Properties

Label 6025.2.a.h
Level 6025
Weight 2
Character orbit 6025.a
Self dual Yes
Analytic conductor 48.110
Analytic rank 0
Dimension 12
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 6025 = 5^{2} \cdot 241 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 6025.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(48.1098672178\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

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

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{12}\mathstrut -\mathstrut \) \(3\) \(x^{11}\mathstrut -\mathstrut \) \(14\) \(x^{10}\mathstrut +\mathstrut \) \(44\) \(x^{9}\mathstrut +\mathstrut \) \(65\) \(x^{8}\mathstrut -\mathstrut \) \(219\) \(x^{7}\mathstrut -\mathstrut \) \(123\) \(x^{6}\mathstrut +\mathstrut \) \(444\) \(x^{5}\mathstrut +\mathstrut \) \(105\) \(x^{4}\mathstrut -\mathstrut \) \(328\) \(x^{3}\mathstrut -\mathstrut \) \(45\) \(x^{2}\mathstrut +\mathstrut \) \(18\) \(x\mathstrut -\mathstrut \) \(1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 3 \)
\(\beta_{3}\)\(=\)\((\)\( \nu^{11} + \nu^{10} - 23 \nu^{9} - 13 \nu^{8} + 184 \nu^{7} + 54 \nu^{6} - 611 \nu^{5} - 94 \nu^{4} + 768 \nu^{3} + 94 \nu^{2} - 213 \nu - 8 \)\()/8\)
\(\beta_{4}\)\(=\)\((\)\( -4 \nu^{11} + 9 \nu^{10} + 57 \nu^{9} - 119 \nu^{8} - 273 \nu^{7} + 488 \nu^{6} + 538 \nu^{5} - 663 \nu^{4} - 422 \nu^{3} + 168 \nu^{2} + 34 \nu + 7 \)\()/8\)
\(\beta_{5}\)\(=\)\((\)\( 7 \nu^{11} - 10 \nu^{10} - 130 \nu^{9} + 172 \nu^{8} + 869 \nu^{7} - 1046 \nu^{6} - 2491 \nu^{5} + 2625 \nu^{4} + 2774 \nu^{3} - 2230 \nu^{2} - 745 \nu + 85 \)\()/16\)
\(\beta_{6}\)\(=\)\((\)\( -13 \nu^{11} + 48 \nu^{10} + 160 \nu^{9} - 690 \nu^{8} - 569 \nu^{7} + 3330 \nu^{6} + 597 \nu^{5} - 6481 \nu^{4} - 470 \nu^{3} + 4634 \nu^{2} + 951 \nu - 169 \)\()/16\)
\(\beta_{7}\)\(=\)\((\)\( 13 \nu^{11} - 44 \nu^{10} - 180 \nu^{9} + 662 \nu^{8} + 829 \nu^{7} - 3426 \nu^{6} - 1629 \nu^{5} + 7333 \nu^{4} + 1798 \nu^{3} - 5698 \nu^{2} - 1335 \nu + 141 \)\()/16\)
\(\beta_{8}\)\(=\)\((\)\( 11 \nu^{11} - 30 \nu^{10} - 158 \nu^{9} + 432 \nu^{8} + 773 \nu^{7} - 2086 \nu^{6} - 1631 \nu^{5} + 4025 \nu^{4} + 1654 \nu^{3} - 2750 \nu^{2} - 741 \nu + 93 \)\()/8\)
\(\beta_{9}\)\(=\)\((\)\( -11 \nu^{11} + 34 \nu^{10} + 150 \nu^{9} - 492 \nu^{8} - 669 \nu^{7} + 2398 \nu^{6} + 1223 \nu^{5} - 4717 \nu^{4} - 1178 \nu^{3} + 3354 \nu^{2} + 737 \nu - 109 \)\()/8\)
\(\beta_{10}\)\(=\)\((\)\( 25 \nu^{11} - 76 \nu^{10} - 340 \nu^{9} + 1086 \nu^{8} + 1513 \nu^{7} - 5178 \nu^{6} - 2777 \nu^{5} + 9809 \nu^{4} + 2718 \nu^{3} - 6602 \nu^{2} - 1643 \nu + 201 \)\()/16\)
\(\beta_{11}\)\(=\)\((\)\( 31 \nu^{11} - 90 \nu^{10} - 450 \nu^{9} + 1340 \nu^{8} + 2237 \nu^{7} - 6822 \nu^{6} - 4819 \nu^{5} + 14249 \nu^{4} + 5014 \nu^{3} - 10758 \nu^{2} - 2497 \nu + 381 \)\()/16\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2}\mathstrut +\mathstrut \) \(3\)
\(\nu^{3}\)\(=\)\(\beta_{11}\mathstrut -\mathstrut \) \(\beta_{10}\mathstrut +\mathstrut \) \(\beta_{9}\mathstrut -\mathstrut \) \(\beta_{6}\mathstrut -\mathstrut \) \(\beta_{5}\mathstrut -\mathstrut \) \(\beta_{4}\mathstrut +\mathstrut \) \(\beta_{3}\mathstrut +\mathstrut \) \(5\) \(\beta_{1}\mathstrut -\mathstrut \) \(1\)
\(\nu^{4}\)\(=\)\(2\) \(\beta_{11}\mathstrut +\mathstrut \) \(\beta_{9}\mathstrut -\mathstrut \) \(\beta_{8}\mathstrut -\mathstrut \) \(2\) \(\beta_{7}\mathstrut -\mathstrut \) \(\beta_{6}\mathstrut -\mathstrut \) \(\beta_{5}\mathstrut +\mathstrut \) \(\beta_{3}\mathstrut +\mathstrut \) \(8\) \(\beta_{2}\mathstrut +\mathstrut \) \(15\)
