Properties

Label 6016.2.a.o
Level 6016
Weight 2
Character orbit 6016.a
Self dual yes
Analytic conductor 48.038
Analytic rank 0
Dimension 13
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 6016 = 2^{7} \cdot 47 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6016.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(48.0380018560\)
Analytic rank: \(0\)
Dimension: \(13\)
Coefficient field: \(\mathbb{Q}[x]/(x^{13} - \cdots)\)
Defining polynomial: \(x^{13} - 4 x^{12} - 22 x^{11} + 96 x^{10} + 163 x^{9} - 840 x^{8} - 438 x^{7} + 3358 x^{6} + 5 x^{5} - 6230 x^{4} + 1488 x^{3} + 4720 x^{2} - 1440 x - 832\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
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 \(1,\beta_1,\ldots,\beta_{12}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{1} q^{3} -\beta_{8} q^{5} -\beta_{9} q^{7} + ( 2 + \beta_{2} ) q^{9} +O(q^{10})\) \( q + \beta_{1} q^{3} -\beta_{8} q^{5} -\beta_{9} q^{7} + ( 2 + \beta_{2} ) q^{9} + ( 1 - \beta_{12} ) q^{11} + ( -1 + \beta_{1} + \beta_{2} + \beta_{3} + \beta_{6} + \beta_{9} + \beta_{10} ) q^{13} + ( 2 - \beta_{1} + \beta_{2} + \beta_{5} - \beta_{8} + \beta_{10} ) q^{15} + ( 1 - \beta_{3} - \beta_{9} ) q^{17} + ( \beta_{1} + \beta_{4} - \beta_{5} + \beta_{6} ) q^{19} + ( -1 - 2 \beta_{2} - \beta_{3} - \beta_{5} - \beta_{7} - 2 \beta_{9} - \beta_{10} + \beta_{11} - \beta_{12} ) q^{21} + ( 1 + \beta_{6} + \beta_{11} ) q^{23} + ( 1 + \beta_{1} + \beta_{5} + \beta_{7} + \beta_{8} + \beta_{10} ) q^{25} + ( 2 + \beta_{2} - \beta_{3} - \beta_{4} - \beta_{8} + \beta_{9} ) q^{27} + ( -1 + \beta_{1} + \beta_{2} + \beta_{3} + \beta_{7} + \beta_{10} - \beta_{11} ) q^{29} + ( 1 - \beta_{1} - \beta_{4} - \beta_{5} - \beta_{9} - \beta_{10} ) q^{31} + ( 2 + \beta_{5} - \beta_{6} - \beta_{9} - \beta_{10} - \beta_{12} ) q^{33} + ( 2 \beta_{1} + \beta_{3} + \beta_{7} + \beta_{9} + \beta_{10} + \beta_{12} ) q^{35} + ( -1 + \beta_{1} + \beta_{5} + \beta_{7} - \beta_{8} + \beta_{10} - \beta_{11} ) q^{37} + ( 2 + 2 \beta_{2} + \beta_{3} - \beta_{6} + \beta_{7} - 2 \beta_{8} + 2 \beta_{9} + 2 \beta_{10} - 2 \beta_{11} + 2 \beta_{12} ) q^{39} + ( -\beta_{2} + \beta_{4} - \beta_{5} + \beta_{6} - \beta_{7} + \beta_{8} - \beta_{9} - \beta_{10} - \beta_{12} ) q^{41} + ( 2 - \beta_{1} + \beta_{2} + \beta_{3} + \beta_{5} - \beta_{6} - \beta_{7} - \beta_{8} + \beta_{9} - \beta_{11} ) q^{43} + ( -1 - \beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} - \beta_{7} - \beta_{10} + 2 \beta_{11} - \beta_{12} ) q^{45} - q^{47} + ( \beta_{1} - 2 \beta_{5} + \beta_{6} + \beta_{11} ) q^{49} + ( 2 - \beta_{2} - \beta_{3} - \beta_{5} - 2 \beta_{7} - 2 \beta_{9} - 3 \beta_{10} + 2 \beta_{11} - \beta_{12} ) q^{51} + ( -2 - \beta_{2} - \beta_{7} - \beta_{9} - 2 \beta_{10} ) q^{53} + ( -\beta_{1} - \beta_{3} + 2 \beta_{4} + \beta_{5} - 3 \beta_{7} - \beta_{8} + \beta_{9} - \beta_{10} + \beta_{11} ) q^{55} + ( 1 + 2 \beta_{1} + 2 \beta_{2} + \beta_{3} - 2 \beta_{4} - \beta_{6} + \beta_{9} + 2 \beta_{10} - 2 \beta_{11} + 2 \beta_{12} ) q^{57} + ( 2 + \beta_{2} + \beta_{3} + \beta_{6} + \beta_{7} - \beta_{8} + 2 \beta_{9} + 2 \beta_{10} + \beta_{12} ) q^{59} + ( -1 + \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} - \beta_{6} + \beta_{7} - \beta_{11} + \beta_{12} ) q^{61} + ( 