Properties

Label 8001.2.a.o
Level 8001
Weight 2
Character orbit 8001.a
Self dual Yes
Analytic conductor 63.888
Analytic rank 1
Dimension 13
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: \(1\)
Dimension: \(13\)
Coefficient field: \(\mathbb{Q}[x]/(x^{13} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{2} \)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{12}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \( -\beta_{1} q^{2} \) \( + ( 1 + \beta_{2} ) q^{4} \) \( + ( -1 + \beta_{10} ) q^{5} \) \(+ q^{7}\) \( + ( -1 - \beta_{1} - \beta_{6} + \beta_{7} ) q^{8} \) \(+O(q^{10})\) \( q\) \( -\beta_{1} q^{2} \) \( + ( 1 + \beta_{2} ) q^{4} \) \( + ( -1 + \beta_{10} ) q^{5} \) \(+ q^{7}\) \( + ( -1 - \beta_{1} - \beta_{6} + \beta_{7} ) q^{8} \) \( + ( \beta_{1} - \beta_{11} ) q^{10} \) \( + \beta_{8} q^{11} \) \( + ( 2 + \beta_{4} - \beta_{7} - \beta_{12} ) q^{13} \) \( -\beta_{1} q^{14} \) \( + ( 1 + \beta_{1} + \beta_{4} + \beta_{6} - \beta_{7} - \beta_{9} - \beta_{12} ) q^{16} \) \( + ( -1 - \beta_{4} + \beta_{5} + \beta_{9} ) q^{17} \) \( + ( -\beta_{4} - \beta_{8} + \beta_{12} ) q^{19} \) \( + ( -3 - \beta_{1} - \beta_{2} - \beta_{5} - \beta_{6} ) q^{20} \) \( + ( -\beta_{3} - \beta_{4} + \beta_{5} + \beta_{9} + \beta_{12} ) q^{22} \) \( + ( 3 \beta_{1} - \beta_{2} + \beta_{3} + \beta_{5} + \beta_{6} - \beta_{8} - \beta_{9} - 2 \beta_{10} + 2 \beta_{11} ) q^{23} \) \( + ( \beta_{1} - \beta_{2} + \beta_{3} + \beta_{7} - 3 \beta_{10} ) q^{25} \) \( + ( -2 - \beta_{1} - \beta_{2} - \beta_{3} + \beta_{4} + \beta_{7} + \beta_{12} ) q^{26} \) \( + ( 1 + \beta_{2} ) q^{28} \) \( + ( -2 + \beta_{1} - \beta_{4} + \beta_{6} + \beta_{10} - \beta_{11} ) q^{29} \) \( + ( -1 - \beta_{1} + \beta_{2} - \beta_{5} - \beta_{6} + \beta_{7} + \beta_{9} ) q^{31} \) \( + ( -2 - 2 \beta_{1} - 2 \beta_{3} + \beta_{4} - \beta_{5} - \beta_{6} + \beta_{9} + 2 \beta_{10} - \beta_{11} + \beta_{12} ) q^{32} \) \( + ( -1 + 3 \beta_{1} - \beta_{2} - \beta_{4} - \beta_{5} + \beta_{6} + \beta_{8} + \beta_{12} ) q^{34} \) \( + ( -1 + \beta_{10} ) q^{35} \) \( + ( 1 + \beta_{4} + \beta_{11} + \beta_{12} ) q^{37} \) \( + ( 1 + 2 \beta_{3} - \beta_{4} - \beta_{5} - \beta_{7} - \beta_{9} - \beta_{12} ) q^{38} \) \( + ( 4 + 5 \beta_{1} + \beta_{3} + 2 \beta_{5} + 3 \beta_{6} - 2 \beta_{7} - \beta_{8} - \beta_{9} - 2 \beta_{10} + 3 \beta_{11} ) q^{40} \) \( + ( -1 + \beta_{4} + \beta_{6} ) q^{41} \) \( + ( -2 - \beta_{4} - 2 \beta_{5} - \beta_{9} - \beta_{12} ) q^{43} \) \( + ( 2 \beta_{1} - \beta_{2} + 2 \beta_{3} - \beta_{4} - \beta_{5} + \beta_{6} - \beta_{7} - \beta_{8} - \beta_{9} - \beta_{10} + \beta_{11} - \beta_{12} ) q^{44} \) \( + ( -4 - \beta_{1} - 2 \beta_{3} - 2 \beta_{5} - \beta_{6} + \beta_{8} + \beta_{9} + \beta_{10} - \beta_{11} ) q^{46} \) \( + ( -2 + \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} - \beta_{7} - 2 \beta_{9} - \beta_{10} - \beta_{12} ) q^{47} \) \(+ q^{49}\) \( + ( -1 + \beta_{1} - \beta_{3} + \beta_{6} - \beta_{7} + \beta_{9} + \beta_{10} + 2 \beta_{11} ) q^{50} \) \( + ( 2 + 3 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} + \beta_{6} - \beta_{9} - \beta_{10} + \beta_{11} - 2 \beta_{12} ) q^{52} \) \( + ( -3 - \beta_{1} - 2 \beta_{3} + \beta_{4} - \beta_{5} - \beta_{7} + \beta_{8} + \beta_{9} + 2 \beta_{10} - \beta_{11} ) q^{53} \) \( + ( -1 + \beta_{1} - \beta_{3} + \beta_{4} - \beta_{7} - \beta_{8} - \beta_{9} + \beta_{11} + \beta_{12} ) q^{55} \) \( + ( -1 - \beta_{1} - \beta_{6} + \beta_{7} ) q^{56} \) \( + ( -4 - \beta_{2} - \beta_{4} - \beta_{5} - 2 \beta_{6} + \beta_{7} + \beta_{9} + 2 \beta_{10} - \beta_{11} + \beta_{12} ) q^{58} \) \( + ( -3 + \beta_{1} - 2 \beta_{2} - \beta_{3} - \beta_{4} + \beta_{6} - \beta_{8} - 3 \beta_{10} + \beta_{11} + \beta_{12} ) q^{59} \) \( + ( 3 + \beta_{4} + \beta_{6} + \beta_{8} - \beta_{12} ) q^{61} \) \( + ( 3 + 3 \beta_{1} - \beta_{2} + 2 \beta_{3} + \beta_{4} + 3 \beta_{5} + 3 \beta_{6} - \beta_{8} - \beta_{9} - 4 \beta_{10} + 2 \beta_{11} - \beta_{12} ) q^{62} \) \( + ( 3 + 4 \beta_{1} - \beta_{2} + 5 \beta_{3} + 2 \beta_{5} + \beta_{6} - \beta_{8} - \beta_{9} - 4 \beta_{10} + 2 \beta_{11} - 2 \beta_{12} ) q^{64} \) \( + ( -3 - 3 \beta_{1} + \beta_{2} - 2 \beta_{3} - \beta_{4} - \beta_{6} + \beta_{7} + \beta_{8} + 2 \beta_{9} + 3 \beta_{10} - 2 \beta_{11} + \beta_{12} ) q^{65} \) \( + ( -2 - \beta_{1} - \beta_{3} + \beta_{5} - 2 \beta_{6} - \beta_{8} + \beta_{9} - \beta_{11} + 2 \beta_{12} ) q^{67} \) \( + ( -3 + 3 \beta_{1} - 4 \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} + \beta_{6} - \beta_{7} - \beta_{8} - 2 \beta_{10} + \beta_{11} + \beta_{12} ) q^{68} \) \( + ( \beta_{1} - \beta_{11} ) q^{70} \) \( + ( -1 + \beta_{1} + \beta_{3} - \beta_{5} - \beta_{7} - 2 \beta_{8} - \beta_{9} - \beta_{10} + \beta_{11} + 2 \beta_{12} ) q^{71} \) \( + ( 1 + \beta_{2} - 2 \beta_{3} - 2 \beta_{4} + \beta_{5} + \beta_{7} + \beta_{8} + \beta_{9} + \beta_{10} - \beta_{11} ) q^{73} \) \( + ( 3 + \beta_{3} + \beta_{5} + \beta_{6} - \beta_{7} - 2 \beta_{10} - 2 \beta_{12} ) q^{74} \) \( + ( -1 - 2 \beta_{1} - 3 \beta_{3} - \beta_{4} + \beta_{5} - \beta_{6} + \beta_{7} + \beta_{8} + 2 \beta_{9} + 2 \beta_{10} - 2 \beta_{11} + 3 \beta_{12} ) q^{76} \) \( + \beta_{8} q^{77} \) \( + ( -4 + 2 \beta_{1} - 3 \beta_{2} - \beta_{3} - \beta_{4} + \beta_{6} + \beta_{7} - \beta_{9} + \beta_{12} ) q^{79} \) \( + ( -4 - 5 \beta_{1} - \beta_{2} - 3 \beta_{3} - 2 \beta_{4} - \beta_{5} - 2 \beta_{6} + 3 \beta_{7} + 2 \beta_{8} + 3 \beta_{9} + \beta_{10} - 2 \beta_{11} + 2 \beta_{12} ) q^{80} \) \( + ( 1 + \beta_{4} - \beta_{6} + \beta_{7} + \beta_{9} - \beta_{12} ) q^{82} \) \( + ( -2 - 4 \beta_{1} + 2 \beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} - 2 \beta_{6} + \beta_{7} + \beta_{8} - 2 \beta_{11} + \beta_{12} ) q^{83} \) \( + ( 1 - \beta_{1} + \beta_{4} - \beta_{5} - \beta_{6} - \beta_{7} - \beta_{8} - \beta_{9} - \beta_{10} - \beta_{11} - \beta_{12} ) q^{85} \) \( + ( 1 + \beta_{1} + 3 \beta_{5} + \beta_{7} - 2 \beta_{8} - 2 \beta_{10} + \beta_{11} + 2 \beta_{12} ) q^{86} \) \( + ( -3 + \beta_{1} - 2 \beta_{2} - \beta_{5} + \beta_{7} - \beta_{8} - \beta_{11} + 2 \beta_{12} ) q^{88} \) \( + ( -5 - 2 \beta_{1} - 2 \beta_{3} - 2 \beta_{5} - \beta_{6} + 2 \beta_{8} + 2 \beta_{10} - 2 \beta_{11} ) q^{89} \) \( + ( 2 + \beta_{4} - \beta_{7} - \beta_{12} ) q^{91} \) \( + ( 2 + 3 \beta_{1} - \beta_{2} + 2 \beta_{3} + \beta_{4} + 3 \beta_{5} + 2 \beta_{6} - \beta_{7} - 2 \beta_{10} - \beta_{12} ) q^{92} \) \( + ( -3 + \beta_{2} - 3 \beta_{3} - 2 \beta_{6} - \beta_{8} + \beta_{9} + 3 \beta_{10} - \beta_{11} + 2 \beta_{12} ) q^{94} \) \( + ( 1 + 2 \beta_{3} + \beta_{7} + \beta_{8} - \beta_{10} - 2 \beta_{12} ) q^{95} \) \( + ( -1 - \beta_{2} - 2 \beta_{3} + \beta_{4} - \beta_{5} - \beta_{6} + 2 \beta_{7} + \beta_{8} + \beta_{10} - 2 \beta_{11} + \beta_{12} ) q^{97} \) \( -\beta_{1} q^{98} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(13q \) \(\mathstrut -\mathstrut 4q^{2} \) \(\mathstrut +\mathstrut 10q^{4} \) \(\mathstrut -\mathstrut 12q^{5} \) \(\mathstrut +\mathstrut 13q^{7} \) \(\mathstrut -\mathstrut 9q^{8} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(13q \) \(\mathstrut -\mathstrut 4q^{2} \) \(\mathstrut +\mathstrut 10q^{4} \) \(\mathstrut -\mathstrut 12q^{5} \) \(\mathstrut +\mathstrut 13q^{7} \) \(\mathstrut -\mathstrut 9q^{8} \) \(\mathstrut +\mathstrut 6q^{10} \) \(\mathstrut -\mathstrut 3q^{11} \) \(\mathstrut +\mathstrut 21q^{13} \) \(\mathstrut -\mathstrut 4q^{14} \) \(\mathstrut +\mathstrut 8q^{16} \) \(\mathstrut -\mathstrut 17q^{17} \) \(\mathstrut +\mathstrut 5q^{19} \) \(\mathstrut -\mathstrut 29q^{20} \) \(\mathstrut +\mathstrut q^{22} \) \(\mathstrut -\mathstrut 4q^{23} \) \(\mathstrut +\mathstrut q^{25} \) \(\mathstrut -\mathstrut 22q^{26} \) \(\mathstrut +\mathstrut 