Properties

Label 6026.2.a.f
Level 6026
Weight 2
Character orbit 6026.a
Self dual Yes
Analytic conductor 48.118
Analytic rank 1
Dimension 20
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 6026 = 2 \cdot 23 \cdot 131 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 6026.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(48.117852258\)
Analytic rank: \(1\)
Dimension: \(20\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
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 \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{20} - 5 x^{19} - 17 x^{18} + 115 x^{17} + 78 x^{16} - 1083 x^{15} + 248 x^{14} + 5359 x^{13} - 3723 x^{12} - 14776 x^{11} + 14837 x^{10} + 21886 x^{9} - 28084 x^{8} - 14682 x^{7} + 25315 x^{6} + 2042 x^{5} - 9137 x^{4} + 576 x^{3} + 1159 x^{2} - 78 x - 18\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 3 \)
\(\beta_{3}\)\(=\)\((\)\(99716990 \nu^{19} - 165884442 \nu^{18} - 2663674000 \nu^{17} + 3599346675 \nu^{16} + 30194840133 \nu^{15} - 31119918252 \nu^{14} - 187493827325 \nu^{13} + 136061253901 \nu^{12} + 686826922760 \nu^{11} - 311582977642 \nu^{10} - 1492751196985 \nu^{9} + 330166097514 \nu^{8} + 1839263430391 \nu^{7} - 50250467098 \nu^{6} - 1142544646218 \nu^{5} - 154167082409 \nu^{4} + 266688463738 \nu^{3} + 80734359230 \nu^{2} - 6210157485 \nu - 5029558767\)\()/ 1711992393 \)
\(\beta_{4}\)\(=\)\((\)\(152897921 \nu^{19} - 788021802 \nu^{18} - 2581287385 \nu^{17} + 17942676336 \nu^{16} + 12991637169 \nu^{15} - 168614176551 \nu^{14} + 12131055271 \nu^{13} + 845435323468 \nu^{12} - 353315039944 \nu^{11} - 2433681484336 \nu^{10} + 1387964422985 \nu^{9} + 4014750146469 \nu^{8} - 2433986096297 \nu^{7} - 3564239814652 \nu^{6} + 1927564243203 \nu^{5} + 1484823148105 \nu^{4} - 514352194880 \nu^{3} - 239014121539 \nu^{2} + 19091308350 \nu - 1903874094\)\()/ 1711992393 \)
\(\beta_{5}\)\(=\)\((\)\(309851458 \nu^{19} - 1383955827 \nu^{18} - 6233624999 \nu^{17} + 33136669317 \nu^{16} + 46406228355 \nu^{15} - 328492457904 \nu^{14} - 134696224735 \nu^{13} + 1740985774982 \nu^{12} - 79154083532 \nu^{11} - 5302318876388 \nu^{10} + 1465883080024 \nu^{9} + 9252549939678 \nu^{8} - 3483690211072 \nu^{7} - 8681537826470 \nu^{6} + 3181938636582 \nu^{5} + 3828596607116 \nu^{4} - 868959764302 \nu^{3} - 655658120918 \nu^{2} + 12667611897 \nu + 10210385484\)\()/ 1711992393 \)
\(\beta_{6}\)\(=\)\((\)\(322669970 \nu^{19} - 1360457148 \nu^{18} - 6450050938 \nu^{17} + 31439546223 \nu^{16} + 48925034667 \nu^{15} - 299751293436 \nu^{14} - 160904820680 \nu^{13} + 1522754372488 \nu^{12} + 94977561293 \nu^{11} - 4429190235145 \nu^{10} + 822808635896 \nu^{9} + 7345735584573 \nu^{8} - 2230153018763 \nu^{7} - 6497668918675 \nu^{6} + 2022451755138 \nu^{5} + 2669559331153 \nu^{4} - 479591138117 \nu^{3} - 444981502009 \nu^{2} - 13818490434 \nu + 11277017919\)\()/ 1711992393 \)
