Properties

Label 7500.2.a.m
Level $7500$
Weight $2$
Character orbit 7500.a
Self dual yes
Analytic conductor $59.888$
Analytic rank $1$
Dimension $12$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 7500 = 2^{2} \cdot 3 \cdot 5^{4} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7500.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(59.8878015160\)
Analytic rank: \(1\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Defining polynomial: \(x^{12} - 6 x^{11} - 11 x^{10} + 94 x^{9} + 27 x^{8} - 460 x^{7} + 55 x^{6} + 812 x^{5} - 127 x^{4} - 512 x^{3} + 40 x^{2} + 72 x - 4\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 5^{3} \)
Twist minimal: no (minimal twist has level 300)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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.

\(f(q)\) \(=\) \( q - q^{3} + ( -1 + \beta_{7} ) q^{7} + q^{9} +O(q^{10})\) \( q - q^{3} + ( -1 + \beta_{7} ) q^{7} + q^{9} + \beta_{8} q^{11} -\beta_{10} q^{13} + ( -1 - \beta_{2} - \beta_{11} ) q^{17} + ( 1 + \beta_{2} - \beta_{5} - \beta_{8} + \beta_{10} ) q^{19} + ( 1 - \beta_{7} ) q^{21} + ( -1 + \beta_{5} - \beta_{6} - \beta_{7} ) q^{23} - q^{27} + ( 1 - \beta_{2} + \beta_{3} ) q^{29} + ( -\beta_{1} + \beta_{2} - \beta_{3} + \beta_{6} - \beta_{7} - \beta_{8} - \beta_{9} + \beta_{10} + \beta_{11} ) q^{31} -\beta_{8} q^{33} + ( 1 + \beta_{1} + \beta_{4} + \beta_{5} + \beta_{9} - \beta_{10} + 2 \beta_{11} ) q^{37} + \beta_{10} q^{39} + ( 1 + \beta_{1} + \beta_{5} + \beta_{6} + \beta_{8} + \beta_{9} + \beta_{11} ) q^{41} + ( -3 + 2 \beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} + 2 \beta_{6} - \beta_{8} - \beta_{9} + \beta_{10} + \beta_{11} ) q^{43} + ( -1 + \beta_{3} + \beta_{4} - \beta_{5} + \beta_{10} ) q^{47} + ( 3 - \beta_{6} - 2 \beta_{7} - \beta_{8} + \beta_{9} - \beta_{10} - \beta_{11} ) q^{49} + ( 1 + \beta_{2} + \beta_{11} ) q^{51} + ( -3 - \beta_{1} - \beta_{2} - 3 \beta_{4} - \beta_{7} + \beta_{9} + \beta_{10} - \beta_{11} ) q^{53} + ( -1 - \beta_{2} + \beta_{5} + \beta_{8} - \beta_{10} ) q^{57} + ( -1 + 2 \beta_{1} - \beta_{2} - 2 \beta_{4} + 2 \beta_{5} - \beta_{6} + 2 \beta_{8} - \beta_{10} ) q^{59} + ( 3 - \beta_{2} - \beta_{3} - \beta_{6} - \beta_{7} - \beta_{10} ) q^{61} + ( -1 + \beta_{7} ) q^{63} + ( -3 - 2 \beta_{1} - \beta_{3} + \beta_{4} - \beta_{5} - \beta_{8} - \beta_{9} + \beta_{10} - \beta_{11} ) q^{67} + ( 1 - \beta_{5} + \beta_{6} + \beta_{7} ) q^{69} + ( 2 - 2 \beta_{1} - \beta_{2} + \beta_{3} + 3 \beta_{4} - \beta_{5} - 2 \beta_{6} - 2 \beta_{7} - \beta_{8} - \beta_{10} - 2 \beta_{11} ) q^{71} + ( -2 + \beta_{1} + 2 \beta_{3} - \beta_{4} + \beta_{7} + \beta_{10} ) q^{73} + ( 1 + 2 \beta_{2} + 4 \beta_{4} - \beta_{5} + \beta_{6} - \beta_{7} - 2 \beta_{8} - 2 \beta_{9} + 2 \beta_{10} ) q^{77} + ( 1 + 2 \beta_{1} + \beta_{2} + \beta_{5} + \beta_{6} - \beta_{9} + \beta_{11} ) q^{79} + q^{81} + ( -2 + \beta_{2} - \beta_{3} - \beta_{4} - 2 \beta_{7} - \beta_{8} - \beta_{9} + \beta_{11} ) q^{83} + ( -1 + \beta_{2} - \beta_{3} ) q^{87} + ( -2 \beta_{1} + \beta_{2} - 2 \beta_{3} - \beta_{4} - \beta_{5} + \beta_{6} - 2 \beta_{7} - \beta_{8} - 2 \beta_{9} ) q^{89} + ( 3 - \beta_{1} - 2 \beta_{2} + 3 \beta_{3} + 2 \beta_{4} + \beta_{5} - 2 \beta_{6} + \beta_{7} - \beta_{8} + 2 \beta_{9} + \beta_{10} - 2 \beta_{11} ) q^{91} + ( \beta_{1} - \beta_{2} + \beta_{3} - \beta_{6} + \beta_{7} + \beta_{8} + \beta_{9} - \beta_{10} - \beta_{11} ) q^{93} + ( 1 - 2 \beta_{1} + \beta_{3} + 3 \beta_{4} - \beta_{5} - 2 \beta_{6} - 2 \beta_{7} - \beta_{8} - \beta_{10} - \beta_{11} ) q^{97} + \beta_{8} q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12q - 12q^{3} - 8q^{7} + 12q^{9} + O(q^{10}) \) \( 12q - 12q^{3} - 8q^{7} + 12q^{9} + 2q^{11} - 8q^{17} + 10q^{19} + 8q^{21} - 18q^{23} - 12q^{27} + 8q^{29} - 2q^{31} - 2q^{33} - 4q^{37} + 10q^{41} - 28q^{43} - 22q^{47} + 28q^{49} + 8q^{51} - 16q^{53} - 10q^{57} - 2q^{59} + 34q^{61} - 8q^{63} - 32q^{67} + 18q^{69} - 24q^{73} - 18q^{77} + 6q^{79} + 12q^{81} - 28q^{83} - 8q^{87} + 10q^{89} + 20q^{91} + 2q^{93} - 16q^{97} + 2q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{12} - 6 x^{11} - 11 x^{10} + 94 x^{9} + 27 x^{8} - 460 x^{7} + 55 x^{6} + 812 x^{5} - 127 x^{4} - 512 x^{3} + 40 x^{2} + 72 x - 4\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\(1902023 \nu^{11} - 15346545 \nu^{10} + 3038089 \nu^{9} + 220227853 \nu^{8} - 329926105 \nu^{7} - 909974979 \nu^{6} + 1944407257 \nu^{5} + 799087297 \nu^{4} - 3160553209 \nu^{3} + 477033175 \nu^{2} + 1283973348 \nu - 427658178\)\()/ 168665204 \)
\(\beta_{2}\)\(=\)\((\)\(573781 \nu^{11} + 249144 \nu^{10} - 30517201 \nu^{9} + 28884678 \nu^{8} + 364863104 \nu^{7} - 383566399 \nu^{6} - 1429176716 \nu^{5} + 1551855632 \nu^{4} + 1644633434 \nu^{3} - 1558387780 \nu^{2} - 533575158 \nu + 193537713\)\()/42166301\)
\(\beta_{3}\)\(=\)\((\)\(4446263 \nu^{11} - 31885257 \nu^{10} - 15534203 \nu^{9} + 460483725 \nu^{8} - 384922893 \nu^{7} - 1955450995 \nu^{6} + 2622496817 \nu^{5} + 2115679805 \nu^{4} - 4596778413 \nu^{3} + 543616735 \nu^{2} + 2251682668 \nu - 471915530\)\()/ 168665204 \)
