Properties

Label 241.2.a.b
Level $241$
Weight $2$
Character orbit 241.a
Self dual yes
Analytic conductor $1.924$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 241 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 241.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(1.92439468871\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Defining polynomial: \(x^{12} - 3 x^{11} - 14 x^{10} + 44 x^{9} + 65 x^{8} - 219 x^{7} - 123 x^{6} + 444 x^{5} + 105 x^{4} - 328 x^{3} - 45 x^{2} + 18 x - 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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.

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

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

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

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
−2.59703
−2.02418
−1.32986
−1.28632
−0.342147
0.0822506
0.115670
1.54879
1.63125
2.01020
2.49073
2.70063
−2.59703 −1.20534 4.74454 3.49051 3.13029 −0.744578 −7.12764 −1.54716 −9.06493
1.2 −2.02418 2.93498 2.09729 1.44091 −5.94092 0.381245 −0.196936 5.61411 −2.91665
1.3 −1.32986 −2.18147 −0.231473 −3.40432 2.90104 −3.83334 2.96755 1.75880 4.52727
1.4 −1.28632 −0.126224 −0.345373 0.612768 0.162365 1.03110 3.01691 −2.98407 −0.788217
1.5 −0.342147 2.18519 −1.88294 −0.548903 −0.747658 1.82459 1.32853 1.77508 0.187805
1.6 0.0822506 1.81824 −1.99323 4.31963 0.149552 0.690569 −0.328446 0.306010 0.355292
1.7 0.115670 −3.28295 −1.98662 −1.31091 −0.379739 3.19647 −0.461133 7.77775 −0.151633
1.8 1.54879 2.81087 0.398765 0.334961 4.35346 −4.24623 −2.47998 4.90098 0.518786
1.9 1.63125 −1.16790 0.660992 1.75438 −1.90514 5.06139 −2.18426 −1.63601 2.86183
1.10 2.01020 0.500591 2.04092 1.92585 1.00629 −0.852319 0.0822476 −2.74941 3.87135
1.11 2.49073 1.22208 4.20371 −3.14843 3.04385 0.136122 5.48885 −1.50653 −7.84189
1.12 2.70063 −2.50808 5.29342 0.533570 −6.77340 0.354992 8.89432 3.29045 1.44098
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.12
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(241\) \(-1\)

