Properties

Label 6042.2.a.bf
Level 6042
Weight 2
Character orbit 6042.a
Self dual yes
Analytic conductor 48.246
Analytic rank 0
Dimension 12
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 6042 = 2 \cdot 3 \cdot 19 \cdot 53 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6042.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(48.2456129013\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Defining polynomial: \(x^{12} - 5 x^{11} - 27 x^{10} + 134 x^{9} + 294 x^{8} - 1313 x^{7} - 1685 x^{6} + 5910 x^{5} + 5237 x^{4} - 12206 x^{3} - 7876 x^{2} + 9264 x + 4048\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q - q^{2} + q^{3} + q^{4} + ( -1 + \beta_{1} ) q^{5} - q^{6} -\beta_{7} q^{7} - q^{8} + q^{9} +O(q^{10})\) \( q - q^{2} + q^{3} + q^{4} + ( -1 + \beta_{1} ) q^{5} - q^{6} -\beta_{7} q^{7} - q^{8} + q^{9} + ( 1 - \beta_{1} ) q^{10} + \beta_{2} q^{11} + q^{12} + ( -\beta_{5} - \beta_{7} ) q^{13} + \beta_{7} q^{14} + ( -1 + \beta_{1} ) q^{15} + q^{16} + ( 1 - \beta_{1} + \beta_{3} - \beta_{6} + \beta_{7} + \beta_{8} + \beta_{11} ) q^{17} - q^{18} + q^{19} + ( -1 + \beta_{1} ) q^{20} -\beta_{7} q^{21} -\beta_{2} q^{22} + ( \beta_{4} + \beta_{10} - \beta_{11} ) q^{23} - q^{24} + ( 2 - \beta_{1} + \beta_{3} + \beta_{4} ) q^{25} + ( \beta_{5} + \beta_{7} ) q^{26} + q^{27} -\beta_{7} q^{28} + ( \beta_{2} + \beta_{3} - \beta_{9} ) q^{29} + ( 1 - \beta_{1} ) q^{30} + ( \beta_{6} - \beta_{7} - \beta_{11} ) q^{31} - q^{32} + \beta_{2} q^{33} + ( -1 + \beta_{1} - \beta_{3} + \beta_{6} - \beta_{7} - \beta_{8} - \beta_{11} ) q^{34} + ( -2 - 2 \beta_{4} - \beta_{6} + \beta_{7} ) q^{35} + q^{36} + ( 2 + \beta_{4} + \beta_{5} + \beta_{10} ) q^{37} - q^{38} + ( -\beta_{5} - \beta_{7} ) q^{39} + ( 1 - \beta_{1} ) q^{40} + ( 2 \beta_{1} - \beta_{2} - \beta_{3} + \beta_{4} + \beta_{9} ) q^{41} + \beta_{7} q^{42} + ( 2 - \beta_{4} - \beta_{10} + \beta_{11} ) q^{43} + \beta_{2} q^{44} + ( -1 + \beta_{1} ) q^{45} + ( -\beta_{4} - \beta_{10} + \beta_{11} ) q^{46} + ( 2 \beta_{1} - \beta_{3} + \beta_{6} - \beta_{7} - \beta_{10} ) q^{47} + q^{48} + ( 1 - \beta_{4} - 2 \beta_{5} - \beta_{6} - \beta_{7} + \beta_{8} - \beta_{10} + \beta_{11} ) q^{49} + ( -2 + \beta_{1} - \beta_{3} - \beta_{4} ) q^{50} + ( 1 - \beta_{1} + \beta_{3} - \beta_{6} + \beta_{7} + \beta_{8} + \beta_{11} ) q^{51} + ( -\beta_{5} - \beta_{7} ) q^{52} + q^{53} - q^{54} + ( -\beta_{2} - \beta_{3} + 2 \beta_{4} - \beta_{8} + \beta_{10} ) q^{55} + \beta_{7} q^{56} + q^{57} + ( -\beta_{2} - \beta_{3} + \beta_{9} ) q^{58} + ( -\beta_{5} + \beta_{11} ) q^{59} + ( -1 + \beta_{1} ) q^{60} + ( 2 - 2 \beta_{4} + \beta_{5} - \beta_{8} ) q^{61} + ( -\beta_{6} + \beta_{7} + \beta_{11} ) q^{62} -\beta_{7} q^{63} + q^{64} + ( -2 + 2 \beta_{1} - 2 \beta_{3} - \beta_{4} - \beta_{8} + \beta_{9} ) q^{65} -\beta_{2} q^{66} + ( 3 + \beta_{1} + \beta_{2} + \beta_{9} ) q^{67} + ( 1 - \beta_{1} + \beta_{3} - \beta_{6} + \beta_{7} + \beta_{8} + \beta_{11} ) q^{68} + ( \beta_{4} + \beta_{10} - \beta_{11} ) q^{69} + ( 2 + 2 \beta_{4} + \beta_{6} - \beta_{7} ) q^{70} + ( \beta_{2} + \beta_{3} - 2 \beta_{4} + 2 \beta_{5} - \beta_{8} - \beta_{10} + \beta_{11} ) q^{71} - q^{72} + ( 1 + \beta_{1} + \beta_{2} - 2 \beta_{3} - 2 \beta_{4} + \beta_{6} - \beta_{7} - \beta_{10} ) q^{73} + ( -2 - \beta_{4} - \beta_{5} - \beta_{10} ) q^{74} + ( 2 - \beta_{1} + \beta_{3} + \beta_{4} ) q^{75} + q^{76} + ( -3 + \beta_{1} - \beta_{3} + 2 \beta_{6} - 2 \beta_{7} - \beta_{9} - \beta_{10} - \beta_{11} ) q^{77} + ( \beta_{5} + \beta_{7} ) q^{78} + ( 3 + \beta_{1} - \beta_{2} + \beta_{4} - \beta_{6} + \beta_{7} + \beta_{8} - \beta_{9} ) q^{79} + ( -1 + \beta_{1} ) q^{80} + q^{81} + ( -2 \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} - \beta_{9} ) q^{82} + ( 2 - \beta_{2} + \beta_{3} + 2 \beta_{4} - \beta_{5} + \beta_{7} + \beta_{8} + \beta_{10} ) q^{83} -\beta_{7} q^{84} + ( -3 + 3 \beta_{1} - \beta_{3} + 2 \beta_{5} + 2 \beta_{6} - \beta_{7} - 3 \beta_{8} - 3 \beta_{11} ) q^{85} + ( -2 + \beta_{4} + \beta_{10} - \beta_{11} ) q^{86} + ( \beta_{2} + \beta_{3} - \beta_{9} ) q^{87} -\beta_{2} q^{88} + ( -3 - \beta_{1} + \beta_{2} - \beta_{4} + \beta_{5} + \beta_{6} - \beta_{7} - \beta_{8} ) q^{89} + ( 1 - \beta_{1} ) q^{90} + ( 4 - \beta_{2} - \beta_{3} - 2 \beta_{5} - \beta_{6} - 3 \beta_{7} + 2 \beta_{8} - \beta_{10} ) q^{91} + ( \beta_{4} + \beta_{10} - \beta_{11} ) q^{92} + ( \beta_{6} - \beta_{7} - \beta_{11} ) q^{93} + ( -2 \beta_{1} + \beta_{3} - \beta_{6} + \beta_{7} + \beta_{10} ) q^{94} + ( -1 + \beta_{1} ) q^{95} - q^{96} + ( 1 - \beta_{1} - \beta_{3} - 2 \beta_{4} + 2 \beta_{7} + \beta_{9} + \beta_{10} - \beta_{11} ) q^{97} + ( -1 + \beta_{4} + 2 \beta_{5} + \beta_{6} + \beta_{7} - \beta_{8} + \beta_{10} - \beta_{11} ) q^{98} + \beta_{2} q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12q - 12q^{2} + 12q^{3} + 12q^{4} - 7q^{5} - 12q^{6} + 5q^{7} - 12q^{8} + 12q^{9} + O(q^{10}) \) \( 12q - 12q^{2} + 12q^{3} + 12q^{4} - 7q^{5} - 12q^{6} + 5q^{7} - 12q^{8} + 12q^{9} + 7q^{10} - 5q^{11} + 12q^{12} + 10q^{13} - 5q^{14} - 7q^{15} + 12q^{16} + 6q^{17} - 12q^{18} + 12q^{19} - 7q^{20} + 5q^{21} + 5q^{22} - 12q^{24} + 21q^{25} - 10q^{26} + 12q^{27} + 5q^{28} + 7q^{30} + 2q^{31} - 12q^{32} - 5q^{33} - 6q^{34} - 17q^{35} + 12q^{36} + 16q^{37} - 12q^{38} + 10q^{39} + 7q^{40} + 7q^{41} - 5q^{42} + 24q^{43} - 5q^{44} - 7q^{45} + 4q^{47} + 12q^{48} + 29q^{49} - 21q^{50} + 6q^{51} + 10q^{52} + 12q^{53} - 12q^{54} - 2q^{55} - 5q^{56} + 12q^{57} + 2q^{59} - 7q^{60} + 29q^{61} - 2q^{62} + 5q^{63} + 12q^{64} - 17q^{65} + 5q^{66} + 36q^{67} + 6q^{68} + 17q^{70} - 3q^{71} - 12q^{72} + 7q^{73} - 16q^{74} + 21q^{75} + 12q^{76} - 35q^{77} - 10q^{78} + 40q^{79} - 7q^{80} + 12q^{81} - 7q^{82} + 24q^{83} + 5q^{84} - 22q^{85} - 24q^{86} + 5q^{88} - 45q^{89} + 7q^{90} + 71q^{91} + 