Properties

Label 8030.2.a.ba
Level $8030$
Weight $2$
Character orbit 8030.a
Self dual yes
Analytic conductor $64.120$
Analytic rank $1$
Dimension $7$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 8030 = 2 \cdot 5 \cdot 11 \cdot 73 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8030.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(64.1198728231\)
Analytic rank: \(1\)
Dimension: \(7\)
Coefficient field: \(\mathbb{Q}[x]/(x^{7} - \cdots)\)
Defining polynomial: \(x^{7} - x^{6} - 11 x^{5} + 17 x^{4} + 9 x^{3} - 15 x^{2} + 2\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
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 \(1,\beta_1,\ldots,\beta_{6}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{7} - x^{6} - 11 x^{5} + 17 x^{4} + 9 x^{3} - 15 x^{2} + 2\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{5} - 10 \nu^{3} + 7 \nu^{2} + 8 \nu - 2 \)\()/2\)
\(\beta_{3}\)\(=\)\((\)\( \nu^{6} - 12 \nu^{4} + 5 \nu^{3} + 24 \nu^{2} - 8 \)\()/2\)
\(\beta_{4}\)\(=\)\((\)\( 2 \nu^{6} - \nu^{5} - 22 \nu^{4} + 22 \nu^{3} + 23 \nu^{2} - 6 \nu - 2 \)\()/2\)
\(\beta_{5}\)\(=\)\((\)\( -3 \nu^{6} + 2 \nu^{5} + 34 \nu^{4} - 39 \nu^{3} - 42 \nu^{2} + 28 \nu + 8 \)\()/2\)
\(\beta_{6}\)\(=\)\( 2 \nu^{6} - \nu^{5} - 22 \nu^{4} + 23 \nu^{3} + 25 \nu^{2} - 13 \nu - 5 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{6} + \beta_{5} - \beta_{4} + \beta_{3} - \beta_{2} + 3\)
\(\nu^{3}\)\(=\)\(-\beta_{6} - 2 \beta_{5} - 2 \beta_{3} + 2 \beta_{2} + 7 \beta_{1} - 3\)
\(\nu^{4}\)\(=\)\(10 \beta_{6} + 11 \beta_{5} - 8 \beta_{4} + 9 \beta_{3} - 10 \beta_{2} - 8 \beta_{1} + 24\)
\(\nu^{5}\)\(=\)\(-17 \beta_{6} - 27 \beta_{5} + 7 \beta_{4} - 27 \beta_{3} + 29 \beta_{2} + 62 \beta_{1} - 49\)
\(\nu^{6}\)\(=\)\(101 \beta_{6} + 118 \beta_{5} - 72 \beta_{4} + 96 \beta_{3} - 106 \beta_{2} - 131 \beta_{1} + 239\)

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
0.725391
−0.902407
0.528993
2.45630
−0.358554
1.93660
−3.38633
−1.00000 −1.93518 1.00000 1.00000 1.93518 −4.53467 −1.00000 0.744937 −1.00000
1.2 −1.00000 −1.61566 1.00000 1.00000 1.61566 3.03677 −1.00000 −0.389633 −1.00000
1.3 −1.00000 −1.37595 1.00000 1.00000 1.37595 −1.05505 −1.00000 −1.10676 −1.00000
1.4 −1.00000 −0.550071 1.00000 1.00000 0.550071 −3.11640 −1.00000 −2.69742 −1.00000
1.5 −1.00000 1.75674 1.00000 1.00000 −1.75674 2.75784 −1.00000 0.0861198 −1.00000
1.6 −1.00000 2.82268 1.00000 1.00000 −2.82268 0.448361 −1.00000 4.96752 −1.00000
1.7 −1.00000 2.89745 1.00000 1.00000 −2.89745 −4.53686 −1.00000 5.39524 −1.00000
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.7
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(5\) \(-1\)
\(11\) \(1\)
\(73\) \(-1\)

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 8030.2.a.ba 7
