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: \(\mathbb{Q}[x]/(x^{14} - \cdots)\)
Defining polynomial: \(x^{14} - 5 x^{13} - 9 x^{12} + 76 x^{11} - 12 x^{10} - 414 x^{9} + 331 x^{8} + 959 x^{7} - 1067 x^{6} - 875 x^{5} + 1134 x^{4} + 301 x^{3} - 418 x^{2} - 42 x + 44\)
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:

Atkin-Lehner signs

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

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 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

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

$p$ $F_p(T)$
$2$ \( 44 - 42 T - 418 T^{2} + 301 T^{3} + 1134 T^{4} - 875 T^{5} - 1067 T^{6} + 959 T^{7} + 331 T^{8} - 414 T^{9} - 12 T^{10} + 76 T^{11} - 9 T^{12} - 5 T^{13} + T^{14} \)
$3$ \( T^{14} \)
$5$ \( 844 - 4012 T + 646 T^{2} + 14710 T^{3} - 5956 T^{4} - 14288 T^{5} + 5316 T^{6} + 5839 T^{7} - 1908 T^{8} - 1158 T^{9} + 335 T^{10} + 110 T^{11} - 29 T^{12} - 4 T^{13} + T^{14} \)
$7$ \( ( 1 + T )^{14} \)
$11$ \( -256 - 2944 T + 33504 T^{2} + 49040 T^{3} - 94576 T^{4} - 55952 T^{5} + 73088 T^{6} + 20410 T^{7} - 19505 T^{8} - 3172 T^{9} + 1843 T^{10} + 173 T^{11} - 72 T^{12} - 3 T^{13} + T^{14} \)
$13$ \( -512 - 50944 T - 84736 T^{2} + 524256 T^{3} + 623984 T^{4} - 95188 T^{5} - 268652 T^{6} - 30483 T^{7} + 37427 T^{8} + 8317 T^{9} - 1771 T^{10} - 612 T^{11} + 13 T^{13} + T^{14} \)
$17$ \( 1964992 + 1777152 T - 5168176 T^{2} - 1060960 T^{3} + 4938204 T^{4} - 1787364 T^{5} - 600797 T^{6} + 418580 T^{7} - 14898 T^{8} - 26878 T^{9} + 3376 T^{10} + 637 T^{11} - 108 T^{12} - 5 T^{13} + T^{14} \)
$19$ \( 11651104 - 30961916 T + 26973876 T^{2} - 3637042 T^{3} - 7571698 T^{4} + 3919626 T^{5} + 71704 T^{6} - 516244 T^{7} + 107263 T^{8} + 15026 T^{9} - 7848 T^{10} + 618 T^{11} + 108 T^{12} - 21 T^{13} + T^{14} \)
$23$ \( -14984800 - 499200 T + 43538564 T^{2} + 23254216 T^{3} - 13540804 T^{4} - 8493672 T^{5} + 1613786 T^{6} + 1067683 T^{7} - 108151 T^{8} - 57499 T^{9} + 5004 T^{10} + 1339 T^{11} - 125 T^{12} - 10 T^{13} + T^{14} \)
$29$ \( 11323600 + 3534720 T - 28565872 T^{2} - 1687784 T^{3} + 19272510 T^{4} - 1896891 T^{5} - 4522989 T^{6} + 964530 T^{7} + 271582 T^{8} - 79379 T^{9} - 3253 T^{10} + 2113 T^{11} - 81 T^{12} - 15 T^{13} + T^{14} \)
$31$ \( -614457824 - 49438640 T + 564197398 T^{2} - 139229932 T^{3} - 105212858 T^{4} + 42678692 T^{5} + 4290348 T^{6} - 4025057 T^{7} + 325445 T^{8} + 123828 T^{9} - 24641 T^{10} + 149 T^{11} + 338 T^{12} - 33 T^{13} + T^{14} \)
$37$ \( -299125948 - 442800288 T + 73414612 T^{2} + 257809128 T^{3} + 23141442 T^{4} - 48073719 T^{5} - 9679053 T^{6} + 3100959 T^{7} + 987928 T^{8} - 13785 T^{9} - 28453 T^{10} - 2228 T^{11} + 175 T^{12} + 29 T^{13} + T^{14} \)
$41$ \( 3726826112 - 10518493440 T + 3338888416 T^{2} + 2866470992 T^{3} - 1099107232 T^{4} - 222790978 T^{5} + 95419651 T^{6} + 6508093 T^{7} - 3237495 T^{8} - 84603 T^{9} + 50685 T^{10} + 491 T^{11} - 368 T^{12} - T^{13} + T^{14} \)