\(\nu^{5}\)\(=\)\(11\) \(\beta_{11}\mathstrut -\mathstrut \) \(9\) \(\beta_{10}\mathstrut +\mathstrut \) \(10\) \(\beta_{9}\mathstrut -\mathstrut \) \(\beta_{7}\mathstrut -\mathstrut \) \(8\) \(\beta_{6}\mathstrut -\mathstrut \) \(11\) \(\beta_{5}\mathstrut -\mathstrut \) \(9\) \(\beta_{4}\mathstrut +\mathstrut \) \(9\) \(\beta_{3}\mathstrut +\mathstrut \) \(\beta_{2}\mathstrut +\mathstrut \) \(29\) \(\beta_{1}\mathstrut -\mathstrut \) \(10\)
\(\nu^{6}\)\(=\)\(20\) \(\beta_{11}\mathstrut +\mathstrut \) \(2\) \(\beta_{10}\mathstrut +\mathstrut \) \(10\) \(\beta_{9}\mathstrut -\mathstrut \) \(14\) \(\beta_{8}\mathstrut -\mathstrut \) \(19\) \(\beta_{7}\mathstrut -\mathstrut \) \(10\) \(\beta_{6}\mathstrut -\mathstrut \) \(9\) \(\beta_{5}\mathstrut -\mathstrut \) \(2\) \(\beta_{4}\mathstrut +\mathstrut \) \(11\) \(\beta_{3}\mathstrut +\mathstrut \) \(56\) \(\beta_{2}\mathstrut +\mathstrut \) \(88\)
\(\nu^{7}\)\(=\)\(93\) \(\beta_{11}\mathstrut -\mathstrut \) \(67\) \(\beta_{10}\mathstrut +\mathstrut \) \(78\) \(\beta_{9}\mathstrut -\mathstrut \) \(3\) \(\beta_{8}\mathstrut -\mathstrut \) \(12\) \(\beta_{7}\mathstrut -\mathstrut \) \(54\) \(\beta_{6}\mathstrut -\mathstrut \) \(94\) \(\beta_{5}\mathstrut -\mathstrut \) \(68\) \(\beta_{4}\mathstrut +\mathstrut \) \(71\) \(\beta_{3}\mathstrut +\mathstrut \) \(11\) \(\beta_{2}\mathstrut +\mathstrut \) \(181\) \(\beta_{1}\mathstrut -\mathstrut \) \(77\)
\(\nu^{8}\)\(=\)\(163\) \(\beta_{11}\mathstrut +\mathstrut \) \(24\) \(\beta_{10}\mathstrut +\mathstrut \) \(76\) \(\beta_{9}\mathstrut -\mathstrut \) \(138\) \(\beta_{8}\mathstrut -\mathstrut \) \(146\) \(\beta_{7}\mathstrut -\mathstrut \) \(78\) \(\beta_{6}\mathstrut -\mathstrut \) \(71\) \(\beta_{5}\mathstrut -\mathstrut \) \(30\) \(\beta_{4}\mathstrut +\mathstrut \) \(98\) \(\beta_{3}\mathstrut +\mathstrut \) \(379\) \(\beta_{2}\mathstrut +\mathstrut \) \(4\) \(\beta_{1}\mathstrut +\mathstrut \) \(558\)
\(\nu^{9}\)\(=\)\(720\) \(\beta_{11}\mathstrut -\mathstrut \) \(478\) \(\beta_{10}\mathstrut +\mathstrut \) \(557\) \(\beta_{9}\mathstrut -\mathstrut \) \(59\) \(\beta_{8}\mathstrut -\mathstrut \) \(106\) \(\beta_{7}\mathstrut -\mathstrut \) \(355\) \(\beta_{6}\mathstrut -\mathstrut \) \(733\) \(\beta_{5}\mathstrut -\mathstrut \) \(499\) \(\beta_{4}\mathstrut +\mathstrut \) \(542\) \(\beta_{3}\mathstrut +\mathstrut \) \(88\) \(\beta_{2}\mathstrut +\mathstrut \) \(1179\) \(\beta_{1}\mathstrut -\mathstrut \) \(544\)
\(\nu^{10}\)\(=\)\(1256\) \(\beta_{11}\mathstrut +\mathstrut \) \(191\) \(\beta_{10}\mathstrut +\mathstrut \) \(522\) \(\beta_{9}\mathstrut -\mathstrut \) \(1189\) \(\beta_{8}\mathstrut -\mathstrut \) \(1056\) \(\beta_{7}\mathstrut -\mathstrut \) \(566\) \(\beta_{6}\mathstrut -\mathstrut \) \(561\) \(\beta_{5}\mathstrut -\mathstrut \) \(323\) \(\beta_{4}\mathstrut +\mathstrut \) \(822\) \(\beta_{3}\mathstrut +\mathstrut \) \(2542\) \(\beta_{2}\mathstrut +\mathstrut \) \(76\) \(\beta_{1}\mathstrut +\mathstrut \) \(3665\)
\(\nu^{11}\)\(=\)\(5372\) \(\beta_{11}\mathstrut -\mathstrut \) \(3384\) \(\beta_{10}\mathstrut +\mathstrut \) \(3821\) \(\beta_{9}\mathstrut -\mathstrut \) \(748\) \(\beta_{8}\mathstrut -\mathstrut \) \(845\) \(\beta_{7}\mathstrut -\mathstrut \) \(2351\) \(\beta_{6}\mathstrut -\mathstrut \) \(5486\) \(\beta_{5}\mathstrut -\mathstrut \) \(3655\) \(\beta_{4}\mathstrut +\mathstrut \) \(4093\) \(\beta_{3}\mathstrut +\mathstrut \) \(630\) \(\beta_{2}\mathstrut +\mathstrut \) \(7881\) \(\beta_{1}\mathstrut -\mathstrut \) \(3713\)

