Properties

Label 6027.2.a.bj
Level 6027
Weight 2
Character orbit 6027.a
Self dual Yes
Analytic conductor 48.126
Analytic rank 0
Dimension 14
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 6027 = 3 \cdot 7^{2} \cdot 41 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 6027.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(48.1258372982\)
Analytic rank: \(0\)
Dimension: \(14\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
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 \(1,\beta_1,\ldots,\beta_{13}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \( -\beta_{1} q^{2} \) \(- q^{3}\) \( + ( 1 + \beta_{2} ) q^{4} \) \( + ( 1 + \beta_{3} ) q^{5} \) \( + \beta_{1} q^{6} \) \( + ( -1 - \beta_{1} - \beta_{2} - \beta_{5} - \beta_{6} ) q^{8} \) \(+ q^{9}\) \(+O(q^{10})\) \( q\) \( -\beta_{1} q^{2} \) \(- q^{3}\) \( + ( 1 + \beta_{2} ) q^{4} \) \( + ( 1 + \beta_{3} ) q^{5} \) \( + \beta_{1} q^{6} \) \( + ( -1 - \beta_{1} - \beta_{2} - \beta_{5} - \beta_{6} ) q^{8} \) \(+ q^{9}\) \( + ( -2 \beta_{1} - \beta_{2} - \beta_{5} - \beta_{6} + \beta_{7} - \beta_{11} - \beta_{13} ) q^{10} \) \( + ( -\beta_{1} + \beta_{4} + \beta_{6} + \beta_{7} + \beta_{8} - \beta_{11} - \beta_{12} ) q^{11} \) \( + ( -1 - \beta_{2} ) q^{12} \) \( + ( 2 - \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} + \beta_{6} + \beta_{7} + \beta_{10} + \beta_{13} ) q^{13} \) \( + ( -1 - \beta_{3} ) q^{15} \) \( + ( 1 + \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} + \beta_{5} - \beta_{6} - \beta_{8} - \beta_{9} + \beta_{11} + \beta_{12} ) q^{16} \) \( + ( 2 - \beta_{1} + \beta_{6} + \beta_{8} - \beta_{10} - \beta_{12} + \beta_{13} ) q^{17} \) \( -\beta_{1} q^{18} \) \( + ( -1 - \beta_{2} - 2 \beta_{6} - \beta_{9} + \beta_{12} - \beta_{13} ) q^{19} \) \( + ( 2 + 2 \beta_{2} + \beta_{3} + \beta_{4} - \beta_{6} - \beta_{8} - \beta_{9} + \beta_{10} + \beta_{11} + \beta_{12} - \beta_{13} ) q^{20} \) \( + ( 3 \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} + \beta_{9} + \beta_{10} + \beta_{11} + \beta_{12} - \beta_{13} ) q^{22} \) \( + ( -1 + \beta_{1} + \beta_{3} - \beta_{6} + \beta_{10} + \beta_{12} ) q^{23} \) \( + ( 1 + \beta_{1} + \beta_{2} + \beta_{5} + \beta_{6} ) q^{24} \) \( + ( 2 + \beta_{5} + 2 \beta_{7} - 2 \beta_{8} - \beta_{9} + \beta_{10} + \beta_{11} + \beta_{12} ) q^{25} \) \( + ( -2 \beta_{1} + \beta_{2} - \beta_{3} - \beta_{7} + \beta_{9} + \beta_{11} + \beta_{13} ) q^{26} \) \(- q^{27}\) \( + ( -1 - 2 \beta_{2} + \beta_{3} - \beta_{5} + \beta_{8} - \beta_{9} ) q^{29} \) \( + ( 2 \beta_{1} + \beta_{2} + \beta_{5} + \beta_{6} - \beta_{7} + \beta_{11} + \beta_{13} ) q^{30} \) \( + ( 1 + \beta_{4} + 2 \beta_{7} + 2 \beta_{10} - \beta_{11} - \beta_{13} ) q^{31} \) \( + ( -3 - 3 \beta_{1} - 2 \beta_{2} - 2 \beta_{5} - 2 \beta_{6} - \beta_{10} - 2 \beta_{11} + \beta_{12} - \beta_{13} ) q^{32} \) \( + ( \beta_{1} - \beta_{4} - \beta_{6} - \beta_{7} - \beta_{8} + \beta_{11} + \beta_{12} ) q^{33} \) \( + ( 2 + \beta_{1} + \beta_{2} - \beta_{4} + \beta_{5} - \beta_{6} - \beta_{7} + 2 \beta_{11} + \beta_{12} ) q^{34} \) \( + ( 1 + \beta_{2} ) q^{36} \) \( + ( \beta_{1} - \beta_{4} - \beta_{5} + \beta_{6} + \beta_{10} - \beta_{11} ) q^{37} \) \( + ( 2 - \beta_{1} + \beta_{2} + 2 \beta_{3} + \beta_{4} - \beta_{5} - \beta_{7} - \beta_{10} ) q^{38} \) \( + ( -2 + \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} - \beta_{6} - \beta_{7} - \beta_{10} - \beta_{13} ) q^{39} \) \( + ( -2 - 5 \beta_{1} - 2 \beta_{2} - \beta_{3} - 