Properties

Label 349.2.a.a
Level 349
Weight 2
Character orbit 349.a
Self dual Yes
Analytic conductor 2.787
Analytic rank 1
Dimension 11
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 349 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 349.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(2.78677903054\)
Analytic rank: \(1\)
Dimension: \(11\)
Coefficient field: \(\mathbb{Q}[x]/(x^{11} - \cdots)\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
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 \(1,\beta_1,\ldots,\beta_{10}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{11} - 5 x^{10} - x^{9} + 35 x^{8} - 24 x^{7} - 80 x^{6} + 66 x^{5} + 77 x^{4} - 56 x^{3} - 31 x^{2} + 15 x + 4\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - \nu - 2 \)
\(\beta_{3}\)\(=\)\((\)\( -\nu^{10} + 6 \nu^{9} - 4 \nu^{8} - 33 \nu^{7} + 49 \nu^{6} + 42 \nu^{5} - 85 \nu^{4} - 2 \nu^{3} + 33 \nu^{2} - 12 \nu \)\()/2\)
\(\beta_{4}\)\(=\)\( -\nu^{10} + 5 \nu^{9} - 30 \nu^{7} + 24 \nu^{6} + 50 \nu^{5} - 42 \nu^{4} - 27 \nu^{3} + 13 \nu^{2} + 5 \nu + 2 \)
\(\beta_{5}\)\(=\)\((\)\( 2 \nu^{10} - 9 \nu^{9} - 6 \nu^{8} + 64 \nu^{7} - 15 \nu^{6} - 147 \nu^{5} + 36 \nu^{4} + 131 \nu^{3} + 4 \nu^{2} - 39 \nu - 14 \)\()/2\)
\(\beta_{6}\)\(=\)\((\)\( 2 \nu^{10} - 9 \nu^{9} - 6 \nu^{8} + 64 \nu^{7} - 15 \nu^{6} - 149 \nu^{5} + 40 \nu^{4} + 141 \nu^{3} - 14 \nu^{2} - 47 \nu - 2 \)\()/2\)
\(\beta_{7}\)\(=\)\( -\nu^{10} + 5 \nu^{9} - 31 \nu^{7} + 27 \nu^{6} + 55 \nu^{5} - 58 \nu^{4} - 33 \nu^{3} + 31 \nu^{2} + 6 \nu - 1 \)
\(\beta_{8}\)\(=\)\( \nu^{10} - 5 \nu^{9} + 31 \nu^{7} - 27 \nu^{6} - 55 \nu^{5} + 59 \nu^{4} + 32 \nu^{3} - 36 \nu^{2} - 4 \nu + 5 \)
\(\beta_{9}\)\(=\)\((\)\( \nu^{10} - 2 \nu^{9} - 16 \nu^{8} + 33 \nu^{7} + 73 \nu^{6} - 144 \nu^{5} - 125 \nu^{4} + 202 \nu^{3} + 87 \nu^{2} - 78 \nu - 20 \)\()/2\)
\(\beta_{10}\)\(=\)\((\)\( -2 \nu^{10} + 7 \nu^{9} + 16 \nu^{8} - 64 \nu^{7} - 47 \nu^{6} + 201 \nu^{5} + 72 \nu^{4} - 243 \nu^{3} - 60 \nu^{2} + 87 \nu + 16 \)\()/2\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + \beta_{1} + 2\)
\(\nu^{3}\)\(=\)\(-\beta_{9} + \beta_{7} + \beta_{6} + \beta_{5} + \beta_{3} + \beta_{2} + 5 \beta_{1} + 1\)
\(\nu^{4}\)\(=\)\(-\beta_{9} + \beta_{8} + 2 \beta_{7} + \beta_{6} + \beta_{5} + \beta_{3} + 6 \beta_{2} + 8 \beta_{1} + 7\)
\(\nu^{5}\)\(=\)\(-7 \beta_{9} + 2 \beta_{8} + 9 \beta_{7} + 6 \beta_{6} + 8 \beta_{5} + 7 \beta_{3} + 8 \beta_{2} + 28 \beta_{1} + 7\)
\(\nu^{6}\)\(=\)\(-2 \beta_{10} - 13 \beta_{9} + 9 \beta_{8} + 21 \beta_{7} + 9 \beta_{6} + 13 \beta_{5} + 11 \beta_{3} + 34 \beta_{2} + 55 \beta_{1} + 30\)
\(\nu^{7}\)\(=\)\(-6 \beta_{10} - 52 \beta_{9} + 21 \beta_{8} + 69 \beta_{7} + 35 \beta_{6} + 57 \beta_{5} + \beta_{4} + 46 \beta_{3} + 58 \beta_{2} + 166 \beta_{1} + 40\)
\(\nu^{8}\)\(=\)\(-29 \beta_{10} - 121 \beta_{9} + 68 \beta_{8} + 171 \beta_{7} + 65 \beta_{6} + 118 \beta_{5} + 2 \beta_{4} + 93 \beta_{3} + 199 \beta_{2} + 364 \beta_{1} + 138\)
\(\nu^{9}\)\(=\)\(-84 \beta_{10} - 389 \beta_{9} + 169 \beta_{8} + 499 \beta_{7} + 206 \beta_{6} + 400 \beta_{5} + 10 \beta_{4} + 311 \beta_{3} + 397 \beta_{2} + 1019 \beta_{1} + 216\)
\(\nu^{10}\)\(=\)\(-288 \beta_{10} - 978 \beta_{9} + 489 \beta_{8} + 1268 \beta_{7} + 427 \beta_{6} + 933 \beta_{5} + 19 \beta_{4} + 720 \beta_{3} + 1195 \beta_{2} + 2382 \beta_{1} + 657\)

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
2.60070
2.42122
2.17018
1.31062
0.831463
0.734924
−0.216390
−0.767986
−0.916892
−1.25205
−1.91579
−2.60070 −0.209864 4.76365 1.72230 0.545795 −2.16571 −7.18744 −2.95596 −4.47919
1.2 −2.42122 2.15404 3.86232 −2.99362 −5.21540 0.559596 −4.50909 1.63987 7.24821
1.3 −2.17018 −2.21125 2.70967 −2.88560 4.79881 3.08193 −1.54011 1.88963 6.26228
1.4 −1.31062 0.477277 −0.282282 −1.24067 −0.625527 0.925788 2.99120 −2.77221 1.62604
1.5 −0.831463 −3.29989 −1.30867 1.20940 2.74374 3.20765 2.75104 7.88930 −1.00557
1.6 −0.734924 0.314167 −1.45989 2.34643 −0.230889 −4.45589 2.54275 −2.90130 −1.72445
1.7 0.216390 2.05625 −1.95318 −4.07164 0.444951 −1.74178 −0.855428 1.22815 −0.881063
1.8 0.767986 0.193628 −1.41020 −0.953573 0.148703 −2.43623 −2.61898 −2.96251 −0.732330
1.9 0.916892 −2.19816 −1.15931 2.45207 −2.01547 −1.41213 −2.89675 1.83189 2.24828
1.10 1.25205 −1.02718 −0.432368 −3.34750 −1.28608 4.81118 −3.04545 −1.94490 −4.19124
1.11 1.91579 −2.24901 1.67025 −1.23759 −4.30862 −3.37442 −0.631741 2.05803 −2.37097
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.11
Significant digits:
Format:

Inner twists

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

Atkin-Lehner signs

\( p \) Sign
\(349\) \(1\)

Hecke kernels

This newform can be constructed as the kernel of the linear operator \(T_{2}^{11} + \cdots\) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(349))\).