Properties

Label 8034.2.a.o
Level 8034
Weight 2
Character orbit 8034.a
Self dual yes
Analytic conductor 64.152
Analytic rank 1
Dimension 7
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: \(7\)
Coefficient field: \(\mathbb{Q}[x]/(x^{7} - \cdots)\)
Defining polynomial: \(x^{7} - 3 x^{6} - 4 x^{5} + 14 x^{4} + 3 x^{3} - 12 x^{2} - 3 x + 1\)
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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{7} - 3 x^{6} - 4 x^{5} + 14 x^{4} + 3 x^{3} - 12 x^{2} - 3 x + 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{4} - \nu^{3} - 5 \nu^{2} + 3 \nu + 3 \)
\(\beta_{3}\)\(=\)\( -\nu^{6} + 3 \nu^{5} + 3 \nu^{4} - 12 \nu^{3} + \nu^{2} + 6 \nu \)
\(\beta_{4}\)\(=\)\( -\nu^{6} + 3 \nu^{5} + 3 \nu^{4} - 12 \nu^{3} + 2 \nu^{2} + 5 \nu - 2 \)
\(\beta_{5}\)\(=\)\( \nu^{6} - 3 \nu^{5} - 3 \nu^{4} + 13 \nu^{3} - 2 \nu^{2} - 9 \nu \)
\(\beta_{6}\)\(=\)\( \nu^{6} - 4 \nu^{5} - \nu^{4} + 17 \nu^{3} - 9 \nu^{2} - 10 \nu + 2 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{4} - \beta_{3} + \beta_{1} + 2\)
\(\nu^{3}\)\(=\)\(\beta_{5} + \beta_{4} + 4 \beta_{1} + 2\)
\(\nu^{4}\)\(=\)\(\beta_{5} + 6 \beta_{4} - 5 \beta_{3} + \beta_{2} + 6 \beta_{1} + 9\)
\(\nu^{5}\)\(=\)\(-\beta_{6} + 7 \beta_{5} + 9 \beta_{4} - 3 \beta_{3} + 2 \beta_{2} + 20 \beta_{1} + 14\)
\(\nu^{6}\)\(=\)\(-3 \beta_{6} + 12 \beta_{5} + 34 \beta_{4} - 26 \beta_{3} + 9 \beta_{2} + 37 \beta_{1} + 47\)

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
−0.519850
2.16681
1.27539
−1.86678
2.51101
−0.761570
0.194986
−1.00000 −1.00000 1.00000 −2.94146 1.00000 −3.14882 −1.00000 1.00000 2.94146
1.2 −1.00000 −1.00000 1.00000 −2.61702 1.00000 −0.977840 −1.00000 1.00000 2.61702
1.3 −1.00000 −1.00000 1.00000 −2.55242 1.00000 1.37822 −1.00000 1.00000 2.55242
1.4 −1.00000 −1.00000 1.00000 1.72171 1.00000 2.39003 −1.00000 1.00000 −1.72171
1.5 −1.00000 −1.00000 1.00000 1.77686 1.00000 −3.99413 −1.00000 1.00000 −1.77686
1.6 −1.00000 −1.00000 1.00000 3.28276 1.00000 −3.26302 −1.00000 1.00000 −3.28276
1.7 −1.00000 −1.00000 1.00000 3.32958 1.00000 −1.38443 −1.00000 1.00000 −3.32958
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.7
Significant digits:
Format:

Atkin-Lehner signs

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

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 8034.2.a.o 7
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8034.2.a.o 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8034))\):

