Properties

Label 8016.2.a.bg
Level 8016
Weight 2
Character orbit 8016.a
Self dual yes
Analytic conductor 64.008
Analytic rank 0
Dimension 13
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 8016 = 2^{4} \cdot 3 \cdot 167 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8016.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(64.0080822603\)
Analytic rank: \(0\)
Dimension: \(13\)
Coefficient field: \(\mathbb{Q}[x]/(x^{13} - \cdots)\)
Defining polynomial: \(x^{13} - 2 x^{12} - 49 x^{11} + 99 x^{10} + 901 x^{9} - 1879 x^{8} - 7582 x^{7} + 16968 x^{6} + 26911 x^{5} - 72240 x^{4} - 14532 x^{3} + 112850 x^{2} - 72184 x + 12144\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 4008)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{12}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{3} + \beta_{1} q^{5} + \beta_{9} q^{7} + q^{9} +O(q^{10})\) \( q - q^{3} + \beta_{1} q^{5} + \beta_{9} q^{7} + q^{9} + ( -1 - \beta_{5} ) q^{11} + ( 1 + \beta_{3} ) q^{13} -\beta_{1} q^{15} + ( 1 - \beta_{2} ) q^{17} + ( -1 + \beta_{9} - \beta_{10} ) q^{19} -\beta_{9} q^{21} + ( -2 - \beta_{2} - \beta_{4} - \beta_{5} + \beta_{6} + \beta_{7} + \beta_{8} - \beta_{9} + \beta_{10} - \beta_{11} - \beta_{12} ) q^{23} + ( 3 - \beta_{5} + \beta_{6} ) q^{25} - q^{27} -\beta_{7} q^{29} + ( 1 - \beta_{4} - \beta_{10} ) q^{31} + ( 1 + \beta_{5} ) q^{33} + ( -\beta_{1} + \beta_{3} + \beta_{4} - \beta_{7} - \beta_{8} + \beta_{10} ) q^{35} + ( 1 - \beta_{5} + \beta_{7} - \beta_{10} - \beta_{11} ) q^{37} + ( -1 - \beta_{3} ) q^{39} + ( \beta_{1} - \beta_{4} + \beta_{10} - \beta_{12} ) q^{41} + ( \beta_{3} + \beta_{4} + \beta_{5} - \beta_{7} - \beta_{8} + \beta_{9} + \beta_{11} + \beta_{12} ) q^{43} + \beta_{1} q^{45} + ( 1 + \beta_{4} - \beta_{6} - \beta_{7} + \beta_{11} + \beta_{12} ) q^{47} + ( \beta_{1} - \beta_{4} - \beta_{6} + 2 \beta_{7} + \beta_{8} + \beta_{9} - \beta_{10} - \beta_{12} ) q^{49} + ( -1 + \beta_{2} ) q^{51} + ( -1 + \beta_{1} + \beta_{9} + \beta_{10} ) q^{53} + ( 2 \beta_{4} + \beta_{7} + 2 \beta_{9} ) q^{55} + ( 1 - \beta_{9} + \beta_{10} ) q^{57} + ( -2 + \beta_{1} - \beta_{4} + \beta_{8} - \beta_{9} ) q^{59} + ( 3 + \beta_{2} - \beta_{7} - \beta_{8} + \beta_{9} + \beta_{11} + \beta_{12} ) q^{61} + \beta_{9} q^{63} + ( 2 \beta_{1} - 2 \beta_{3} + \beta_{4} - \beta_{5} + \beta_{7} + 2 \beta_{8} + \beta_{9} - \beta_{10} - \beta_{11} + \beta_{12} ) q^{65} + ( -2 + 2 \beta_{1} - \beta_{4} - \beta_{5} - \beta_{6} + 2 \beta_{7} + \beta_{8} - \beta_{10} - \beta_{11} ) q^{67} + ( 2 + \beta_{2} + \beta_{4} + \beta_{5} - \beta_{6} - \beta_{7} - \beta_{8} + \beta_{9} - \beta_{10} + \beta_{11} + \beta_{12} ) q^{69} + ( -1 + \beta_{1} + \beta_{5} - \beta_{8} + \beta_{9} + \beta_{10} + \beta_{11} ) q^{71} + ( 4 - \beta_{1} + \beta_{2} + \beta_{4} - \beta_{7} + \beta_{9} + \beta_{12} ) q^{73} + ( -3 + \beta_{5} - \beta_{6} ) q^{75} + ( 2 + \beta_{1} + \beta_{4} - \beta_{5} - \beta_{6} + \beta_{7} - \beta_{8} - \beta_{10} + \beta_{11} ) q^{77} + ( -1 + \beta_{1} - \beta_{4} + \beta_{5} - \beta_{7} + \beta_{10} + \beta_{11} ) q^{79} + q^{81} + ( -4 + 2 \beta_{1} - \beta_{4} + \beta_{5} + \beta_{7} - \beta_{9} - \beta_{11} ) q^{83} + ( 3 + 2 \beta_{4} + \beta_{5} - 2 \beta_{7} + 