Properties

Label 24.4.d.a
Level 24
Weight 4
Character orbit 24.d
Analytic conductor 1.416
Analytic rank 0
Dimension 6
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) \(=\) \( 24 = 2^{3} \cdot 3 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 24.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.41604584014\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.8248384.1
Defining polynomial: \(x^{6} + x^{4} - 12 x^{3} + 4 x^{2} + 64\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{5}\cdot 3^{2} \)
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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{3} q^{2} + \beta_{1} q^{3} + ( 3 - \beta_{5} ) q^{4} + ( 2 \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} ) q^{5} + ( -1 + \beta_{2} ) q^{6} + ( 6 - \beta_{2} - 3 \beta_{3} + 2 \beta_{4} + \beta_{5} ) q^{7} + ( -14 - 4 \beta_{1} + 4 \beta_{3} - \beta_{4} - \beta_{5} ) q^{8} -9 q^{9} +O(q^{10})\) \( q + \beta_{3} q^{2} + \beta_{1} q^{3} + ( 3 - \beta_{5} ) q^{4} + ( 2 \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} ) q^{5} + ( -1 + \beta_{2} ) q^{6} + ( 6 - \beta_{2} - 3 \beta_{3} + 2 \beta_{4} + \beta_{5} ) q^{7} + ( -14 - 4 \beta_{1} + 4 \beta_{3} - \beta_{4} - \beta_{5} ) q^{8} -9 q^{9} + ( 10 - 8 \beta_{1} + 2 \beta_{2} + 2 \beta_{4} + 2 \beta_{5} ) q^{10} + ( 4 \beta_{2} - 4 \beta_{3} + 4 \beta_{5} ) q^{11} + ( -3 + 4 \beta_{1} - 3 \beta_{4} ) q^{12} + ( -4 \beta_{1} - 4 \beta_{2} - 2 \beta_{4} - 2 \beta_{5} ) q^{13} + ( -16 + 8 \beta_{1} + 2 \beta_{3} + 6 \beta_{4} + 2 \beta_{5} ) q^{14} + ( -14 - \beta_{2} + 9 \beta_{3} + 3 \beta_{5} ) q^{15} + ( 14 - 8 \beta_{1} - 4 \beta_{2} - 12 \beta_{3} - 2 \beta_{4} - 4 \beta_{5} ) q^{16} + ( 14 + 4 \beta_{2} - 12 \beta_{3} - 4 \beta_{4} - 8 \beta_{5} ) q^{17} -9 \beta_{3} q^{18} + ( -4 \beta_{1} + 4 \beta_{2} + 12 \beta_{3} + 8 \beta_{4} - 4 \beta_{5} ) q^{19} + ( 6 + 24 \beta_{1} - 8 \beta_{2} + 8 \beta_{3} - 2 \beta_{4} ) q^{20} + ( 2 \beta_{1} - 3 \beta_{2} + 9 \beta_{3} + 3 \beta_{4} - 6 \beta_{5} ) q^{21} + ( 32 + 32 \beta_{1} - 8 \beta_{4} + 8 \beta_{5} ) q^{22} + ( 52 + 2 \beta_{2} + 6 \beta_{3} - 4 \beta_{4} - 2 \beta_{5} ) q^{23} + ( 32 - 12 \beta_{1} + 4 \beta_{2} - 3 \beta_{4} + 3 \beta_{5} ) q^{24} + ( -39 + 48 \beta_{3} - 8 \beta_{4} + 8 \beta_{5} ) q^{25} + ( 12 - 32 \beta_{1} - 4 \beta_{2} + 8 \beta_{4} ) q^{26} -9 \beta_{1} q^{27} + ( -46 + 32 \beta_{1} + 8 \beta_{2} - 24 \beta_{3} + 8 \beta_{4} - 6 \beta_{5} ) q^{28} + ( -22 \beta_{1} + 7 \beta_{2} - \beta_{3} + 3 \beta_{4} + 4 \beta_{5} ) q^{29} + ( 72 + 8 \beta_{1} - 18 \beta_{3} + 6 \beta_{4} - 6 \beta_{5} ) q^{30} + ( -86 + \beta_{2} - 45 \beta_{3} + 6 \beta_{4} - 9 \beta_{5} ) q^{31} + ( -48 - 40 \beta_{1} - 8 \beta_{2} + 16 \beta_{3} + 6 \beta_{4} + 10 \beta_{5} ) q^{32} + ( 16 - 4 \beta_{2} - 36 \beta_{3} + 12 \beta_{4} ) q^{33} + ( -112 - 32 \beta_{1} + 30 \beta_{3} - 24 \beta_{4} + 8 \beta_{5} ) q^{34} + ( 4 \beta_{1} - 12 \beta_{2} - 20 \beta_{3} - 16 \beta_{4} + 4 \beta_{5} ) q^{35} + ( -27 + 9 \beta_{5} ) q^{36} + ( 40 \beta_{1} + 14 \beta_{2} - 30 \beta_{3} - 8 \beta_{4} + 22 \beta_{5} ) q^{37} + ( -124 + 32 \beta_{1} - 4 \beta_{2} - 8 \beta_{4} - 24 \beta_{5} ) q^{38} + ( 36 + 36 \beta_{3} - 6 \beta_{4} + 6 \beta_{5} ) q^{39} + ( 48 - 40 \beta_{1} + 24 \beta_{2} + 22 \beta_{4} - 6 \beta_{5} ) q^{40} + ( 66 - 12 \beta_{2} - 60 \beta_{3} + 28 \beta_{4} + 8 \beta_{5} ) q^{41} + ( -86 - 24 \beta_{1} + 2 \beta_{2} + 6 \beta_{4} - 18 \beta_{5} ) q^{42} + ( 36 \beta_{1} - 28 \beta_{2} + 12 \beta_{3} - 8 \beta_{4} - 20 \beta_{5} ) q^{43} + ( 176 + 32 \beta_{2} + 32 \beta_{3} + 16 \beta_{5} ) q^{44} + ( -18 \beta_{1} + 9 \beta_{2} + 9 \beta_{3} + 9 \beta_{4} ) q^{45} + ( 32 - 16 \beta_{1} + 60 \beta_{3} - 12 \beta_{4} - 4 \beta_{5} ) q^{46} + ( -92 + 2 \beta_{2} + 54 \beta_{3} - 12 \beta_{4} + 6 \beta_{5} ) q^{47} + ( 78 + 16 \beta_{1} - 12 \beta_{2} + 36 \beta_{3} - 12 \beta_{4} + 6 \beta_{5} ) q^{48} + ( 77 - 8 \beta_{2} + 72 \beta_{3} + 24 \beta_{5} ) q^{49} + ( 352 - 39 \beta_{3} - 32 \beta_{5} ) q^{50} + ( 30 \beta_{1} - 12 \beta_{2} - 36 \beta_{3} - 24 \beta_{4} + 12 \beta_{5} ) q^{51} + ( -52 + 16 \beta_{1} - 32 \beta_{2} + 20 \beta_{4} - 8 \beta_{5} ) q^{52} + ( 102 \beta_{1} + \beta_{2} + 41 \beta_{3} + 21 \beta_{4} - 20 \beta_{5} ) q^{53} + ( 9 - 9 \beta_{2} ) q^{54} + ( 224 + 8 \beta_{2} - 120 \beta_{3} + 8 \beta_{4} - 32 \beta_{5} ) q^{55} + ( -308 + 40 \beta_{1} + 32 \beta_{2} - 40 \beta_{3} - 22 \beta_{4} + 10 \beta_{5} ) q^{56} + ( -12 + 12 \beta_{2} - 36 \beta_{3} - 12 \beta_{4} - 24 \beta_{5} ) q^{57} + ( 18 + 56 \beta_{1} - 22 \beta_{2} - 14 \beta_{4} + 2 \beta_{5} ) q^{58} + ( -36 \beta_{1} + 32 \beta_{3} + 16 \beta_{4} - 16 \beta_{5} ) q^{59} + ( -218 + 8 \beta_{2} + 72 \beta_{3} + 6 \beta_{5} ) q^{60} + ( -160 \beta_{1} + 6 \beta_{2} - 30 \beta_{3} - 12 \beta_{4} + 18 \beta_{5} ) q^{61} + ( -336 - 8 \beta_{1} - 82 \beta_{3} - 6 \beta_{4} + 30 \beta_{5} ) q^{62} + ( -54 + 9 \beta_{2} + 27 \beta_{3} - 