\(\beta_{4}\)\(=\)\((\)\(-391865 \nu^{11} + 2401909 \nu^{10} + 4018773 \nu^{9} - 37312825 \nu^{8} - 6589229 \nu^{7} + 179797043 \nu^{6} - 35115263 \nu^{5} - 298851685 \nu^{4} + 57237739 \nu^{3} + 154217921 \nu^{2} - 9043168 \nu - 14339954\)\()/8877116\)
\(\beta_{5}\)\(=\)\((\)\(-8420182 \nu^{11} + 49252135 \nu^{10} + 92384342 \nu^{9} - 744704695 \nu^{8} - 195693826 \nu^{7} + 3339948103 \nu^{6} - 991891852 \nu^{5} - 4996541349 \nu^{4} + 2864832352 \nu^{3} + 2549576465 \nu^{2} - 2134099514 \nu - 334004182\)\()/ 168665204 \)
\(\beta_{6}\)\(=\)\((\)\(-8914978 \nu^{11} + 50577467 \nu^{10} + 111939570 \nu^{9} - 797245563 \nu^{8} - 415423858 \nu^{7} + 3889973671 \nu^{6} - 89286696 \nu^{5} - 6961200305 \nu^{4} + 1445986192 \nu^{3} + 3891072669 \nu^{2} - 847833314 \nu + 220528906\)\()/ 168665204 \)
\(\beta_{7}\)\(=\)\((\)\(-2517509 \nu^{11} + 15676974 \nu^{10} + 25167924 \nu^{9} - 246751910 \nu^{8} - 30334377 \nu^{7} + 1223880325 \nu^{6} - 277813878 \nu^{5} - 2168660915 \nu^{4} + 455108743 \nu^{3} + 1335522859 \nu^{2} - 183251934 \nu - 138029335\)\()/42166301\)
\(\beta_{8}\)\(=\)\((\)\(-5284275 \nu^{11} + 34049245 \nu^{10} + 40352913 \nu^{9} - 495464015 \nu^{8} + 90173559 \nu^{7} + 2100765259 \nu^{6} - 1072808513 \nu^{5} - 2447662093 \nu^{4} + 903666945 \nu^{3} + 283268561 \nu^{2} + 311995052 \nu + 357264474\)\()/84332602\)
\(\beta_{9}\)\(=\)\((\)\(11430516 \nu^{11} - 55973135 \nu^{10} - 195330276 \nu^{9} + 909529415 \nu^{8} + 1380545088 \nu^{7} - 4533486639 \nu^{6} - 4245380734 \nu^{5} + 8480947257 \nu^{4} + 5630984674 \nu^{3} - 5809628005 \nu^{2} - 1986390078 \nu + 843366226\)\()/ 168665204 \)
\(\beta_{10}\)\(=\)\((\)\(-13569902 \nu^{11} + 91413839 \nu^{10} + 93467242 \nu^{9} - 1398305819 \nu^{8} + 463244558 \nu^{7} + 6739023411 \nu^{6} - 4315055896 \nu^{5} - 11511925689 \nu^{4} + 6443638716 \nu^{3} + 7278045201 \nu^{2} - 1706190394 \nu - 898531966\)\()/ 168665204 \)
\(\beta_{11}\)\(=\)\((\)\(26459466 \nu^{11} - 158726547 \nu^{10} - 270630514 \nu^{9} + 2376011335 \nu^{8} + 452955598 \nu^{7} - 10563029603 \nu^{6} + 2595314028 \nu^{5} + 15054853845 \nu^{4} - 4362974516 \nu^{3} - 6271292845 \nu^{2} + 1093845846 \nu + 260029058\)\()/ 168665204 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-\beta_{11} + 2 \beta_{9} + \beta_{8} - 2 \beta_{6} + \beta_{5} - \beta_{4} - 2 \beta_{2} + \beta_{1} + 2\)\()/5\)
\(\nu^{2}\)\(=\)\((\)\(-4 \beta_{11} - 2 \beta_{10} + 6 \beta_{9} - 2 \beta_{8} + \beta_{7} - 6 \beta_{6} + 2 \beta_{5} + 6 \beta_{4} + \beta_{3} - 3 \beta_{2} + 5 \beta_{1} + 28\)\()/5\)
