Properties

Label 6018.2.a.s
Level $6018$
Weight $2$
Character orbit 6018.a
Self dual yes
Analytic conductor $48.054$
Analytic rank $0$
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: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Defining polynomial: \(x^{8} - 3 x^{7} - 17 x^{6} + 37 x^{5} + 105 x^{4} - 117 x^{3} - 238 x^{2} + 42 x + 90\)
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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - 3 x^{7} - 17 x^{6} + 37 x^{5} + 105 x^{4} - 117 x^{3} - 238 x^{2} + 42 x + 90\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( 23 \nu^{7} - 126 \nu^{6} - 82 \nu^{5} + 1079 \nu^{4} - 276 \nu^{3} - 2202 \nu^{2} + 184 \nu + 810 \)\()/30\)
\(\beta_{3}\)\(=\)\((\)\( 34 \nu^{7} - 183 \nu^{6} - 146 \nu^{5} + 1612 \nu^{4} - 213 \nu^{3} - 3456 \nu^{2} - 58 \nu + 1200 \)\()/30\)
\(\beta_{4}\)\(=\)\((\)\( 34 \nu^{7} - 183 \nu^{6} - 146 \nu^{5} + 1612 \nu^{4} - 213 \nu^{3} - 3486 \nu^{2} - 28 \nu + 1350 \)\()/30\)
\(\beta_{5}\)\(=\)\((\)\( 43 \nu^{7} - 231 \nu^{6} - 182 \nu^{5} + 2029 \nu^{4} - 321 \nu^{3} - 4362 \nu^{2} + 164 \nu + 1710 \)\()/30\)
\(\beta_{6}\)\(=\)\((\)\( -77 \nu^{7} + 414 \nu^{6} + 328 \nu^{5} - 3641 \nu^{4} + 549 \nu^{3} + 7803 \nu^{2} - 196 \nu - 2925 \)\()/15\)
\(\beta_{7}\)\(=\)\((\)\( 169 \nu^{7} - 903 \nu^{6} - 746 \nu^{5} + 7957 \nu^{4} - 993 \nu^{3} - 17106 \nu^{2} + 92 \nu + 6420 \)\()/30\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(-\beta_{4} + \beta_{3} + \beta_{1} + 5\)
\(\nu^{3}\)\(=\)\(\beta_{6} + 2 \beta_{5} - \beta_{4} + 3 \beta_{3} + 7 \beta_{1} + 6\)
\(\nu^{4}\)\(=\)\(-\beta_{7} + 2 \beta_{6} + 9 \beta_{5} - 12 \beta_{4} + 16 \beta_{3} - 2 \beta_{2} + 12 \beta_{1} + 45\)
\(\nu^{5}\)\(=\)\(-2 \beta_{7} + 17 \beta_{6} + 49 \beta_{5} - 31 \beta_{4} + 58 \beta_{3} - 3 \beta_{2} + 62 \beta_{1} + 106\)
\(\nu^{6}\)\(=\)\(-10 \beta_{7} + 59 \beta_{6} + 206 \beta_{5} - 175 \beta_{4} + 249 \beta_{3} - 26 \beta_{2} + 153 \beta_{1} + 520\)
\(\nu^{7}\)\(=\)\(-15 \beta_{7} + 302 \beta_{6} + 905 \beta_{5} - 614 \beta_{4} + 952 \beta_{3} - 58 \beta_{2} + 668 \beta_{1} + 1631\)

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
−2.36472
2.14759
−2.36258
3.92135
0.672052
3.02133
−1.32287
−0.712152
−1.00000 −1.00000 1.00000 −3.85158 1.00000 4.10228 −1.00000 1.00000 3.85158
1.2 −1.00000 −1.00000 1.00000 −2.25098 1.00000 0.744783 −1.00000 1.00000 2.25098
1.3 −1.00000 −1.00000 1.00000 −1.89711 1.00000 1.64158 −1.00000 1.00000 1.89711
1.4 −1.00000 −1.00000 1.00000 −1.03120 1.00000 −2.85327 −1.00000 1.00000 1.03120
1.5 −1.00000 −1.00000 1.00000 −0.462452 1.00000 −3.09978 −1.00000 1.00000 0.462452
1.6 −1.00000 −1.00000 1.00000 2.51738 1.00000 4.42797 −1.00000 1.00000 −2.51738
1.7 −1.00000 −1.00000 1.00000 2.86769 1.00000 0.636539 −1.00000 1.00000 −2.86769
1.8 −1.00000 −1.00000 1.00000 3.10826 1.00000 0.399898 −1.00000 1.00000 −3.10826
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.8
Significant digits:
Format:

Atkin-Lehner signs

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

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 6018.2.a.s 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6018.2.a.s 8 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6018))\):

