Properties

Label 108.4.b.a
Level 108
Weight 4
Character orbit 108.b
Analytic conductor 6.372
Analytic rank 0
Dimension 12
CM no
Inner twists 4

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Newspace parameters

Level: \( N \) \(=\) \( 108 = 2^{2} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 108.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(6.37220628062\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{18}\cdot 3^{12} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(f(q)\) \(=\) \( q + \beta_{5} q^{2} + ( -1 - \beta_{4} ) q^{4} -\beta_{9} q^{5} + ( \beta_{2} + \beta_{4} ) q^{7} + ( -\beta_{5} - \beta_{9} + \beta_{10} ) q^{8} +O(q^{10})\) \( q + \beta_{5} q^{2} + ( -1 - \beta_{4} ) q^{4} -\beta_{9} q^{5} + ( \beta_{2} + \beta_{4} ) q^{7} + ( -\beta_{5} - \beta_{9} + \beta_{10} ) q^{8} + ( 2 + \beta_{3} - \beta_{6} ) q^{10} + ( \beta_{5} + \beta_{10} + \beta_{11} ) q^{11} + ( 3 - \beta_{2} + 2 \beta_{3} - \beta_{4} - \beta_{6} - \beta_{7} ) q^{13} + ( \beta_{1} + \beta_{5} + \beta_{8} - \beta_{9} - \beta_{10} - \beta_{11} ) q^{14} + ( 2 + 2 \beta_{2} + \beta_{3} + 2 \beta_{4} + \beta_{6} - \beta_{7} ) q^{16} + ( -2 \beta_{1} + 3 \beta_{5} - \beta_{8} + \beta_{9} - 2 \beta_{10} ) q^{17} + ( 2 \beta_{2} - 6 \beta_{4} - \beta_{6} + 2 \beta_{7} ) q^{19} + ( \beta_{5} - \beta_{8} + 4 \beta_{9} - 2 \beta_{11} ) q^{20} + ( 10 + 4 \beta_{2} - \beta_{3} - 3 \beta_{6} ) q^{22} + ( 2 \beta_{1} - 6 \beta_{5} - \beta_{8} + 2 \beta_{9} - 3 \beta_{10} + 3 \beta_{11} ) q^{23} + ( -11 - 8 \beta_{4} - 2 \beta_{7} ) q^{25} + ( \beta_{8} + 10 \beta_{9} + 2 \beta_{10} - 2 \beta_{11} ) q^{26} + ( 35 - 2 \beta_{2} + \beta_{3} - \beta_{4} + 5 \beta_{6} + 3 \beta_{7} ) q^{28} + ( -4 \beta_{1} - 30 \beta_{5} + 2 \beta_{8} - 2 \beta_{9} - 4 \beta_{10} ) q^{29} + ( 8 \beta_{4} + 2 \beta_{6} - 2 \beta_{7} ) q^{31} + ( 4 \beta_{1} + 3 \beta_{5} + 5 \beta_{8} + 4 \beta_{9} - 2 \beta_{11} ) q^{32} + ( -30 - 8 \beta_{2} + \beta_{3} - 8 \beta_{4} - 9 \beta_{6} - 4 \beta_{7} ) q^{34} + ( -21 \beta_{5} - 4 \beta_{8} + 7 \beta_{10} + 7 \beta_{11} ) q^{35} + ( 43 + \beta_{2} - 2 \beta_{3} + 25 \beta_{4} + \beta_{6} + 7 \beta_{7} ) q^{37} + ( \beta_{1} + 2 \beta_{5} - 6 \beta_{8} - 11 \beta_{9} + 5 \beta_{10} - 3 \beta_{11} ) q^{38} + ( -96 - 4 \beta_{2} - 2 \beta_{3} + 14 \beta_{6} - 2 \beta_{7} ) q^{40} + ( -4 \beta_{1} + 42 \beta_{5} - 6 \beta_{8} - 10 \beta_{9} - 4 \beta_{10} ) q^{41} + ( -8 \beta_{2} + 16 \beta_{4} - 2 \beta_{6} - 6 \beta_{7} ) q^{43} + ( 15 \beta_{5} + 5 \beta_{8} - 16 \beta_{9} - 4 \beta_{10} - 6 \beta_{11} ) q^{44} + ( -58 + 4 \beta_{2} - 7 \beta_{3} + 8 \beta_{4} - 13 \beta_{6} + 12 \beta_{7} ) q^{46} + ( 2 \beta_{1} + 80 \beta_{5} + 11 \beta_{8} + 2 \beta_{9} - \beta_{10} + 5 \beta_{11} ) q^{47} + ( -60 + 6 \beta_{2} - 12 \beta_{3} - 10 \beta_{4} + 6 \beta_{6} + 2 \beta_{7} ) q^{49} + ( -11 \beta_{5} + 8 \beta_{8} - 8 \beta_{9} + 8 \beta_{10} ) q^{50} + ( 17 - 8 \beta_{3} + \beta_{4} + 24 \beta_{6} - 4 \beta_{7} ) q^{52} + ( 36 \beta_{5} - 4 \beta_{8} - 14 \beta_{9} ) q^{53} + ( -16 \beta_{2} - 40 \beta_{4} + 8 \beta_{6} + 6 \beta_{7} ) q^{55} + ( 4 \beta_{1} + 32 \beta_{5} - 15 \beta_{8} + 13 \beta_{9} + 7 \beta_{10} + 6 \beta_{11} ) q^{56} + ( 236 - 16 \beta_{2} + 6 \beta_{3} + 16 \beta_{4} - 22 \beta_{6} - 8 \beta_{7} ) q^{58} + ( -4 \beta_{1} - 95 \beta_{5} - 14 \beta_{8} - 4 \beta_{9} + 11 \beta_{10} - \beta_{11} ) q^{59} + ( -81 + \beta_{2} - 2 \beta_{3} - 15 \beta_{4} + \beta_{6} - 3 \beta_{7} ) q^{61} + ( 2 \beta_{1} + 8 \beta_{8} + 10 \beta_{9} - 6 \beta_{10} + 2 \beta_{11} ) q^{62} + ( 208 + 4 \beta_{2} - 6 \beta_{3} + 26 \beta_{6} + 10 \beta_{7} ) q^{64} + ( 6 \beta_{1} - 189 \beta_{5} + 23 \beta_{8} + 3 \beta_{9} + 6 \beta_{10} ) q^{65} + ( -4 \beta_{2} - 4 \beta_{4} - 15 \beta_{6} ) q^{67} + ( -16 \beta_{1} - 39 \beta_{5} + 7 \beta_{8} + 4 \beta_{9} - 2 \beta_{11} ) q^{68} + ( -154 + 28 \beta_{2} - 7 \beta_{3} + 32 \beta_{4} - 21 \beta_{6} ) q^{70} + ( 4 \beta_{1} - 34 \beta_{5} - 2 \beta_{8} + 4 \beta_{9} - 28 \beta_{10} - 16 \beta_{11} ) q^{71} + ( 55 - 6 \beta_{2} + 12 \beta_{3} + 26 \beta_{4} - 6 \beta_{6} + 2 \beta_{7} ) q^{73} + ( 46 \beta_{5} - 25 \beta_{8} + 14 \beta_{9} - 26 \beta_{10} + 2 \beta_{11} ) q^{74} + ( -417 + 6 \beta_{2} + 13 \beta_{3} + 11 \beta_{4} + 17 \beta_{6} - 5 \beta_{7} ) q^{76} + ( 4 \beta_{1} + 174 \beta_{5} - 18 \beta_{8} + 57 \beta_{9} + 4 \beta_{10} ) q^{77} + ( 23 \beta_{2} + 31 \beta_{4} + 18 \beta_{6} - 2 \beta_{7} ) q^{79} + ( 8 \beta_{1} - 98 \beta_{5} + 6 \beta_{8} + 12 \beta_{9} + 12 \beta_{10} + 20 \beta_{11} ) q^{80} + ( -324 - 16 \beta_{2} + 14 \beta_{3} - 48 \beta_{4} - 30 \beta_{6} - 8 \beta_{7} ) q^{82} + ( -4 \beta_{1} + 196 \beta_{5} + 30 \beta_{8} - 4 \beta_{9} - 6 \beta_{10} - 18 \beta_{11} ) q^{83} + ( 88 - 8 \beta_{2} + 16 \beta_{3} - 16 \beta_{4} - 8 \beta_{6} - 10 \beta_{7} ) q^{85} + ( -10 \beta_{1} - 8 \beta_{5} + 16 \beta_{8} + 30 \beta_{9} - 18 \beta_{10} + 6 \beta_{11} ) q^{86} + ( 264 - 20 \beta_{2} + 22 \beta_{3} - 24 \beta_{4} + 6 \beta_{6} - 2 \beta_{7} ) q^{88} + ( 14 \beta_{1} + 