1 - 3 \beta_{1} - 3 \beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} - 2 \beta_{7} + 2 \beta_{8} - 4 \beta_{9} - 3 \beta_{10} + 2 \beta_{11} - 2 \beta_{12} ) q^{63} + ( -1 + \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} - 2 \beta_{6} - \beta_{7} + \beta_{8} - \beta_{9} ) q^{65} + ( 1 + \beta_{1} + \beta_{2} + \beta_{3} - 2 \beta_{4} + \beta_{5} - \beta_{6} + \beta_{7} - \beta_{8} + \beta_{10} - 2 \beta_{11} + \beta_{12} ) q^{67} + ( 1 - \beta_{3} + \beta_{5} - 2 \beta_{6} - 2 \beta_{8} + \beta_{10} + \beta_{12} ) q^{69} + ( 4 + \beta_{2} - \beta_{4} + 2 \beta_{7} - \beta_{8} + \beta_{9} - \beta_{11} ) q^{71} + ( -1 - 3 \beta_{2} - \beta_{3} + \beta_{4} - 2 \beta_{7} + 3 \beta_{8} - 2 \beta_{9} - 2 \beta_{10} + \beta_{11} - 2 \beta_{12} ) q^{73} + ( 3 - 2 \beta_{2} + \beta_{4} - \beta_{6} - 2 \beta_{10} + 2 \beta_{11} - \beta_{12} ) q^{75} + ( -4 + \beta_{1} + \beta_{2} + \beta_{5} - \beta_{7} + \beta_{8} - \beta_{11} + \beta_{12} ) q^{77} + ( 1 + \beta_{2} + \beta_{3} - \beta_{5} + \beta_{6} + 3 \beta_{7} - 2 \beta_{8} + 2 \beta_{9} + \beta_{10} - 2 \beta_{11} + \beta_{12} ) q^{79} + ( 3 + 2 \beta_{2} + \beta_{5} - 2 \beta_{8} + 2 \beta_{9} - \beta_{10} + \beta_{12} ) q^{81} + ( 1 + 2 \beta_{1} + 2 \beta_{2} + \beta_{5} + \beta_{6} - \beta_{8} + \beta_{9} + \beta_{10} - 2 \beta_{11} ) q^{83} + ( \beta_{1} - \beta_{2} - \beta_{6} + \beta_{7} - \beta_{8} + \beta_{9} - \beta_{10} + \beta_{11} ) q^{85} + ( \beta_{1} + \beta_{3} - \beta_{4} - \beta_{5} + \beta_{7} + \beta_{9} + \beta_{10} - \beta_{11} ) q^{87} + ( 2 + 2 \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} + \beta_{6} + \beta_{7} + \beta_{8} + \beta_{9} - \beta_{10} + \beta_{11} + \beta_{12} ) q^{89} + ( -1 - 2 \beta_{2} - \beta_{3} + 2 \beta_{4} + \beta_{6} + \beta_{7} - \beta_{9} + \beta_{10} + \beta_{11} - \beta_{12} ) q^{91} + ( -5 + 2 \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} - 2 \beta_{5} + \beta_{8} - 2 \beta_{9} - \beta_{11} ) q^{93} + ( 1 - \beta_{1} + \beta_{3} - \beta_{6} + \beta_{7} + 2 \beta_{10} - \beta_{11} ) q^{95} + ( -1 + 2 \beta_{1} + \beta_{2} + \beta_{3} - 2 \beta_{4} - \beta_{5} + \beta_{6} + \beta_{7} - \beta_{8} + \beta_{9} + \beta_{10} - \beta_{11} ) q^{97} + ( 3 - 2 \beta_{2} - 3 \beta_{3} + 2 \beta_{5} - \beta_{6} - \beta_{7} - 2 \beta_{9} - 2 \beta_{10} + 2 \beta_{11} - \beta_{12} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 13q + 4q^{3} - 6q^{5} - 2q^{7} + 21q^{9} + O(q^{10}) \) \( 13q + 4q^{3} - 6q^{5} - 2q^{7} + 21q^{9} + 10q^{11} - 4q^{13} + 14q^{15} + 10q^{17} + 8q^{19} - 10q^{21} + 18q^{23} + 23q^{25} + 16q^{27} - 14q^{29} + 4q^{31} + 14q^{33} + 14q^{35} - 16q^{37} + 12q^{39} + 10q^{41} + 12q^{43} - 10q^{45} - 13q^{47} + 9q^{49} + 22q^{51} - 26q^{53} - 2q^{55} + 20q^{57} + 30q^{59} - 18q^{61} + 12q^{63} - 4q^{65} + 4q^{67} - 2q^{69} + 36q^{71} + 10q^{73} + 38q^{75} - 42q^{77} + 21q^{81} + 12q^{83} - 4q^{85} + 6q^{87} + 50q^{89} - 4q^{91} - 52q^{93} + 8q^{95} - 10q^{97} + 34q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{13} - 4 x^{12} - 22 x^{11} + 96 x^{10} + 163 x^{9} - 840 x^{8} - 438 x^{7} + 3358 x^{6} + 5 x^{5} - 6230 x^{4} + 1488 x^{3} + 4720 x^{2} - 1440 x - 832\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 5 \)
