Properties

Label 4010.2.a.j
Level 4010
Weight 2
Character orbit 4010.a
Self dual Yes
Analytic conductor 32.020
Analytic rank 1
Dimension 12
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 4010 = 2 \cdot 5 \cdot 401 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 4010.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(32.0200112105\)
Analytic rank: \(1\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
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.

\(f(q)\) \(=\) \( q\) \(+ q^{2}\) \( -\beta_{1} q^{3} \) \(+ q^{4}\) \(- q^{5}\) \( -\beta_{1} q^{6} \) \( + ( -1 - \beta_{11} ) q^{7} \) \(+ q^{8}\) \( + \beta_{2} q^{9} \) \(+O(q^{10})\) \( q\) \(+ q^{2}\) \( -\beta_{1} q^{3} \) \(+ q^{4}\) \(- q^{5}\) \( -\beta_{1} q^{6} \) \( + ( -1 - \beta_{11} ) q^{7} \) \(+ q^{8}\) \( + \beta_{2} q^{9} \) \(- q^{10}\) \( + ( \beta_{1} - \beta_{2} - \beta_{6} - \beta_{7} + \beta_{10} + \beta_{11} ) q^{11} \) \( -\beta_{1} q^{12} \) \( + ( -1 + \beta_{1} - \beta_{2} - \beta_{3} + \beta_{5} + \beta_{7} + \beta_{9} + \beta_{11} ) q^{13} \) \( + ( -1 - \beta_{11} ) q^{14} \) \( + \beta_{1} q^{15} \) \(+ q^{16}\) \( + ( -1 + \beta_{2} + \beta_{3} - \beta_{5} + \beta_{6} + \beta_{7} - \beta_{8} - \beta_{9} ) q^{17} \) \( + \beta_{2} q^{18} \) \( + ( \beta_{2} + \beta_{3} - \beta_{5} + \beta_{6} - \beta_{7} + \beta_{8} - \beta_{9} - \beta_{10} - \beta_{11} ) q^{19} \) \(- q^{20}\) \( + ( -1 + 2 \beta_{1} + \beta_{3} - \beta_{5} + \beta_{6} - \beta_{10} ) q^{21} \) \( + ( \beta_{1} - \beta_{2} - \beta_{6} - \beta_{7} + \beta_{10} + \beta_{11} ) q^{22} \) \( + ( -1 + \beta_{1} + \beta_{4} + \beta_{5} - 2 \beta_{10} ) q^{23} \) \( -\beta_{1} q^{24} \) \(+ q^{25}\) \( + ( -1 + \beta_{1} - \beta_{2} - \beta_{3} + \beta_{5} + \beta_{7} + \beta_{9} + \beta_{11} ) q^{26} \) \( + ( -1 + \beta_{1} - \beta_{2} - \beta_{4} - \beta_{6} + \beta_{10} + \beta_{11} ) q^{27} \) \( + ( -1 - \beta_{11} ) q^{28} \) \( + ( -1 + \beta_{1} - \beta_{2} - 2 \beta_{3} + \beta_{7} + \beta_{8} + \beta_{9} - \beta_{10} + \beta_{11} ) q^{29} \) \( + \beta_{1} q^{30} \) \( + ( -2 - \beta_{2} - 2 \beta_{3} - 2 \beta_{4} + \beta_{7} + \beta_{10} + \beta_{11} ) q^{31} \) \(+ q^{32}\) \( + ( -1 + \beta_{1} - \beta_{3} + 2 \beta_{4} + \beta_{5} - \beta_{9} ) q^{33} \) \( + ( -1 + \beta_{2} + \beta_{3} - \beta_{5} + \beta_{6} + \beta_{7} - \beta_{8} - \beta_{9} ) q^{34} \) \( + ( 1 + \beta_{11} ) q^{35} \) \( + \beta_{2} q^{36} \) \( + ( -2 + 2 \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} + 