2 \beta_{9} + 2 \beta_{11} ) q^{85} + \beta_{7} q^{87} + ( 3 + \beta_{2} + \beta_{6} - \beta_{7} - \beta_{8} + \beta_{10} ) q^{89} + ( -2 + \beta_{1} + \beta_{2} - \beta_{4} + \beta_{5} - \beta_{7} + \beta_{10} - \beta_{11} ) q^{91} + ( -1 + \beta_{4} + \beta_{10} ) q^{93} + ( 2 - 2 \beta_{1} + \beta_{4} - \beta_{7} - 2 \beta_{10} ) q^{95} + ( 3 - \beta_{1} - \beta_{2} - \beta_{4} + \beta_{5} - \beta_{7} - 2 \beta_{9} + 2 \beta_{10} - \beta_{11} ) q^{97} + ( -1 - \beta_{5} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 13q - 13q^{3} + 2q^{5} - q^{7} + 13q^{9} + O(q^{10}) \) \( 13q - 13q^{3} + 2q^{5} - q^{7} + 13q^{9} - 11q^{11} + 12q^{13} - 2q^{15} + 15q^{17} - 14q^{19} + q^{21} - 9q^{23} + 37q^{25} - 13q^{27} - 3q^{29} + 17q^{31} + 11q^{33} - 15q^{35} + 16q^{37} - 12q^{39} + 12q^{41} - 20q^{43} + 2q^{45} + 6q^{47} + 26q^{49} - 15q^{51} - 12q^{53} - 7q^{55} + 14q^{57} - 14q^{59} + 24q^{61} - q^{63} + 8q^{65} - 3q^{67} + 9q^{69} - 17q^{71} + 34q^{73} - 37q^{75} + 30q^{77} - 10q^{79} + 13q^{81} - 44q^{83} + 25q^{85} + 3q^{87} + 25q^{89} - 29q^{91} - 17q^{93} + 15q^{95} + 38q^{97} - 11q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{13} - 2 x^{12} - 49 x^{11} + 99 x^{10} + 901 x^{9} - 1879 x^{8} - 7582 x^{7} + 16968 x^{6} + 26911 x^{5} - 72240 x^{4} - 14532 x^{3} + 112850 x^{2} - 72184 x + 12144\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\(-6207348777 \nu^{12} - 12640530894 \nu^{11} + 576861363065 \nu^{10} + 295838273673 \nu^{9} - 16627326024925 \nu^{8} + 5016057310831 \nu^{7} + 201417980697138 \nu^{6} - 196205928405368 \nu^{5} - 1037037612348531 \nu^{4} + 1724063123712248 \nu^{3} + 1650853947740980 \nu^{2} - 4373968597429142 \nu + 1211669325949820\)\()/ 111451400959900 \)
\(\beta_{3}\)\(=\)\((\)\(126516435801 \nu^{12} - 351643713690 \nu^{11} - 4532098501229 \nu^{10} + 14015645726951 \nu^{9} + 48247617829913 \nu^{8} - 193085396549923 \nu^{7} - 90263735212158 \nu^{6} + 1012871433621172 \nu^{5} - 714691137288825 \nu^{4} - 993417294727940 \nu^{3} + 598916455112988 \nu^{2} - 3707736754746094 \nu + 3605634089719788\)\()/ 557257004799500 \)
\(\beta_{4}\)\(=\)\((\)\(8155241826 \nu^{12} - 24439858116 \nu^{11} - 363301449706 \nu^{10} + 1102737562371 \nu^{9} + 5918945839452 \nu^{8} - 18714943276923 \nu^{7} - 42939671149010 \nu^{6} + 149114772131121 \nu^{5} + 129168526434186 \nu^{4} - 552823809736903 \nu^{3} - 61985566741568 \nu^{2} + 719430531278666 \nu - 261781210799648\)\()/ 27862850239975 \)
\(\beta_{5}\)\(=\)\((\)\(42253210649 \nu^{12} - 42904580362 \nu^{11} - 2192557864275 \nu^{10} + 1857640717639 \nu^{9} + 43420560752365 \nu^{8} - 30214877144177 \nu^{7} - 406570644479846 \nu^{6} + 236986143576626 \nu^{5} + 1772258179746447 \nu^{4} - 967840084997036 \nu^{3} - 2780474603245950 \nu^{2} + 1829362384230014 \nu - 9603799562310\)\()/ 55725700479950 \)
\(\beta_{6}\)\(=\)\((\)\(42253210649 \nu^{12} - 42904580362 \nu^{11} - 2192557864275 \nu^{10} + 1857640717639 \nu^{9} + 43420560752365 \nu^{8} - 30214877144177 \nu^{7} - 406570644479846 \nu^{6} + 236986143576626 \nu^{5} + 1772258179746447 \nu^{4} - 967840084997036 \nu^{3} - 2724748902766000 \nu^{2} + 1829362384230014 \nu - 455409403401910\)\()/ 55725700479950 \)