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8030.2.a.ba 7 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8030))\):

\( T_{3}^{7} - 2 T_{3}^{6} - 12 T_{3}^{5} + 14 T_{3}^{4} + 54 T_{3}^{3} - 13 T_{3}^{2} - 82 T_{3} - 34 \)
\( T_{7}^{7} + 7 T_{7}^{6} - 10 T_{7}^{5} - 128 T_{7}^{4} - 27 T_{7}^{3} + 615 T_{7}^{2} + 308 T_{7} - 254 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T )^{7} \)
$3$ \( 1 - 2 T + 9 T^{2} - 22 T^{3} + 63 T^{4} - 115 T^{5} + 269 T^{6} - 436 T^{7} + 807 T^{8} - 1035 T^{9} + 1701 T^{10} - 1782 T^{11} + 2187 T^{12} - 1458 T^{13} + 2187 T^{14} \)
$5$ \( ( 1 - T )^{7} \)
$7$ \( 1 + 7 T + 39 T^{2} + 166 T^{3} + 652 T^{4} + 2176 T^{5} + 6846 T^{6} + 18744 T^{7} + 47922 T^{8} + 106624 T^{9} + 223636 T^{10} + 398566 T^{11} + 655473 T^{12} + 823543 T^{13} + 823543 T^{14} \)
$11$ \( ( 1 + T )^{7} \)
$13$ \( 1 + 11 T + 93 T^{2} + 568 T^{3} + 3072 T^{4} + 14293 T^{5} + 61112 T^{6} + 230772 T^{7} + 794456 T^{8} + 2415517 T^{9} + 6749184 T^{10} + 16222648 T^{11} + 34530249 T^{12} + 53094899 T^{13} + 62748517 T^{14} \)
$17$ \( 1 + 7 T + 94 T^{2} + 490 T^{3} + 3972 T^{4} + 16851 T^{5} + 103025 T^{6} + 358903 T^{7} + 1751425 T^{8} + 4869939 T^{9} + 19514436 T^{10} + 40925290 T^{11} + 133466558 T^{12} + 168962983 T^{13} + 410338673 T^{14} \)
$19$ \( 1 + 76 T^{2} + 110 T^{3} + 2518 T^{4} + 7775 T^{5} + 53783 T^{6} + 216241 T^{7} + 1021877 T^{8} + 2806775 T^{9} + 17270962 T^{10} + 14335310 T^{11} + 188183524 T^{12} + 893871739 T^{14} \)
$23$ \( 1 + 4 T + 136 T^{2} + 510 T^{3} + 8393 T^{4} + 27884 T^{5} + 304527 T^{6} + 840418 T^{7} + 7004121 T^{8} + 14750636 T^{9} + 102117631 T^{10} + 142718910 T^{11} + 875342648 T^{12} + 592143556 T^{13} + 3404825447 T^{14} \)
$29$ \( 1 - 4 T + 149 T^{2} - 544 T^{3} + 10665 T^{4} - 34093 T^{5} + 468282 T^{6} - 1258015 T^{7} + 13580178 T^{8} - 28672213 T^{9} + 260108685 T^{10} - 384760864 T^{11} + 3056161201 T^{12} - 2379293284 T^{13} + 17249876309 T^{14} \)
$31$ \( 1 - 3 T + 68 T^{2} - 129 T^{3} + 1896 T^{4} - 2585 T^{5} + 47805 T^{6} - 89814 T^{7} + 1481955 T^{8} - 2484185 T^{9} + 56483736 T^{10} - 119134209 T^{11} + 1946782268 T^{12} - 2662511043 T^{13} + 27512614111 T^{14} \)
$37$ \( 1 + 22 T + 370 T^{2} + 4316 T^{3} + 44172 T^{4} + 368389 T^{5} + 2770751 T^{6} + 17670967 T^{7} + 102517787 T^{8} + 504324541 T^{9} + 2237444316 T^{10} + 8088878876 T^{11} + 25657264090 T^{12} + 56445980998 T^{13} + 94931877133 T^{14} \)
$41$ \( 1 + 8 T + 208 T^{2} + 1686 T^{3} + 21129 T^{4} + 156272 T^{5} + 1321753 T^{6} + 8256054 T^{7} + 54191873 T^{8} + 262693232 T^{9} + 1456231809 T^{10} + 4764233046 T^{11} + 24098089808 T^{12} + 38000833928 T^{13} + 194754273881 T^{14} \)