2 \beta_{5} - \beta_{7} - \beta_{8} ) q^{37} \) \( + ( \beta_{2} + \beta_{3} - \beta_{5} + \beta_{6} - \beta_{7} + \beta_{8} - \beta_{9} - \beta_{10} - \beta_{11} ) q^{38} \) \( + ( -1 - \beta_{2} - 2 \beta_{4} + \beta_{5} + \beta_{9} ) q^{39} \) \(- q^{40}\) \( + ( -3 \beta_{1} + 3 \beta_{2} + 3 \beta_{3} - 2 \beta_{4} - 3 \beta_{5} - 2 \beta_{7} + \beta_{8} - \beta_{9} - 3 \beta_{11} ) q^{41} \) \( + ( -1 + 2 \beta_{1} + \beta_{3} - \beta_{5} + \beta_{6} - \beta_{10} ) q^{42} \) \( + ( -2 + \beta_{6} + \beta_{7} ) q^{43} \) \( + ( \beta_{1} - \beta_{2} - \beta_{6} - \beta_{7} + \beta_{10} + \beta_{11} ) q^{44} \) \( -\beta_{2} q^{45} \) \( + ( -1 + \beta_{1} + \beta_{4} + \beta_{5} - 2 \beta_{10} ) q^{46} \) \( + ( -1 + 5 \beta_{1} - 3 \beta_{2} - \beta_{3} + 3 \beta_{4} + 3 \beta_{5} + 2 \beta_{9} + 2 \beta_{11} ) q^{47} \) \( -\beta_{1} q^{48} \) \( + ( -1 - \beta_{2} - \beta_{4} - 3 \beta_{6} - \beta_{7} + \beta_{8} + \beta_{9} + 2 \beta_{10} + 3 \beta_{11} ) q^{49} \) \(+ q^{50}\) \( + ( -2 + \beta_{1} + 2 \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} - 2 \beta_{11} ) q^{51} \) \( + ( -1 + \beta_{1} - \beta_{2} - \beta_{3} + \beta_{5} + \beta_{7} + \beta_{9} + \beta_{11} ) q^{52} \) \( + ( -4 - 2 \beta_{1} + \beta_{2} - 2 \beta_{4} + 2 \beta_{6} - \beta_{8} - \beta_{9} - 2 \beta_{11} ) q^{53} \) \( + ( -1 + \beta_{1} - \beta_{2} - \beta_{4} - \beta_{6} + \beta_{10} + \beta_{11} ) q^{54} \) \( + ( -\beta_{1} + \beta_{2} + \beta_{6} + \beta_{7} - \beta_{10} - \beta_{11} ) q^{55} \) \( + ( -1 - \beta_{11} ) q^{56} \) \( + ( -1 - \beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} - \beta_{6} - \beta_{9} + \beta_{10} ) q^{57} \) \( + ( -1 + \beta_{1} - \beta_{2} - 2 \beta_{3} + \beta_{7} + \beta_{8} + \beta_{9} - \beta_{10} + \beta_{11} ) q^{58} \) \( + ( 1 + 3 \beta_{2} + 2 \beta_{4} - 2 \beta_{5} - \beta_{6} + \beta_{7} - \beta_{8} + \beta_{11} ) q^{59} \) \( + \beta_{1} q^{60} \) \( + ( -2 - \beta_{1} + 4 \beta_{2} + 2 \beta_{3} - 2 \beta_{5} - \beta_{6} - \beta_{7} - \beta_{9} - 2 \beta_{11} ) q^{61} \) \( + ( -2 - \beta_{2} - 2 \beta_{3} - 2 \beta_{4} + \beta_{7} + \beta_{10} + \beta_{11} ) q^{62} \) \( + ( -2 + 2 \beta_{1} - \beta_{3} + 2 \beta_{4} + \beta_{6} + \beta_{7} - \beta_{8} - \beta_{10} ) q^{63} \) \(+ q^{64}\) \( + ( 1 - \beta_{1} + \beta_{2} + \beta_{3} - \beta_{5} - \beta_{7} - \beta_{9} - \beta_{11} ) q^{65} \) \( + ( -1 + \beta_{1} - \beta_{3} + 2 \beta_{4} + \beta_{5} - \beta_{9} ) q^{66} \) \( + ( -2 - 4 \beta_{1} + 5 \beta_{2} + 2 \beta_{3} + \beta_{4} - 4 \beta_{5} + \beta_{6} + \beta_{7} - 3 \beta_{8} - 3 \beta_{9} + \beta_{10} - \beta_{11} ) q^{67} \) \( + ( -1 + \beta_{2} + \beta_{3} - \beta_{5} + \beta_{6} + \beta_{7} - \beta_{8} - \beta_{9} ) q^{68} \) \( + ( -1 + 3 \beta_{1} - 4 \beta_{2} - 2 \beta_{3} - \beta_{4} + 2 \beta_{5} - \beta_{7} - \beta_{8} + 2 \beta_{9} + \beta_{10} + \beta_{11} ) q^{69} \) \( + ( 1 + \beta_{11} ) q^{70} \) \( + ( -2 - \beta_{1} + 4 \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} + 2 \beta_{6} + 2 \beta_{7} + \beta_{8} - \beta_{9} + \beta_{10} - \beta_{11} ) q^{71} \) \( + \beta_{2} q^{72} \) \( + ( 3 \beta_{1} - \beta_{2} - \beta_{3} + 3 \beta_{4} + 2 \beta_{5} - 2 \beta_{6} - \beta_{7} + 2 \beta_{9} + \beta_{11} ) q^{73} \) \( + ( -2 + 2 \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} + 2 \beta_{5} - \beta_{7} - \beta_{8} ) q^{74} \) \( -\beta_{1} q^{75} \) \( + ( \beta_{2} + \beta_{3} - \beta_{5} + \beta_{6} - \beta_{7} + \beta_{8} - \beta_{9} - \beta_{10} - \beta_{11} ) q^{76} \) \( + ( -3 - 2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} - 2 \beta_{6} + 2 \beta_{9} + 2 \beta_{10} - \beta_{11} ) q^{77} \) \( + ( -1 - \beta_{2} - 2 \beta_{4} + \beta_{5} + \beta_{9} ) q^{78} \) \( + ( -3 + \beta_{1} - 2 \beta_{2} - \beta_{3} + \beta_{4} - \beta_{8} + 2 \beta_{9} - \beta_{10} + 2 \beta_{11} ) q^{79} \) \(- q^{80}\) \( + ( -4 + \beta_{1} - 3 \beta_{2} - \beta_{3} + \beta_{5} + \beta_{7} + \beta_{8} ) q^{81} \) \( + ( -3 \beta_{1} + 3 \beta_{2} + 3 \beta_{3} - 2 \beta_{4} - 3 \beta_{5} - 2 \beta_{7} + \beta_{8} - \beta_{9} - 3 \beta_{11} ) q^{82} \) \( + ( -2 - 3 \beta_{1} + 4 \beta_{2} + 2 \beta_{3} - 2 \beta_{6} + \beta_{7} + \beta_{9} + \beta_{10} - 2 \beta_{11} ) q^{83} \) \( + ( -1 + 2 \beta_{1} + \beta_{3} - \beta_{5} + \beta_{6} - \beta_{10} ) q^{84} \) \( + ( 1 - \beta_{2} - \beta_{3} + \beta_{5} - \beta_{6} - \beta_{7} + \beta_{8} + \beta_{9} ) q^{85} \) \( + ( -2 + \beta_{6} + \beta_{7} ) q^{86} \) \( + ( -1 - \beta_{1} - \beta_{2} + \beta_{3} - 3 \beta_{4} - \beta_{7} + \beta_{8} - \beta_{11} ) q^{87} \) \( + ( \beta_{1} - \beta_{2} - \beta_{6} - \beta_{7} + \beta_{10} + \beta_{11} ) q^{88} \) \( + ( -6 \beta_{2} - 3 \beta_{3} + 3 \beta_{5} - \beta_{6} - 2 \beta_{7} + \beta_{8} + 3 \beta_{10} + 2 \beta_{11} ) q^{89} \) \( -\beta_{2} q^{90} \) \( + ( 1 + 2 \beta_{2} + 2 \beta_{3} + 2 \beta_{4} - \beta_{5} + \beta_{6} - 2 \beta_{7} - 2 \beta_{9} - \beta_{11} ) q^{91} \) \( + ( -1 + \beta_{1} + \beta_{4} + \beta_{5} - 2 \beta_{10} ) q^{92} \) \( + ( -3 + \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} - 3 \beta_{6} + 4 \beta_{8} + \beta_{9} + 2 \beta_{11} ) q^{93} \) \( + ( -1 + 5 \beta_{1} - 3 \beta_{2} - \beta_{3} + 3 \beta_{4} + 3 \beta_{5} + 2 \beta_{9} + 2 \beta_{11} ) q^{94} \) \( + ( -\beta_{2} - \beta_{3} + \beta_{5} - \beta_{6} + \beta_{7} - \beta_{8} + \beta_{9} + \beta_{10} + \beta_{11} ) q^{95} \) \( -\beta_{1} q^{96} \) \( + ( -1 - 2 \beta_{1} + 2 \beta_{2} + 4 \beta_{3} - \beta_{4} - 3 \beta_{5} + \beta_{6} - \beta_{7} + \beta_{8} - 2 \beta_{9} - 3 \beta_{10} - \beta_{11} ) q^{97} \) \( + ( -1 - \beta_{2} - \beta_{4} - 3 \beta_{6} - \beta_{7} + \beta_{8} + \beta_{9} + 2 \beta_{10} + 3 \beta_{11} ) q^{98} \) \( + ( -\beta_{1} + 2 \beta_{3} - \beta_{4} - \beta_{5} + 2 \beta_{6} - \beta_{7} - \beta_{9} - \beta_{10} - \beta_{11} ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(12q \) \(\mathstrut +\mathstrut 12q^{2} \) \(\mathstrut -\mathstrut 2q^{3} \) \(\mathstrut +\mathstrut 12q^{4} \) \(\mathstrut -\mathstrut 12q^{5} \) \(\mathstrut -\mathstrut 2q^{6} \) \(\mathstrut -\mathstrut 9q^{7} \) \(\mathstrut +\mathstrut 12q^{8} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(12q \) \(\mathstrut +\mathstrut 12q^{2} \) \(\mathstrut -\mathstrut 2q^{3} \) \(\mathstrut +\mathstrut 12q^{4} \) \(\mathstrut -\mathstrut 12q^{5} \) \(\mathstrut -\mathstrut 2q^{6} \) \(\mathstrut -\mathstrut 9q^{7} \) \(\mathstrut +\mathstrut 12q^{8} \) \(\mathstrut -\mathstrut 12q^{10} \) \(\mathstrut +\mathstrut q^{11} \) \(\mathstrut -\mathstrut 2q^{12} \) \(\mathstrut -\mathstrut 6q^{13} \) \(\mathstrut -\mathstrut 9q^{14} \) \(\mathstrut +\mathstrut 2q^{15} \) \(\mathstrut +\mathstrut 12q^{16} \) \(\mathstrut -\mathstrut 11q^{17} \) \(\mathstrut -\mathstrut 13q^{19} \) \(\mathstrut -\mathstrut 12q^{20} \) \(\mathstrut -\mathstrut 14q^{21} \) \(\mathstrut +\mathstrut q^{22} \) \(\mathstrut -\mathstrut 21q^{23} \) \(\mathstrut -\mathstrut 2q^{24} \) \(\mathstrut +\mathstrut 12q^{25} \) \(\mathstrut -\mathstrut 6q^{26} \) \(\mathstrut -\mathstrut 2q^{27} \) \(\mathstrut -\mathstrut 9q^{28} \) \(\mathstrut -\mathstrut 10q^{29} \) \(\mathstrut +\mathstrut 2q^{30} \) \(\mathstrut -\mathstrut 11q^{31} \) \(\mathstrut +\mathstrut 12q^{32} \) \(\mathstrut -\mathstrut 22q^{33} \) \(\mathstrut -\mathstrut 11q^{34} \) \(\mathstrut +\mathstrut 9q^{35} \) \(\mathstrut -\mathstrut 29q^{37} \) \(\mathstrut -\mathstrut 