2 \beta_{5} - 2 \beta_{6} - \beta_{10} - \beta_{11} + \beta_{13} ) q^{40} \) \(+ q^{41}\) \( + ( 2 - \beta_{1} + \beta_{4} + \beta_{5} + 2 \beta_{7} - \beta_{8} - \beta_{11} - \beta_{12} ) q^{43} \) \( + ( 2 - 2 \beta_{1} - 2 \beta_{3} + 2 \beta_{4} + 5 \beta_{6} + 2 \beta_{7} + 4 \beta_{8} + \beta_{9} - 2 \beta_{10} - 3 \beta_{11} - 6 \beta_{12} + 2 \beta_{13} ) q^{44} \) \( + ( 1 + \beta_{3} ) q^{45} \) \( + ( -4 \beta_{1} + 2 \beta_{4} - 2 \beta_{5} + \beta_{6} + \beta_{7} + \beta_{8} + \beta_{9} - \beta_{10} - 2 \beta_{11} - 2 \beta_{12} ) q^{46} \) \( + ( 2 + \beta_{1} - \beta_{4} + 2 \beta_{5} - \beta_{9} - \beta_{10} + \beta_{11} - \beta_{12} ) q^{47} \) \( + ( -1 - \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} + \beta_{6} + \beta_{8} + \beta_{9} - \beta_{11} - \beta_{12} ) q^{48} \) \( + ( -1 - 4 \beta_{1} - \beta_{2} - 3 \beta_{5} - \beta_{6} - 2 \beta_{11} - \beta_{12} - \beta_{13} ) q^{50} \) \( + ( -2 + \beta_{1} - \beta_{6} - \beta_{8} + \beta_{10} + \beta_{12} - \beta_{13} ) q^{51} \) \( + ( 5 + 3 \beta_{2} + \beta_{5} - 2 \beta_{7} + 2 \beta_{8} - 3 \beta_{10} - 2 \beta_{12} + \beta_{13} ) q^{52} \) \( + ( -1 - \beta_{4} + \beta_{5} + \beta_{11} + \beta_{12} ) q^{53} \) \( + \beta_{1} q^{54} \) \( + ( 1 - 2 \beta_{1} + \beta_{2} - 3 \beta_{3} + 3 \beta_{6} + 2 \beta_{8} + \beta_{9} - \beta_{10} - 2 \beta_{11} - 3 \beta_{12} + \beta_{13} ) q^{55} \) \( + ( 1 + \beta_{2} + 2 \beta_{6} + \beta_{9} - \beta_{12} + \beta_{13} ) q^{57} \) \( + ( 1 + 4 \beta_{1} + \beta_{2} - \beta_{4} + 2 \beta_{5} - \beta_{6} - \beta_{8} - \beta_{10} + \beta_{11} + \beta_{12} - \beta_{13} ) q^{58} \) \( + ( 2 - 2 \beta_{1} - \beta_{4} - \beta_{5} + \beta_{7} - \beta_{8} + \beta_{9} + \beta_{10} - \beta_{11} ) q^{59} \) \( + ( -2 - 2 \beta_{2} - \beta_{3} - \beta_{4} + \beta_{6} + \beta_{8} + \beta_{9} - \beta_{10} - \beta_{11} - \beta_{12} + \beta_{13} ) q^{60} \) \( + ( 2 - 3 \beta_{1} + 2 \beta_{2} - \beta_{3} + 2 \beta_{4} + \beta_{6} + 2 \beta_{7} - 2 \beta_{8} + 2 \beta_{10} - \beta_{11} - \beta_{13} ) q^{61} \) \( + ( -3 \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} - 2 \beta_{5} + 3 \beta_{6} + 2 \beta_{7} + \beta_{8} + 2 \beta_{9} + \beta_{10} - 2 \beta_{11} - 2 \beta_{12} - \beta_{13} ) q^{62} \) \( + ( 3 \beta_{1} + \beta_{2} + 3 \beta_{3} - \beta_{4} - 3 \beta_{6} - 3 \beta_{7} - 2 \beta_{8} - \beta_{9} + \beta_{10} + 2 \beta_{11} + 3 \beta_{12} - \beta_{13} ) q^{64} \) \( + ( \beta_{1} + \beta_{2} + 2 \beta_{3} + \beta_{4} - \beta_{7} + \beta_{8} + \beta_{9} - \beta_{10} - \beta_{12} + 2 \beta_{13} ) q^{65} \) \( + ( -3 \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} - \beta_{9} - \beta_{10} - \beta_{11} - \beta_{12} + \beta_{13} ) q^{66} \) \( + ( -1 - \beta_{1} + \beta_{2} + \beta_{4} - \beta_{5} + \beta_{6} - \beta_{7} + 2 \beta_{8} - \beta_{11} - 2 \beta_{12} + \beta_{13} ) q^{67} \) \( + ( 3 - 7 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} + 3 \beta_{4} - 2 \beta_{5} + 2 \beta_{6} + \beta_{7} + 3 \beta_{8} + \beta_{9} - \beta_{10} - 3 \beta_{11} - 4 \beta_{12} + \beta_{13} ) q^{68} \) \( + ( 1 - \beta_{1} - \beta_{3} + \beta_{6} - \beta_{10} - \beta_{12} ) q^{69} \) \( + ( -1 + \beta_{1} - \beta_{2} + \beta_{3} + \beta_{5} + \beta_{6} + \beta_{7} + \beta_{9} + \beta_{10} + \beta_{11} + \beta_{13} ) q^{71} \) \( + ( -1 - \beta_{1} - \beta_{2} - \beta_{5} - \beta_{6} ) q^{72} \) \( + ( 2 - 2 \beta_{1} - 3 \beta_{2} + \beta_{3} - 3 \beta_{5} + 2 \beta_{7} + 2 \beta_{10} - \beta_{11} + \beta_{12} - 