Properties

Label 8001.2.a.p
Level 8001
Weight 2
Character orbit 8001.a
Self dual yes
Analytic conductor 63.888
Analytic rank 0
Dimension 14
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 8001 = 3^{2} \cdot 7 \cdot 127 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 8001.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(63.8883066572\)
Analytic rank: \(0\)
Dimension: \(14\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 5 \)
Twist minimal: no (minimal twist has level 2667)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{13}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{14} - 5 x^{13} - 9 x^{12} + 76 x^{11} - 12 x^{10} - 414 x^{9} + 331 x^{8} + 959 x^{7} - 1067 x^{6} - 875 x^{5} + 1134 x^{4} + 301 x^{3} - 418 x^{2} - 42 x + 44\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 3 \)
\(\beta_{3}\)\(=\)\((\)\( -2 \nu^{13} + 34 \nu^{12} + 21 \nu^{11} - 541 \nu^{10} - 60 \nu^{9} + 3055 \nu^{8} - 58 \nu^{7} - 7387 \nu^{6} + 495 \nu^{5} + 7866 \nu^{4} - 486 \nu^{3} - 5045 \nu^{2} + 274 \nu + 1454 \)\()/274\)
\(\beta_{4}\)\(=\)\((\)\( -12 \nu^{13} + 67 \nu^{12} + 126 \nu^{11} - 917 \nu^{10} - 223 \nu^{9} + 3945 \nu^{8} - 1581 \nu^{7} - 4044 \nu^{6} + 6669 \nu^{5} - 7878 \nu^{4} - 7437 \nu^{3} + 11241 \nu^{2} + 1918 \nu - 2510 \)\()/274\)
\(\beta_{5}\)\(=\)\((\)\( 20 \nu^{13} - 66 \nu^{12} - 347 \nu^{11} + 1163 \nu^{10} + 2244 \nu^{9} - 7671 \nu^{8} - 6544 \nu^{7} + 23317 \nu^{6} + 8065 \nu^{5} - 32354 \nu^{4} - 2812 \nu^{3} + 17433 \nu^{2} - 2210 \)\()/274\)
\(\beta_{6}\)\(=\)\((\)\( 13 \nu^{13} - 84 \nu^{12} - 68 \nu^{11} + 1256 \nu^{10} - 843 \nu^{9} - 6637 \nu^{8} + 7501 \nu^{7} + 14382 \nu^{6} - 18767 \nu^{5} - 10851 \nu^{4} + 13845 \nu^{3} + 2173 \nu^{2} - 2055 \nu - 135 \)\()/137\)
\(\beta_{7}\)\(=\)\((\)\( -67 \nu^{13} + 180 \nu^{12} + 909 \nu^{11} - 2711 \nu^{10} - 3791 \nu^{9} + 14731 \nu^{8} + 2852 \nu^{7} - 34361 \nu^{6} + 12130 \nu^{5} + 31707 \nu^{4} - 17651 \nu^{3} - 9882 \nu^{2} + 5480 \nu + 896 \)\()/274\)
\(\beta_{8}\)\(=\)\((\)\( -13 \nu^{13} - 53 \nu^{12} + 479 \nu^{11} + 662 \nu^{10} - 5322 \nu^{9} - 2268 \nu^{8} + 25379 \nu^{7} - 819 \nu^{6} - 54117 \nu^{5} + 13591 \nu^{4} + 44380 \nu^{3} - 12311 \nu^{2} - 9864 \nu + 2190 \)\()/274\)
\(\beta_{9}\)\(=\)\((\)\( -69 \nu^{13} + 214 \nu^{12} + 930 \nu^{11} - 3252 \nu^{10} - 3851 \nu^{9} + 17786 \nu^{8} + 2794 \nu^{7} - 41748 \nu^{6} + 12625 \nu^{5} + 39299 \nu^{4} - 17863 \nu^{3} - 13283 \nu^{2} + 4658 \nu + 1254 \)\()/274\)
\(\beta_{10}\)\(=\)\((\)\( 27 \nu^{13} - 48 \nu^{12} - 489 \nu^{11} + 796 \nu^{10} + 3413 \nu^{9} - 5006 \nu^{8} - 11410 \nu^{7} + 14853 \nu^{6} + 18183 \nu^{5} - 21114 \nu^{4} - 11386 \nu^{3} + 12965 \nu^{2} + 1370 \nu - 2230 \)\()/137\)
