Properties

Label 224.3.o.c
Level 224
Weight 3
Character orbit 224.o
Analytic conductor 6.104
Analytic rank 0
Dimension 12
CM no
Inner twists 4

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Newspace parameters

Level: \( N \) = \( 224 = 2^{5} \cdot 7 \)
Weight: \( k \) = \( 3 \)
Character orbit: \([\chi]\) = 224.o (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(6.10355792167\)
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: \( 2^{6} \)
Twist minimal: no (minimal twist has level 56)
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.

\(f(q)\) \(=\) \( q + ( -1 + \beta_{2} - \beta_{8} ) q^{3} + ( \beta_{1} - \beta_{4} + \beta_{5} + \beta_{9} ) q^{5} + ( -\beta_{1} + \beta_{5} + \beta_{6} - \beta_{10} ) q^{7} + ( -\beta_{2} + \beta_{3} - \beta_{7} + \beta_{8} ) q^{9} +O(q^{10})\) \( q + ( -1 + \beta_{2} - \beta_{8} ) q^{3} + ( \beta_{1} - \beta_{4} + \beta_{5} + \beta_{9} ) q^{5} + ( -\beta_{1} + \beta_{5} + \beta_{6} - \beta_{10} ) q^{7} + ( -\beta_{2} + \beta_{3} - \beta_{7} + \beta_{8} ) q^{9} + ( 2 - 2 \beta_{2} + 2 \beta_{8} - \beta_{11} ) q^{11} + ( \beta_{1} - \beta_{6} - \beta_{9} ) q^{13} + ( -4 \beta_{1} + \beta_{5} - 6 \beta_{6} + 4 \beta_{9} ) q^{15} + ( -14 + 14 \beta_{2} + 3 \beta_{8} - \beta_{11} ) q^{17} + ( 16 \beta_{2} + \beta_{3} + 2 \beta_{7} - 2 \beta_{8} ) q^{19} + ( \beta_{1} + \beta_{4} - 2 \beta_{5} - \beta_{6} - 5 \beta_{10} ) q^{21} + ( 5 \beta_{1} + \beta_{4} - \beta_{5} + 3 \beta_{6} - \beta_{9} - 3 \beta_{10} ) q^{23} + ( 18 - 18 \beta_{2} - 6 \beta_{8} - 4 \beta_{11} ) q^{25} + ( 4 - 3 \beta_{3} - 4 \beta_{7} - 3 \beta_{11} ) q^{27} + ( -9 \beta_{1} + 4 \beta_{5} - 5 \beta_{6} + 9 \beta_{9} ) q^{29} + ( -5 \beta_{4} + \beta_{9} - 2 \beta_{10} ) q^{31} + ( 23 \beta_{2} - 4 \beta_{3} + 6 \beta_{7} - 6 \beta_{8} ) q^{33} + ( -11 - 22 \beta_{2} + 4 \beta_{3} + 7 \beta_{7} - \beta_{11} ) q^{35} + ( -6 \beta_{1} - \beta_{4} + \beta_{5} + 5 \beta_{6} + \beta_{9} - 5 \beta_{10} ) q^{37} + ( 2 \beta_{4} - \beta_{9} + \beta_{10} ) q^{39} + ( 11 + 3 \beta_{3} + 3 \beta_{7} + 3 \beta_{11} ) q^{41} + ( -2 + 4 \beta_{3} - 6 \beta_{7} + 4 \beta_{11} ) q^{43} + ( 4 \beta_{4} - 7 \beta_{9} + 11 \beta_{10} ) q^{45} + ( 12 \beta_{1} - \beta_{4} + \beta_{5} - 6 \beta_{6} + \beta_{9} + 6 \beta_{10} ) q^{47} + ( -8 - 16 \beta_{2} + \beta_{3} - 7 \beta_{7} + 14 \beta_{8} + 5 \beta_{11} ) q^{49} + ( 16 \beta_{2} - 5 \beta_{3} - 10 \beta_{7} + 10 \beta_{8} ) q^{51} + ( -5 \beta_{4} - 9 \beta_{9} + 9 \beta_{10} ) q^{53} + ( 7 \beta_{1} + 3 \beta_{5} + 17 \beta_{6} - 7 \beta_{9} ) q^{55} + ( -31 - 12 \beta_{7} ) q^{57} + ( -9 + 9 \beta_{2} + 11 \beta_{8} + 4 \beta_{11} ) q^{59} + ( 32 \beta_{1} + 3 \beta_{4} - 3 \beta_{5} + 11 \beta_{6} - 3 \beta_{9} - 11 \beta_{10} ) q^{61} + ( -7 \beta_{1} + 2 \beta_{4} - 2 \beta_{5} - 7 \beta_{6} - 14 \beta_{9} + 9 \beta_{10} ) q^{63} + ( -11 \beta_{2} - \beta_{3} + 7 \beta_{7} - 7 \beta_{8} ) q^{65} + ( 33 - 33 \beta_{2} - 9 \beta_{8} + 10 \beta_{11} ) q^{67} + ( -11 \beta_{1} - 7 \beta_{5} - 6 \beta_{6} + 11 \beta_{9} ) q^{69} + ( -22 \beta_{1} - 8 \beta_{5} + 8 \beta_{6} + 22 \beta_{9} ) q^{71} + ( 11 - 11 \beta_{2} + 8 \beta_{8} + 6 \beta_{11} ) q^{73} + ( -24 \beta_{2} - 2 \beta_{3} + 34 \beta_{7} - 34 \beta_{8} ) q^{75} + ( 11 \beta_{1} + \beta_{4} + 2 \beta_{5} + 3 \beta_{6} - 21 \beta_{9} + 12 \beta_{10} ) q^{77} + ( -17 \beta_{1} + 5 \beta_{4} - 5 \beta_{5} - 3 \beta_{6} - 5 \beta_{9} + 3 \beta_{10} ) q^{79} + ( 32 - 32 \beta_{2} - 7 \beta_{8} - 7 \beta_{11} ) q^{81} + ( 28 - 14 \beta_{3} - 6 \beta_{7} - 14 \beta_{11} ) q^{83} + ( -6 \beta_{1} - 15 \beta_{5} + 23 \beta_{6} + 6 \beta_{9} ) q^{85} + ( 14 \beta_{4} - 29 \beta_{9} - \beta_{10} ) q^{87} + ( -7 \beta_{2} - 8 \beta_{3} + 4 \beta_{7} - 4 \beta_{8} ) q^{89} + ( -3 + 22 \beta_{2} + 3 \beta_{3} + 7 \beta_{7} + \beta_{11} ) q^{91} + ( -4 \beta_{1} - \beta_{4} + \beta_{5} - 25 \beta_{6} + \beta_{9} + 25 \beta_{10} ) q^{93} + ( -7 \beta_{4} + 30 \beta_{9} - 7 \beta_{10} ) q^{95} + ( -11 + 13 \beta_{3} - \beta_{7} + 13 \beta_{11} ) q^{97} + ( -71 + 5 \beta_{3} - 21 \beta_{7} + 5 \beta_{11} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12q - 6q^{3} - 8q^{9} + O(q^{10}) \) \( 12q - 6q^{3} - 8q^{9} + 14q^{11} - 82q^{17} + 94q^{19} + 116q^{25} + 60q^{27} + 146q^{33} - 270q^{35} + 120q^{41} - 40q^{43} - 204q^{49} + 106q^{51} - 372q^{57} - 62q^{59} - 64q^{65} + 178q^{67} + 54q^{73} - 140q^{75} + 206q^{81} + 392q^{83} - 26q^{89} + 88q^{91} - 184q^{97} - 872q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{12} + 2 x^{10} - 12 x^{9} + 12 x^{8} - 12 x^{7} + 148 x^{6} - 48 x^{5} + 192 x^{4} - 768 x^{3} + 512 x^{2} + 4096\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( -11 \nu^{11} + 48 \nu^{10} + 298 \nu^{9} + 420 \nu^{8} + 1212 \nu^{7} - 1980 \nu^{6} + 7012 \nu^{5} - 4656 \nu^{4} + 30912 \nu^{3} - 18432 \nu^{2} + 57856 \nu - 49152 \)\()/93184\)