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T )^{8} \)
$3$ \( ( 1 + T )^{8} \)
$5$ \( 1 + T + 16 T^{2} + 13 T^{3} + 155 T^{4} + 169 T^{5} + 1120 T^{6} + 1205 T^{7} + 6024 T^{8} + 6025 T^{9} + 28000 T^{10} + 21125 T^{11} + 96875 T^{12} + 40625 T^{13} + 250000 T^{14} + 78125 T^{15} + 390625 T^{16} \)
$7$ \( 1 - 6 T + 45 T^{2} - 192 T^{3} + 870 T^{4} - 2981 T^{5} + 10569 T^{6} - 30258 T^{7} + 88824 T^{8} - 211806 T^{9} + 517881 T^{10} - 1022483 T^{11} + 2088870 T^{12} - 3226944 T^{13} + 5294205 T^{14} - 4941258 T^{15} + 5764801 T^{16} \)
$11$ \( 1 + 40 T^{2} - 37 T^{3} + 887 T^{4} - 1413 T^{5} + 13552 T^{6} - 26460 T^{7} + 165104 T^{8} - 291060 T^{9} + 1639792 T^{10} - 1880703 T^{11} + 12986567 T^{12} - 5958887 T^{13} + 70862440 T^{14} + 214358881 T^{16} \)
$13$ \( 1 - 6 T + 89 T^{2} - 368 T^{3} + 3167 T^{4} - 9602 T^{5} + 64903 T^{6} - 156108 T^{7} + 950952 T^{8} - 2029404 T^{9} + 10968607 T^{10} - 21095594 T^{11} + 90452687 T^{12} - 136635824 T^{13} + 429586001 T^{14} - 376491102 T^{15} + 815730721 T^{16} \)
$17$ \( ( 1 + T )^{8} \)
$19$ \( 1 + 7 T + 81 T^{2} + 298 T^{3} + 2413 T^{4} + 6649 T^{5} + 65062 T^{6} + 201479 T^{7} + 1577194 T^{8} + 3828101 T^{9} + 23487382 T^{10} + 45605491 T^{11} + 314464573 T^{12} + 737877502 T^{13} + 3810716361 T^{14} + 6257102173 T^{15} + 16983563041 T^{16} \)
$23$ \( 1 + 5 T + 103 T^{2} + 429 T^{3} + 5130 T^{4} + 18025 T^{5} + 171911 T^{6} + 528498 T^{7} + 4431424 T^{8} + 12155454 T^{9} + 90940919 T^{10} + 219310175 T^{11} + 1435584330 T^{12} + 2761191147 T^{13} + 15247696567 T^{14} + 17024127235 T^{15} + 78310985281 T^{16} \)
$29$ \( 1 + 15 T + 234 T^{2} + 2369 T^{3} + 22509 T^{4} + 173855 T^{5} + 1246686 T^{6} + 7705949 T^{7} + 44401236 T^{8} + 223472521 T^{9} + 1048462926 T^{10} + 4240149595 T^{11} + 15920188029 T^{12} + 48590911981 T^{13} + 139188657114 T^{14} + 258748144635 T^{15} + 500246412961 T^{16} \)
$31$ \( 1 - 21 T + 356 T^{2} - 4191 T^{3} + 42775 T^{4} - 358287 T^{5} + 2698408 T^{6} - 17567237 T^{7} + 104455208 T^{8} - 544584347 T^{9} + 2593170088 T^{10} - 10673728017 T^{11} + 39503610775 T^{12} - 119984771841 T^{13} + 315951310436 T^{14} - 577764896331 T^{15} + 852891037441 T^{16} \)
$37$ \( 1 - 7 T + 226 T^{2} - 1661 T^{3} + 24233 T^{4} - 171303 T^{5} + 1611578 T^{6} - 10128697 T^{7} + 72053348 T^{8} - 374761789 T^{9} + 2206250282 T^{10} - 8677010859 T^{11} + 45416543513 T^{12} - 115180312577 T^{13} + 579854168434 T^{14} - 664523139931 T^{15} + 3512479453921 T^{16} \)
$41$ \( 1 + T + 173 T^{2} - 95 T^{3} + 12846 T^{4} - 36877 T^{5} + 583411 T^{6} - 3265996 T^{7} + 22849452 T^{8} - 133905836 T^{9} + 980713891 T^{10} - 2541599717 T^{11} + 36299725806 T^{12} - 11006339095 T^{13} + 821768033693 T^{14} + 194754273881 T^{15} + 7984925229121 T^{16} \)
$43$ \( 1 - 14 T + 378 T^{2} - 3989 T^{3} + 59851 T^{4} - 496697 T^{5} + 5279166 T^{6} - 34984358 T^{7} + 285945288 T^{8} - 1504327394 T^{9} + 9761177934 T^{10} - 39490888379 T^{11} + 204618658651 T^{12} - 586416679127 T^{13} + 2389475232522 T^{14} - 3805460555498 T^{15} + 11688200277601 T^{16} \)