2q^{93} - 4q^{94} - 7q^{95} - 12q^{96} + q^{97} - 29q^{98} - 5q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{12} - 5 x^{11} - 27 x^{10} + 134 x^{9} + 294 x^{8} - 1313 x^{7} - 1685 x^{6} + 5910 x^{5} + 5237 x^{4} - 12206 x^{3} - 7876 x^{2} + 9264 x + 4048\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\(-33394 \nu^{11} - 28307 \nu^{10} + 3132030 \nu^{9} - 8852987 \nu^{8} - 42987049 \nu^{7} + 171770673 \nu^{6} + 137572104 \nu^{5} - 955165179 \nu^{4} + 154476765 \nu^{3} + 1678479792 \nu^{2} - 725209880 \nu - 434774552\)\()/29065268\)
\(\beta_{3}\)\(=\)\((\)\(-240415 \nu^{11} + 881782 \nu^{10} + 9164831 \nu^{9} - 32921225 \nu^{8} - 111040173 \nu^{7} + 392181836 \nu^{6} + 492479259 \nu^{5} - 1794924157 \nu^{4} - 605896466 \nu^{3} + 3031359368 \nu^{2} - 389196712 \nu - 1413490944\)\()/ 116261072 \)
\(\beta_{4}\)\(=\)\((\)\(240415 \nu^{11} - 881782 \nu^{10} - 9164831 \nu^{9} + 32921225 \nu^{8} + 111040173 \nu^{7} - 392181836 \nu^{6} - 492479259 \nu^{5} + 1794924157 \nu^{4} + 605896466 \nu^{3} - 2915098296 \nu^{2} + 272935640 \nu + 715924512\)\()/ 116261072 \)
\(\beta_{5}\)\(=\)\((\)\(189989 \nu^{11} - 1107305 \nu^{10} - 4357903 \nu^{9} + 30837332 \nu^{8} + 28149920 \nu^{7} - 307621629 \nu^{6} + 14597997 \nu^{5} + 1326709990 \nu^{4} - 619570675 \nu^{3} - 2303774536 \nu^{2} + 1274870756 \nu + 975911880\)\()/58130536\)
\(\beta_{6}\)\(=\)\((\)\(-706435 \nu^{11} + 1519886 \nu^{10} + 27452799 \nu^{9} - 41360357 \nu^{8} - 388277645 \nu^{7} + 336991472 \nu^{6} + 2397828747 \nu^{5} - 796795369 \nu^{4} - 6357025590 \nu^{3} - 304376060 \nu^{2} + 5708543024 \nu + 1249035152\)\()/ 116261072 \)
\(\beta_{7}\)\(=\)\((\)\(-719715 \nu^{11} + 3372970 \nu^{10} + 19188627 \nu^{9} - 87318673 \nu^{8} - 187114117 \nu^{7} + 778788496 \nu^{6} + 765347575 \nu^{5} - 2840645421 \nu^{4} - 976094510 \nu^{3} + 3639608632 \nu^{2} - 466097312 \nu - 413025456\)\()/ 116261072 \)
\(\beta_{8}\)\(=\)\((\)\(-933169 \nu^{11} + 5051586 \nu^{10} + 21846301 \nu^{9} - 130167559 \nu^{8} - 171505199 \nu^{7} + 1161069336 \nu^{6} + 449085225 \nu^{5} - 4213225091 \nu^{4} + 228301822 \nu^{3} + 5271985324 \nu^{2} - 1466460072 \nu - 579260304\)\()/ 116261072 \)
\(\beta_{9}\)\(=\)\((\)\(-486498 \nu^{11} + 2895903 \nu^{10} + 10801602 \nu^{9} - 78900597 \nu^{8} - 65415103 \nu^{7} + 747354843 \nu^{6} - 42256088 \nu^{5} - 2879670749 \nu^{4} + 1260969879 \nu^{3} + 4021894868 \nu^{2} - 2403458836 \nu - 1147126304\)\()/58130536\)
\(\beta_{10}\)\(=\)\((\)\(-1217761 \nu^{11} + 8309250 \nu^{10} + 15914015 \nu^{9} - 180804043 \nu^{8} + 3535843 \nu^{7} + 1331413850 \nu^{6} - 552351777 \nu^{5} - 4140276495 \nu^{4} + 1747051468 \nu^{3} + 5090328594 \nu^{2} - 1451589104 \nu - 1441942312\)\()/58130536\)
\(\beta_{11}\)\(=\)\((\)\(-7036535 \nu^{11} + 49323472 \nu^{10} + 87625363 \nu^{9} - 1101810507 \nu^{8} + 209647641 \nu^{7} + 8450311842 \nu^{6} - 5568913397 \nu^{5} - 27852411711 \nu^{4} + 20572041196 \nu^{3} + 37691484484 \nu^{2} - 22532365104 \nu - 13604819008\)\()/ 116261072 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{4} + \beta_{3} + \beta_{1} + 6\)