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
2.70063
2.49073
2.01020
1.63125
1.54879
0.115670
0.0822506
−0.342147
−1.28632
−1.32986
−2.02418
−2.59703
−2.70063 2.50808 5.29342 0 −6.77340 −0.354992 −8.89432 3.29045 0
1.2 −2.49073 −1.22208 4.20371 0 3.04385 −0.136122 −5.48885 −1.50653 0
1.3 −2.01020 −0.500591 2.04092 0 1.00629 0.852319 −0.0822476 −2.74941 0
1.4 −1.63125 1.16790 0.660992 0 −1.90514 −5.06139 2.18426 −1.63601 0
1.5 −1.54879 −2.81087 0.398765 0 4.35346 4.24623 2.47998 4.90098 0
1.6 −0.115670 3.28295 −1.98662 0 −0.379739 −3.19647 0.461133 7.77775 0
1.7 −0.0822506 −1.81824 −1.99323 0 0.149552 −0.690569 0.328446 0.306010 0
1.8 0.342147 −2.18519 −1.88294 0 −0.747658 −1.82459 −1.32853 1.77508 0
1.9 1.28632 0.126224 −0.345373 0 0.162365 −1.03110 −3.01691 −2.98407 0
1.10 1.32986 2.18147 −0.231473 0 2.90104 3.83334 −2.96755 1.75880 0
1.11 2.02418 −2.93498 2.09729 0 −5.94092 −0.381245 0.196936 5.61411 0
1.12 2.59703 1.20534 4.74454 0 3.13029 0.744578 7.12764 −1.54716 0
\(n\): 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
\(5\) \(1\)
\(241\) \(-1\)

Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6025))\):

\(T_{2}^{12} + \cdots\)
\(T_{3}^{12} + \cdots\)