\(\beta_{7}\)\(=\)\((\)\(138385219703 \nu^{12} - 105435570025 \nu^{11} - 6791649579897 \nu^{10} + 5771355044103 \nu^{9} + 127066606088584 \nu^{8} - 121899916781369 \nu^{7} - 1131795821929059 \nu^{6} + 1217996115089036 \nu^{5} + 4728856566083805 \nu^{4} - 5762733210647085 \nu^{3} - 6707972583182866 \nu^{2} + 10417867358098218 \nu - 2901404480931816\)\()/ 139314251199875 \)
\(\beta_{8}\)\(=\)\((\)\(111023509273 \nu^{12} - 148746957474 \nu^{11} - 5533191476745 \nu^{10} + 7117312024823 \nu^{9} + 104000386651945 \nu^{8} - 129559753690019 \nu^{7} - 912294162144422 \nu^{6} + 1109495790415652 \nu^{5} + 3686070397711199 \nu^{4} - 4408101894281592 \nu^{3} - 5320985228785100 \nu^{2} + 6347787481933558 \nu - 629228580752360\)\()/ 111451400959900 \)
\(\beta_{9}\)\(=\)\((\)\(-323942240303 \nu^{12} + 655210507370 \nu^{11} + 16238231918487 \nu^{10} - 30174775561853 \nu^{9} - 309203328646339 \nu^{8} + 524566352898419 \nu^{7} + 2761108370208174 \nu^{6} - 4297098906328266 \nu^{5} - 11231871461142475 \nu^{4} + 16707098673252820 \nu^{3} + 14675109344372636 \nu^{2} - 25352491816897518 \nu + 6802024064231936\)\()/ 278628502399750 \)
\(\beta_{10}\)\(=\)\((\)\(208422301596 \nu^{12} + 101272023955 \nu^{11} - 9810853937919 \nu^{10} - 3015289235129 \nu^{9} + 173349897433318 \nu^{8} + 10485608743617 \nu^{7} - 1435122721494978 \nu^{6} + 348943217873482 \nu^{5} + 5522030332256130 \nu^{4} - 3276596851987855 \nu^{3} - 7596810398586632 \nu^{2} + 7530986189108326 \nu - 1313465392184157\)\()/ 139314251199875 \)
\(\beta_{11}\)\(=\)\((\)\(889692540773 \nu^{12} - 1710596413190 \nu^{11} - 43690158347557 \nu^{10} + 79678278406623 \nu^{9} + 807810238699029 \nu^{8} - 1397243777337779 \nu^{7} - 6930554296860074 \nu^{6} + 11473935236033836 \nu^{5} + 26745397139049795 \nu^{4} - 44471075457654780 \nu^{3} - 31856351450553896 \nu^{2} + 67210663867287138 \nu - 20950363454455596\)\()/ 557257004799500 \)
\(\beta_{12}\)\(=\)\((\)\(224267911141 \nu^{12} + 63390661414 \nu^{11} - 11028950583521 \nu^{10} - 1496522255749 \nu^{9} + 205661421513157 \nu^{8} - 13254732421323 \nu^{7} - 1812887391077730 \nu^{6} + 558179494985656 \nu^{5} + 7475418034130291 \nu^{4} - 4406202874244528 \nu^{3} - 10980877487070868 \nu^{2} + 10312125439960266 \nu - 2219186123392628\)\()/ 111451400959900 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{6} - \beta_{5} + 8\)
\(\nu^{3}\)\(=\)\(-\beta_{12} - \beta_{11} + \beta_{10} + \beta_{8} + 2 \beta_{6} - \beta_{5} + 2 \beta_{4} - \beta_{3} + 11 \beta_{1} - 1\)
\(\nu^{4}\)\(=\)\(-5 \beta_{12} - \beta_{11} + 5 \beta_{10} - \beta_{9} + 2 \beta_{8} + 19 \beta_{6} - 17 \beta_{5} - 4 \beta_{4} + 2 \beta_{3} - 4 \beta_{2} + 2 \beta_{1} + 91\)
\(\nu^{5}\)\(=\)\(-30 \beta_{12} - 18 \beta_{11} + 29 \beta_{10} + 7 \beta_{9} + 20 \beta_{8} + 3 \beta_{7} + 46 \beta_{6} - 16 \beta_{5} + 43 \beta_{4} - 21 \beta_{3} - 7 \beta_{2} + 143 \beta_{1} - 7\)
\(\nu^{6}\)\(=\)\(-137 \beta_{12} - 28 \beta_{11} + 144 \beta_{10} - 42 \beta_{9} + 41 \beta_{8} - 12 \beta_{7} + 335 \beta_{6} - 280 \beta_{5} - 100 \beta_{4} + 40 \beta_{3} - 102 \beta_{2} + 49 \beta_{1} + 1184\)
\(\nu^{7}\)\(=\)\(-665 \beta_{12} - 281 \beta_{11} + 632 \beta_{10} + 213 \beta_{9} + 350 \beta_{8} + 97 \beta_{7} + 905 \beta_{6} - 247 \beta_{5} + 753 \beta_{4} - 381 \beta_{3} - 199 \beta_{2} + 2061 \beta_{1} + 6\)
