Properties

Label 1336.2.a.d
Level $1336$
Weight $2$
Character orbit 1336.a
Self dual yes
Analytic conductor $10.668$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1336 = 2^{3} \cdot 167 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1336.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(10.6680137100\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Defining polynomial: \(x^{12} - x^{11} - 25 x^{10} + 20 x^{9} + 224 x^{8} - 135 x^{7} - 865 x^{6} + 330 x^{5} + 1341 x^{4} - 127 x^{3} - 652 x^{2} - 48 x + 64\)
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.

\(f(q)\) \(=\) \( q -\beta_{1} q^{3} + ( 1 - \beta_{9} ) q^{5} + \beta_{11} q^{7} + ( 2 - \beta_{3} + \beta_{7} - \beta_{9} ) q^{9} +O(q^{10})\) \( q -\beta_{1} q^{3} + ( 1 - \beta_{9} ) q^{5} + \beta_{11} q^{7} + ( 2 - \beta_{3} + \beta_{7} - \beta_{9} ) q^{9} + ( -\beta_{3} + \beta_{5} + \beta_{6} ) q^{11} + ( 1 - \beta_{8} ) q^{13} + ( 1 - 2 \beta_{1} - \beta_{2} - \beta_{6} ) q^{15} + ( 1 + \beta_{3} - \beta_{5} ) q^{17} + ( \beta_{3} + \beta_{8} ) q^{19} + ( 1 + \beta_{2} + \beta_{8} + \beta_{10} - \beta_{11} ) q^{21} + ( -1 + \beta_{2} + \beta_{6} - \beta_{7} + \beta_{9} ) q^{23} + ( 1 - \beta_{4} - \beta_{7} - \beta_{9} - \beta_{10} ) q^{25} + ( -\beta_{1} + \beta_{3} - \beta_{5} - 2 \beta_{6} - \beta_{8} - \beta_{10} + \beta_{11} ) q^{27} + ( 3 - \beta_{2} + \beta_{4} - \beta_{6} - \beta_{10} + \beta_{11} ) q^{29} + ( -1 + \beta_{2} + \beta_{4} + \beta_{6} - \beta_{11} ) q^{31} + ( -1 + \beta_{2} + \beta_{3} + \beta_{4} + \beta_{5} - \beta_{7} + \beta_{9} + \beta_{10} - \beta_{11} ) q^{33} + ( \beta_{3} - \beta_{5} - \beta_{7} + 2 \beta_{9} + 2 \beta_{10} ) q^{35} + ( 3 - \beta_{2} + \beta_{6} + \beta_{8} + \beta_{9} ) q^{37} + ( 1 - 2 \beta_{1} - 2 \beta_{4} + \beta_{7} + \beta_{8} - \beta_{9} - 2 \beta_{11} ) q^{39} + ( 2 - \beta_{2} + \beta_{3} + \beta_{5} - \beta_{6} ) q^{41} + ( -\beta_{3} - \beta_{5} - \beta_{8} ) q^{43} + ( 4 - \beta_{3} - 2 \beta_{4} - \beta_{5} + 2 \beta_{7} - \beta_{9} - 2 \beta_{10} ) q^{45} + ( \beta_{3} - \beta_{4} + \beta_{7} - \beta_{8} + \beta_{9} - \beta_{10} ) q^{47} + ( 1 - \beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} + \beta_{6} - \beta_{7} + \beta_{9} + 2 \beta_{10} - \beta_{11} ) q^{49} + ( 2 - 2 \beta_{1} - \beta_{2} - \beta_{3} - \beta_{5} - \beta_{6} ) q^{51} + ( 3 + \beta_{3} - 2 \beta_{4} - \beta_{5} + \beta_{8} + \beta_{9} ) q^{53} + ( -\beta_{2} + 2 \beta_{4} + 2 \beta_{5} + \beta_{6} + \beta_{8} + 2 \beta_{10} - 2 \beta_{11} ) q^{55} + ( \beta_{1} - \beta_{3} + 2 \beta_{4} + \beta_{5} - \beta_{7} - \beta_{8} + \beta_{9} + \beta_{10} + \beta_{11} ) q^{57} + ( 1 + 2 \beta_{1} + \beta_{2} - \beta_{6} + \beta_{7} - \beta_{8} - 2 \beta_{10} + 2 \beta_{11} ) q^{59} + ( 1 + \beta_{1} + \beta_{2} - \beta_{3} + \beta_{5} + \beta_{6} - \beta_{8} - 2 \beta_{11} ) q^{61} + ( -1 + 2 \beta_{4} + \beta_{5} + \beta_{6} - 2 \beta_{8} + 2 \beta_{9} + \beta_{11} ) q^{63} + ( 