\(\nu^{3}\)\(=\)\((\)\(-14 \beta_{11} - 8 \beta_{10} + 32 \beta_{9} + 10 \beta_{8} + 14 \beta_{7} - 25 \beta_{6} + 17 \beta_{5} - 9 \beta_{4} - \beta_{3} - 23 \beta_{2} + 28 \beta_{1} + 48\)\()/5\)
\(\nu^{4}\)\(=\)\((\)\(-74 \beta_{11} - 43 \beta_{10} + 126 \beta_{9} - 5 \beta_{8} + 54 \beta_{7} - 100 \beta_{6} + 39 \beta_{5} + 26 \beta_{4} + 4 \beta_{3} - 73 \beta_{2} + 108 \beta_{1} + 323\)\()/5\)
\(\nu^{5}\)\(=\)\((\)\(-312 \beta_{11} - 178 \beta_{10} + 574 \beta_{9} + 88 \beta_{8} + 324 \beta_{7} - 431 \beta_{6} + 203 \beta_{5} - 182 \beta_{4} - 16 \beta_{3} - 379 \beta_{2} + 486 \beta_{1} + 949\)\()/5\)
\(\nu^{6}\)\(=\)\((\)\(-1499 \beta_{11} - 814 \beta_{10} + 2379 \beta_{9} + 17 \beta_{8} + 1412 \beta_{7} - 1814 \beta_{6} + 540 \beta_{5} - 559 \beta_{4} - 68 \beta_{3} - 1478 \beta_{2} + 1961 \beta_{1} + 4733\)\()/5\)
\(\nu^{7}\)\(=\)\((\)\(-6711 \beta_{11} - 3528 \beta_{10} + 10503 \beta_{9} + 627 \beta_{8} + 7104 \beta_{7} - 7924 \beta_{6} + 2210 \beta_{5} - 5136 \beta_{4} - 601 \beta_{3} - 6877 \beta_{2} + 8405 \beta_{1} + 17267\)\()/5\)
\(\nu^{8}\)\(=\)\((\)\(-31128 \beta_{11} - 15717 \beta_{10} + 44816 \beta_{9} - 298 \beta_{8} + 32406 \beta_{7} - 34139 \beta_{6} + 5782 \beta_{5} - 24183 \beta_{4} - 3274 \beta_{3} - 28976 \beta_{2} + 35019 \beta_{1} + 77161\)\()/5\)
\(\nu^{9}\)\(=\)\((\)\(-141469 \beta_{11} - 69421 \beta_{10} + 196849 \beta_{9} - 194 \beta_{8} + 154368 \beta_{7} - 149967 \beta_{6} + 18264 \beta_{5} - 136579 \beta_{4} - 18267 \beta_{3} - 130423 \beta_{2} + 149322 \beta_{1} + 307868\)\()/5\)
\(\nu^{10}\)\(=\)\((\)\(-648941 \beta_{11} - 309058 \beta_{10} + 855413 \beta_{9} - 30978 \beta_{8} + 711629 \beta_{7} - 655659 \beta_{6} + 29115 \beta_{5} - 665216 \beta_{4} - 94011 \beta_{3} - 567892 \beta_{2} + 634165 \beta_{1} + 1329277\)\()/5\)
\(\nu^{11}\)\(=\)\((\)\(-2958644 \beta_{11} - 1377117 \beta_{10} + 3769058 \beta_{9} - 169657 \beta_{8} + 3319976 \beta_{7} - 2896281 \beta_{6} - 18872 \beta_{5} - 3348094 \beta_{4} - 474289 \beta_{3} - 2536718 \beta_{2} + 2722525 \beta_{1} + 5525588\)\()/5\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−2.61083
0.0550688
1.62661
3.89742
1.09160
0.434428
1.83182
−0.879783
−0.446694
−1.20648
−2.34221
4.54905
0 −1.00000 0 0 0 −4.62675 0 1.00000 0
1.2 0 −1.00000 0 0 0 −4.41540 0 1.00000 0
1.3 0 −1.00000 0 0 0 −3.80992 0 1.00000 0
1.4 0 −1.00000 0 0 0 −3.78808 0 1.00000 0
1.5 0 −1.00000 0 0 0 −2.44380 0 1.00000 0
1.6 0 −1.00000 0 0 0 −1.04684 0 1.00000 0
1.7 0 −1.00000 0 0 0 0.595901 0 1.00000 0
1.8 0 −1.00000 0 0 0 0.957526 0 1.00000 0
1.9 0 −1.00000 0 0 0 1.31873 0 1.00000 0
1.10 0 −1.00000 0 0 0 1.57893 0 1.00000 0
