209 0 4.50627i 5.71184 + 5.60133i 0 11.3643 11.4386i
163.5 −0.247102 1.98468i 0 −3.87788 + 0.980835i 2.20185 0 8.35824i 2.90487 + 7.45397i 0 −0.544081 4.36996i
163.6 −0.247102 + 1.98468i 0 −3.87788 0.980835i 2.20185 0 8.35824i 2.90487 7.45397i 0 −0.544081 + 4.36996i
163.7 1.97436 0.319229i 0 3.79619 1.26055i 2.71218 0 11.5967i 7.09263 3.70062i 0 5.35481 0.865806i
163.8 1.97436 + 0.319229i 0 3.79619 + 1.26055i 2.71218 0 11.5967i 7.09263 + 3.70062i 0 5.35481 + 0.865806i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 163.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 324.3.d.g 8
3.b odd 2 1 324.3.d.i 8
4.b odd 2 1 inner 324.3.d.g 8
9.c even 3 2 108.3.f.c 16
9.d odd 6 2 36.3.f.c 16
12.b even 2 1 324.3.d.i 8
36.f odd 6 2 108.3.f.c 16
36.h even 6 2 36.3.f.c 16
72.j odd 6 2 576.3.o.g 16
72.l even 6 2 576.3.o.g 16
72.n even 6 2 1728.3.o.g 16
72.p odd 6 2 1728.3.o.g 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
36.3.f.c 16 9.d odd 6 2
36.3.f.c 16 36.h even 6 2
108.3.f.c 16 9.c even 3 2
108.3.f.c 16 36.f odd 6 2
324.3.d.g 8 1.a even 1 1 trivial
324.3.d.g 8 4.b odd 2 1 inner
324.3.d.i 8 3.b odd 2 1
324.3.d.i 8 12.b even 2 1
576.3.o.g 16 72.j odd 6 2
576.3.o.g 16 72.l even 6 2
1728.3.o.g 16 72.n even 6 2
1728.3.o.g 16 72.p odd 6 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{4} - 3 T_{5}^{3} - 53 T_{5}^{2} + 255 T_{5} - 296 \) acting on \(S_{3}^{\mathrm{new}}(324, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + 3 T + 2 T^{2} - 12 T^{3} - 36 T^{4} - 48 T^{5} + 32 T^{6} + 192 T^{7} + 256 T^{8} \)
$3$ 1
$5$ \( ( 1 - 3 T + 47 T^{2} + 30 T^{3} + 804 T^{4} + 750 T^{5} + 29375 T^{6} - 46875 T^{7} + 390625 T^{8} )^{2} \)
$7$ \( 1 - 167 T^{2} + 14701 T^{4} - 959570 T^{6} + 51275386 T^{8} - 2303927570 T^{10} + 84748339501 T^{12} - 2311494962567 T^{14} + 33232930569601 T^{16} \)
$11$ \( 1 - 524 T^{2} + 134482 T^{4} - 22990832 T^{6} + 3065351611 T^{8} - 336608771312 T^{10} + 28827411034642 T^{12} - 1644536469401804 T^{14} + 45949729863572161 T^{16} \)
$13$ \( ( 1 - 23 T + 805 T^{2} - 11762 T^{3} + 214858 T^{4} - 1987778 T^{5} + 22991605 T^{6} - 111016607 T^{7} + 815730721 T^{8} )^{2} \)
$17$ \( ( 1 + 3 T + 578 T^{2} + 6549 T^{3} + 169242 T^{4} + 1892661 T^{5} + 48275138 T^{6} + 72412707 T^{7} + 6975757441 T^{8} )^{2} \)
$19$ \( 1 - 1673 T^{2} + 1559890 T^{4} - 937716023 T^{6} + 401371470970 T^{8} - 122204089833383 T^{10} + 26492490152025490 T^{12} - 3702875859597687353 T^{14} + \)\(28\!\cdots\!81\)\( T^{16} \)
$23$ \( 1 - 2687 T^{2} + 3500317 T^{4} - 2971858562 T^{6} + 1824468134410 T^{8} - 831647871848642 T^{10} + 274113273065834077 T^{12} - 58884595848838602527 T^{14} + \)\(61\!\cdots\!61\)\( T^{16} \)
$29$ \( ( 1 - 21 T + 2873 T^{2} - 39750 T^{3} + 3357342 T^{4} - 33429750 T^{5} + 2032018313 T^{6} - 12491289741 T^{7} + 500246412961 T^{8} )^{2} \)
