Properties

Label 8046.2.a.i
Level 8046
Weight 2
Character orbit 8046.a
Self dual yes
Analytic conductor 64.248
Analytic rank 1
Dimension 12
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 8046 = 2 \cdot 3^{3} \cdot 149 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 8046.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(64.2476334663\)
Analytic rank: \(1\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
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.

\(f(q)\) \(=\) \( q - q^{2} + q^{4} -\beta_{1} q^{5} + ( 1 - \beta_{5} ) q^{7} - q^{8} +O(q^{10})\) \( q - q^{2} + q^{4} -\beta_{1} q^{5} + ( 1 - \beta_{5} ) q^{7} - q^{8} + \beta_{1} q^{10} + ( -1 + \beta_{2} + \beta_{3} + \beta_{8} ) q^{11} + ( 1 - \beta_{5} - \beta_{6} + \beta_{11} ) q^{13} + ( -1 + \beta_{5} ) q^{14} + q^{16} + ( -\beta_{4} + \beta_{6} - \beta_{7} ) q^{17} + ( \beta_{4} - \beta_{10} ) q^{19} -\beta_{1} q^{20} + ( 1 - \beta_{2} - \beta_{3} - \beta_{8} ) q^{22} + ( -1 + \beta_{5} + \beta_{6} + \beta_{10} ) q^{23} + ( 1 + \beta_{1} - \beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} - \beta_{8} + \beta_{10} ) q^{25} + ( -1 + \beta_{5} + \beta_{6} - \beta_{11} ) q^{26} + ( 1 - \beta_{5} ) q^{28} + ( -3 - \beta_{3} + \beta_{4} + \beta_{5} - \beta_{9} + \beta_{10} + \beta_{11} ) q^{29} + ( \beta_{1} + \beta_{10} ) q^{31} - q^{32} + ( \beta_{4} - \beta_{6} + \beta_{7} ) q^{34} + ( -1 - \beta_{1} - 2 \beta_{2} + \beta_{5} - \beta_{6} - \beta_{8} - 2 \beta_{10} - 2 \beta_{11} ) q^{35} + ( 2 \beta_{1} - \beta_{2} - \beta_{3} - \beta_{5} - 2 \beta_{6} - \beta_{7} + \beta_{8} + 2 \beta_{9} - \beta_{10} - \beta_{11} ) q^{37} + ( -\beta_{4} + \beta_{10} ) q^{38} + \beta_{1} q^{40} + ( -3 + \beta_{1} - \beta_{6} + \beta_{7} - \beta_{10} - \beta_{11} ) q^{41} + ( -\beta_{1} - \beta_{2} + \beta_{5} - \beta_{6} - \beta_{8} - \beta_{9} - \beta_{10} - \beta_{11} ) q^{43} + ( -1 + \beta_{2} + \beta_{3} + \beta_{8} ) q^{44} + ( 1 - \beta_{5} - \beta_{6} - \beta_{10} ) q^{46} + ( -2 - 2 \beta_{3} + \beta_{4} + \beta_{5} - \beta_{6} + 2 \beta_{7} - \beta_{8} + \beta_{9} - \beta_{10} ) q^{47} + ( -1 + \beta_{2} + \beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} - \beta_{10} - 2 \beta_{11} ) q^{49} + ( -1 - \beta_{1} + \beta_{4} - \beta_{5} - \beta_{6} - \beta_{7} + \beta_{8} - \beta_{10} ) q^{50} + ( 1 - \beta_{5} - \beta_{6} + \beta_{11} ) q^{52} + ( -1 + \beta_{1} - \beta_{2} + \beta_{3} - \beta_{8} + \beta_{9} - \beta_{11} ) q^{53} + ( \beta_{1} - 2 \beta_{3} + 3 \beta_{5} + 2 \beta_{6} + \beta_{7} - \beta_{8} - \beta_{9} + \beta_{10} + \beta_{11} ) q^{55} + ( -1 + \beta_{5} ) q^{56} + ( 3 + \beta_{3} - \beta_{4} - \beta_{5} + \beta_{9} - \beta_{10} - \beta_{11} ) q^{58} + ( -2 + 2 \beta_{1} + \beta_{2} - \beta_{4} - 2 \beta_{5} + \beta_{10} + \beta_{11} ) q^{59} + ( -\beta_{1} - \beta_{2} - \beta_{3} + \beta_{5} + \beta_{6} - 2 \beta_{7} - \beta_{8} - \beta_{9} + \beta_{10} ) q^{61} + ( -\beta_{1} - \beta_{10} ) q^{62} + q^{64} + ( -1 - \beta_{2} - \beta_{3} + \beta_{4} - 2 \beta_{6} + \beta_{9} ) q^{65} + ( -1 - \beta_{2} + \beta_{3} - \beta_{4} + 2 \beta_{5} + 