Properties

Label 8037.2.a.m
Level 8037
Weight 2
Character orbit 8037.a
Self dual Yes
Analytic conductor 64.176
Analytic rank 1
Dimension 12
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 8037 = 3^{2} \cdot 19 \cdot 47 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 8037.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(64.1757681045\)
Analytic rank: \(1\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{12}\mathstrut -\mathstrut \) \(x^{11}\mathstrut -\mathstrut \) \(15\) \(x^{10}\mathstrut +\mathstrut \) \(14\) \(x^{9}\mathstrut +\mathstrut \) \(84\) \(x^{8}\mathstrut -\mathstrut \) \(76\) \(x^{7}\mathstrut -\mathstrut \) \(213\) \(x^{6}\mathstrut +\mathstrut \) \(196\) \(x^{5}\mathstrut +\mathstrut \) \(225\) \(x^{4}\mathstrut -\mathstrut \) \(229\) \(x^{3}\mathstrut -\mathstrut \) \(49\) \(x^{2}\mathstrut +\mathstrut \) \(83\) \(x\mathstrut -\mathstrut \) \(17\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 3 \)
\(\beta_{3}\)\(=\)\( \nu^{7} - \nu^{6} - 9 \nu^{5} + 7 \nu^{4} + 23 \nu^{3} - 14 \nu^{2} - 14 \nu + 7 \)
\(\beta_{4}\)\(=\)\( -\nu^{11} + 12 \nu^{9} + \nu^{8} - 48 \nu^{7} - 5 \nu^{6} + 73 \nu^{5} + 8 \nu^{4} - 33 \nu^{3} - 11 \nu^{2} + 2 \nu + 4 \)
\(\beta_{5}\)\(=\)\( 2 \nu^{9} - 24 \nu^{7} - \nu^{6} + 95 \nu^{5} - 139 \nu^{3} + 16 \nu^{2} + 53 \nu - 13 \)
\(\beta_{6}\)\(=\)\( -\nu^{11} - \nu^{10} + 13 \nu^{9} + 13 \nu^{8} - 59 \nu^{7} - 53 \nu^{6} + 116 \nu^{5} + 76 \nu^{4} - 99 \nu^{3} - 24 \nu^{2} + 28 \nu - 4 \)
\(\beta_{7}\)\(=\)\( -\nu^{11} + \nu^{10} + 13 \nu^{9} - 11 \nu^{8} - 61 \nu^{7} + 43 \nu^{6} + 125 \nu^{5} - 70 \nu^{4} - 106 \nu^{3} + 43 \nu^{2} + 29 \nu - 10 \)
\(\beta_{8}\)\(=\)\( \nu^{11} - 14 \nu^{9} + 72 \nu^{7} - 5 \nu^{6} - 170 \nu^{5} + 31 \nu^{4} + 182 \nu^{3} - 53 \nu^{2} - 65 \nu + 21 \)
\(\beta_{9}\)\(=\)\( \nu^{11} + \nu^{10} - 14 \nu^{9} - 12 \nu^{8} + 71 \nu^{7} + 43 \nu^{6} - 165 \nu^{5} - 42 \nu^{4} + 175 \nu^{3} - 20 \nu^{2} - 59 \nu + 18 \)
\(\beta_{10}\)\(=\)\( \nu^{11} + \nu^{10} - 14 \nu^{9} - 12 \nu^{8} + 71 \nu^{7} + 43 \nu^{6} - 165 \nu^{5} - 42 \nu^{4} + 176 \nu^{3} - 21 \nu^{2} - 63 \nu + 21 \)
\(\beta_{11}\)\(=\)\( \nu^{11} - 15 \nu^{9} + 84 \nu^{7} - 5 \nu^{6} - 217 \nu^{5} + 35 \nu^{4} + 249 \nu^{3} - 70 \nu^{2} - 89 \nu + 32 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2}\mathstrut +\mathstrut \) \(3\)
\(\nu^{3}\)\(=\)\(\beta_{10}\mathstrut -\mathstrut \) \(\beta_{9}\mathstrut +\mathstrut \) \(\beta_{2}\mathstrut +\mathstrut \) \(4\) \(\beta_{1}\)
\(\nu^{4}\)\(=\)\(-\)\(\beta_{11}\mathstrut +\mathstrut \) \(\beta_{10}\mathstrut -\mathstrut \) \(\beta_{7}\mathstrut +\mathstrut \) \(\beta_{4}\mathstrut +\mathstrut \) \(5\) \(\beta_{2}\mathstrut +\mathstrut \) \(\beta_{1}\mathstrut +\mathstrut \) \(12\)
\(\nu^{5}\)\(=\)\(7\) \(\beta_{10}\mathstrut -\mathstrut \) \(5\) \(\beta_{9}\mathstrut -\mathstrut \) \(2\) \(\beta_{8}\mathstrut -\mathstrut \) \(\beta_{7}\mathstrut +\mathstrut \) \(\beta_{6}\mathstrut +\mathstrut \) \(8\) \(\beta_{2}\mathstrut +\mathstrut \) \(17\) \(\beta_{1}\mathstrut +\mathstrut \) \(3\)