13q^{38} \) \(\mathstrut -\mathstrut 2q^{39} \) \(\mathstrut -\mathstrut 12q^{40} \) \(\mathstrut -\mathstrut q^{41} \) \(\mathstrut -\mathstrut 14q^{42} \) \(\mathstrut -\mathstrut 23q^{43} \) \(\mathstrut +\mathstrut q^{44} \) \(\mathstrut -\mathstrut 21q^{46} \) \(\mathstrut -\mathstrut 17q^{47} \) \(\mathstrut -\mathstrut 2q^{48} \) \(\mathstrut -\mathstrut 3q^{49} \) \(\mathstrut +\mathstrut 12q^{50} \) \(\mathstrut -\mathstrut 19q^{51} \) \(\mathstrut -\mathstrut 6q^{52} \) \(\mathstrut -\mathstrut 47q^{53} \) \(\mathstrut -\mathstrut 2q^{54} \) \(\mathstrut -\mathstrut q^{55} \) \(\mathstrut -\mathstrut 9q^{56} \) \(\mathstrut -\mathstrut 11q^{57} \) \(\mathstrut -\mathstrut 10q^{58} \) \(\mathstrut +\mathstrut 14q^{59} \) \(\mathstrut +\mathstrut 2q^{60} \) \(\mathstrut -\mathstrut 22q^{61} \) \(\mathstrut -\mathstrut 11q^{62} \) \(\mathstrut -\mathstrut 28q^{63} \) \(\mathstrut +\mathstrut 12q^{64} \) \(\mathstrut +\mathstrut 6q^{65} \) \(\mathstrut -\mathstrut 22q^{66} \) \(\mathstrut -\mathstrut 28q^{67} \) \(\mathstrut -\mathstrut 11q^{68} \) \(\mathstrut -\mathstrut q^{69} \) \(\mathstrut +\mathstrut 9q^{70} \) \(\mathstrut -\mathstrut 18q^{71} \) \(\mathstrut -\mathstrut 2q^{73} \) \(\mathstrut -\mathstrut 29q^{74} \) \(\mathstrut -\mathstrut 2q^{75} \) \(\mathstrut -\mathstrut 13q^{76} \) \(\mathstrut -\mathstrut 11q^{77} \) \(\mathstrut -\mathstrut 2q^{78} \) \(\mathstrut -\mathstrut 39q^{79} \) \(\mathstrut -\mathstrut 12q^{80} \) \(\mathstrut -\mathstrut 44q^{81} \) \(\mathstrut -\mathstrut q^{82} \) \(\mathstrut -\mathstrut 5q^{83} \) \(\mathstrut -\mathstrut 14q^{84} \) \(\mathstrut +\mathstrut 11q^{85} \) \(\mathstrut -\mathstrut 23q^{86} \) \(\mathstrut -\mathstrut 6q^{87} \) \(\mathstrut +\mathstrut q^{88} \) \(\mathstrut -\mathstrut 8q^{89} \) \(\mathstrut -\mathstrut 12q^{91} \) \(\mathstrut -\mathstrut 21q^{92} \) \(\mathstrut -\mathstrut 30q^{93} \) \(\mathstrut -\mathstrut 17q^{94} \) \(\mathstrut +\mathstrut 13q^{95} \) \(\mathstrut -\mathstrut 2q^{96} \) \(\mathstrut -\mathstrut 32q^{97} \) \(\mathstrut -\mathstrut 3q^{98} \) \(\mathstrut -\mathstrut 13q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{12}\mathstrut -\mathstrut \) \(2\) \(x^{11}\mathstrut -\mathstrut \) \(16\) \(x^{10}\mathstrut +\mathstrut \) \(30\) \(x^{9}\mathstrut +\mathstrut \) \(93\) \(x^{8}\mathstrut -\mathstrut \) \(162\) \(x^{7}\mathstrut -\mathstrut \) \(238\) \(x^{6}\mathstrut +\mathstrut \) \(391\) \(x^{5}\mathstrut +\mathstrut \) \(240\) \(x^{4}\mathstrut -\mathstrut \) \(408\) \(x^{3}\mathstrut -\mathstrut \) \(42\) \(x^{2}\mathstrut +\mathstrut \) \(120\) \(x\mathstrut -\mathstrut \) \(8\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 3 \)