18 \beta_{4} - 9 \beta_{5} ) q^{63} + ( 180 + 32 \beta_{1} - 40 \beta_{2} - 72 \beta_{3} + 40 \beta_{4} - 12 \beta_{5} ) q^{64} + ( -328 + 4 \beta_{2} + 84 \beta_{3} - 20 \beta_{4} + 8 \beta_{5} ) q^{65} + ( -240 + 32 \beta_{1} + 24 \beta_{4} + 24 \beta_{5} ) q^{66} + ( -116 \beta_{1} + 48 \beta_{2} + 48 \beta_{3} + 48 \beta_{4} ) q^{67} + ( 522 - 64 \beta_{1} - 32 \beta_{2} - 96 \beta_{3} - 16 \beta_{4} + 2 \beta_{5} ) q^{68} + ( 60 \beta_{1} + 6 \beta_{2} - 18 \beta_{3} - 6 \beta_{4} + 12 \beta_{5} ) q^{69} + ( 220 - 96 \beta_{1} + 4 \beta_{2} + 24 \beta_{4} + 40 \beta_{5} ) q^{70} + ( -324 - 18 \beta_{2} + 90 \beta_{3} + 12 \beta_{4} + 42 \beta_{5} ) q^{71} + ( 126 + 36 \beta_{1} - 36 \beta_{3} + 9 \beta_{4} + 9 \beta_{5} ) q^{72} + ( 170 + 24 \beta_{2} - 24 \beta_{3} - 32 \beta_{4} - 40 \beta_{5} ) q^{73} + ( 232 + 112 \beta_{1} + 40 \beta_{2} - 28 \beta_{4} + 60 \beta_{5} ) q^{74} + ( -39 \beta_{1} + 48 \beta_{2} + 24 \beta_{4} + 24 \beta_{5} ) q^{75} + ( -260 - 144 \beta_{1} + 32 \beta_{2} - 96 \beta_{3} - 20 \beta_{4} - 16 \beta_{5} ) q^{76} + ( -256 \beta_{1} - 40 \beta_{2} + 40 \beta_{3} - 40 \beta_{5} ) q^{77} + ( 264 + 36 \beta_{3} - 24 \beta_{5} ) q^{78} + ( -14 - 11 \beta_{2} + 15 \beta_{3} + 14 \beta_{4} + 19 \beta_{5} ) q^{79} + ( -380 + 160 \beta_{1} - 40 \beta_{2} + 56 \beta_{3} - 56 \beta_{4} - 28 \beta_{5} ) q^{80} + 81 q^{81} + ( -368 + 96 \beta_{1} + 18 \beta_{3} + 72 \beta_{4} + 40 \beta_{5} ) q^{82} + ( 112 \beta_{1} + 12 \beta_{2} + 20 \beta_{3} + 16 \beta_{4} - 4 \beta_{5} ) q^{83} + ( -306 - 40 \beta_{1} - 24 \beta_{2} - 72 \beta_{3} - 18 \beta_{4} - 24 \beta_{5} ) q^{84} + ( 316 \beta_{1} - 70 \beta_{2} + 42 \beta_{3} - 14 \beta_{4} - 56 \beta_{5} ) q^{85} + ( -100 - 224 \beta_{1} + 36 \beta_{2} + 56 \beta_{4} - 24 \beta_{5} ) q^{86} + ( 202 - \beta_{2} - 63 \beta_{3} + 12 \beta_{4} - 9 \beta_{5} ) q^{87} + ( 224 + 192 \beta_{1} + 192 \beta_{3} - 80 \beta_{4} - 16 \beta_{5} ) q^{88} + ( -26 + 8 \beta_{2} - 24 \beta_{3} - 8 \beta_{4} - 16 \beta_{5} ) q^{89} + ( -90 + 72 \beta_{1} - 18 \beta_{2} - 18 \beta_{4} - 18 \beta_{5} ) q^{90} + ( 120 \beta_{1} + 20 \beta_{2} - 132 \beta_{3} - 56 \beta_{4} + 76 \beta_{5} ) q^{91} + ( 284 - 64 \beta_{1} - 16 \beta_{2} + 48 \beta_{3} - 16 \beta_{4} - 52 \beta_{5} ) q^{92} + ( -82 \beta_{1} - 45 \beta_{2} - 9 \beta_{3} - 27 \beta_{4} - 18 \beta_{5} ) q^{93} + ( 384 - 16 \beta_{1} - 84 \beta_{3} - 12 \beta_{4} - 36 \beta_{5} ) q^{94} + ( 920 - 20 \beta_{2} - 156 \beta_{3} + 