\(\nu^{8}\)\(=\)\(-2931 \beta_{12} - 676 \beta_{11} + 3142 \beta_{10} - 1097 \beta_{9} + 716 \beta_{8} - 320 \beta_{7} + 5857 \beta_{6} - 4654 \beta_{5} - 1956 \beta_{4} + 668 \beta_{3} - 2055 \beta_{2} + 1017 \beta_{1} + 16711\)
\(\nu^{9}\)\(=\)\(-13266 \beta_{12} - 4390 \beta_{11} + 12482 \beta_{10} + 4644 \beta_{9} + 6123 \beta_{8} + 2314 \beta_{7} + 16957 \beta_{6} - 4186 \beta_{5} + 12395 \beta_{4} - 6687 \beta_{3} - 4266 \beta_{2} + 31753 \beta_{1} + 1836\)
\(\nu^{10}\)\(=\)\(-57319 \beta_{12} - 14661 \beta_{11} + 61925 \beta_{10} - 23592 \beta_{9} + 12118 \beta_{8} - 6102 \beta_{7} + 102280 \beta_{6} - 78127 \beta_{5} - 34966 \beta_{4} + 10790 \beta_{3} - 38346 \beta_{2} + 20530 \beta_{1} + 249558\)
\(\nu^{11}\)\(=\)\(-251329 \beta_{12} - 70391 \beta_{11} + 235453 \beta_{10} + 89347 \beta_{9} + 107880 \beta_{8} + 48175 \beta_{7} + 310864 \beta_{6} - 76539 \beta_{5} + 199760 \beta_{4} - 116067 \beta_{3} - 83231 \beta_{2} + 510061 \beta_{1} + 60252\)
\(\nu^{12}\)\(=\)\(-1073038 \beta_{12} - 294937 \beta_{11} + 1162932 \beta_{10} - 459487 \beta_{9} + 205019 \beta_{8} - 103543 \beta_{7} + 1786032 \beta_{6} - 1320651 \beta_{5} - 597828 \beta_{4} + 173012 \beta_{3} - 691732 \beta_{2} + 410169 \beta_{1} + 3880740\)

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
−4.02595
−3.60065
−3.05600
−2.71099
−2.04519
0.284022
0.649069
1.25667
1.68837
2.79706
3.18209
3.38131
4.20017
0 −1.00000 0 −4.02595 0 −0.910115 0 1.00000 0
1.2 0 −1.00000 0 −3.60065 0 4.85776 0 1.00000 0
1.3 0 −1.00000 0 −3.05600 0 −2.59989 0 1.00000 0
1.4 0 −1.00000 0 −2.71099 0 0.775439 0 1.00000 0
1.5 0 −1.00000 0 −2.04519 0 −1.81275 0 1.00000 0
1.6 0 −1.00000 0 0.284022 0 3.90646 0 1.00000 0
1.7 0 −1.00000 0 0.649069 0 −4.19646 0 1.00000 0
1.8 0 −1.00000 0 1.25667 0 4.55521 0 1.00000 0
1.9 0 −1.00000 0 1.68837 0 −0.722089 0 1.00000 0
1.10 0 −1.00000 0 2.79706 0 −4.56518 0 1.00000 0
1.11 0 −1.00000 0 3.18209 0 1.39408 0 1.00000 0
1.12 0 −1.00000 0 3.38131 0 −2.18014 0 1.00000 0
1.13 0 −1.00000 0 4.20017 0 0.497670 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.13
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 8016.2.a.bg 13
4.b odd 2 1 4008.2.a.m 13
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4008.2.a.m 13 4.b odd 2 1
8016.2.a.bg 13 1.a even 1 1 trivial

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(3\) \(1\)
\(167\) \(-1\)

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8016))\):

\(T_{5}^{13} - \cdots\)
\(T_{7}^{13} + \cdots\)
\(T_{11}^{13} + \cdots\)
\(T_{13}^{13} - \cdots\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( ( 1 + T )^{13} \)
$5$ \( 1 - 2 T + 16 T^{2} - 21 T^{3} + 156 T^{4} - 229 T^{5} + 1338 T^{6} - 1817 T^{7} + 8691 T^{8} - 12250 T^{9} + 53818 T^{10} - 78200 T^{11} + 288836 T^{12} - 394106 T^{13} + 1444180 T^{14} - 1955000 T^{15} + 6727250 T^{16} - 7656250 T^{17} + 27159375 T^{18} - 28390625 T^{19} + 104531250 T^{20} - 89453125 T^{21} + 304687500 T^{22} - 205078125 T^{23} + 781250000 T^{24} - 488281250 T^{25} + 1220703125 T^{26} \)
