Properties

Label 4002.2.a.bi
Level 4002
Weight 2
Character orbit 4002.a
Self dual yes
Analytic conductor 31.956
Analytic rank 1
Dimension 8
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 4002 = 2 \cdot 3 \cdot 23 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4002.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(31.9561308889\)
Analytic rank: \(1\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Defining polynomial: \(x^{8} - 24 x^{6} - 3 x^{5} + 194 x^{4} + 39 x^{3} - 607 x^{2} - 104 x + 600\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - 24 x^{6} - 3 x^{5} + 194 x^{4} + 39 x^{3} - 607 x^{2} - 104 x + 600\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( -11 \nu^{7} - 79 \nu^{6} + 173 \nu^{5} + 1466 \nu^{4} - 180 \nu^{3} - 8105 \nu^{2} - 2752 \nu + 12248 \)\()/1048\)
\(\beta_{2}\)\(=\)\((\)\( 9 \nu^{7} + 17 \nu^{6} - 213 \nu^{5} - 342 \nu^{4} + 1362 \nu^{3} + 1701 \nu^{2} - 1726 \nu - 732 \)\()/524\)
\(\beta_{3}\)\(=\)\((\)\( -21 \nu^{7} + 135 \nu^{6} + 235 \nu^{5} - 2346 \nu^{4} + 228 \nu^{3} + 11489 \nu^{2} - 3920 \nu - 15584 \)\()/1048\)
\(\beta_{4}\)\(=\)\((\)\( 57 \nu^{7} - 67 \nu^{6} - 1087 \nu^{5} + 978 \nu^{4} + 6268 \nu^{3} - 3637 \nu^{2} - 11368 \nu + 4272 \)\()/1048\)
\(\beta_{5}\)\(=\)\((\)\( 57 \nu^{7} - 67 \nu^{6} - 1087 \nu^{5} + 978 \nu^{4} + 6268 \nu^{3} - 3637 \nu^{2} - 9272 \nu + 4272 \)\()/1048\)
\(\beta_{6}\)\(=\)\((\)\( -20 \nu^{7} + 35 \nu^{6} + 386 \nu^{5} - 681 \nu^{4} - 2197 \nu^{3} + 3949 \nu^{2} + 3690 \nu - 6408 \)\()/262\)
\(\beta_{7}\)\(=\)\((\)\( 103 \nu^{7} - 213 \nu^{6} - 2001 \nu^{5} + 3422 \nu^{4} + 11308 \nu^{3} - 14331 \nu^{2} - 17104 \nu + 15552 \)\()/1048\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{5} - \beta_{4}\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{7} - \beta_{4} + \beta_{3} - 2 \beta_{2} - \beta_{1} + 13\)\()/2\)
\(\nu^{3}\)\(=\)\((\)\(-\beta_{7} + 8 \beta_{5} - 5 \beta_{4} + \beta_{3} - 2 \beta_{2} + \beta_{1} + 3\)\()/2\)
\(\nu^{4}\)\(=\)\((\)\(11 \beta_{7} - 4 \beta_{6} + \beta_{5} - 16 \beta_{4} + 13 \beta_{3} - 26 \beta_{2} - 13 \beta_{1} + 109\)\()/2\)
\(\nu^{5}\)\(=\)\((\)\(-20 \beta_{7} + 77 \beta_{5} - 27 \beta_{4} + 6 \beta_{3} - 32 \beta_{2} + 8 \beta_{1} + 44\)\()/2\)
\(\nu^{6}\)\(=\)\((\)\(99 \beta_{7} - 72 \beta_{6} + 23 \beta_{5} - 188 \beta_{4} + 147 \beta_{3} - 286 \beta_{2} - 153 \beta_{1} + 1017\)\()/2\)
\(\nu^{7}\)\(=\)\(-140 \beta_{7} - 8 \beta_{6} + 399 \beta_{5} - 69 \beta_{4} + 9 \beta_{3} - 204 \beta_{2} + 11 \beta_{1} + 257\)

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
−2.45069
−1.57534
−2.15995
1.98746
3.34107
2.85533
−3.18819
1.19031
−1.00000 1.00000 1.00000 −4.40973 −1.00000 −0.551242 −1.00000 1.00000 4.40973
1.2 −1.00000 1.00000 1.00000 −2.69359 −1.00000 −3.87906 −1.00000 1.00000 2.69359
1.3 −1.00000 1.00000 1.00000 −1.69873 −1.00000 3.41134 −1.00000 1.00000 1.69873
1.4 −1.00000 1.00000 1.00000 −0.881455 −1.00000 0.253486 −1.00000 1.00000 0.881455
1.5 −1.00000 1.00000 1.00000 −0.464023 −1.00000 −2.35800 −1.00000 1.00000 0.464023
1.6 −1.00000 1.00000 1.00000 1.27098 −1.00000 3.55617 −1.00000 1.00000 −1.27098
1.7 −1.00000 1.00000 1.00000 1.60812 −1.00000 −3.37648 −1.00000 1.00000 −1.60812
1.8 −1.00000 1.00000 1.00000 4.26843 −1.00000 −3.05622 −1.00000 1.00000 −4.26843
\(n\): 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 4002.2.a.bi 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4002.2.a.bi 8 1.a even 1 1 trivial

