Properties

Label 8034.2.a.q
Level 8034
Weight 2
Character orbit 8034.a
Self dual yes
Analytic conductor 64.152
Analytic rank 1
Dimension 8
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 8034 = 2 \cdot 3 \cdot 13 \cdot 103 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8034.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(64.1518129839\)
Analytic rank: \(1\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Defining polynomial: \(x^{8} - 2 x^{7} - 11 x^{6} + 21 x^{5} + 23 x^{4} - 29 x^{3} - 27 x^{2} + x + 3\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - 2 x^{7} - 11 x^{6} + 21 x^{5} + 23 x^{4} - 29 x^{3} - 27 x^{2} + x + 3\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{7} - 3 \nu^{6} - 9 \nu^{5} + 31 \nu^{4} + 3 \nu^{3} - 41 \nu^{2} - 6 \nu + 8 \)
\(\beta_{3}\)\(=\)\( 2 \nu^{7} - 6 \nu^{6} - 17 \nu^{5} + 60 \nu^{4} - 4 \nu^{3} - 63 \nu^{2} - 3 \nu + 6 \)
\(\beta_{4}\)\(=\)\( -2 \nu^{7} + 6 \nu^{6} + 17 \nu^{5} - 60 \nu^{4} + 4 \nu^{3} + 64 \nu^{2} + 3 \nu - 9 \)
\(\beta_{5}\)\(=\)\( -2 \nu^{7} + 5 \nu^{6} + 19 \nu^{5} - 51 \nu^{4} - 15 \nu^{3} + 61 \nu^{2} + 14 \nu - 9 \)
\(\beta_{6}\)\(=\)\( -6 \nu^{7} + 17 \nu^{6} + 53 \nu^{5} - 171 \nu^{4} - 8 \nu^{3} + 188 \nu^{2} + 26 \nu - 24 \)
\(\beta_{7}\)\(=\)\( 8 \nu^{7} - 22 \nu^{6} - 71 \nu^{5} + 221 \nu^{4} + 13 \nu^{3} - 240 \nu^{2} - 28 \nu + 33 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{4} + \beta_{3} + 3\)
\(\nu^{3}\)\(=\)\(-\beta_{6} + \beta_{5} + \beta_{4} - \beta_{3} + 6 \beta_{1}\)
\(\nu^{4}\)\(=\)\(\beta_{7} + \beta_{6} + \beta_{5} + 10 \beta_{4} + 9 \beta_{3} + 2 \beta_{2} - 3 \beta_{1} + 20\)
\(\nu^{5}\)\(=\)\(2 \beta_{7} - 8 \beta_{6} + 12 \beta_{5} + 11 \beta_{4} - 10 \beta_{3} + 2 \beta_{2} + 45 \beta_{1} - 7\)
\(\nu^{6}\)\(=\)\(13 \beta_{7} + 12 \beta_{6} + 13 \beta_{5} + 91 \beta_{4} + 77 \beta_{3} + 22 \beta_{2} - 40 \beta_{1} + 157\)
\(\nu^{7}\)\(=\)\(26 \beta_{7} - 64 \beta_{6} + 113 \beta_{5} + 100 \beta_{4} - 94 \beta_{3} + 23 \beta_{2} + 366 \beta_{1} - 97\)

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
−2.96271
−0.811382
−0.749852
−0.501271
0.315468
1.79755
2.03483
2.87737
1.00000 1.00000 1.00000 −3.96271 1.00000 0.133176 1.00000 1.00000 −3.96271
1.2 1.00000 1.00000 1.00000 −1.81138 1.00000 4.37785 1.00000 1.00000 −1.81138
1.3 1.00000 1.00000 1.00000 −1.74985 1.00000 −2.02962 1.00000 1.00000 −1.74985
1.4 1.00000 1.00000 1.00000 −1.50127 1.00000 1.44399 1.00000 1.00000 −1.50127
1.5 1.00000 1.00000 1.00000 −0.684532 1.00000 −1.46314 1.00000 1.00000 −0.684532
1.6 1.00000 1.00000 1.00000 0.797548 1.00000 −4.52438 1.00000 1.00000 0.797548
1.7 1.00000 1.00000 1.00000 1.03483 1.00000 0.662784 1.00000 1.00000 1.03483
1.8 1.00000 1.00000 1.00000 1.87737 1.00000 −1.60066 1.00000 1.00000 1.87737
\(n\): 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 8034.2.a.q 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8034.2.a.q 8 1.a even 1 1 trivial

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)
\(13\) \(1\)
\(103\) \(1\)

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8034))\):

