Properties

Label 6042.2.a.be
Level 6042
Weight 2
Character orbit 6042.a
Self dual yes
Analytic conductor 48.246
Analytic rank 1
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: \(1\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Defining polynomial: \(x^{12} - 3 x^{11} - 36 x^{10} + 105 x^{9} + 465 x^{8} - 1287 x^{7} - 2668 x^{6} + 6443 x^{5} + 7140 x^{4} - 10858 x^{3} - 10086 x^{2} + 2072 x + 1496\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{12} - 3 x^{11} - 36 x^{10} + 105 x^{9} + 465 x^{8} - 1287 x^{7} - 2668 x^{6} + 6443 x^{5} + 7140 x^{4} - 10858 x^{3} - 10086 x^{2} + 2072 x + 1496\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\(-8627 \nu^{11} - 153659 \nu^{10} + 429814 \nu^{9} + 5289977 \nu^{8} - 7829211 \nu^{7} - 62240161 \nu^{6} + 64541040 \nu^{5} + 291739913 \nu^{4} - 208168678 \nu^{3} - 436949806 \nu^{2} + 56806586 \nu + 66361412\)\()/2955272\)
\(\beta_{3}\)\(=\)\((\)\(-44133 \nu^{11} + 357911 \nu^{10} + 1189180 \nu^{9} - 12276379 \nu^{8} - 7457875 \nu^{7} + 144429857 \nu^{6} - 26871376 \nu^{5} - 669511535 \nu^{4} + 276730430 \nu^{3} + 945017570 \nu^{2} - 82642718 \nu - 138362652\)\()/2955272\)
\(\beta_{4}\)\(=\)\((\)\(-153659 \nu^{11} + 580219 \nu^{10} + 4906370 \nu^{9} - 19687587 \nu^{8} - 49855477 \nu^{7} + 228244687 \nu^{6} + 153700554 \nu^{5} - 1021791637 \nu^{4} + 93884262 \nu^{3} + 1277688810 \nu^{2} - 86183202 \nu - 168446612\)\()/2955272\)
\(\beta_{5}\)\(=\)\((\)\(-189500 \nu^{11} + 192809 \nu^{10} + 6681248 \nu^{9} - 5889366 \nu^{8} - 81842389 \nu^{7} + 58062170 \nu^{6} + 408639469 \nu^{5} - 171960525 \nu^{4} - 686539810 \nu^{3} - 122170844 \nu^{2} + 95475990 \nu + 26647108\)\()/2955272\)
\(\beta_{6}\)\(=\)\((\)\(123929 \nu^{11} + 136741 \nu^{10} - 4654614 \nu^{9} - 5213807 \nu^{8} + 62549233 \nu^{7} + 69405275 \nu^{6} - 362487140 \nu^{5} - 395937951 \nu^{4} + 789802582 \nu^{3} + 853386646 \nu^{2} - 141548970 \nu - 130878392\)\()/1477636\)
\(\beta_{7}\)\(=\)\((\)\(-436706 \nu^{11} - 514389 \nu^{10} + 16556874 \nu^{9} + 19588198 \nu^{8} - 225402809 \nu^{7} - 260559368 \nu^{6} + 1330188249 \nu^{5} + 1479064815 \nu^{4} - 2983182366 \nu^{3} - 3143496692 \nu^{2} + 632768662 \nu + 484423404\)\()/2955272\)
\(\beta_{8}\)\(=\)\((\)\(-492933 \nu^{11} - 163655 \nu^{10} + 18247374 \nu^{9} + 7660263 \nu^{8} - 240145831 \nu^{7} - 121976379 \nu^{6} + 1346156238 \nu^{5} + 849811789 \nu^{4} - 2788215142 \nu^{3} - 2297594618 \nu^{2} + 537284210 \nu + 348290980\)\()/2955272\)
