Properties

Label 6018.2.a.t
Level 6018
Weight 2
Character orbit 6018.a
Self dual yes
Analytic conductor 48.054
Analytic rank 1
Dimension 8
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 6018 = 2 \cdot 3 \cdot 17 \cdot 59 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 6018.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(48.0539719364\)
Analytic rank: \(1\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - 2 x^{7} - 15 x^{6} + 14 x^{5} + 84 x^{4} + 9 x^{3} - 158 x^{2} - 142 x - 35\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - \nu - 4 \)
\(\beta_{3}\)\(=\)\( \nu^{7} - 4 \nu^{6} - 8 \nu^{5} + 33 \nu^{4} + 27 \nu^{3} - 66 \nu^{2} - 53 \nu - 6 \)
\(\beta_{4}\)\(=\)\( 3 \nu^{7} - 11 \nu^{6} - 27 \nu^{5} + 89 \nu^{4} + 105 \nu^{3} - 166 \nu^{2} - 200 \nu - 50 \)
\(\beta_{5}\)\(=\)\( 4 \nu^{7} - 14 \nu^{6} - 39 \nu^{5} + 115 \nu^{4} + 163 \nu^{3} - 214 \nu^{2} - 311 \nu - 87 \)
\(\beta_{6}\)\(=\)\( -8 \nu^{7} + 28 \nu^{6} + 78 \nu^{5} - 229 \nu^{4} - 328 \nu^{3} + 420 \nu^{2} + 630 \nu + 188 \)
\(\beta_{7}\)\(=\)\( -8 \nu^{7} + 28 \nu^{6} + 78 \nu^{5} - 229 \nu^{4} - 329 \nu^{3} + 422 \nu^{2} + 635 \nu + 185 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + \beta_{1} + 4\)
\(\nu^{3}\)\(=\)\(-\beta_{7} + \beta_{6} + 2 \beta_{2} + 7 \beta_{1} + 5\)
\(\nu^{4}\)\(=\)\(-2 \beta_{7} + 3 \beta_{6} + 2 \beta_{5} + 12 \beta_{2} + 14 \beta_{1} + 28\)
\(\nu^{5}\)\(=\)\(-13 \beta_{7} + 16 \beta_{6} + 5 \beta_{5} + 2 \beta_{4} - 2 \beta_{3} + 36 \beta_{2} + 60 \beta_{1} + 64\)
\(\nu^{6}\)\(=\)\(-35 \beta_{7} + 54 \beta_{6} + 35 \beta_{5} + 7 \beta_{4} - 9 \beta_{3} + 148 \beta_{2} + 161 \beta_{1} + 256\)
\(\nu^{7}\)\(=\)\(-151 \beta_{7} + 218 \beta_{6} + 114 \beta_{5} + 44 \beta_{4} - 51 \beta_{3} + 496 \beta_{2} + 592 \beta_{1} + 747\)

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
2.12828
−1.50609
−0.502246
3.51784
−1.48295
2.70183
−2.13380
−0.722864
−1.00000 1.00000 1.00000 −3.78816 −1.00000 0.659879 −1.00000 1.00000 3.78816
1.2 −1.00000 1.00000 1.00000 −3.00979 −1.00000 3.51588 −1.00000 1.00000 3.00979
1.3 −1.00000 1.00000 1.00000 −2.59811 −1.00000 2.10035 −1.00000 1.00000 2.59811
1.4 −1.00000 1.00000 1.00000 −1.98195 −1.00000 −2.53589 −1.00000 1.00000 1.98195
1.5 −1.00000 1.00000 1.00000 0.137387 −1.00000 0.345564 −1.00000 1.00000 −0.137387
1.6 −1.00000 1.00000 1.00000 0.876977 −1.00000 −4.57880 −1.00000 1.00000 −0.876977
1.7 −1.00000 1.00000 1.00000 1.28626 −1.00000 −0.152464 −1.00000 1.00000 −1.28626
1.8 −1.00000 1.00000 1.00000 3.07738 −1.00000 −3.35452 −1.00000 1.00000 −3.07738
\(n\): 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 6018.2.a.t 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6018.2.a.t 8 1.a even 1 1 trivial

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(3\) \(-1\)
\(17\) \(-1\)
\(59\) \(1\)

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6018))\):