2 \beta_{13} ) q^{73} \) \( + ( -5 + \beta_{1} - \beta_{2} - \beta_{3} - 2 \beta_{4} + 2 \beta_{5} - 3 \beta_{8} + \beta_{10} + \beta_{11} + \beta_{12} - \beta_{13} ) q^{74} \) \( + ( -2 - \beta_{5} - 2 \beta_{7} + 2 \beta_{8} + \beta_{9} - \beta_{10} - \beta_{11} - \beta_{12} ) q^{75} \) \( + ( -2 - 3 \beta_{1} - 2 \beta_{2} - \beta_{4} - \beta_{5} - 4 \beta_{6} + \beta_{7} - 2 \beta_{8} + 2 \beta_{12} ) q^{76} \) \( + ( 2 \beta_{1} - \beta_{2} + \beta_{3} + \beta_{7} - \beta_{9} - \beta_{11} - \beta_{13} ) q^{78} \) \( + ( 2 + 2 \beta_{1} + \beta_{2} - \beta_{3} - 2 \beta_{4} + 2 \beta_{6} - \beta_{7} + 2 \beta_{8} + \beta_{9} - 3 \beta_{10} - \beta_{12} + \beta_{13} ) q^{79} \) \( + ( 5 + 5 \beta_{1} + 5 \beta_{2} + 2 \beta_{3} - \beta_{4} + 3 \beta_{5} - 3 \beta_{6} - 4 \beta_{7} - 3 \beta_{8} - \beta_{9} - \beta_{10} + 4 \beta_{11} + 4 \beta_{12} + 2 \beta_{13} ) q^{80} \) \(+ q^{81}\) \( -\beta_{1} q^{82} \) \( + ( 1 + \beta_{1} + \beta_{2} - \beta_{4} + \beta_{5} - \beta_{6} - 3 \beta_{7} + 2 \beta_{8} - \beta_{9} - 2 \beta_{10} - \beta_{12} ) q^{83} \) \( + ( 5 - 2 \beta_{1} - 2 \beta_{2} + \beta_{4} + \beta_{6} + 2 \beta_{7} + \beta_{8} - 2 \beta_{10} - \beta_{11} - 2 \beta_{12} + 2 \beta_{13} ) q^{85} \) \( + ( -3 - 2 \beta_{2} + 3 \beta_{3} - \beta_{4} - 3 \beta_{5} - 2 \beta_{6} - 2 \beta_{8} + 3 \beta_{10} + 3 \beta_{12} - 3 \beta_{13} ) q^{86} \) \( + ( 1 + 2 \beta_{2} - \beta_{3} + \beta_{5} - \beta_{8} + \beta_{9} ) q^{87} \) \( + ( -4 + 10 \beta_{1} + \beta_{2} - 4 \beta_{4} + 5 \beta_{5} + \beta_{6} - 2 \beta_{7} - 2 \beta_{8} + 3 \beta_{10} + 6 \beta_{11} + 4 \beta_{12} ) q^{88} \) \( + ( 3 - \beta_{1} - 2 \beta_{2} - \beta_{4} + \beta_{7} - 2 \beta_{9} - \beta_{11} + \beta_{12} ) q^{89} \) \( + ( -2 \beta_{1} - \beta_{2} - \beta_{5} - \beta_{6} + \beta_{7} - \beta_{11} - \beta_{13} ) q^{90} \) \( + ( 2 + 4 \beta_{1} + \beta_{3} - \beta_{4} + 2 \beta_{5} - 4 \beta_{6} - 3 \beta_{8} - 2 \beta_{9} + \beta_{10} + 4 \beta_{11} + 4 \beta_{12} - 2 \beta_{13} ) q^{92} \) \( + ( -1 - \beta_{4} - 2 \beta_{7} - 2 \beta_{10} + \beta_{11} + \beta_{13} ) q^{93} \) \( + ( 2 - \beta_{1} - \beta_{2} - 2 \beta_{4} - 2 \beta_{5} + 2 \beta_{6} + \beta_{8} + \beta_{9} - \beta_{11} - \beta_{12} - \beta_{13} ) q^{94} \) \( + ( -7 - 3 \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} - 2 \beta_{5} - 3 \beta_{6} - 5 \beta_{7} + \beta_{8} - \beta_{10} + \beta_{12} - \beta_{13} ) q^{95} \) \( + ( 3 + 3 \beta_{1} + 2 \beta_{2} + 2 \beta_{5} + 2 \beta_{6} + \beta_{10} + 2 \beta_{11} - \beta_{12} + \beta_{13} ) q^{96} \) \( + ( 1 + 3 \beta_{1} + \beta_{3} - \beta_{4} + \beta_{5} - \beta_{7} - 2 \beta_{9} + \beta_{11} + \beta_{12} ) q^{97} \) \( + ( -\beta_{1} + \beta_{4} + \beta_{6} + \beta_{7} + \beta_{8} - \beta_{11} - \beta_{12} ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(14q \) \(\mathstrut -\mathstrut 2q^{2} \) \(\mathstrut -\mathstrut 14q^{3} \) \(\mathstrut +\mathstrut 14q^{4} \) \(\mathstrut +\mathstrut 10q^{5} \) \(\mathstrut +\mathstrut 2q^{6} \) \(\mathstrut -\mathstrut 6q^{8} \) \(\mathstrut +\mathstrut 14q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(14q \) \(\mathstrut -\mathstrut 2q^{2} \) \(\mathstrut -\mathstrut 14q^{3} \) \(\mathstrut +\mathstrut 14q^{4} \) \(\mathstrut +\mathstrut 10q^{5} \) \(\mathstrut +\mathstrut 2q^{6} \) \(\mathstrut -\mathstrut 6q^{8} \) \(\mathstrut +\mathstrut 14q^{9} \) \(\mathstrut +\mathstrut 3q^{10} \) \(\mathstrut -\mathstrut 16q^{11} \) \(\mathstrut -\mathstrut 14q^{12} \) \(\mathstrut +\mathstrut 21q^{13} \) \(\mathstrut -\mathstrut 10q^{15} \) \(\mathstrut +\mathstrut 22q^{16} \) \(\mathstrut +\mathstrut 12q^{17} \) \(\mathstrut -\mathstrut 2q^{18} \) \(\mathstrut +\mathstrut 2q^{19} \) \(\mathstrut +\mathstrut 40q^{20} \) \(\mathstrut +\mathstrut q^{22} \) \(\mathstrut -\mathstrut 7q^{23} \) \(\mathstrut +\mathstrut 6q^{24} \) \(\mathstrut +\mathstrut 22q^{25} \) \(\mathstrut +\mathstrut 2q^{26} \) \(\mathstrut -\mathstrut 14q^{27} \) \(\mathstrut -\mathstrut 16q^{29} \) \(\mathstrut -\mathstrut 3q^{30} \) \(\mathstrut +\mathstrut 8q^{31} \) \(\mathstrut -\mathstrut 19q^{32} \) \(\mathstrut +\mathstrut 16q^{33} \) \(\mathstrut +\mathstrut 33q^{34} \) \(\mathstrut +\mathstrut 14q^{36} \) \(\mathstrut +\mathstrut q^{37} \) \(\mathstrut +\mathstrut 32q^{38} \) \(\mathstrut -\mathstrut 21q^{39} \) \(\mathstrut -\mathstrut 13q^{40} \) \(\mathstrut +\mathstrut 14q^{41} \) \(\mathstrut +\mathstrut 14q^{43} \) \(\mathstrut -\mathstrut 36q^{44} \) \(\mathstrut +\mathstrut 10q^{45} \) \(\mathstrut -\mathstrut 12q^{46} \) \(\mathstrut +\mathstrut 12q^{47} \) \(\mathstrut -\mathstrut 22q^{48} \) \(\mathstrut -\mathstrut q^{50} \) \(\mathstrut -\mathstrut 12q^{51} \) \(\mathstrut +\mathstrut 60q^{52} \) \(\mathstrut -\mathstrut 20q^{53} \) \(\mathstrut +\mathstrut 2q^{54} \) \(\mathstrut -\mathstrut 11q^{55} \) \(\mathstrut -\mathstrut 2q^{57} \) \(\mathstrut +\mathstrut 21q^{58} \) \(\mathstrut +\mathstrut 25q^{59} \) \(\mathstrut -\mathstrut 40q^{60} \) \(\mathstrut +\mathstrut 26q^{61} \) \(\mathstrut -\mathstrut 33q^{62} \) \(\mathstrut +\mathstrut 42q^{64} \) \(\mathstrut -\mathstrut 8q^{65} \) \(\mathstrut -\mathstrut q^{66} \) \(\mathstrut -\mathstrut 22q^{67} \) \(\mathstrut +\mathstrut 15q^{68} \) \(\mathstrut +\mathstrut 7q^{69} \) \(\mathstrut -\mathstrut 36q^{71} \) \(\mathstrut -\mathstrut 6q^{72} \) \(\mathstrut +\mathstrut 31q^{73} \) \(\mathstrut -\mathstrut 65q^{74} \) \(\mathstrut -\mathstrut 22q^{75} \) \(\mathstrut -\mathstrut 2q^{76} \) \(\mathstrut -\mathstrut 2q^{78} \) \(\mathstrut +\mathstrut 12q^{79} \) \(\mathstrut +\mathstrut 112q^{80} \) \(\mathstrut +\mathstrut 14q^{81} \) \(\mathstrut -\mathstrut 2q^{82} \) \(\mathstrut +\mathstrut 20q^{83} \) \(\mathstrut +\mathstrut 40q^{85} \) \(\mathstrut -\mathstrut 9q^{86} \) \(\mathstrut +\mathstrut 16q^{87} \) \(\mathstrut -\mathstrut 54q^{88} \) \(\mathstrut +\mathstrut 39q^{89} \) \(\mathstrut +\mathstrut 3q^{90} \) \(\mathstrut +\mathstrut 63q^{92} \) \(\mathstrut -\mathstrut 8q^{93} \) \(\mathstrut +\mathstrut 14q^{94} \) \(\mathstrut -\mathstrut 55q^{95} \) \(\mathstrut +\mathstrut 19q^{96} \) \(\mathstrut +\mathstrut 18q^{97} \) \(\mathstrut -\mathstrut 16q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{14}\mathstrut -\mathstrut \) \(2\) \(x^{13}\mathstrut -\mathstrut \) \(19\) \(x^{12}\mathstrut +\mathstrut \) \(36\) \(x^{11}\mathstrut +\mathstrut \) \(134\) \(x^{10}\mathstrut -\mathstrut \) \(237\) \(x^{9}\mathstrut -\mathstrut \) \(438\) \(x^{8}\mathstrut +\mathstrut \) \(716\) \(x^{7}\mathstrut +\mathstrut \) \(662\) \(x^{6}\mathstrut -\mathstrut \) \(1007\) \(x^{5}\mathstrut -\mathstrut \) \(384\) \(x^{4}\mathstrut +\mathstrut \) \(579\) \(x^{3}\mathstrut +\mathstrut \) \(44\) \(x^{2}\mathstrut -\mathstrut \) \(112\) \(x\mathstrut +\mathstrut \) \(16\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 3 \)