\(\beta_{3}\)\(=\)\((\)\(195435 \nu^{12} - 562332 \nu^{11} - 4303150 \nu^{10} + 11599664 \nu^{9} + 31578537 \nu^{8} - 76866624 \nu^{7} - 79869654 \nu^{6} + 168487210 \nu^{5} - 20578969 \nu^{4} + 4239486 \nu^{3} + 231304564 \nu^{2} - 180048216 \nu - 78029720\)\()/16452824\)
\(\beta_{4}\)\(=\)\((\)\(-623713 \nu^{12} + 2092838 \nu^{11} + 13458666 \nu^{10} - 45195340 \nu^{9} - 97777627 \nu^{8} + 330991582 \nu^{7} + 270408522 \nu^{6} - 962998794 \nu^{5} - 174049953 \nu^{4} + 918870696 \nu^{3} - 126028448 \nu^{2} + 13662280 \nu - 58768544\)\()/32905648\)
\(\beta_{5}\)\(=\)\((\)\(-1685943 \nu^{12} + 3467516 \nu^{11} + 45355618 \nu^{10} - 79576928 \nu^{9} - 458324917 \nu^{8} + 647371992 \nu^{7} + 2158227746 \nu^{6} - 2309620130 \nu^{5} - 4697624803 \nu^{4} + 3627295322 \nu^{3} + 4041025704 \nu^{2} - 2024987008 \nu - 787062208\)\()/32905648\)
\(\beta_{6}\)\(=\)\((\)\(3310971 \nu^{12} - 7613086 \nu^{11} - 84660434 \nu^{10} + 168699740 \nu^{9} + 808510705 \nu^{8} - 1304694094 \nu^{7} - 3604828890 \nu^{6} + 4342745078 \nu^{5} + 7523189915 \nu^{4} - 6285692652 \nu^{3} - 6526418140 \nu^{2} + 3317853968 \nu + 1617289984\)\()/32905648\)
\(\beta_{7}\)\(=\)\((\)\(-5277677 \nu^{12} + 12120046 \nu^{11} + 135579006 \nu^{10} - 271736444 \nu^{9} - 1296809943 \nu^{8} + 2135000694 \nu^{7} + 5748539542 \nu^{6} - 7237641474 \nu^{5} - 11777706325 \nu^{4} + 10612170176 \nu^{3} + 9846735948 \nu^{2} - 5579893712 \nu - 2287713712\)\()/32905648\)
\(\beta_{8}\)\(=\)\((\)\(-2662843 \nu^{12} + 5722128 \nu^{11} + 69595980 \nu^{10} - 128084672 \nu^{9} - 680556885 \nu^{8} + 1003716556 \nu^{7} + 3098280608 \nu^{6} - 3390849962 \nu^{5} - 6541061235 \nu^{4} + 4970241098 \nu^{3} + 5597835874 \nu^{2} - 2653342116 \nu - 1239592192\)\()/16452824\)
\(\beta_{9}\)\(=\)\((\)\(-5558529 \nu^{12} + 12412430 \nu^{11} + 144044326 \nu^{10} - 278165356 \nu^{9} - 1395734323 \nu^{8} + 2184691446 \nu^{7} + 6307230430 \nu^{6} - 7407724298 \nu^{5} - 13297330361 \nu^{4} + 10900737512 \nu^{3} + 11499346780 \nu^{2} - 5850552272 \nu - 2595295424\)\()/32905648\)
\(\beta_{10}\)\(=\)\((\)\(6218745 \nu^{12} - 14050128 \nu^{11} - 161260170 \nu^{10} + 316434536 \nu^{9} + 1563354275 \nu^{8} - 2503849628 \nu^{7} - 7063620194 \nu^{6} + 8583414246 \nu^{5} + 14854043917 \nu^{4} - 12807105642 \nu^{3} - 12708813292 \nu^{2} + 6899380648 \nu + 2837565520\)\()/32905648\)
\(\beta_{11}\)\(=\)\((\)\(6720997 \nu^{12} - 15973050 \nu^{11} - 172617142 \nu^{10} + 361687364 \nu^{9} + 1655301055 \nu^{8} - 2884160314 \nu^{7} - 7403133806 \nu^{6} + 9972299306 \nu^{5} + 15486782437 \nu^{4} - 14883035676 \nu^{3} - 13398793452 \nu^{2} + 7844979872 \nu + 3128024784\)\()/32905648\)
\(\beta_{12}\)\(=\)\((\)\(4185187 \nu^{12} - 9726996 \nu^{11} - 108160260 \nu^{10} + 220001744 \nu^{9} + 1045460149 \nu^{8} - 1752869144 \nu^{7} - 4721593184 \nu^{6} + 6072541562 \nu^{5} + 10007495075 \nu^{4} - 9177455798 \nu^{3} - 8859575594 \nu^{2} + 5006051868 \nu + 2191670088\)\()/16452824\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + 5\)
\(\nu^{3}\)\(=\)\(\beta_{9} - \beta_{8} - \beta_{4} - \beta_{3} + \beta_{2} + 6 \beta_{1} + 2\)
\(\nu^{4}\)\(=\)\(\beta_{12} - \beta_{10} + 2 \beta_{9} - 2 \beta_{8} + \beta_{5} + 11 \beta_{2} + 39\)