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

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T )^{8} \)
$3$ \( ( 1 - T )^{8} \)
$5$ \( 1 + 6 T + 35 T^{2} + 133 T^{3} + 494 T^{4} + 1440 T^{5} + 4131 T^{6} + 10000 T^{7} + 24138 T^{8} + 50000 T^{9} + 103275 T^{10} + 180000 T^{11} + 308750 T^{12} + 415625 T^{13} + 546875 T^{14} + 468750 T^{15} + 390625 T^{16} \)
$7$ \( 1 + 4 T + 36 T^{2} + 127 T^{3} + 645 T^{4} + 1945 T^{5} + 7427 T^{6} + 19357 T^{7} + 60672 T^{8} + 135499 T^{9} + 363923 T^{10} + 667135 T^{11} + 1548645 T^{12} + 2134489 T^{13} + 4235364 T^{14} + 3294172 T^{15} + 5764801 T^{16} \)
$11$ \( 1 - T + 28 T^{2} - 67 T^{3} + 592 T^{4} - 1536 T^{5} + 8997 T^{6} - 24749 T^{7} + 110558 T^{8} - 272239 T^{9} + 1088637 T^{10} - 2044416 T^{11} + 8667472 T^{12} - 10790417 T^{13} + 49603708 T^{14} - 19487171 T^{15} + 214358881 T^{16} \)
$13$ \( 1 + 2 T + 53 T^{2} + 96 T^{3} + 1523 T^{4} + 2314 T^{5} + 29187 T^{6} + 38268 T^{7} + 427896 T^{8} + 497484 T^{9} + 4932603 T^{10} + 5083858 T^{11} + 43498403 T^{12} + 35644128 T^{13} + 255820877 T^{14} + 125497034 T^{15} + 815730721 T^{16} \)
$17$ \( ( 1 - T )^{8} \)
$19$ \( 1 - 4 T + 108 T^{2} - 298 T^{3} + 5251 T^{4} - 10444 T^{5} + 161108 T^{6} - 249190 T^{7} + 3550216 T^{8} - 4734610 T^{9} + 58159988 T^{10} - 71635396 T^{11} + 684315571 T^{12} - 737877502 T^{13} + 5080955148 T^{14} - 3575486956 T^{15} + 16983563041 T^{16} \)
$23$ \( 1 + 11 T + 100 T^{2} + 489 T^{3} + 2444 T^{4} + 10110 T^{5} + 71157 T^{6} + 433905 T^{7} + 2508440 T^{8} + 9979815 T^{9} + 37642053 T^{10} + 123008370 T^{11} + 683931404 T^{12} + 3147371727 T^{13} + 14803588900 T^{14} + 37453079917 T^{15} + 78310985281 T^{16} \)
$29$ \( 1 + 12 T + 179 T^{2} + 1423 T^{3} + 12700 T^{4} + 76480 T^{5} + 525951 T^{6} + 2682170 T^{7} + 16473666 T^{8} + 77782930 T^{9} + 442324791 T^{10} + 1865270720 T^{11} + 8982468700 T^{12} + 29187365027 T^{13} + 106473374459 T^{14} + 206998515708 T^{15} + 500246412961 T^{16} \)
$31$ \( 1 + 9 T + 176 T^{2} + 1501 T^{3} + 16335 T^{4} + 115369 T^{5} + 934532 T^{6} + 5484469 T^{7} + 35127160 T^{8} + 170018539 T^{9} + 898085252 T^{10} + 3436957879 T^{11} + 15085715535 T^{12} + 42972355651 T^{13} + 156200647856 T^{14} + 247613526999 T^{15} + 852891037441 T^{16} \)
$37$ \( 1 + 22 T + 391 T^{2} + 5141 T^{3} + 56638 T^{4} + 533086 T^{5} + 4392001 T^{6} + 31785128 T^{7} + 205785776 T^{8} + 1176049736 T^{9} + 6012649369 T^{10} + 27002405158 T^{11} + 106148730718 T^{12} + 356497282937 T^{13} + 1003199025919 T^{14} + 2088501296926 T^{15} + 3512479453921 T^{16} \)
$41$ \( 1 + 19 T + 260 T^{2} + 2775 T^{3} + 26784 T^{4} + 221780 T^{5} + 1723117 T^{6} + 12215681 T^{7} + 82121514 T^{8} + 500842921 T^{9} + 2896559677 T^{10} + 15285299380 T^{11} + 75685182624 T^{12} + 321500957775 T^{13} + 1235027102660 T^{14} + 3700331203739 T^{15} + 7984925229121 T^{16} \)
$43$ \( 1 + 5 T + 168 T^{2} + 557 T^{3} + 11432 T^{4} + 20412 T^{5} + 451285 T^{6} + 228877 T^{7} + 16424272 T^{8} + 9841711 T^{9} + 834425965 T^{10} + 1622896884 T^{11} + 39083733032 T^{12} + 81883702751 T^{13} + 1061988992232 T^{14} + 1359093055535 T^{15} + 11688200277601 T^{16} \)
