Properties

Label 4730.2.a.w
Level 4730
Weight 2
Character orbit 4730.a
Self dual yes
Analytic conductor 37.769
Analytic rank 1
Dimension 8
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 4730 = 2 \cdot 5 \cdot 11 \cdot 43 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 4730.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(37.7692401561\)
Analytic rank: \(1\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - x^{7} - 12 x^{6} + 7 x^{5} + 41 x^{4} - 6 x^{3} - 28 x^{2} + 2 x + 4\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{6} - 10 \nu^{4} - \nu^{3} + 24 \nu^{2} + 2 \nu - 8 \)\()/2\)
\(\beta_{3}\)\(=\)\((\)\( \nu^{7} - \nu^{6} - 12 \nu^{5} + 7 \nu^{4} + 39 \nu^{3} - 4 \nu^{2} - 16 \nu - 2 \)\()/2\)
\(\beta_{4}\)\(=\)\((\)\( -\nu^{7} + \nu^{6} + 12 \nu^{5} - 5 \nu^{4} - 39 \nu^{3} - 8 \nu^{2} + 12 \nu + 4 \)\()/2\)
\(\beta_{5}\)\(=\)\((\)\( \nu^{7} - \nu^{6} - 12 \nu^{5} + 7 \nu^{4} + 39 \nu^{3} - 6 \nu^{2} - 16 \nu + 4 \)\()/2\)
\(\beta_{6}\)\(=\)\((\)\( \nu^{7} - \nu^{6} - 12 \nu^{5} + 7 \nu^{4} + 41 \nu^{3} - 6 \nu^{2} - 26 \nu + 2 \)\()/2\)
\(\beta_{7}\)\(=\)\((\)\( \nu^{7} - 2 \nu^{6} - 10 \nu^{5} + 17 \nu^{4} + 26 \nu^{3} - 32 \nu^{2} - 8 \nu + 8 \)\()/2\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(-\beta_{5} + \beta_{3} + 3\)
\(\nu^{3}\)\(=\)\(\beta_{6} - \beta_{5} + 5 \beta_{1} + 1\)
\(\nu^{4}\)\(=\)\(-6 \beta_{5} + \beta_{4} + 7 \beta_{3} + 2 \beta_{1} + 17\)
\(\nu^{5}\)\(=\)\(\beta_{7} + 7 \beta_{6} - 9 \beta_{5} + \beta_{3} + \beta_{2} + 30 \beta_{1} + 12\)
\(\nu^{6}\)\(=\)\(\beta_{6} - 37 \beta_{5} + 10 \beta_{4} + 46 \beta_{3} + 2 \beta_{2} + 23 \beta_{1} + 107\)
\(\nu^{7}\)\(=\)\(12 \beta_{7} + 46 \beta_{6} - 68 \beta_{5} + 3 \beta_{4} + 15 \beta_{3} + 14 \beta_{2} + 190 \beta_{1} + 107\)

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
−2.42597
−1.86277
−0.843708
−0.402785
0.484643
0.771947
2.57663
2.70201
−1.00000 −3.42597 1.00000 1.00000 3.42597 −2.60155 −1.00000 8.73725 −1.00000
1.2 −1.00000 −2.86277 1.00000 1.00000 2.86277 −1.78911 −1.00000 5.19548 −1.00000
1.3 −1.00000 −1.84371 1.00000 1.00000 1.84371 0.526782 −1.00000 0.399257 −1.00000
1.4 −1.00000 −1.40279 1.00000 1.00000 1.40279 3.56264 −1.00000 −1.03219 −1.00000
1.5 −1.00000 −0.515357 1.00000 1.00000 0.515357 −4.64210 −1.00000 −2.73441 −1.00000
1.6 −1.00000 −0.228053 1.00000 1.00000 0.228053 −2.81890 −1.00000 −2.94799 −1.00000
1.7 −1.00000 1.57663 1.00000 1.00000 −1.57663 0.800428 −1.00000 −0.514223 −1.00000
1.8 −1.00000 1.70201 1.00000 1.00000 −1.70201 0.961819 −1.00000 −0.103166 −1.00000
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.8
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 4730.2.a.w 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4730.2.a.w 8 1.a even 1 1 trivial

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(5\) \(-1\)
\(11\) \(-1\)
\(43\) \(1\)

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4730))\):