\(\beta_{7}\)\(=\)\((\)\(322669970 \nu^{19} - 1360457148 \nu^{18} - 6450050938 \nu^{17} + 31439546223 \nu^{16} + 48925034667 \nu^{15} - 299751293436 \nu^{14} - 160904820680 \nu^{13} + 1522754372488 \nu^{12} + 94977561293 \nu^{11} - 4429190235145 \nu^{10} + 822808635896 \nu^{9} + 7345735584573 \nu^{8} - 2230153018763 \nu^{7} - 6497668918675 \nu^{6} + 2022451755138 \nu^{5} + 2669559331153 \nu^{4} - 481303130510 \nu^{3} - 443269509616 \nu^{2} - 6970520862 \nu + 7853033133\)\()/ 1711992393 \)
\(\beta_{8}\)\(=\)\((\)\(146343330 \nu^{19} - 627942667 \nu^{18} - 2750855046 \nu^{17} + 13927919237 \nu^{16} + 19013317515 \nu^{15} - 126771603270 \nu^{14} - 50992377522 \nu^{13} + 611952043660 \nu^{12} - 27928299023 \nu^{11} - 1685927227003 \nu^{10} + 474253792532 \nu^{9} + 2647408333697 \nu^{8} - 1049940919284 \nu^{7} - 2228055002861 \nu^{6} + 917906878673 \nu^{5} + 880564478145 \nu^{4} - 263557449692 \nu^{3} - 141470821499 \nu^{2} + 13425493694 \nu + 3805064997\)\()/ 570664131 \)
\(\beta_{9}\)\(=\)\((\)\(-477486794 \nu^{19} + 2377631685 \nu^{18} + 8657638597 \nu^{17} - 56024761614 \nu^{16} - 50354083845 \nu^{15} + 545581112568 \nu^{14} + 14470325336 \nu^{13} - 2832716706445 \nu^{12} + 1050361806526 \nu^{11} + 8410243049992 \nu^{10} - 4771815292994 \nu^{9} - 14170725778413 \nu^{8} + 9136296553166 \nu^{7} + 12597173558332 \nu^{6} - 7746619103055 \nu^{5} - 5106166101469 \nu^{4} + 2213572376246 \nu^{3} + 862543526110 \nu^{2} - 123020445429 \nu - 21496095405\)\()/ 1711992393 \)
\(\beta_{10}\)\(=\)\((\)\(550477613 \nu^{19} - 2275865718 \nu^{18} - 11409071752 \nu^{17} + 54163977555 \nu^{16} + 89385080193 \nu^{15} - 531636343350 \nu^{14} - 296476727960 \nu^{13} + 2775395086510 \nu^{12} + 96329231267 \nu^{11} - 8265873305065 \nu^{10} + 2096214081833 \nu^{9} + 13944905779887 \nu^{8} - 5708901648527 \nu^{7} - 12398618288461 \nu^{6} + 5657252453124 \nu^{5} + 5021707486708 \nu^{4} - 1822112773976 \nu^{3} - 827450554963 \nu^{2} + 129553986273 \nu + 18210231207\)\()/ 1711992393 \)
\(\beta_{11}\)\(=\)\((\)\(562516600 \nu^{19} - 2270260821 \nu^{18} - 11766008126 \nu^{17} + 53784538836 \nu^{16} + 94239617940 \nu^{15} - 525192157785 \nu^{14} - 334836535351 \nu^{13} + 2725240933223 \nu^{12} + 284298537844 \nu^{11} - 8055235042100 \nu^{10} + 1516003317697 \nu^{9} + 13445110060605 \nu^{8} - 4598705287915 \nu^{7} - 11746626266969 \nu^{6} + 4417753791675 \nu^{5} + 4609864588088 \nu^{4} - 1130822268817 \nu^{3} - 741450867083 \nu^{2} + 7395908349 \nu + 12725619606\)\()/ 1711992393 \)
\(\beta_{12}\)\(=\)\((\)\(-840747436 \nu^{19} + 3451389795 \nu^{18} + 17340713522 \nu^{17} - 81372015369 \nu^{16} - 135968571153 \nu^{15} + 791153125932 \nu^{14} + 460823919703 \nu^{13} - 4091073927500 \nu^{12} - 252098034391 \nu^{11} + 12065066105960 \nu^{10} - 2722915606537 \nu^{9} - 20128396990494 \nu^{8} + 7662576443257 \nu^{7} + 17620952974694 \nu^{6} - 7487287629810 \nu^{5} - 6931059756389 \nu^{4} + 2217972391426 \nu^{3} + 1077516056294 \nu^{2} - 98272405995 \nu - 15043298778\)\()/ 1711992393 \)