\(\beta_{11}\)\(=\)\((\)\( 40 \nu^{13} - 132 \nu^{12} - 557 \nu^{11} + 2052 \nu^{10} + 2570 \nu^{9} - 11643 \nu^{8} - 3909 \nu^{7} + 29235 \nu^{6} - 584 \nu^{5} - 31828 \nu^{4} + 2185 \nu^{3} + 14453 \nu^{2} + 411 \nu - 2091 \)\()/137\)
\(\beta_{12}\)\(=\)\((\)\( 51 \nu^{13} - 319 \nu^{12} - 193 \nu^{11} + 4548 \nu^{10} - 4498 \nu^{9} - 21938 \nu^{8} + 36551 \nu^{7} + 37737 \nu^{6} - 92973 \nu^{5} - 5769 \nu^{4} + 77194 \nu^{3} - 17463 \nu^{2} - 16166 \nu + 4160 \)\()/274\)
\(\beta_{13}\)\(=\)\((\)\( 120 \nu^{13} - 259 \nu^{12} - 1945 \nu^{11} + 3964 \nu^{10} + 11683 \nu^{9} - 21914 \nu^{8} - 32003 \nu^{7} + 52359 \nu^{6} + 40444 \nu^{5} - 51096 \nu^{4} - 23037 \nu^{3} + 18836 \nu^{2} + 4110 \nu - 2026 \)\()/274\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + 3\)
\(\nu^{3}\)\(=\)\(-\beta_{11} + \beta_{10} - \beta_{9} + \beta_{7} + \beta_{6} + \beta_{3} + \beta_{2} + 4 \beta_{1} + 1\)
\(\nu^{4}\)\(=\)\(-\beta_{11} + \beta_{10} - 2 \beta_{9} + 2 \beta_{7} + \beta_{6} + 2 \beta_{3} + 7 \beta_{2} + 15\)
\(\nu^{5}\)\(=\)\(\beta_{13} - \beta_{12} - 8 \beta_{11} + 8 \beta_{10} - 9 \beta_{9} + 2 \beta_{8} + 10 \beta_{7} + 9 \beta_{6} + 10 \beta_{3} + 10 \beta_{2} + 20 \beta_{1} + 9\)
\(\nu^{6}\)\(=\)\(\beta_{13} - 11 \beta_{11} + 12 \beta_{10} - 21 \beta_{9} + 2 \beta_{8} + 23 \beta_{7} + 11 \beta_{6} + \beta_{4} + 22 \beta_{3} + 47 \beta_{2} + 3 \beta_{1} + 84\)
\(\nu^{7}\)\(=\)\(11 \beta_{13} - 10 \beta_{12} - 57 \beta_{11} + 59 \beta_{10} - 69 \beta_{9} + 23 \beta_{8} + 82 \beta_{7} + 67 \beta_{6} + \beta_{5} + 83 \beta_{3} + 84 \beta_{2} + 109 \beta_{1} + 73\)
\(\nu^{8}\)\(=\)\(15 \beta_{13} - 95 \beta_{11} + 109 \beta_{10} - 171 \beta_{9} + 29 \beta_{8} + 202 \beta_{7} + 96 \beta_{6} + 3 \beta_{5} + 12 \beta_{4} + 193 \beta_{3} + 322 \beta_{2} + 44 \beta_{1} + 496\)
\(\nu^{9}\)\(=\)\(92 \beta_{13} - 74 \beta_{12} - 397 \beta_{11} + 429 \beta_{10} - 504 \beta_{9} + 196 \beta_{8} + 630 \beta_{7} + 474 \beta_{6} + 15 \beta_{5} + 2 \beta_{4} + 645 \beta_{3} + 659 \beta_{2} + 630 \beta_{1} + 568\)
\(\nu^{10}\)\(=\)\(154 \beta_{13} - 3 \beta_{12} - 753 \beta_{11} + 892 \beta_{10} - 1285 \beta_{9} + 295 \beta_{8} + 1609 \beta_{7} + 773 \beta_{6} + 49 \beta_{5} + 104 \beta_{4} + 1556 \beta_{3} + 2241 \beta_{2} + 448 \beta_{1} + 3048\)
\(\nu^{11}\)\(=\)\(703 \beta_{13} - 494 \beta_{12} - 2755 \beta_{11} + 3097 \beta_{10} - 3614 \beta_{9} + 1508 \beta_{8} + 4694 \beta_{7} + 3299 \beta_{6} + 156 \beta_{5} + 39 \beta_{4} + 4847 \beta_{3} + 4987 \beta_{2} + 3815 \beta_{1} + 4304\)
\(\nu^{12}\)\(=\)\(1357 \beta_{13} - 62 \beta_{12} - 5725 \beta_{11} + 6926 \beta_{10} - 9353 \beta_{9} + 2601 \beta_{8} + 12230 \beta_{7} + 5974 \beta_{6} + 524 \beta_{5} + 802 \beta_{4} + 11998 \beta_{3} + 15734 \beta_{2} + 3937 \beta_{1} + 19350\)