\(\nu^{3}\)\(=\)\(-\beta_{8} + 3 \beta_{7} - \beta_{6} + 2 \beta_{5} + 2 \beta_{4} + 3 \beta_{3} + 10 \beta_{1} + 7\)
\(\nu^{4}\)\(=\)\(-2 \beta_{9} + \beta_{8} + 7 \beta_{7} - 3 \beta_{6} + 6 \beta_{5} + 20 \beta_{4} + 21 \beta_{3} + \beta_{2} + 22 \beta_{1} + 69\)
\(\nu^{5}\)\(=\)\(2 \beta_{11} - 3 \beta_{10} - 8 \beta_{9} - 16 \beta_{8} + 61 \beta_{7} - 21 \beta_{6} + 54 \beta_{5} + 50 \beta_{4} + 78 \beta_{3} + 5 \beta_{2} + 140 \beta_{1} + 173\)
\(\nu^{6}\)\(=\)\(7 \beta_{11} - 14 \beta_{10} - 65 \beta_{9} + 16 \beta_{8} + 195 \beta_{7} - 83 \beta_{6} + 190 \beta_{5} + 334 \beta_{4} + 419 \beta_{3} + 24 \beta_{2} + 429 \beta_{1} + 1060\)
\(\nu^{7}\)\(=\)\(65 \beta_{11} - 120 \beta_{10} - 269 \beta_{9} - 177 \beta_{8} + 1154 \beta_{7} - 432 \beta_{6} + 1176 \beta_{5} + 1002 \beta_{4} + 1706 \beta_{3} + 105 \beta_{2} + 2272 \beta_{1} + 3579\)
\(\nu^{8}\)\(=\)\(259 \beta_{11} - 553 \beta_{10} - 1565 \beta_{9} + 301 \beta_{8} + 4335 \beta_{7} - 1881 \beta_{6} + 4642 \beta_{5} + 5495 \beta_{4} + 8353 \beta_{3} + 388 \beta_{2} + 8155 \beta_{1} + 18406\)
\(\nu^{9}\)\(=\)\(1600 \beta_{11} - 3295 \beta_{10} - 6760 \beta_{9} - 1248 \beta_{8} + 22111 \beta_{7} - 9011 \beta_{6} + 24562 \beta_{5} + 18679 \beta_{4} + 35673 \beta_{3} + 1558 \beta_{2} + 39436 \beta_{1} + 70877\)
\(\nu^{10}\)\(=\)\(6855 \beta_{11} - 15257 \beta_{10} - 34617 \beta_{9} + 7268 \beta_{8} + 90252 \beta_{7} - 40470 \beta_{6} + 103732 \beta_{5} + 91781 \beta_{4} + 167454 \beta_{3} + 5090 \beta_{2} + 154062 \beta_{1} + 337587\)
\(\nu^{11}\)\(=\)\(36164 \beta_{11} - 79401 \beta_{10} - 153606 \beta_{9} + 7995 \beta_{8} + 430534 \beta_{7} - 188334 \beta_{6} + 508700 \beta_{5} + 338956 \beta_{4} + 733361 \beta_{3} + 18361 \beta_{2} + 710461 \beta_{1} + 1387136\)

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
−3.22349
−2.24876
−2.19598
−1.72799
−1.60709
−0.392682
1.26637
1.70127
2.22012
2.70802
4.00527
4.49493
−1.00000 1.00000 1.00000 −4.22349 −1.00000 3.89357 −1.00000 1.00000 4.22349
1.2 −1.00000 1.00000 1.00000 −3.24876 −1.00000 1.74785 −1.00000 1.00000 3.24876
1.3 −1.00000 1.00000 1.00000 −3.19598 −1.00000 2.75709 −1.00000 1.00000 3.19598
1.4 −1.00000 1.00000 1.00000 −2.72799 −1.00000 −0.326815 −1.00000 1.00000 2.72799
1.5 −1.00000 1.00000 1.00000 −2.60709 −1.00000 −4.26018 −1.00000 1.00000 2.60709
1.6 −1.00000 1.00000 1.00000 −1.39268 −1.00000 −2.74135 −1.00000 1.00000 1.39268
1.7 −1.00000 1.00000 1.00000 0.266375 −1.00000 1.01292 −1.00000 1.00000 −0.266375
1.8 −1.00000 1.00000 1.00000 0.701272 −1.00000 5.18729 −1.00000 1.00000 −0.701272
1.9 −1.00000 1.00000 1.00000 1.22012 −1.00000 −3.11298 −1.00000 1.00000 −1.22012
1.10 −1.00000 1.00000 1.00000 1.70802 −1.00000 3.64339 −1.00000 1.00000 −1.70802