3 - 2 \beta_{6} - \beta_{8} - 2 \beta_{9} - 2 \beta_{10} + 2 \beta_{11} ) q^{65} + ( -1 + 2 \beta_{1} + \beta_{2} + \beta_{6} + \beta_{8} - \beta_{9} ) q^{67} + ( 2 + 2 \beta_{1} - \beta_{3} + 2 \beta_{4} + \beta_{5} + 2 \beta_{6} + \beta_{8} + 2 \beta_{9} + 2 \beta_{10} ) q^{69} + ( 3 - 3 \beta_{3} + \beta_{5} + \beta_{7} - \beta_{9} ) q^{71} + ( -1 + 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{6} - \beta_{7} - \beta_{8} + \beta_{9} ) q^{73} + ( 1 - \beta_{1} - 3 \beta_{2} - \beta_{6} ) q^{75} + ( 3 + \beta_{1} - \beta_{5} - 2 \beta_{6} + \beta_{7} - \beta_{8} + \beta_{9} + 2 \beta_{11} ) q^{77} + ( \beta_{2} - \beta_{3} - 2 \beta_{4} - \beta_{5} - \beta_{6} - 2 \beta_{9} - 2 \beta_{10} ) q^{79} + ( 2 - \beta_{2} - 3 \beta_{4} - \beta_{5} + \beta_{6} + \beta_{7} + 2 \beta_{8} - \beta_{9} - \beta_{10} - \beta_{11} ) q^{81} + ( -1 + 2 \beta_{1} + \beta_{2} + \beta_{3} + \beta_{5} - \beta_{6} - \beta_{7} ) q^{83} + ( 5 - \beta_{3} - 2 \beta_{4} - \beta_{5} - 2 \beta_{6} - 2 \beta_{9} - 2 \beta_{10} + 2 \beta_{11} ) q^{85} + ( -2 \beta_{1} - \beta_{3} - \beta_{5} + \beta_{6} + 2 \beta_{8} - 2 \beta_{9} - \beta_{10} - \beta_{11} ) q^{87} + ( -1 + 3 \beta_{1} + 2 \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} + 2 \beta_{6} - \beta_{7} + \beta_{9} + \beta_{10} ) q^{89} + ( 2 + 2 \beta_{1} - \beta_{2} - \beta_{3} + \beta_{5} - \beta_{6} + 2 \beta_{7} - 2 \beta_{9} + 2 \beta_{11} ) q^{91} + ( 3 - \beta_{2} - 2 \beta_{3} + 3 \beta_{4} + \beta_{5} + \beta_{6} + 2 \beta_{9} + \beta_{10} + \beta_{11} ) q^{93} + ( -\beta_{3} + \beta_{5} + 2 \beta_{6} - \beta_{7} + \beta_{8} + \beta_{9} + 2 \beta_{10} - 2 \beta_{11} ) q^{95} + ( 1 + 3 \beta_{3} - \beta_{4} - \beta_{5} - 3 \beta_{6} + 2 \beta_{7} - \beta_{8} - 2 \beta_{9} - \beta_{10} ) q^{97} + ( 4 \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} + \beta_{5} + 2 \beta_{6} + \beta_{7} + \beta_{8} + \beta_{9} + 2 \beta_{10} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12q - q^{3} + 8q^{5} - 4q^{7} + 15q^{9} + O(q^{10}) \) \( 12q - q^{3} + 8q^{5} - 4q^{7} + 15q^{9} + 6q^{11} + 13q^{13} + 4q^{15} + 10q^{17} - q^{19} + 13q^{21} + 3q^{23} + 18q^{25} - 10q^{27} + 29q^{29} - 3q^{31} + 3q^{35} + 41q^{37} + 10q^{39} + 20q^{41} - q^{43} + 42q^{45} + 5q^{47} + 20q^{49} + 14q^{51} + 39q^{53} + 3q^{55} + 3q^{57} + 8q^{59} + 30q^{61} - 2q^{63} + 21q^{65} - 9q^{67} + 33q^{69} + 29q^{71} + 12q^{73} + q^{75} + 19q^{77} - 2q^{79} + 24q^{81} - 5q^{83} + 44q^{85} - 2q^{87} + 7q^{89} - 4q^{91} + 37q^{93} + 18q^{95} - 14q^{97} + 5q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{12} - x^{11} - 25 x^{10} + 20 x^{9} + 224 x^{8} - 135 x^{7} - 865 x^{6} + 330 x^{5} + 1341 x^{4} - 127 x^{3} - 652 x^{2} - 48 x + 64\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( -513 \nu^{11} + 769 \nu^{10} + 12409 \nu^{9} - 16400 \nu^{8} - 105696 \nu^{7} + 120839 \nu^{6} + 372809 \nu^{5} - 347946 \nu^{4} - 475249 \nu^{3} + 288703 \nu^{2} + 151904 \nu - 47568 \)\()/4136\)