\(\beta_{13}\)\(=\)\((\)\(-306455260 \nu^{19} + 1193388237 \nu^{18} + 6440353895 \nu^{17} - 27863463012 \nu^{16} - 52656595317 \nu^{15} + 267919145877 \nu^{14} + 200567866756 \nu^{13} - 1368544326728 \nu^{12} - 271585338031 \nu^{11} + 3983590226324 \nu^{10} - 438853391215 \nu^{9} - 6556627302303 \nu^{8} + 1860096940774 \nu^{7} + 5660449894751 \nu^{6} - 1963791412764 \nu^{5} - 2191104403268 \nu^{4} + 583750175422 \nu^{3} + 334887564545 \nu^{2} - 25266898338 \nu - 6583539255\)\()/ 570664131 \)
\(\beta_{14}\)\(=\)\((\)\(306841820 \nu^{19} - 1142759006 \nu^{18} - 6603875713 \nu^{17} + 26675284060 \nu^{16} + 56303390241 \nu^{15} - 256415042700 \nu^{14} - 234380734790 \nu^{13} + 1309458273684 \nu^{12} + 436085177527 \nu^{11} - 3812158841505 \nu^{10} - 5754095250 \nu^{9} + 6281885236696 \nu^{8} - 1212682461482 \nu^{7} - 5441991275711 \nu^{6} + 1522471869796 \nu^{5} + 2119661708299 \nu^{4} - 494972035686 \nu^{3} - 316169845004 \nu^{2} + 28438633600 \nu + 3636815862\)\()/ 570664131 \)
\(\beta_{15}\)\(=\)\((\)\(1084667243 \nu^{19} - 4546314924 \nu^{18} - 21817310038 \nu^{17} + 105828203667 \nu^{16} + 165046457061 \nu^{15} - 1015296823950 \nu^{14} - 521211788708 \nu^{13} + 5180055874786 \nu^{12} + 90802281113 \nu^{11} - 15082153681294 \nu^{10} + 3787701103535 \nu^{9} + 24888060042309 \nu^{8} - 9712589542436 \nu^{7} - 21643300706941 \nu^{6} + 9087254168172 \nu^{5} + 8535466217248 \nu^{4} - 2585383086992 \nu^{3} - 1351112921560 \nu^{2} + 110179494222 \nu + 21903959223\)\()/ 1711992393 \)
\(\beta_{16}\)\(=\)\((\)\(1089425851 \nu^{19} - 4793379879 \nu^{18} - 21455653031 \nu^{17} + 112365319416 \nu^{16} + 154602262989 \nu^{15} - 1087065350931 \nu^{14} - 412087129414 \nu^{13} + 5600687059850 \nu^{12} - 493304121437 \nu^{11} - 16492312061762 \nu^{10} + 5529256958347 \nu^{9} + 27580460910462 \nu^{8} - 12573673222303 \nu^{7} - 24392018763977 \nu^{6} + 11437902014430 \nu^{5} + 9847996099748 \nu^{4} - 3344500909942 \nu^{3} - 1588589428406 \nu^{2} + 167597701761 \nu + 26342084538\)\()/ 1711992393 \)
\(\beta_{17}\)\(=\)\((\)\(4456799 \nu^{19} - 18879387 \nu^{18} - 90366943 \nu^{17} + 445558719 \nu^{16} + 685127538 \nu^{15} - 4338879945 \nu^{14} - 2109921236 \nu^{13} + 22488696304 \nu^{12} - 294234073 \nu^{11} - 66542626897 \nu^{10} + 18907936892 \nu^{9} + 111562121574 \nu^{8} - 47391522458 \nu^{7} - 98477264995 \nu^{6} + 44859980994 \nu^{5} + 39427171861 \nu^{4} - 13420672217 \nu^{3} - 6391055815 \nu^{2} + 740541705 \nu + 133089489\)\()/6661449\)
\(\beta_{18}\)\(=\)\((\)\(1571933387 \nu^{19} - 6592015260 \nu^{18} - 31890712168 \nu^{17} + 154671482871 \nu^{16} + 243686192967 \nu^{15} - 1496966561592 \nu^{14} - 779896124435 \nu^{13} + 7710757987687 \nu^{12} + 160415769779 \nu^{11} - 22682866537195 \nu^{10} + 5678248146920 \nu^{9} + 37854401980263 \nu^{8} - 14792189063120 \nu^{7} - 33358718318455 \nu^{6} + 14054638663002 \nu^{5} + 13409940203227 \nu^{4} - 4131055705850 \nu^{3} - 2180302356475 \nu^{2} + 204713127345 \nu + 44758625463\)\()/ 1711992393 \)