$43$ \( 3855391552 - 21006556816 T - 25963039600 T^{2} - 1570642884 T^{3} + 3579566092 T^{4} + 406480512 T^{5} - 163084176 T^{6} - 22639084 T^{7} + 3127655 T^{8} + 545758 T^{9} - 20695 T^{10} - 6043 T^{11} - 72 T^{12} + 25 T^{13} + T^{14} \)
$47$ \( -637784464 + 3792587608 T + 2626073772 T^{2} - 1607675010 T^{3} - 806189588 T^{4} + 144461060 T^{5} + 86978208 T^{6} + 711498 T^{7} - 2782921 T^{8} - 124613 T^{9} + 41000 T^{10} + 1917 T^{11} - 308 T^{12} - 9 T^{13} + T^{14} \)
$53$ \( -19702384400 + 44716709040 T - 4410030288 T^{2} - 9488110276 T^{3} + 1721428082 T^{4} + 667317045 T^{5} - 163522368 T^{6} - 14775971 T^{7} + 6067549 T^{8} - 138937 T^{9} - 78698 T^{10} + 6140 T^{11} + 192 T^{12} - 35 T^{13} + T^{14} \)
$59$ \( 49597767424 + 15812110464 T - 65738450560 T^{2} - 19940579520 T^{3} + 5553599512 T^{4} + 2330680298 T^{5} + 5110778 T^{6} - 67472505 T^{7} - 3875309 T^{8} + 825241 T^{9} + 66996 T^{10} - 4637 T^{11} - 437 T^{12} + 10 T^{13} + T^{14} \)
$61$ \( -181770499328 + 18863026304 T + 47707459136 T^{2} - 4013527904 T^{3} - 4705179456 T^{4} + 275490694 T^{5} + 228420602 T^{6} - 7608501 T^{7} - 5779885 T^{8} + 75892 T^{9} + 73985 T^{10} - 97 T^{11} - 444 T^{12} - T^{13} + T^{14} \)
$67$ \( 2652304192 - 749950560 T - 5396726848 T^{2} - 2068540000 T^{3} + 2196650464 T^{4} + 2194456220 T^{5} + 810554998 T^{6} + 146416972 T^{7} + 10209159 T^{8} - 712019 T^{9} - 175578 T^{10} - 9164 T^{11} + 236 T^{12} + 38 T^{13} + T^{14} \)
$71$ \( -54609728 + 153483232 T + 183072352 T^{2} - 888220400 T^{3} + 840079856 T^{4} - 227341564 T^{5} - 32309010 T^{6} + 21148092 T^{7} - 1081343 T^{8} - 476019 T^{9} + 41348 T^{10} + 3819 T^{11} - 373 T^{12} - 10 T^{13} + T^{14} \)
$73$ \( -4865757632 - 40747784800 T - 69837463744 T^{2} + 52419557120 T^{3} - 3382890764 T^{4} - 2793446094 T^{5} + 314466960 T^{6} + 60751421 T^{7} - 7747703 T^{8} - 665403 T^{9} + 87618 T^{10} + 3653 T^{11} - 475 T^{12} - 8 T^{13} + T^{14} \)
$79$ \( 37260800 + 727549120 T - 1492270112 T^{2} - 844560592 T^{3} + 725753752 T^{4} + 144509328 T^{5} - 107107111 T^{6} - 859253 T^{7} + 5515183 T^{8} - 579109 T^{9} - 38788 T^{10} + 7814 T^{11} - 111 T^{12} - 26 T^{13} + T^{14} \)
$83$ \( -84701372800 + 328851835200 T - 231043881824 T^{2} - 35301592528 T^{3} + 27214464552 T^{4} + 56918588 T^{5} - 1090178362 T^{6} + 68482809 T^{7} + 16325975 T^{8} - 1732914 T^{9} - 69336 T^{10} + 13694 T^{11} - 193 T^{12} - 30 T^{13} + T^{14} \)
$89$ \( 208920272 + 833364960 T + 282365714 T^{2} - 915101288 T^{3} - 276449502 T^{4} + 310306594 T^{5} + 37098768 T^{6} - 22519427 T^{7} - 2676117 T^{8} + 492951 T^{9} + 66233 T^{10} - 2747 T^{11} - 466 T^{12} + 4 T^{13} + T^{14} \)
$97$ \( 1491113902028 - 1269848612822 T + 53897901606 T^{2} + 166588284422 T^{3} - 18340948134 T^{4} - 8566070042 T^{5} + 873360603 T^{6} + 203899300 T^{7} - 16711128 T^{8} - 2166556 T^{9} + 156100 T^{10} + 9841 T^{11} - 670 T^{12} - 15 T^{13} + T^{14} \)
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