2 \beta_{6} + \beta_{8} - \beta_{9} - 2 \beta_{11} ) q^{67} + ( -\beta_{4} + \beta_{6} - \beta_{7} ) q^{68} + ( 1 + \beta_{1} + 2 \beta_{2} - \beta_{5} + \beta_{6} + \beta_{8} + 2 \beta_{10} + 2 \beta_{11} ) q^{70} + ( 2 \beta_{1} - \beta_{2} - 2 \beta_{3} - \beta_{4} - \beta_{5} - \beta_{7} - \beta_{8} + \beta_{9} + 2 \beta_{10} + \beta_{11} ) q^{71} + ( -1 - \beta_{1} - \beta_{2} + \beta_{4} + 2 \beta_{5} + \beta_{6} - \beta_{7} - 3 \beta_{9} + \beta_{10} + \beta_{11} ) q^{73} + ( -2 \beta_{1} + \beta_{2} + \beta_{3} + \beta_{5} + 2 \beta_{6} + \beta_{7} - \beta_{8} - 2 \beta_{9} + \beta_{10} + \beta_{11} ) q^{74} + ( \beta_{4} - \beta_{10} ) q^{76} + ( -6 + 2 \beta_{1} + 2 \beta_{2} + \beta_{3} + \beta_{4} + \beta_{5} - \beta_{6} + 2 \beta_{7} - \beta_{9} - \beta_{10} ) q^{77} + ( 4 - 2 \beta_{2} - 2 \beta_{4} - \beta_{5} - \beta_{6} - \beta_{7} + 2 \beta_{9} + \beta_{10} + \beta_{11} ) q^{79} -\beta_{1} q^{80} + ( 3 - \beta_{1} + \beta_{6} - \beta_{7} + \beta_{10} + \beta_{11} ) q^{82} + ( -1 + \beta_{3} + \beta_{4} + \beta_{5} + \beta_{6} + \beta_{9} + \beta_{10} ) q^{83} + ( -3 + 2 \beta_{2} + \beta_{3} + \beta_{4} + 4 \beta_{5} + 4 \beta_{6} + 3 \beta_{7} - 2 \beta_{9} - \beta_{11} ) q^{85} + ( \beta_{1} + \beta_{2} - \beta_{5} + \beta_{6} + \beta_{8} + \beta_{9} + \beta_{10} + \beta_{11} ) q^{86} + ( 1 - \beta_{2} - \beta_{3} - \beta_{8} ) q^{88} + ( -\beta_{2} - 2 \beta_{3} + 3 \beta_{5} + 2 \beta_{6} + 2 \beta_{7} - 2 \beta_{8} + \beta_{10} ) q^{89} + ( 2 + 3 \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} - 2 \beta_{5} - 2 \beta_{6} - \beta_{7} + \beta_{8} + 4 \beta_{9} ) q^{91} + ( -1 + \beta_{5} + \beta_{6} + \beta_{10} ) q^{92} + ( 2 + 2 \beta_{3} - \beta_{4} - \beta_{5} + \beta_{6} - 2 \beta_{7} + \beta_{8} - \beta_{9} + \beta_{10} ) q^{94} + ( -6 - \beta_{2} + \beta_{3} - \beta_{5} - \beta_{6} - 3 \beta_{7} + \beta_{8} - 2 \beta_{9} ) q^{95} + ( 1 + \beta_{1} - 2 \beta_{2} - \beta_{4} - 2 \beta_{5} - \beta_{6} - 2 \beta_{7} + \beta_{8} + 2 \beta_{9} ) q^{97} + ( 1 - \beta_{2} - \beta_{4} - \beta_{5} - \beta_{6} - \beta_{7} + \beta_{10} + 2 \beta_{11} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12q - 12q^{2} + 12q^{4} - 5q^{5} + 6q^{7} - 12q^{8} + O(q^{10}) \) \( 12q - 12q^{2} + 12q^{4} - 5q^{5} + 6q^{7} - 12q^{8} + 5q^{10} - 6q^{11} + 3q^{13} - 6q^{14} + 12q^{16} - 6q^{17} + 8q^{19} - 5q^{20} + 6q^{22} - 11q^{23} + 11q^{25} - 3q^{26} + 6q^{28} - 29q^{29} + 2q^{31} - 12q^{32} + 6q^{34} - 4q^{35} + 5q^{37} - 8q^{38} + 5q^{40} - 22q^{41} + 9q^{43} - 6q^{44} + 11q^{46} - 15q^{47} + 14q^{49} - 11q^{50} + 3q^{52} - 12q^{53} + 13q^{55} - 6q^{56} + 29q^{58} - 34q^{59} - 4q^{61} - 2q^{62} + 12q^{64} - 12q^{65} + q^{67} - 6q^{68} + 4q^{70} - 21q^{71} - 2q^{73} - 5q^{74} + 8q^{76} - 34q^{77} + 9q^{79} - 5q^{80} + 22q^{82} - 10q^{83} + 5q^{85} - 9q^{86} + 6q^{88} + 2q^{89} + 17q^{91} - 11q^{92} + 15q^{94} - 69q^{95} - 13q^{97} - 14q^{98} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{12} - 5 x^{11} - 23 x^{10} + 142 x^{9} + 104 x^{8} - 1302 x^{7} + 607 x^{6} + 4323 x^{5} - 4461 x^{4} - 3333 x^{3} + 4805 x^{2} - 224 x - 553\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\(39753790711 \nu^{11} - 198200510956 \nu^{10} - 967064971419 \nu^{9} + 5714838266656 \nu^{8} + 5759628424968 \nu^{7} - 53850580778217 \nu^{6} + 7072339723081 \nu^{5} + 189124537678876 \nu^{4} - 106414471211396 \nu^{3} - 177474875785820 \nu^{2} + 99233618007315 \nu + 17827377683725\)\()/ 7247830497003 \)