\(\beta_{3}\)\(=\)\((\)\(53519 \nu^{11} - 83931 \nu^{10} - 1289331 \nu^{9} + 1796400 \nu^{8} + 10951328 \nu^{7} - 13323097 \nu^{6} - 38624171 \nu^{5} + 38888922 \nu^{4} + 49322147 \nu^{3} - 33553237 \nu^{2} - 15089256 \nu + 5721888\)\()/57904\)
\(\beta_{4}\)\(=\)\((\)\(-107109 \nu^{11} + 169593 \nu^{10} + 2577481 \nu^{9} - 3638784 \nu^{8} - 21858800 \nu^{7} + 27096771 \nu^{6} + 76927913 \nu^{5} - 79711206 \nu^{4} - 97956257 \nu^{3} + 70284111 \nu^{2} + 30054808 \nu - 12293136\)\()/57904\)
\(\beta_{5}\)\(=\)\((\)\(125227 \nu^{11} - 198647 \nu^{10} - 3012143 \nu^{9} + 4262080 \nu^{8} + 25523536 \nu^{7} - 31727693 \nu^{6} - 89660807 \nu^{5} + 93203714 \nu^{4} + 113631231 \nu^{3} - 81664489 \nu^{2} - 34529208 \nu + 14118640\)\()/57904\)
\(\beta_{6}\)\(=\)\((\)\(-32139 \nu^{11} + 50481 \nu^{10} + 774057 \nu^{9} - 1084624 \nu^{8} - 6567776 \nu^{7} + 8092065 \nu^{6} + 23103625 \nu^{5} - 23842512 \nu^{4} - 29323889 \nu^{3} + 20965555 \nu^{2} + 8898792 \nu - 3617364\)\()/14476\)
\(\beta_{7}\)\(=\)\((\)\(-367753 \nu^{11} + 578405 \nu^{10} + 8865733 \nu^{9} - 12442928 \nu^{8} - 75314336 \nu^{7} + 92952287 \nu^{6} + 265325861 \nu^{5} - 274170838 \nu^{4} - 337567853 \nu^{3} + 241205875 \nu^{2} + 103577096 \nu - 41904752\)\()/115808\)
\(\beta_{8}\)\(=\)\((\)\(56675 \nu^{11} - 88801 \nu^{10} - 1366609 \nu^{9} + 1910016 \nu^{8} + 11611044 \nu^{7} - 14266689 \nu^{6} - 40900337 \nu^{5} + 42073316 \nu^{4} + 51973173 \nu^{3} - 36979827 \nu^{2} - 15826216 \nu + 6392900\)\()/14476\)
\(\beta_{9}\)\(=\)\((\)\(-474791 \nu^{11} + 746267 \nu^{10} + 11444395 \nu^{9} - 16035728 \nu^{8} - 97216992 \nu^{7} + 119598481 \nu^{6} + 342574203 \nu^{5} - 351948682 \nu^{4} - 436212147 \nu^{3} + 308196541 \nu^{2} + 133755608 \nu - 52769488\)\()/115808\)
\(\beta_{10}\)\(=\)\((\)\(540349 \nu^{11} - 847241 \nu^{10} - 13023209 \nu^{9} + 18193152 \nu^{8} + 110627008 \nu^{7} - 135607883 \nu^{6} - 389945641 \nu^{5} + 398980686 \nu^{4} + 497150241 \nu^{3} - 349874159 \nu^{2} - 152716536 \nu + 60098064\)\()/115808\)
\(\beta_{11}\)\(=\)\((\)\(56631 \nu^{11} - 89035 \nu^{10} - 1365163 \nu^{9} + 1913696 \nu^{8} + 11597760 \nu^{7} - 14279777 \nu^{6} - 40869419 \nu^{5} + 42065146 \nu^{4} + 52023251 \nu^{3} - 36953309 \nu^{2} - 15910440 \nu + 6377904\)\()/10528\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(-\beta_{9} + \beta_{7} - \beta_{3} + 5\)
\(\nu^{3}\)\(=\)\(-\beta_{11} + \beta_{10} + \beta_{8} + 2 \beta_{6} + \beta_{5} - \beta_{3} + 7 \beta_{1}\)
\(\nu^{4}\)\(=\)\(-\beta_{11} - \beta_{10} - 10 \beta_{9} + 2 \beta_{8} + 10 \beta_{7} + \beta_{6} - \beta_{5} - 3 \beta_{4} - 9 \beta_{3} - \beta_{2} + 38\)
\(\nu^{5}\)\(=\)\(-13 \beta_{11} + 11 \beta_{10} - 2 \beta_{9} + 16 \beta_{8} + 4 \beta_{7} + 26 \beta_{6} + 12 \beta_{5} - 2 \beta_{4} - 12 \beta_{3} + 3 \beta_{2} + 58 \beta_{1} + 4\)
\(\nu^{6}\)\(=\)\(-22 \beta_{11} - 16 \beta_{10} - 103 \beta_{9} + 35 \beta_{8} + 97 \beta_{7} + 19 \beta_{6} - 15 \beta_{5} - 48 \beta_{4} - 80 \beta_{3} - 12 \beta_{2} + 8 \beta_{1} + 322\)