$47$ \( 1 + 26 T + 492 T^{2} + 6323 T^{3} + 68189 T^{4} + 596113 T^{5} + 4739424 T^{6} + 33725672 T^{7} + 237394796 T^{8} + 1585106584 T^{9} + 10469387616 T^{10} + 61890239999 T^{11} + 332740567709 T^{12} + 1450148479261 T^{13} + 5303373941868 T^{14} + 13172201132038 T^{15} + 23811286661761 T^{16} \)
$53$ \( 1 + 21 T + 455 T^{2} + 6256 T^{3} + 82279 T^{4} + 859472 T^{5} + 8462593 T^{6} + 70691735 T^{7} + 554252848 T^{8} + 3746661955 T^{9} + 23771423737 T^{10} + 127955612944 T^{11} + 649220886199 T^{12} + 2616231004208 T^{13} + 10084784313695 T^{14} + 24668933936577 T^{15} + 62259690411361 T^{16} \)
$59$ \( ( 1 + T )^{8} \)
$61$ \( 1 - 9 T + 300 T^{2} - 1619 T^{3} + 39328 T^{4} - 155036 T^{5} + 3707793 T^{6} - 13147211 T^{7} + 268087570 T^{8} - 801979871 T^{9} + 13796697753 T^{10} - 35190226316 T^{11} + 544529234848 T^{12} - 1367401411319 T^{13} + 15456112308300 T^{14} - 28284685524189 T^{15} + 191707312997281 T^{16} \)
$67$ \( 1 - 26 T + 514 T^{2} - 7659 T^{3} + 97111 T^{4} - 1045991 T^{5} + 10331237 T^{6} - 92351405 T^{7} + 781128642 T^{8} - 6187544135 T^{9} + 46376922893 T^{10} - 314595391133 T^{11} + 1956895511431 T^{12} - 10340608194513 T^{13} + 46495608434866 T^{14} - 157578501738398 T^{15} + 406067677556641 T^{16} \)
$71$ \( 1 + 14 T + 306 T^{2} + 2819 T^{3} + 43863 T^{4} + 356263 T^{5} + 4613774 T^{6} + 32461330 T^{7} + 363915232 T^{8} + 2304754430 T^{9} + 23258034734 T^{10} + 127510446593 T^{11} + 1114632563703 T^{12} + 5086122540469 T^{13} + 39198686879826 T^{14} + 127331682217474 T^{15} + 645753531245761 T^{16} \)
$73$ \( 1 - 17 T + 546 T^{2} - 7255 T^{3} + 130006 T^{4} - 1408330 T^{5} + 18097581 T^{6} - 161460869 T^{7} + 1624660828 T^{8} - 11786643437 T^{9} + 96442009149 T^{10} - 547864311610 T^{11} + 3691941719446 T^{12} - 15040134407215 T^{13} + 82628487553794 T^{14} - 187805774824649 T^{15} + 806460091894081 T^{16} \)
$79$ \( 1 + 39 T + 710 T^{2} + 7712 T^{3} + 55699 T^{4} + 247027 T^{5} - 615726 T^{6} - 31339559 T^{7} - 384243014 T^{8} - 2475825161 T^{9} - 3842745966 T^{10} + 121793945053 T^{11} + 2169480561619 T^{12} + 23730258949088 T^{13} + 172592093419910 T^{14} + 748952450460201 T^{15} + 1517108809906561 T^{16} \)
$83$ \( 1 + 11 T + 588 T^{2} + 5647 T^{3} + 156887 T^{4} + 1294249 T^{5} + 24761240 T^{6} + 171771317 T^{7} + 2526281208 T^{8} + 14257019311 T^{9} + 170580182360 T^{10} + 740034752963 T^{11} + 7445593606727 T^{12} + 22243762511021 T^{13} + 192240939540972 T^{14} + 298496560885897 T^{15} + 2252292232139041 T^{16} \)
$89$ \( 1 + 469 T^{2} + 917 T^{3} + 105570 T^{4} + 309532 T^{5} + 15627273 T^{6} + 46220456 T^{7} + 1644960196 T^{8} + 4113620584 T^{9} + 123783629433 T^{10} + 218210464508 T^{11} + 6623698382370 T^{12} + 5120582514733 T^{13} + 233084225460709 T^{14} + 3936588805702081 T^{16} \)
$97$ \( 1 - 16 T + 605 T^{2} - 8438 T^{3} + 172985 T^{4} - 2098320 T^{5} + 30307115 T^{6} - 314316010 T^{7} + 3553639420 T^{8} - 30488652970 T^{9} + 285159645035 T^{10} - 1915080009360 T^{11} + 15314237673785 T^{12} - 72459977088566 T^{13} + 503948062982045 T^{14} - 1292772551649808 T^{15} + 7837433594376961 T^{16} \)
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