56 \beta_{4} + 4 \beta_{5} ) q^{95} + ( 356 - 72 \beta_{1} + 16 \beta_{2} + 72 \beta_{3} + 30 \beta_{4} - 18 \beta_{5} ) q^{96} + ( -354 + 8 \beta_{2} - 120 \beta_{3} + 8 \beta_{4} - 32 \beta_{5} ) q^{97} + ( 576 + 64 \beta_{1} + 45 \beta_{3} + 48 \beta_{4} - 48 \beta_{5} ) q^{98} + ( -36 \beta_{2} + 36 \beta_{3} - 36 \beta_{5} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q + 2q^{2} + 16q^{4} - 6q^{6} + 28q^{7} - 76q^{8} - 54q^{9} + O(q^{10}) \) \( 6q + 2q^{2} + 16q^{4} - 6q^{6} + 28q^{7} - 76q^{8} - 54q^{9} + 60q^{10} - 12q^{12} - 100q^{14} - 60q^{15} + 56q^{16} + 52q^{17} - 18q^{18} + 56q^{20} + 224q^{22} + 328q^{23} + 204q^{24} - 106q^{25} + 56q^{26} - 352q^{28} + 372q^{30} - 636q^{31} - 248q^{32} - 548q^{34} - 144q^{36} - 776q^{38} + 312q^{39} + 232q^{40} + 236q^{41} - 564q^{42} + 1152q^{44} + 328q^{46} - 408q^{47} + 576q^{48} + 654q^{49} + 1970q^{50} - 368q^{52} + 54q^{54} + 1024q^{55} - 1864q^{56} - 168q^{57} + 140q^{58} - 1152q^{60} - 2108q^{62} - 252q^{63} + 832q^{64} - 1744q^{65} - 1440q^{66} + 2976q^{68} + 1352q^{70} - 1704q^{71} + 684q^{72} + 956q^{73} + 1568q^{74} - 1744q^{76} + 1608q^{78} - 44q^{79} - 2112q^{80} + 486q^{81} - 2236q^{82} - 1992q^{84} - 760q^{86} + 1044q^{87} + 1856q^{88} - 220q^{89} - 540q^{90} + 1728q^{92} + 2088q^{94} + 5104q^{95} + 2184q^{96} - 2444q^{97} + 3354q^{98} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6} + x^{4} - 12 x^{3} + 4 x^{2} + 64\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( -3 \nu^{5} - 6 \nu^{4} + 9 \nu^{3} + 6 \nu^{2} + 24 \nu - 96 \)\()/32\)
\(\beta_{2}\)\(=\)\((\)\( -3 \nu^{4} - 6 \nu^{3} + 9 \nu^{2} + 6 \nu + 32 \)\()/8\)
\(\beta_{3}\)\(=\)\((\)\( -\nu^{5} + \nu^{4} + \nu^{3} + 9 \nu^{2} - 6 \nu - 8 \)\()/8\)
\(\beta_{4}\)\(=\)\((\)\( -\nu^{5} + 2 \nu^{4} + 3 \nu^{3} + 6 \nu^{2} - 40 \nu - 24 \)\()/8\)
\(\beta_{5}\)\(=\)\((\)\( -\nu^{5} + 4 \nu^{4} - 9 \nu^{3} - 28 \nu + 56 \)\()/8\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-3 \beta_{4} + 3 \beta_{3} - \beta_{2} - 2\)\()/12\)
\(\nu^{2}\)\(=\)\((\)\(-3 \beta_{5} + 9 \beta_{3} + 3 \beta_{2} - 8 \beta_{1} - 6\)\()/12\)
\(\nu^{3}\)\(=\)\((\)\(-6 \beta_{5} + 3 \beta_{4} + 3 \beta_{3} - 5 \beta_{2} + 74\)\()/12\)
\(\nu^{4}\)\(=\)\((\)\(\beta_{5} - 4 \beta_{4} + 9 \beta_{3} - 5 \beta_{2} - 8 \beta_{1} - 14\)\()/4\)
\(\nu^{5}\)\(=\)\((\)\(-30 \beta_{5} + 9 \beta_{4} - 3 \beta_{3} + 13 \beta_{2} - 96 \beta_{1} - 106\)\()/12\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/24\mathbb{Z}\right)^\times\).