123 \beta_{5} - 9 \beta_{8} + 41 \beta_{9} + 14 \beta_{10} ) q^{89} + ( 46 \beta_{2} + 38 \beta_{4} - 27 \beta_{6} + 2 \beta_{7} ) q^{91} + ( -16 \beta_{1} - 47 \beta_{5} - 37 \beta_{8} - 48 \beta_{9} - 28 \beta_{10} - 10 \beta_{11} ) q^{92} + ( 634 + 12 \beta_{2} - 9 \beta_{3} - 88 \beta_{4} - 19 \beta_{6} + 12 \beta_{7} ) q^{94} + ( -10 \beta_{1} - 140 \beta_{5} - 19 \beta_{8} - 10 \beta_{9} + 13 \beta_{10} - 17 \beta_{11} ) q^{95} + ( 211 - 6 \beta_{2} + 12 \beta_{3} + 2 \beta_{4} - 6 \beta_{6} - 4 \beta_{7} ) q^{97} + ( -42 \beta_{5} + 10 \beta_{8} - 76 \beta_{9} + 4 \beta_{10} + 12 \beta_{11} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12q - 12q^{4} + O(q^{10}) \) \( 12q - 12q^{4} + 24q^{10} + 36q^{13} + 24q^{16} + 120q^{22} - 132q^{25} + 420q^{28} - 360q^{34} + 516q^{37} - 1152q^{40} - 696q^{46} - 720q^{49} + 204q^{52} + 2832q^{58} - 972q^{61} + 2496q^{64} - 1848q^{70} + 660q^{73} - 5004q^{76} - 3888q^{82} + 1056q^{85} + 3168q^{88} + 7608q^{94} + 2532q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{12} - 12 x^{10} + 112 x^{8} - 368 x^{6} + 928 x^{4} - 256 x^{2} + 64\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( -34 \nu^{11} + 383 \nu^{9} - 2918 \nu^{7} + 3680 \nu^{5} + 28168 \nu^{3} - 112656 \nu \)\()/3152\)
\(\beta_{2}\)\(=\)\((\)\( -149 \nu^{10} + 1916 \nu^{8} - 16832 \nu^{6} + 54600 \nu^{4} - 87232 \nu^{2} - 1152 \)\()/6304\)
\(\beta_{3}\)\(=\)\((\)\( -128 \nu^{10} + 1523 \nu^{8} - 13952 \nu^{6} + 40368 \nu^{4} - 69448 \nu^{2} - 94640 \)\()/3152\)
\(\beta_{4}\)\(=\)\((\)\( 283 \nu^{10} - 3298 \nu^{8} + 30256 \nu^{6} - 92280 \nu^{4} + 215120 \nu^{2} - 17824 \)\()/6304\)
\(\beta_{5}\)\(=\)\((\)\( 347 \nu^{11} - 4158 \nu^{9} + 38808 \nu^{7} - 126648 \nu^{5} + 315248 \nu^{3} - 50880 \nu \)\()/12608\)
\(\beta_{6}\)\(=\)\((\)\( 252 \nu^{10} - 2943 \nu^{8} + 27468 \nu^{6} - 85680 \nu^{4} + 227592 \nu^{2} - 34416 \)\()/3152\)
\(\beta_{7}\)\(=\)\((\)\( -211 \nu^{10} + 2626 \nu^{8} - 24772 \nu^{6} + 86712 \nu^{4} - 213584 \nu^{2} + 47616 \)\()/1576\)
\(\beta_{8}\)\(=\)\((\)\( -1261 \nu^{11} + 14002 \nu^{9} - 128584 \nu^{7} + 350728 \nu^{5} - 853776 \nu^{3} - 352704 \nu \)\()/12608\)
\(\beta_{9}\)\(=\)\((\)\( -190 \nu^{11} + 2233 \nu^{9} - 20710 \nu^{7} + 64600 \nu^{5} - 159552 \nu^{3} + 4560 \nu \)\()/1576\)
\(\beta_{10}\)\(=\)\((\)\( -2105 \nu^{11} + 25294 \nu^{9} - 235552 \nu^{7} + 773224 \nu^{5} - 1928752 \nu^{3} + 632064 \nu \)\()/12608\)