$7$ \( 1 + T + 33 T^{2} + 3 T^{3} + 503 T^{4} - 354 T^{5} + 5636 T^{6} - 5895 T^{7} + 58851 T^{8} - 56585 T^{9} + 539397 T^{10} - 532940 T^{11} + 4140879 T^{12} - 4388332 T^{13} + 28986153 T^{14} - 26114060 T^{15} + 185013171 T^{16} - 135860585 T^{17} + 989108757 T^{18} - 693540855 T^{19} + 4641488348 T^{20} - 2040739554 T^{21} + 20297864321 T^{22} + 847425747 T^{23} + 65251782519 T^{24} + 13841287201 T^{25} + 96889010407 T^{26} \)
$11$ \( 1 + 11 T + 114 T^{2} + 748 T^{3} + 4668 T^{4} + 22829 T^{5} + 109910 T^{6} + 452044 T^{7} + 1901898 T^{8} + 7177201 T^{9} + 28190517 T^{10} + 100465816 T^{11} + 367612018 T^{12} + 1210062606 T^{13} + 4043732198 T^{14} + 12156363736 T^{15} + 37521578127 T^{16} + 105081399841 T^{17} + 306302574798 T^{18} + 800823520684 T^{19} + 2141834964610 T^{20} + 4893598894349 T^{21} + 11006899821588 T^{22} + 19401193601548 T^{23} + 32525530449654 T^{24} + 34522712143931 T^{25} + 34522712143931 T^{26} \)
$13$ \( 1 - 12 T + 108 T^{2} - 768 T^{3} + 4901 T^{4} - 27045 T^{5} + 139081 T^{6} - 660978 T^{7} + 2991458 T^{8} - 12740044 T^{9} + 52566005 T^{10} - 207026958 T^{11} + 790583670 T^{12} - 2888715078 T^{13} + 10277587710 T^{14} - 34987555902 T^{15} + 115487512985 T^{16} - 363868396684 T^{17} + 1110707415194 T^{18} - 3190414559202 T^{19} + 8727126492877 T^{20} - 22061437349445 T^{21} + 51972651427073 T^{22} - 105875321740032 T^{23} + 193553322555996 T^{24} - 279577021469772 T^{25} + 302875106592253 T^{26} \)
$17$ \( 1 - 15 T + 167 T^{2} - 1346 T^{3} + 9671 T^{4} - 58701 T^{5} + 325131 T^{6} - 1572070 T^{7} + 7060135 T^{8} - 28216201 T^{9} + 107281433 T^{10} - 374119928 T^{11} + 1367940006 T^{12} - 5174549350 T^{13} + 23254980102 T^{14} - 108120659192 T^{15} + 527073680329 T^{16} - 2356645323721 T^{17} + 10024382100695 T^{18} - 37945948097830 T^{19} + 133413823091163 T^{20} - 409483937544141 T^{21} + 1146863353602487 T^{22} - 2713527790004354 T^{23} + 5723406683374711 T^{24} - 8739333558446415 T^{25} + 9904578032905937 T^{26} \)
$19$ \( 1 + 14 T + 222 T^{2} + 2068 T^{3} + 19253 T^{4} + 135689 T^{5} + 932679 T^{6} + 5301112 T^{7} + 29447836 T^{8} + 141620870 T^{9} + 680098931 T^{10} + 2942567796 T^{11} + 13246753430 T^{12} + 55845390246 T^{13} + 251688315170 T^{14} + 1062266974356 T^{15} + 4664798567729 T^{16} + 18456173399270 T^{17} + 72915757271764 T^{18} + 249395484319672 T^{19} + 833695399658781 T^{20} + 2304482685470249 T^{21} + 6212706245339087 T^{22} + 12679045021132468 T^{23} + 25860837475404618 T^{24} + 30986408866926254 T^{25} + 42052983462257059 T^{26} \)
$23$ \( 1 + 9 T + 90 T^{2} + 376 T^{3} + 2002 T^{4} + 2087 T^{5} + 13880 T^{6} - 88608 T^{7} + 177222 T^{8} - 623197 T^{9} + 21435133 T^{10} + 90849960 T^{11} + 949979326 T^{12} + 3321483482 T^{13} + 21849524498 T^{14} + 48059628840 T^{15} + 260801263211 T^{16} - 174396071677 T^{17} + 1140661579146 T^{18} - 13117164052512 T^{19} + 47258977204360 T^{20} + 163435026281447 T^{21} + 3605907628248926 T^{22} + 15576368216332024 T^{23} + 85752878212253430 T^{24} + 197231619888182889 T^{25} + 504036361936467383 T^{26} \)
$29$ \( 1 + 3 T + 226 T^{2} + 744 T^{3} + 26600 T^{4} + 88729 T^{5} + 2101656 T^{6} + 6843616 T^{7} + 122885736 T^{8} + 380232137 T^{9} + 5575564329 T^{10} + 16029028280 T^{11} + 201175820032 T^{12} + 525255655926 T^{13} + 5834098780928 T^{14} + 13480412783480 T^{15} + 135982438419981 T^{16} + 268930966089497 T^{17} + 2520527641070664 T^{18} + 4070742396768736 T^{19} + 36253306044067704 T^{20} + 44386363975616569 T^{21} + 385890082958115400 T^{22} + 313006181575349544 T^{23} + 2757315207049517354 T^{24} + 1061444349616407123 T^{25} + 10260628712958602189 T^{26} \)