\( T_{5}^{7} - 2 T_{5}^{6} - 23 T_{5}^{5} + 41 T_{5}^{4} + 173 T_{5}^{3} - 279 T_{5}^{2} - 417 T_{5} + 657 \)
\( T_{7}^{7} + 9 T_{7}^{6} + 17 T_{7}^{5} - 51 T_{7}^{4} - 178 T_{7}^{3} - 32 T_{7}^{2} + 270 T_{7} + 183 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T )^{7} \)
$3$ \( ( 1 + T )^{7} \)
$5$ \( 1 - 2 T + 12 T^{2} - 19 T^{3} + 123 T^{4} - 209 T^{5} + 803 T^{6} - 983 T^{7} + 4015 T^{8} - 5225 T^{9} + 15375 T^{10} - 11875 T^{11} + 37500 T^{12} - 31250 T^{13} + 78125 T^{14} \)
$7$ \( 1 + 9 T + 66 T^{2} + 327 T^{3} + 1446 T^{4} + 5155 T^{5} + 16867 T^{6} + 46481 T^{7} + 118069 T^{8} + 252595 T^{9} + 495978 T^{10} + 785127 T^{11} + 1109262 T^{12} + 1058841 T^{13} + 823543 T^{14} \)
$11$ \( 1 + 50 T^{2} - 5 T^{3} + 1221 T^{4} - 112 T^{5} + 19364 T^{6} - 1257 T^{7} + 213004 T^{8} - 13552 T^{9} + 1625151 T^{10} - 73205 T^{11} + 8052550 T^{12} + 19487171 T^{14} \)
$13$ \( ( 1 - T )^{7} \)
$17$ \( 1 - 3 T + 48 T^{2} - 210 T^{3} + 1728 T^{4} - 6112 T^{5} + 41181 T^{6} - 130549 T^{7} + 700077 T^{8} - 1766368 T^{9} + 8489664 T^{10} - 17539410 T^{11} + 68153136 T^{12} - 72412707 T^{13} + 410338673 T^{14} \)
$19$ \( 1 + 16 T + 225 T^{2} + 2028 T^{3} + 16324 T^{4} + 101598 T^{5} + 571970 T^{6} + 2616215 T^{7} + 10867430 T^{8} + 36676878 T^{9} + 111966316 T^{10} + 264290988 T^{11} + 557122275 T^{12} + 752734096 T^{13} + 893871739 T^{14} \)
$23$ \( 1 - 6 T + 124 T^{2} - 579 T^{3} + 6515 T^{4} - 24568 T^{5} + 206736 T^{6} - 663561 T^{7} + 4754928 T^{8} - 12996472 T^{9} + 79268005 T^{10} - 162027939 T^{11} + 798106532 T^{12} - 888215334 T^{13} + 3404825447 T^{14} \)
$29$ \( 1 + 5 T + 102 T^{2} + 612 T^{3} + 5141 T^{4} + 30066 T^{5} + 189821 T^{6} + 958991 T^{7} + 5504809 T^{8} + 25285506 T^{9} + 125383849 T^{10} + 432855972 T^{11} + 2092137198 T^{12} + 2974116605 T^{13} + 17249876309 T^{14} \)
$31$ \( 1 + 16 T + 225 T^{2} + 2052 T^{3} + 18352 T^{4} + 128552 T^{5} + 886488 T^{6} + 4975541 T^{7} + 27481128 T^{8} + 123538472 T^{9} + 546724432 T^{10} + 1895065092 T^{11} + 6441558975 T^{12} + 14200058896 T^{13} + 27512614111 T^{14} \)
$37$ \( 1 - 17 T + 306 T^{2} - 3124 T^{3} + 32871 T^{4} - 245430 T^{5} + 1913427 T^{6} - 11334611 T^{7} + 70796799 T^{8} - 335993670 T^{9} + 1665014763 T^{10} - 5854878964 T^{11} + 21219250842 T^{12} - 43617348953 T^{13} + 94931877133 T^{14} \)
$41$ \( 1 - 12 T + 164 T^{2} - 964 T^{3} + 8573 T^{4} - 47076 T^{5} + 482781 T^{6} - 2693347 T^{7} + 19794021 T^{8} - 79134756 T^{9} + 590859733 T^{10} - 2724033604 T^{11} + 19000416964 T^{12} - 57001250892 T^{13} + 194754273881 T^{14} \)
$43$ \( 1 + 22 T + 325 T^{2} + 3220 T^{3} + 24121 T^{4} + 136680 T^{5} + 650694 T^{6} + 3418661 T^{7} + 27979842 T^{8} + 252721320 T^{9} + 1917788347 T^{10} + 11008539220 T^{11} + 47777743975 T^{12} + 139069987078 T^{13} + 271818611107 T^{14} \)