\(\beta_{9}\)\(=\)\((\)\(590365 \nu^{11} - 65830 \nu^{10} - 21463244 \nu^{9} + 99389 \nu^{8} + 275258286 \nu^{7} + 32314681 \nu^{6} - 1483888803 \nu^{5} - 457273178 \nu^{4} + 2889298104 \nu^{3} + 1862852610 \nu^{2} - 546585460 \nu - 295289888\)\()/2955272\)
\(\beta_{10}\)\(=\)\((\)\(853346 \nu^{11} - 302055 \nu^{10} - 30694620 \nu^{9} + 7336736 \nu^{8} + 387502119 \nu^{7} - 39850156 \nu^{6} - 2037578341 \nu^{5} - 239539453 \nu^{4} + 3800163318 \nu^{3} + 2015976264 \nu^{2} - 689742890 \nu - 339214300\)\()/2955272\)
\(\beta_{11}\)\(=\)\((\)\(1215147 \nu^{11} - 86525 \nu^{10} - 44279466 \nu^{9} - 1709593 \nu^{8} + 569886495 \nu^{7} + 92646333 \nu^{6} - 3089711460 \nu^{5} - 1083764241 \nu^{4} + 6075452558 \nu^{3} + 4086959558 \nu^{2} - 1162833098 \nu - 669738660\)\()/2955272\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(-\beta_{9} - \beta_{7} - \beta_{4} + 7\)
\(\nu^{3}\)\(=\)\(\beta_{10} + \beta_{8} + \beta_{5} + \beta_{4} + 2 \beta_{2} + 10 \beta_{1}\)
\(\nu^{4}\)\(=\)\(-4 \beta_{11} + 3 \beta_{10} - 13 \beta_{9} - \beta_{8} - 17 \beta_{7} - \beta_{6} - \beta_{5} - 14 \beta_{4} + 4 \beta_{2} - 5 \beta_{1} + 76\)
\(\nu^{5}\)\(=\)\(-3 \beta_{11} + 21 \beta_{10} - 7 \beta_{9} + 14 \beta_{8} - 5 \beta_{7} + 5 \beta_{6} + 23 \beta_{5} + 15 \beta_{4} - 8 \beta_{3} + 41 \beta_{2} + 113 \beta_{1} - 4\)
\(\nu^{6}\)\(=\)\(-92 \beta_{11} + 67 \beta_{10} - 163 \beta_{9} - 8 \beta_{8} - 270 \beta_{7} - 21 \beta_{6} - 28 \beta_{5} - 196 \beta_{4} + 7 \beta_{3} + 106 \beta_{2} - 109 \beta_{1} + 923\)
\(\nu^{7}\)\(=\)\(-79 \beta_{11} + 362 \beta_{10} - 181 \beta_{9} + 196 \beta_{8} - 151 \beta_{7} + 118 \beta_{6} + 406 \beta_{5} + 200 \beta_{4} - 206 \beta_{3} + 702 \beta_{2} + 1390 \beta_{1} - 3\)
\(\nu^{8}\)\(=\)\(-1661 \beta_{11} + 1193 \beta_{10} - 2137 \beta_{9} - 5 \beta_{8} - 4212 \beta_{7} - 349 \beta_{6} - 514 \beta_{5} - 2823 \beta_{4} + 157 \beta_{3} + 2051 \beta_{2} - 1795 \beta_{1} + 12107\)
\(\nu^{9}\)\(=\)\(-1606 \beta_{11} + 5892 \beta_{10} - 3531 \beta_{9} + 2862 \beta_{8} - 3312 \beta_{7} + 2126 \beta_{6} + 6525 \beta_{5} + 2547 \beta_{4} - 3893 \beta_{3} + 11395 \beta_{2} + 18218 \beta_{1} + 1626\)
\(\nu^{10}\)\(=\)\(-27558 \beta_{11} + 19854 \beta_{10} - 29471 \beta_{9} + 1313 \beta_{8} - 65131 \beta_{7} - 5320 \beta_{6} - 8040 \beta_{5} - 41397 \beta_{4} + 2506 \beta_{3} + 35695 \beta_{2} - 26734 \beta_{1} + 167757\)
\(\nu^{11}\)\(=\)\(-29950 \beta_{11} + 93985 \beta_{10} - 62489 \beta_{9} + 42771 \beta_{8} - 64361 \beta_{7} + 34683 \beta_{6} + 100789 \beta_{5} + 31056 \beta_{4} - 65724 \beta_{3} + 181164 \beta_{2} + 249669 \beta_{1} + 53884\)

