Properties

Label 6027.2.a.bi
Level 6027
Weight 2
Character orbit 6027.a
Self dual Yes
Analytic conductor 48.126
Analytic rank 1
Dimension 13
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: \(1\)
Dimension: \(13\)
Coefficient field: \(\mathbb{Q}[x]/(x^{13} - \cdots)\)
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 \(1,\beta_1,\ldots,\beta_{12}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \( -\beta_{1} q^{2} \) \(+ q^{3}\) \( + ( 1 + \beta_{2} ) q^{4} \) \( + ( -1 + \beta_{8} ) q^{5} \) \( -\beta_{1} q^{6} \) \( + ( -\beta_{1} - \beta_{8} - \beta_{9} ) q^{8} \) \(+ q^{9}\) \(+O(q^{10})\) \( q\) \( -\beta_{1} q^{2} \) \(+ q^{3}\) \( + ( 1 + \beta_{2} ) q^{4} \) \( + ( -1 + \beta_{8} ) q^{5} \) \( -\beta_{1} q^{6} \) \( + ( -\beta_{1} - \beta_{8} - \beta_{9} ) q^{8} \) \(+ q^{9}\) \( + ( -1 + \beta_{1} - \beta_{2} - \beta_{4} - \beta_{5} - \beta_{7} - \beta_{8} + \beta_{9} + \beta_{10} + \beta_{12} ) q^{10} \) \( + ( -1 + \beta_{1} - \beta_{2} + \beta_{6} - \beta_{10} - \beta_{12} ) q^{11} \) \( + ( 1 + \beta_{2} ) q^{12} \) \( + ( -1 - \beta_{6} - \beta_{7} - \beta_{8} + \beta_{10} ) q^{13} \) \( + ( -1 + \beta_{8} ) q^{15} \) \( + ( 1 + \beta_{1} + \beta_{2} + \beta_{5} + \beta_{7} + \beta_{8} + \beta_{9} + \beta_{11} ) q^{16} \) \( + ( -2 - \beta_{3} + \beta_{6} + \beta_{8} - \beta_{9} + \beta_{11} + 2 \beta_{12} ) q^{17} \) \( -\beta_{1} q^{18} \) \( + ( 1 - \beta_{3} + 2 \beta_{4} + \beta_{7} - \beta_{9} - \beta_{10} - \beta_{12} ) q^{19} \) \( + ( -1 + 2 \beta_{1} - \beta_{2} + \beta_{4} - \beta_{5} + \beta_{6} + \beta_{7} + 2 \beta_{8} - 2 \beta_{9} - 2 \beta_{10} + \beta_{12} ) q^{20} \) \( + ( -1 + \beta_{1} - \beta_{2} + 2 \beta_{3} - \beta_{6} - \beta_{7} + 2 \beta_{9} - \beta_{11} - \beta_{12} ) q^{22} \) \( + ( -2 + 2 \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} + \beta_{7} + \beta_{9} + \beta_{10} - \beta_{12} ) q^{23} \) \( + ( -\beta_{1} - \beta_{8} - \beta_{9} ) q^{24} \) \( + ( 1 + \beta_{2} + \beta_{3} + \beta_{5} - \beta_{8} - \beta_{11} ) q^{25} \) \( + ( -3 + 2 \beta_{1} + \beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} + 2 \beta_{8} - \beta_{10} + \beta_{11} ) q^{26} \) \(+ q^{27}\) \( + ( 1 - \beta_{1} + 2 \beta_{2} - \beta_{4} + \beta_{5} - \beta_{6} + 2 \beta_{7} - \beta_{8} + \beta_{9} - \beta_{12} ) q^{29} \) \( + ( -1 + \beta_{1} - \beta_{2} - \beta_{4} - \beta_{5} - \beta_{7} - \beta_{8} + \beta_{9} + \beta_{10} + \beta_{12} ) q^{30} \) \( + ( -2 \beta_{2} + 2 \beta_{4} - \beta_{5} - \beta_{11} ) q^{31} \) \( + ( 1 - 2 \beta_{1} - \beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} - \beta_{6} - \beta_{8} - \beta_{9} - 2 \beta_{11} - 2 \beta_{12} ) q^{32} \) \( + ( -1 + \beta_{1} - \beta_{2} + \beta_{6} - \beta_{10} - \beta_{12} ) q^{33} \) \( + ( \beta_{1} + \beta_{2} - 2 \beta_{4} - 2 \beta_{6} - \beta_{7} - \beta_{8} + \beta_{9} + 