\(\beta_{3}\)\(=\)\((\)\(-\)\(115\) \(\nu^{13}\mathstrut +\mathstrut \) \(8468\) \(\nu^{12}\mathstrut -\mathstrut \) \(24595\) \(\nu^{11}\mathstrut -\mathstrut \) \(121798\) \(\nu^{10}\mathstrut +\mathstrut \) \(409674\) \(\nu^{9}\mathstrut +\mathstrut \) \(523923\) \(\nu^{8}\mathstrut -\mathstrut \) \(2228088\) \(\nu^{7}\mathstrut -\mathstrut \) \(530532\) \(\nu^{6}\mathstrut +\mathstrut \) \(4814290\) \(\nu^{5}\mathstrut -\mathstrut \) \(797891\) \(\nu^{4}\mathstrut -\mathstrut \) \(3817394\) \(\nu^{3}\mathstrut +\mathstrut \) \(1001107\) \(\nu^{2}\mathstrut +\mathstrut \) \(965410\) \(\nu\mathstrut -\mathstrut \) \(272668\)\()/18364\)
\(\beta_{4}\)\(=\)\((\)\(-\)\(66\) \(\nu^{13}\mathstrut -\mathstrut \) \(2845\) \(\nu^{12}\mathstrut +\mathstrut \) \(8640\) \(\nu^{11}\mathstrut +\mathstrut \) \(47189\) \(\nu^{10}\mathstrut -\mathstrut \) \(131524\) \(\nu^{9}\mathstrut -\mathstrut \) \(270514\) \(\nu^{8}\mathstrut +\mathstrut \) \(741431\) \(\nu^{7}\mathstrut +\mathstrut \) \(636556\) \(\nu^{6}\mathstrut -\mathstrut \) \(1814842\) \(\nu^{5}\mathstrut -\mathstrut \) \(509978\) \(\nu^{4}\mathstrut +\mathstrut \) \(1847591\) \(\nu^{3}\mathstrut -\mathstrut \) \(89430\) \(\nu^{2}\mathstrut -\mathstrut \) \(575125\) \(\nu\mathstrut +\mathstrut \) \(145640\)\()/9182\)
\(\beta_{5}\)\(=\)\((\)\(433\) \(\nu^{13}\mathstrut -\mathstrut \) \(186\) \(\nu^{12}\mathstrut -\mathstrut \) \(11191\) \(\nu^{11}\mathstrut +\mathstrut \) \(11712\) \(\nu^{10}\mathstrut +\mathstrut \) \(97154\) \(\nu^{9}\mathstrut -\mathstrut \) \(152093\) \(\nu^{8}\mathstrut -\mathstrut \) \(324642\) \(\nu^{7}\mathstrut +\mathstrut \) \(730612\) \(\nu^{6}\mathstrut +\mathstrut \) \(274678\) \(\nu^{5}\mathstrut -\mathstrut \) \(1345959\) \(\nu^{4}\mathstrut +\mathstrut \) \(323980\) \(\nu^{3}\mathstrut +\mathstrut \) \(678535\) \(\nu^{2}\mathstrut -\mathstrut \) \(234844\) \(\nu\mathstrut -\mathstrut \) \(25044\)\()/18364\)
\(\beta_{6}\)\(=\)\((\)\(-\)\(433\) \(\nu^{13}\mathstrut +\mathstrut \) \(186\) \(\nu^{12}\mathstrut +\mathstrut \) \(11191\) \(\nu^{11}\mathstrut -\mathstrut \) \(11712\) \(\nu^{10}\mathstrut -\mathstrut \) \(97154\) \(\nu^{9}\mathstrut +\mathstrut \) \(152093\) \(\nu^{8}\mathstrut +\mathstrut \) \(324642\) \(\nu^{7}\mathstrut -\mathstrut \) \(730612\) \(\nu^{6}\mathstrut -\mathstrut \) \(274678\) \(\nu^{5}\mathstrut +\mathstrut \) \(1345959\) \(\nu^{4}\mathstrut -\mathstrut \) \(305616\) \(\nu^{3}\mathstrut -\mathstrut \) \(696899\) \(\nu^{2}\mathstrut +\mathstrut \) \(143024\) \(\nu\mathstrut +\mathstrut \) \(61772\)\()/18364\)
\(\beta_{7}\)\(=\)\((\)\(-\)\(657\) \(\nu^{13}\mathstrut +\mathstrut \) \(8616\) \(\nu^{12}\mathstrut -\mathstrut \) \(9569\) \(\nu^{11}\mathstrut -\mathstrut \) \(132302\) \(\nu^{10}\mathstrut +\mathstrut \) \(265034\) \(\nu^{9}\mathstrut +\mathstrut \) \(661925\) \(\nu^{8}\mathstrut -\mathstrut \) \(1635072\) \(\nu^{7}\mathstrut -\mathstrut \) \(1196788\) \(\nu^{6}\mathstrut +\mathstrut \) \(3821874\) \(\nu^{5}\mathstrut +\mathstrut \) \(481015\) \(\nu^{4}\mathstrut -\mathstrut \) \(3322458\) \(\nu^{3}\mathstrut +\mathstrut \) \(193241\) \(\nu^{2}\mathstrut +\mathstrut \) \(922034\) \(\nu\mathstrut -\mathstrut \) \(166252\)\()/18364\)
\(\beta_{8}\)\(=\)\((\)\(-\)\(1101\) \(\nu^{13}\mathstrut +\mathstrut \) \(4502\) \(\nu^{12}\mathstrut +\mathstrut \) \(16835\) \(\nu^{11}\mathstrut -\mathstrut \) \(80292\) \(\nu^{10}\mathstrut -\mathstrut \) \(80530\) \(\nu^{9}\mathstrut +\mathstrut \) \(514897\) \(\nu^{8}\mathstrut +\mathstrut \) \(103978\) \(\nu^{7}\mathstrut -\mathstrut \) \(1457088\) \(\nu^{6}\mathstrut +\mathstrut \) \(148858\) \(\nu^{5}\mathstrut +\mathstrut \) \(1748099\) \(\nu^{4}\mathstrut -\mathstrut \) \(330636\) \(\nu^{3}\mathstrut -\mathstrut \) \(628747\) \(\nu^{2}\mathstrut +\mathstrut \) \(56360\) \(\nu\mathstrut -\mathstrut \) \(12872\)\()/18364\)