\(\beta_{3}\)\(=\)\((\)\(80534503868 \nu^{11} - 351696237482 \nu^{10} - 2108871784872 \nu^{9} + 10131881046614 \nu^{8} + 15896674895505 \nu^{7} - 95422133728626 \nu^{6} - 23898794925802 \nu^{5} + 335975682458783 \nu^{4} - 90437371366366 \nu^{3} - 327059747358370 \nu^{2} + 92295543950502 \nu + 42821455278854\)\()/ 7247830497003 \)
\(\beta_{4}\)\(=\)\((\)\(-82493402629 \nu^{11} + 263830494832 \nu^{10} + 2411694496521 \nu^{9} - 7581331666612 \nu^{8} - 23166629157747 \nu^{7} + 71744854896003 \nu^{6} + 84556908978098 \nu^{5} - 260250839987092 \nu^{4} - 94448906762320 \nu^{3} + 286297569815276 \nu^{2} + 36333927662739 \nu - 37905558611308\)\()/ 7247830497003 \)
\(\beta_{5}\)\(=\)\((\)\(111684601157 \nu^{11} - 439380667640 \nu^{10} - 3018114108333 \nu^{9} + 12565700557217 \nu^{8} + 24513574501275 \nu^{7} - 117331699982658 \nu^{6} - 53450947539877 \nu^{5} + 411530092181318 \nu^{4} - 68568106482415 \nu^{3} - 416256290933614 \nu^{2} + 100786856841096 \nu + 78764454042977\)\()/ 7247830497003 \)
\(\beta_{6}\)\(=\)\((\)\(126477824479 \nu^{11} - 465007218553 \nu^{10} - 3547935397485 \nu^{9} + 13375681875472 \nu^{8} + 31285095157641 \nu^{7} - 125748469351326 \nu^{6} - 90255138053357 \nu^{5} + 441397343073052 \nu^{4} + 6081982929931 \nu^{3} - 427114040209208 \nu^{2} + 79483670481720 \nu + 52979135676436\)\()/ 7247830497003 \)
\(\beta_{7}\)\(=\)\((\)\(-233966606153 \nu^{11} + 906289859315 \nu^{10} + 6340426420320 \nu^{9} - 25919528717069 \nu^{8} - 51693664769592 \nu^{7} + 241755270906525 \nu^{6} + 112822364381107 \nu^{5} - 841128794307659 \nu^{4} + 152460063612613 \nu^{3} + 807815616826948 \nu^{2} - 241244419540353 \nu - 116176930526963\)\()/ 7247830497003 \)
\(\beta_{8}\)\(=\)\((\)\(-264188216893 \nu^{11} + 1052591859745 \nu^{10} + 7122140689275 \nu^{9} - 30110515868080 \nu^{8} - 57550285913988 \nu^{7} + 280571600276976 \nu^{6} + 122706115800737 \nu^{5} - 970382672201788 \nu^{4} + 176585646871547 \nu^{3} + 895224641034665 \nu^{2} - 259184076615561 \nu - 75969812453317\)\()/ 7247830497003 \)
\(\beta_{9}\)\(=\)\((\)\(265780092172 \nu^{11} - 1045158248329 \nu^{10} - 7226529730506 \nu^{9} + 29933540825302 \nu^{8} + 59704300931763 \nu^{7} - 279842271576060 \nu^{6} - 140534451251591 \nu^{5} + 978788449854322 \nu^{4} - 122917256587556 \nu^{3} - 959282014883780 \nu^{2} + 236241029700939 \nu + 135370428068599\)\()/ 7247830497003 \)
\(\beta_{10}\)\(=\)\((\)\(-116959146335 \nu^{11} + 438173460485 \nu^{10} + 3253152757098 \nu^{9} - 12571233750104 \nu^{8} - 28273973320353 \nu^{7} + 117880451200146 \nu^{6} + 79382248663654 \nu^{5} - 414144051045197 \nu^{4} - 2612399983634 \nu^{3} + 408108251887606 \nu^{2} - 56374695744096 \nu - 57643004413031\)\()/ 2415943499001 \)