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
3.97453
2.89261
2.87876
2.73947
2.45965
0.427666
−0.365902
−1.11996
−1.13903
−2.89280
−3.06562
−3.78937
−1.00000 −1.00000 1.00000 −3.97453 1.00000 2.10909 −1.00000 1.00000 3.97453
1.2 −1.00000 −1.00000 1.00000 −2.89261 1.00000 −4.19999 −1.00000 1.00000 2.89261
1.3 −1.00000 −1.00000 1.00000 −2.87876 1.00000 3.95461 −1.00000 1.00000 2.87876
1.4 −1.00000 −1.00000 1.00000 −2.73947 1.00000 3.23526 −1.00000 1.00000 2.73947
1.5 −1.00000 −1.00000 1.00000 −2.45965 1.00000 −4.70163 −1.00000 1.00000 2.45965
1.6 −1.00000 −1.00000 1.00000 −0.427666 1.00000 −1.56626 −1.00000 1.00000 0.427666
1.7 −1.00000 −1.00000 1.00000 0.365902 1.00000 1.87719 −1.00000 1.00000 −0.365902
1.8 −1.00000 −1.00000 1.00000 1.11996 1.00000 0.414381 −1.00000 1.00000 −1.11996
1.9 −1.00000 −1.00000 1.00000 1.13903 1.00000 4.65781 −1.00000 1.00000 −1.13903
1.10 −1.00000 −1.00000 1.00000 2.89280 1.00000 −4.43415 −1.00000 1.00000 −2.89280
1.11 −1.00000 −1.00000 1.00000 3.06562 1.00000 −0.702849 −1.00000 1.00000 −3.06562
1.12 −1.00000 −1.00000 1.00000 3.78937 1.00000 −1.64348 −1.00000 1.00000 −3.78937
\(n\): 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.be 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6042.2.a.be 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

$p$ $F_p(T)$
$2$ \( ( 1 + T )^{12} \)
$3$ \( ( 1 + T )^{12} \)
$5$ \( 1 + 3 T + 24 T^{2} + 60 T^{3} + 315 T^{4} + 687 T^{5} + 2932 T^{6} + 5977 T^{7} + 21975 T^{8} + 41708 T^{9} + 137214 T^{10} + 243173 T^{11} + 732886 T^{12} + 1215865 T^{13} + 3430350 T^{14} + 5213500 T^{15} + 13734375 T^{16} + 18678125 T^{17} + 45812500 T^{18} + 53671875 T^{19} + 123046875 T^{20} + 117187500 T^{21} + 234375000 T^{22} + 146484375 T^{23} + 244140625 T^{24} \)
$7$ \( 1 + T + 24 T^{2} + 37 T^{3} + 340 T^{4} + 613 T^{5} + 3948 T^{6} + 6444 T^{7} + 39516 T^{8} + 58027 T^{9} + 327844 T^{10} + 483768 T^{11} + 2379822 T^{12} + 3386376 T^{13} + 16064356 T^{14} + 19903261 T^{15} + 94877916 T^{16} + 108304308 T^{17} + 464478252 T^{18} + 504831859 T^{19} + 1960032340 T^{20} + 1493083459 T^{21} + 6779405976 T^{22} + 1977326743 T^{23} + 13841287201 T^{24} \)
$11$ \( 1 + 7 T + 68 T^{2} + 402 T^{3} + 2494 T^{4} + 12378 T^{5} + 61262 T^{6} + 268473 T^{7} + 1140523 T^{8} + 4506422 T^{9} + 16999038 T^{10} + 60867808 T^{11} + 206716044 T^{12} + 669545888 T^{13} + 2056883598 T^{14} + 5998047682 T^{15} + 16698397243 T^{16} + 43237845123 T^{17} + 108529369982 T^{18} + 241212202638 T^{19} + 534611049214 T^{20} + 947894971782 T^{21} + 1763744872868 T^{22} + 1997181694277 T^{23} + 3138428376721 T^{24} \)