\(\beta_{9}\)\(=\)\((\)\(2301\) \(\nu^{13}\mathstrut -\mathstrut \) \(7032\) \(\nu^{12}\mathstrut -\mathstrut \) \(42039\) \(\nu^{11}\mathstrut +\mathstrut \) \(138006\) \(\nu^{10}\mathstrut +\mathstrut \) \(266526\) \(\nu^{9}\mathstrut -\mathstrut \) \(1000485\) \(\nu^{8}\mathstrut -\mathstrut \) \(633748\) \(\nu^{7}\mathstrut +\mathstrut \) \(3272368\) \(\nu^{6}\mathstrut +\mathstrut \) \(78546\) \(\nu^{5}\mathstrut -\mathstrut \) \(4549267\) \(\nu^{4}\mathstrut +\mathstrut \) \(1321658\) \(\nu^{3}\mathstrut +\mathstrut \) \(1777283\) \(\nu^{2}\mathstrut -\mathstrut \) \(620446\) \(\nu\mathstrut -\mathstrut \) \(9076\)\()/18364\)
\(\beta_{10}\)\(=\)\((\)\( 720 \nu^{13} - 1518 \nu^{12} - 13286 \nu^{11} + 28201 \nu^{10} + 85888 \nu^{9} - 191269 \nu^{8} - 216860 \nu^{7} + 579567 \nu^{6} + 93287 \nu^{5} - 735873 \nu^{4} + 299871 \nu^{3} + 236449 \nu^{2} - 161239 \nu + 13459 \)\()/4591\)
\(\beta_{11}\)\(=\)\((\)\(-\)\(2470\) \(\nu^{13}\mathstrut +\mathstrut \) \(4825\) \(\nu^{12}\mathstrut +\mathstrut \) \(46216\) \(\nu^{11}\mathstrut -\mathstrut \) \(85969\) \(\nu^{10}\mathstrut -\mathstrut \) \(315048\) \(\nu^{9}\mathstrut +\mathstrut \) \(554328\) \(\nu^{8}\mathstrut +\mathstrut \) \(953861\) \(\nu^{7}\mathstrut -\mathstrut \) \(1600106\) \(\nu^{6}\mathstrut -\mathstrut \) \(1192380\) \(\nu^{5}\mathstrut +\mathstrut \) \(2012748\) \(\nu^{4}\mathstrut +\mathstrut \) \(343411\) \(\nu^{3}\mathstrut -\mathstrut \) \(821800\) \(\nu^{2}\mathstrut +\mathstrut \) \(66047\) \(\nu\mathstrut +\mathstrut \) \(57572\)\()/9182\)
\(\beta_{12}\)\(=\)\((\)\(5521\) \(\nu^{13}\mathstrut -\mathstrut \) \(14586\) \(\nu^{12}\mathstrut -\mathstrut \) \(87939\) \(\nu^{11}\mathstrut +\mathstrut \) \(233648\) \(\nu^{10}\mathstrut +\mathstrut \) \(475158\) \(\nu^{9}\mathstrut -\mathstrut \) \(1272953\) \(\nu^{8}\mathstrut -\mathstrut \) \(1042982\) \(\nu^{7}\mathstrut +\mathstrut \) \(2811688\) \(\nu^{6}\mathstrut +\mathstrut \) \(878202\) \(\nu^{5}\mathstrut -\mathstrut \) \(2298535\) \(\nu^{4}\mathstrut -\mathstrut \) \(203184\) \(\nu^{3}\mathstrut +\mathstrut \) \(465907\) \(\nu^{2}\mathstrut -\mathstrut \) \(151836\) \(\nu\mathstrut +\mathstrut \) \(41296\)\()/18364\)
\(\beta_{13}\)\(=\)\((\)\(12521\) \(\nu^{13}\mathstrut -\mathstrut \) \(27814\) \(\nu^{12}\mathstrut -\mathstrut \) \(219659\) \(\nu^{11}\mathstrut +\mathstrut \) \(464720\) \(\nu^{10}\mathstrut +\mathstrut \) \(1391798\) \(\nu^{9}\mathstrut -\mathstrut \) \(2725189\) \(\nu^{8}\mathstrut -\mathstrut \) \(3990986\) \(\nu^{7}\mathstrut +\mathstrut \) \(6893844\) \(\nu^{6}\mathstrut +\mathstrut \) \(5292938\) \(\nu^{5}\mathstrut -\mathstrut \) \(7406035\) \(\nu^{4}\mathstrut -\mathstrut \) \(2959952\) \(\nu^{3}\mathstrut +\mathstrut \) \(2807311\) \(\nu^{2}\mathstrut +\mathstrut \) \(577848\) \(\nu\mathstrut -\mathstrut \) \(261192\)\()/18364\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2}\mathstrut +\mathstrut \) \(3\)
\(\nu^{3}\)\(=\)\(\beta_{6}\mathstrut +\mathstrut \) \(\beta_{5}\mathstrut +\mathstrut \) \(\beta_{2}\mathstrut +\mathstrut \) \(5\) \(\beta_{1}\mathstrut +\mathstrut \) \(1\)