\(\beta_{11}\)\(=\)\((\)\(376299976007 \nu^{11} - 1405118384054 \nu^{10} - 10441796596398 \nu^{9} + 40199942268095 \nu^{8} + 90410848803024 \nu^{7} - 375246438055287 \nu^{6} - 252528829047475 \nu^{5} + 1308736876472261 \nu^{4} + 10639560659018 \nu^{3} - 1271104499897389 \nu^{2} + 158243198361369 \nu + 171474493496885\)\()/ 7247830497003 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{10} - \beta_{8} + \beta_{7} + \beta_{6} + \beta_{5} - \beta_{4} + \beta_{1} + 6\)
\(\nu^{3}\)\(=\)\(-\beta_{11} - \beta_{10} - \beta_{9} - \beta_{8} - \beta_{6} + \beta_{5} + 2 \beta_{3} - 3 \beta_{2} + 8 \beta_{1}\)
\(\nu^{4}\)\(=\)\(2 \beta_{11} + 17 \beta_{10} - 12 \beta_{8} + 15 \beta_{7} + 18 \beta_{6} + 19 \beta_{5} - 13 \beta_{4} + \beta_{3} + 14 \beta_{1} + 61\)
\(\nu^{5}\)\(=\)\(-22 \beta_{11} - 15 \beta_{10} - 6 \beta_{9} - 18 \beta_{8} - \beta_{7} - 16 \beta_{6} + 9 \beta_{5} - 2 \beta_{4} + 36 \beta_{3} - 61 \beta_{2} + 86 \beta_{1} + 16\)
\(\nu^{6}\)\(=\)\(55 \beta_{11} + 270 \beta_{10} - 4 \beta_{9} - 149 \beta_{8} + 211 \beta_{7} + 292 \beta_{6} + 286 \beta_{5} - 168 \beta_{4} + 28 \beta_{3} + 3 \beta_{2} + 190 \beta_{1} + 742\)
\(\nu^{7}\)\(=\)\(-334 \beta_{11} - 165 \beta_{10} + 18 \beta_{9} - 258 \beta_{8} - 31 \beta_{7} - 224 \beta_{6} + 23 \beta_{5} - 63 \beta_{4} + 588 \beta_{3} - 986 \beta_{2} + 1062 \beta_{1} + 437\)
\(\nu^{8}\)\(=\)\(1032 \beta_{11} + 4150 \beta_{10} - 61 \beta_{9} - 1902 \beta_{8} + 2902 \beta_{7} + 4487 \beta_{6} + 4003 \beta_{5} - 2315 \beta_{4} + 664 \beta_{3} + 37 \beta_{2} + 2643 \beta_{1} + 9899\)
\(\nu^{9}\)\(=\)\(-4527 \beta_{11} - 1300 \beta_{10} + 1303 \beta_{9} - 3491 \beta_{8} - 706 \beta_{7} - 2872 \beta_{6} - 945 \beta_{5} - 1461 \beta_{4} + 9351 \beta_{3} - 14872 \beta_{2} + 14198 \beta_{1} + 8796\)
\(\nu^{10}\)\(=\)\(17114 \beta_{11} + 62852 \beta_{10} - 375 \beta_{9} - 24898 \beta_{8} + 39584 \beta_{7} + 67054 \beta_{6} + 54592 \beta_{5} - 33297 \beta_{4} + 13472 \beta_{3} - 329 \beta_{2} + 37757 \beta_{1} + 138910\)
\(\nu^{11}\)\(=\)\(-58540 \beta_{11} - 1621 \beta_{10} + 28961 \beta_{9} - 46940 \beta_{8} - 13406 \beta_{7} - 33588 \beta_{6} - 27030 \beta_{5} - 29463 \beta_{4} + 146119 \beta_{3} - 218819 \beta_{2} + 198673 \beta_{1} + 158591\)

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
3.89381
3.31930
2.74383
2.27289
1.33540
1.20508
0.521020
−0.302744
−1.09224
−2.38357
−2.78478
−3.72799
−1.00000 0 1.00000 −3.89381 0 0.649105 −1.00000 0 3.89381
1.2 −1.00000 0 1.00000 −3.31930 0 −3.41918 −1.00000 0 3.31930
1.3 −1.00000 0 1.00000 −2.74383 0 0.881706 −1.00000 0 2.74383
1.4 −1.00000 0 1.00000 −2.27289 0 3.85060 −1.00000 0 2.27289
1.5 −1.00000 0 1.00000 −1.33540 0 2.26685 −1.00000 0 1.33540
1.6 −1.00000 0 1.00000 −1.20508 0 4.22185 −1.00000 0 1.20508
1.7 −1.00000 0 1.00000 −0.521020 0 −3.80450 −1.00000 0 0.521020
1.8 −1.00000 0 1.00000 0.302744 0 −1.13869 −1.00000 0 −0.302744
1.9 −1.00000 0 1.00000 1.09224 0 −1.25253 −1.00000 0 −1.09224
1.10 −1.00000 0 1.00000 2.38357 0 4.24377 −1.00000 0 −2.38357
1.11 −1.00000 0 1.00000 2.78478 0 2.18432 −1.00000 0 −2.78478
1.12 −1.00000 0 1.00000 3.72799 0 −2.68331 −1.00000 0 −3.72799
\(n\): 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 8046.2.a.i 12