10q^{28} \) \(\mathstrut -\mathstrut 21q^{29} \) \(\mathstrut -\mathstrut 7q^{31} \) \(\mathstrut -\mathstrut 12q^{32} \) \(\mathstrut +\mathstrut 2q^{34} \) \(\mathstrut -\mathstrut 12q^{35} \) \(\mathstrut +\mathstrut 7q^{37} \) \(\mathstrut +\mathstrut 9q^{38} \) \(\mathstrut +\mathstrut 29q^{40} \) \(\mathstrut -\mathstrut 21q^{41} \) \(\mathstrut -\mathstrut 9q^{43} \) \(\mathstrut +\mathstrut 2q^{44} \) \(\mathstrut -\mathstrut 28q^{46} \) \(\mathstrut -\mathstrut 23q^{47} \) \(\mathstrut +\mathstrut 13q^{49} \) \(\mathstrut -\mathstrut 15q^{50} \) \(\mathstrut +\mathstrut 15q^{52} \) \(\mathstrut -\mathstrut 31q^{53} \) \(\mathstrut -\mathstrut 8q^{55} \) \(\mathstrut -\mathstrut 9q^{56} \) \(\mathstrut -\mathstrut 25q^{58} \) \(\mathstrut -\mathstrut 28q^{59} \) \(\mathstrut +\mathstrut 29q^{61} \) \(\mathstrut +\mathstrut 3q^{62} \) \(\mathstrut +\mathstrut 9q^{64} \) \(\mathstrut -\mathstrut 30q^{65} \) \(\mathstrut -\mathstrut 18q^{67} \) \(\mathstrut -\mathstrut 34q^{68} \) \(\mathstrut +\mathstrut 6q^{70} \) \(\mathstrut -\mathstrut 10q^{71} \) \(\mathstrut +\mathstrut 24q^{73} \) \(\mathstrut +\mathstrut 19q^{74} \) \(\mathstrut -\mathstrut 3q^{77} \) \(\mathstrut -\mathstrut 28q^{79} \) \(\mathstrut -\mathstrut 26q^{80} \) \(\mathstrut +\mathstrut 18q^{82} \) \(\mathstrut -\mathstrut 26q^{83} \) \(\mathstrut +\mathstrut 20q^{85} \) \(\mathstrut +\mathstrut 2q^{86} \) \(\mathstrut -\mathstrut 17q^{88} \) \(\mathstrut -\mathstrut 44q^{89} \) \(\mathstrut +\mathstrut 21q^{91} \) \(\mathstrut -\mathstrut 6q^{92} \) \(\mathstrut -\mathstrut 9q^{94} \) \(\mathstrut +\mathstrut 2q^{95} \) \(\mathstrut +\mathstrut 17q^{97} \) \(\mathstrut -\mathstrut 4q^{98} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{13}\mathstrut -\mathstrut \) \(4\) \(x^{12}\mathstrut -\mathstrut \) \(10\) \(x^{11}\mathstrut +\mathstrut \) \(53\) \(x^{10}\mathstrut +\mathstrut \) \(19\) \(x^{9}\mathstrut -\mathstrut \) \(242\) \(x^{8}\mathstrut +\mathstrut \) \(61\) \(x^{7}\mathstrut +\mathstrut \) \(467\) \(x^{6}\mathstrut -\mathstrut \) \(211\) \(x^{5}\mathstrut -\mathstrut \) \(372\) \(x^{4}\mathstrut +\mathstrut \) \(146\) \(x^{3}\mathstrut +\mathstrut \) \(116\) \(x^{2}\mathstrut -\mathstrut \) \(12\) \(x\mathstrut -\mathstrut \) \(8\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 3 \)
\(\beta_{3}\)\(=\)\((\)\( \nu^{12} - 2 \nu^{11} - 18 \nu^{10} + 29 \nu^{9} + 125 \nu^{8} - 144 \nu^{7} - 411 \nu^{6} + 285 \nu^{5} + 615 \nu^{4} - 190 \nu^{3} - 334 \nu^{2} - 12 \nu + 16 \)\()/4\)