\(n\) \(7\) \(13\) \(17\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
13.1
−1.24181 + 1.56777i
−1.24181 1.56777i
−0.641412 + 1.89436i
−0.641412 1.89436i
1.88322 0.673417i
1.88322 + 0.673417i
−2.80958 0.325969i 3.00000i 7.78749 + 1.83167i 18.5422i 0.977907 8.42874i 9.32669 −21.2825 7.68472i −9.00000 6.04419 52.0958i
13.2 −2.80958 + 0.325969i 3.00000i 7.78749 1.83167i 18.5422i 0.977907 + 8.42874i 9.32669 −21.2825 + 7.68472i −9.00000 6.04419 + 52.0958i
13.3 1.25295 2.53577i 3.00000i −4.86025 6.35436i 9.15486i −7.60731 3.75884i 27.4175 −22.2028 + 4.36281i −9.00000 23.2146 + 11.4705i
13.4 1.25295 + 2.53577i 3.00000i −4.86025 + 6.35436i 9.15486i −7.60731 + 3.75884i 27.4175 −22.2028 4.36281i −9.00000 23.2146 11.4705i
13.5 2.55664 1.20980i 3.00000i 5.07277 6.18604i 0.612661i 3.62940 + 7.66991i −22.7441 5.48534 21.9525i −9.00000 0.741198 + 1.56635i
13.6 2.55664 + 1.20980i 3.00000i 5.07277 + 6.18604i 0.612661i 3.62940 7.66991i −22.7441 5.48534 + 21.9525i −9.00000 0.741198 1.56635i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 13.6
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 24.4.d.a 6
3.b odd 2 1 72.4.d.d 6
4.b odd 2 1 96.4.d.a 6
8.b even 2 1 inner 24.4.d.a 6
8.d odd 2 1 96.4.d.a 6
12.b even 2 1 288.4.d.d 6
16.e even 4 1 768.4.a.r 3
16.e even 4 1 768.4.a.s 3
16.f odd 4 1 768.4.a.q 3
16.f odd 4 1 768.4.a.t 3
24.f even 2 1 288.4.d.d 6
24.h odd 2 1 72.4.d.d 6
48.i odd 4 1 2304.4.a.bt 3
48.i odd 4 1 2304.4.a.bv 3
48.k even 4 1 2304.4.a.bu 3
48.k even 4 1 2304.4.a.bw 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
24.4.d.a 6 1.a even 1 1 trivial
24.4.d.a 6 8.b even 2 1 inner
72.4.d.d 6 3.b odd 2 1
72.4.d.d 6 24.h odd 2 1
96.4.d.a 6 4.b odd 2 1
96.4.d.a 6 8.d odd 2 1
288.4.d.d 6 12.b even 2 1
288.4.d.d 6 24.f even 2 1
768.4.a.q 3 16.f odd 4 1
768.4.a.r 3 16.e even 4 1
768.4.a.s 3 16.e even 4 1
768.4.a.t 3 16.f odd 4 1
2304.4.a.bt 3 48.i odd 4 1
2304.4.a.bu 3 48.k even 4 1
2304.4.a.bv 3 48.i odd 4 1
2304.4.a.bw 3 48.k even 4 1

Hecke kernels

This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(24, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - 2 T - 6 T^{2} + 40 T^{3} - 48 T^{4} - 128 T^{5} + 512 T^{6} \)
$3$ \( ( 1 + 9 T^{2} )^{3} \)
$5$ \( 1 - 322 T^{2} + 49351 T^{4} - 6170684 T^{6} + 771109375 T^{8} - 78613281250 T^{10} + 3814697265625 T^{12} \)
$7$ \( ( 1 - 14 T + 449 T^{2} - 3788 T^{3} + 154007 T^{4} - 1647086 T^{5} + 40353607 T^{6} )^{2} \)
$11$ \( 1 - 2354 T^{2} + 4518503 T^{4} - 5987722076 T^{6} + 8004803693183 T^{8} - 7387860398801234 T^{10} + 5559917313492231481 T^{12} \)
$13$ \( 1 - 8270 T^{2} + 36264983 T^{4} - 97600232804 T^{6} + 175044146329247 T^{8} - 192675163962917870 T^{10} + \)\(11\!\cdots\!29\)\( T^{12} \)
$17$ \( ( 1 - 26 T + 3615 T^{2} + 222100 T^{3} + 17760495 T^{4} - 627576794 T^{5} + 118587876497 T^{6} )^{2} \)