\(\beta_{19}\)\(=\)\((\)\(600429471 \nu^{19} - 2519966401 \nu^{18} - 12156366042 \nu^{17} + 59005753619 \nu^{16} + 92842904574 \nu^{15} - 570148648821 \nu^{14} - 298853019066 \nu^{13} + 2934376722508 \nu^{12} + 81393002008 \nu^{11} - 8637298145164 \nu^{10} + 2074169438390 \nu^{9} + 14459887772984 \nu^{8} - 5460601016910 \nu^{7} - 12843632042951 \nu^{6} + 5219898632999 \nu^{5} + 5245158178278 \nu^{4} - 1558745920940 \nu^{3} - 859091649656 \nu^{2} + 81393303662 \nu + 16459377834\)\()/ 570664131 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + 3\)
\(\nu^{3}\)\(=\)\(-\beta_{7} + \beta_{6} + \beta_{2} + 4 \beta_{1} + 1\)
\(\nu^{4}\)\(=\)\(-\beta_{17} + \beta_{14} - \beta_{9} - \beta_{7} + \beta_{6} - \beta_{4} - \beta_{3} + 7 \beta_{2} + \beta_{1} + 16\)
\(\nu^{5}\)\(=\)\(\beta_{19} - \beta_{18} - \beta_{17} + \beta_{16} + 3 \beta_{14} + 2 \beta_{13} + \beta_{12} + 2 \beta_{11} - \beta_{10} - \beta_{9} - 11 \beta_{7} + 9 \beta_{6} - \beta_{5} - \beta_{4} - 2 \beta_{3} + 9 \beta_{2} + 21 \beta_{1} + 15\)
\(\nu^{6}\)\(=\)\(\beta_{19} + 2 \beta_{18} - 12 \beta_{17} + \beta_{16} - 2 \beta_{15} + 12 \beta_{14} + \beta_{13} + 2 \beta_{11} - 3 \beta_{10} - 10 \beta_{9} - 15 \beta_{7} + 10 \beta_{6} - \beta_{5} - 9 \beta_{4} - 12 \beta_{3} + 47 \beta_{2} + 13 \beta_{1} + 100\)
\(\nu^{7}\)\(=\)\(14 \beta_{19} - 9 \beta_{18} - 17 \beta_{17} + 13 \beta_{16} - 4 \beta_{15} + 37 \beta_{14} + 22 \beta_{13} + 9 \beta_{12} + 25 \beta_{11} - 16 \beta_{10} - 12 \beta_{9} - 95 \beta_{7} + 66 \beta_{6} - 14 \beta_{5} - 13 \beta_{4} - 30 \beta_{3} + 70 \beta_{2} + 127 \beta_{1} + 145\)
\(\nu^{8}\)\(=\)\(22 \beta_{19} + 28 \beta_{18} - 114 \beta_{17} + 17 \beta_{16} - 31 \beta_{15} + 109 \beta_{14} + 15 \beta_{13} - 3 \beta_{12} + 31 \beta_{11} - 49 \beta_{10} - 78 \beta_{9} - 2 \beta_{8} - 155 \beta_{7} + 79 \beta_{6} - 22 \beta_{5} - 71 \beta_{4} - 117 \beta_{3} + 317 \beta_{2} + 131 \beta_{1} + 673\)
\(\nu^{9}\)\(=\)\(150 \beta_{19} - 51 \beta_{18} - 202 \beta_{17} + 129 \beta_{16} - 73 \beta_{15} + 343 \beta_{14} + 184 \beta_{13} + 58 \beta_{12} + 236 \beta_{11} - 186 \beta_{10} - 109 \beta_{9} - 4 \beta_{8} - 766 \beta_{7} + 454 \beta_{6} - 150 \beta_{5} - 127 \beta_{4} - 320 \beta_{3} + 524 \beta_{2} + 841 \beta_{1} + 1229\)
\(\nu^{10}\)\(=\)\(300 \beta_{19} + 281 \beta_{18} - 1009 \beta_{17} + 209 \beta_{16} - 349 \beta_{15} + 917 \beta_{14} + 164 \beta_{13} - 55 \beta_{12} + 353 \beta_{11} - 569 \beta_{10} - 568 \beta_{9} - 38 \beta_{8} - 1421 \beta_{7} + 578 \beta_{6} - 298 \beta_{5} - 553 \beta_{4} - 1064 \beta_{3} + 2153 \beta_{2} + 1190 \beta_{1} + 4730\)