\(\nu^{13}\)\(=\)\(5181 \beta_{13} - 3167 \beta_{12} - 19157 \beta_{11} + 22239 \beta_{10} - 25724 \beta_{9} + 11112 \beta_{8} + 34387 \beta_{7} + 22867 \beta_{6} + 1395 \beta_{5} + 488 \beta_{4} + 35716 \beta_{3} + 36946 \beta_{2} + 23986 \beta_{1} + 32026\)

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
−2.33388
−2.16813
−1.87889
−1.00753
−0.579209
−0.477041
0.369092
0.789779
1.09407
1.75826
1.86702
2.23233
2.66571
2.66839
−2.33388 0 3.44698 0.335557 0 −1.00000 −3.37706 0 −0.783148
1.2 −2.16813 0 2.70077 1.98599 0 −1.00000 −1.51936 0 −4.30589
1.3 −1.87889 0 1.53022 −2.15857 0 −1.00000 0.882670 0 4.05571
1.4 −1.00753 0 −0.984892 −2.84275 0 −1.00000 3.00736 0 2.86415
1.5 −0.579209 0 −1.66452 1.12937 0 −1.00000 2.12252 0 −0.654144
1.6 −0.477041 0 −1.77243 1.98652 0 −1.00000 1.79961 0 −0.947652
1.7 0.369092 0 −1.86377 3.55869 0 −1.00000 −1.42609 0 1.31348
1.8 0.789779 0 −1.37625 0.366860 0 −1.00000 −2.66649 0 0.289738
1.9 1.09407 0 −0.803001 −2.53877 0 −1.00000 −3.06669 0 −2.77760
1.10 1.75826 0 1.09149 −1.44353 0 −1.00000 −1.59740 0 −2.53810
1.11 1.86702 0 1.48577 4.06976 0 −1.00000 −0.960078 0 7.59833
1.12 2.23233 0 2.98331 −2.48067 0 −1.00000 2.19508 0 −5.53768
1.13 2.66571 0 5.10602 −0.697799 0 −1.00000 8.27977 0 −1.86013
1.14 2.66839 0 5.12030 2.72934 0 −1.00000 8.32617 0 7.28293
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.14
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 8001.2.a.p 14
3.b odd 2 1 2667.2.a.m 14
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2667.2.a.m 14 3.b odd 2 1
8001.2.a.p 14 1.a even 1 1 trivial

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(7\) \(1\)
\(127\) \(1\)

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8001))\):

\(T_{2}^{14} - \cdots\)
\(T_{5}^{14} - \cdots\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 - 5 T + 19 T^{2} - 54 T^{3} + 136 T^{4} - 302 T^{5} + 627 T^{6} - 1213 T^{7} + 2245 T^{8} - 3945 T^{9} + 6666 T^{10} - 10741 T^{11} + 16698 T^{12} - 24876 T^{13} + 35892 T^{14} - 49752 T^{15} + 66792 T^{16} - 85928 T^{17} + 106656 T^{18} - 126240 T^{19} + 143680 T^{20} - 155264 T^{21} + 160512 T^{22} - 154624 T^{23} + 139264 T^{24} - 110592 T^{25} + 77824 T^{26} - 40960 T^{27} + 16384 T^{28} \)
$3$ 1
$5$ \( 1 - 4 T + 41 T^{2} - 150 T^{3} + 870 T^{4} - 2908 T^{5} + 12492 T^{6} - 38021 T^{7} + 133996 T^{8} - 370873 T^{9} + 1127299 T^{10} - 2836015 T^{11} + 7634651 T^{12} - 17439737 T^{13} + 42178904 T^{14} - 87198685 T^{15} + 190866275 T^{16} - 354501875 T^{17} + 704561875 T^{18} - 1158978125 T^{19} + 2093687500 T^{20} - 2970390625 T^{21} + 4879687500 T^{22} - 5679687500 T^{23} + 8496093750 T^{24} - 7324218750 T^{25} + 10009765625 T^{26} - 4882812500 T^{27} + 6103515625 T^{28} \)
$7$ \( ( 1 + T )^{14} \)