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(3\) \(-1\)
\(23\) \(1\)
\(29\) \(-1\)

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4002))\):

\(T_{5}^{8} + \cdots\)
\(T_{7}^{8} + \cdots\)
\(T_{11}^{8} + \cdots\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T )^{8} \)
$3$ \( ( 1 - T )^{8} \)
$5$ \( 1 + 3 T + 18 T^{2} + 39 T^{3} + 107 T^{4} + 156 T^{5} + 72 T^{6} - 114 T^{7} - 1452 T^{8} - 570 T^{9} + 1800 T^{10} + 19500 T^{11} + 66875 T^{12} + 121875 T^{13} + 281250 T^{14} + 234375 T^{15} + 390625 T^{16} \)
$7$ \( 1 + 6 T + 41 T^{2} + 150 T^{3} + 662 T^{4} + 1978 T^{5} + 7343 T^{6} + 19418 T^{7} + 61090 T^{8} + 135926 T^{9} + 359807 T^{10} + 678454 T^{11} + 1589462 T^{12} + 2521050 T^{13} + 4823609 T^{14} + 4941258 T^{15} + 5764801 T^{16} \)
$11$ \( 1 + 7 T + 49 T^{2} + 224 T^{3} + 1080 T^{4} + 4308 T^{5} + 17831 T^{6} + 63439 T^{7} + 228190 T^{8} + 697829 T^{9} + 2157551 T^{10} + 5733948 T^{11} + 15812280 T^{12} + 36075424 T^{13} + 86806489 T^{14} + 136410197 T^{15} + 214358881 T^{16} \)
$13$ \( 1 + 3 T + 47 T^{2} + 148 T^{3} + 1078 T^{4} + 3916 T^{5} + 16797 T^{6} + 70033 T^{7} + 223018 T^{8} + 910429 T^{9} + 2838693 T^{10} + 8603452 T^{11} + 30788758 T^{12} + 54951364 T^{13} + 226860023 T^{14} + 188245551 T^{15} + 815730721 T^{16} \)
$17$ \( 1 + 12 T + 89 T^{2} + 390 T^{3} + 1214 T^{4} + 146 T^{5} - 19777 T^{6} - 165076 T^{7} - 756206 T^{8} - 2806292 T^{9} - 5715553 T^{10} + 717298 T^{11} + 101394494 T^{12} + 553744230 T^{13} + 2148243641 T^{14} + 4924064076 T^{15} + 6975757441 T^{16} \)
$19$ \( 1 + 4 T + 41 T^{2} + 174 T^{3} + 1178 T^{4} + 3534 T^{5} + 17143 T^{6} + 49756 T^{7} + 281498 T^{8} + 945364 T^{9} + 6188623 T^{10} + 24239706 T^{11} + 153518138 T^{12} + 430841226 T^{13} + 1928881121 T^{14} + 3575486956 T^{15} + 16983563041 T^{16} \)
$23$ \( ( 1 + T )^{8} \)
$29$ \( ( 1 - T )^{8} \)
$31$ \( 1 + T + 89 T^{2} + 6 T^{3} + 3372 T^{4} - 9830 T^{5} + 76567 T^{6} - 684371 T^{7} + 1711558 T^{8} - 21215501 T^{9} + 73580887 T^{10} - 292845530 T^{11} + 3114112812 T^{12} + 171774906 T^{13} + 78987827609 T^{14} + 27512614111 T^{15} + 852891037441 T^{16} \)
$37$ \( 1 + 11 T + 226 T^{2} + 1803 T^{3} + 20923 T^{4} + 134628 T^{5} + 1171400 T^{6} + 6559782 T^{7} + 48561940 T^{8} + 242711934 T^{9} + 1603646600 T^{10} + 6819312084 T^{11} + 39213070603 T^{12} + 125027154471 T^{13} + 579854168434 T^{14} + 1044250648463 T^{15} + 3512479453921 T^{16} \)
$41$ \( 1 + 17 T + 298 T^{2} + 3055 T^{3} + 31995 T^{4} + 245016 T^{5} + 1994164 T^{6} + 12858016 T^{7} + 91409676 T^{8} + 527178656 T^{9} + 3352189684 T^{10} + 16886747736 T^{11} + 90410223195 T^{12} + 353940694055 T^{13} + 1415531063818 T^{14} + 3310822655977 T^{15} + 7984925229121 T^{16} \)
$43$ \( 1 + 10 T + 185 T^{2} + 1458 T^{3} + 14574 T^{4} + 95914 T^{5} + 710911 T^{6} + 4244050 T^{7} + 29600770 T^{8} + 182494150 T^{9} + 1314474439 T^{10} + 7625834398 T^{11} + 49825605774 T^{12} + 214338309894 T^{13} + 1169452164065 T^{14} + 2718186111070 T^{15} + 11688200277601 T^{16} \)
$47$ \( 1 + 26 T + 457 T^{2} + 5922 T^{3} + 65470 T^{4} + 628070 T^{5} + 5499191 T^{6} + 43324526 T^{7} + 312994370 T^{8} + 2036252722 T^{9} + 12147712919 T^{10} + 65208111610 T^{11} + 319472715070 T^{12} + 1358181131454 T^{13} + 4926101405353 T^{14} + 13172201132038 T^{15} + 23811286661761 T^{16} \)