\(\beta_{3}\)\(=\)\((\)\( 11 \nu^{11} - 6 \nu^{10} - 190 \nu^{9} + 80 \nu^{8} + 1171 \nu^{7} - 374 \nu^{6} - 3046 \nu^{5} + 809 \nu^{4} + 2952 \nu^{3} - 964 \nu^{2} - 604 \nu + 278 \)\()/58\)
\(\beta_{4}\)\(=\)\((\)\( 53 \nu^{11} - 108 \nu^{10} - 752 \nu^{9} + 1382 \nu^{8} + 3765 \nu^{7} - 5804 \nu^{6} - 8254 \nu^{5} + 9487 \nu^{4} + 7490 \nu^{3} - 5404 \nu^{2} - 2578 \nu + 596 \)\()/232\)
\(\beta_{5}\)\(=\)\((\)\( 25 \nu^{11} + 76 \nu^{10} - 648 \nu^{9} - 994 \nu^{8} + 5361 \nu^{7} + 4196 \nu^{6} - 17542 \nu^{5} - 5965 \nu^{4} + 21130 \nu^{3} - 356 \nu^{2} - 5786 \nu + 732 \)\()/232\)
\(\beta_{6}\)\(=\)\((\)\( -105 \nu^{11} + 52 \nu^{10} + 1840 \nu^{9} - 558 \nu^{8} - 11705 \nu^{7} + 1540 \nu^{6} + 32334 \nu^{5} - 467 \nu^{4} - 35154 \nu^{3} + 196 \nu^{2} + 10010 \nu - 12 \)\()/232\)
\(\beta_{7}\)\(=\)\((\)\( -113 \nu^{11} + 88 \nu^{10} + 1936 \nu^{9} - 1038 \nu^{8} - 12177 \nu^{7} + 3784 \nu^{6} + 34022 \nu^{5} - 5147 \nu^{4} - 39062 \nu^{3} + 3892 \nu^{2} + 13402 \nu + 60 \)\()/232\)
\(\beta_{8}\)\(=\)\((\)\( 33 \nu^{11} - 47 \nu^{10} - 512 \nu^{9} + 588 \nu^{8} + 2875 \nu^{7} - 2369 \nu^{6} - 7166 \nu^{5} + 3645 \nu^{4} + 7435 \nu^{3} - 2196 \nu^{2} - 2566 \nu + 312 \)\()/58\)
\(\beta_{9}\)\(=\)\((\)\( 223 \nu^{11} - 148 \nu^{10} - 3952 \nu^{9} + 1954 \nu^{8} + 25511 \nu^{7} - 8684 \nu^{6} - 72138 \nu^{5} + 16253 \nu^{4} + 82734 \nu^{3} - 14924 \nu^{2} - 27446 \nu + 1908 \)\()/232\)
\(\beta_{10}\)\(=\)\((\)\( 139 \nu^{11} - 176 \nu^{10} - 2248 \nu^{9} + 2250 \nu^{8} + 13247 \nu^{7} - 9424 \nu^{6} - 34578 \nu^{5} + 15601 \nu^{4} + 36886 \nu^{3} - 10684 \nu^{2} - 11666 \nu + 1388 \)\()/116\)
\(\beta_{11}\)\(=\)\((\)\( -165 \nu^{11} + 148 \nu^{10} + 2792 \nu^{9} - 1838 \nu^{8} - 17217 \nu^{7} + 7292 \nu^{6} + 46618 \nu^{5} - 11091 \nu^{4} - 50834 \nu^{3} + 8196 \nu^{2} + 15962 \nu - 1328 \)\()/116\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2}\mathstrut +\mathstrut \) \(3\)
\(\nu^{3}\)\(=\)\(-\)\(\beta_{11}\mathstrut -\mathstrut \) \(\beta_{10}\mathstrut +\mathstrut \) \(\beta_{6}\mathstrut +\mathstrut \) \(\beta_{4}\mathstrut +\mathstrut \) \(\beta_{2}\mathstrut +\mathstrut \) \(5\) \(\beta_{1}\mathstrut +\mathstrut \) \(1\)