1.11 −1.00000 1.00000 1.00000 3.00527 −1.00000 −3.21582 −1.00000 1.00000 −3.00527
1.12 −1.00000 1.00000 1.00000 3.49493 −1.00000 0.415035 −1.00000 1.00000 −3.49493
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.12
Significant digits:
Format:

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 6042.2.a.bf 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6042.2.a.bf 12 1.a even 1 1 trivial

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(3\) \(-1\)
\(19\) \(-1\)
\(53\) \(-1\)

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6042))\):

\(T_{5}^{12} + \cdots\)
\(T_{7}^{12} - \cdots\)
\(T_{11}^{12} + \cdots\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T )^{12} \)
$3$ \( ( 1 - T )^{12} \)
$5$ \( 1 + 7 T + 44 T^{2} + 194 T^{3} + 805 T^{4} + 2795 T^{5} + 9256 T^{6} + 27503 T^{7} + 78751 T^{8} + 208262 T^{9} + 532232 T^{10} + 1272391 T^{11} + 2938046 T^{12} + 6361955 T^{13} + 13305800 T^{14} + 26032750 T^{15} + 49219375 T^{16} + 85946875 T^{17} + 144625000 T^{18} + 218359375 T^{19} + 314453125 T^{20} + 378906250 T^{21} + 429687500 T^{22} + 341796875 T^{23} + 244140625 T^{24} \)
$7$ \( 1 - 5 T + 40 T^{2} - 161 T^{3} + 812 T^{4} - 2959 T^{5} + 11794 T^{6} - 39286 T^{7} + 131886 T^{8} - 404199 T^{9} + 1199194 T^{10} - 3408652 T^{11} + 9158914 T^{12} - 23860564 T^{13} + 58760506 T^{14} - 138640257 T^{15} + 316658286 T^{16} - 660279802 T^{17} + 1387552306 T^{18} - 2436863737 T^{19} + 4681018412 T^{20} - 6496930727 T^{21} + 11299009960 T^{22} - 9886633715 T^{23} + 13841287201 T^{24} \)
$11$ \( 1 + 5 T + 48 T^{2} + 276 T^{3} + 1724 T^{4} + 7934 T^{5} + 41032 T^{6} + 174373 T^{7} + 750391 T^{8} + 2840340 T^{9} + 11183368 T^{10} + 38312114 T^{11} + 133334840 T^{12} + 421433254 T^{13} + 1353187528 T^{14} + 3780492540 T^{15} + 10986474631 T^{16} + 28082946023 T^{17} + 72690690952 T^{18} + 154611214714 T^{19} + 369554710844 T^{20} + 650793562716 T^{21} + 1244996380848 T^{22} + 1426558353055 T^{23} + 3138428376721 T^{24} \)
$13$ \( 1 - 10 T + 116 T^{2} - 808 T^{3} + 5586 T^{4} - 30232 T^{5} + 160674 T^{6} - 734165 T^{7} + 3354249 T^{8} - 13754493 T^{9} + 56950666 T^{10} - 214131440 T^{11} + 810386728 T^{12} - 2783708720 T^{13} + 9624662554 T^{14} - 30218621121 T^{15} + 95800705689 T^{16} - 272590325345 T^{17} + 775542709266 T^{18} - 1897013165944 T^{19} + 4556671807506 T^{20} - 8568435493384 T^{21} + 15991585054484 T^{22} - 17921603940370 T^{23} + 23298085122481 T^{24} \)
$17$ \( 1 - 6 T + 76 T^{2} - 408 T^{3} + 2974 T^{4} - 12234 T^{5} + 71662 T^{6} - 227597 T^{7} + 1209477 T^{8} - 3092811 T^{9} + 17848174 T^{10} - 39620144 T^{11} + 282368744 T^{12} - 673542448 T^{13} + 5158122286 T^{14} - 15194980443 T^{15} + 101016728517 T^{16} - 323155193629 T^{17} + 1729746469678 T^{18} - 5020083325482 T^{19} + 20745902629534 T^{20} - 48383853610776 T^{21} + 153215536434124 T^{22} - 205631377845798 T^{23} + 582622237229761 T^{24} \)
$19$ \( ( 1 - T )^{12} \)