\(\nu^{7}\)\(=\)\(-156 \beta_{11} + 110 \beta_{10} - 50 \beta_{9} + 202 \beta_{8} + 74 \beta_{7} + 294 \beta_{6} + 122 \beta_{5} - 45 \beta_{4} - 132 \beta_{3} + 59 \beta_{2} + 521 \beta_{1} + 96\)
\(\nu^{8}\)\(=\)\(-335 \beta_{11} - 187 \beta_{10} - 1076 \beta_{9} + 477 \beta_{8} + 958 \beta_{7} + 282 \beta_{6} - 171 \beta_{5} - 602 \beta_{4} - 749 \beta_{3} - 100 \beta_{2} + 190 \beta_{1} + 2910\)
\(\nu^{9}\)\(=\)\(-1827 \beta_{11} + 1073 \beta_{10} - 849 \beta_{9} + 2372 \beta_{8} + 1049 \beta_{7} + 3189 \beta_{6} + 1191 \beta_{5} - 721 \beta_{4} - 1454 \beta_{3} + 811 \beta_{2} + 4920 \beta_{1} + 1597\)
\(\nu^{10}\)\(=\)\(-4456 \beta_{11} - 1910 \beta_{10} - 11292 \beta_{9} + 5969 \beta_{8} + 9670 \beta_{7} + 3806 \beta_{6} - 1739 \beta_{5} - 6950 \beta_{4} - 7357 \beta_{3} - 637 \beta_{2} + 3121 \beta_{1} + 27578\)
\(\nu^{11}\)\(=\)\(-21047 \beta_{11} + 10383 \beta_{10} - 12259 \beta_{9} + 27045 \beta_{8} + 13533 \beta_{7} + 34094 \beta_{6} + 11472 \beta_{5} - 10067 \beta_{4} - 16155 \beta_{3} + 9727 \beta_{2} + 48116 \beta_{1} + 22864\)

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
3.34132
2.68363
2.31477
1.79442
0.830621
0.295455
−0.569286
−0.571891
−1.17540
−2.42555
−2.49668
−3.02141
0 −3.34132 0 2.46644 0 −2.50015 0 8.16441 0
1.2 0 −2.68363 0 0.670957 0 4.03610 0 4.20190 0
1.3 0 −2.31477 0 −1.78615 0 −4.41953 0 2.35817 0
1.4 0 −1.79442 0 3.50243 0 −0.885520 0 0.219933 0
1.5 0 −0.830621 0 −1.03423 0 −1.08177 0 −2.31007 0
1.6 0 −0.295455 0 −3.76247 0 1.37107 0 −2.91271 0
1.7 0 0.569286 0 3.51919 0 3.94871 0 −2.67591 0
1.8 0 0.571891 0 −2.42234 0 −3.89487 0 −2.67294 0
1.9 0 1.17540 0 3.77550 0 −4.01031 0 −1.61844 0
1.10 0 2.42555 0 −0.247339 0 2.30877 0 2.88329 0
1.11 0 2.49668 0 0.494819 0 2.18883 0 3.23342 0
1.12 0 3.02141 0 2.82320 0 −1.06132 0 6.12894 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
\(2\) \(-1\)
\(167\) \(1\)

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 1336.2.a.d 12
4.b odd 2 1 2672.2.a.p 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1336.2.a.d 12 1.a even 1 1 trivial
2672.2.a.p 12 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{3}^{12} + \cdots\) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1336))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( 1 + T + 11 T^{2} + 13 T^{3} + 68 T^{4} + 90 T^{5} + 326 T^{6} + 480 T^{7} + 1314 T^{8} + 2062 T^{9} + 4559 T^{10} + 7212 T^{11} + 14242 T^{12} + 21636 T^{13} + 41031 T^{14} + 55674 T^{15} + 106434 T^{16} + 116640 T^{17} + 237654 T^{18} + 196830 T^{19} + 446148 T^{20} + 255879 T^{21} + 649539 T^{22} + 177147 T^{23} + 531441 T^{24} \)
$5$ \( 1 - 8 T + 53 T^{2} - 248 T^{3} + 1047 T^{4} - 3688 T^{5} + 12156 T^{6} - 35512 T^{7} + 99309 T^{8} - 254088 T^{9} + 634687 T^{10} - 1479832 T^{11} + 3411222 T^{12} - 7399160 T^{13} + 15867175 T^{14} - 31761000 T^{15} + 62068125 T^{16} - 110975000 T^{17} + 189937500 T^{18} - 288125000 T^{19} + 408984375 T^{20} - 484375000 T^{21} + 517578125 T^{22} - 390625000 T^{23} + 244140625 T^{24} \)