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - T )^{8} \)
$3$ \( ( 1 - T )^{8} \)
$5$ \( 1 + 6 T + 43 T^{2} + 179 T^{3} + 753 T^{4} + 2414 T^{5} + 7440 T^{6} + 19070 T^{7} + 46230 T^{8} + 95350 T^{9} + 186000 T^{10} + 301750 T^{11} + 470625 T^{12} + 559375 T^{13} + 671875 T^{14} + 468750 T^{15} + 390625 T^{16} \)
$7$ \( 1 + 3 T + 35 T^{2} + 82 T^{3} + 520 T^{4} + 980 T^{5} + 4627 T^{6} + 7598 T^{7} + 33038 T^{8} + 53186 T^{9} + 226723 T^{10} + 336140 T^{11} + 1248520 T^{12} + 1378174 T^{13} + 4117715 T^{14} + 2470629 T^{15} + 5764801 T^{16} \)
$11$ \( 1 + 15 T + 163 T^{2} + 1267 T^{3} + 8202 T^{4} + 43956 T^{5} + 204875 T^{6} + 823935 T^{7} + 2926889 T^{8} + 9063285 T^{9} + 24789875 T^{10} + 58505436 T^{11} + 120085482 T^{12} + 204051617 T^{13} + 288764443 T^{14} + 292307565 T^{15} + 214358881 T^{16} \)
$13$ \( ( 1 + T )^{8} \)
$17$ \( 1 + 11 T + 121 T^{2} + 865 T^{3} + 6128 T^{4} + 34226 T^{5} + 187199 T^{6} + 859993 T^{7} + 3847800 T^{8} + 14619881 T^{9} + 54100511 T^{10} + 168152338 T^{11} + 511816688 T^{12} + 1228176305 T^{13} + 2920645849 T^{14} + 4513725403 T^{15} + 6975757441 T^{16} \)
$19$ \( 1 + 15 T + 176 T^{2} + 1577 T^{3} + 11879 T^{4} + 77654 T^{5} + 446604 T^{6} + 2291803 T^{7} + 10538805 T^{8} + 43544257 T^{9} + 161224044 T^{10} + 532628786 T^{11} + 1548083159 T^{12} + 3904808123 T^{13} + 8280075056 T^{14} + 13408076085 T^{15} + 16983563041 T^{16} \)
$23$ \( 1 - T + 69 T^{2} - 79 T^{3} + 2418 T^{4} - 4870 T^{5} + 57771 T^{6} - 174833 T^{7} + 1243603 T^{8} - 4021159 T^{9} + 30560859 T^{10} - 59253290 T^{11} + 676655538 T^{12} - 508471097 T^{13} + 10214476341 T^{14} - 3404825447 T^{15} + 78310985281 T^{16} \)
$29$ \( 1 + 10 T + 132 T^{2} + 1191 T^{3} + 10091 T^{4} + 69737 T^{5} + 488836 T^{6} + 2845164 T^{7} + 16378047 T^{8} + 82509756 T^{9} + 411111076 T^{10} + 1700815693 T^{11} + 7137172571 T^{12} + 24428778459 T^{13} + 78516678372 T^{14} + 172498763090 T^{15} + 500246412961 T^{16} \)
$31$ \( 1 + 3 T + 140 T^{2} + 303 T^{3} + 9749 T^{4} + 17084 T^{5} + 472412 T^{6} + 743553 T^{7} + 17040609 T^{8} + 23050143 T^{9} + 453987932 T^{10} + 508949444 T^{11} + 9003406229 T^{12} + 8674632753 T^{13} + 124250515340 T^{14} + 82537842333 T^{15} + 852891037441 T^{16} \)
$37$ \( 1 + 26 T + 534 T^{2} + 7527 T^{3} + 90799 T^{4} + 887857 T^{5} + 7663020 T^{6} + 56248822 T^{7} + 368805321 T^{8} + 2081206414 T^{9} + 10490674380 T^{10} + 44972620621 T^{11} + 170171944639 T^{12} + 521951964339 T^{13} + 1370097902406 T^{14} + 2468228805458 T^{15} + 3512479453921 T^{16} \)
$41$ \( 1 + 12 T + 263 T^{2} + 2656 T^{3} + 31913 T^{4} + 265520 T^{5} + 2357812 T^{6} + 16122221 T^{7} + 116649126 T^{8} + 661011061 T^{9} + 3963481972 T^{10} + 18299903920 T^{11} + 90178510793 T^{12} + 307714069856 T^{13} + 1249277415383 T^{14} + 2337051286572 T^{15} + 7984925229121 T^{16} \)
$43$ \( 1 + 4 T + 122 T^{2} + 62 T^{3} + 9102 T^{4} + 2034 T^{5} + 597587 T^{6} + 22235 T^{7} + 28402378 T^{8} + 956105 T^{9} + 1104938363 T^{10} + 161717238 T^{11} + 31117926702 T^{12} + 9114523466 T^{13} + 771206291978 T^{14} + 1087274444428 T^{15} + 11688200277601 T^{16} \)
$47$ \( 1 + 6 T + 220 T^{2} + 1226 T^{3} + 26865 T^{4} + 133374 T^{5} + 2109734 T^{6} + 9205719 T^{7} + 117697226 T^{8} + 432668793 T^{9} + 4660402406 T^{10} + 13847288802 T^{11} + 131092630065 T^{12} + 281176978582 T^{13} + 2371427372380 T^{14} + 3039738722778 T^{15} + 23811286661761 T^{16} \)