$31$ \( 1 - 17 T + 354 T^{2} - 4502 T^{3} + 57000 T^{4} - 582684 T^{5} + 5671184 T^{6} - 48655573 T^{7} + 394190897 T^{8} - 2915699897 T^{9} + 20375864258 T^{10} - 132029977589 T^{11} + 809685079210 T^{12} - 4632244947956 T^{13} + 25100237455510 T^{14} - 126880808463029 T^{15} + 607017372110078 T^{16} - 2692710084577337 T^{17} + 11285350713038447 T^{18} - 43182000138664213 T^{19} + 156029096944477424 T^{20} - 496965961260271644 T^{21} + 1507058463158247000 T^{22} - 3689966547987566102 T^{23} + 8994600821327310174 T^{24} - 13390267324405345937 T^{25} + 24417546297445042591 T^{26} \)
$37$ \( 1 - 16 T + 384 T^{2} - 4711 T^{3} + 66546 T^{4} - 675545 T^{5} + 7204628 T^{6} - 63112033 T^{7} + 556641667 T^{8} - 4304114984 T^{9} + 32771180488 T^{10} - 226104266132 T^{11} + 1519031839710 T^{12} - 9379771706774 T^{13} + 56204178069270 T^{14} - 309536740334708 T^{15} + 1659958605258664 T^{16} - 8066604442528424 T^{17} + 38599735820856319 T^{18} - 161928209793779497 T^{19} + 683948860084971524 T^{20} - 2372837932699061945 T^{21} + 8648433936403194042 T^{22} - 22653240978460486639 T^{23} + 68320366763312798592 T^{24} - \)\(10\!\cdots\!96\)\( T^{25} + \)\(24\!\cdots\!97\)\( T^{26} \)
$41$ \( 1 - 12 T + 323 T^{2} - 2549 T^{3} + 41774 T^{4} - 231699 T^{5} + 3266838 T^{6} - 13225111 T^{7} + 194237287 T^{8} - 576726116 T^{9} + 9649549285 T^{10} - 20359747964 T^{11} + 418471414556 T^{12} - 720964412298 T^{13} + 17157327996796 T^{14} - 34224736327484 T^{15} + 665056586271485 T^{16} - 1629690166274276 T^{17} + 22503594164366687 T^{18} - 62820655848795751 T^{19} + 636230662576858278 T^{20} - 1850099190662106579 T^{21} + 13676052927373326814 T^{22} - 34214358581578470149 T^{23} + \)\(17\!\cdots\!43\)\( T^{24} - \)\(27\!\cdots\!72\)\( T^{25} + \)\(92\!\cdots\!21\)\( T^{26} \)
$43$ \( 1 + 20 T + 447 T^{2} + 6085 T^{3} + 81758 T^{4} + 883477 T^{5} + 9136296 T^{6} + 83940113 T^{7} + 732178595 T^{8} + 5916935844 T^{9} + 45657111985 T^{10} + 332352060170 T^{11} + 2333446642614 T^{12} + 15546056453590 T^{13} + 100338205632402 T^{14} + 614518959254330 T^{15} + 3630060002591395 T^{16} + 20228826180403044 T^{17} + 107636435248877585 T^{18} + 530615928647084537 T^{19} + 2483415289382439672 T^{20} + 10326256116654098677 T^{21} + 41090966766732409994 T^{22} + \)\(13\!\cdots\!65\)\( T^{23} + \)\(41\!\cdots\!29\)\( T^{24} + \)\(79\!\cdots\!20\)\( T^{25} + \)\(17\!\cdots\!43\)\( T^{26} \)
$47$ \( 1 - 6 T + 260 T^{2} - 1150 T^{3} + 34866 T^{4} - 115419 T^{5} + 3258908 T^{6} - 8213299 T^{7} + 240130821 T^{8} - 465844244 T^{9} + 14793482024 T^{10} - 23168213451 T^{11} + 789796106976 T^{12} - 1097130074574 T^{13} + 37120417027872 T^{14} - 51178583513259 T^{15} + 1535903684177752 T^{16} - 2273171306406164 T^{17} + 55072804823160747 T^{18} - 88532918482460371 T^{19} + 1651038140261834404 T^{20} - 2748274895213792859 T^{21} + 39019603075201074222 T^{22} - 60489002071204556350 T^{23} + \)\(64\!\cdots\!80\)\( T^{24} - \)\(69\!\cdots\!46\)\( T^{25} + \)\(54\!\cdots\!27\)\( T^{26} \)