$7$ \( 1 + 4 T + 40 T^{2} + 136 T^{3} + 833 T^{4} + 2529 T^{5} + 12154 T^{6} + 33386 T^{7} + 137681 T^{8} + 347839 T^{9} + 1274382 T^{10} + 2958957 T^{11} + 9776074 T^{12} + 20712699 T^{13} + 62444718 T^{14} + 119308777 T^{15} + 330572081 T^{16} + 561118502 T^{17} + 1429905946 T^{18} + 2082740247 T^{19} + 4802079233 T^{20} + 5488090552 T^{21} + 11299009960 T^{22} + 7909306972 T^{23} + 13841287201 T^{24} \)
$11$ \( 1 - 6 T + 78 T^{2} - 368 T^{3} + 2890 T^{4} - 11755 T^{5} + 71010 T^{6} - 258818 T^{7} + 1309339 T^{8} - 4348186 T^{9} + 19257904 T^{10} - 58566749 T^{11} + 232768628 T^{12} - 644234239 T^{13} + 2330206384 T^{14} - 5787435566 T^{15} + 19170032299 T^{16} - 41682897718 T^{17} + 125798546610 T^{18} - 229071695105 T^{19} + 619497166090 T^{20} - 867724750288 T^{21} + 2023119118878 T^{22} - 1711870023666 T^{23} + 3138428376721 T^{24} \)
$13$ \( 1 - 13 T + 166 T^{2} - 1386 T^{3} + 10959 T^{4} - 70182 T^{5} + 426682 T^{6} - 2255577 T^{7} + 11375923 T^{8} - 51440742 T^{9} + 222705776 T^{10} - 876391344 T^{11} + 3307744314 T^{12} - 11393087472 T^{13} + 37637276144 T^{14} - 113015310174 T^{15} + 324907736803 T^{16} - 837479951061 T^{17} + 2059512517738 T^{18} - 4403816420094 T^{19} + 8939592971439 T^{20} - 14697836130978 T^{21} + 22884509646934 T^{22} - 23298085122481 T^{23} + 23298085122481 T^{24} \)
$17$ \( 1 - 10 T + 165 T^{2} - 1192 T^{3} + 11741 T^{4} - 68630 T^{5} + 517025 T^{6} - 2584468 T^{7} + 16278147 T^{8} - 71577210 T^{9} + 392465306 T^{10} - 1537406410 T^{11} + 7474717374 T^{12} - 26135908970 T^{13} + 113422473434 T^{14} - 351658832730 T^{15} + 1359567115587 T^{16} - 3669574981076 T^{17} + 12479726612225 T^{18} - 28161543127990 T^{19} + 81902368114781 T^{20} - 141356748784424 T^{21} + 332638993574085 T^{22} - 342718963076330 T^{23} + 582622237229761 T^{24} \)
$19$ \( 1 + T + 143 T^{2} + 81 T^{3} + 10035 T^{4} + 1401 T^{5} + 458986 T^{6} - 103028 T^{7} + 15376081 T^{8} - 7311047 T^{9} + 401594583 T^{10} - 231239358 T^{11} + 8453969526 T^{12} - 4393547802 T^{13} + 144975644463 T^{14} - 50146471373 T^{15} + 2003826252001 T^{16} - 255107527772 T^{17} + 21593400736666 T^{18} + 1252314306339 T^{19} + 170430055116435 T^{20} + 26137703520099 T^{21} + 876742474865543 T^{22} + 116490258898219 T^{23} + 2213314919066161 T^{24} \)
$23$ \( 1 - 3 T + 156 T^{2} - 544 T^{3} + 12859 T^{4} - 46134 T^{5} + 722980 T^{6} - 2512287 T^{7} + 30281675 T^{8} - 98657940 T^{9} + 984601984 T^{10} - 2937072550 T^{11} + 25361223538 T^{12} - 67552668650 T^{13} + 520854449536 T^{14} - 1200371155980 T^{15} + 8474054213675 T^{16} - 16169940846441 T^{17} + 107026987029220 T^{18} - 157078217171898 T^{19} + 1007000959728379 T^{20} - 979827047835872 T^{21} + 6462535749329244 T^{22} - 2858429273741781 T^{23} + 21914624432020321 T^{24} \)