\(\beta_{4}\)\(=\)\((\)\( \nu^{12} - 2 \nu^{11} - 15 \nu^{10} + 26 \nu^{9} + 84 \nu^{8} - 114 \nu^{7} - 224 \nu^{6} + 198 \nu^{5} + 287 \nu^{4} - 109 \nu^{3} - 150 \nu^{2} - 10 \nu + 10 \)\()/2\)
\(\beta_{5}\)\(=\)\((\)\( \nu^{12} + 2 \nu^{11} - 22 \nu^{10} - 27 \nu^{9} + 169 \nu^{8} + 120 \nu^{7} - 571 \nu^{6} - 203 \nu^{5} + 871 \nu^{4} + 118 \nu^{3} - 502 \nu^{2} - 28 \nu + 48 \)\()/4\)
\(\beta_{6}\)\(=\)\((\)\( 3 \nu^{12} - 8 \nu^{11} - 40 \nu^{10} + 105 \nu^{9} + 187 \nu^{8} - 470 \nu^{7} - 387 \nu^{6} + 863 \nu^{5} + 363 \nu^{4} - 586 \nu^{3} - 154 \nu^{2} + 72 \nu + 8 \)\()/4\)
\(\beta_{7}\)\(=\)\((\)\( 3 \nu^{12} - 8 \nu^{11} - 40 \nu^{10} + 105 \nu^{9} + 187 \nu^{8} - 470 \nu^{7} - 387 \nu^{6} + 863 \nu^{5} + 363 \nu^{4} - 590 \nu^{3} - 154 \nu^{2} + 92 \nu + 12 \)\()/4\)
\(\beta_{8}\)\(=\)\((\)\( -\nu^{12} + 4 \nu^{11} + 8 \nu^{10} - 51 \nu^{9} + 9 \nu^{8} + 220 \nu^{7} - 193 \nu^{6} - 387 \nu^{5} + 455 \nu^{4} + 246 \nu^{3} - 302 \nu^{2} - 44 \nu + 28 \)\()/2\)
\(\beta_{9}\)\(=\)\((\)\( -2 \nu^{12} + 5 \nu^{11} + 27 \nu^{10} - 65 \nu^{9} - 128 \nu^{8} + 285 \nu^{7} + 269 \nu^{6} - 498 \nu^{5} - 265 \nu^{4} + 298 \nu^{3} + 138 \nu^{2} - 30 \nu - 16 \)\()/2\)
\(\beta_{10}\)\(=\)\((\)\( 3 \nu^{12} - 8 \nu^{11} - 44 \nu^{10} + 109 \nu^{9} + 243 \nu^{8} - 510 \nu^{7} - 655 \nu^{6} + 975 \nu^{5} + 883 \nu^{4} - 670 \nu^{3} - 518 \nu^{2} + 60 \nu + 48 \)\()/4\)
\(\beta_{11}\)\(=\)\((\)\( 2 \nu^{12} - 7 \nu^{11} - 25 \nu^{10} + 93 \nu^{9} + 108 \nu^{8} - 419 \nu^{7} - 213 \nu^{6} + 758 \nu^{5} + 223 \nu^{4} - 478 \nu^{3} - 144 \nu^{2} + 42 \nu + 12 \)\()/2\)
\(\beta_{12}\)\(=\)\((\)\( 3 \nu^{12} - 7 \nu^{11} - 42 \nu^{10} + 91 \nu^{9} + 212 \nu^{8} - 399 \nu^{7} - 493 \nu^{6} + 696 \nu^{5} + 550 \nu^{4} - 405 \nu^{3} - 276 \nu^{2} + 12 \nu + 18 \)\()/2\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2}\mathstrut +\mathstrut \) \(3\)
\(\nu^{3}\)\(=\)\(-\)\(\beta_{7}\mathstrut +\mathstrut \) \(\beta_{6}\mathstrut +\mathstrut \) \(5\) \(\beta_{1}\mathstrut +\mathstrut \) \(1\)
\(\nu^{4}\)\(=\)\(-\)\(\beta_{12}\mathstrut -\mathstrut \) \(\beta_{9}\mathstrut -\mathstrut \) \(\beta_{7}\mathstrut +\mathstrut \) \(\beta_{6}\mathstrut +\mathstrut \) \(\beta_{4}\mathstrut +\mathstrut \) \(6\) \(\beta_{2}\mathstrut +\mathstrut \) \(\beta_{1}\mathstrut +\mathstrut \) \(15\)
\(\nu^{5}\)\(=\)\(-\)\(\beta_{12}\mathstrut +\mathstrut \) \(\beta_{11}\mathstrut -\mathstrut \) \(2\) \(\beta_{10}\mathstrut -\mathstrut \) \(\beta_{9}\mathstrut -\mathstrut \) \(8\) \(\beta_{7}\mathstrut +\mathstrut \) \(9\) \(\beta_{6}\mathstrut +\mathstrut \) \(\beta_{5}\mathstrut -\mathstrut \) \(\beta_{4}\mathstrut +\mathstrut \) \(2\) \(\beta_{3}\mathstrut +\mathstrut \) \(30\) \(\beta_{1}\mathstrut +\mathstrut \) \(10\)