\(\beta_{11}\)\(=\)\((\)\( 1453 \nu^{11} - 17764 \nu^{9} + 166060 \nu^{7} - 567304 \nu^{5} + 1428384 \nu^{3} - 616416 \nu \)\()/6304\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(2 \beta_{11} + 6 \beta_{10} - 6 \beta_{9} + \beta_{8} - 3 \beta_{5}\)\()/72\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{7} + 6 \beta_{6} - 8 \beta_{4} + 2 \beta_{3} - 4 \beta_{2} + 72\)\()/36\)
\(\nu^{3}\)\(=\)\((\)\(2 \beta_{10} - 16 \beta_{9} + 5 \beta_{8} - 39 \beta_{5} + 2 \beta_{1}\)\()/18\)
\(\nu^{4}\)\(=\)\((\)\(5 \beta_{7} + 26 \beta_{6} - 46 \beta_{4} - 8 \beta_{3} - 14 \beta_{2} - 240\)\()/18\)
\(\nu^{5}\)\(=\)\((\)\(-21 \beta_{11} - 28 \beta_{10} - 28 \beta_{9} + 20 \beta_{8} - 42 \beta_{5} + 5 \beta_{1}\)\()/9\)
\(\nu^{6}\)\(=\)\((\)\(-2 \beta_{7} + 32 \beta_{6} - 104 \beta_{4} - 64 \beta_{3} + 32 \beta_{2} - 1800\)\()/9\)
\(\nu^{7}\)\(=\)\((\)\(-58 \beta_{11} - 82 \beta_{10} + 90 \beta_{9} + 3 \beta_{8} + 395 \beta_{5} + 4 \beta_{1}\)\()/3\)
\(\nu^{8}\)\(=\)\((\)\(-130 \beta_{7} - 492 \beta_{6} + 704 \beta_{4} - 260 \beta_{3} + 856 \beta_{2} - 7104\)\()/9\)
\(\nu^{9}\)\(=\)\((\)\(-32 \beta_{10} + 1352 \beta_{9} - 460 \beta_{8} + 4044 \beta_{5} - 32 \beta_{1}\)\()/3\)
\(\nu^{10}\)\(=\)\((\)\(-676 \beta_{7} - 6056 \beta_{6} + 13544 \beta_{4} + 2128 \beta_{3} + 5032 \beta_{2} + 57408\)\()/9\)
\(\nu^{11}\)\(=\)\((\)\(11832 \beta_{11} + 15344 \beta_{10} + 15344 \beta_{9} - 12496 \beta_{8} + 15504 \beta_{5} - 1576 \beta_{1}\)\()/9\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/108\mathbb{Z}\right)^\times\).

\(n\) \(29\) \(55\)
\(\chi(n)\) \(-1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
107.1
−1.61829 0.934317i
−1.61829 + 0.934317i
0.456937 0.263813i
0.456937 + 0.263813i
2.48442 + 1.43438i
2.48442 1.43438i
−2.48442 + 1.43438i
−2.48442 1.43438i
−0.456937 0.263813i
−0.456937 + 0.263813i
1.61829 0.934317i
1.61829 + 0.934317i
−2.72087 0.772562i 0 6.80630 + 4.20408i 3.33155i 0 16.9016i −15.2712 16.6971i 0 −2.57383 + 9.06473i
107.2 −2.72087 + 0.772562i 0 6.80630 4.20408i 3.33155i 0 16.9016i −15.2712 + 16.6971i 0 −2.57383 9.06473i
107.3 −1.51859 2.38619i 0 −3.38780 + 7.24726i 13.1987i 0 4.49091i 22.4380 2.92167i 0 −31.4945 + 20.0433i
107.4 −1.51859 + 2.38619i 0 −3.38780 7.24726i 13.1987i 0 4.49091i 22.4380 + 2.92167i 0 −31.4945 20.0433i
107.5 −0.889241 2.68500i 0 −6.41850 + 4.77523i 14.9230i 0 30.0528i 18.5291 + 12.9874i 0 40.0683 13.2701i
107.6 −0.889241 + 2.68500i 0 −6.41850 4.77523i 14.9230i 0 30.0528i 18.5291 12.9874i 0 40.0683 + 13.2701i
107.7 0.889241 2.68500i 0 −6.41850 4.77523i 14.9230i 0 30.0528i −18.5291 + 12.9874i 0 40.0683 + 13.2701i