$11$ \( 1 - 3 T + 82 T^{2} - 256 T^{3} + 3350 T^{4} - 10553 T^{5} + 92717 T^{6} - 284301 T^{7} + 1964384 T^{8} - 5713164 T^{9} + 33629114 T^{10} - 91772019 T^{11} + 477433889 T^{12} - 1214718248 T^{13} + 5707030766 T^{14} - 13361900728 T^{15} + 57769500569 T^{16} - 122148557289 T^{17} + 492363858074 T^{18} - 920110775364 T^{19} + 3480026083424 T^{20} - 5540222202471 T^{21} + 19874712369677 T^{22} - 24883421983123 T^{23} + 86890372413350 T^{24} - 73039787676416 T^{25} + 257351126891122 T^{26} - 103568136431793 T^{27} + 379749833583241 T^{28} \)
$13$ \( 1 + 13 T + 182 T^{2} + 1585 T^{3} + 13608 T^{4} + 92167 T^{5} + 606905 T^{6} + 3422512 T^{7} + 18744862 T^{8} + 91352922 T^{9} + 433195838 T^{10} + 1864962208 T^{11} + 7828834193 T^{12} + 30141395677 T^{13} + 113300440326 T^{14} + 391838143801 T^{15} + 1323072978617 T^{16} + 4097321970976 T^{17} + 12372506329118 T^{18} + 33918700468146 T^{19} + 90477868605358 T^{20} + 214757552414704 T^{21} + 495071053228505 T^{22} + 977384893711291 T^{23} + 1875978357081192 T^{24} + 2840574224548645 T^{25} + 4240251492291542 T^{26} + 3937376385699289 T^{27} + 3937376385699289 T^{28} \)
$17$ \( 1 - 5 T + 130 T^{2} - 468 T^{3} + 7643 T^{4} - 20469 T^{5} + 287362 T^{6} - 594229 T^{7} + 8149596 T^{8} - 13821766 T^{9} + 190977908 T^{10} - 284790841 T^{11} + 3876597272 T^{12} - 5340393118 T^{13} + 69746292016 T^{14} - 90786683006 T^{15} + 1120336611608 T^{16} - 1399177401833 T^{17} + 15950665854068 T^{18} - 19624931207462 T^{19} + 196711435772124 T^{20} - 243835139318117 T^{21} + 2004567609760642 T^{22} - 2427375244017093 T^{23} + 15408241381131707 T^{24} - 16039247471972244 T^{25} + 75740890839868930 T^{26} - 49522890164529685 T^{27} + 168377826559400929 T^{28} \)
$19$ \( 1 - 21 T + 374 T^{2} - 4569 T^{3} + 49627 T^{4} - 447130 T^{5} + 3686027 T^{6} - 26871562 T^{7} + 182306081 T^{8} - 1126820515 T^{9} + 6549395980 T^{10} - 35231461213 T^{11} + 179334900367 T^{12} - 851666403346 T^{13} + 3838575491950 T^{14} - 16181661663574 T^{15} + 64739899032487 T^{16} - 241652592459967 T^{17} + 853523833509580 T^{18} - 2790119150370985 T^{19} + 8576750192302361 T^{20} - 24019729854586318 T^{21} + 62601871925328107 T^{22} - 144283350307924270 T^{23} + 304266425175890227 T^{24} - 532243992905962611 T^{25} + 827779779730744214 T^{26} - 883112652707398239 T^{27} + 799006685782884121 T^{28} \)
$23$ \( 1 - 10 T + 197 T^{2} - 1651 T^{3} + 18643 T^{4} - 131352 T^{5} + 1107307 T^{6} - 6674025 T^{7} + 46362563 T^{8} - 244354670 T^{9} + 1483666023 T^{10} - 7040607329 T^{11} + 39336210461 T^{12} - 174731022341 T^{13} + 936647851466 T^{14} - 4018813513843 T^{15} + 20808855333869 T^{16} - 85663069371943 T^{17} + 415190583542343 T^{18} - 1572750469771810 T^{19} + 6863323230023507 T^{20} - 22723890153914175 T^{21} + 86714302178548267 T^{22} - 236585004388487976 T^{23} + 772314448556058307 T^{24} - 1573088910315893477 T^{25} + 4317181013108003237 T^{26} - 5040363619364673830 T^{27} + 11592836324538749809 T^{28} \)