2 \beta_{10} - \beta_{11} + \beta_{12} ) q^{34} \) \( + ( 1 + \beta_{2} ) q^{36} \) \( + ( 2 - \beta_{1} - \beta_{3} - \beta_{6} - 2 \beta_{8} - \beta_{10} - \beta_{11} ) q^{37} \) \( + ( -1 + \beta_{3} - \beta_{4} + 2 \beta_{5} - 2 \beta_{6} - 2 \beta_{8} + 2 \beta_{9} + \beta_{10} - \beta_{11} - 2 \beta_{12} ) q^{38} \) \( + ( -1 - \beta_{6} - \beta_{7} - \beta_{8} + \beta_{10} ) q^{39} \) \( + ( -4 + \beta_{1} - \beta_{3} - 3 \beta_{4} - 2 \beta_{6} - \beta_{7} - 2 \beta_{8} + 3 \beta_{9} + 2 \beta_{10} ) q^{40} \) \(+ q^{41}\) \( + ( -1 + \beta_{2} - \beta_{3} - \beta_{4} - \beta_{6} - \beta_{7} - \beta_{10} + \beta_{11} + 2 \beta_{12} ) q^{43} \) \( + ( -1 - \beta_{1} - 2 \beta_{2} - \beta_{3} + 2 \beta_{4} - \beta_{7} - \beta_{9} ) q^{44} \) \( + ( -1 + \beta_{8} ) q^{45} \) \( + ( -1 + 4 \beta_{1} - \beta_{3} + \beta_{4} + \beta_{6} + 2 \beta_{7} + 2 \beta_{8} - \beta_{9} - \beta_{10} ) q^{46} \) \( + ( -2 - \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} - 2 \beta_{6} + \beta_{9} + 2 \beta_{10} - 2 \beta_{11} + \beta_{12} ) q^{47} \) \( + ( 1 + \beta_{1} + \beta_{2} + \beta_{5} + \beta_{7} + \beta_{8} + \beta_{9} + \beta_{11} ) q^{48} \) \( + ( 1 - 3 \beta_{1} + \beta_{2} - \beta_{3} + 2 \beta_{4} + 2 \beta_{5} + \beta_{6} + \beta_{7} - 2 \beta_{9} - \beta_{10} + \beta_{11} - 2 \beta_{12} ) q^{50} \) \( + ( -2 - \beta_{3} + \beta_{6} + \beta_{8} - \beta_{9} + \beta_{11} + 2 \beta_{12} ) q^{51} \) \( + ( -2 \beta_{2} - \beta_{4} + 2 \beta_{5} - \beta_{6} - 3 \beta_{8} + \beta_{9} - 3 \beta_{11} - 3 \beta_{12} ) q^{52} \) \( + ( -2 + 3 \beta_{1} - \beta_{2} + 2 \beta_{3} - \beta_{4} - 2 \beta_{5} + \beta_{6} - \beta_{7} - \beta_{8} + \beta_{9} + \beta_{11} + \beta_{12} ) q^{53} \) \( -\beta_{1} q^{54} \) \( + ( -\beta_{2} + 2 \beta_{3} - 2 \beta_{5} - 2 \beta_{6} - 2 \beta_{7} - 2 \beta_{8} + 2 \beta_{9} + \beta_{10} ) q^{55} \) \( + ( 1 - \beta_{3} + 2 \beta_{4} + \beta_{7} - \beta_{9} - \beta_{10} - \beta_{12} ) q^{57} \) \( + ( 7 - \beta_{1} + 2 \beta_{2} - 2 \beta_{3} + 3 \beta_{4} - \beta_{5} + 2 \beta_{6} + 3 \beta_{7} - 6 \beta_{9} - 2 \beta_{10} + \beta_{11} ) q^{58} \) \( + ( -2 \beta_{1} + \beta_{5} - \beta_{7} + \beta_{9} - \beta_{10} ) q^{59} \) \( + ( -1 + 2 \beta_{1} - \beta_{2} + \beta_{4} - \beta_{5} + \beta_{6} + \beta_{7} + 2 \beta_{8} - 2 \beta_{9} - 2 \beta_{10} + \beta_{12} ) q^{60} \) \( + ( -5 + 3 \beta_{1} + \beta_{2} - \beta_{4} + \beta_{8} + 2 \beta_{9} + 2 \beta_{10} + 2 \beta_{11} + \beta_{12} ) q^{61} \) \( + ( -2 + 3 \beta_{1} - \beta_{2} - \beta_{4} + 2 \beta_{5} - \beta_{6} + \beta_{8} + 2 \beta_{9} - \beta_{11} - 2 \beta_{12} ) q^{62} \) \( + ( -4 + 4 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} + \beta_{4} - 2 \beta_{5} + 2 \beta_{6} - \beta_{7} + 2 \beta_{8} + 2 \beta_{9} + \beta_{11} + \beta_{12} ) q^{64} \) \( + ( -1 - \beta_{3} - \beta_{4} + 3 \beta_{5} + \beta_{6} + 2 \beta_{7} - \beta_{9} - 2 \beta_{10} - \beta_{11} - \beta_{12} ) q^{65} \) \( + ( -1 + \beta_{1} - \beta_{2} + 2 \beta_{3} - \beta_{6} - \beta_{7} + 2 \beta_{9} - \beta_{11} - \beta_{12} ) q^{66} \) \( + ( -2 + 2 \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} - \beta_{7} + 2 \beta_{10} + \beta_{12} ) q^{67} \) \( + ( -5 + 3 \beta_{1} - 3 \beta_{2} + 2 \beta_{4} - 4 \beta_{5} + 3 \beta_{6} + 2 \beta_{7} + 3 \beta_{8} - 2 \beta_{9} - 2 \beta_{10} + 2 \beta_{11} + \beta_{12} ) q^{68} \) \( + ( -2 + 2 \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} + \beta_{7} + \beta_{9} + \beta_{10} - \beta_{12} ) q^{69} \) \( + ( -2 - \beta_{1} + \beta_{2} - 2 \beta_{5} + \beta_{6} - \beta_{8} - 3 \beta_{9} + 2 \beta_{12} ) q^{71} \) \( + ( -\beta_{1} - \beta_{8} - \beta_{9} ) q^{72} \) \( + ( -1 + \beta_{1} - 2 \beta_{2} + 3 \beta_{3} + \beta_{4} + \beta_{6} + \beta_{7} + \beta_{8} - \beta_{10} - \beta_{11} - \beta_{12} ) q^{73} \) \( + ( -1 + 2 \beta_{1} - \beta_{2} + 2 \beta_{4} - 2 \beta_{5} + 2 \beta_{6} + \beta_{7} + 3 \beta_{8} - 2 \beta_{9} - 2 \beta_{10} + 2 \beta_{11} + \beta_{12} ) q^{74} \) \( + ( 1 + \beta_{2} + \beta_{3} + \beta_{5} - \beta_{8} - \beta_{11} ) q^{75} \) \( + ( -2 + 4 \beta_{1} - \beta_{2} + \beta_{3} + 2 \beta_{4} + \beta_{5} + 4 \beta_{6} + \beta_{7} + 4 \beta_{8} - 2 \beta_{9} - 2 \beta_{10} + 2 \beta_{11} ) q^{76} \) \( + ( -3 + 2 \beta_{1} + \beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} + 2 \beta_{8} - \beta_{10} + \beta_{11} ) q^{78} \) \( + ( -\beta_{1} - \beta_{2} + 2 \beta_{3} - \beta_{5} - \beta_{7} + 2 \beta_{9} ) q^{79} \) \( + ( -4 + 4 \beta_{1} - 2 \beta_{2} + 3 \beta_{4} - 3 \beta_{5} + 3 \beta_{6} + \beta_{7} + 4 \beta_{8} - 3 \beta_{9} - \beta_{10} + 2 \beta_{11} + 2 \beta_{12} ) q^{80} \) \(+ q^{81}\) \( -\beta_{1} q^{82} \) \( + ( -2 + 2 \beta_{1} + \beta_{2} - 2 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} + \beta_{6} + \beta_{7} - \beta_{9} + 2 \beta_{11} ) q^{83} \) \( + ( 3 - 2 \beta_{1} + 2 \beta_{2} + \beta_{3} - \beta_{6} - \beta_{7} - \beta_{8} + 3 \beta_{9} + 4 \beta_{10} + \beta_{11} ) q^{85} \) \( + ( -4 + \beta_{1} - 3 \beta_{2} - \beta_{3} - 4 \beta_{5} + \beta_{6} - 2 \beta_{7} - \beta_{8} - 2 \beta_{9} + \beta_{11} + 3 \beta_{12} ) q^{86} \) \( + ( 1 - \beta_{1} + 2 \beta_{2} - \beta_{4} + \beta_{5} - \beta_{6} + 2 \beta_{7} - \beta_{8} + \beta_{9} - \beta_{12} ) q^{87} \) \( + ( 2 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} - \beta_{4} + 3 \beta_{5} + \beta_{7} + \beta_{8} + \beta_{9} + \beta_{10} + \beta_{11} ) q^{88} \) \( + ( -6 + \beta_{1} + 2 \beta_{2} - 3 \beta_{4} + 3 \beta_{5} + \beta_{6} + 2 \beta_{8} + \beta_{10} - \beta_{11} + \beta_{12} ) q^{89} \) \( + ( -1 + \beta_{1} - \beta_{2} - \beta_{4} - \beta_{5} - \beta_{7} - \beta_{8} + \beta_{9} + \beta_{10} + \beta_{12} ) q^{90} \) \( + ( -7 - 2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} - \beta_{4} - \beta_{5} - 2 \beta_{6} - 3 \beta_{7} - 3 \beta_{8} + \beta_{10} - \beta_{11} + 3 \beta_{12} ) q^{92} \) \( + ( -2 \beta_{2} + 2 \beta_{4} - \beta_{5} - \beta_{11} ) q^{93} \) \( + ( -2 + 5 \beta_{1} - \beta_{2} - 4 \beta_{3} + \beta_{4} - 3 \beta_{5} + 5 \beta_{6} + 2 \beta_{7} + 3 \beta_{8} - 4 \beta_{9} - \beta_{10} + 4 \beta_{11} + 4 \beta_{12} ) q^{94} \) \( + ( -2 + 5 \beta_{1} + \beta_{3} - \beta_{4} + \beta_{6} + \beta_{7} + \beta_{9} - \beta_{10} + \beta_{12} ) q^{95} \) \( + ( 1 - 2 \beta_{1} - \beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} - \beta_{6} - \beta_{8} - \beta_{9} - 2 \beta_{11} - 2 \beta_{12} ) q^{96} \) \( + ( -3 + \beta_{1} - \beta_{3} + 2 \beta_{4} - \beta_{6} - \beta_{8} - 2 \beta_{9} - 2 \beta_{10} ) q^{97} \) \( + ( -1 + \beta_{1} - \beta_{2} + \beta_{6} - \beta_{10} - \beta_{12} ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(13q \) \(\mathstrut -\mathstrut 4q^{2} \) \(\mathstrut +\mathstrut 13q^{3} \) \(\mathstrut +\mathstrut 12q^{4} \) \(\mathstrut -\mathstrut 8q^{5} \) \(\mathstrut -\mathstrut 4q^{6} \) \(\mathstrut -\mathstrut 12q^{8} \) \(\mathstrut +\mathstrut 13q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(13q \) \(\mathstrut -\mathstrut 4q^{2} \) \(\mathstrut +\mathstrut 13q^{3} \) \(\mathstrut +\mathstrut 12q^{4} \) \(\mathstrut -\mathstrut 8q^{5} \) \(\mathstrut -\mathstrut 4q^{6} \) \(\mathstrut -\mathstrut 12q^{8} \) \(\mathstrut +\mathstrut 13q^{9} \) \(\mathstrut -\mathstrut q^{10} \) \(\mathstrut -\mathstrut 10q^{11} \) \(\mathstrut +\mathstrut 12q^{12} \) \(\mathstrut -\mathstrut 16q^{13} \) \(\mathstrut -\mathstrut 8q^{15} \) \(\mathstrut +\mathstrut 26q^{16} \) \(\mathstrut -\mathstrut 12q^{17} \) \(\mathstrut -\mathstrut 4q^{18} \) \(\mathstrut -\mathstrut 11q^{19} \) \(\mathstrut -\mathstrut 6q^{20} \) \(\mathstrut +\mathstrut q^{22} \) \(\mathstrut -\mathstrut 15q^{23} \) \(\mathstrut -\mathstrut 12q^{24} \) \(\mathstrut +\mathstrut 15q^{25} \) \(\mathstrut -\mathstrut 18q^{26} \) \(\mathstrut +\mathstrut 13q^{27} \) \(\mathstrut -\mathstrut 8q^{29} \) \(\mathstrut -\mathstrut q^{30} \) \(\mathstrut -\mathstrut 9q^{31} \) \(\mathstrut -\mathstrut 23q^{32} \) \(\mathstrut -\mathstrut 10q^{33} \) \(\mathstrut +\mathstrut 7q^{34} \) \(\mathstrut +\mathstrut 12q^{36} \) \(\mathstrut -\mathstrut 2q^{37} \) \(\mathstrut -\mathstrut 20q^{38} \) \(\mathstrut -\mathstrut 16q^{39} \) \(\mathstrut -\mathstrut 49q^{40} \) \(\mathstrut +\mathstrut 13q^{41} \) \(\mathstrut -\mathstrut 7q^{43} \) \(\mathstrut -\mathstrut 22q^{44} \) \(\mathstrut -\mathstrut 8q^{45} \) \(\mathstrut -\mathstrut 4q^{46} \) \(\mathstrut -\mathstrut 26q^{47} \) \(\mathstrut +\mathstrut 26q^{48} \) \(\mathstrut -\mathstrut 