$23$ \( 1 + 125 T^{2} + 231 T^{3} + 8357 T^{4} + 26107 T^{5} + 401809 T^{6} + 1556582 T^{7} + 15274147 T^{8} + 62253302 T^{9} + 475254850 T^{10} + 1850702332 T^{11} + 12085943150 T^{12} + 42566153636 T^{13} + 251409815650 T^{14} + 757435925434 T^{15} + 4274332570627 T^{16} + 10018695659626 T^{17} + 59482152523201 T^{18} + 88889777944829 T^{19} + 654444903993317 T^{20} + 416066264797953 T^{21} + 5178313901706125 T^{22} + 21914624432020321 T^{24} \)
$29$ \( 1 + 97 T^{2} + 426 T^{3} + 5573 T^{4} + 40856 T^{5} + 315915 T^{6} + 2176096 T^{7} + 15405568 T^{8} + 93892080 T^{9} + 571504484 T^{10} + 3484118014 T^{11} + 17450169620 T^{12} + 101039422406 T^{13} + 480635271044 T^{14} + 2289933939120 T^{15} + 10896065540608 T^{16} + 44634229294304 T^{17} + 187913609453715 T^{18} + 704760946480504 T^{19} + 2787873259431653 T^{20} + 6180044185720194 T^{21} + 40808601630119497 T^{22} + 353814783205469041 T^{24} \)
$31$ \( 1 - 2 T + 136 T^{2} - 11 T^{3} + 9693 T^{4} + 12510 T^{5} + 518776 T^{6} + 1058553 T^{7} + 23553227 T^{8} + 51092647 T^{9} + 934297024 T^{10} + 1842012917 T^{11} + 31605417006 T^{12} + 57102400427 T^{13} + 897859440064 T^{14} + 1522101046777 T^{15} + 21751899752267 T^{16} + 30305473678503 T^{17} + 460415609614456 T^{18} + 344182802528610 T^{19} + 8267072825915613 T^{20} - 290835843767381 T^{21} + 111469447029388936 T^{22} - 50816953792809662 T^{23} + 787662783788549761 T^{24} \)
$37$ \( 1 - 16 T + 404 T^{2} - 4976 T^{3} + 72318 T^{4} - 729844 T^{5} + 7866424 T^{6} - 67315069 T^{7} + 591051315 T^{8} - 4375833075 T^{9} + 32735047808 T^{10} - 212008905968 T^{11} + 1380843555700 T^{12} - 7844329520816 T^{13} + 44814280449152 T^{14} - 221649072747975 T^{15} + 1107725323571715 T^{16} - 4667893250188033 T^{17} + 20183091801191416 T^{18} - 69285460934257252 T^{19} + 254015489148658878 T^{20} - 646689617220303152 T^{21} + 1942668086456810996 T^{22} - 2846681948471366608 T^{23} + 6582952005840035281 T^{24} \)
$41$ \( 1 - 7 T + 166 T^{2} - 1154 T^{3} + 13572 T^{4} - 75580 T^{5} + 627785 T^{6} - 1861539 T^{7} + 10614593 T^{8} + 75160310 T^{9} - 595293375 T^{10} + 8714916874 T^{11} - 45677427724 T^{12} + 357311591834 T^{13} - 1000688163375 T^{14} + 5180123725510 T^{15} + 29994302930273 T^{16} - 215670836553339 T^{17} + 2982044190936185 T^{18} - 14719528019925980 T^{19} + 108371405209630212 T^{20} - 377798752290630994 T^{21} + 2228161445485298566 T^{22} - 3852303222013739087 T^{23} + 22563490300366186081 T^{24} \)
$43$ \( 1 - 24 T + 629 T^{2} - 10323 T^{3} + 163695 T^{4} - 2065163 T^{5} + 24644365 T^{6} - 252817374 T^{7} + 2445485667 T^{8} - 21009717358 T^{9} + 170284725302 T^{10} - 1243500740060 T^{11} + 8572892064154 T^{12} - 53470531822580 T^{13} + 314856457083398 T^{14} - 1670419597982506 T^{15} + 8360628843825267 T^{16} - 37166288515088682 T^{17} + 155785978277068885 T^{18} - 561349738369565441 T^{19} + 1913299944441895695 T^{20} - 5188263533024030289 T^{21} + 13593622375055792621 T^{22} - 22303049747309344968 T^{23} + 39959630797262576401 T^{24} \)