$29$ \( 1 - 15 T + 325 T^{2} - 3542 T^{3} + 45090 T^{4} - 389302 T^{5} + 3709822 T^{6} - 26645384 T^{7} + 208750551 T^{8} - 1291861725 T^{9} + 8811883923 T^{10} - 48651787808 T^{11} + 303330222126 T^{12} - 1554455012568 T^{13} + 9203796607924 T^{14} - 45079195364472 T^{15} + 255100716807966 T^{16} - 1186568452849312 T^{17} + 6232478072943363 T^{18} - 26497568328872025 T^{19} + 124169696006399871 T^{20} - 459629578205807656 T^{21} + 1855825148223802942 T^{22} - 5647660942697753438 T^{23} + 18969689149506063090 T^{24} - 43214205590130046318 T^{25} + \)\(11\!\cdots\!25\)\( T^{26} - \)\(15\!\cdots\!35\)\( T^{27} + \)\(29\!\cdots\!81\)\( T^{28} \)
$31$ \( 1 - 33 T + 772 T^{2} - 13150 T^{3} + 188546 T^{4} - 2298977 T^{5} + 24968647 T^{6} - 242769108 T^{7} + 2159103944 T^{8} - 17604871449 T^{9} + 133190121022 T^{10} - 935380216908 T^{11} + 6142523519475 T^{12} - 37677429544483 T^{13} + 216850565998226 T^{14} - 1168000315878973 T^{15} + 5902965102215475 T^{16} - 27865912041906228 T^{17} + 123003873756358462 T^{18} - 504012523049009799 T^{19} + 1916212697961617864 T^{20} - 6679212786475682988 T^{21} + 21295535243328112327 T^{22} - 60784083236072933567 T^{23} + \)\(15\!\cdots\!46\)\( T^{24} - \)\(33\!\cdots\!50\)\( T^{25} + \)\(60\!\cdots\!92\)\( T^{26} - \)\(80\!\cdots\!03\)\( T^{27} + \)\(75\!\cdots\!21\)\( T^{28} \)
$37$ \( 1 + 29 T + 693 T^{2} + 11721 T^{3} + 173826 T^{4} + 2176097 T^{5} + 24709960 T^{6} + 250869276 T^{7} + 2356076231 T^{8} + 20315367197 T^{9} + 163971869231 T^{10} + 1232026880055 T^{11} + 8724657176274 T^{12} + 57914924180169 T^{13} + 363474458224772 T^{14} + 2142852194666253 T^{15} + 11944055674319106 T^{16} + 62405857555425915 T^{17} + 307309682409840191 T^{18} + 1408747949347978529 T^{19} + 6045047007493884479 T^{20} + 23815491285676665708 T^{21} + 86793226807209753160 T^{22} + \)\(28\!\cdots\!69\)\( T^{23} + \)\(83\!\cdots\!74\)\( T^{24} + \)\(20\!\cdots\!73\)\( T^{25} + \)\(45\!\cdots\!33\)\( T^{26} + \)\(70\!\cdots\!13\)\( T^{27} + \)\(90\!\cdots\!89\)\( T^{28} \)
$41$ \( 1 - T + 206 T^{2} - 42 T^{3} + 22600 T^{4} + 5720 T^{5} + 1802471 T^{6} + 973585 T^{7} + 116330717 T^{8} + 88412265 T^{9} + 6386509096 T^{10} + 5890073146 T^{11} + 308288137130 T^{12} + 303314192679 T^{13} + 13329630557542 T^{14} + 12435881899839 T^{15} + 518232358515530 T^{16} + 405949731295466 T^{17} + 18046748329622056 T^{18} + 10243109144705265 T^{19} + 552583032180270797 T^{20} + 189609839736433385 T^{21} + 14392596162658957991 T^{22} + 1872624664733456920 T^{23} + \)\(30\!\cdots\!00\)\( T^{24} - 23113819332082434522 T^{25} + \)\(46\!\cdots\!86\)\( T^{26} - \)\(92\!\cdots\!21\)\( T^{27} + \)\(37\!\cdots\!61\)\( T^{28} \)