15q^{50} \) \(\mathstrut -\mathstrut 12q^{51} \) \(\mathstrut -\mathstrut 24q^{52} \) \(\mathstrut +\mathstrut 4q^{53} \) \(\mathstrut -\mathstrut 4q^{54} \) \(\mathstrut -\mathstrut q^{55} \) \(\mathstrut -\mathstrut 11q^{57} \) \(\mathstrut +\mathstrut 39q^{58} \) \(\mathstrut +\mathstrut 3q^{59} \) \(\mathstrut -\mathstrut 6q^{60} \) \(\mathstrut -\mathstrut 28q^{61} \) \(\mathstrut -\mathstrut 7q^{62} \) \(\mathstrut +\mathstrut 2q^{64} \) \(\mathstrut -\mathstrut 20q^{65} \) \(\mathstrut +\mathstrut q^{66} \) \(\mathstrut +\mathstrut 7q^{67} \) \(\mathstrut -\mathstrut 55q^{68} \) \(\mathstrut -\mathstrut 15q^{69} \) \(\mathstrut -\mathstrut 40q^{71} \) \(\mathstrut -\mathstrut 12q^{72} \) \(\mathstrut +\mathstrut 2q^{73} \) \(\mathstrut +\mathstrut q^{74} \) \(\mathstrut +\mathstrut 15q^{75} \) \(\mathstrut +\mathstrut 26q^{76} \) \(\mathstrut -\mathstrut 18q^{78} \) \(\mathstrut +\mathstrut 13q^{79} \) \(\mathstrut -\mathstrut 22q^{80} \) \(\mathstrut +\mathstrut 13q^{81} \) \(\mathstrut -\mathstrut 4q^{82} \) \(\mathstrut -\mathstrut 14q^{83} \) \(\mathstrut +\mathstrut 48q^{85} \) \(\mathstrut -\mathstrut 49q^{86} \) \(\mathstrut -\mathstrut 8q^{87} \) \(\mathstrut +\mathstrut 20q^{88} \) \(\mathstrut -\mathstrut 35q^{89} \) \(\mathstrut -\mathstrut q^{90} \) \(\mathstrut -\mathstrut 105q^{92} \) \(\mathstrut -\mathstrut 9q^{93} \) \(\mathstrut +\mathstrut 2q^{94} \) \(\mathstrut +\mathstrut 7q^{95} \) \(\mathstrut -\mathstrut 23q^{96} \) \(\mathstrut -\mathstrut 64q^{97} \) \(\mathstrut -\mathstrut 10q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{13}\mathstrut -\mathstrut \) \(4\) \(x^{12}\mathstrut -\mathstrut \) \(11\) \(x^{11}\mathstrut +\mathstrut \) \(56\) \(x^{10}\mathstrut +\mathstrut \) \(26\) \(x^{9}\mathstrut -\mathstrut \) \(263\) \(x^{8}\mathstrut +\mathstrut \) \(50\) \(x^{7}\mathstrut +\mathstrut \) \(478\) \(x^{6}\mathstrut -\mathstrut \) \(174\) \(x^{5}\mathstrut -\mathstrut \) \(311\) \(x^{4}\mathstrut +\mathstrut \) \(84\) \(x^{3}\mathstrut +\mathstrut \) \(69\) \(x^{2}\mathstrut -\mathstrut \) \(12\) \(x\mathstrut -\mathstrut \) \(4\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 3 \)
\(\beta_{3}\)\(=\)\((\)\( -105 \nu^{12} + 402 \nu^{11} + 1471 \nu^{10} - 6122 \nu^{9} - 7424 \nu^{8} + 32643 \nu^{7} + 19248 \nu^{6} - 71846 \nu^{5} - 35248 \nu^{4} + 56201 \nu^{3} + 35530 \nu^{2} - 1463 \nu - 4612 \)\()/2162\)
\(\beta_{4}\)\(=\)\((\)\( 944 \nu^{12} - 3120 \nu^{11} - 12772 \nu^{10} + 44868 \nu^{9} + 58087 \nu^{8} - 219999 \nu^{7} - 110536 \nu^{6} + 428762 \nu^{5} + 119516 \nu^{4} - 298916 \nu^{3} - 91598 \nu^{2} + 44430 \nu + 11577 \)\()/1081\)
\(\beta_{5}\)\(=\)\((\)\( 2617 \nu^{12} - 8228 \nu^{11} - 37363 \nu^{10} + 120936 \nu^{9} + 187114 \nu^{8} - 614931 \nu^{7} - 418518 \nu^{6} + 1279188 \nu^{5} + 499382 \nu^{4} - 1003841 \nu^{3} - 321158 \nu^{2} + 182903 \nu + 44488 \)\()/2162\)