\(\nu^{5}\)\(=\)\(2 \beta_{12} - 2 \beta_{11} + 5 \beta_{10} + 17 \beta_{9} - 14 \beta_{8} + \beta_{7} - \beta_{6} + 5 \beta_{5} - 11 \beta_{4} - 11 \beta_{3} + 17 \beta_{2} + 47 \beta_{1} + 30\)
\(\nu^{6}\)\(=\)\(13 \beta_{12} + 3 \beta_{11} - 11 \beta_{10} + 38 \beta_{9} - 33 \beta_{8} + \beta_{7} + 3 \beta_{6} + 17 \beta_{5} + 3 \beta_{4} + 2 \beta_{3} + 113 \beta_{2} + 4 \beta_{1} + 351\)
\(\nu^{7}\)\(=\)\(37 \beta_{12} - 34 \beta_{11} + 89 \beta_{10} + 217 \beta_{9} - 169 \beta_{8} + 24 \beta_{7} - 22 \beta_{6} + 93 \beta_{5} - 111 \beta_{4} - 106 \beta_{3} + 218 \beta_{2} + 412 \beta_{1} + 374\)
\(\nu^{8}\)\(=\)\(139 \beta_{12} + 59 \beta_{11} - 92 \beta_{10} + 513 \beta_{9} - 424 \beta_{8} + 30 \beta_{7} + 53 \beta_{6} + 232 \beta_{5} + 51 \beta_{4} + 29 \beta_{3} + 1157 \beta_{2} + 79 \beta_{1} + 3368\)
\(\nu^{9}\)\(=\)\(473 \beta_{12} - 396 \beta_{11} + 1134 \beta_{10} + 2495 \beta_{9} - 1925 \beta_{8} + 346 \beta_{7} - 321 \beta_{6} + 1262 \beta_{5} - 1098 \beta_{4} - 1027 \beta_{3} + 2517 \beta_{2} + 3803 \beta_{1} + 4367\)
\(\nu^{10}\)\(=\)\(1385 \beta_{12} + 853 \beta_{11} - 705 \beta_{10} + 6094 \beta_{9} - 4987 \beta_{8} + 477 \beta_{7} + 641 \beta_{6} + 2927 \beta_{5} + 631 \beta_{4} + 246 \beta_{3} + 11866 \beta_{2} + 1104 \beta_{1} + 33412\)
\(\nu^{11}\)\(=\)\(5193 \beta_{12} - 3906 \beta_{11} + 12697 \beta_{10} + 27252 \beta_{9} - 21274 \beta_{8} + 4102 \beta_{7} - 4006 \beta_{6} + 15289 \beta_{5} - 10752 \beta_{4} - 10257 \beta_{3} + 27677 \beta_{2} + 36058 \beta_{1} + 49510\)
\(\nu^{12}\)\(=\)\(13150 \beta_{12} + 11091 \beta_{11} - 5337 \beta_{10} + 68027 \beta_{9} - 56328 \beta_{8} + 5908 \beta_{7} + 6511 \beta_{6} + 35369 \beta_{5} + 6959 \beta_{4} + 889 \beta_{3} + 121784 \beta_{2} + 13525 \beta_{1} + 337959\)

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
−3.10994
−2.44796
−2.08298
−1.49302
−1.22559
−0.319482
0.904794
1.18016
1.73608
1.76446
2.61905
3.18041
3.29402
0 −3.10994 0 −0.407310 0 3.05265 0 6.67175 0
1.2 0 −2.44796 0 −2.61725 0 0.969862 0 2.99250 0
1.3 0 −2.08298 0 −4.13384 0 1.67766 0 1.33879 0
1.4 0 −1.49302 0 1.02737 0 −3.87698 0 −0.770878 0
1.5 0 −1.22559 0 −0.177756 0 −3.50603 0 −1.49794 0
1.6 0 −0.319482 0 2.22340 0 0.484368 0 −2.89793 0
1.7 0 0.904794 0 −3.10347 0 −1.70842 0 −2.18135 0
1.8 0 1.18016 0 2.06312 0 0.279748 0 −1.60723 0
1.9 0 1.73608 0 3.07388 0 4.60716 0 0.0139658 0
1.10 0 1.76446 0 −4.26056 0 −2.37797 0 0.113336 0
1.11 0 2.61905 0 −1.63183 0 3.46539 0 3.85941 0
1.12 0 3.18041 0 −1.45922 0 −4.05725 0 7.11498 0
1.13 0 3.29402 0 3.40346 0 −1.01019 0 7.85059 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 6016.2.a.o yes 13
4.b odd 2 1 6016.2.a.m 13
8.b even 2 1 6016.2.a.n yes 13
8.d odd 2 1 6016.2.a.p yes 13
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6016.2.a.m 13 4.b odd 2 1
6016.2.a.n yes 13 8.b even 2 1
6016.2.a.o yes 13 1.a even 1 1 trivial
6016.2.a.p yes 13 8.d odd 2 1

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(47\) \(1\)

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6016))\):

\(T_{3}^{13} - \cdots\)
\(T_{5}^{13} + \cdots\)
\(T_{13}^{13} + \cdots\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( 1 - 4 T + 17 T^{2} - 48 T^{3} + 139 T^{4} - 336 T^{5} + 795 T^{6} - 1682 T^{7} + 3524 T^{8} - 6806 T^{9} + 13146 T^{10} - 23654 T^{11} + 43182 T^{12} - 74140 T^{13} + 129546 T^{14} - 212886 T^{15} + 354942 T^{16} - 551286 T^{17} + 856332 T^{18} - 1226178 T^{19} + 1738665 T^{20} - 2204496 T^{21} + 2735937 T^{22} - 2834352 T^{23} + 3011499 T^{24} - 2125764 T^{25} + 1594323 T^{26} \)