\(\beta_{2}\)\(=\)\((\)\( 6 \nu^{11} - 51 \nu^{10} + 36 \nu^{9} - 14 \nu^{8} + 828 \nu^{7} - 12 \nu^{6} + 444 \nu^{5} - 3516 \nu^{4} + 1008 \nu^{3} + 2112 \nu^{2} + 28800 \nu + 52224 \)\()/46592\)
\(\beta_{3}\)\(=\)\((\)\( -10 \nu^{11} - 97 \nu^{10} - 60 \nu^{9} - 826 \nu^{8} + 804 \nu^{7} + 748 \nu^{6} + 7268 \nu^{5} - 3604 \nu^{4} - 10416 \nu^{3} + 13952 \nu^{2} + 21888 \nu + 99328 \)\()/46592\)
\(\beta_{4}\)\(=\)\((\)\( 16 \nu^{11} + 319 \nu^{10} + 96 \nu^{9} - 98 \nu^{8} - 340 \nu^{7} - 396 \nu^{6} + 1548 \nu^{5} + 11372 \nu^{4} + 4144 \nu^{3} - 192 \nu^{2} - 28032 \nu - 512 \)\()/46592\)
\(\beta_{5}\)\(=\)\((\)\( 3 \nu^{11} + 20 \nu^{10} - 86 \nu^{9} + 84 \nu^{8} - 132 \nu^{7} + 956 \nu^{6} - 324 \nu^{5} + 2064 \nu^{4} - 9584 \nu^{3} + 3968 \nu^{2} - 3072 \nu + 46080 \)\()/6656\)
\(\beta_{6}\)\(=\)\((\)\( -8 \nu^{11} - 23 \nu^{10} + 43 \nu^{9} - 42 \nu^{8} + 170 \nu^{7} + 16 \nu^{6} + 136 \nu^{5} - 2592 \nu^{4} - 2436 \nu^{3} - 2816 \nu^{2} + 8192 \nu + 11904 \)\()/11648\)
\(\beta_{7}\)\(=\)\((\)\( -\nu^{11} + 6 \nu^{9} + 12 \nu^{8} + 4 \nu^{7} - 84 \nu^{6} - 52 \nu^{5} - 48 \nu^{4} + 480 \nu^{3} + 384 \nu^{2} + 1024 \nu - 3072 \)\()/1024\)
\(\beta_{8}\)\(=\)\((\)\( 51 \nu^{11} - 24 \nu^{10} + 306 \nu^{9} - 756 \nu^{8} + 668 \nu^{7} - 3924 \nu^{6} + 7596 \nu^{5} - 4224 \nu^{4} + 23856 \nu^{3} - 43200 \nu^{2} + 64256 \nu - 115200 \)\()/46592\)
\(\beta_{9}\)\(=\)\((\)\( 51 \nu^{11} - 24 \nu^{10} + 306 \nu^{9} - 756 \nu^{8} + 668 \nu^{7} - 3924 \nu^{6} + 7596 \nu^{5} - 4224 \nu^{4} + 23856 \nu^{3} - 43200 \nu^{2} - 28928 \nu - 115200 \)\()/46592\)
\(\beta_{10}\)\(=\)\((\)\( 75 \nu^{11} - 46 \nu^{10} + 450 \nu^{9} - 448 \nu^{8} + 1796 \nu^{7} - 1788 \nu^{6} + 7188 \nu^{5} - 14648 \nu^{4} + 19152 \nu^{3} + 192 \nu^{2} + 86272 \nu + 512 \)\()/46592\)
\(\beta_{11}\)\(=\)\((\)\( -81 \nu^{11} - 267 \nu^{10} - 486 \nu^{9} - 266 \nu^{8} + 1744 \nu^{7} - 2568 \nu^{6} - 3264 \nu^{5} - 35708 \nu^{4} - 2688 \nu^{3} - 72192 \nu^{2} + 71296 \nu - 192512 \)\()/46592\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-\beta_{9} + \beta_{8}\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{10} - \beta_{8} + \beta_{7} - \beta_{6} + \beta_{3} - \beta_{2} - \beta_{1}\)\()/2\)
\(\nu^{3}\)\(=\)\(-\beta_{9} - \beta_{7} - \beta_{6} - \beta_{5} + \beta_{1} + 3\)
\(\nu^{4}\)\(=\)\(-\beta_{11} - 3 \beta_{10} + 2 \beta_{8} + 7 \beta_{2} - 7\)