\(\beta_{6}\)\(=\)\((\)\( 1537 \nu^{12} - 5483 \nu^{11} - 19453 \nu^{10} + 78197 \nu^{9} + 75852 \nu^{8} - 378472 \nu^{7} - 93135 \nu^{6} + 723044 \nu^{5} + 43093 \nu^{4} - 495112 \nu^{3} - 63961 \nu^{2} + 80294 \nu + 9559 \)\()/1081\)
\(\beta_{7}\)\(=\)\((\)\( 1680 \nu^{12} - 6432 \nu^{11} - 19212 \nu^{10} + 89304 \nu^{9} + 55005 \nu^{8} - 412026 \nu^{7} + 21737 \nu^{6} + 714974 \nu^{5} - 146249 \nu^{4} - 398713 \nu^{3} + 5531 \nu^{2} + 50433 \nu + 7851 \)\()/1081\)
\(\beta_{8}\)\(=\)\((\)\( -4041 \nu^{12} + 14730 \nu^{11} + 49447 \nu^{10} - 207380 \nu^{9} - 177062 \nu^{8} + 981221 \nu^{7} + 151906 \nu^{6} - 1793936 \nu^{5} + 9716 \nu^{4} + 1126967 \nu^{3} + 153280 \nu^{2} - 171323 \nu - 31592 \)\()/2162\)
\(\beta_{9}\)\(=\)\((\)\( 4041 \nu^{12} - 14730 \nu^{11} - 49447 \nu^{10} + 207380 \nu^{9} + 177062 \nu^{8} - 981221 \nu^{7} - 151906 \nu^{6} + 1793936 \nu^{5} - 9716 \nu^{4} - 1124805 \nu^{3} - 153280 \nu^{2} + 160513 \nu + 31592 \)\()/2162\)
\(\beta_{10}\)\(=\)\((\)\( -2764 \nu^{12} + 10088 \nu^{11} + 33585 \nu^{10} - 141614 \nu^{9} - 117946 \nu^{8} + 666901 \nu^{7} + 90681 \nu^{6} - 1209623 \nu^{5} + 23336 \nu^{4} + 751304 \nu^{3} + 104974 \nu^{2} - 113389 \nu - 23055 \)\()/1081\)
\(\beta_{11}\)\(=\)\((\)\(-\)\(5977\) \(\nu^{12}\mathstrut +\mathstrut \) \(21092\) \(\nu^{11}\mathstrut +\mathstrut \) \(75787\) \(\nu^{10}\mathstrut -\mathstrut \) \(299544\) \(\nu^{9}\mathstrut -\mathstrut \) \(297124\) \(\nu^{8}\mathstrut +\mathstrut \) \(1438983\) \(\nu^{7}\mathstrut +\mathstrut \) \(375044\) \(\nu^{6}\mathstrut -\mathstrut \) \(2709136\) \(\nu^{5}\mathstrut -\mathstrut \) \(204722\) \(\nu^{4}\mathstrut +\mathstrut \) \(1799105\) \(\nu^{3}\mathstrut +\mathstrut \) \(294962\) \(\nu^{2}\mathstrut -\mathstrut \) \(275121\) \(\nu\mathstrut -\mathstrut \) \(47218\)\()/2162\)
\(\beta_{12}\)\(=\)\((\)\(6745\) \(\nu^{12}\mathstrut -\mathstrut \) \(23044\) \(\nu^{11}\mathstrut -\mathstrut \) \(88523\) \(\nu^{10}\mathstrut +\mathstrut \) \(329744\) \(\nu^{9}\mathstrut +\mathstrut \) \(376628\) \(\nu^{8}\mathstrut -\mathstrut \) \(1604297\) \(\nu^{7}\mathstrut -\mathstrut \) \(611328\) \(\nu^{6}\mathstrut +\mathstrut \) \(3091452\) \(\nu^{5}\mathstrut +\mathstrut \) \(548460\) \(\nu^{4}\mathstrut -\mathstrut \) \(2142659\) \(\nu^{3}\mathstrut -\mathstrut \) \(502024\) \(\nu^{2}\mathstrut +\mathstrut \) \(347655\) \(\nu\mathstrut +\mathstrut \) \(78110\)\()/2162\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2}\mathstrut +\mathstrut \) \(3\)
\(\nu^{3}\)\(=\)\(\beta_{9}\mathstrut +\mathstrut \) \(\beta_{8}\mathstrut +\mathstrut \) \(5\) \(\beta_{1}\)
\(\nu^{4}\)\(=\)\(\beta_{11}\mathstrut +\mathstrut \) \(\beta_{9}\mathstrut +\mathstrut \) \(\beta_{8}\mathstrut +\mathstrut \) \(\beta_{7}\mathstrut +\mathstrut \) \(\beta_{5}\mathstrut +\mathstrut \) \(7\) \(\beta_{2}\mathstrut +\mathstrut \) \(\beta_{1}\mathstrut +\mathstrut \) \(15\)