$47$ \( 1 - 4 T + 359 T^{2} - 1723 T^{3} + 60907 T^{4} - 345403 T^{5} + 6540414 T^{6} - 42888383 T^{7} + 506613775 T^{8} - 3689170755 T^{9} + 30846703871 T^{10} - 231920333144 T^{11} + 1568159043250 T^{12} - 10900255657768 T^{13} + 68140368851039 T^{14} - 383020775296365 T^{15} + 2472113612205775 T^{16} - 9836236499353681 T^{17} + 70500530846806206 T^{18} - 174989145677281589 T^{19} + 1450274036707877227 T^{20} - 1928261805156067541 T^{21} + 18883088472662987591 T^{22} - 9888636860336049212 T^{23} + \)\(11\!\cdots\!41\)\( T^{24} \)
$53$ \( ( 1 - T )^{12} \)
$59$ \( 1 - 2 T + 425 T^{2} - 782 T^{3} + 88853 T^{4} - 148438 T^{5} + 12276169 T^{6} - 18637146 T^{7} + 1262304326 T^{8} - 1749724342 T^{9} + 102239323896 T^{10} - 129170121742 T^{11} + 6688633977524 T^{12} - 7621037182778 T^{13} + 355895086481976 T^{14} - 359356635635618 T^{15} + 15295797210003686 T^{16} - 13324148539410654 T^{17} + 517815359487101329 T^{18} - 369410449103562722 T^{19} + 13046324872456733813 T^{20} - 6774462730188162298 T^{21} + \)\(21\!\cdots\!25\)\( T^{22} - 60311776889475685318 T^{23} + \)\(17\!\cdots\!81\)\( T^{24} \)
$61$ \( 1 - 29 T + 806 T^{2} - 13091 T^{3} + 205941 T^{4} - 2302884 T^{5} + 26195435 T^{6} - 222631979 T^{7} + 2099401209 T^{8} - 14830563372 T^{9} + 132037301575 T^{10} - 863273221809 T^{11} + 7927226452226 T^{12} - 52659666530349 T^{13} + 491310799160575 T^{14} - 3366256104739932 T^{15} + 29067975335021769 T^{16} - 188034145947709679 T^{17} + 1349598617749242035 T^{18} - 7237372193187384564 T^{19} + 39480395745973046421 T^{20} - \)\(15\!\cdots\!31\)\( T^{21} + \)\(57\!\cdots\!06\)\( T^{22} - \)\(12\!\cdots\!69\)\( T^{23} + \)\(26\!\cdots\!21\)\( T^{24} \)
$67$ \( 1 - 36 T + 1051 T^{2} - 21585 T^{3} + 382737 T^{4} - 5657721 T^{5} + 74944779 T^{6} - 877186598 T^{7} + 9449792731 T^{8} - 93112067098 T^{9} + 865637371538 T^{10} - 7558574919308 T^{11} + 63468339732006 T^{12} - 506424519593636 T^{13} + 3885846160834082 T^{14} - 28004664636595774 T^{15} + 190423916747301451 T^{16} - 1184311649483715986 T^{17} + 6779383460353245651 T^{18} - 34289815324379648883 T^{19} + \)\(15\!\cdots\!17\)\( T^{20} - \)\(58\!\cdots\!95\)\( T^{21} + \)\(19\!\cdots\!99\)\( T^{22} - \)\(43\!\cdots\!88\)\( T^{23} + \)\(81\!\cdots\!61\)\( T^{24} \)
$71$ \( 1 + 3 T + 335 T^{2} + 41 T^{3} + 56367 T^{4} - 79705 T^{5} + 7198087 T^{6} - 10517987 T^{7} + 764841091 T^{8} - 879412558 T^{9} + 66280842426 T^{10} - 76862900650 T^{11} + 4935011632922 T^{12} - 5457265946150 T^{13} + 334121726669466 T^{14} - 314751428046338 T^{15} + 19435897820183971 T^{16} - 18976860858836437 T^{17} + 922076988388059127 T^{18} - 724926552224554655 T^{19} + 36399189295729810287 T^{20} + 1879788529456410271 T^{21} + \)\(10\!\cdots\!35\)\( T^{22} + \)\(69\!\cdots\!13\)\( T^{23} + \)\(16\!\cdots\!41\)\( T^{24} \)