$29$ \( 1 - 29 T + 548 T^{2} - 7552 T^{3} + 87152 T^{4} - 863504 T^{5} + 7675970 T^{6} - 61582806 T^{7} + 455093348 T^{8} - 3101918876 T^{9} + 19739486944 T^{10} - 117092823849 T^{11} + 652152687522 T^{12} - 3395691891621 T^{13} + 16600908519904 T^{14} - 75652699466764 T^{15} + 321878878266788 T^{16} - 1263134109704094 T^{17} + 4565845967296370 T^{18} - 14895337192326736 T^{19} + 43597475382377072 T^{20} - 109557966409762688 T^{21} + 230547563848510148 T^{22} - 353814783205469041 T^{23} + 353814783205469041 T^{24} \)
$31$ \( 1 + 3 T + 184 T^{2} + 496 T^{3} + 17088 T^{4} + 38208 T^{5} + 1061590 T^{6} + 1787402 T^{7} + 49574412 T^{8} + 57911964 T^{9} + 1884191468 T^{10} + 1580225069 T^{11} + 61992076274 T^{12} + 48986977139 T^{13} + 1810708000748 T^{14} + 1725255319524 T^{15} + 45783010544652 T^{16} + 51171801755702 T^{17} + 942165032712790 T^{18} + 1051201959953088 T^{19} + 14574202047791808 T^{20} + 13114052591692816 T^{21} + 150811604804467384 T^{22} + 76225430689214493 T^{23} + 787662783788549761 T^{24} \)
$37$ \( 1 - 41 T + 1033 T^{2} - 18759 T^{3} + 275331 T^{4} - 3406272 T^{5} + 36890664 T^{6} - 356083210 T^{7} + 3115607565 T^{8} - 24911189221 T^{9} + 183516644463 T^{10} - 1249548077529 T^{11} + 7892527209182 T^{12} - 46233278868573 T^{13} + 251234286269847 T^{14} - 1261826467611313 T^{15} + 5839150189627965 T^{16} - 24692218802661970 T^{17} + 94651350870345576 T^{18} - 323363794985578176 T^{19} + 967094480527522851 T^{20} - 2437952276815849443 T^{21} + 4967267656707638017 T^{22} - 7294622492957876933 T^{23} + 6582952005840035281 T^{24} \)
$41$ \( 1 - 20 T + 462 T^{2} - 6324 T^{3} + 87975 T^{4} - 937816 T^{5} + 9887402 T^{6} - 87412528 T^{7} + 762361555 T^{8} - 5818085772 T^{9} + 44034155752 T^{10} - 298034708708 T^{11} + 2010901084426 T^{12} - 12219423057028 T^{13} + 74021415819112 T^{14} - 400988289492012 T^{15} + 2154251550018355 T^{16} - 10127283413886128 T^{17} + 46966190172671882 T^{18} - 182643674113983896 T^{19} + 702473797031919975 T^{20} - 2070363353107409364 T^{21} + 6201268601290409262 T^{22} - 11006580634324968820 T^{23} + 22563490300366186081 T^{24} \)
$43$ \( 1 + T + 322 T^{2} + 210 T^{3} + 52081 T^{4} + 20518 T^{5} + 5574006 T^{6} + 1198663 T^{7} + 438696187 T^{8} + 47048986 T^{9} + 26694033368 T^{10} + 1531431188 T^{11} + 1285689958822 T^{12} + 65851541084 T^{13} + 49357267697432 T^{14} + 3740723729902 T^{15} + 1499814962811787 T^{16} + 176213581311709 T^{17} + 35235315563304294 T^{18} + 5577174262693426 T^{19} + 608733158657737681 T^{20} + 105544448506737030 T^{21} + 6958897304877528178 T^{22} + 929293739471222707 T^{23} + 39959630797262576401 T^{24} \)
$47$ \( 1 - 5 T + 272 T^{2} - 1154 T^{3} + 38220 T^{4} - 142892 T^{5} + 3793456 T^{6} - 12969688 T^{7} + 294553462 T^{8} - 924023442 T^{9} + 18438929550 T^{10} - 52757098657 T^{11} + 949938052894 T^{12} - 2479583636879 T^{13} + 40731595375950 T^{14} - 95934885818766 T^{15} + 1437326932005622 T^{16} - 2974533185147816 T^{17} + 40890479065087024 T^{18} - 72392390929198996 T^{19} + 910067376212505420 T^{20} - 1291476565960593118 T^{21} + 14306963968145773328 T^{22} - 12360796075420061515 T^{23} + \)\(11\!\cdots\!41\)\( T^{24} \)