\(T_{3}^{8} + \cdots\)
\(T_{7}^{8} + \cdots\)
\(T_{13}^{8} + \cdots\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T )^{8} \)
$3$ \( 1 + 7 T + 33 T^{2} + 117 T^{3} + 345 T^{4} + 882 T^{5} + 1996 T^{6} + 4052 T^{7} + 7394 T^{8} + 12156 T^{9} + 17964 T^{10} + 23814 T^{11} + 27945 T^{12} + 28431 T^{13} + 24057 T^{14} + 15309 T^{15} + 6561 T^{16} \)
$5$ \( ( 1 - T )^{8} \)
$7$ \( 1 + 6 T + 47 T^{2} + 201 T^{3} + 938 T^{4} + 3162 T^{5} + 11103 T^{6} + 31303 T^{7} + 91046 T^{8} + 219121 T^{9} + 544047 T^{10} + 1084566 T^{11} + 2252138 T^{12} + 3378207 T^{13} + 5529503 T^{14} + 4941258 T^{15} + 5764801 T^{16} \)
$11$ \( ( 1 - T )^{8} \)
$13$ \( 1 + 2 T + 62 T^{2} + 92 T^{3} + 1736 T^{4} + 1868 T^{5} + 30866 T^{6} + 25246 T^{7} + 430990 T^{8} + 328198 T^{9} + 5216354 T^{10} + 4103996 T^{11} + 49581896 T^{12} + 34158956 T^{13} + 299262158 T^{14} + 125497034 T^{15} + 815730721 T^{16} \)
$17$ \( 1 + 8 T + 117 T^{2} + 781 T^{3} + 6252 T^{4} + 34310 T^{5} + 199135 T^{6} + 896479 T^{7} + 4139526 T^{8} + 15240143 T^{9} + 57550015 T^{10} + 168565030 T^{11} + 522173292 T^{12} + 1108908317 T^{13} + 2824095573 T^{14} + 3282709384 T^{15} + 6975757441 T^{16} \)
$19$ \( 1 + 69 T^{2} - 17 T^{3} + 2296 T^{4} + 54 T^{5} + 50801 T^{6} + 17155 T^{7} + 953410 T^{8} + 325945 T^{9} + 18339161 T^{10} + 370386 T^{11} + 299217016 T^{12} - 42093683 T^{13} + 3246165789 T^{14} + 16983563041 T^{16} \)
$23$ \( 1 + 18 T + 282 T^{2} + 2920 T^{3} + 26876 T^{4} + 197944 T^{5} + 1320702 T^{6} + 7469186 T^{7} + 38622038 T^{8} + 171791278 T^{9} + 698651358 T^{10} + 2408384648 T^{11} + 7521006716 T^{12} + 18794121560 T^{13} + 41746120698 T^{14} + 61286858046 T^{15} + 78310985281 T^{16} \)
$29$ \( 1 - 8 T + 98 T^{2} - 882 T^{3} + 7209 T^{4} - 48058 T^{5} + 342856 T^{6} - 2001388 T^{7} + 11097800 T^{8} - 58040252 T^{9} + 288341896 T^{10} - 1172086562 T^{11} + 5098788729 T^{12} - 18090833418 T^{13} + 58292685458 T^{14} - 137999010472 T^{15} + 500246412961 T^{16} \)
$31$ \( 1 + 11 T + 223 T^{2} + 1996 T^{3} + 22686 T^{4} + 165386 T^{5} + 1356577 T^{6} + 8071481 T^{7} + 51961410 T^{8} + 250215911 T^{9} + 1303670497 T^{10} + 4927014326 T^{11} + 20950997406 T^{12} + 57143785396 T^{13} + 197913320863 T^{14} + 302638755221 T^{15} + 852891037441 T^{16} \)
$37$ \( 1 + 17 T + 307 T^{2} + 3294 T^{3} + 36336 T^{4} + 298428 T^{5} + 2493261 T^{6} + 16600665 T^{7} + 112085390 T^{8} + 614224605 T^{9} + 3413274309 T^{10} + 15116273484 T^{11} + 68099514096 T^{12} + 228418994358 T^{13} + 787678007563 T^{14} + 1613841911261 T^{15} + 3512479453921 T^{16} \)
$41$ \( 1 - 12 T + 244 T^{2} - 2378 T^{3} + 28932 T^{4} - 230138 T^{5} + 2108972 T^{6} - 14013792 T^{7} + 103822454 T^{8} - 574565472 T^{9} + 3545181932 T^{10} - 15861341098 T^{11} + 81754917252 T^{12} - 275506045978 T^{13} + 1159025434804 T^{14} - 2337051286572 T^{15} + 7984925229121 T^{16} \)
$43$ \( ( 1 + T )^{8} \)
$47$ \( 1 + 19 T + 237 T^{2} + 1565 T^{3} + 9168 T^{4} + 27117 T^{5} + 86891 T^{6} - 1482653 T^{7} - 10377266 T^{8} - 69684691 T^{9} + 191942219 T^{10} + 2815368291 T^{11} + 44736915408 T^{12} + 358924935955 T^{13} + 2554674032973 T^{14} + 9625839288797 T^{15} + 23811286661761 T^{16} \)
