Properties

Label 8030.2.a.z
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: \(x^{7} - 2 x^{6} - 11 x^{5} + 17 x^{4} + 25 x^{3} - 9 x^{2} - 14 x - 3\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{7} - 2 x^{6} - 11 x^{5} + 17 x^{4} + 25 x^{3} - 9 x^{2} - 14 x - 3\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( -2 \nu^{6} + 5 \nu^{5} + 19 \nu^{4} - 43 \nu^{3} - 23 \nu^{2} + 25 \nu + 7 \)\()/2\)
\(\beta_{3}\)\(=\)\((\)\( 2 \nu^{6} - 5 \nu^{5} - 19 \nu^{4} + 43 \nu^{3} + 25 \nu^{2} - 27 \nu - 15 \)\()/2\)
\(\beta_{4}\)\(=\)\((\)\( 3 \nu^{6} - 8 \nu^{5} - 28 \nu^{4} + 70 \nu^{3} + 32 \nu^{2} - 50 \nu - 15 \)\()/2\)
\(\beta_{5}\)\(=\)\((\)\( 3 \nu^{6} - 7 \nu^{5} - 31 \nu^{4} + 61 \nu^{3} + 57 \nu^{2} - 43 \nu - 26 \)\()/2\)
\(\beta_{6}\)\(=\)\((\)\( 4 \nu^{6} - 9 \nu^{5} - 41 \nu^{4} + 79 \nu^{3} + 73 \nu^{2} - 61 \nu - 33 \)\()/2\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{3} + \beta_{2} + \beta_{1} + 4\)
\(\nu^{3}\)\(=\)\(\beta_{6} - \beta_{5} + \beta_{4} - \beta_{3} + \beta_{2} + 8 \beta_{1}\)
\(\nu^{4}\)\(=\)\(\beta_{6} - 2 \beta_{5} + 9 \beta_{3} + 8 \beta_{2} + 9 \beta_{1} + 30\)
\(\nu^{5}\)\(=\)\(12 \beta_{6} - 13 \beta_{5} + 7 \beta_{4} - 7 \beta_{3} + 8 \beta_{2} + 67 \beta_{1} + 1\)
\(\nu^{6}\)\(=\)\(18 \beta_{6} - 30 \beta_{5} - 4 \beta_{4} + 78 \beta_{3} + 62 \beta_{2} + 82 \beta_{1} + 245\)

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
3.07415
2.55939
0.914270
−0.335989
−0.496547
−0.882415
−2.83286
−1.00000 −3.07415 1.00000 1.00000 3.07415 −2.88946 −1.00000 6.45040 −1.00000
1.2 −1.00000 −2.55939 1.00000 1.00000 2.55939 2.15359 −1.00000 3.55049 −1.00000
1.3 −1.00000 −0.914270 1.00000 1.00000 0.914270 2.28524 −1.00000 −2.16411 −1.00000
1.4 −1.00000 0.335989 1.00000 1.00000 −0.335989 −3.89614 −1.00000 −2.88711 −1.00000
1.5 −1.00000 0.496547 1.00000 1.00000 −0.496547 0.154508 −1.00000 −2.75344 −1.00000
1.6 −1.00000 0.882415 1.00000 1.00000 −0.882415 0.390090 −1.00000 −2.22134 −1.00000
1.7 −1.00000 2.83286 1.00000 1.00000 −2.83286 −1.19783 −1.00000 5.02511 −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.z 7
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8030.2.a.z 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8030))\):

\( T_{3}^{7} + 2 T_{3}^{6} - 11 T_{3}^{5} - 17 T_{3}^{4} + 25 T_{3}^{3} + 9 T_{3}^{2} - 14 T_{3} + 3 \)
\( T_{7}^{7} + 3 T_{7}^{6} - 13 T_{7}^{5} - 27 T_{7}^{4} + 53 T_{7}^{3} + 45 T_{7}^{2} - 34 T_{7} + 4 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T )^{7} \)