$53$ \( 1 - 39 T + 1040 T^{2} - 20272 T^{3} + 328087 T^{4} - 4495172 T^{5} + 54394544 T^{6} - 587035227 T^{7} + 5781909051 T^{8} - 52318711658 T^{9} + 441970840512 T^{10} - 3498408843716 T^{11} + 26210432476442 T^{12} - 185415668716948 T^{13} + 1241496090998208 T^{14} - 7789052835508066 T^{15} + 45622043510643531 T^{16} - 245495486163631911 T^{17} + 1205620316663280176 T^{18} - 5280528623883366964 T^{19} + 20426595047992196407 T^{20} - 66892807533012840176 T^{21} + \)\(18\!\cdots\!60\)\( T^{22} - \)\(36\!\cdots\!83\)\( T^{23} + \)\(49\!\cdots\!41\)\( T^{24} \)
$59$ \( 1 - 8 T + 360 T^{2} - 2574 T^{3} + 64925 T^{4} - 419628 T^{5} + 7794256 T^{6} - 46045554 T^{7} + 706940547 T^{8} - 3846500746 T^{9} + 52187047304 T^{10} - 263973152262 T^{11} + 3297051899198 T^{12} - 15574415983458 T^{13} + 181663111665224 T^{14} - 789990476712734 T^{15} + 8566253813536467 T^{16} - 32919085415516646 T^{17} + 328765877414566096 T^{18} - 1044307845271627332 T^{19} + 9532966161460540925 T^{20} - 22298551237217812986 T^{21} + \)\(18\!\cdots\!60\)\( T^{22} - \)\(24\!\cdots\!72\)\( T^{23} + \)\(17\!\cdots\!81\)\( T^{24} \)
$61$ \( 1 - 30 T + 843 T^{2} - 15790 T^{3} + 274782 T^{4} - 3898493 T^{5} + 51874541 T^{6} - 601780665 T^{7} + 6604262409 T^{8} - 65123828282 T^{9} + 611139914150 T^{10} - 5226719073036 T^{11} + 42670329733292 T^{12} - 318829863455196 T^{13} + 2274051620552150 T^{14} - 14781871667276642 T^{15} + 91441567237290969 T^{16} - 508261723672320165 T^{17} + 2672595772125043301 T^{18} - 12251960947028016353 T^{19} + 52677718880018867742 T^{20} - \)\(18\!\cdots\!90\)\( T^{21} + \)\(60\!\cdots\!43\)\( T^{22} - \)\(13\!\cdots\!30\)\( T^{23} + \)\(26\!\cdots\!21\)\( T^{24} \)
$67$ \( 1 + 9 T + 601 T^{2} + 3775 T^{3} + 157069 T^{4} + 604972 T^{5} + 23942342 T^{6} + 34337308 T^{7} + 2438929425 T^{8} - 2442927869 T^{9} + 188022122801 T^{10} - 539005678755 T^{11} + 12811914209986 T^{12} - 36113380476585 T^{13} + 844031309253689 T^{14} - 734742314664047 T^{15} + 49147161953635425 T^{16} + 46359661637591956 T^{17} + 2165785522656899798 T^{18} + 3666560821295465956 T^{19} + 63780644046144045229 T^{20} + \)\(10\!\cdots\!25\)\( T^{21} + \)\(10\!\cdots\!49\)\( T^{22} + \)\(10\!\cdots\!47\)\( T^{23} + \)\(81\!\cdots\!61\)\( T^{24} \)
$71$ \( 1 - 29 T + 933 T^{2} - 18131 T^{3} + 349216 T^{4} - 5237441 T^{5} + 75956737 T^{6} - 938754783 T^{7} + 11147389591 T^{8} - 117660839402 T^{9} + 1191836294858 T^{10} - 10951471090550 T^{11} + 96586453776592 T^{12} - 777554447429050 T^{13} + 6008046762379178 T^{14} - 42112108691209222 T^{15} + 283273908269212471 T^{16} - 1693728932880235833 T^{17} + 9730079575412725777 T^{18} - 47635155217483517431 T^{19} + \)\(22\!\cdots\!76\)\( T^{20} - \)\(83\!\cdots\!61\)\( T^{21} + \)\(30\!\cdots\!33\)\( T^{22} - \)\(67\!\cdots\!59\)\( T^{23} + \)\(16\!\cdots\!41\)\( T^{24} \)