$13$ \( 1 - 4 T + 64 T^{2} - 190 T^{3} + 2096 T^{4} - 4714 T^{5} + 47766 T^{6} - 86875 T^{7} + 900769 T^{8} - 1467767 T^{9} + 14884558 T^{10} - 22766494 T^{11} + 211030308 T^{12} - 295964422 T^{13} + 2515490302 T^{14} - 3224684099 T^{15} + 25726863409 T^{16} - 32256079375 T^{17} + 230557358694 T^{18} - 295796509138 T^{19} + 1709771591216 T^{20} - 2014854880870 T^{21} + 8822943478336 T^{22} - 7168641576148 T^{23} + 23298085122481 T^{24} \)
$17$ \( 1 + 4 T + 126 T^{2} + 478 T^{3} + 7928 T^{4} + 29268 T^{5} + 332220 T^{6} + 1182395 T^{7} + 10277153 T^{8} + 34538639 T^{9} + 246444654 T^{10} + 762843696 T^{11} + 4688561020 T^{12} + 12968342832 T^{13} + 71222505006 T^{14} + 169688333407 T^{15} + 858358095713 T^{16} + 1678831817515 T^{17} + 8018983173180 T^{18} + 12009792281364 T^{19} + 55303804992248 T^{20} + 56685004965566 T^{21} + 254015231456574 T^{22} + 137087585230532 T^{23} + 582622237229761 T^{24} \)
$19$ \( ( 1 + T )^{12} \)
$23$ \( 1 + 18 T + 277 T^{2} + 2849 T^{3} + 26965 T^{4} + 206287 T^{5} + 1504353 T^{6} + 9486742 T^{7} + 58464451 T^{8} + 322192772 T^{9} + 1766231674 T^{10} + 8812006758 T^{11} + 44173025294 T^{12} + 202676155434 T^{13} + 934336555546 T^{14} + 3920119456924 T^{15} + 16360750432291 T^{16} + 61059925464506 T^{17} + 222698233724817 T^{18} + 702371226985289 T^{19} + 2111655718102165 T^{20} + 5131483932508087 T^{21} + 11475143606180773 T^{22} + 17150575642450686 T^{23} + 21914624432020321 T^{24} \)
$29$ \( 1 - 10 T + 201 T^{2} - 1510 T^{3} + 17045 T^{4} - 103782 T^{5} + 859615 T^{6} - 4372612 T^{7} + 29965380 T^{8} - 128207764 T^{9} + 820476684 T^{10} - 3150875206 T^{11} + 22069839308 T^{12} - 91375380974 T^{13} + 690020891244 T^{14} - 3126859156196 T^{15} + 21193943931780 T^{16} - 89687296251188 T^{17} + 511319049081415 T^{18} - 1790226663100638 T^{19} + 8526700108920245 T^{20} - 21905790423562190 T^{21} + 84562153893340401 T^{22} - 122005097657058290 T^{23} + 353814783205469041 T^{24} \)
$31$ \( 1 - 14 T + 262 T^{2} - 2649 T^{3} + 31175 T^{4} - 264176 T^{5} + 2441356 T^{6} - 18016911 T^{7} + 139806331 T^{8} - 912412221 T^{9} + 6160616494 T^{10} - 35877413715 T^{11} + 214389199162 T^{12} - 1112199825165 T^{13} + 5920352450734 T^{14} - 27181672475811 T^{15} + 129114082611451 T^{16} - 515808865572561 T^{17} + 2166712436631436 T^{18} - 7268172345387536 T^{19} + 26588878092223175 T^{20} - 70038559103617479 T^{21} + 214742611188969862 T^{22} - 355718676549667634 T^{23} + 787662783788549761 T^{24} \)