107.8 0.889241 + 2.68500i 0 −6.41850 + 4.77523i 14.9230i 0 30.0528i −18.5291 12.9874i 0 40.0683 13.2701i
107.9 1.51859 2.38619i 0 −3.38780 7.24726i 13.1987i 0 4.49091i −22.4380 2.92167i 0 −31.4945 20.0433i
107.10 1.51859 + 2.38619i 0 −3.38780 + 7.24726i 13.1987i 0 4.49091i −22.4380 + 2.92167i 0 −31.4945 + 20.0433i
107.11 2.72087 0.772562i 0 6.80630 4.20408i 3.33155i 0 16.9016i 15.2712 16.6971i 0 −2.57383 9.06473i
107.12 2.72087 + 0.772562i 0 6.80630 + 4.20408i 3.33155i 0 16.9016i 15.2712 + 16.6971i 0 −2.57383 + 9.06473i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 107.12
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 108.4.b.a 12
3.b odd 2 1 inner 108.4.b.a 12
4.b odd 2 1 inner 108.4.b.a 12
8.b even 2 1 1728.4.c.i 12
8.d odd 2 1 1728.4.c.i 12
12.b even 2 1 inner 108.4.b.a 12
24.f even 2 1 1728.4.c.i 12
24.h odd 2 1 1728.4.c.i 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
108.4.b.a 12 1.a even 1 1 trivial
108.4.b.a 12 3.b odd 2 1 inner
108.4.b.a 12 4.b odd 2 1 inner
108.4.b.a 12 12.b even 2 1 inner
1728.4.c.i 12 8.b even 2 1
1728.4.c.i 12 8.d odd 2 1
1728.4.c.i 12 24.f even 2 1
1728.4.c.i 12 24.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{6} + 408 T_{5}^{4} + 43200 T_{5}^{2} + 430592 \) acting on \(S_{4}^{\mathrm{new}}(108, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + 6 T^{2} + 12 T^{4} - 416 T^{6} + 768 T^{8} + 24576 T^{10} + 262144 T^{12} \)
$3$ 1
$5$ \( ( 1 - 342 T^{2} + 73575 T^{4} - 11181908 T^{6} + 1149609375 T^{8} - 83496093750 T^{10} + 3814697265625 T^{12} )^{2} \)
$7$ \( ( 1 - 849 T^{2} + 387966 T^{4} - 141880421 T^{6} + 45643811934 T^{8} - 11751252833649 T^{10} + 1628413597910449 T^{12} )^{2} \)
$11$ \( ( 1 + 4074 T^{2} + 8337351 T^{4} + 12032309932 T^{6} + 14770125874911 T^{8} + 12785957206761354 T^{10} + 5559917313492231481 T^{12} )^{2} \)
$13$ \( ( 1 - 9 T + 1962 T^{2} + 66427 T^{3} + 4310514 T^{4} - 43441281 T^{5} + 10604499373 T^{6} )^{4} \)
$17$ \( ( 1 - 13134 T^{2} + 78982671 T^{4} - 357919940036 T^{6} + 1906449671066799 T^{8} - 7652160463775680974 T^{10} + \)\(14\!\cdots\!09\)\( T^{12} )^{2} \)
$19$ \( ( 1 - 18825 T^{2} + 202841862 T^{4} - 1656627349277 T^{6} + 9542874101470422 T^{8} - 41665653351420480825 T^{10} + \)\(10\!\cdots\!41\)\( T^{12} )^{2} \)
$23$ \( ( 1 + 6306 T^{2} + 110681535 T^{4} + 2149854095644 T^{6} + 16384839429609615 T^{8} + \)\(13\!\cdots\!26\)\( T^{10} + \)\(32\!\cdots\!69\)\( T^{12} )^{2} \)
$29$ \( ( 1 - 43326 T^{2} + 1075658103 T^{4} - 29470174509188 T^{6} + 639826525087020063 T^{8} - \)\(15\!\cdots\!66\)\( T^{10} + \)\(21\!\cdots\!61\)\( T^{12} )^{2} \)