\(\nu^{5}\)\(=\)\(\beta_{10} + 2 \beta_{9} - \beta_{8} + \beta_{7} - \beta_{6} + 2 \beta_{5} - 2 \beta_{4} + 3 \beta_{3} - 9 \beta_{2} + 7 \beta_{1}\)
\(\nu^{6}\)\(=\)\(-2 \beta_{11} - 10 \beta_{9} - 14 \beta_{7} + 4 \beta_{6} - 6 \beta_{5} - 2 \beta_{3} + 10 \beta_{1} - 28\)
\(\nu^{7}\)\(=\)\(-12 \beta_{10} + 14 \beta_{9} - 10 \beta_{8} + 12 \beta_{4} + 84 \beta_{2} - 84\)
\(\nu^{8}\)\(=\)\(6 \beta_{10} + 24 \beta_{9} - 18 \beta_{8} + 18 \beta_{7} - 6 \beta_{6} + 24 \beta_{5} - 24 \beta_{4} - 18 \beta_{3} - 22 \beta_{2} + 54 \beta_{1}\)
\(\nu^{9}\)\(=\)\(-36 \beta_{11} + 8 \beta_{9} - 8 \beta_{7} + 120 \beta_{6} - 12 \beta_{5} - 36 \beta_{3} - 8 \beta_{1} - 120\)
\(\nu^{10}\)\(=\)\(44 \beta_{11} + 76 \beta_{10} - 40 \beta_{9} - 40 \beta_{8} + 168 \beta_{4} - 124 \beta_{2} + 124\)
\(\nu^{11}\)\(=\)\(308 \beta_{10} + 136 \beta_{9} + 20 \beta_{8} - 20 \beta_{7} - 308 \beta_{6} + 136 \beta_{5} - 136 \beta_{4} - 228 \beta_{3} + 12 \beta_{2} - 316 \beta_{1}\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/224\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(129\) \(197\)
\(\chi(n)\) \(-1\) \(-\beta_{2}\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
79.1
1.71059 + 1.03628i
0.0421483 + 1.99956i
1.83041 0.805972i
−1.61320 + 1.18220i
−0.685878 1.87872i
−1.28408 1.53335i
1.71059 1.03628i
0.0421483 1.99956i
1.83041 + 0.805972i
−1.61320 1.18220i
−0.685878 + 1.87872i
−1.28408 + 1.53335i
0 −2.25274 3.90186i 0 −6.07099 3.50509i 0 −2.51181 6.53382i 0 −5.64968 + 9.78553i 0
79.2 0 −2.25274 3.90186i 0 6.07099 + 3.50509i 0 2.51181 + 6.53382i 0 −5.64968 + 9.78553i 0
79.3 0 −0.717214 1.24225i 0 −2.27256 1.31206i 0 5.39122 + 4.46483i 0 3.47121 6.01231i 0
79.4 0 −0.717214 1.24225i 0 2.27256 + 1.31206i 0 −5.39122 4.46483i 0 3.47121 6.01231i 0
79.5 0 1.46995 + 2.54604i 0 −7.59793 4.38667i 0 3.55324 6.03112i 0 0.178469 0.309118i 0
79.6 0 1.46995 + 2.54604i 0 7.59793 + 4.38667i 0 −3.55324 + 6.03112i 0 0.178469 0.309118i 0
207.1 0 −2.25274 + 3.90186i 0 −6.07099 + 3.50509i 0 −2.51181 + 6.53382i 0 −5.64968 9.78553i 0
207.2 0 −2.25274 + 3.90186i 0 6.07099 3.50509i 0 2.51181 6.53382i 0 −5.64968 9.78553i 0
207.3 0 −0.717214 + 1.24225i 0 −2.27256 + 1.31206i 0 5.39122 4.46483i 0 3.47121 + 6.01231i 0
207.4 0 −0.717214 + 1.24225i 0 2.27256 1.31206i 0 −5.39122 + 4.46483i 0 3.47121 + 6.01231i 0
207.5 0 1.46995 2.54604i 0 −7.59793 + 4.38667i 0 3.55324 + 6.03112i 0 0.178469 + 0.309118i 0