$31$ \( 1 - 4715 T^{2} + 11501641 T^{4} - 18322192394 T^{6} + 20727776670694 T^{8} - 16920929441899274 T^{10} + 9809646524763940681 T^{12} - \)\(37\!\cdots\!15\)\( T^{14} + \)\(72\!\cdots\!81\)\( T^{16} \)
$37$ \( ( 1 - 14 T + 3040 T^{2} - 66050 T^{3} + 4581118 T^{4} - 90422450 T^{5} + 5697449440 T^{6} - 35920169726 T^{7} + 3512479453921 T^{8} )^{2} \)
$41$ \( ( 1 - 42 T + 6188 T^{2} - 192468 T^{3} + 15263853 T^{4} - 323538708 T^{5} + 17485809068 T^{6} - 199504378122 T^{7} + 7984925229121 T^{8} )^{2} \)
$43$ \( 1 - 10292 T^{2} + 53105962 T^{4} - 172233596384 T^{6} + 381742267864051 T^{8} - 588832391551215584 T^{10} + \)\(62\!\cdots\!62\)\( T^{12} - \)\(41\!\cdots\!92\)\( T^{14} + \)\(13\!\cdots\!01\)\( T^{16} \)
$47$ \( 1 - 12983 T^{2} + 78335149 T^{4} - 294042673394 T^{6} + 766851321282202 T^{8} - 1434834446549907314 T^{10} + \)\(18\!\cdots\!89\)\( T^{12} - \)\(15\!\cdots\!03\)\( T^{14} + \)\(56\!\cdots\!21\)\( T^{16} \)
$53$ \( ( 1 - 18 T + 10016 T^{2} - 126558 T^{3} + 40472766 T^{4} - 355501422 T^{5} + 79031057696 T^{6} - 398958500322 T^{7} + 62259690411361 T^{8} )^{2} \)
$59$ \( 1 - 18092 T^{2} + 151416802 T^{4} - 803198812400 T^{6} + 3155407101414283 T^{8} - 9732649964622076400 T^{10} + \)\(22\!\cdots\!42\)\( T^{12} - \)\(32\!\cdots\!52\)\( T^{14} + \)\(21\!\cdots\!41\)\( T^{16} \)
$61$ \( ( 1 - 17 T + 3871 T^{2} - 95678 T^{3} + 31663708 T^{4} - 356017838 T^{5} + 53597250511 T^{6} - 875846364137 T^{7} + 191707312997281 T^{8} )^{2} \)
$67$ \( 1 - 25148 T^{2} + 302080258 T^{4} - 2289407522192 T^{6} + 12118574906350123 T^{8} - 46134127998001177232 T^{10} + \)\(12\!\cdots\!78\)\( T^{12} - \)\(20\!\cdots\!28\)\( T^{14} + \)\(16\!\cdots\!81\)\( T^{16} \)
$71$ \( 1 - 12968 T^{2} + 137492380 T^{4} - 912038324888 T^{6} + 5454839368725190 T^{8} - 23176426971828216728 T^{10} + \)\(88\!\cdots\!80\)\( T^{12} - \)\(21\!\cdots\!88\)\( T^{14} + \)\(41\!\cdots\!21\)\( T^{16} \)
$73$ \( ( 1 - 29 T + 7774 T^{2} - 620747 T^{3} + 46170850 T^{4} - 3307960763 T^{5} + 220767925534 T^{6} - 4388692562381 T^{7} + 806460091894081 T^{8} )^{2} \)
$79$ \( 1 - 23147 T^{2} + 207724873 T^{4} - 836021205386 T^{6} + 2483172469984678 T^{8} - 32563093667502336266 T^{10} + \)\(31\!\cdots\!53\)\( T^{12} - \)\(13\!\cdots\!27\)\( T^{14} + \)\(23\!\cdots\!21\)\( T^{16} \)
$83$ \( 1 - 17795 T^{2} + 288209353 T^{4} - 2684318937722 T^{6} + 22778145105251062 T^{8} - \)\(12\!\cdots\!62\)\( T^{10} + \)\(64\!\cdots\!73\)\( T^{12} - \)\(19\!\cdots\!95\)\( T^{14} + \)\(50\!\cdots\!81\)\( T^{16} \)
$89$ \( ( 1 - 96 T + 27716 T^{2} - 1940016 T^{3} + 308634966 T^{4} - 15366866736 T^{5} + 1738963951556 T^{6} - 47710203932256 T^{7} + 3936588805702081 T^{8} )^{2} \)
$97$ \( ( 1 - 74 T + 35344 T^{2} - 2077892 T^{3} + 488109505 T^{4} - 19550885828 T^{5} + 3128978907664 T^{6} - 61639928364746 T^{7} + 7837433594376961 T^{8} )^{2} \)
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