\(\nu^{11}\)\(=\)\(1457 \beta_{19} - 161 \beta_{18} - 2077 \beta_{17} + 1166 \beta_{16} - 898 \beta_{15} + 2892 \beta_{14} + 1416 \beta_{13} + 295 \beta_{12} + 2033 \beta_{11} - 1898 \beta_{10} - 901 \beta_{9} - 84 \beta_{8} - 6038 \beta_{7} + 3034 \beta_{6} - 1453 \beta_{5} - 1131 \beta_{4} - 3015 \beta_{3} + 3856 \beta_{2} + 5903 \beta_{1} + 9882\)
\(\nu^{12}\)\(=\)\(3342 \beta_{19} + 2525 \beta_{18} - 8679 \beta_{17} + 2212 \beta_{16} - 3470 \beta_{15} + 7512 \beta_{14} + 1590 \beta_{13} - 694 \beta_{12} + 3546 \beta_{11} - 5751 \beta_{10} - 4058 \beta_{9} - 481 \beta_{8} - 12353 \beta_{7} + 4077 \beta_{6} - 3300 \beta_{5} - 4329 \beta_{4} - 9340 \beta_{3} + 14710 \beta_{2} + 10239 \beta_{1} + 34160\)
\(\nu^{13}\)\(=\)\(13444 \beta_{19} + 740 \beta_{18} - 19767 \beta_{17} + 10096 \beta_{16} - 9373 \beta_{15} + 23467 \beta_{14} + 10612 \beta_{13} + 935 \beta_{12} + 16881 \beta_{11} - 18032 \beta_{10} - 7152 \beta_{9} - 1147 \beta_{8} - 47285 \beta_{7} + 19942 \beta_{6} - 13352 \beta_{5} - 9674 \beta_{4} - 26801 \beta_{3} + 28087 \beta_{2} + 42988 \beta_{1} + 77594\)
\(\nu^{14}\)\(=\)\(33431 \beta_{19} + 21740 \beta_{18} - 73552 \beta_{17} + 21449 \beta_{16} - 32345 \beta_{15} + 60853 \beta_{14} + 14464 \beta_{13} - 7510 \beta_{12} + 33229 \beta_{11} - 54015 \beta_{10} - 28963 \beta_{9} - 5143 \beta_{8} - 104348 \beta_{7} + 28091 \beta_{6} - 32865 \beta_{5} - 34068 \beta_{4} - 80170 \beta_{3} + 100996 \beta_{2} + 85379 \beta_{1} + 251135\)
\(\nu^{15}\)\(=\)\(120065 \beta_{19} + 20697 \beta_{18} - 179378 \beta_{17} + 85390 \beta_{16} - 89704 \beta_{15} + 187312 \beta_{14} + 79155 \beta_{13} - 3253 \beta_{12} + 137936 \beta_{11} - 163762 \beta_{10} - 55697 \beta_{9} - 12964 \beta_{8} - 370131 \beta_{7} + 129516 \beta_{6} - 118726 \beta_{5} - 80930 \beta_{4} - 230736 \beta_{3} + 203180 \beta_{2} + 320473 \beta_{1} + 602574\)
\(\nu^{16}\)\(=\)\(313513 \beta_{19} + 183868 \beta_{18} - 617232 \beta_{17} + 196626 \beta_{16} - 289842 \beta_{15} + 490002 \beta_{14} + 126411 \beta_{13} - 74835 \beta_{12} + 297878 \beta_{11} - 485299 \beta_{10} - 208001 \beta_{9} - 50324 \beta_{8} - 865965 \beta_{7} + 189820 \beta_{6} - 307192 \beta_{5} - 268940 \beta_{4} - 677205 \beta_{3} + 696274 \beta_{2} + 698414 \beta_{1} + 1869005\)
\(\nu^{17}\)\(=\)\(1047731 \beta_{19} + 267647 \beta_{18} - 1576771 \beta_{17} + 711696 \beta_{16} - 815151 \beta_{15} + 1484509 \beta_{14} + 592816 \beta_{13} - 102869 \beta_{12} + 1118662 \beta_{11} - 1442100 \beta_{10} - 429821 \beta_{9} - 131962 \beta_{8} - 2902042 \beta_{7} + 832041 \beta_{6} - 1031854 \beta_{5} - 667258 \beta_{4} - 1947334 \beta_{3} + 1462671 \beta_{2} + 2425748 \beta_{1} + 4656292\)