$5$ \( 1 + 6 T + 39 T^{2} + 170 T^{3} + 718 T^{4} + 2584 T^{5} + 8834 T^{6} + 27538 T^{7} + 81729 T^{8} + 226546 T^{9} + 602183 T^{10} + 1506988 T^{11} + 3635352 T^{12} + 8270864 T^{13} + 18176760 T^{14} + 37674700 T^{15} + 75272875 T^{16} + 141591250 T^{17} + 255403125 T^{18} + 430281250 T^{19} + 690156250 T^{20} + 1009375000 T^{21} + 1402343750 T^{22} + 1660156250 T^{23} + 1904296875 T^{24} + 1464843750 T^{25} + 1220703125 T^{26} \)
$7$ \( 1 + 2 T + 43 T^{2} + 74 T^{3} + 941 T^{4} + 1356 T^{5} + 14049 T^{6} + 16748 T^{7} + 161544 T^{8} + 159150 T^{9} + 1523588 T^{10} + 1270786 T^{11} + 12247846 T^{12} + 9186328 T^{13} + 85734922 T^{14} + 62268514 T^{15} + 522590684 T^{16} + 382119150 T^{17} + 2715070008 T^{18} + 1970385452 T^{19} + 11569955607 T^{20} + 7817070156 T^{21} + 37972744187 T^{22} + 20903168426 T^{23} + 85025049949 T^{24} + 27682574402 T^{25} + 96889010407 T^{26} \)
$11$ \( 1 - 10 T + 99 T^{2} - 650 T^{3} + 4058 T^{4} - 20652 T^{5} + 99178 T^{6} - 414690 T^{7} + 1652187 T^{8} - 5930846 T^{9} + 20771961 T^{10} - 67868140 T^{11} + 226158528 T^{12} - 731794280 T^{13} + 2487743808 T^{14} - 8212044940 T^{15} + 27647480091 T^{16} - 86833516286 T^{17} + 266086368537 T^{18} - 734648631090 T^{19} + 1932698645438 T^{20} - 4426939610412 T^{21} + 9568551730078 T^{22} - 16859325990650 T^{23} + 28245855390489 T^{24} - 31384283767210 T^{25} + 34522712143931 T^{26} \)
$13$ \( 1 + 4 T + 57 T^{2} + 162 T^{3} + 1706 T^{4} + 4756 T^{5} + 40298 T^{6} + 107526 T^{7} + 758747 T^{8} + 1984020 T^{9} + 12573747 T^{10} + 31907560 T^{11} + 181279940 T^{12} + 434738504 T^{13} + 2356639220 T^{14} + 5392377640 T^{15} + 27624522159 T^{16} + 56665595220 T^{17} + 281717449871 T^{18} + 519007464534 T^{19} + 2528639738066 T^{20} + 3879615309076 T^{21} + 18091275930338 T^{22} + 22333075679538 T^{23} + 102153142460109 T^{24} + 93192340489924 T^{25} + 302875106592253 T^{26} \)
$17$ \( 1 - 10 T + 159 T^{2} - 1144 T^{3} + 10655 T^{4} - 60322 T^{5} + 422229 T^{6} - 1960636 T^{7} + 11409516 T^{8} - 44670952 T^{9} + 233071850 T^{10} - 805433516 T^{11} + 4082487178 T^{12} - 13519828736 T^{13} + 69402282026 T^{14} - 232770286124 T^{15} + 1145081999050 T^{16} - 3730962581992 T^{17} + 16199881159212 T^{18} - 47324986733884 T^{19} + 173256887562117 T^{20} - 420791640356002 T^{21} + 1263553824075535 T^{22} - 2306297022113656 T^{23} + 5449231512913647 T^{24} - 5826222372297610 T^{25} + 9904578032905937 T^{26} \)
$19$ \( 1 - 8 T + 101 T^{2} - 542 T^{3} + 4080 T^{4} - 16068 T^{5} + 98348 T^{6} - 295602 T^{7} + 1949009 T^{8} - 5458848 T^{9} + 43927245 T^{10} - 135331040 T^{11} + 1016846988 T^{12} - 3006254856 T^{13} + 19320092772 T^{14} - 48854505440 T^{15} + 301296973455 T^{16} - 711402530208 T^{17} + 4825939235891 T^{18} - 13906856515362 T^{19} + 87910497787172 T^{20} - 272891890942788 T^{21} + 1316565806938320 T^{22} - 3323037911728142 T^{23} + 11765516148720119 T^{24} - 17706519352529288 T^{25} + 42052983462257059 T^{26} \)