207.6 0 1.46995 2.54604i 0 7.59793 4.38667i 0 −3.55324 6.03112i 0 0.178469 + 0.309118i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 207.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner
8.d odd 2 1 inner
56.k odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 224.3.o.c 12
4.b odd 2 1 56.3.k.c 12
7.c even 3 1 inner 224.3.o.c 12
7.c even 3 1 1568.3.g.k 6
7.d odd 6 1 1568.3.g.i 6
8.b even 2 1 56.3.k.c 12
8.d odd 2 1 inner 224.3.o.c 12
28.d even 2 1 392.3.k.k 12
28.f even 6 1 392.3.g.l 6
28.f even 6 1 392.3.k.k 12
28.g odd 6 1 56.3.k.c 12
28.g odd 6 1 392.3.g.k 6
56.h odd 2 1 392.3.k.k 12
56.j odd 6 1 392.3.g.l 6
56.j odd 6 1 392.3.k.k 12
56.k odd 6 1 inner 224.3.o.c 12
56.k odd 6 1 1568.3.g.k 6
56.m even 6 1 1568.3.g.i 6
56.p even 6 1 56.3.k.c 12
56.p even 6 1 392.3.g.k 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
56.3.k.c 12 4.b odd 2 1
56.3.k.c 12 8.b even 2 1
56.3.k.c 12 28.g odd 6 1
56.3.k.c 12 56.p even 6 1
224.3.o.c 12 1.a even 1 1 trivial
224.3.o.c 12 7.c even 3 1 inner
224.3.o.c 12 8.d odd 2 1 inner
224.3.o.c 12 56.k odd 6 1 inner
392.3.g.k 6 28.g odd 6 1
392.3.g.k 6 56.p even 6 1
392.3.g.l 6 28.f even 6 1
392.3.g.l 6 56.j odd 6 1
392.3.k.k 12 28.d even 2 1
392.3.k.k 12 28.f even 6 1
392.3.k.k 12 56.h odd 2 1
392.3.k.k 12 56.j odd 6 1
1568.3.g.i 6 7.d odd 6 1
1568.3.g.i 6 56.m even 6 1
1568.3.g.k 6 7.c even 3 1
1568.3.g.k 6 56.k odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(224, [\chi])\):

\( T_{3}^{6} + 3 T_{3}^{5} + 20 T_{3}^{4} + 5 T_{3}^{3} + 178 T_{3}^{2} + 209 T_{3} + 361 \)
\( T_{5}^{12} - 133 T_{5}^{10} + 13038 T_{5}^{8} - 566489 T_{5}^{6} + 18167550 T_{5}^{4} - 121144597 T_{5}^{2} + 678446209 \)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( ( 1 + 3 T - 7 T^{2} - 22 T^{3} + 7 T^{4} - 61 T^{5} - 350 T^{6} - 549 T^{7} + 567 T^{8} - 16038 T^{9} - 45927 T^{10} + 177147 T^{11} + 531441 T^{12} )^{2} \)
$5$ \( 1 + 17 T^{2} - 437 T^{4} - 28164 T^{6} - 270975 T^{8} + 5916803 T^{10} + 456152134 T^{12} + 3698001875 T^{14} - 105849609375 T^{16} - 6875976562500 T^{18} - 66680908203125 T^{20} + 1621246337890625 T^{22} + 59604644775390625 T^{24} \)
$7$ \( 1 + 102 T^{2} + 8463 T^{4} + 426692 T^{6} + 20319663 T^{8} + 588009702 T^{10} + 13841287201 T^{12} \)
$11$ \( ( 1 - 7 T - 193 T^{2} + 2268 T^{3} + 15415 T^{4} - 171941 T^{5} - 336170 T^{6} - 20804861 T^{7} + 225691015 T^{8} + 4017900348 T^{9} - 41371264033 T^{10} - 181561972207 T^{11} + 3138428376721 T^{12} )^{2} \)