$73$ \( 1 - 12 T + 635 T^{2} - 7394 T^{3} + 194695 T^{4} - 2147414 T^{5} + 38452515 T^{6} - 390830824 T^{7} + 5462001571 T^{8} - 50006017692 T^{9} + 586344593106 T^{10} - 4769156371240 T^{11} + 48581803711146 T^{12} - 348148415100520 T^{13} + 3124630336661874 T^{14} - 19453190984488764 T^{15} + 155111236955636611 T^{16} - 810220278903182632 T^{17} + 5819181606391166835 T^{18} - 23723338243488165158 T^{19} + \)\(15\!\cdots\!95\)\( T^{20} - \)\(43\!\cdots\!22\)\( T^{21} + \)\(27\!\cdots\!15\)\( T^{22} - \)\(37\!\cdots\!24\)\( T^{23} + \)\(22\!\cdots\!21\)\( T^{24} \)
$79$ \( 1 + 2 T + 478 T^{2} + 662 T^{3} + 116823 T^{4} + 163500 T^{5} + 19383770 T^{6} + 31924712 T^{7} + 2440536275 T^{8} + 4660284782 T^{9} + 248660799624 T^{10} + 498568629982 T^{11} + 21281094687146 T^{12} + 39386921768578 T^{13} + 1551892050453384 T^{14} + 2297702148632498 T^{15} + 95059085594688275 T^{16} + 98234139345832088 T^{17} + 4711951327704294170 T^{18} + 3139839119236996500 T^{19} + \)\(17\!\cdots\!03\)\( T^{20} + 79341756540493327178 T^{21} + \)\(45\!\cdots\!78\)\( T^{22} + \)\(14\!\cdots\!58\)\( T^{23} + \)\(59\!\cdots\!41\)\( T^{24} \)
$83$ \( 1 + 5 T + 575 T^{2} + 3737 T^{3} + 167083 T^{4} + 1276028 T^{5} + 32443144 T^{6} + 271988078 T^{7} + 4692313117 T^{8} + 40595516297 T^{9} + 533788509385 T^{10} + 4470477865035 T^{11} + 49073769984782 T^{12} + 371049662797905 T^{13} + 3677269041153265 T^{14} + 23211988476912739 T^{15} + 222689302139096557 T^{16} + 1071372093653454154 T^{17} + 10606973612624232136 T^{18} + 34626360872191761556 T^{19} + \)\(37\!\cdots\!03\)\( T^{20} + \)\(69\!\cdots\!11\)\( T^{21} + \)\(89\!\cdots\!75\)\( T^{22} + \)\(64\!\cdots\!35\)\( T^{23} + \)\(10\!\cdots\!61\)\( T^{24} \)
$89$ \( 1 - 7 T + 525 T^{2} - 2629 T^{3} + 131137 T^{4} - 350743 T^{5} + 21160946 T^{6} - 7265218 T^{7} + 2583323699 T^{8} + 4194750279 T^{9} + 267349298145 T^{10} + 731344991270 T^{11} + 24912791136118 T^{12} + 65089704223030 T^{13} + 2117673790606545 T^{14} + 2957168909436351 T^{15} + 162083518103669459 T^{16} - 40569409221944882 T^{17} + 10516594261036009106 T^{18} - 15513831095262528047 T^{19} + \)\(51\!\cdots\!97\)\( T^{20} - \)\(92\!\cdots\!61\)\( T^{21} + \)\(16\!\cdots\!25\)\( T^{22} - \)\(19\!\cdots\!23\)\( T^{23} + \)\(24\!\cdots\!21\)\( T^{24} \)
$97$ \( 1 + 14 T + 471 T^{2} + 4178 T^{3} + 85936 T^{4} + 546301 T^{5} + 9377607 T^{6} + 39835307 T^{7} + 587492441 T^{8} - 333339196 T^{9} + 5928057546 T^{10} - 362031342048 T^{11} - 1474830902652 T^{12} - 35117040178656 T^{13} + 55777093450314 T^{14} - 304229684030908 T^{15} + 52010283394664921 T^{16} + 342079335451053899 T^{17} + 7811284104226224903 T^{18} + 44140183608677610013 T^{19} + \)\(67\!\cdots\!96\)\( T^{20} + \)\(31\!\cdots\!26\)\( T^{21} + \)\(34\!\cdots\!79\)\( T^{22} + \)\(10\!\cdots\!42\)\( T^{23} + \)\(69\!\cdots\!41\)\( T^{24} \)
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