$23$ \( 1 - 18 T + 305 T^{2} - 3514 T^{3} + 37136 T^{4} - 325668 T^{5} + 2646392 T^{6} - 19051934 T^{7} + 128728281 T^{8} - 794783542 T^{9} + 4658817553 T^{10} - 25389597448 T^{11} + 132491485696 T^{12} - 648297296808 T^{13} + 3047304171008 T^{14} - 13431097049992 T^{15} + 56683833167351 T^{16} - 222413021176822 T^{17} + 828539370316383 T^{18} - 2820369986859326 T^{19} + 9010502824337224 T^{20} - 25503381954492708 T^{21} + 66887605236089968 T^{22} - 145572760404762586 T^{23} + 290606976163747735 T^{24} - 394463239776365778 T^{25} + 504036361936467383 T^{26} \)
$29$ \( 1 + 14 T + 333 T^{2} + 3734 T^{3} + 51010 T^{4} + 475156 T^{5} + 4815362 T^{6} + 38263902 T^{7} + 315851919 T^{8} + 2180821162 T^{9} + 15336511931 T^{10} + 93085872148 T^{11} + 570906321012 T^{12} + 3061660588728 T^{13} + 16556283309348 T^{14} + 78285218476468 T^{15} + 374042189485159 T^{16} + 1542453372280522 T^{17} + 6478485772544931 T^{18} + 22760261262058542 T^{19} + 83064398883058858 T^{20} + 237695084596896916 T^{21} + 740009516229077690 T^{22} + 1570920809142950534 T^{23} + 4062769751980041057 T^{24} + 4953406964876566574 T^{25} + 10260628712958602189 T^{26} \)
$31$ \( 1 - 4 T + 239 T^{2} - 906 T^{3} + 28368 T^{4} - 99112 T^{5} + 2237064 T^{6} - 7142474 T^{7} + 131575251 T^{8} - 384647452 T^{9} + 6112924669 T^{10} - 16368870892 T^{11} + 230736218000 T^{12} - 562852408048 T^{13} + 7152822758000 T^{14} - 15730484927212 T^{15} + 182110138814179 T^{16} - 355229999518492 T^{17} + 3766887728741901 T^{18} - 6338971966446794 T^{19} + 61547478573610104 T^{20} - 84531736502852392 T^{21} + 750039201453914928 T^{22} - 742583228004605706 T^{23} + 6072625978240754609 T^{24} - 3150651135154199044 T^{25} + 24417546297445042591 T^{26} \)
$37$ \( 1 + 16 T + 387 T^{2} + 4842 T^{3} + 69421 T^{4} + 723480 T^{5} + 7862893 T^{6} + 70420010 T^{7} + 633309636 T^{8} + 4959474450 T^{9} + 38427836622 T^{10} + 265750638756 T^{11} + 1809250029880 T^{12} + 11098073086620 T^{13} + 66942251105560 T^{14} + 363812624456964 T^{15} + 1946485208414166 T^{16} + 9294853594686450 T^{17} + 43916196166469652 T^{18} + 180678479379044090 T^{19} + 746439192185925769 T^{20} + 2541208635322765080 T^{21} + 9022073938314040417 T^{22} + 23283165531247224858 T^{23} + 68854119628651179831 T^{24} + \)\(10\!\cdots\!96\)\( T^{25} + \)\(24\!\cdots\!97\)\( T^{26} \)
$41$ \( 1 - 10 T + 283 T^{2} - 2126 T^{3} + 35838 T^{4} - 195204 T^{5} + 2615990 T^{6} - 8556290 T^{7} + 115921277 T^{8} + 2704026 T^{9} + 2770228559 T^{10} + 23541984656 T^{11} + 16764701796 T^{12} + 1447866430888 T^{13} + 687352773636 T^{14} + 39574076206736 T^{15} + 190926922514839 T^{16} + 7640931213786 T^{17} + 13430198768288677 T^{18} - 40643269416225890 T^{19} + 509475232929957190 T^{20} - 1558689344425335684 T^{21} + 11732713764810774318 T^{22} - 28536573693384004526 T^{23} + \)\(15\!\cdots\!03\)\( T^{24} - \)\(22\!\cdots\!10\)\( T^{25} + \)\(92\!\cdots\!21\)\( T^{26} \)
$43$ \( 1 - 12 T + 339 T^{2} - 3606 T^{3} + 55398 T^{4} - 530800 T^{5} + 5898998 T^{6} - 51617958 T^{7} + 465699971 T^{8} - 3758156420 T^{9} + 29192394481 T^{10} - 217849810828 T^{11} + 1504990594496 T^{12} - 10324539522064 T^{13} + 64714595563328 T^{14} - 402804300220972 T^{15} + 2320999708000867 T^{16} - 12848388926852420 T^{17} + 68461827641855153 T^{18} - 326295852366033942 T^{19} + 1603457443282970786 T^{20} - 6204096707350610800 T^{21} + 27842625516077228514 T^{22} - 77931005221703001894 T^{23} + \)\(31\!\cdots\!73\)\( T^{24} - \)\(47\!\cdots\!12\)\( T^{25} + \)\(17\!\cdots\!43\)\( T^{26} \)