$53$ \( 1 + 2 T + 184 T^{2} + 298 T^{3} + 20580 T^{4} + 25998 T^{5} + 1608712 T^{6} + 1735078 T^{7} + 97850646 T^{8} + 91959134 T^{9} + 4518872008 T^{10} + 3870504246 T^{11} + 162386098980 T^{12} + 124622256914 T^{13} + 4078242447736 T^{14} + 2349422279674 T^{15} + 62259690411361 T^{16} \)
$59$ \( 1 + 25 T + 382 T^{2} + 4743 T^{3} + 54617 T^{4} + 590176 T^{5} + 5671518 T^{6} + 48070994 T^{7} + 379541188 T^{8} + 2836188646 T^{9} + 19742554158 T^{10} + 121209756704 T^{11} + 661813905737 T^{12} + 3390885950157 T^{13} + 16112963850862 T^{14} + 62216287120475 T^{15} + 146830437604321 T^{16} \)
$61$ \( 1 - 3 T + 257 T^{2} - 320 T^{3} + 31936 T^{4} - 19322 T^{5} + 2892351 T^{6} - 2207999 T^{7} + 204160846 T^{8} - 134687939 T^{9} + 10762438071 T^{10} - 4385726882 T^{11} + 442180778176 T^{12} - 270270816320 T^{13} + 13240736210777 T^{14} - 9428228508063 T^{15} + 191707312997281 T^{16} \)
$67$ \( 1 - T + 169 T^{2} + 160 T^{3} + 17638 T^{4} + 14422 T^{5} + 1486335 T^{6} + 266089 T^{7} + 109608914 T^{8} + 17827963 T^{9} + 6672157815 T^{10} + 4337603986 T^{11} + 355425472198 T^{12} + 216020017120 T^{13} + 15287466586561 T^{14} - 6060711605323 T^{15} + 406067677556641 T^{16} \)
$71$ \( 1 + 27 T + 501 T^{2} + 5942 T^{3} + 59960 T^{4} + 486414 T^{5} + 4067907 T^{6} + 31459219 T^{7} + 274745214 T^{8} + 2233604549 T^{9} + 20506319187 T^{10} + 174092921154 T^{11} + 1523684392760 T^{12} + 10720730803642 T^{13} + 64178242244421 T^{14} + 245568244276557 T^{15} + 645753531245761 T^{16} \)
$73$ \( 1 - 16 T + 428 T^{2} - 4396 T^{3} + 69276 T^{4} - 520252 T^{5} + 6654516 T^{6} - 41620584 T^{7} + 510058182 T^{8} - 3038302632 T^{9} + 35461915764 T^{10} - 202386872284 T^{11} + 1967316543516 T^{12} - 9113222722828 T^{13} + 64771048851692 T^{14} - 176758376305552 T^{15} + 806460091894081 T^{16} \)
$79$ \( 1 + 6 T + 350 T^{2} + 1334 T^{3} + 57376 T^{4} + 152006 T^{5} + 6748130 T^{6} + 15515606 T^{7} + 617416446 T^{8} + 1225732874 T^{9} + 42115079330 T^{10} + 74944886234 T^{11} + 2234799847456 T^{12} + 4104793236266 T^{13} + 85080609432350 T^{14} + 115223453916954 T^{15} + 1517108809906561 T^{16} \)
$83$ \( 1 + 44 T + 1156 T^{2} + 21892 T^{3} + 330292 T^{4} + 4169940 T^{5} + 46199964 T^{6} + 463933500 T^{7} + 4353903798 T^{8} + 38506480500 T^{9} + 318271551996 T^{10} + 2384317482780 T^{11} + 15675103759732 T^{12} + 86233477756556 T^{13} + 377943071614564 T^{14} + 1193986243543588 T^{15} + 2252292232139041 T^{16} \)
$89$ \( 1 + 52 T + 1736 T^{2} + 41896 T^{3} + 809700 T^{4} + 12939536 T^{5} + 175733048 T^{6} + 2052055476 T^{7} + 20775355382 T^{8} + 182632937364 T^{9} + 1391981473208 T^{10} + 9121971754384 T^{11} + 50802392537700 T^{12} + 233949754675304 T^{13} + 862759521108296 T^{14} + 2300029414567508 T^{15} + 3936588805702081 T^{16} \)
$97$ \( 1 + 4 T + 152 T^{2} - 664 T^{3} + 7084 T^{4} - 93824 T^{5} + 482312 T^{6} - 2061036 T^{7} + 58267366 T^{8} - 199920492 T^{9} + 4538073608 T^{10} - 85630631552 T^{11} + 627141426604 T^{12} - 5701993930648 T^{13} + 126611744749208 T^{14} + 323193137912452 T^{15} + 7837433594376961 T^{16} \)
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