1.11 0 −1.00000 0 0 0 3.54704 0 1.00000 0
1.12 0 −1.00000 0 0 0 4.13266 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.12
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(1\)
\(5\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 7500.2.a.m 12
5.b even 2 1 7500.2.a.n 12
5.c odd 4 2 7500.2.d.g 24
25.d even 5 2 1500.2.m.d 24
25.e even 10 2 1500.2.m.c 24
25.f odd 20 2 300.2.o.a 24
25.f odd 20 2 1500.2.o.c 24
75.l even 20 2 900.2.w.c 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
300.2.o.a 24 25.f odd 20 2
900.2.w.c 24 75.l even 20 2
1500.2.m.c 24 25.e even 10 2
1500.2.m.d 24 25.d even 5 2
1500.2.o.c 24 25.f odd 20 2
7500.2.a.m 12 1.a even 1 1 trivial
7500.2.a.n 12 5.b even 2 1
7500.2.d.g 24 5.c odd 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{7}^{12} + \cdots\) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(7500))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \)
$3$ \( ( 1 + T )^{12} \)
$5$ \( T^{12} \)
$7$ \( 13136 - 30272 T - 6744 T^{2} + 44640 T^{3} - 7175 T^{4} - 19212 T^{5} + 2289 T^{6} + 3768 T^{7} + 10 T^{8} - 300 T^{9} - 24 T^{10} + 8 T^{11} + T^{12} \)
$11$ \( 11216 - 30832 T - 142804 T^{2} - 42780 T^{3} + 114645 T^{4} + 24138 T^{5} - 25266 T^{6} - 2922 T^{7} + 2180 T^{8} + 130 T^{9} - 79 T^{10} - 2 T^{11} + T^{12} \)
$13$ \( 45125 + 45000 T - 260125 T^{2} + 41000 T^{3} + 187175 T^{4} - 21800 T^{5} - 42025 T^{6} + 2000 T^{7} + 3310 T^{8} - 40 T^{9} - 100 T^{10} + T^{12} \)
$17$ \( -1075799 + 3273818 T - 2618254 T^{2} - 423720 T^{3} + 1131890 T^{4} - 194902 T^{5} - 101251 T^{6} + 20328 T^{7} + 4295 T^{8} - 690 T^{9} - 99 T^{10} + 8 T^{11} + T^{12} \)
$19$ \( 50000 - 3250000 T - 5362500 T^{2} - 1605500 T^{3} + 897025 T^{4} + 383650 T^{5} - 63225 T^{6} - 28800 T^{7} + 2945 T^{8} + 900 T^{9} - 85 T^{10} - 10 T^{11} + T^{12} \)
$23$ \( 15235856 + 17299328 T - 2800424 T^{2} - 9593360 T^{3} - 3222195 T^{4} + 296908 T^{5} + 310054 T^{6} + 30698 T^{7} - 7740 T^{8} - 1530 T^{9} + 11 T^{10} + 18 T^{11} + T^{12} \)
$29$ \( -8040059 + 15235032 T - 3651184 T^{2} - 5373620 T^{3} + 1526100 T^{4} + 697152 T^{5} - 164201 T^{6} - 39458 T^{7} + 7155 T^{8} + 950 T^{9} - 139 T^{10} - 8 T^{11} + T^{12} \)
$31$ \( 5989136 + 29646192 T - 6230964 T^{2} - 12022460 T^{3} + 3661485 T^{4} + 929412 T^{5} - 372156 T^{6} - 19738 T^{7} + 13690 T^{8} - 40 T^{9} - 199 T^{10} + 2 T^{11} + T^{12} \)
$37$ \( 69436261 - 318432924 T - 36360416 T^{2} + 158406220 T^{3} + 33050690 T^{4} - 7893624 T^{5} - 1806129 T^{6} + 151344 T^{7} + 36735 T^{8} - 1280 T^{9} - 321 T^{10} + 4 T^{11} + T^{12} \)