\(\nu^{4}\)\(=\)\(\beta_{12}\mathstrut +\mathstrut \) \(\beta_{11}\mathstrut -\mathstrut \) \(\beta_{9}\mathstrut -\mathstrut \) \(\beta_{8}\mathstrut -\mathstrut \) \(\beta_{6}\mathstrut +\mathstrut \) \(\beta_{5}\mathstrut +\mathstrut \) \(\beta_{4}\mathstrut +\mathstrut \) \(\beta_{3}\mathstrut +\mathstrut \) \(7\) \(\beta_{2}\mathstrut +\mathstrut \) \(\beta_{1}\mathstrut +\mathstrut \) \(15\)
\(\nu^{5}\)\(=\)\(\beta_{13}\mathstrut -\mathstrut \) \(\beta_{12}\mathstrut +\mathstrut \) \(2\) \(\beta_{11}\mathstrut +\mathstrut \) \(\beta_{10}\mathstrut +\mathstrut \) \(10\) \(\beta_{6}\mathstrut +\mathstrut \) \(10\) \(\beta_{5}\mathstrut +\mathstrut \) \(10\) \(\beta_{2}\mathstrut +\mathstrut \) \(31\) \(\beta_{1}\mathstrut +\mathstrut \) \(11\)
\(\nu^{6}\)\(=\)\(-\)\(\beta_{13}\mathstrut +\mathstrut \) \(13\) \(\beta_{12}\mathstrut +\mathstrut \) \(12\) \(\beta_{11}\mathstrut +\mathstrut \) \(\beta_{10}\mathstrut -\mathstrut \) \(11\) \(\beta_{9}\mathstrut -\mathstrut \) \(12\) \(\beta_{8}\mathstrut -\mathstrut \) \(3\) \(\beta_{7}\mathstrut -\mathstrut \) \(13\) \(\beta_{6}\mathstrut +\mathstrut \) \(10\) \(\beta_{5}\mathstrut +\mathstrut \) \(9\) \(\beta_{4}\mathstrut +\mathstrut \) \(13\) \(\beta_{3}\mathstrut +\mathstrut \) \(47\) \(\beta_{2}\mathstrut +\mathstrut \) \(13\) \(\beta_{1}\mathstrut +\mathstrut \) \(86\)
\(\nu^{7}\)\(=\)\(11\) \(\beta_{13}\mathstrut -\mathstrut \) \(9\) \(\beta_{12}\mathstrut +\mathstrut \) \(26\) \(\beta_{11}\mathstrut +\mathstrut \) \(13\) \(\beta_{10}\mathstrut +\mathstrut \) \(\beta_{9}\mathstrut +\mathstrut \) \(\beta_{8}\mathstrut -\mathstrut \) \(3\) \(\beta_{7}\mathstrut +\mathstrut \) \(80\) \(\beta_{6}\mathstrut +\mathstrut \) \(80\) \(\beta_{5}\mathstrut -\mathstrut \) \(\beta_{4}\mathstrut +\mathstrut \) \(3\) \(\beta_{3}\mathstrut +\mathstrut \) \(80\) \(\beta_{2}\mathstrut +\mathstrut \) \(210\) \(\beta_{1}\mathstrut +\mathstrut \) \(90\)
\(\nu^{8}\)\(=\)\(-\)\(16\) \(\beta_{13}\mathstrut +\mathstrut \) \(127\) \(\beta_{12}\mathstrut +\mathstrut \) \(112\) \(\beta_{11}\mathstrut +\mathstrut \) \(18\) \(\beta_{10}\mathstrut -\mathstrut \) \(94\) \(\beta_{9}\mathstrut -\mathstrut \) \(113\) \(\beta_{8}\mathstrut -\mathstrut \) \(41\) \(\beta_{7}\mathstrut -\mathstrut \) \(123\) \(\beta_{6}\mathstrut +\mathstrut \) \(80\) \(\beta_{5}\mathstrut +\mathstrut \) \(63\) \(\beta_{4}\mathstrut +\mathstrut \) \(122\) \(\beta_{3}\mathstrut +\mathstrut \) \(320\) \(\beta_{2}\mathstrut +\mathstrut \) \(123\) \(\beta_{1}\mathstrut +\mathstrut \) \(530\)
\(\nu^{9}\)\(=\)\(90\) \(\beta_{13}\mathstrut -\mathstrut \) \(53\) \(\beta_{12}\mathstrut +\mathstrut \) \(254\) \(\beta_{11}\mathstrut +\mathstrut \) \(126\) \(\beta_{10}\mathstrut +\mathstrut \) \(15\) \(\beta_{9}\mathstrut +\mathstrut \) \(8\) \(\beta_{8}\mathstrut -\mathstrut \) \(47\) \(\beta_{7}\mathstrut +\mathstrut \) \(595\) \(\beta_{6}\mathstrut +\mathstrut \) \(604\) \(\beta_{5}\mathstrut -\mathstrut \) \(20\) \(\beta_{4}\mathstrut +\mathstrut \) \(48\) \(\beta_{3}\mathstrut +\mathstrut \) \(598\) \(\beta_{2}\mathstrut +\mathstrut \) \(1484\) \(\beta_{1}\mathstrut +\mathstrut \) \(671\)
\(\nu^{10}\)\(=\)\(-\)\(176\) \(\beta_{13}\mathstrut +\mathstrut \) \(1114\) \(\beta_{12}\mathstrut +\mathstrut \) \(957\) \(\beta_{11}\mathstrut +\mathstrut \) \(209\) \(\beta_{10}\mathstrut -\mathstrut \) \(738\) \(\beta_{9}\mathstrut -\mathstrut \) \(968\) \(\beta_{8}\mathstrut -\mathstrut \) \(406\) \(\beta_{7}\mathstrut -\mathstrut \) \(1044\) \(\beta_{6}\mathstrut +\mathstrut \) \(608\) \(\beta_{5}\mathstrut +\mathstrut \) \(403\) \(\beta_{4}\mathstrut +\mathstrut \) \(1022\) \(\beta_{3}\mathstrut +\mathstrut \) \(2212\) \(\beta_{2}\mathstrut +\mathstrut \) \(1048\) \(\beta_{1}\mathstrut +\mathstrut \) \(3419\)