$43$ \( 1 + 19 T + 374 T^{2} + 4641 T^{3} + 52833 T^{4} + 481849 T^{5} + 3922753 T^{6} + 27307106 T^{7} + 168678379 T^{8} + 890938801 T^{9} + 4200593331 T^{10} + 15866655441 T^{11} + 54981157682 T^{12} + 120105897931 T^{13} + 271818611107 T^{14} \)
$47$ \( 1 - 3 T + 139 T^{2} - 538 T^{3} + 10398 T^{4} - 37815 T^{5} + 550904 T^{6} - 1984392 T^{7} + 25892488 T^{8} - 83533335 T^{9} + 1079551554 T^{10} - 2625268378 T^{11} + 31878955973 T^{12} - 32337645987 T^{13} + 506623120463 T^{14} \)
$53$ \( 1 + 126 T^{2} - 188 T^{3} + 10881 T^{4} - 32605 T^{5} + 699732 T^{6} - 1920568 T^{7} + 37085796 T^{8} - 91587445 T^{9} + 1619930637 T^{10} - 1483410428 T^{11} + 52692632118 T^{12} + 1174711139837 T^{14} \)
$59$ \( 1 - 23 T + 374 T^{2} - 3601 T^{3} + 19801 T^{4} + 27693 T^{5} - 1821044 T^{6} + 19026766 T^{7} - 107441596 T^{8} + 96399333 T^{9} + 4066709579 T^{10} - 43634616961 T^{11} + 267381687826 T^{12} - 970152273743 T^{13} + 2488651484819 T^{14} \)
$61$ \( 1 + 25 T + 513 T^{2} + 7285 T^{3} + 93267 T^{4} + 967572 T^{5} + 9246463 T^{6} + 74924432 T^{7} + 564034243 T^{8} + 3600335412 T^{9} + 21169836927 T^{10} + 100866951685 T^{11} + 433277902413 T^{12} + 1288009359025 T^{13} + 3142742836021 T^{14} \)
$67$ \( 1 - 27 T + 545 T^{2} - 7212 T^{3} + 87120 T^{4} - 870707 T^{5} + 8567014 T^{6} - 71525228 T^{7} + 573989938 T^{8} - 3908603723 T^{9} + 26202472560 T^{10} - 145329884652 T^{11} + 735818183315 T^{12} - 2442376318563 T^{13} + 6060711605323 T^{14} \)
$71$ \( 1 - 11 T + 295 T^{2} - 2327 T^{3} + 39159 T^{4} - 229466 T^{5} + 3345361 T^{6} - 16799656 T^{7} + 237520631 T^{8} - 1156738106 T^{9} + 14015436849 T^{10} - 59132981687 T^{11} + 532247658545 T^{12} - 1409103123131 T^{13} + 9095120158391 T^{14} \)
$73$ \( ( 1 - T )^{7} \)
$79$ \( 1 - 34 T + 696 T^{2} - 8936 T^{3} + 76739 T^{4} - 331626 T^{5} - 933038 T^{6} + 24628668 T^{7} - 73710002 T^{8} - 2069677866 T^{9} + 37835319821 T^{10} - 348057923816 T^{11} + 2141631253704 T^{12} - 8264973487714 T^{13} + 19203908986159 T^{14} \)
$83$ \( 1 + 27 T + 666 T^{2} + 10295 T^{3} + 155589 T^{4} + 1808263 T^{5} + 20746600 T^{6} + 190851998 T^{7} + 1721967800 T^{8} + 12457123807 T^{9} + 88963767543 T^{10} + 488583414695 T^{11} + 2623401068238 T^{12} + 8827390080963 T^{13} + 27136050989627 T^{14} \)
$89$ \( 1 + 16 T + 461 T^{2} + 4552 T^{3} + 75197 T^{4} + 473441 T^{5} + 6807496 T^{6} + 34707885 T^{7} + 605867144 T^{8} + 3750126161 T^{9} + 53011553893 T^{10} + 285602681032 T^{11} + 2574251405989 T^{12} + 7951700655376 T^{13} + 44231334895529 T^{14} \)
$97$ \( 1 + 25 T + 412 T^{2} + 3637 T^{3} + 25311 T^{4} - 71105 T^{5} - 3211384 T^{6} - 51762810 T^{7} - 311504248 T^{8} - 669026945 T^{9} + 23100666303 T^{10} + 321980994997 T^{11} + 3537984185884 T^{12} + 20824300123225 T^{13} + 80798284478113 T^{14} \)
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