$53$ \( 1 + 12 T + 546 T^{2} + 5632 T^{3} + 139434 T^{4} + 1252903 T^{5} + 22234644 T^{6} + 175977017 T^{7} + 2496799673 T^{8} + 17567770072 T^{9} + 211114344916 T^{10} + 1329580708237 T^{11} + 13992947725186 T^{12} + 79044335106166 T^{13} + 741626229434858 T^{14} + 3734792209437733 T^{15} + 31430070328059332 T^{16} + 138618155965484632 T^{17} + 1044150370172473789 T^{18} + 3900418155192172193 T^{19} + 26119283997109913028 T^{20} + 78005352895465430983 T^{21} + \)\(46\!\cdots\!22\)\( T^{22} + \)\(98\!\cdots\!68\)\( T^{23} + \)\(50\!\cdots\!62\)\( T^{24} + \)\(58\!\cdots\!92\)\( T^{25} + \)\(26\!\cdots\!73\)\( T^{26} \)
$59$ \( 1 + 14 T + 580 T^{2} + 7718 T^{3} + 167442 T^{4} + 2023305 T^{5} + 31301076 T^{6} + 336139055 T^{7} + 4167954839 T^{8} + 39566719048 T^{9} + 414256794384 T^{10} + 3477881140059 T^{11} + 31532600144230 T^{12} + 233818570878866 T^{13} + 1860423408509570 T^{14} + 12106504248545379 T^{15} + 85079646173791536 T^{16} + 479444218290192328 T^{17} + 2979772191535732861 T^{18} + 14178524717481449255 T^{19} + 77897469263832365244 T^{20} + \)\(29\!\cdots\!05\)\( T^{21} + \)\(14\!\cdots\!38\)\( T^{22} + \)\(39\!\cdots\!18\)\( T^{23} + \)\(17\!\cdots\!20\)\( T^{24} + \)\(24\!\cdots\!34\)\( T^{25} + \)\(10\!\cdots\!79\)\( T^{26} \)
$61$ \( 1 - 24 T + 706 T^{2} - 12276 T^{3} + 214433 T^{4} - 2938037 T^{5} + 38645919 T^{6} - 439777380 T^{7} + 4761251434 T^{8} - 46772112756 T^{9} + 438402378815 T^{10} - 3835108282184 T^{11} + 32216109760932 T^{12} - 256351733098142 T^{13} + 1965182695416852 T^{14} - 14270437918006664 T^{15} + 99509010345807515 T^{16} - 647599236453647796 T^{17} + 4021335349287345634 T^{18} - 22657495253099754180 T^{19} + \)\(12\!\cdots\!99\)\( T^{20} - \)\(56\!\cdots\!97\)\( T^{21} + \)\(25\!\cdots\!53\)\( T^{22} - \)\(87\!\cdots\!76\)\( T^{23} + \)\(30\!\cdots\!66\)\( T^{24} - \)\(63\!\cdots\!04\)\( T^{25} + \)\(16\!\cdots\!81\)\( T^{26} \)
$67$ \( 1 + 3 T + 355 T^{2} + 1048 T^{3} + 65889 T^{4} + 183700 T^{5} + 8652330 T^{6} + 21924177 T^{7} + 907926505 T^{8} + 2115795481 T^{9} + 80535141351 T^{10} + 178166589169 T^{11} + 6181969561833 T^{12} + 12957587379252 T^{13} + 414191960642811 T^{14} + 799789818779641 T^{15} + 24221990718150813 T^{16} + 42635650748884201 T^{17} + 1225814369711261035 T^{18} + 1983225581806799913 T^{19} + 52439276844084352590 T^{20} + 74594632367154951700 T^{21} + \)\(17\!\cdots\!83\)\( T^{22} + \)\(19\!\cdots\!52\)\( T^{23} + \)\(43\!\cdots\!65\)\( T^{24} + \)\(24\!\cdots\!83\)\( T^{25} + \)\(54\!\cdots\!87\)\( T^{26} \)
$71$ \( 1 + 17 T + 649 T^{2} + 9605 T^{3} + 199299 T^{4} + 2563593 T^{5} + 38342131 T^{6} + 432635821 T^{7} + 5216366827 T^{8} + 52468153659 T^{9} + 542623256987 T^{10} + 4955284901374 T^{11} + 45797136160416 T^{12} + 384369572541270 T^{13} + 3251596667389536 T^{14} + 24979591187826334 T^{15} + 194210832531474157 T^{16} + 1333303983441490779 T^{17} + 9411522134856139277 T^{18} + 55420771504494934141 T^{19} + \)\(34\!\cdots\!21\)\( T^{20} + \)\(16\!\cdots\!73\)\( T^{21} + \)\(91\!\cdots\!69\)\( T^{22} + \)\(31\!\cdots\!05\)\( T^{23} + \)\(14\!\cdots\!79\)\( T^{24} + \)\(27\!\cdots\!97\)\( T^{25} + \)\(11\!\cdots\!11\)\( T^{26} \)
