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: \( 5 \)
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.

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

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

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

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.

Atkin-Lehner signs

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

Hecke kernels

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

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