\(\nu^{5}\)\(=\)\(2\) \(\beta_{12}\mathstrut +\mathstrut \) \(2\) \(\beta_{11}\mathstrut +\mathstrut \) \(9\) \(\beta_{9}\mathstrut +\mathstrut \) \(9\) \(\beta_{8}\mathstrut +\mathstrut \) \(\beta_{6}\mathstrut -\mathstrut \) \(\beta_{5}\mathstrut -\mathstrut \) \(\beta_{4}\mathstrut +\mathstrut \) \(\beta_{3}\mathstrut +\mathstrut \) \(\beta_{2}\mathstrut +\mathstrut \) \(30\) \(\beta_{1}\mathstrut -\mathstrut \) \(1\)
\(\nu^{6}\)\(=\)\(\beta_{12}\mathstrut +\mathstrut \) \(11\) \(\beta_{11}\mathstrut +\mathstrut \) \(12\) \(\beta_{9}\mathstrut +\mathstrut \) \(12\) \(\beta_{8}\mathstrut +\mathstrut \) \(9\) \(\beta_{7}\mathstrut +\mathstrut \) \(2\) \(\beta_{6}\mathstrut +\mathstrut \) \(8\) \(\beta_{5}\mathstrut +\mathstrut \) \(\beta_{4}\mathstrut +\mathstrut \) \(2\) \(\beta_{3}\mathstrut +\mathstrut \) \(44\) \(\beta_{2}\mathstrut +\mathstrut \) \(14\) \(\beta_{1}\mathstrut +\mathstrut \) \(82\)
\(\nu^{7}\)\(=\)\(25\) \(\beta_{12}\mathstrut +\mathstrut \) \(26\) \(\beta_{11}\mathstrut +\mathstrut \) \(66\) \(\beta_{9}\mathstrut +\mathstrut \) \(70\) \(\beta_{8}\mathstrut +\mathstrut \) \(3\) \(\beta_{7}\mathstrut +\mathstrut \) \(14\) \(\beta_{6}\mathstrut -\mathstrut \) \(13\) \(\beta_{5}\mathstrut -\mathstrut \) \(8\) \(\beta_{4}\mathstrut +\mathstrut \) \(10\) \(\beta_{3}\mathstrut +\mathstrut \) \(14\) \(\beta_{2}\mathstrut +\mathstrut \) \(191\) \(\beta_{1}\mathstrut -\mathstrut \) \(6\)
\(\nu^{8}\)\(=\)\(17\) \(\beta_{12}\mathstrut +\mathstrut \) \(98\) \(\beta_{11}\mathstrut -\mathstrut \) \(4\) \(\beta_{10}\mathstrut +\mathstrut \) \(104\) \(\beta_{9}\mathstrut +\mathstrut \) \(112\) \(\beta_{8}\mathstrut +\mathstrut \) \(67\) \(\beta_{7}\mathstrut +\mathstrut \) \(32\) \(\beta_{6}\mathstrut +\mathstrut \) \(49\) \(\beta_{5}\mathstrut +\mathstrut \) \(17\) \(\beta_{4}\mathstrut +\mathstrut \) \(24\) \(\beta_{3}\mathstrut +\mathstrut \) \(274\) \(\beta_{2}\mathstrut +\mathstrut \) \(141\) \(\beta_{1}\mathstrut +\mathstrut \) \(471\)
\(\nu^{9}\)\(=\)\(229\) \(\beta_{12}\mathstrut +\mathstrut \) \(251\) \(\beta_{11}\mathstrut -\mathstrut \) \(8\) \(\beta_{10}\mathstrut +\mathstrut \) \(454\) \(\beta_{9}\mathstrut +\mathstrut \) \(529\) \(\beta_{8}\mathstrut +\mathstrut \) \(49\) \(\beta_{7}\mathstrut +\mathstrut \) \(147\) \(\beta_{6}\mathstrut -\mathstrut \) \(121\) \(\beta_{5}\mathstrut -\mathstrut \) \(41\) \(\beta_{4}\mathstrut +\mathstrut \) \(76\) \(\beta_{3}\mathstrut +\mathstrut \) \(140\) \(\beta_{2}\mathstrut +\mathstrut \) \(1259\) \(\beta_{1}\mathstrut -\mathstrut \) \(11\)