$37$ \( 1 - 14 T + 340 T^{2} - 3986 T^{3} + 56288 T^{4} - 559922 T^{5} + 5935740 T^{6} - 50824307 T^{7} + 442524299 T^{8} - 3295759793 T^{9} + 24531661484 T^{10} - 159907974214 T^{11} + 1036081982592 T^{12} - 5916595045918 T^{13} + 33583844571596 T^{14} - 166940120794829 T^{15} + 829361782738139 T^{16} - 3524358559162799 T^{17} + 15229484874957660 T^{18} - 53154446508063626 T^{19} + 197710443502305248 T^{20} - 518027494823176922 T^{21} + 1634918686622068660 T^{22} - 2490846704912445782 T^{23} + 6582952005840035281 T^{24} \)
$41$ \( 1 + 9 T + 206 T^{2} + 1144 T^{3} + 18788 T^{4} + 66254 T^{5} + 1119533 T^{6} + 1821259 T^{7} + 52493749 T^{8} + 5197468 T^{9} + 2300048053 T^{10} - 1680157150 T^{11} + 96255910220 T^{12} - 68886443150 T^{13} + 3866380777093 T^{14} + 358214692028 T^{15} + 148334788667989 T^{16} + 211004148777059 T^{17} + 5317898451239453 T^{18} + 12903249661711774 T^{19} + 150020775204725348 T^{20} + 374524932946691384 T^{21} + 2765067817891394606 T^{22} + 4952961285446235969 T^{23} + 22563490300366186081 T^{24} \)
$43$ \( 1 + 4 T + 153 T^{2} + 151 T^{3} + 11075 T^{4} - 8019 T^{5} + 696245 T^{6} - 573064 T^{7} + 42683803 T^{8} - 26391044 T^{9} + 2192892010 T^{10} - 2234107318 T^{11} + 96420458114 T^{12} - 96066614674 T^{13} + 4054657326490 T^{14} - 2098272735308 T^{15} + 145927428380203 T^{16} - 84245246379352 T^{17} + 4401217416051005 T^{18} - 2179713442467033 T^{19} + 129446818074431075 T^{20} + 75891484402463293 T^{21} + 3306556793932490097 T^{22} + 3717174957884890828 T^{23} + 39959630797262576401 T^{24} \)
$47$ \( 1 + 14 T + 397 T^{2} + 4457 T^{3} + 71645 T^{4} + 670595 T^{5} + 7928806 T^{6} + 63746013 T^{7} + 617137951 T^{8} + 4390698577 T^{9} + 37237575967 T^{10} + 241901661008 T^{11} + 1881552850466 T^{12} + 11369378067376 T^{13} + 82257805311103 T^{14} + 455855498359871 T^{15} + 3011436333873631 T^{16} + 14619829797707091 T^{17} + 85466307175867174 T^{18} + 339738931466885485 T^{19} + 1705959632881866845 T^{20} + 4987964518619032519 T^{21} + 20881855497624529453 T^{22} + 34610229011176172242 T^{23} + \)\(11\!\cdots\!41\)\( T^{24} \)
$53$ \( ( 1 + T )^{12} \)