$47$ \( ( 1 + T )^{13} \)
$53$ \( 1 + 26 T + 685 T^{2} + 11638 T^{3} + 192613 T^{4} + 2550640 T^{5} + 32708011 T^{6} + 361005744 T^{7} + 3860065380 T^{8} + 36727550772 T^{9} + 339012997604 T^{10} + 2830531783778 T^{11} + 22951742789562 T^{12} + 169519617790676 T^{13} + 1216442367846786 T^{14} + 7950963780632402 T^{15} + 50471238044290708 T^{16} + 289798041543001332 T^{17} + 1614261944601332340 T^{18} + 8001461679659324976 T^{19} + 38422464883611134207 T^{20} + \)\(15\!\cdots\!40\)\( T^{21} + \)\(63\!\cdots\!29\)\( T^{22} + \)\(20\!\cdots\!62\)\( T^{23} + \)\(63\!\cdots\!45\)\( T^{24} + \)\(12\!\cdots\!66\)\( T^{25} + \)\(26\!\cdots\!73\)\( T^{26} \)
$59$ \( 1 - 30 T + 913 T^{2} - 17338 T^{3} + 309975 T^{4} - 4354138 T^{5} + 57090923 T^{6} - 638072946 T^{7} + 6691277736 T^{8} - 62616621148 T^{9} + 558068596550 T^{10} - 4615099515516 T^{11} + 37220300810970 T^{12} - 287476961586248 T^{13} + 2195997747847230 T^{14} - 16065161413511196 T^{15} + 114615570290842450 T^{16} - 758748203050550428 T^{17} + 4783757044824107064 T^{18} - 26914257364164976386 T^{19} + \)\(14\!\cdots\!37\)\( T^{20} - \)\(63\!\cdots\!98\)\( T^{21} + \)\(26\!\cdots\!25\)\( T^{22} - \)\(88\!\cdots\!38\)\( T^{23} + \)\(27\!\cdots\!67\)\( T^{24} - \)\(53\!\cdots\!30\)\( T^{25} + \)\(10\!\cdots\!79\)\( T^{26} \)
$61$ \( 1 + 18 T + 649 T^{2} + 9986 T^{3} + 201257 T^{4} + 2667564 T^{5} + 39440763 T^{6} + 453983044 T^{7} + 5448608768 T^{8} + 54852935628 T^{9} + 560216304972 T^{10} + 4958943015746 T^{11} + 44095318019126 T^{12} + 344010714552380 T^{13} + 2689814399166686 T^{14} + 18452226961590866 T^{15} + 127158457118849532 T^{16} + 759485025088523148 T^{17} + 4601874811048967168 T^{18} + 23389376380426334884 T^{19} + \)\(12\!\cdots\!23\)\( T^{20} + \)\(51\!\cdots\!84\)\( T^{21} + \)\(23\!\cdots\!37\)\( T^{22} + \)\(71\!\cdots\!86\)\( T^{23} + \)\(28\!\cdots\!89\)\( T^{24} + \)\(47\!\cdots\!78\)\( T^{25} + \)\(16\!\cdots\!81\)\( T^{26} \)
$67$ \( 1 - 4 T + 495 T^{2} - 954 T^{3} + 111592 T^{4} - 9264 T^{5} + 15716876 T^{6} + 25271854 T^{7} + 1605262459 T^{8} + 4976176156 T^{9} + 132022621149 T^{10} + 552382727860 T^{11} + 9534631107900 T^{12} + 43065297690560 T^{13} + 638820284229300 T^{14} + 2479646065363540 T^{15} + 39707519604636687 T^{16} + 100275527836870876 T^{17} + 2167305149220458113 T^{18} + 2286051027251171326 T^{19} + 95255452772622530948 T^{20} - 3761810964884722224 T^{21} + \)\(30\!\cdots\!24\)\( T^{22} - \)\(17\!\cdots\!46\)\( T^{23} + \)\(60\!\cdots\!85\)\( T^{24} - \)\(32\!\cdots\!44\)\( T^{25} + \)\(54\!\cdots\!87\)\( T^{26} \)
$71$ \( 1 - 36 T + 1137 T^{2} - 24680 T^{3} + 483457 T^{4} - 7852114 T^{5} + 117670391 T^{6} - 1555822768 T^{7} + 19259706764 T^{8} - 216508036136 T^{9} + 2300410602674 T^{10} - 22528250174488 T^{11} + 209677334974696 T^{12} - 1810640137650212 T^{13} + 14887090783203416 T^{14} - 113564909129594008 T^{15} + 823342259213654014 T^{16} - 5501833148224504616 T^{17} + 34748928235262030164 T^{18} - \)\(19\!\cdots\!28\)\( T^{19} + \)\(10\!\cdots\!81\)\( T^{20} - \)\(50\!\cdots\!54\)\( T^{21} + \)\(22\!\cdots\!67\)\( T^{22} - \)\(80\!\cdots\!80\)\( T^{23} + \)\(26\!\cdots\!27\)\( T^{24} - \)\(59\!\cdots\!76\)\( T^{25} + \)\(11\!\cdots\!11\)\( T^{26} \)