$47$ \( 1 + 8 T + 174 T^{2} + 1355 T^{3} + 18659 T^{4} + 133669 T^{5} + 1408814 T^{6} + 8757326 T^{7} + 75891024 T^{8} + 411594322 T^{9} + 3112070126 T^{10} + 13877916587 T^{11} + 91049967779 T^{12} + 310762484485 T^{13} + 1875583467246 T^{14} + 4052984963704 T^{15} + 23811286661761 T^{16} \)
$53$ \( 1 - 8 T + 257 T^{2} - 2258 T^{3} + 34400 T^{4} - 280917 T^{5} + 3115751 T^{6} - 21346436 T^{7} + 197739264 T^{8} - 1131361108 T^{9} + 8752144559 T^{10} - 41822080209 T^{11} + 271432546400 T^{12} - 944285423194 T^{13} + 5696240810153 T^{14} - 9397689118696 T^{15} + 62259690411361 T^{16} \)
$59$ \( ( 1 - T )^{8} \)
$61$ \( 1 + 246 T^{2} + 81 T^{3} + 33189 T^{4} + 4535 T^{5} + 3088722 T^{6} - 6804 T^{7} + 215176596 T^{8} - 415044 T^{9} + 11493134562 T^{10} + 1029358835 T^{11} + 459529616949 T^{12} + 68412300381 T^{13} + 12674012092806 T^{14} + 191707312997281 T^{16} \)
$67$ \( 1 - 15 T + 353 T^{2} - 3872 T^{3} + 57887 T^{4} - 521506 T^{5} + 6166835 T^{6} - 47777769 T^{7} + 478658792 T^{8} - 3201110523 T^{9} + 27682922315 T^{10} - 156849709078 T^{11} + 1166487941327 T^{12} - 5227684414304 T^{13} + 31931808905657 T^{14} - 90910674079845 T^{15} + 406067677556641 T^{16} \)
$71$ \( 1 + 22 T + 514 T^{2} + 7845 T^{3} + 114881 T^{4} + 1373943 T^{5} + 15251750 T^{6} + 147989716 T^{7} + 1321412100 T^{8} + 10507269836 T^{9} + 76884071750 T^{10} + 491749313073 T^{11} + 2919319324961 T^{12} + 14154179258595 T^{13} + 65843545935394 T^{14} + 200092643484602 T^{15} + 645753531245761 T^{16} \)
$73$ \( 1 - 13 T + 299 T^{2} - 2613 T^{3} + 38508 T^{4} - 247425 T^{5} + 3210865 T^{6} - 16961992 T^{7} + 236132828 T^{8} - 1238225416 T^{9} + 17110699585 T^{10} - 96252531225 T^{11} + 1093559464428 T^{12} - 5416936072509 T^{13} + 45248933660411 T^{14} - 143616180748261 T^{15} + 806460091894081 T^{16} \)
$79$ \( 1 - 26 T + 640 T^{2} - 9656 T^{3} + 142699 T^{4} - 1596408 T^{5} + 18223156 T^{6} - 169570774 T^{7} + 1652607816 T^{8} - 13396091146 T^{9} + 113730716596 T^{10} - 787091403912 T^{11} + 5558137608619 T^{12} - 29712056588744 T^{13} + 155575971533440 T^{14} - 499301633640134 T^{15} + 1517108809906561 T^{16} \)
$83$ \( 1 - 30 T + 740 T^{2} - 12480 T^{3} + 194182 T^{4} - 2481911 T^{5} + 29776469 T^{6} - 306563522 T^{7} + 2989193580 T^{8} - 25444772326 T^{9} + 205130094941 T^{10} - 1419124444957 T^{11} + 9215551688422 T^{12} - 49159227224640 T^{13} + 241935876293060 T^{14} - 814081529688810 T^{15} + 2252292232139041 T^{16} \)
$89$ \( 1 + 6 T + 412 T^{2} + 2490 T^{3} + 80192 T^{4} + 540605 T^{5} + 10393669 T^{6} + 74826734 T^{7} + 1035585538 T^{8} + 6659579326 T^{9} + 82328252149 T^{10} + 381109766245 T^{11} + 5031425790272 T^{12} + 13904308028010 T^{13} + 204756291875932 T^{14} + 265388009373174 T^{15} + 3936588805702081 T^{16} \)
$97$ \( 1 - 23 T + 706 T^{2} - 10817 T^{3} + 195228 T^{4} - 2300379 T^{5} + 31448105 T^{6} - 308141370 T^{7} + 3535616038 T^{8} - 29889712890 T^{9} + 295895219945 T^{10} - 2099493803067 T^{11} + 17283394471068 T^{12} - 92889259559969 T^{13} + 588078235479874 T^{14} - 1858360542996599 T^{15} + 7837433594376961 T^{16} \)
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