3.b odd 2 1 8046.2.a.p yes 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8046.2.a.i 12 1.a even 1 1 trivial
8046.2.a.p yes 12 3.b odd 2 1

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(3\) \(-1\)
\(149\) \(-1\)

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8046))\):

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

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T )^{12} \)
$3$ \( \)
$5$ \( 1 + 5 T + 37 T^{2} + 133 T^{3} + 604 T^{4} + 1787 T^{5} + 6392 T^{6} + 16572 T^{7} + 50924 T^{8} + 119058 T^{9} + 327460 T^{10} + 701969 T^{11} + 1770847 T^{12} + 3509845 T^{13} + 8186500 T^{14} + 14882250 T^{15} + 31827500 T^{16} + 51787500 T^{17} + 99875000 T^{18} + 139609375 T^{19} + 235937500 T^{20} + 259765625 T^{21} + 361328125 T^{22} + 244140625 T^{23} + 244140625 T^{24} \)
$7$ \( 1 - 6 T + 53 T^{2} - 242 T^{3} + 1336 T^{4} - 5118 T^{5} + 22241 T^{6} - 74476 T^{7} + 275221 T^{8} - 821625 T^{9} + 2668448 T^{10} - 7159148 T^{11} + 20764979 T^{12} - 50114036 T^{13} + 130753952 T^{14} - 281817375 T^{15} + 660805621 T^{16} - 1251718132 T^{17} + 2616631409 T^{18} - 4214893074 T^{19} + 7701774136 T^{20} - 9765572894 T^{21} + 14971188197 T^{22} - 11863960458 T^{23} + 13841287201 T^{24} \)
$11$ \( 1 + 6 T + 83 T^{2} + 400 T^{3} + 3275 T^{4} + 13795 T^{5} + 85490 T^{6} + 324432 T^{7} + 1657369 T^{8} + 5718173 T^{9} + 25187716 T^{10} + 78968378 T^{11} + 307925583 T^{12} + 868652158 T^{13} + 3047713636 T^{14} + 7610888263 T^{15} + 24265539529 T^{16} + 52250098032 T^{17} + 151450749890 T^{18} + 268825523945 T^{19} + 702025335275 T^{20} + 943179076400 T^{21} + 2152806241883 T^{22} + 1711870023666 T^{23} + 3138428376721 T^{24} \)
$13$ \( 1 - 3 T + 93 T^{2} - 185 T^{3} + 4117 T^{4} - 5292 T^{5} + 120357 T^{6} - 96192 T^{7} + 2648158 T^{8} - 1305941 T^{9} + 46450531 T^{10} - 15709542 T^{11} + 665668655 T^{12} - 204224046 T^{13} + 7850139739 T^{14} - 2869152377 T^{15} + 75634040638 T^{16} - 35715416256 T^{17} + 580940250813 T^{18} - 332065151964 T^{19} + 3358363378357 T^{20} - 1961832384005 T^{21} + 12820839741957 T^{22} - 5376481182111 T^{23} + 23298085122481 T^{24} \)
$17$ \( 1 + 6 T + 105 T^{2} + 597 T^{3} + 6053 T^{4} + 32404 T^{5} + 242061 T^{6} + 1194171 T^{7} + 7302766 T^{8} + 32901614 T^{9} + 173184434 T^{10} + 705647403 T^{11} + 3282052285 T^{12} + 11996005851 T^{13} + 50050301426 T^{14} + 161645629582 T^{15} + 609934319086 T^{16} + 1695552053547 T^{17} + 5842764089709 T^{18} + 13296614359892 T^{19} + 42224259790373 T^{20} + 70796962268709 T^{21} + 211679359547145 T^{22} + 205631377845798 T^{23} + 582622237229761 T^{24} \)
$19$ \( 1 - 8 T + 129 T^{2} - 782 T^{3} + 7183 T^{4} - 38111 T^{5} + 260210 T^{6} - 1339878 T^{7} + 7370283 T^{8} - 37496113 T^{9} + 172035206 T^{10} - 849459602 T^{11} + 3458078643 T^{12} - 16139732438 T^{13} + 62104709366 T^{14} - 257185839067 T^{15} + 960502650843 T^{16} - 3317670575922 T^{17} + 12241808695010 T^{18} - 34066345845029 T^{19} + 121992933323503 T^{20} - 252341779663178 T^{21} + 790907547256329 T^{22} - 931922071185752 T^{23} + 2213314919066161 T^{24} \)