$59$ \( 1 + 18 T + 663 T^{2} + 9600 T^{3} + 200429 T^{4} + 2424716 T^{5} + 36988255 T^{6} + 382245750 T^{7} + 4673888770 T^{8} + 41762236780 T^{9} + 427749299366 T^{10} + 3319410455900 T^{11} + 29134332721224 T^{12} + 195845216898100 T^{13} + 1488995311093046 T^{14} + 8577086427639620 T^{15} + 56635197499935970 T^{16} + 273276774864479250 T^{17} + 1560184334349386455 T^{18} + 6034273073664386404 T^{19} + 29429077778596453709 T^{20} + 83164759859087414400 T^{21} + \)\(33\!\cdots\!63\)\( T^{22} + \)\(54\!\cdots\!62\)\( T^{23} + \)\(17\!\cdots\!81\)\( T^{24} \)
$61$ \( 1 + 7 T + 324 T^{2} + 1973 T^{3} + 52579 T^{4} + 273984 T^{5} + 5885543 T^{6} + 26544167 T^{7} + 519606493 T^{8} + 2094148788 T^{9} + 38811764973 T^{10} + 143665347805 T^{11} + 2528142756094 T^{12} + 8763586216105 T^{13} + 144418577464533 T^{14} + 475331986049028 T^{15} + 7194388884645613 T^{16} + 22419105261326267 T^{17} + 303225378677763023 T^{18} + 861061253184377664 T^{19} + 10079778810084037699 T^{20} + 23072550241161760193 T^{21} + \)\(23\!\cdots\!24\)\( T^{22} + \)\(30\!\cdots\!27\)\( T^{23} + \)\(26\!\cdots\!21\)\( T^{24} \)
$67$ \( 1 - 8 T + 489 T^{2} - 4391 T^{3} + 124115 T^{4} - 1136315 T^{5} + 21277365 T^{6} - 186397314 T^{7} + 2695009947 T^{8} - 21722177986 T^{9} + 261837198506 T^{10} - 1897963105728 T^{11} + 19826358261602 T^{12} - 127163528083776 T^{13} + 1175387184093434 T^{14} - 6533227417603318 T^{15} + 54307471538200587 T^{16} - 251659693508762598 T^{17} + 1924716014719304685 T^{18} - 6886877507802604745 T^{19} + 50399089799942497715 T^{20} - \)\(11\!\cdots\!77\)\( T^{21} + \)\(89\!\cdots\!61\)\( T^{22} - \)\(97\!\cdots\!64\)\( T^{23} + \)\(81\!\cdots\!61\)\( T^{24} \)
$71$ \( 1 + T + 517 T^{2} + 1289 T^{3} + 130947 T^{4} + 438001 T^{5} + 22169645 T^{6} + 79615669 T^{7} + 2800760467 T^{8} + 9815291020 T^{9} + 275988974622 T^{10} + 906340894864 T^{11} + 21776891780370 T^{12} + 64350203535344 T^{13} + 1391260421069502 T^{14} + 3513000624259220 T^{15} + 71172031544815027 T^{16} + 143644926809300819 T^{17} + 2839937818927778045 T^{18} + 3983671724495416391 T^{19} + 84559487656038665667 T^{20} + 59098717426080800959 T^{21} + \)\(16\!\cdots\!17\)\( T^{22} + \)\(23\!\cdots\!71\)\( T^{23} + \)\(16\!\cdots\!41\)\( T^{24} \)
$73$ \( 1 + 31 T + 834 T^{2} + 16630 T^{3} + 296136 T^{4} + 4532874 T^{5} + 63727178 T^{6} + 807398767 T^{7} + 9514617687 T^{8} + 103108757858 T^{9} + 1047266275604 T^{10} + 9867616205064 T^{11} + 87411789698432 T^{12} + 720335982969672 T^{13} + 5580881982693716 T^{14} + 40111059655645586 T^{15} + 270198406098288567 T^{16} + 1673795448090925831 T^{17} + 9644103176211382442 T^{18} + 50076465514853294778 T^{19} + \)\(23\!\cdots\!16\)\( T^{20} + \)\(97\!\cdots\!90\)\( T^{21} + \)\(35\!\cdots\!66\)\( T^{22} + \)\(97\!\cdots\!87\)\( T^{23} + \)\(22\!\cdots\!21\)\( T^{24} \)