$41$ \( 59575625 + 49848750 T - 35081625 T^{2} - 28208750 T^{3} + 5689275 T^{4} + 3374300 T^{5} - 442450 T^{6} - 129200 T^{7} + 14595 T^{8} + 1950 T^{9} - 205 T^{10} - 10 T^{11} + T^{12} \)
$43$ \( -31628144 + 50062848 T + 91425996 T^{2} - 56283880 T^{3} - 24491855 T^{4} + 3669708 T^{5} + 1986774 T^{6} + 78528 T^{7} - 38345 T^{8} - 4120 T^{9} + 86 T^{10} + 28 T^{11} + T^{12} \)
$47$ \( 834896 + 4201152 T - 222004 T^{2} - 17246320 T^{3} - 6770555 T^{4} + 849772 T^{5} + 670449 T^{6} + 43222 T^{7} - 17420 T^{8} - 2580 T^{9} + 26 T^{10} + 22 T^{11} + T^{12} \)
$53$ \( 300763681 - 997363266 T + 70959334 T^{2} + 223936660 T^{3} - 18363930 T^{4} - 16126786 T^{5} + 489901 T^{6} + 500156 T^{7} + 12895 T^{8} - 5310 T^{9} - 271 T^{10} + 16 T^{11} + T^{12} \)
$59$ \( -866673664 + 1444159232 T - 189582304 T^{2} - 323991720 T^{3} + 92907945 T^{4} + 7718152 T^{5} - 3856241 T^{6} - 32078 T^{7} + 60555 T^{8} - 330 T^{9} - 409 T^{10} + 2 T^{11} + T^{12} \)
$61$ \( 4582416901 - 3075579686 T - 44054021 T^{2} + 418684210 T^{3} - 66403385 T^{4} - 14093136 T^{5} + 4016886 T^{6} - 23104 T^{7} - 67205 T^{8} + 5030 T^{9} + 199 T^{10} - 34 T^{11} + T^{12} \)
$67$ \( 4750542736 + 4379425312 T + 841227356 T^{2} - 300987640 T^{3} - 111128475 T^{4} + 1759412 T^{5} + 3912794 T^{6} + 237532 T^{7} - 49565 T^{8} - 5300 T^{9} + 126 T^{10} + 32 T^{11} + T^{12} \)
$71$ \( 5618722000 - 8944126000 T - 4440858500 T^{2} + 511772500 T^{3} + 306792325 T^{4} - 10648150 T^{5} - 8046125 T^{6} + 96250 T^{7} + 95440 T^{8} - 290 T^{9} - 510 T^{10} + T^{12} \)
$73$ \( -540439819 + 867360576 T + 1586430889 T^{2} + 189858360 T^{3} - 136209670 T^{4} - 23505904 T^{5} + 3307261 T^{6} + 752584 T^{7} - 11870 T^{8} - 7760 T^{9} - 191 T^{10} + 24 T^{11} + T^{12} \)
$79$ \( 2241715456 + 4381631616 T + 3113091984 T^{2} + 960597600 T^{3} + 80347385 T^{4} - 22694744 T^{5} - 4670604 T^{6} + 87294 T^{7} + 66450 T^{8} + 980 T^{9} - 411 T^{10} - 6 T^{11} + T^{12} \)
$83$ \( -354151984 - 47196672 T + 232385016 T^{2} + 32986080 T^{3} - 29541855 T^{4} - 4032672 T^{5} + 1385544 T^{6} + 196508 T^{7} - 23930 T^{8} - 4120 T^{9} + 56 T^{10} + 28 T^{11} + T^{12} \)
$89$ \( -6367139875 - 7826798500 T - 2432776875 T^{2} + 345801500 T^{3} + 249288875 T^{4} + 11314600 T^{5} - 6514225 T^{6} - 436200 T^{7} + 79100 T^{8} + 4110 T^{9} - 480 T^{10} - 10 T^{11} + T^{12} \)
$97$ \( -4393476419 - 23755582136 T + 2072387039 T^{2} + 2548037720 T^{3} - 61934765 T^{4} - 98833736 T^{5} - 1772814 T^{6} + 1584216 T^{7} + 69815 T^{8} - 9280 T^{9} - 521 T^{10} + 16 T^{11} + T^{12} \)
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