\(\nu^{10}\)\(=\)\(196\) \(\beta_{12}\mathstrut +\mathstrut \) \(811\) \(\beta_{11}\mathstrut -\mathstrut \) \(75\) \(\beta_{10}\mathstrut +\mathstrut \) \(797\) \(\beta_{9}\mathstrut +\mathstrut \) \(963\) \(\beta_{8}\mathstrut +\mathstrut \) \(488\) \(\beta_{7}\mathstrut +\mathstrut \) \(357\) \(\beta_{6}\mathstrut +\mathstrut \) \(266\) \(\beta_{5}\mathstrut +\mathstrut \) \(196\) \(\beta_{4}\mathstrut +\mathstrut \) \(207\) \(\beta_{3}\mathstrut +\mathstrut \) \(1727\) \(\beta_{2}\mathstrut +\mathstrut \) \(1254\) \(\beta_{1}\mathstrut +\mathstrut \) \(2806\)
\(\nu^{11}\)\(=\)\(1867\) \(\beta_{12}\mathstrut +\mathstrut \) \(2161\) \(\beta_{11}\mathstrut -\mathstrut \) \(166\) \(\beta_{10}\mathstrut +\mathstrut \) \(3042\) \(\beta_{9}\mathstrut +\mathstrut \) \(3979\) \(\beta_{8}\mathstrut +\mathstrut \) \(550\) \(\beta_{7}\mathstrut +\mathstrut \) \(1364\) \(\beta_{6}\mathstrut -\mathstrut \) \(995\) \(\beta_{5}\mathstrut -\mathstrut \) \(100\) \(\beta_{4}\mathstrut +\mathstrut \) \(534\) \(\beta_{3}\mathstrut +\mathstrut \) \(1226\) \(\beta_{2}\mathstrut +\mathstrut \) \(8497\) \(\beta_{1}\mathstrut +\mathstrut \) \(202\)
\(\nu^{12}\)\(=\)\(1927\) \(\beta_{12}\mathstrut +\mathstrut \) \(6467\) \(\beta_{11}\mathstrut -\mathstrut \) \(937\) \(\beta_{10}\mathstrut +\mathstrut \) \(5748\) \(\beta_{9}\mathstrut +\mathstrut \) \(7966\) \(\beta_{8}\mathstrut +\mathstrut \) \(3595\) \(\beta_{7}\mathstrut +\mathstrut \) \(3425\) \(\beta_{6}\mathstrut +\mathstrut \) \(1281\) \(\beta_{5}\mathstrut +\mathstrut \) \(1932\) \(\beta_{4}\mathstrut +\mathstrut \) \(1587\) \(\beta_{3}\mathstrut +\mathstrut \) \(11075\) \(\beta_{2}\mathstrut +\mathstrut \) \(10469\) \(\beta_{1}\mathstrut +\mathstrut \) \(17210\)

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
2.73393
2.58904
2.20098
1.84587
1.21702
0.467051
0.441481
−0.196565
−0.558375
−0.646272
−1.46649
−2.15509
−2.47258
−2.73393 1.00000 5.47440 2.29019 −2.73393 0 −9.49877 1.00000 −6.26122
1.2 −2.58904 1.00000 4.70310 −3.45213 −2.58904 0 −6.99843 1.00000 8.93769
1.3 −2.20098 1.00000 2.84432 2.77621 −2.20098 0 −1.85834 1.00000 −6.11039
1.4 −1.84587 1.00000 1.40722 −1.41311 −1.84587 0 1.09419 1.00000 2.60841
1.5 −1.21702 1.00000 −0.518850 −3.45145 −1.21702 0 3.06550 1.00000 4.20050
1.6 −0.467051 1.00000 −1.78186 0.377698 −0.467051 0 1.76632 1.00000 −0.176404
1.7 −0.441481 1.00000 −1.80509 −3.83895 −0.441481 0 1.67988 1.00000 1.69482
1.8 0.196565 1.00000 −1.96136 −1.00666 0.196565 0 −0.778664 1.00000 −0.197874
1.9 0.558375 1.00000 −1.68822 −0.297196 0.558375 0 −2.05941 1.00000 −0.165947
1.10 0.646272 1.00000 −1.58233 2.20733 0.646272 0 −2.31516 1.00000 1.42653
1.11 1.46649 1.00000 0.150578 1.27945 1.46649 0 −2.71215 1.00000 1.87630
1.12 2.15509 1.00000 2.64442 0.784696 2.15509 0 1.38879 1.00000 1.69109
1.13 2.47258 1.00000 4.11367 −4.25608 2.47258 0 5.22624 1.00000 −10.5235
\(n\): 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
\(3\) \(-1\)
\(7\) \(1\)
\(41\) \(-1\)

Hecke kernels

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

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