\(\nu^{6}\)\(=\)\(-\)\(10\) \(\beta_{11}\mathstrut +\mathstrut \) \(10\) \(\beta_{10}\mathstrut -\mathstrut \) \(9\) \(\beta_{7}\mathstrut +\mathstrut \) \(\beta_{6}\mathstrut -\mathstrut \) \(\beta_{5}\mathstrut +\mathstrut \) \(8\) \(\beta_{4}\mathstrut +\mathstrut \) \(25\) \(\beta_{2}\mathstrut +\mathstrut \) \(10\) \(\beta_{1}\mathstrut +\mathstrut \) \(54\)
\(\nu^{7}\)\(=\)\(-\)\(3\) \(\beta_{11}\mathstrut +\mathstrut \) \(43\) \(\beta_{10}\mathstrut -\mathstrut \) \(22\) \(\beta_{9}\mathstrut -\mathstrut \) \(18\) \(\beta_{8}\mathstrut -\mathstrut \) \(11\) \(\beta_{7}\mathstrut +\mathstrut \) \(10\) \(\beta_{6}\mathstrut -\mathstrut \) \(\beta_{5}\mathstrut +\mathstrut \) \(\beta_{4}\mathstrut +\mathstrut \) \(\beta_{3}\mathstrut +\mathstrut \) \(53\) \(\beta_{2}\mathstrut +\mathstrut \) \(78\) \(\beta_{1}\mathstrut +\mathstrut \) \(32\)
\(\nu^{8}\)\(=\)\(-\)\(71\) \(\beta_{11}\mathstrut +\mathstrut \) \(75\) \(\beta_{10}\mathstrut -\mathstrut \) \(3\) \(\beta_{8}\mathstrut -\mathstrut \) \(62\) \(\beta_{7}\mathstrut +\mathstrut \) \(13\) \(\beta_{6}\mathstrut -\mathstrut \) \(10\) \(\beta_{5}\mathstrut +\mathstrut \) \(50\) \(\beta_{4}\mathstrut +\mathstrut \) \(134\) \(\beta_{2}\mathstrut +\mathstrut \) \(75\) \(\beta_{1}\mathstrut +\mathstrut \) \(264\)
\(\nu^{9}\)\(=\)\(-\)\(41\) \(\beta_{11}\mathstrut +\mathstrut \) \(258\) \(\beta_{10}\mathstrut -\mathstrut \) \(96\) \(\beta_{9}\mathstrut -\mathstrut \) \(121\) \(\beta_{8}\mathstrut -\mathstrut \) \(89\) \(\beta_{7}\mathstrut +\mathstrut \) \(73\) \(\beta_{6}\mathstrut -\mathstrut \) \(12\) \(\beta_{5}\mathstrut +\mathstrut \) \(16\) \(\beta_{4}\mathstrut +\mathstrut \) \(12\) \(\beta_{3}\mathstrut +\mathstrut \) \(330\) \(\beta_{2}\mathstrut +\mathstrut \) \(385\) \(\beta_{1}\mathstrut +\mathstrut \) \(251\)
\(\nu^{10}\)\(=\)\(-\)\(448\) \(\beta_{11}\mathstrut +\mathstrut \) \(508\) \(\beta_{10}\mathstrut -\mathstrut \) \(3\) \(\beta_{9}\mathstrut -\mathstrut \) \(45\) \(\beta_{8}\mathstrut -\mathstrut \) \(391\) \(\beta_{7}\mathstrut +\mathstrut \) \(113\) \(\beta_{6}\mathstrut -\mathstrut \) \(73\) \(\beta_{5}\mathstrut +\mathstrut \) \(290\) \(\beta_{4}\mathstrut +\mathstrut \) \(\beta_{3}\mathstrut +\mathstrut \) \(760\) \(\beta_{2}\mathstrut +\mathstrut \) \(508\) \(\beta_{1}\mathstrut +\mathstrut \) \(1373\)
\(\nu^{11}\)\(=\)\(-\)\(377\) \(\beta_{11}\mathstrut +\mathstrut \) \(1543\) \(\beta_{10}\mathstrut -\mathstrut \) \(428\) \(\beta_{9}\mathstrut -\mathstrut \) \(737\) \(\beta_{8}\mathstrut -\mathstrut \) \(638\) \(\beta_{7}\mathstrut +\mathstrut \) \(477\) \(\beta_{6}\mathstrut -\mathstrut \) \(101\) \(\beta_{5}\mathstrut +\mathstrut \) \(161\) \(\beta_{4}\mathstrut +\mathstrut \) \(96\) \(\beta_{3}\mathstrut +\mathstrut \) \(2005\) \(\beta_{2}\mathstrut +\mathstrut \) \(2020\) \(\beta_{1}\mathstrut +\mathstrut \) \(1756\)