$43$ \( 1 + 25 T + 530 T^{2} + 7932 T^{3} + 110412 T^{4} + 1292969 T^{5} + 14382905 T^{6} + 142501427 T^{7} + 1353730590 T^{8} + 11754987450 T^{9} + 98445954790 T^{10} + 763920391599 T^{11} + 5734234384189 T^{12} + 40147595965818 T^{13} + 272318392470142 T^{14} + 1726346626530174 T^{15} + 10602599376365461 T^{16} + 60737018574861693 T^{17} + 336567128682006790 T^{18} + 1728082402509040350 T^{19} + 8557422529926968910 T^{20} + 38734539967905549689 T^{21} + \)\(16\!\cdots\!05\)\( T^{22} + \)\(64\!\cdots\!67\)\( T^{23} + \)\(23\!\cdots\!88\)\( T^{24} + \)\(73\!\cdots\!24\)\( T^{25} + \)\(21\!\cdots\!30\)\( T^{26} + \)\(42\!\cdots\!75\)\( T^{27} + \)\(73\!\cdots\!49\)\( T^{28} \)
$47$ \( 1 - 9 T + 350 T^{2} - 3582 T^{3} + 68307 T^{4} - 684242 T^{5} + 9374099 T^{6} - 86334288 T^{7} + 965719113 T^{8} - 8092193530 T^{9} + 77594134130 T^{10} - 590988285775 T^{11} + 4991471177695 T^{12} - 34417211240088 T^{13} + 260228788002610 T^{14} - 1617608928284136 T^{15} + 11026159831528255 T^{16} - 61358176794017825 T^{17} + 378634622025612530 T^{18} - 1855904181783204710 T^{19} + 10409694266357883177 T^{20} - 43738946389511335344 T^{21} + \)\(22\!\cdots\!39\)\( T^{22} - \)\(76\!\cdots\!14\)\( T^{23} + \)\(35\!\cdots\!43\)\( T^{24} - \)\(88\!\cdots\!46\)\( T^{25} + \)\(40\!\cdots\!50\)\( T^{26} - \)\(49\!\cdots\!43\)\( T^{27} + \)\(25\!\cdots\!69\)\( T^{28} \)
$53$ \( 1 - 35 T + 934 T^{2} - 17975 T^{3} + 299033 T^{4} - 4227887 T^{5} + 54144485 T^{6} - 622708390 T^{7} + 6648233679 T^{8} - 65496431709 T^{9} + 608126212712 T^{10} - 5291692221571 T^{11} + 43805743758863 T^{12} - 342748023160961 T^{13} + 2563077085008698 T^{14} - 18165645227530933 T^{15} + 123050334218646167 T^{16} - 787811262870825767 T^{17} + 4798408327005994472 T^{18} - 27390312548286087537 T^{19} + \)\(14\!\cdots\!91\)\( T^{20} - \)\(73\!\cdots\!30\)\( T^{21} + \)\(33\!\cdots\!85\)\( T^{22} - \)\(13\!\cdots\!71\)\( T^{23} + \)\(52\!\cdots\!17\)\( T^{24} - \)\(16\!\cdots\!75\)\( T^{25} + \)\(45\!\cdots\!94\)\( T^{26} - \)\(91\!\cdots\!55\)\( T^{27} + \)\(13\!\cdots\!69\)\( T^{28} \)
$59$ \( 1 + 10 T + 389 T^{2} + 3033 T^{3} + 74371 T^{4} + 531008 T^{5} + 10011285 T^{6} + 70337571 T^{7} + 1054894651 T^{8} + 7383068244 T^{9} + 90904277275 T^{10} + 631193665181 T^{11} + 6652047058913 T^{12} + 44814617366759 T^{13} + 421301398427094 T^{14} + 2644062424638781 T^{15} + 23155775812076153 T^{16} + 129633923761208599 T^{17} + 1101519944185271275 T^{18} + 5278334888810860956 T^{19} + 44496019314216454291 T^{20} + \)\(17\!\cdots\!49\)\( T^{21} + \)\(14\!\cdots\!85\)\( T^{22} + \)\(46\!\cdots\!12\)\( T^{23} + \)\(38\!\cdots\!71\)\( T^{24} + \)\(91\!\cdots\!47\)\( T^{25} + \)\(69\!\cdots\!09\)\( T^{26} + \)\(10\!\cdots\!90\)\( T^{27} + \)\(61\!\cdots\!61\)\( T^{28} \)