\(\nu^{6}\)\(=\)\(-\)\(12\) \(\beta_{12}\mathstrut +\mathstrut \) \(2\) \(\beta_{11}\mathstrut -\mathstrut \) \(4\) \(\beta_{10}\mathstrut -\mathstrut \) \(11\) \(\beta_{9}\mathstrut -\mathstrut \) \(\beta_{8}\mathstrut -\mathstrut \) \(10\) \(\beta_{7}\mathstrut +\mathstrut \) \(11\) \(\beta_{6}\mathstrut +\mathstrut \) \(2\) \(\beta_{5}\mathstrut +\mathstrut \) \(10\) \(\beta_{4}\mathstrut +\mathstrut \) \(5\) \(\beta_{3}\mathstrut +\mathstrut \) \(35\) \(\beta_{2}\mathstrut +\mathstrut \) \(14\) \(\beta_{1}\mathstrut +\mathstrut \) \(89\)
\(\nu^{7}\)\(=\)\(-\)\(16\) \(\beta_{12}\mathstrut +\mathstrut \) \(14\) \(\beta_{11}\mathstrut -\mathstrut \) \(27\) \(\beta_{10}\mathstrut -\mathstrut \) \(15\) \(\beta_{9}\mathstrut -\mathstrut \) \(2\) \(\beta_{8}\mathstrut -\mathstrut \) \(58\) \(\beta_{7}\mathstrut +\mathstrut \) \(68\) \(\beta_{6}\mathstrut +\mathstrut \) \(14\) \(\beta_{5}\mathstrut -\mathstrut \) \(8\) \(\beta_{4}\mathstrut +\mathstrut \) \(29\) \(\beta_{3}\mathstrut +\mathstrut \) \(193\) \(\beta_{1}\mathstrut +\mathstrut \) \(86\)
\(\nu^{8}\)\(=\)\(-\)\(109\) \(\beta_{12}\mathstrut +\mathstrut \) \(31\) \(\beta_{11}\mathstrut -\mathstrut \) \(59\) \(\beta_{10}\mathstrut -\mathstrut \) \(95\) \(\beta_{9}\mathstrut -\mathstrut \) \(14\) \(\beta_{8}\mathstrut -\mathstrut \) \(84\) \(\beta_{7}\mathstrut +\mathstrut \) \(98\) \(\beta_{6}\mathstrut +\mathstrut \) \(31\) \(\beta_{5}\mathstrut +\mathstrut \) \(77\) \(\beta_{4}\mathstrut +\mathstrut \) \(72\) \(\beta_{3}\mathstrut +\mathstrut \) \(207\) \(\beta_{2}\mathstrut +\mathstrut \) \(141\) \(\beta_{1}\mathstrut +\mathstrut \) \(571\)
\(\nu^{9}\)\(=\)\(-\)\(174\) \(\beta_{12}\mathstrut +\mathstrut \) \(139\) \(\beta_{11}\mathstrut -\mathstrut \) \(262\) \(\beta_{10}\mathstrut -\mathstrut \) \(156\) \(\beta_{9}\mathstrut -\mathstrut \) \(31\) \(\beta_{8}\mathstrut -\mathstrut \) \(414\) \(\beta_{7}\mathstrut +\mathstrut \) \(495\) \(\beta_{6}\mathstrut +\mathstrut \) \(140\) \(\beta_{5}\mathstrut -\mathstrut \) \(44\) \(\beta_{4}\mathstrut +\mathstrut \) \(293\) \(\beta_{3}\mathstrut +\mathstrut \) \(4\) \(\beta_{2}\mathstrut +\mathstrut \) \(1284\) \(\beta_{1}\mathstrut +\mathstrut \) \(694\)
\(\nu^{10}\)\(=\)\(-\)\(894\) \(\beta_{12}\mathstrut +\mathstrut \) \(327\) \(\beta_{11}\mathstrut -\mathstrut \) \(607\) \(\beta_{10}\mathstrut -\mathstrut \) \(757\) \(\beta_{9}\mathstrut -\mathstrut \) \(140\) \(\beta_{8}\mathstrut -\mathstrut \) \(673\) \(\beta_{7}\mathstrut +\mathstrut \) \(812\) \(\beta_{6}\mathstrut +\mathstrut \) \(328\) \(\beta_{5}\mathstrut +\mathstrut \) \(546\) \(\beta_{4}\mathstrut +\mathstrut \) \(732\) \(\beta_{3}\mathstrut +\mathstrut \) \(1246\) \(\beta_{2}\mathstrut +\mathstrut \) \(1252\) \(\beta_{1}\mathstrut +\mathstrut \) \(3811\)