\(\nu^{11}\)\(=\)\(657\) \(\beta_{13}\mathstrut -\mathstrut \) \(200\) \(\beta_{12}\mathstrut +\mathstrut \) \(2234\) \(\beta_{11}\mathstrut +\mathstrut \) \(1103\) \(\beta_{10}\mathstrut +\mathstrut \) \(151\) \(\beta_{9}\mathstrut +\mathstrut \) \(6\) \(\beta_{8}\mathstrut -\mathstrut \) \(512\) \(\beta_{7}\mathstrut +\mathstrut \) \(4282\) \(\beta_{6}\mathstrut +\mathstrut \) \(4480\) \(\beta_{5}\mathstrut -\mathstrut \) \(265\) \(\beta_{4}\mathstrut +\mathstrut \) \(528\) \(\beta_{3}\mathstrut +\mathstrut \) \(4357\) \(\beta_{2}\mathstrut +\mathstrut \) \(10718\) \(\beta_{1}\mathstrut +\mathstrut \) \(4817\)
\(\nu^{12}\)\(=\)\(-\)\(1663\) \(\beta_{13}\mathstrut +\mathstrut \) \(9261\) \(\beta_{12}\mathstrut +\mathstrut \) \(7839\) \(\beta_{11}\mathstrut +\mathstrut \) \(2040\) \(\beta_{10}\mathstrut -\mathstrut \) \(5589\) \(\beta_{9}\mathstrut -\mathstrut \) \(7893\) \(\beta_{8}\mathstrut -\mathstrut \) \(3559\) \(\beta_{7}\mathstrut -\mathstrut \) \(8437\) \(\beta_{6}\mathstrut +\mathstrut \) \(4571\) \(\beta_{5}\mathstrut +\mathstrut \) \(2452\) \(\beta_{4}\mathstrut +\mathstrut \) \(8131\) \(\beta_{3}\mathstrut +\mathstrut \) \(15469\) \(\beta_{2}\mathstrut +\mathstrut \) \(8542\) \(\beta_{1}\mathstrut +\mathstrut \) \(22732\)
\(\nu^{13}\)\(=\)\(4513\) \(\beta_{13}\mathstrut +\mathstrut \) \(238\) \(\beta_{12}\mathstrut +\mathstrut \) \(18655\) \(\beta_{11}\mathstrut +\mathstrut \) \(9207\) \(\beta_{10}\mathstrut +\mathstrut \) \(1282\) \(\beta_{9}\mathstrut -\mathstrut \) \(650\) \(\beta_{8}\mathstrut -\mathstrut \) \(4823\) \(\beta_{7}\mathstrut +\mathstrut \) \(30292\) \(\beta_{6}\mathstrut +\mathstrut \) \(33049\) \(\beta_{5}\mathstrut -\mathstrut \) \(2907\) \(\beta_{4}\mathstrut +\mathstrut \) \(5001\) \(\beta_{3}\mathstrut +\mathstrut \) \(31423\) \(\beta_{2}\mathstrut +\mathstrut \) \(78322\) \(\beta_{1}\mathstrut +\mathstrut \) \(34017\)

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
2.73044
2.55282
2.13215
1.64093
1.21066
0.766034
0.442251
0.181424
−0.712717
−0.766355
−1.67187
−1.68292
−2.13271
−2.69016
−2.73044 −1.00000 5.45531 3.67975 2.73044 0 −9.43451 1.00000 −10.0473
1.2 −2.55282 −1.00000 4.51690 1.95019 2.55282 0 −6.42520 1.00000 −4.97848
1.3 −2.13215 −1.00000 2.54608 −2.91322 2.13215 0 −1.16433 1.00000 6.21144
1.4 −1.64093 −1.00000 0.692668 −0.0817657 1.64093 0 2.14525 1.00000 0.134172
1.5 −1.21066 −1.00000 −0.534291 0.826979 1.21066 0 3.06818 1.00000 −1.00119
1.6 −0.766034 −1.00000 −1.41319 −0.561773 0.766034 0 2.61462 1.00000 0.430337
1.7 −0.442251 −1.00000 −1.80441 4.29274 0.442251 0 1.68250 1.00000 −1.89847
1.8 −0.181424 −1.00000 −1.96709 −3.75467 0.181424 0 0.719724 1.00000 0.681187
1.9 0.712717 −1.00000 −1.49203 0.415066 −0.712717 0 −2.48883 1.00000 0.295824
1.10 0.766355 −1.00000 −1.41270 1.12787 −0.766355 0 −2.61534 1.00000 0.864345
1.11 1.67187 −1.00000 0.795136 −2.68319 −1.67187 0 −2.01437 1.00000 −4.48594
1.12 1.68292 −1.00000 0.832222 3.40481 −1.68292 0 −1.96528 1.00000 5.73002
1.13 2.13271 −1.00000 2.54845 0.889988 −2.13271 0 1.16968 1.00000 1.89809
1.14 2.69016 −1.00000 5.23695 3.40724 −2.69016 0 8.70791 1.00000 9.16601
\(n\): 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
\(3\) \(1\)
\(7\) \(1\)
\(41\) \(-1\)

Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6027))\):

\(T_{2}^{14} + \cdots\)
\(T_{5}^{14} - \cdots\)
\(T_{13}^{14} - \cdots\)