$73$ \( 1 - 10 T + 421 T^{2} - 2978 T^{3} + 77486 T^{4} - 339084 T^{5} + 8288554 T^{6} - 10520782 T^{7} + 609481099 T^{8} + 1640739850 T^{9} + 39305346439 T^{10} + 258779684912 T^{11} + 2764318667232 T^{12} + 22000169901656 T^{13} + 201795262707936 T^{14} + 1379036940896048 T^{15} + 15290447955660463 T^{16} + 46594125678603850 T^{17} + 1263497952807320707 T^{18} - 1592154403925237998 T^{19} + 91566959185055515738 T^{20} - \)\(27\!\cdots\!04\)\( T^{21} + \)\(45\!\cdots\!18\)\( T^{22} - \)\(12\!\cdots\!22\)\( T^{23} + \)\(13\!\cdots\!17\)\( T^{24} - \)\(22\!\cdots\!10\)\( T^{25} + \)\(16\!\cdots\!33\)\( T^{26} \)
$79$ \( 1 + 149 T^{2} - 564 T^{3} + 26745 T^{4} - 110714 T^{5} + 3945667 T^{6} - 13776944 T^{7} + 438680120 T^{8} - 1893670824 T^{9} + 46008636418 T^{10} - 189544266748 T^{11} + 4101925241336 T^{12} - 14824119105532 T^{13} + 324052094065544 T^{14} - 1182945768774268 T^{15} + 22684052090894302 T^{16} - 73758631982136744 T^{17} + 1349843470360087880 T^{18} - 3349002261815307824 T^{19} + 75772229957691023053 T^{20} - \)\(16\!\cdots\!54\)\( T^{21} + \)\(32\!\cdots\!55\)\( T^{22} - \)\(53\!\cdots\!64\)\( T^{23} + \)\(11\!\cdots\!71\)\( T^{24} + \)\(46\!\cdots\!39\)\( T^{26} \)
$83$ \( 1 - 12 T + 319 T^{2} - 4520 T^{3} + 72838 T^{4} - 966784 T^{5} + 12561522 T^{6} - 149129032 T^{7} + 1730470995 T^{8} - 18449656564 T^{9} + 196333746981 T^{10} - 1931733794768 T^{11} + 18847316508148 T^{12} - 172459787424384 T^{13} + 1564327270176284 T^{14} - 13307714112156752 T^{15} + 112261084185025047 T^{16} - 875589723554069044 T^{17} + 6816395580837649785 T^{18} - 48756301402237548808 T^{19} + \)\(34\!\cdots\!94\)\( T^{20} - \)\(21\!\cdots\!44\)\( T^{21} + \)\(13\!\cdots\!14\)\( T^{22} - \)\(70\!\cdots\!80\)\( T^{23} + \)\(41\!\cdots\!73\)\( T^{24} - \)\(12\!\cdots\!32\)\( T^{25} + \)\(88\!\cdots\!63\)\( T^{26} \)
$89$ \( 1 - 50 T + 1587 T^{2} - 36660 T^{3} + 698583 T^{4} - 11246054 T^{5} + 158804785 T^{6} - 1979141704 T^{7} + 22241935380 T^{8} - 226371590856 T^{9} + 2138323619974 T^{10} - 19097674945796 T^{11} + 169672090960558 T^{12} - 1552349392519880 T^{13} + 15100816095489662 T^{14} - 151272683245650116 T^{15} + 1507451864049450806 T^{16} - 14203060909040548296 T^{17} + \)\(12\!\cdots\!20\)\( T^{18} - \)\(98\!\cdots\!44\)\( T^{19} + \)\(70\!\cdots\!65\)\( T^{20} - \)\(44\!\cdots\!74\)\( T^{21} + \)\(24\!\cdots\!47\)\( T^{22} - \)\(11\!\cdots\!60\)\( T^{23} + \)\(44\!\cdots\!43\)\( T^{24} - \)\(12\!\cdots\!50\)\( T^{25} + \)\(21\!\cdots\!69\)\( T^{26} \)
$97$ \( 1 + 10 T + 707 T^{2} + 7892 T^{3} + 262079 T^{4} + 2920210 T^{5} + 66548437 T^{6} + 700888200 T^{7} + 12650340544 T^{8} + 123586793152 T^{9} + 1883565030914 T^{10} + 16883461405644 T^{11} + 225253830634038 T^{12} + 1830944139026712 T^{13} + 21849621571501686 T^{14} + 158856488365704396 T^{15} + 1719078947459373122 T^{16} + 10941049938842283712 T^{17} + \)\(10\!\cdots\!08\)\( T^{18} + \)\(58\!\cdots\!00\)\( T^{19} + \)\(53\!\cdots\!81\)\( T^{20} + \)\(22\!\cdots\!10\)\( T^{21} + \)\(19\!\cdots\!43\)\( T^{22} + \)\(58\!\cdots\!08\)\( T^{23} + \)\(50\!\cdots\!71\)\( T^{24} + \)\(69\!\cdots\!10\)\( T^{25} + \)\(67\!\cdots\!77\)\( T^{26} \)
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