\(\nu^{18}\)\(=\)\(2819553 \beta_{19} + 1541732 \beta_{18} - 5140078 \beta_{17} + 1735208 \beta_{16} - 2528798 \beta_{15} + 3929546 \beta_{14} + 1074956 \beta_{13} - 708412 \beta_{12} + 2590321 \beta_{11} - 4234197 \beta_{10} - 1507157 \beta_{9} - 466977 \beta_{8} - 7101037 \beta_{7} + 1257823 \beta_{6} - 2755579 \beta_{5} - 2126030 \beta_{4} - 5651200 \beta_{3} + 4817306 \beta_{2} + 5643656 \beta_{1} + 14033583\)
\(\nu^{19}\)\(=\)\(8984716 \beta_{19} + 2827347 \beta_{18} - 13552001 \beta_{17} + 5871457 \beta_{16} - 7162439 \beta_{15} + 11730170 \beta_{14} + 4471866 \beta_{13} - 1358968 \beta_{12} + 9037024 \beta_{11} - 12420347 \beta_{10} - 3304008 \beta_{9} - 1258618 \beta_{8} - 22801754 \beta_{7} + 5281303 \beta_{6} - 8816784 \beta_{5} - 5443768 \beta_{4} - 16213139 \beta_{3} + 10493112 \beta_{2} + 18549725 \beta_{1} + 35913688\)

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
2.80474
2.65160
2.60929
1.87740
1.76837
1.71702
1.51806
1.09509
0.700104
0.686421
0.178107
−0.100365
−0.435939
−0.590088
−1.29878
−1.82988
−1.91911
−1.95611
−2.02882
−2.44709
1.00000 −2.80474 1.00000 −0.947662 −2.80474 2.44948 1.00000 4.86656 −0.947662
1.2 1.00000 −2.65160 1.00000 0.347024 −2.65160 −2.28847 1.00000 4.03099 0.347024
1.3 1.00000 −2.60929 1.00000 1.96845 −2.60929 −1.34895 1.00000 3.80840 1.96845
1.4 1.00000 −1.87740 1.00000 3.35129 −1.87740 −2.95261 1.00000 0.524644 3.35129
1.5 1.00000 −1.76837 1.00000 1.49209 −1.76837 3.65436 1.00000 0.127121 1.49209
1.6 1.00000 −1.71702 1.00000 −4.22974 −1.71702 −1.91461 1.00000 −0.0518545 −4.22974
1.7 1.00000 −1.51806 1.00000 −3.34509 −1.51806 1.42565 1.00000 −0.695496 −3.34509
1.8 1.00000 −1.09509 1.00000 −0.379683 −1.09509 −3.84052 1.00000 −1.80078 −0.379683
1.9 1.00000 −0.700104 1.00000 1.01839 −0.700104 0.757569 1.00000 −2.50985 1.01839
1.10 1.00000 −0.686421 1.00000 1.74349 −0.686421 −1.83413 1.00000 −2.52883 1.74349
1.11 1.00000 −0.178107 1.00000 −1.41285 −0.178107 −1.81585 1.00000 −2.96828 −1.41285
1.12 1.00000 0.100365 1.00000 −2.25871 0.100365 1.56060 1.00000 −2.98993 −2.25871
1.13 1.00000 0.435939 1.00000 −0.796234 0.435939 3.50309 1.00000 −2.80996 −0.796234
1.14 1.00000 0.590088 1.00000 0.936829 0.590088 1.40799 1.00000 −2.65180 0.936829
1.15 1.00000 1.29878 1.00000 2.87965 1.29878 −3.57798 1.00000 −1.31316 2.87965
1.16 1.00000 1.82988 1.00000 0.255245 1.82988 −4.20483 1.00000 0.348474 0.255245
1.17 1.00000 1.91911 1.00000 −2.55109 1.91911 0.891228 1.00000 0.682993 −2.55109
1.18 1.00000 1.95611 1.00000 −3.19910 1.95611 0.785940 1.00000 0.826384 −3.19910
1.19 1.00000 2.02882 1.00000 −0.795313 2.02882 −4.29820 1.00000 1.11611 −0.795313
1.20 1.00000 2.44709 1.00000 −0.0770010 2.44709 −0.359767 1.00000 2.98826 −0.0770010
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.20
Significant digits:
Format:

Inner twists

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

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(23\) \(-1\)
\(131\) \(1\)

Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6026))\):

\(T_{3}^{20} + \cdots\)
\(T_{5}^{20} + \cdots\)