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
−2.13808
−1.93999
−1.75952
−1.67181
−0.792001
0.332508
0.531564
0.821767
1.38739
1.87636
1.88968
2.46212
−2.13808 0 2.57138 −1.02698 0 −3.43362 −1.22166 0 2.19577
1.2 −1.93999 0 1.76355 −1.45338 0 −4.17438 0.458705 0 2.81953
1.3 −1.75952 0 1.09589 1.58069 0 4.03503 1.59079 0 −2.78124
1.4 −1.67181 0 0.794958 2.50226 0 −1.08348 2.01460 0 −4.18331
1.5 −0.792001 0 −1.37273 1.55768 0 −2.36074 2.67121 0 −1.23369
1.6 0.332508 0 −1.88944 1.32217 0 1.64871 −1.29327 0 0.439633
1.7 0.531564 0 −1.71744 −0.555438 0 1.54136 −1.97606 0 −0.295251
1.8 0.821767 0 −1.32470 0.409207 0 −1.21348 −2.73213 0 0.336273
1.9 1.38739 0 −0.0751478 4.34713 0 −2.61174 −2.87904 0 6.03117
1.10 1.87636 0 1.52073 −3.86690 0 −2.13974 −0.899279 0 −7.25570
1.11 1.88968 0 1.57091 0.784971 0 −0.254031 −0.810846 0 1.48335
1.12 2.46212 0 4.06203 1.39858 0 −2.95389 5.07697 0 3.44347
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.12
Significant digits:
Format:

Inner twists

This newform does not admit any (nontrivial) inner twists.

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(19\) \(-1\)
\(47\) \(1\)

Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8037))\):

\(T_{2}^{12} - \cdots\)
\(T_{5}^{12} - \cdots\)