$23$ \( 1 + 11 T + 262 T^{2} + 2269 T^{3} + 30685 T^{4} + 221080 T^{5} + 2185501 T^{6} + 13469165 T^{7} + 106742913 T^{8} + 570337343 T^{9} + 3787749917 T^{10} + 17621235363 T^{11} + 100387404119 T^{12} + 405288413349 T^{13} + 2003719706093 T^{14} + 6939294452281 T^{15} + 29871043516833 T^{16} + 86692165863595 T^{17} + 323532583445389 T^{18} + 752738809822760 T^{19} + 2402972583347485 T^{20} + 4086815388859547 T^{21} + 10853745937976038 T^{22} + 10480907337053197 T^{23} + 21914624432020321 T^{24} \)
$29$ \( 1 + 29 T + 609 T^{2} + 9253 T^{3} + 117590 T^{4} + 1265253 T^{5} + 12043636 T^{6} + 102332654 T^{7} + 791357894 T^{8} + 5594431404 T^{9} + 36508906376 T^{10} + 220065448467 T^{11} + 1231373716743 T^{12} + 6381898005543 T^{13} + 30703990262216 T^{14} + 136442587512156 T^{15} + 559712402626214 T^{16} + 2098960313759446 T^{17} + 7163835562435156 T^{18} + 21825457749591177 T^{19} + 58823975700083990 T^{20} + 134234621714715857 T^{21} + 256210705079822409 T^{22} + 353814783205469041 T^{23} + 353814783205469041 T^{24} \)
$31$ \( 1 - 2 T + 316 T^{2} - 629 T^{3} + 47111 T^{4} - 90807 T^{5} + 4384473 T^{6} - 7978244 T^{7} + 283776859 T^{8} - 475465865 T^{9} + 13457602963 T^{10} - 20221270731 T^{11} + 479490551393 T^{12} - 626859392661 T^{13} + 12932756447443 T^{14} - 14164603584215 T^{15} + 262073888600539 T^{16} - 228410352190844 T^{17} + 3891235926745113 T^{18} - 2498337949577577 T^{19} + 40180549664882951 T^{20} - 16630522339062059 T^{21} + 259002538685933116 T^{22} - 50816953792809662 T^{23} + 787662783788549761 T^{24} \)
$37$ \( 1 - 5 T + 119 T^{2} - 115 T^{3} + 7645 T^{4} - 1649 T^{5} + 440841 T^{6} + 321802 T^{7} + 19993330 T^{8} + 6656905 T^{9} + 950071077 T^{10} + 125428093 T^{11} + 34857558493 T^{12} + 4640839441 T^{13} + 1300647304413 T^{14} + 337192208965 T^{15} + 37470719346130 T^{16} + 22315024050514 T^{17} + 1131077395869969 T^{18} - 156542665392317 T^{19} + 26852905425226045 T^{20} - 14945600076433855 T^{21} + 572221540317724031 T^{22} - 889588108897302065 T^{23} + 6582952005840035281 T^{24} \)
$41$ \( 1 + 22 T + 570 T^{2} + 8654 T^{3} + 134326 T^{4} + 1580751 T^{5} + 18404440 T^{6} + 177542004 T^{7} + 1677920778 T^{8} + 13669761819 T^{9} + 108769892738 T^{10} + 759716186158 T^{11} + 5178296674430 T^{12} + 31148363632478 T^{13} + 182842189692578 T^{14} + 942133654327299 T^{15} + 4741403095562058 T^{16} + 20569342101366804 T^{17} + 87423008497230040 T^{18} + 307858013191664631 T^{19} + 1072583066326907446 T^{20} + 2833163260245338494 T^{21} + 7650915806786868570 T^{22} + 12107238697757465702 T^{23} + 22563490300366186081 T^{24} \)
$43$ \( 1 - 9 T + 340 T^{2} - 3067 T^{3} + 59853 T^{4} - 506558 T^{5} + 6943178 T^{6} - 53812233 T^{7} + 581760563 T^{8} - 4070801777 T^{9} + 36810465985 T^{10} - 229950733232 T^{11} + 1796356034379 T^{12} - 9887881528976 T^{13} + 68062551606265 T^{14} - 323657236883939 T^{15} + 1988923594544963 T^{16} - 7910852587683219 T^{17} + 43890348851829722 T^{18} - 137691892005139706 T^{19} + 699573851215252653 T^{20} - 1541451540810297481 T^{21} + 7347903986516644660 T^{22} - 8363643655241004363 T^{23} + 39959630797262576401 T^{24} \)