$19$ \( 1 - 18194 T^{2} + 183315287 T^{4} - 1372700323292 T^{6} + 8624229177682847 T^{8} - 40269051637489733234 T^{10} + \)\(10\!\cdots\!41\)\( T^{12} \)
$23$ \( ( 1 - 164 T + 42885 T^{2} - 4036280 T^{3} + 521781795 T^{4} - 24277885796 T^{5} + 1801152661463 T^{6} )^{2} \)
$29$ \( 1 - 123986 T^{2} + 6779237687 T^{4} - 212188653261788 T^{6} + 4032448674829698527 T^{8} - \)\(43\!\cdots\!26\)\( T^{10} + \)\(21\!\cdots\!61\)\( T^{12} \)
$31$ \( ( 1 + 318 T + 93849 T^{2} + 15197452 T^{3} + 2795855559 T^{4} + 282226170558 T^{5} + 26439622160671 T^{6} )^{2} \)
$37$ \( 1 - 124142 T^{2} + 4233590471 T^{4} - 51775900405988 T^{6} + 10862234876335448639 T^{8} - \)\(81\!\cdots\!02\)\( T^{10} + \)\(16\!\cdots\!29\)\( T^{12} \)
$41$ \( ( 1 - 118 T + 89463 T^{2} + 3720620 T^{3} + 6165879423 T^{4} - 560512300438 T^{5} + 327381934393961 T^{6} )^{2} \)
$43$ \( 1 - 247490 T^{2} + 32846327015 T^{4} - 3025278566260412 T^{6} + \)\(20\!\cdots\!35\)\( T^{8} - \)\(98\!\cdots\!90\)\( T^{10} + \)\(25\!\cdots\!49\)\( T^{12} \)
$47$ \( ( 1 + 204 T + 283677 T^{2} + 40395048 T^{3} + 29452197171 T^{4} + 2198959927116 T^{5} + 1119130473102767 T^{6} )^{2} \)
$53$ \( 1 - 292802 T^{2} + 13793539751 T^{4} + 2699981198933572 T^{6} + \)\(30\!\cdots\!79\)\( T^{8} - \)\(14\!\cdots\!82\)\( T^{10} + \)\(10\!\cdots\!89\)\( T^{12} \)
$59$ \( 1 - 1093858 T^{2} + 524838290887 T^{4} - 140555838506313212 T^{6} + \)\(22\!\cdots\!67\)\( T^{8} - \)\(19\!\cdots\!98\)\( T^{10} + \)\(75\!\cdots\!21\)\( T^{12} \)
$61$ \( 1 - 459870 T^{2} + 162687674679 T^{4} - 38865284671151684 T^{6} + \)\(83\!\cdots\!19\)\( T^{8} - \)\(12\!\cdots\!70\)\( T^{10} + \)\(13\!\cdots\!81\)\( T^{12} \)
$67$ \( 1 - 750066 T^{2} + 226548162807 T^{4} - 53949461413257884 T^{6} + \)\(20\!\cdots\!83\)\( T^{8} - \)\(61\!\cdots\!26\)\( T^{10} + \)\(74\!\cdots\!09\)\( T^{12} \)
$71$ \( ( 1 + 852 T + 1006773 T^{2} + 524795352 T^{3} + 360335131203 T^{4} + 109141441900692 T^{5} + 45848500718449031 T^{6} )^{2} \)
$73$ \( ( 1 - 478 T + 911095 T^{2} - 251066948 T^{3} + 354431443615 T^{4} - 72337760166142 T^{5} + 58871586708267913 T^{6} )^{2} \)
$79$ \( ( 1 + 22 T + 1407593 T^{2} + 13791100 T^{3} + 693998245127 T^{4} + 5347924021462 T^{5} + 119851595982618319 T^{6} )^{2} \)
$83$ \( 1 - 2910274 T^{2} + 3775777045015 T^{4} - 2787348361783974908 T^{6} + \)\(12\!\cdots\!35\)\( T^{8} - \)\(31\!\cdots\!14\)\( T^{10} + \)\(34\!\cdots\!09\)\( T^{12} \)
$89$ \( ( 1 + 110 T + 2073543 T^{2} + 156516836 T^{3} + 1461783535167 T^{4} + 54667942005710 T^{5} + 350356403707485209 T^{6} )^{2} \)
$97$ \( ( 1 + 1222 T + 2989679 T^{2} + 2155770388 T^{3} + 2728599301967 T^{4} + 1017891790023238 T^{5} + 760231058654565217 T^{6} )^{2} \)
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