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 241.2.a.b 12
3.b odd 2 1 2169.2.a.h 12
4.b odd 2 1 3856.2.a.n 12
5.b even 2 1 6025.2.a.h 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
241.2.a.b 12 1.a even 1 1 trivial
2169.2.a.h 12 3.b odd 2 1
3856.2.a.n 12 4.b odd 2 1
6025.2.a.h 12 5.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{2}^{12} - \cdots\) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(241))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -1 + 18 T - 45 T^{2} - 328 T^{3} + 105 T^{4} + 444 T^{5} - 123 T^{6} - 219 T^{7} + 65 T^{8} + 44 T^{9} - 14 T^{10} - 3 T^{11} + T^{12} \)
$3$ \( 64 + 400 T - 992 T^{2} - 960 T^{3} + 1540 T^{4} + 725 T^{5} - 888 T^{6} - 210 T^{7} + 224 T^{8} + 25 T^{9} - 25 T^{10} - T^{11} + T^{12} \)
$5$ \( 62 - 347 T + 339 T^{2} + 1071 T^{3} - 2193 T^{4} + 497 T^{5} + 1301 T^{6} - 797 T^{7} - 68 T^{8} + 134 T^{9} - 14 T^{10} - 6 T^{11} + T^{12} \)
$7$ \( 4 - 53 T + 200 T^{2} - 131 T^{3} - 588 T^{4} + 855 T^{5} + 263 T^{6} - 854 T^{7} + 245 T^{8} + 96 T^{9} - 33 T^{10} - 3 T^{11} + T^{12} \)
$11$ \( 128 - 1460 T + 5900 T^{2} - 9672 T^{3} + 3100 T^{4} + 9811 T^{5} - 12739 T^{6} + 5545 T^{7} - 215 T^{8} - 553 T^{9} + 177 T^{10} - 22 T^{11} + T^{12} \)
$13$ \( -52672 - 441248 T - 90512 T^{2} + 288472 T^{3} + 69802 T^{4} - 64645 T^{5} - 15049 T^{6} + 6470 T^{7} + 1425 T^{8} - 296 T^{9} - 62 T^{10} + 5 T^{11} + T^{12} \)
$17$ \( 154144 - 302576 T - 261264 T^{2} + 295792 T^{3} + 159474 T^{4} - 87027 T^{5} - 35221 T^{6} + 9972 T^{7} + 2997 T^{8} - 370 T^{9} - 97 T^{10} + 4 T^{11} + T^{12} \)
$19$ \( -3556280 - 3617301 T + 903711 T^{2} + 1614089 T^{3} + 58969 T^{4} - 247081 T^{5} - 28947 T^{6} + 16891 T^{7} + 2538 T^{8} - 524 T^{9} - 86 T^{10} + 6 T^{11} + T^{12} \)
$23$ \( -116949436 + 276824423 T - 182523158 T^{2} + 31479187 T^{3} + 11455889 T^{4} - 5191210 T^{5} + 372702 T^{6} + 138011 T^{7} - 28306 T^{8} + 627 T^{9} + 304 T^{10} - 32 T^{11} + T^{12} \)
$29$ \( 58109390 - 179077009 T + 164527164 T^{2} - 50865568 T^{3} - 3169769 T^{4} + 4236578 T^{5} - 355685 T^{6} - 116722 T^{7} + 15216 T^{8} + 1375 T^{9} - 213 T^{10} - 6 T^{11} + T^{12} \)
$31$ \( -318193616 + 468437780 T + 77270368 T^{2} - 108211396 T^{3} + 841016 T^{4} + 8192365 T^{5} - 688338 T^{6} - 208450 T^{7} + 22930 T^{8} + 2167 T^{9} - 262 T^{10} - 8 T^{11} + T^{12} \)
$37$ \( 50796928 - 18033984 T - 29177888 T^{2} + 5067040 T^{3} + 4698614 T^{4} - 619307 T^{5} - 323567 T^{6} + 36249 T^{7} + 10466 T^{8} - 928 T^{9} - 159 T^{10} + 8 T^{11} + T^{12} \)
$41$ \( -63338 - 4034251 T - 930334 T^{2} + 9341574 T^{3} - 976432 T^{4} - 2920817 T^{5} - 471938 T^{6} + 92737 T^{7} + 21111 T^{8} - 708 T^{9} - 262 T^{10} + T^{11} + T^{12} \)
$43$ \( 12503272 - 1488169 T - 25360675 T^{2} - 4519982 T^{3} + 6500883 T^{4} + 657869 T^{5} - 569920 T^{6} - 18272 T^{7} + 18808 T^{8} - 26 T^{9} - 237 T^{10} + 2 T^{11} + T^{12} \)
$47$ \( 53297792 + 37689968 T - 69239488 T^{2} + 11362328 T^{3} + 11628792 T^{4} - 4772881 T^{5} + 233524 T^{6} + 179952 T^{7} - 34508 T^{8} + 851 T^{9} + 332 T^{10} - 34 T^{11} + T^{12} \)
$53$ \( -3014 + 298853 T + 6874728 T^{2} - 11787807 T^{3} + 1527793 T^{4} + 1538756 T^{5} - 270515 T^{6} - 65448 T^{7} + 12170 T^{8} + 1019 T^{9} - 195 T^{10} - 5 T^{11} + T^{12} \)
$59$ \( -25476160 + 96501648 T - 59605584 T^{2} - 36571744 T^{3} + 47289076 T^{4} - 17302787 T^{5} + 2412075 T^{6} + 67258 T^{7} - 57444 T^{8} + 5338 T^{9} + 22 T^{10} - 26 T^{11} + T^{12} \)
$61$ \( 10893274 - 29191311 T + 16419914 T^{2} + 10617016 T^{3} - 8091392 T^{4} - 1560634 T^{5} + 801476 T^{6} + 117122 T^{7} - 22505 T^{8} - 3955 T^{9} + 20 T^{10} + 26 T^{11} + T^{12} \)
$67$ \( 4538509504 + 4714883120 T - 678403520 T^{2} - 669419464 T^{3} + 75258180 T^{4} + 29459511 T^{5} - 3555172 T^{6} - 474596 T^{7} + 62307 T^{8} + 2947 T^{9} - 429 T^{10} - 6 T^{11} + T^{12} \)
$71$ \( -12017198348 + 39886445545 T - 42976926619 T^{2} + 21246049133 T^{3} - 5606812300 T^{4} + 801669175 T^{5} - 46063490 T^{6} - 3647013 T^{7} + 918997 T^{8} - 79770 T^{9} + 3737 T^{10} - 94 T^{11} + T^{12} \)
$73$ \( 2219968 + 25741920 T + 64034288 T^{2} - 76860088 T^{3} - 83908922 T^{4} - 7168229 T^{5} + 3607447 T^{6} + 533877 T^{7} - 28097 T^{8} - 7860 T^{9} - 208 T^{10} + 22 T^{11} + T^{12} \)
$79$ \( -1277319040 + 3418562576 T + 1017318496 T^{2} - 2331826216 T^{3} + 163831840 T^{4} + 90986869 T^{5} - 7904508 T^{6} - 1163461 T^{7} + 109307 T^{8} + 5783 T^{9} - 581 T^{10} - 9 T^{11} + T^{12} \)
$83$ \( 98860915136 - 10813065520 T - 10895951696 T^{2} + 1151271176 T^{3} + 452061900 T^{4} - 42544271 T^{5} - 9197429 T^{6} + 669374 T^{7} + 100342 T^{8} - 4386 T^{9} - 548 T^{10} + 8 T^{11} + T^{12} \)
$89$ \( -1500609440 - 3427797584 T - 829552176 T^{2} + 490376448 T^{3} + 124536598 T^{4} - 21479633 T^{5} - 5233031 T^{6} + 305681 T^{7} + 79002 T^{8} - 1663 T^{9} - 479 T^{10} + 3 T^{11} + T^{12} \)
$97$ \( 107861318 - 194920563 T - 162786822 T^{2} + 64432913 T^{3} + 85302025 T^{4} + 27963948 T^{5} + 3390903 T^{6} - 118070 T^{7} - 70766 T^{8} - 5577 T^{9} + 85 T^{10} + 29 T^{11} + T^{12} \)
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