$47$ \( 1 + 15 T + 362 T^{2} + 4070 T^{3} + 57742 T^{4} + 536591 T^{5} + 5843321 T^{6} + 47584927 T^{7} + 441055404 T^{8} + 3260908938 T^{9} + 26980598283 T^{10} + 183650618078 T^{11} + 1382094670383 T^{12} + 8631579049666 T^{13} + 59600141607147 T^{14} + 338557348669974 T^{15} + 2152209674846124 T^{16} + 10913365415909489 T^{17} + 62986415295467609 T^{18} + 271849406832361633 T^{19} + 1374911314423403662 T^{20} + 4554861025528261690 T^{21} + 19040885869370477738 T^{22} + 37082388226260184545 T^{23} + \)\(11\!\cdots\!41\)\( T^{24} \)
$53$ \( 1 + 12 T + 401 T^{2} + 4001 T^{3} + 70543 T^{4} + 575383 T^{5} + 7018763 T^{6} + 45343642 T^{7} + 427988438 T^{8} + 2060058947 T^{9} + 17268440727 T^{10} + 61388213430 T^{11} + 693724505451 T^{12} + 3253575311790 T^{13} + 48507050002143 T^{14} + 306695395852519 T^{15} + 3377034638258678 T^{16} + 18962506720605506 T^{17} + 155566397810863427 T^{18} + 675908819772832571 T^{19} + 4391985340688639023 T^{20} + 13202354130800334133 T^{21} + 70129875616570732649 T^{22} + \)\(11\!\cdots\!64\)\( T^{23} + \)\(49\!\cdots\!41\)\( T^{24} \)
$59$ \( 1 + 34 T + 991 T^{2} + 19524 T^{3} + 340902 T^{4} + 4900505 T^{5} + 64248442 T^{6} + 743642222 T^{7} + 7996728095 T^{8} + 78396120401 T^{9} + 720435496810 T^{10} + 6116626005439 T^{11} + 48799744500633 T^{12} + 360880934320901 T^{13} + 2507835964395610 T^{14} + 16100916811836979 T^{15} + 96899241145957295 T^{16} + 531647894270152378 T^{17} + 2710033569162837322 T^{18} + 12195649044612933595 T^{19} + 50054789840188237542 T^{20} + \)\(16\!\cdots\!36\)\( T^{21} + \)\(50\!\cdots\!91\)\( T^{22} + \)\(10\!\cdots\!06\)\( T^{23} + \)\(17\!\cdots\!81\)\( T^{24} \)
$61$ \( 1 + 4 T + 406 T^{2} + 2298 T^{3} + 82658 T^{4} + 590587 T^{5} + 11299823 T^{6} + 92930194 T^{7} + 1165100774 T^{8} + 10151363202 T^{9} + 95808777573 T^{10} + 817946848303 T^{11} + 6441572717915 T^{12} + 49894757746483 T^{13} + 356504461349133 T^{14} + 2304166570953162 T^{15} + 16131800065780934 T^{16} + 78488498103612394 T^{17} + 582171111173038103 T^{18} + 1856063063297134327 T^{19} + 15846143077729252898 T^{20} + 26873147721332856018 T^{21} + \)\(28\!\cdots\!06\)\( T^{22} + \)\(17\!\cdots\!44\)\( T^{23} + \)\(26\!\cdots\!21\)\( T^{24} \)
$67$ \( 1 - T + 257 T^{2} + 45 T^{3} + 40286 T^{4} + 24251 T^{5} + 5084197 T^{6} + 4140364 T^{7} + 503637946 T^{8} + 544064397 T^{9} + 42191060791 T^{10} + 45687110740 T^{11} + 3079301020221 T^{12} + 3061036419580 T^{13} + 189395671890799 T^{14} + 163634440234911 T^{15} + 10148869190037466 T^{16} + 5590009388518948 T^{17} + 459908235248483293 T^{18} + 146978317140688073 T^{19} + 16358842458046839326 T^{20} + 1224294047833272615 T^{21} + \)\(46\!\cdots\!93\)\( T^{22} - \)\(12\!\cdots\!83\)\( T^{23} + \)\(81\!\cdots\!61\)\( T^{24} \)
$71$ \( 1 + 21 T + 815 T^{2} + 13439 T^{3} + 298431 T^{4} + 4057110 T^{5} + 66280222 T^{6} + 764199396 T^{7} + 10056987962 T^{8} + 99917777007 T^{9} + 1105322506763 T^{10} + 9528643307964 T^{11} + 90523489032239 T^{12} + 676533674865444 T^{13} + 5571930756592283 T^{14} + 35761671486352377 T^{15} + 255564969911184122 T^{16} + 1378790980279671996 T^{17} + 8490515256546910462 T^{18} + 36899902945809710010 T^{19} + \)\(19\!\cdots\!91\)\( T^{20} + \)\(61\!\cdots\!09\)\( T^{21} + \)\(26\!\cdots\!15\)\( T^{22} + \)\(48\!\cdots\!91\)\( T^{23} + \)\(16\!\cdots\!41\)\( T^{24} \)