$79$ \( 1 + 358 T^{2} - 1145 T^{3} + 79396 T^{4} - 338120 T^{5} + 13001361 T^{6} - 64727295 T^{7} + 1670112273 T^{8} - 8441409187 T^{9} + 176988554865 T^{10} - 852293310995 T^{11} + 15293019262212 T^{12} - 67331171568605 T^{13} + 1104585570912465 T^{14} - 4161943944149293 T^{15} + 65051008312444113 T^{16} - 199169537269710705 T^{17} + 3160467763799964081 T^{18} - 6493225706400081080 T^{19} + \)\(12\!\cdots\!56\)\( T^{20} - \)\(13\!\cdots\!55\)\( T^{21} + \)\(33\!\cdots\!58\)\( T^{22} + \)\(59\!\cdots\!41\)\( T^{24} \)
$83$ \( 1 + 48 T + 1548 T^{2} + 36442 T^{3} + 709298 T^{4} + 11656196 T^{5} + 168584118 T^{6} + 2168483725 T^{7} + 25427798483 T^{8} + 274419632687 T^{9} + 2784745224590 T^{10} + 26813220647848 T^{11} + 248918594206548 T^{12} + 2225497313771384 T^{13} + 19184109852200510 T^{14} + 156909578515201669 T^{15} + 1206760622729527043 T^{16} + 8541745526459035175 T^{17} + 55116954483003553542 T^{18} + \)\(31\!\cdots\!92\)\( T^{19} + \)\(15\!\cdots\!18\)\( T^{20} + \)\(68\!\cdots\!26\)\( T^{21} + \)\(24\!\cdots\!52\)\( T^{22} + \)\(61\!\cdots\!16\)\( T^{23} + \)\(10\!\cdots\!61\)\( T^{24} \)
$89$ \( 1 - 13 T + 649 T^{2} - 7625 T^{3} + 216783 T^{4} - 2304145 T^{5} + 48195601 T^{6} - 463705509 T^{7} + 7882775667 T^{8} - 68571962902 T^{9} + 995382856534 T^{10} - 7793439760542 T^{11} + 99313353481338 T^{12} - 693616138688238 T^{13} + 7884427606605814 T^{14} - 48341108115060038 T^{15} + 494583010647849747 T^{16} - 2589359129084804541 T^{17} + 23952312003621262561 T^{18} - \)\(10\!\cdots\!05\)\( T^{19} + \)\(85\!\cdots\!23\)\( T^{20} - \)\(26\!\cdots\!25\)\( T^{21} + \)\(20\!\cdots\!49\)\( T^{22} - \)\(36\!\cdots\!57\)\( T^{23} + \)\(24\!\cdots\!21\)\( T^{24} \)
$97$ \( 1 - 25 T + 1092 T^{2} - 22092 T^{3} + 552554 T^{4} - 9264404 T^{5} + 171023956 T^{6} - 2425271421 T^{7} + 36117670607 T^{8} - 439125683202 T^{9} + 5491951115368 T^{10} - 57636033718824 T^{11} + 617208445099148 T^{12} - 5590695270725928 T^{13} + 51673768044497512 T^{14} - 400778154665018946 T^{15} + 3197471410232543567 T^{16} - 20826630907704895197 T^{17} + \)\(14\!\cdots\!24\)\( T^{18} - \)\(74\!\cdots\!52\)\( T^{19} + \)\(43\!\cdots\!94\)\( T^{20} - \)\(16\!\cdots\!64\)\( T^{21} + \)\(80\!\cdots\!08\)\( T^{22} - \)\(17\!\cdots\!25\)\( T^{23} + \)\(69\!\cdots\!41\)\( T^{24} \)
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