$73$ \( 1 - 34 T + 1160 T^{2} - 25848 T^{3} + 540301 T^{4} - 9197595 T^{5} + 146223451 T^{6} - 2032969044 T^{7} + 26498553004 T^{8} - 311457231682 T^{9} + 3445866716883 T^{10} - 34878150258260 T^{11} + 333165209029664 T^{12} - 2928902375880834 T^{13} + 24321060259165472 T^{14} - 185865662726267540 T^{15} + 1340500732601674011 T^{16} - 8844837526498271362 T^{17} + 54933397488197215372 T^{18} - \)\(30\!\cdots\!16\)\( T^{19} + \)\(16\!\cdots\!47\)\( T^{20} - \)\(74\!\cdots\!95\)\( T^{21} + \)\(31\!\cdots\!13\)\( T^{22} - \)\(11\!\cdots\!52\)\( T^{23} + \)\(36\!\cdots\!20\)\( T^{24} - \)\(77\!\cdots\!14\)\( T^{25} + \)\(16\!\cdots\!33\)\( T^{26} \)
$79$ \( 1 + 10 T + 727 T^{2} + 6839 T^{3} + 250318 T^{4} + 2246253 T^{5} + 54783194 T^{6} + 472895673 T^{7} + 8629118749 T^{8} + 71522751906 T^{9} + 1046955450079 T^{10} + 8206987443824 T^{11} + 101626477677844 T^{12} + 732512143274958 T^{13} + 8028491736549676 T^{14} + 51219808636905584 T^{15} + 516189868151500081 T^{16} + 2785816980081604386 T^{17} + 26552285064341324851 T^{18} + \)\(11\!\cdots\!33\)\( T^{19} + \)\(10\!\cdots\!46\)\( T^{20} + \)\(34\!\cdots\!33\)\( T^{21} + \)\(30\!\cdots\!42\)\( T^{22} + \)\(64\!\cdots\!39\)\( T^{23} + \)\(54\!\cdots\!33\)\( T^{24} + \)\(59\!\cdots\!10\)\( T^{25} + \)\(46\!\cdots\!39\)\( T^{26} \)
$83$ \( 1 + 44 T + 1370 T^{2} + 31288 T^{3} + 593606 T^{4} + 9457979 T^{5} + 131609258 T^{6} + 1600854901 T^{7} + 17366694499 T^{8} + 167988308462 T^{9} + 1482913205406 T^{10} + 12175793267931 T^{11} + 99368916657416 T^{12} + 857290020851094 T^{13} + 8247620082565528 T^{14} + 83879039822776659 T^{15} + 847910492979480522 T^{16} + 7972443067236612302 T^{17} + 68408115466125522857 T^{18} + \)\(52\!\cdots\!69\)\( T^{19} + \)\(35\!\cdots\!66\)\( T^{20} + \)\(21\!\cdots\!39\)\( T^{21} + \)\(11\!\cdots\!18\)\( T^{22} + \)\(48\!\cdots\!12\)\( T^{23} + \)\(17\!\cdots\!90\)\( T^{24} + \)\(47\!\cdots\!84\)\( T^{25} + \)\(88\!\cdots\!63\)\( T^{26} \)
$89$ \( 1 - 25 T + 1039 T^{2} - 18532 T^{3} + 444053 T^{4} - 6196410 T^{5} + 110229192 T^{6} - 1267330941 T^{7} + 18562991951 T^{8} - 182639744007 T^{9} + 2348123360619 T^{10} - 20519519261863 T^{11} + 242328807802313 T^{12} - 1950518612453148 T^{13} + 21567263894405857 T^{14} - 162535112073216823 T^{15} + 1655354177412215811 T^{16} - 11459226834665499687 T^{17} + \)\(10\!\cdots\!99\)\( T^{18} - \)\(62\!\cdots\!01\)\( T^{19} + \)\(48\!\cdots\!68\)\( T^{20} - \)\(24\!\cdots\!10\)\( T^{21} + \)\(15\!\cdots\!77\)\( T^{22} - \)\(57\!\cdots\!32\)\( T^{23} + \)\(28\!\cdots\!71\)\( T^{24} - \)\(61\!\cdots\!25\)\( T^{25} + \)\(21\!\cdots\!69\)\( T^{26} \)
$97$ \( 1 - 38 T + 1091 T^{2} - 23273 T^{3} + 444655 T^{4} - 7315883 T^{5} + 111741834 T^{6} - 1547021699 T^{7} + 20265183049 T^{8} - 246248874784 T^{9} + 2866188972669 T^{10} - 31400133858964 T^{11} + 331888842256893 T^{12} - 3323054133939038 T^{13} + 32193217698918621 T^{14} - 295443859478992276 T^{15} + 2615893288252734237 T^{16} - 21800235831686550304 T^{17} + \)\(17\!\cdots\!93\)\( T^{18} - \)\(12\!\cdots\!71\)\( T^{19} + \)\(90\!\cdots\!42\)\( T^{20} - \)\(57\!\cdots\!63\)\( T^{21} + \)\(33\!\cdots\!35\)\( T^{22} - \)\(17\!\cdots\!77\)\( T^{23} + \)\(78\!\cdots\!23\)\( T^{24} - \)\(26\!\cdots\!58\)\( T^{25} + \)\(67\!\cdots\!77\)\( T^{26} \)
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