$61$ \( 1 - T + 410 T^{2} - 890 T^{3} + 87588 T^{4} - 279433 T^{5} + 12931865 T^{6} - 50711894 T^{7} + 1484437808 T^{8} - 6339757701 T^{9} + 139727676318 T^{10} - 597738020304 T^{11} + 11017030143027 T^{12} - 44768426473041 T^{13} + 731807583059774 T^{14} - 2730874014855501 T^{15} + 40994369162203467 T^{16} - 135675173586622224 T^{17} + 1934647189598493438 T^{18} - 5354535903500864001 T^{19} + 76478791583782240688 T^{20} - \)\(15\!\cdots\!74\)\( T^{21} + \)\(24\!\cdots\!65\)\( T^{22} - \)\(32\!\cdots\!53\)\( T^{23} + \)\(62\!\cdots\!88\)\( T^{24} - \)\(38\!\cdots\!90\)\( T^{25} + \)\(10\!\cdots\!10\)\( T^{26} - \)\(16\!\cdots\!81\)\( T^{27} + \)\(98\!\cdots\!41\)\( T^{28} \)
$67$ \( 1 + 38 T + 1174 T^{2} + 23934 T^{3} + 422665 T^{4} + 5839509 T^{5} + 71970295 T^{6} + 723216019 T^{7} + 6601917413 T^{8} + 48533153402 T^{9} + 331369870442 T^{10} + 1636952749842 T^{11} + 8113580369697 T^{12} + 14028985989802 T^{13} + 123870851464306 T^{14} + 939942061316734 T^{15} + 36421862279569833 T^{16} + 492334819900729446 T^{17} + 6677474355031065482 T^{18} + 65525828929922664014 T^{19} + \)\(59\!\cdots\!97\)\( T^{20} + \)\(43\!\cdots\!37\)\( T^{21} + \)\(29\!\cdots\!95\)\( T^{22} + \)\(15\!\cdots\!23\)\( T^{23} + \)\(77\!\cdots\!85\)\( T^{24} + \)\(29\!\cdots\!22\)\( T^{25} + \)\(96\!\cdots\!14\)\( T^{26} + \)\(20\!\cdots\!06\)\( T^{27} + \)\(36\!\cdots\!29\)\( T^{28} \)
$71$ \( 1 - 10 T + 621 T^{2} - 5411 T^{3} + 182283 T^{4} - 1425360 T^{5} + 34456003 T^{6} - 247816664 T^{7} + 4799991247 T^{8} - 32264035049 T^{9} + 530496962459 T^{10} - 3348973491864 T^{11} + 48460938627045 T^{12} - 285739722695784 T^{13} + 3734602641050538 T^{14} - 20287520311400664 T^{15} + 244291591618933845 T^{16} - 1198634451446536104 T^{17} + 13480819581477083579 T^{18} - 58211719017098523199 T^{19} + \)\(61\!\cdots\!87\)\( T^{20} - \)\(22\!\cdots\!24\)\( T^{21} + \)\(22\!\cdots\!83\)\( T^{22} - \)\(65\!\cdots\!60\)\( T^{23} + \)\(59\!\cdots\!83\)\( T^{24} - \)\(12\!\cdots\!81\)\( T^{25} + \)\(10\!\cdots\!61\)\( T^{26} - \)\(11\!\cdots\!10\)\( T^{27} + \)\(82\!\cdots\!81\)\( T^{28} \)
$73$ \( 1 - 8 T + 547 T^{2} - 3939 T^{3} + 156457 T^{4} - 1057340 T^{5} + 30751475 T^{6} - 195813211 T^{7} + 4575405639 T^{8} - 27362929950 T^{9} + 541643272461 T^{10} - 3022765696423 T^{11} + 52384471961759 T^{12} - 270009836377377 T^{13} + 4193767875322650 T^{14} - 19710718055548521 T^{15} + 279156851084213711 T^{16} - 1175907242925386191 T^{17} + 15381716187376141101 T^{18} - 56725312780593910350 T^{19} + \)\(69\!\cdots\!71\)\( T^{20} - \)\(21\!\cdots\!67\)\( T^{21} + \)\(24\!\cdots\!75\)\( T^{22} - \)\(62\!\cdots\!20\)\( T^{23} + \)\(67\!\cdots\!93\)\( T^{24} - \)\(12\!\cdots\!03\)\( T^{25} + \)\(12\!\cdots\!87\)\( T^{26} - \)\(13\!\cdots\!64\)\( T^{27} + \)\(12\!\cdots\!09\)\( T^{28} \)