$53$ \( 1 + 7 T + 303 T^{2} + 1683 T^{3} + 42385 T^{4} + 194180 T^{5} + 3728662 T^{6} + 14456650 T^{7} + 231474938 T^{8} + 766202450 T^{9} + 10473811558 T^{10} + 28908935860 T^{11} + 334438037185 T^{12} + 703823014719 T^{13} + 6715801422087 T^{14} + 8222977978859 T^{15} + 62259690411361 T^{16} \)
$59$ \( 1 - T + 297 T^{2} - 83 T^{3} + 41548 T^{4} + 8647 T^{5} + 3752399 T^{6} + 1642201 T^{7} + 251231334 T^{8} + 96889859 T^{9} + 13062100919 T^{10} + 1775912213 T^{11} + 503452114828 T^{12} - 59338716817 T^{13} + 12527618491377 T^{14} - 2488651484819 T^{15} + 146830437604321 T^{16} \)
$61$ \( 1 - 6 T + 316 T^{2} - 758 T^{3} + 38007 T^{4} + 66450 T^{5} + 2364782 T^{6} + 16724690 T^{7} + 121413668 T^{8} + 1020206090 T^{9} + 8799353822 T^{10} + 15082887450 T^{11} + 526238878887 T^{12} - 640203996158 T^{13} + 16280438298076 T^{14} - 18856457016126 T^{15} + 191707312997281 T^{16} \)
$67$ \( 1 + 22 T + 446 T^{2} + 5574 T^{3} + 67696 T^{4} + 622446 T^{5} + 6081474 T^{6} + 48840078 T^{7} + 436862750 T^{8} + 3272285226 T^{9} + 27299736786 T^{10} + 187208726298 T^{11} + 1364150287216 T^{12} + 7525597346418 T^{13} + 40344438447374 T^{14} + 133335655317106 T^{15} + 406067677556641 T^{16} \)
$71$ \( 1 + 14 T + 287 T^{2} + 3393 T^{3} + 49762 T^{4} + 501326 T^{5} + 5711765 T^{6} + 49212093 T^{7} + 477242794 T^{8} + 3494058603 T^{9} + 28793007365 T^{10} + 179430089986 T^{11} + 1264536069922 T^{12} + 6121750187943 T^{13} + 36764781485327 T^{14} + 127331682217474 T^{15} + 645753531245761 T^{16} \)
$73$ \( 1 + 13 T + 514 T^{2} + 5567 T^{3} + 119936 T^{4} + 1079789 T^{5} + 16553022 T^{6} + 123546039 T^{7} + 1481601022 T^{8} + 9018860847 T^{9} + 88211054238 T^{10} + 420056277413 T^{11} + 3405971432576 T^{12} + 11540789558231 T^{13} + 77785792312546 T^{14} + 143616180748261 T^{15} + 806460091894081 T^{16} \)
$79$ \( 1 + 8 T + 424 T^{2} + 2987 T^{3} + 91468 T^{4} + 564333 T^{5} + 12518738 T^{6} + 66589874 T^{7} + 1180934250 T^{8} + 5260600046 T^{9} + 78129443858 T^{10} + 278238177987 T^{11} + 3562686008908 T^{12} + 9191167463813 T^{13} + 103069081140904 T^{14} + 153631271889272 T^{15} + 1517108809906561 T^{16} \)
$83$ \( 1 + 4 T + 452 T^{2} + 1499 T^{3} + 98766 T^{4} + 288133 T^{5} + 13829002 T^{6} + 35768690 T^{7} + 1357063142 T^{8} + 2968801270 T^{9} + 95267994778 T^{10} + 164750703671 T^{11} + 4687268531886 T^{12} + 5904621923857 T^{13} + 147777048762788 T^{14} + 108544203958508 T^{15} + 2252292232139041 T^{16} \)
$89$ \( 1 - 5 T + 299 T^{2} - 508 T^{3} + 49214 T^{4} + 31858 T^{5} + 5570917 T^{6} + 11761287 T^{7} + 540583874 T^{8} + 1046754543 T^{9} + 44127233557 T^{10} + 22458902402 T^{11} + 3087796648574 T^{12} - 2836702200092 T^{13} + 148597405997339 T^{14} - 221156674477645 T^{15} + 3936588805702081 T^{16} \)
$97$ \( 1 + 23 T + 574 T^{2} + 7585 T^{3} + 105528 T^{4} + 830323 T^{5} + 7761410 T^{6} + 29168789 T^{7} + 391255310 T^{8} + 2829372533 T^{9} + 73027106690 T^{10} + 757813383379 T^{11} + 9342317965368 T^{12} + 65134975849345 T^{13} + 478125930829246 T^{14} + 1858360542996599 T^{15} + 7837433594376961 T^{16} \)
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