$73$ \( 1 + 2 T + 398 T^{2} + 1673 T^{3} + 77385 T^{4} + 460592 T^{5} + 10426140 T^{6} + 69484139 T^{7} + 1135186987 T^{8} + 7189648889 T^{9} + 104932563821 T^{10} + 595825738667 T^{11} + 8285671901785 T^{12} + 43495278922691 T^{13} + 559185632602109 T^{14} + 2796895641852113 T^{15} + 32237313636889867 T^{16} + 144045594724963427 T^{17} + 1577831830080794460 T^{18} + 5088343378707925424 T^{19} + 62407914211223458185 T^{20} + 98492164562932218449 T^{21} + \)\(17\!\cdots\!02\)\( T^{22} + \)\(62\!\cdots\!54\)\( T^{23} + \)\(22\!\cdots\!21\)\( T^{24} \)
$79$ \( 1 - 9 T + 518 T^{2} - 3691 T^{3} + 131738 T^{4} - 781364 T^{5} + 22668483 T^{6} - 116720572 T^{7} + 2981526684 T^{8} - 13678163642 T^{9} + 315293662403 T^{10} - 1306179035946 T^{11} + 27391779560869 T^{12} - 103188143839734 T^{13} + 1967747747057123 T^{14} - 6743868123888038 T^{15} + 116130705845461404 T^{16} - 359155782967540228 T^{17} + 5510423852991044643 T^{18} - 15005243141061140876 T^{19} + \)\(19\!\cdots\!18\)\( T^{20} - \)\(44\!\cdots\!29\)\( T^{21} + \)\(49\!\cdots\!18\)\( T^{22} - \)\(67\!\cdots\!11\)\( T^{23} + \)\(59\!\cdots\!41\)\( T^{24} \)
$83$ \( 1 + 10 T + 595 T^{2} + 4428 T^{3} + 153376 T^{4} + 876647 T^{5} + 24166252 T^{6} + 117887750 T^{7} + 2875503543 T^{8} + 14127122419 T^{9} + 295311784318 T^{10} + 1507783663471 T^{11} + 26514609855899 T^{12} + 125146044068093 T^{13} + 2034402882166702 T^{14} + 8077704946592753 T^{15} + 136466570180331303 T^{16} + 464364638561823250 T^{17} + 7900923451809342988 T^{18} + 23788737691903540669 T^{19} + \)\(34\!\cdots\!16\)\( T^{20} + \)\(82\!\cdots\!84\)\( T^{21} + \)\(92\!\cdots\!55\)\( T^{22} + \)\(12\!\cdots\!70\)\( T^{23} + \)\(10\!\cdots\!61\)\( T^{24} \)
$89$ \( 1 - 2 T + 662 T^{2} - 1003 T^{3} + 214775 T^{4} - 220301 T^{5} + 45647850 T^{6} - 27386804 T^{7} + 7148082988 T^{8} - 2124959770 T^{9} + 875420427455 T^{10} - 125970749265 T^{11} + 86394150305655 T^{12} - 11211396684585 T^{13} + 6934205205871055 T^{14} - 1498030764097130 T^{15} + 448486745521096108 T^{16} - 152929541654110996 T^{17} + 22686127422594083850 T^{18} - 9744207308819934229 T^{19} + \)\(84\!\cdots\!75\)\( T^{20} - \)\(35\!\cdots\!27\)\( T^{21} + \)\(20\!\cdots\!62\)\( T^{22} - \)\(55\!\cdots\!78\)\( T^{23} + \)\(24\!\cdots\!21\)\( T^{24} \)
$97$ \( 1 + 13 T + 909 T^{2} + 10619 T^{3} + 399660 T^{4} + 4192016 T^{5} + 111664562 T^{6} + 1049739005 T^{7} + 22001910224 T^{8} + 184572841939 T^{9} + 3213928058406 T^{10} + 23874378565506 T^{11} + 356479471151649 T^{12} + 2315814720854082 T^{13} + 30239849101542054 T^{14} + 168454649370992947 T^{15} + 1947813292757268944 T^{16} + 9014466016979624285 T^{17} + 93013454088658626098 T^{18} + \)\(33\!\cdots\!08\)\( T^{19} + \)\(31\!\cdots\!60\)\( T^{20} + \)\(80\!\cdots\!23\)\( T^{21} + \)\(67\!\cdots\!41\)\( T^{22} + \)\(92\!\cdots\!89\)\( T^{23} + \)\(69\!\cdots\!41\)\( T^{24} \)
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