$73$ \( 1 - 7 T + 308 T^{2} - 1964 T^{3} + 55610 T^{4} - 252832 T^{5} + 6404084 T^{6} - 16265183 T^{7} + 527064463 T^{8} + 98326918 T^{9} + 34540540696 T^{10} + 102268475140 T^{11} + 2261952759980 T^{12} + 7465598685220 T^{13} + 184066541368984 T^{14} + 38250842659606 T^{15} + 14967703642809583 T^{16} - 33718888832246519 T^{17} + 969157097229764276 T^{18} - 2793135862380332704 T^{19} + 44847245710229844410 T^{20} - \)\(11\!\cdots\!32\)\( T^{21} + \)\(13\!\cdots\!92\)\( T^{22} - \)\(21\!\cdots\!39\)\( T^{23} + \)\(22\!\cdots\!21\)\( T^{24} \)
$79$ \( 1 - 40 T + 1230 T^{2} - 26447 T^{3} + 482152 T^{4} - 7208010 T^{5} + 95038213 T^{6} - 1080937965 T^{7} + 11133665041 T^{8} - 102470200985 T^{9} + 896261100749 T^{10} - 7514621189631 T^{11} + 65877706991116 T^{12} - 593655073980849 T^{13} + 5593565529774509 T^{14} - 50521805423443415 T^{15} + 433657155173818321 T^{16} - 3326107082125288035 T^{17} + 23102597375432823973 T^{18} - \)\(13\!\cdots\!90\)\( T^{19} + \)\(73\!\cdots\!72\)\( T^{20} - \)\(31\!\cdots\!93\)\( T^{21} + \)\(11\!\cdots\!30\)\( T^{22} - \)\(29\!\cdots\!60\)\( T^{23} + \)\(59\!\cdots\!41\)\( T^{24} \)
$83$ \( 1 - 24 T + 738 T^{2} - 12284 T^{3} + 220098 T^{4} - 2876126 T^{5} + 38679094 T^{6} - 434381851 T^{7} + 4932031161 T^{8} - 50902096323 T^{9} + 519360248568 T^{10} - 5017245979406 T^{11} + 46761691350344 T^{12} - 416431416290698 T^{13} + 3577872752384952 T^{14} - 29105156950239201 T^{15} + 234065918020740681 T^{16} - 1711047765670570193 T^{17} + 12645757433934647686 T^{18} - 78046701788591945002 T^{19} + \)\(49\!\cdots\!18\)\( T^{20} - \)\(22\!\cdots\!52\)\( T^{21} + \)\(11\!\cdots\!62\)\( T^{22} - \)\(30\!\cdots\!08\)\( T^{23} + \)\(10\!\cdots\!61\)\( T^{24} \)
$89$ \( 1 + 45 T + 1745 T^{2} + 45513 T^{3} + 1055807 T^{4} + 19953139 T^{5} + 343732503 T^{6} + 5143797059 T^{7} + 71283305815 T^{8} + 883744295216 T^{9} + 10245213933384 T^{10} + 107766650683076 T^{11} + 1065025109605522 T^{12} + 9591231910793764 T^{13} + 81152339566334664 T^{14} + 623012332054128304 T^{15} + 4472474352721431415 T^{16} + 28723268571047360491 T^{17} + \)\(17\!\cdots\!83\)\( T^{18} + \)\(88\!\cdots\!31\)\( T^{19} + \)\(41\!\cdots\!67\)\( T^{20} + \)\(15\!\cdots\!17\)\( T^{21} + \)\(54\!\cdots\!45\)\( T^{22} + \)\(12\!\cdots\!05\)\( T^{23} + \)\(24\!\cdots\!21\)\( T^{24} \)
$97$ \( 1 - T + 222 T^{2} - 822 T^{3} + 32042 T^{4} - 246894 T^{5} + 3700022 T^{6} - 24330829 T^{7} + 395877679 T^{8} - 2197148346 T^{9} + 38096152684 T^{10} - 190181060916 T^{11} + 2897024588364 T^{12} - 18447562908852 T^{13} + 358446700603756 T^{14} - 2005277972388858 T^{15} + 35046766285818799 T^{16} - 208937107357883053 T^{17} + 3082014743621408438 T^{18} - 19948611647939231022 T^{19} + \)\(25\!\cdots\!62\)\( T^{20} - \)\(62\!\cdots\!74\)\( T^{21} + \)\(16\!\cdots\!78\)\( T^{22} - \)\(71\!\cdots\!53\)\( T^{23} + \)\(69\!\cdots\!41\)\( T^{24} \)
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