\(\nu^{4}\)\(=\)\(\beta_{8}\mathstrut +\mathstrut \) \(\beta_{7}\mathstrut +\mathstrut \) \(\beta_{5}\mathstrut -\mathstrut \) \(\beta_{3}\mathstrut +\mathstrut \) \(6\) \(\beta_{2}\mathstrut +\mathstrut \) \(\beta_{1}\mathstrut +\mathstrut \) \(14\)
\(\nu^{5}\)\(=\)\(-\)\(8\) \(\beta_{11}\mathstrut -\mathstrut \) \(8\) \(\beta_{10}\mathstrut -\mathstrut \) \(\beta_{8}\mathstrut +\mathstrut \) \(\beta_{7}\mathstrut +\mathstrut \) \(7\) \(\beta_{6}\mathstrut +\mathstrut \) \(\beta_{5}\mathstrut +\mathstrut \) \(11\) \(\beta_{4}\mathstrut -\mathstrut \) \(\beta_{3}\mathstrut +\mathstrut \) \(9\) \(\beta_{2}\mathstrut +\mathstrut \) \(29\) \(\beta_{1}\mathstrut +\mathstrut \) \(10\)
\(\nu^{6}\)\(=\)\(-\)\(\beta_{10}\mathstrut +\mathstrut \) \(\beta_{9}\mathstrut +\mathstrut \) \(10\) \(\beta_{8}\mathstrut +\mathstrut \) \(12\) \(\beta_{7}\mathstrut -\mathstrut \) \(2\) \(\beta_{6}\mathstrut +\mathstrut \) \(10\) \(\beta_{5}\mathstrut +\mathstrut \) \(3\) \(\beta_{4}\mathstrut -\mathstrut \) \(12\) \(\beta_{3}\mathstrut +\mathstrut \) \(37\) \(\beta_{2}\mathstrut +\mathstrut \) \(11\) \(\beta_{1}\mathstrut +\mathstrut \) \(76\)
\(\nu^{7}\)\(=\)\(-\)\(53\) \(\beta_{11}\mathstrut -\mathstrut \) \(56\) \(\beta_{10}\mathstrut +\mathstrut \) \(2\) \(\beta_{9}\mathstrut -\mathstrut \) \(8\) \(\beta_{8}\mathstrut +\mathstrut \) \(14\) \(\beta_{7}\mathstrut +\mathstrut \) \(41\) \(\beta_{6}\mathstrut +\mathstrut \) \(14\) \(\beta_{5}\mathstrut +\mathstrut \) \(93\) \(\beta_{4}\mathstrut -\mathstrut \) \(16\) \(\beta_{3}\mathstrut +\mathstrut \) \(65\) \(\beta_{2}\mathstrut +\mathstrut \) \(182\) \(\beta_{1}\mathstrut +\mathstrut \) \(77\)
\(\nu^{8}\)\(=\)\(-\)\(16\) \(\beta_{10}\mathstrut +\mathstrut \) \(15\) \(\beta_{9}\mathstrut +\mathstrut \) \(79\) \(\beta_{8}\mathstrut +\mathstrut \) \(107\) \(\beta_{7}\mathstrut -\mathstrut \) \(26\) \(\beta_{6}\mathstrut +\mathstrut \) \(82\) \(\beta_{5}\mathstrut +\mathstrut \) \(50\) \(\beta_{4}\mathstrut -\mathstrut \) \(106\) \(\beta_{3}\mathstrut +\mathstrut \) \(238\) \(\beta_{2}\mathstrut +\mathstrut \) \(98\) \(\beta_{1}\mathstrut +\mathstrut \) \(449\)
\(\nu^{9}\)\(=\)\(-\)\(336\) \(\beta_{11}\mathstrut -\mathstrut \) \(384\) \(\beta_{10}\mathstrut +\mathstrut \) \(31\) \(\beta_{9}\mathstrut -\mathstrut \) \(41\) \(\beta_{8}\mathstrut +\mathstrut \) \(141\) \(\beta_{7}\mathstrut +\mathstrut \) \(228\) \(\beta_{6}\mathstrut +\mathstrut \) \(138\) \(\beta_{5}\mathstrut +\mathstrut \) \(718\) \(\beta_{4}\mathstrut -\mathstrut \) \(165\) \(\beta_{3}\mathstrut +\mathstrut \) \(450\) \(\beta_{2}\mathstrut +\mathstrut \) \(1189\) \(\beta_{1}\mathstrut +\mathstrut \) \(550\)