$53$ \( 1 + 4 T + 360 T^{2} + 1339 T^{3} + 58863 T^{4} + 199300 T^{5} + 5761338 T^{6} + 17051412 T^{7} + 371419976 T^{8} + 903724836 T^{9} + 16183598442 T^{10} + 29671186100 T^{11} + 464457383103 T^{12} + 559963765127 T^{13} + 7979170006440 T^{14} + 4698844559348 T^{15} + 62259690411361 T^{16} \)
$59$ \( 1 + 19 T + 429 T^{2} + 5392 T^{3} + 72424 T^{4} + 688510 T^{5} + 7027583 T^{6} + 55252980 T^{7} + 476866142 T^{8} + 3259925820 T^{9} + 24463016423 T^{10} + 141405495290 T^{11} + 877587753064 T^{12} + 3854871820208 T^{13} + 18095448931989 T^{14} + 47284378211561 T^{15} + 146830437604321 T^{16} \)
$61$ \( 1 + 14 T + 314 T^{2} + 2580 T^{3} + 33268 T^{4} + 152612 T^{5} + 1680893 T^{6} + 2492193 T^{7} + 72005300 T^{8} + 152023773 T^{9} + 6254602853 T^{10} + 34640024372 T^{11} + 460623438388 T^{12} + 2179058456580 T^{13} + 16177397549354 T^{14} + 43998399704294 T^{15} + 191707312997281 T^{16} \)
$67$ \( 1 + 13 T + 346 T^{2} + 3293 T^{3} + 51717 T^{4} + 383491 T^{5} + 4693425 T^{6} + 29547058 T^{7} + 333123472 T^{8} + 1979652886 T^{9} + 21068784825 T^{10} + 115339903633 T^{11} + 1042155524757 T^{12} + 4445961977351 T^{13} + 31298600230474 T^{14} + 78789250869199 T^{15} + 406067677556641 T^{16} \)
$71$ \( 1 + 31 T + 855 T^{2} + 15308 T^{3} + 247375 T^{4} + 3153105 T^{5} + 36971771 T^{6} + 363235587 T^{7} + 3317121700 T^{8} + 25789726677 T^{9} + 186374697611 T^{10} + 1128530963655 T^{11} + 6286214587375 T^{12} + 27619142905108 T^{13} + 109525742752455 T^{14} + 281948724910121 T^{15} + 645753531245761 T^{16} \)
$73$ \( 1 + 27 T + 853 T^{2} + 14950 T^{3} + 268502 T^{4} + 3439427 T^{5} + 43709956 T^{6} + 428418795 T^{7} + 4126826152 T^{8} + 31274572035 T^{9} + 232930355524 T^{10} + 1337995573259 T^{11} + 7624984504982 T^{12} + 30992420315350 T^{13} + 129088095024517 T^{14} + 298279760015619 T^{15} + 806460091894081 T^{16} \)
$79$ \( 1 + 13 T + 537 T^{2} + 6547 T^{3} + 131837 T^{4} + 1449160 T^{5} + 19384928 T^{6} + 184019619 T^{7} + 1868852684 T^{8} + 14537549901 T^{9} + 120981335648 T^{10} + 714492397240 T^{11} + 5135061828797 T^{12} + 20145488244253 T^{13} + 130537963614777 T^{14} + 249650816820067 T^{15} + 1517108809906561 T^{16} \)
$83$ \( 1 + 28 T + 775 T^{2} + 11623 T^{3} + 171117 T^{4} + 1590547 T^{5} + 16039995 T^{6} + 104649245 T^{7} + 1094635440 T^{8} + 8685887335 T^{9} + 110499525555 T^{10} + 909454097489 T^{11} + 8120925514557 T^{12} + 45783469393589 T^{13} + 253378789360975 T^{14} + 759809427709556 T^{15} + 2252292232139041 T^{16} \)
$89$ \( 1 + 2 T + 446 T^{2} + 766 T^{3} + 97931 T^{4} + 110033 T^{5} + 13928016 T^{6} + 10186355 T^{7} + 1430090001 T^{8} + 906585595 T^{9} + 110323814736 T^{10} + 77569853977 T^{11} + 6144410403371 T^{12} + 4277389537934 T^{13} + 221653655768606 T^{14} + 88462669791058 T^{15} + 3936588805702081 T^{16} \)
$97$ \( 1 + 30 T + 867 T^{2} + 14706 T^{3} + 245585 T^{4} + 2986073 T^{5} + 37779248 T^{6} + 377683163 T^{7} + 4127980869 T^{8} + 36635266811 T^{9} + 355464944432 T^{10} + 2725308203129 T^{11} + 21741463474385 T^{12} + 126285425819442 T^{13} + 722186728273443 T^{14} + 2423948534343390 T^{15} + 7837433594376961 T^{16} \)
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