$13$ \( ( 1 - 950 T^{2} + 386271 T^{4} - 85941828 T^{6} + 11032286031 T^{8} - 774944184950 T^{10} + 23298085122481 T^{12} )^{2} \)
$17$ \( ( 1 + 41 T + 471 T^{2} + 5456 T^{3} + 209253 T^{4} + 1462951 T^{5} - 27854634 T^{6} + 422792839 T^{7} + 17477019813 T^{8} + 131694576464 T^{9} + 3285581754711 T^{10} + 82655749918409 T^{11} + 582622237229761 T^{12} )^{2} \)
$19$ \( ( 1 - 47 T + 487 T^{2} - 8532 T^{3} + 642195 T^{4} - 10028861 T^{5} + 46257550 T^{6} - 3620418821 T^{7} + 83691494595 T^{8} - 401395456692 T^{9} + 8270995200967 T^{10} - 288160114116647 T^{11} + 2213314919066161 T^{12} )^{2} \)
$23$ \( 1 + 2381 T^{2} + 2947123 T^{4} + 2827993812 T^{6} + 2298643835613 T^{8} + 1531108455942719 T^{10} + 862820921151363886 T^{12} + \)\(42\!\cdots\!79\)\( T^{14} + \)\(18\!\cdots\!53\)\( T^{16} + \)\(61\!\cdots\!52\)\( T^{18} + \)\(18\!\cdots\!03\)\( T^{20} + \)\(40\!\cdots\!81\)\( T^{22} + \)\(48\!\cdots\!41\)\( T^{24} \)
$29$ \( ( 1 - 1838 T^{2} + 2723567 T^{4} - 2423469812 T^{6} + 1926327191327 T^{8} - 919452907022318 T^{10} + 353814783205469041 T^{12} )^{2} \)
$31$ \( 1 + 3329 T^{2} + 4631887 T^{4} + 6475839496 T^{6} + 10686875408317 T^{8} + 11546553377441207 T^{10} + 9928800654054771830 T^{12} + \)\(10\!\cdots\!47\)\( T^{14} + \)\(91\!\cdots\!97\)\( T^{16} + \)\(51\!\cdots\!56\)\( T^{18} + \)\(33\!\cdots\!47\)\( T^{20} + \)\(22\!\cdots\!29\)\( T^{22} + \)\(62\!\cdots\!21\)\( T^{24} \)
$37$ \( 1 + 6401 T^{2} + 22091395 T^{4} + 55331146644 T^{6} + 111395987301489 T^{8} + 188098942339055675 T^{10} + \)\(27\!\cdots\!06\)\( T^{12} + \)\(35\!\cdots\!75\)\( T^{14} + \)\(39\!\cdots\!69\)\( T^{16} + \)\(36\!\cdots\!64\)\( T^{18} + \)\(27\!\cdots\!95\)\( T^{20} + \)\(14\!\cdots\!01\)\( T^{22} + \)\(43\!\cdots\!61\)\( T^{24} \)
$41$ \( ( 1 - 30 T + 4755 T^{2} - 100496 T^{3} + 7993155 T^{4} - 84772830 T^{5} + 4750104241 T^{6} )^{4} \)
$43$ \( ( 1 + 10 T + 3855 T^{2} + 53388 T^{3} + 7127895 T^{4} + 34188010 T^{5} + 6321363049 T^{6} )^{4} \)
$47$ \( 1 + 8313 T^{2} + 34702703 T^{4} + 101154992824 T^{6} + 247422958854189 T^{8} + 586359622432093519 T^{10} + \)\(13\!\cdots\!26\)\( T^{12} + \)\(28\!\cdots\!39\)\( T^{14} + \)\(58\!\cdots\!29\)\( T^{16} + \)\(11\!\cdots\!84\)\( T^{18} + \)\(19\!\cdots\!63\)\( T^{20} + \)\(22\!\cdots\!13\)\( T^{22} + \)\(13\!\cdots\!81\)\( T^{24} \)