$31$ \( ( 1 - 153750 T^{2} + 10409070351 T^{4} - 400107422263412 T^{6} + 9238088252300462031 T^{8} - \)\(12\!\cdots\!50\)\( T^{10} + \)\(69\!\cdots\!41\)\( T^{12} )^{2} \)
$37$ \( ( 1 - 129 T + 39378 T^{2} + 7527427 T^{3} + 1994613834 T^{4} - 330978706761 T^{5} + 129961739795077 T^{6} )^{4} \)
$41$ \( ( 1 - 232950 T^{2} + 25214641887 T^{4} - 1908694392304628 T^{6} + \)\(11\!\cdots\!67\)\( T^{8} - \)\(52\!\cdots\!50\)\( T^{10} + \)\(10\!\cdots\!21\)\( T^{12} )^{2} \)
$43$ \( ( 1 - 262254 T^{2} + 37283382183 T^{4} - 3613899564944228 T^{6} + \)\(23\!\cdots\!67\)\( T^{8} - \)\(10\!\cdots\!54\)\( T^{10} + \)\(25\!\cdots\!49\)\( T^{12} )^{2} \)
$47$ \( ( 1 + 179538 T^{2} + 22658226447 T^{4} + 3067316968586236 T^{6} + \)\(24\!\cdots\!63\)\( T^{8} + \)\(20\!\cdots\!58\)\( T^{10} + \)\(12\!\cdots\!89\)\( T^{12} )^{2} \)
$53$ \( ( 1 - 768750 T^{2} + 259538463591 T^{4} - 49798991602056548 T^{6} + \)\(57\!\cdots\!39\)\( T^{8} - \)\(37\!\cdots\!50\)\( T^{10} + \)\(10\!\cdots\!89\)\( T^{12} )^{2} \)
$59$ \( ( 1 + 523338 T^{2} + 180986988135 T^{4} + 40857650874211948 T^{6} + \)\(76\!\cdots\!35\)\( T^{8} + \)\(93\!\cdots\!78\)\( T^{10} + \)\(75\!\cdots\!21\)\( T^{12} )^{2} \)
$61$ \( ( 1 + 243 T + 648738 T^{2} + 106648063 T^{3} + 147251199978 T^{4} + 12519450969723 T^{5} + 11694146092834141 T^{6} )^{4} \)
$67$ \( ( 1 - 1637409 T^{2} + 1163514522486 T^{4} - 457988207385885221 T^{6} + \)\(10\!\cdots\!34\)\( T^{8} - \)\(13\!\cdots\!49\)\( T^{10} + \)\(74\!\cdots\!09\)\( T^{12} )^{2} \)
$71$ \( ( 1 + 377994 T^{2} + 13348121343 T^{4} - 1180461843095924 T^{6} + \)\(17\!\cdots\!03\)\( T^{8} + \)\(62\!\cdots\!54\)\( T^{10} + \)\(21\!\cdots\!61\)\( T^{12} )^{2} \)
$73$ \( ( 1 - 165 T + 834942 T^{2} - 185551841 T^{3} + 324806632014 T^{4} - 24970147337685 T^{5} + 58871586708267913 T^{6} )^{4} \)
$79$ \( ( 1 - 2108721 T^{2} + 2206739657838 T^{4} - 1369032710710548773 T^{6} + \)\(53\!\cdots\!98\)\( T^{8} - \)\(12\!\cdots\!61\)\( T^{10} + \)\(14\!\cdots\!61\)\( T^{12} )^{2} \)
$83$ \( ( 1 + 290850 T^{2} + 143373074583 T^{4} + 347024073827678524 T^{6} + \)\(46\!\cdots\!27\)\( T^{8} + \)\(31\!\cdots\!50\)\( T^{10} + \)\(34\!\cdots\!09\)\( T^{12} )^{2} \)
$89$ \( ( 1 - 1956702 T^{2} + 2321980636575 T^{4} - 1921108256399844452 T^{6} + \)\(11\!\cdots\!75\)\( T^{8} - \)\(48\!\cdots\!42\)\( T^{10} + \)\(12\!\cdots\!81\)\( T^{12} )^{2} \)
$97$ \( ( 1 - 633 T + 2697438 T^{2} - 1126776701 T^{3} + 2461878831774 T^{4} - 527271279120057 T^{5} + 760231058654565217 T^{6} )^{4} \)
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