$47$ \( 1 + 131 T^{2} - 16 T^{3} + 11688 T^{4} + 6954 T^{5} + 704218 T^{6} + 308813 T^{7} + 33098246 T^{8} + 15361386 T^{9} + 1213483224 T^{10} - 78074896 T^{11} + 30044195917 T^{12} + 506623120463 T^{14} \)
$53$ \( 1 - 2 T + 137 T^{2} + 393 T^{3} + 9678 T^{4} + 48589 T^{5} + 742820 T^{6} + 2496473 T^{7} + 39369460 T^{8} + 136486501 T^{9} + 1440831606 T^{10} + 3100959033 T^{11} + 57292782541 T^{12} - 44328722258 T^{13} + 1174711139837 T^{14} \)
$59$ \( 1 + 3 T + 210 T^{2} + 539 T^{3} + 25292 T^{4} + 53419 T^{5} + 2065249 T^{6} + 3896781 T^{7} + 121849691 T^{8} + 185951539 T^{9} + 5194445668 T^{10} + 6531257579 T^{11} + 150134102790 T^{12} + 126541600923 T^{13} + 2488651484819 T^{14} \)
$61$ \( 1 + 6 T + 279 T^{2} + 1912 T^{3} + 39211 T^{4} + 258784 T^{5} + 3552046 T^{6} + 20060339 T^{7} + 216674806 T^{8} + 962935264 T^{9} + 8900151991 T^{10} + 26473247992 T^{11} + 235642367979 T^{12} + 309122246166 T^{13} + 3142742836021 T^{14} \)
$67$ \( 1 - T + 211 T^{2} + 134 T^{3} + 21876 T^{4} + 41069 T^{5} + 1677249 T^{6} + 3917111 T^{7} + 112375683 T^{8} + 184358741 T^{9} + 6579491388 T^{10} + 2700250214 T^{11} + 284876397577 T^{12} - 90458382169 T^{13} + 6060711605323 T^{14} \)
$71$ \( 1 - 15 T + 452 T^{2} - 4851 T^{3} + 85045 T^{4} - 716014 T^{5} + 9325472 T^{6} - 63589473 T^{7} + 662108512 T^{8} - 3609426574 T^{9} + 30438540995 T^{10} - 123272064531 T^{11} + 815511666652 T^{12} - 1921504258815 T^{13} + 9095120158391 T^{14} \)
$73$ \( 1 - 17 T + 530 T^{2} - 6819 T^{3} + 117094 T^{4} - 1175660 T^{5} + 14247390 T^{6} - 112197599 T^{7} + 1040059470 T^{8} - 6265092140 T^{9} + 45551556598 T^{10} - 193647605379 T^{11} + 1098727944290 T^{12} - 2572681846913 T^{13} + 11047398519097 T^{14} \)
$79$ \( 1 + 27 T + 744 T^{2} + 12364 T^{3} + 197835 T^{4} + 2372854 T^{5} + 27173877 T^{6} + 247348701 T^{7} + 2146736283 T^{8} + 14808981814 T^{9} + 97540370565 T^{10} + 481578801484 T^{11} + 2289329960856 T^{12} + 6563361299067 T^{13} + 19203908986159 T^{14} \)
$83$ \( 1 - 12 T + 446 T^{2} - 4743 T^{3} + 96239 T^{4} - 859806 T^{5} + 12423630 T^{6} - 91258057 T^{7} + 1031161290 T^{8} - 5923203534 T^{9} + 55028209093 T^{10} - 225094816503 T^{11} + 1756812126778 T^{12} - 3923284480428 T^{13} + 27136050989627 T^{14} \)
$89$ \( 1 + 9 T + 340 T^{2} + 1689 T^{3} + 41782 T^{4} - 2472 T^{5} + 2633172 T^{6} - 14512419 T^{7} + 234352308 T^{8} - 19580712 T^{9} + 29455014758 T^{10} + 105971645049 T^{11} + 1898580212660 T^{12} + 4472831618649 T^{13} + 44231334895529 T^{14} \)
$97$ \( 1 + 3 T + 307 T^{2} + 1782 T^{3} + 44866 T^{4} + 386761 T^{5} + 4801989 T^{6} + 46932733 T^{7} + 465792933 T^{8} + 3639034249 T^{9} + 40947986818 T^{10} + 157759178742 T^{11} + 2636313458899 T^{12} + 2498916014787 T^{13} + 80798284478113 T^{14} \)
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