$3$ \( 1 + 2 T + 10 T^{2} + 19 T^{3} + 49 T^{4} + 75 T^{5} + 166 T^{6} + 219 T^{7} + 498 T^{8} + 675 T^{9} + 1323 T^{10} + 1539 T^{11} + 2430 T^{12} + 1458 T^{13} + 2187 T^{14} \)
$5$ \( ( 1 - T )^{7} \)
$7$ \( 1 + 3 T + 36 T^{2} + 99 T^{3} + 627 T^{4} + 1494 T^{5} + 6714 T^{6} + 13276 T^{7} + 46998 T^{8} + 73206 T^{9} + 215061 T^{10} + 237699 T^{11} + 605052 T^{12} + 352947 T^{13} + 823543 T^{14} \)
$11$ \( ( 1 + T )^{7} \)
$13$ \( 1 + 3 T + 51 T^{2} + 213 T^{3} + 1467 T^{4} + 5628 T^{5} + 29676 T^{6} + 87854 T^{7} + 385788 T^{8} + 951132 T^{9} + 3222999 T^{10} + 6083493 T^{11} + 18935943 T^{12} + 14480427 T^{13} + 62748517 T^{14} \)
$17$ \( 1 + 11 T + 118 T^{2} + 933 T^{3} + 6131 T^{4} + 35038 T^{5} + 176338 T^{6} + 760152 T^{7} + 2997746 T^{8} + 10125982 T^{9} + 30121603 T^{10} + 77925093 T^{11} + 167543126 T^{12} + 265513259 T^{13} + 410338673 T^{14} \)
$19$ \( 1 + 8 T + 81 T^{2} + 417 T^{3} + 2629 T^{4} + 11788 T^{5} + 64733 T^{6} + 263970 T^{7} + 1229927 T^{8} + 4255468 T^{9} + 18032311 T^{10} + 54343857 T^{11} + 200564019 T^{12} + 376367048 T^{13} + 893871739 T^{14} \)
$23$ \( 1 - 8 T + 32 T^{2} - 168 T^{3} + 1683 T^{4} - 10398 T^{5} + 42460 T^{6} - 173612 T^{7} + 976580 T^{8} - 5500542 T^{9} + 20477061 T^{10} - 47013288 T^{11} + 205962976 T^{12} - 1184287112 T^{13} + 3404825447 T^{14} \)
$29$ \( 1 + 9 T + 156 T^{2} + 947 T^{3} + 9101 T^{4} + 40136 T^{5} + 306396 T^{6} + 1174192 T^{7} + 8885484 T^{8} + 33754376 T^{9} + 221964289 T^{10} + 669795107 T^{11} + 3199739244 T^{12} + 5353409889 T^{13} + 17249876309 T^{14} \)
$31$ \( 1 + 9 T + 153 T^{2} + 1104 T^{3} + 11517 T^{4} + 68331 T^{5} + 537021 T^{6} + 2644424 T^{7} + 16647651 T^{8} + 65666091 T^{9} + 343102947 T^{10} + 1019567184 T^{11} + 4380260103 T^{12} + 7987533129 T^{13} + 27512614111 T^{14} \)
$37$ \( 1 - 21 T + 286 T^{2} - 3047 T^{3} + 27561 T^{4} - 218540 T^{5} + 1562772 T^{6} - 10003200 T^{7} + 57822564 T^{8} - 299181260 T^{9} + 1396047333 T^{10} - 5710568567 T^{11} + 19832371702 T^{12} - 53880254589 T^{13} + 94931877133 T^{14} \)
$41$ \( 1 + 10 T + 125 T^{2} + 1320 T^{3} + 9455 T^{4} + 74846 T^{5} + 537587 T^{6} + 3074976 T^{7} + 22041067 T^{8} + 125816126 T^{9} + 651648055 T^{10} + 3730004520 T^{11} + 14482025125 T^{12} + 47501042410 T^{13} + 194754273881 T^{14} \)
$43$ \( 1 + 5 T + 140 T^{2} + 606 T^{3} + 10010 T^{4} + 32755 T^{5} + 502547 T^{6} + 1331780 T^{7} + 21609521 T^{8} + 60563995 T^{9} + 795865070 T^{10} + 2071793406 T^{11} + 20581182020 T^{12} + 31606815245 T^{13} + 271818611107 T^{14} \)