\(\nu^{10}\)\(=\)\(-\)\(\beta_{11}\mathstrut -\mathstrut \) \(180\) \(\beta_{10}\mathstrut +\mathstrut \) \(155\) \(\beta_{9}\mathstrut +\mathstrut \) \(584\) \(\beta_{8}\mathstrut +\mathstrut \) \(852\) \(\beta_{7}\mathstrut -\mathstrut \) \(245\) \(\beta_{6}\mathstrut +\mathstrut \) \(632\) \(\beta_{5}\mathstrut +\mathstrut \) \(560\) \(\beta_{4}\mathstrut -\mathstrut \) \(838\) \(\beta_{3}\mathstrut +\mathstrut \) \(1574\) \(\beta_{2}\mathstrut +\mathstrut \) \(820\) \(\beta_{1}\mathstrut +\mathstrut \) \(2799\)
\(\nu^{11}\)\(=\)\(-\)\(2109\) \(\beta_{11}\mathstrut -\mathstrut \) \(2634\) \(\beta_{10}\mathstrut +\mathstrut \) \(332\) \(\beta_{9}\mathstrut -\mathstrut \) \(123\) \(\beta_{8}\mathstrut +\mathstrut \) \(1243\) \(\beta_{7}\mathstrut +\mathstrut \) \(1231\) \(\beta_{6}\mathstrut +\mathstrut \) \(1185\) \(\beta_{5}\mathstrut +\mathstrut \) \(5323\) \(\beta_{4}\mathstrut -\mathstrut \) \(1439\) \(\beta_{3}\mathstrut +\mathstrut \) \(3109\) \(\beta_{2}\mathstrut +\mathstrut \) \(7941\) \(\beta_{1}\mathstrut +\mathstrut \) \(3857\)

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
2.67561
2.34926
2.12920
1.16326
1.11338
0.803872
0.0694414
−0.660223
−1.58108
−1.73269
−1.82424
−2.50579
1.00000 −2.67561 1.00000 −1.00000 −2.67561 0.990876 1.00000 4.15891 −1.00000
1.2 1.00000 −2.34926 1.00000 −1.00000 −2.34926 −0.317129 1.00000 2.51901 −1.00000
1.3 1.00000 −2.12920 1.00000 −1.00000 −2.12920 −4.06902 1.00000 1.53348 −1.00000
1.4 1.00000 −1.16326 1.00000 −1.00000 −1.16326 −3.35475 1.00000 −1.64681 −1.00000
1.5 1.00000 −1.11338 1.00000 −1.00000 −1.11338 3.90936 1.00000 −1.76038 −1.00000
1.6 1.00000 −0.803872 1.00000 −1.00000 −0.803872 1.44710 1.00000 −2.35379 −1.00000
1.7 1.00000 −0.0694414 1.00000 −1.00000 −0.0694414 0.700507 1.00000 −2.99518 −1.00000
1.8 1.00000 0.660223 1.00000 −1.00000 0.660223 0.752610 1.00000 −2.56411 −1.00000
1.9 1.00000 1.58108 1.00000 −1.00000 1.58108 0.389201 1.00000 −0.500177 −1.00000
1.10 1.00000 1.73269 1.00000 −1.00000 1.73269 −1.89534 1.00000 0.00220907 −1.00000
1.11 1.00000 1.82424 1.00000 −1.00000 1.82424 −2.93972 1.00000 0.327838 −1.00000
1.12 1.00000 2.50579 1.00000 −1.00000 2.50579 −4.61369 1.00000 3.27901 −1.00000
\(n\): 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
\(2\) \(-1\)
\(5\) \(1\)
\(401\) \(-1\)

Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4010))\):

\(T_{3}^{12} + \cdots\)
\(T_{7}^{12} + \cdots\)
\(T_{11}^{12} - \cdots\)