\(\nu^{11}\)\(=\)\(-\)\(1613\) \(\beta_{12}\mathstrut +\mathstrut \) \(1210\) \(\beta_{11}\mathstrut -\mathstrut \) \(2248\) \(\beta_{10}\mathstrut -\mathstrut \) \(1404\) \(\beta_{9}\mathstrut -\mathstrut \) \(328\) \(\beta_{8}\mathstrut -\mathstrut \) \(2952\) \(\beta_{7}\mathstrut +\mathstrut \) \(3573\) \(\beta_{6}\mathstrut +\mathstrut \) \(1226\) \(\beta_{5}\mathstrut -\mathstrut \) \(175\) \(\beta_{4}\mathstrut +\mathstrut \) \(2571\) \(\beta_{3}\mathstrut +\mathstrut \) \(83\) \(\beta_{2}\mathstrut +\mathstrut \) \(8714\) \(\beta_{1}\mathstrut +\mathstrut \) \(5431\)
\(\nu^{12}\)\(=\)\(-\)\(6983\) \(\beta_{12}\mathstrut +\mathstrut \) \(2953\) \(\beta_{11}\mathstrut -\mathstrut \) \(5411\) \(\beta_{10}\mathstrut -\mathstrut \) \(5816\) \(\beta_{9}\mathstrut -\mathstrut \) \(1226\) \(\beta_{8}\mathstrut -\mathstrut \) \(5269\) \(\beta_{7}\mathstrut +\mathstrut \) \(6480\) \(\beta_{6}\mathstrut +\mathstrut \) \(2974\) \(\beta_{5}\mathstrut +\mathstrut \) \(3757\) \(\beta_{4}\mathstrut +\mathstrut \) \(6486\) \(\beta_{3}\mathstrut +\mathstrut \) \(7632\) \(\beta_{2}\mathstrut +\mathstrut \) \(10446\) \(\beta_{1}\mathstrut +\mathstrut \) \(26023\)

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
2.70878
2.29288
2.28328
1.45503
1.27342
1.18273
0.307326
−0.305711
−0.423652
−0.910949
−1.58856
−1.82297
−2.45160
−2.70878 0 5.33747 −2.87857 0 1.00000 −9.04047 0 7.79741
1.2 −2.29288 0 3.25728 0.132682 0 1.00000 −2.88278 0 −0.304223
1.3 −2.28328 0 3.21337 −2.58645 0 1.00000 −2.77047 0 5.90560
1.4 −1.45503 0 0.117115 0.420019 0 1.00000 2.73966 0 −0.611141
1.5 −1.27342 0 −0.378389 2.08575 0 1.00000 3.02870 0 −2.65604
1.6 −1.18273 0 −0.601156 −4.39766 0 1.00000 3.07646 0 5.20123
1.7 −0.307326 0 −1.90555 0.988649 0 1.00000 1.20028 0 −0.303838
1.8 0.305711 0 −1.90654 0.276458 0 1.00000 −1.19427 0 0.0845162
1.9 0.423652 0 −1.82052 −2.66196 0 1.00000 −1.61857 0 −1.12774
1.10 0.910949 0 −1.17017 2.06751 0 1.00000 −2.88787 0 1.88339
1.11 1.58856 0 0.523519 −3.57671 0 1.00000 −2.34548 0 −5.68181
1.12 1.82297 0 1.32323 −0.630622 0 1.00000 −1.23374 0 −1.14961
1.13 2.45160 0 4.01034 −1.23909 0 1.00000 4.92856 0 −3.03775
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.13
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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8001))\):

\(T_{2}^{13} + \cdots\)
\(T_{5}^{13} + \cdots\)