$79$ \( 1 - 26 T + 995 T^{2} - 18888 T^{3} + 423915 T^{4} - 6445491 T^{5} + 108617293 T^{6} - 1396649186 T^{7} + 19434091386 T^{8} - 218844443965 T^{9} + 2638999284435 T^{10} - 26567340357667 T^{11} + 284690572139722 T^{12} - 2585314414532307 T^{13} + 24906831120291546 T^{14} - 204239838748052253 T^{15} + 1776753860724005002 T^{16} - 13098734922603780013 T^{17} + \)\(10\!\cdots\!35\)\( T^{18} - \)\(67\!\cdots\!35\)\( T^{19} + \)\(47\!\cdots\!06\)\( T^{20} - \)\(26\!\cdots\!74\)\( T^{21} + \)\(16\!\cdots\!73\)\( T^{22} - \)\(77\!\cdots\!29\)\( T^{23} + \)\(40\!\cdots\!15\)\( T^{24} - \)\(14\!\cdots\!52\)\( T^{25} + \)\(58\!\cdots\!95\)\( T^{26} - \)\(12\!\cdots\!14\)\( T^{27} + \)\(36\!\cdots\!81\)\( T^{28} \)
$83$ \( 1 - 30 T + 969 T^{2} - 18676 T^{3} + 365335 T^{4} - 5350552 T^{5} + 79155481 T^{6} - 943348279 T^{7} + 11483465659 T^{8} - 115946714119 T^{9} + 1228033468687 T^{10} - 10957644918809 T^{11} + 107585686563309 T^{12} - 906460220784007 T^{13} + 8857114926688414 T^{14} - 75236198325072581 T^{15} + 741157794734635701 T^{16} - 6265438915191041683 T^{17} + 58280406555691094527 T^{18} - \)\(45\!\cdots\!17\)\( T^{19} + \)\(37\!\cdots\!71\)\( T^{20} - \)\(25\!\cdots\!33\)\( T^{21} + \)\(17\!\cdots\!21\)\( T^{22} - \)\(10\!\cdots\!56\)\( T^{23} + \)\(56\!\cdots\!15\)\( T^{24} - \)\(24\!\cdots\!92\)\( T^{25} + \)\(10\!\cdots\!09\)\( T^{26} - \)\(26\!\cdots\!90\)\( T^{27} + \)\(73\!\cdots\!29\)\( T^{28} \)
$89$ \( 1 + 4 T + 780 T^{2} + 1881 T^{3} + 289356 T^{4} + 274990 T^{5} + 69261693 T^{6} - 17925425 T^{7} + 12271686510 T^{8} - 13239275906 T^{9} + 1735603354082 T^{10} - 2547188943427 T^{11} + 202853568948719 T^{12} - 309799827574717 T^{13} + 19768290680155150 T^{14} - 27572184654149813 T^{15} + 1606803119642803199 T^{16} - 1795689242258788763 T^{17} + \)\(10\!\cdots\!62\)\( T^{18} - 73928903720817335794 T^{19} + \)\(60\!\cdots\!10\)\( T^{20} - \)\(79\!\cdots\!25\)\( T^{21} + \)\(27\!\cdots\!33\)\( T^{22} + \)\(96\!\cdots\!10\)\( T^{23} + \)\(90\!\cdots\!56\)\( T^{24} + \)\(52\!\cdots\!09\)\( T^{25} + \)\(19\!\cdots\!80\)\( T^{26} + \)\(87\!\cdots\!76\)\( T^{27} + \)\(19\!\cdots\!41\)\( T^{28} \)
$97$ \( 1 - 15 T + 688 T^{2} - 9074 T^{3} + 232439 T^{4} - 2674739 T^{5} + 50852864 T^{6} - 510202963 T^{7} + 8088855856 T^{8} - 71493024766 T^{9} + 1014684615820 T^{10} - 8074806425495 T^{11} + 108805877810262 T^{12} - 810761204923202 T^{13} + 10784014670664272 T^{14} - 78643836877550594 T^{15} + 1023754504316755158 T^{16} - 7369657804775798135 T^{17} + 89829299480305825420 T^{18} - \)\(61\!\cdots\!62\)\( T^{19} + \)\(67\!\cdots\!24\)\( T^{20} - \)\(41\!\cdots\!19\)\( T^{21} + \)\(39\!\cdots\!04\)\( T^{22} - \)\(20\!\cdots\!63\)\( T^{23} + \)\(17\!\cdots\!11\)\( T^{24} - \)\(64\!\cdots\!22\)\( T^{25} + \)\(47\!\cdots\!08\)\( T^{26} - \)\(10\!\cdots\!55\)\( T^{27} + \)\(65\!\cdots\!69\)\( T^{28} \)
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