$47$ \( 1 - 19 T + 344 T^{2} - 3976 T^{3} + 44346 T^{4} - 384357 T^{5} + 3243187 T^{6} - 22473760 T^{7} + 152429789 T^{8} - 849044613 T^{9} + 4604134758 T^{10} - 19401611656 T^{11} + 78894682408 T^{12} - 204805091251 T^{13} + 506623120463 T^{14} \)
$53$ \( 1 + 13 T + 291 T^{2} + 3246 T^{3} + 39721 T^{4} + 377495 T^{5} + 3259579 T^{6} + 25546588 T^{7} + 172757687 T^{8} + 1060383455 T^{9} + 5913543317 T^{10} + 25612501326 T^{11} + 121694888463 T^{12} + 288136694677 T^{13} + 1174711139837 T^{14} \)
$59$ \( 1 - 5 T + 216 T^{2} - 762 T^{3} + 21288 T^{4} - 45067 T^{5} + 1404017 T^{6} - 1955428 T^{7} + 82837003 T^{8} - 156878227 T^{9} + 4372108152 T^{10} - 9233429082 T^{11} + 154423648584 T^{12} - 210902668205 T^{13} + 2488651484819 T^{14} \)
$61$ \( 1 + 12 T + 439 T^{2} + 4147 T^{3} + 81185 T^{4} + 610212 T^{5} + 8278983 T^{6} + 49031670 T^{7} + 505017963 T^{8} + 2270598852 T^{9} + 18427452485 T^{10} + 57418702627 T^{11} + 370777776139 T^{12} + 618244492332 T^{13} + 3142742836021 T^{14} \)
$67$ \( 1 + 23 T + 641 T^{2} + 9711 T^{3} + 151955 T^{4} + 1671970 T^{5} + 18259586 T^{6} + 150693512 T^{7} + 1223392262 T^{8} + 7505473330 T^{9} + 45702441665 T^{10} + 195687536031 T^{11} + 865430193587 T^{12} + 2080542789887 T^{13} + 6060711605323 T^{14} \)
$71$ \( 1 - 22 T + 384 T^{2} - 5255 T^{3} + 66887 T^{4} - 702007 T^{5} + 7071852 T^{6} - 61995037 T^{7} + 502101492 T^{8} - 3538817287 T^{9} + 23939593057 T^{10} - 133538383655 T^{11} + 692824070784 T^{12} - 2818206246262 T^{13} + 9095120158391 T^{14} \)
$73$ \( ( 1 - T )^{7} \)
$79$ \( 1 + 32 T + 889 T^{2} + 15800 T^{3} + 252693 T^{4} + 3128704 T^{5} + 35529917 T^{6} + 328848336 T^{7} + 2806863443 T^{8} + 19526241664 T^{9} + 124587504027 T^{10} + 615411279800 T^{11} + 2735503138711 T^{12} + 7778798576672 T^{13} + 19203908986159 T^{14} \)
$83$ \( 1 - T + 327 T^{2} + 441 T^{3} + 52315 T^{4} + 116476 T^{5} + 5958442 T^{6} + 11988110 T^{7} + 494550686 T^{8} + 802403164 T^{9} + 29913036905 T^{10} + 20929119561 T^{11} + 1288066290261 T^{12} - 326940373369 T^{13} + 27136050989627 T^{14} \)
$89$ \( 1 - 20 T + 418 T^{2} - 6459 T^{3} + 95339 T^{4} - 1113667 T^{5} + 12736072 T^{6} - 124045027 T^{7} + 1133510408 T^{8} - 8821356307 T^{9} + 67211039491 T^{10} - 405252134619 T^{11} + 2334136849682 T^{12} - 9939625819220 T^{13} + 44231334895529 T^{14} \)
$97$ \( 1 + 2 T + 288 T^{2} - 1206 T^{3} + 34063 T^{4} - 476966 T^{5} + 2341844 T^{6} - 67629788 T^{7} + 227158868 T^{8} - 4487773094 T^{9} + 31088380399 T^{10} - 106766312886 T^{11} + 2473153994016 T^{12} + 1665944009858 T^{13} + 80798284478113 T^{14} \)
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