$53$ \( 1 + 9273 T^{2} + 34839939 T^{4} + 130830291652 T^{6} + 620089921398753 T^{8} + 1876428047292173523 T^{10} + \)\(44\!\cdots\!94\)\( T^{12} + \)\(14\!\cdots\!63\)\( T^{14} + \)\(38\!\cdots\!33\)\( T^{16} + \)\(64\!\cdots\!32\)\( T^{18} + \)\(13\!\cdots\!19\)\( T^{20} + \)\(28\!\cdots\!73\)\( T^{22} + \)\(24\!\cdots\!81\)\( T^{24} \)
$59$ \( ( 1 + 31 T - 7567 T^{2} - 176030 T^{3} + 36214647 T^{4} + 412965247 T^{5} - 131958052142 T^{6} + 1437532024807 T^{7} + 438825951186567 T^{8} - 7425039336825230 T^{9} - 1111065921351897007 T^{10} + 15844619352319883431 T^{11} + \)\(17\!\cdots\!81\)\( T^{12} )^{2} \)
$61$ \( 1 - 503 T^{2} - 13520125 T^{4} - 91919186268 T^{6} + 20373908662561 T^{8} + 680714329682975843 T^{10} + \)\(44\!\cdots\!06\)\( T^{12} + \)\(94\!\cdots\!63\)\( T^{14} + \)\(39\!\cdots\!41\)\( T^{16} - \)\(24\!\cdots\!28\)\( T^{18} - \)\(49\!\cdots\!25\)\( T^{20} - \)\(25\!\cdots\!03\)\( T^{22} + \)\(70\!\cdots\!41\)\( T^{24} \)
$67$ \( ( 1 - 89 T - 19 T^{2} + 251662 T^{3} - 15244389 T^{4} + 746405371 T^{5} - 32107721582 T^{6} + 3350613710419 T^{7} - 307191527310069 T^{8} + 22764937373414878 T^{9} - 7715285873576179 T^{10} - \)\(16\!\cdots\!61\)\( T^{11} + \)\(81\!\cdots\!61\)\( T^{12} )^{2} \)
$71$ \( ( 1 - 7830 T^{2} + 72414895 T^{4} - 385201987444 T^{6} + 1840184211388495 T^{8} - 5056250149654308630 T^{10} + \)\(16\!\cdots\!41\)\( T^{12} )^{2} \)
$73$ \( ( 1 - 27 T - 12877 T^{2} + 216188 T^{3} + 104738937 T^{4} - 821298961 T^{5} - 619422939130 T^{6} - 4376702163169 T^{7} + 2974401575009817 T^{8} + 32716643712966332 T^{9} - 10384786603320081037 T^{10} - \)\(11\!\cdots\!23\)\( T^{11} + \)\(22\!\cdots\!21\)\( T^{12} )^{2} \)
$79$ \( 1 + 27485 T^{2} + 389982499 T^{4} + 4285373244996 T^{6} + 40173777270526125 T^{8} + \)\(31\!\cdots\!47\)\( T^{10} + \)\(20\!\cdots\!50\)\( T^{12} + \)\(12\!\cdots\!07\)\( T^{14} + \)\(60\!\cdots\!25\)\( T^{16} + \)\(25\!\cdots\!36\)\( T^{18} + \)\(89\!\cdots\!79\)\( T^{20} + \)\(24\!\cdots\!85\)\( T^{22} + \)\(34\!\cdots\!81\)\( T^{24} \)
$83$ \( ( 1 - 98 T + 12183 T^{2} - 526716 T^{3} + 83928687 T^{4} - 4650915458 T^{5} + 326940373369 T^{6} )^{4} \)
$89$ \( ( 1 + 13 T - 19181 T^{2} - 141748 T^{3} + 218594713 T^{4} + 635752183 T^{5} - 1943770650106 T^{6} + 5035793041543 T^{7} + 13715122164371833 T^{8} - 70446104031139828 T^{9} - 75507709882171615661 T^{10} + \)\(40\!\cdots\!13\)\( T^{11} + \)\(24\!\cdots\!21\)\( T^{12} )^{2} \)
$97$ \( ( 1 + 46 T + 18423 T^{2} + 458352 T^{3} + 173342007 T^{4} + 4072346926 T^{5} + 832972004929 T^{6} )^{4} \)
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