Properties

Label 349.2.a.b
Level 349
Weight 2
Character orbit 349.a
Self dual Yes
Analytic conductor 2.787
Analytic rank 0
Dimension 17
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: \(0\)
Dimension: \(17\)
Coefficient field: \(\mathbb{Q}[x]/(x^{17} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{16}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{17} - 5 x^{16} - 14 x^{15} + 102 x^{14} + 26 x^{13} - 792 x^{12} + 474 x^{11} + 2887 x^{10} - 3021 x^{9} - 4835 x^{8} + 6673 x^{7} + 2880 x^{6} - 5373 x^{5} - 164 x^{4} + 1075 x^{3} + 75 x^{2} - 41 x - 4\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 3 \)
\(\beta_{3}\)\(=\)\( \nu^{3} - 5 \nu \)
\(\beta_{4}\)\(=\)\((\)\(6881 \nu^{16} + 3046496 \nu^{15} - 12143208 \nu^{14} - 53358172 \nu^{13} + 248098852 \nu^{12} + 302832280 \nu^{11} - 1946294566 \nu^{10} - 358261227 \nu^{9} + 7225993314 \nu^{8} - 2198561585 \nu^{7} - 12507143628 \nu^{6} + 7020646464 \nu^{5} + 8001174573 \nu^{4} - 5454831045 \nu^{3} - 528604566 \nu^{2} + 464407341 \nu - 349096\)\()/13854332\)
\(\beta_{5}\)\(=\)\((\)\(129907 \nu^{16} - 673231 \nu^{15} - 842951 \nu^{14} + 11316825 \nu^{13} - 15879081 \nu^{12} - 58049485 \nu^{11} + 202090863 \nu^{10} + 16498676 \nu^{9} - 839916930 \nu^{8} + 693906061 \nu^{7} + 1360000699 \nu^{6} - 1874089153 \nu^{5} - 446520802 \nu^{4} + 1367630449 \nu^{3} - 385718169 \nu^{2} - 18197482 \nu + 33470372\)\()/13854332\)
\(\beta_{6}\)\(=\)\((\)\(342344 \nu^{16} - 1539407 \nu^{15} - 4675055 \nu^{14} + 29695925 \nu^{13} + 6728395 \nu^{12} - 208650675 \nu^{11} + 175130105 \nu^{10} + 610889171 \nu^{9} - 1046385666 \nu^{8} - 440334094 \nu^{7} + 2146286917 \nu^{6} - 1020314197 \nu^{5} - 1394056621 \nu^{4} + 1432645940 \nu^{3} + 26091993 \nu^{2} - 345190511 \nu - 16651472\)\()/13854332\)
\(\beta_{7}\)\(=\)\((\)\(954477 \nu^{16} - 6048190 \nu^{15} - 9436362 \nu^{14} + 122225562 \nu^{13} - 56644828 \nu^{12} - 933243792 \nu^{11} + 1102309566 \nu^{10} + 3290717187 \nu^{9} - 5364723646 \nu^{8} - 5084510469 \nu^{7} + 10918796292 \nu^{6} + 2175158294 \nu^{5} - 8563127797 \nu^{4} + 550576315 \nu^{3} + 1738656858 \nu^{2} + 84175177 \nu - 60860676\)\()/13854332\)
\(\beta_{8}\)\(=\)\((\)\(1423805 \nu^{16} - 5123424 \nu^{15} - 25701018 \nu^{14} + 104173792 \nu^{13} + 157565816 \nu^{12} - 803930404 \nu^{11} - 296651690 \nu^{10} + 2889057543 \nu^{9} - 534122266 \nu^{8} - 4627216617 \nu^{7} + 2410932626 \nu^{6} + 2127467628 \nu^{5} - 2034349445 \nu^{4} + 787373203 \nu^{3} + 301820608 \nu^{2} - 187476637 \nu - 38280072\)\()/13854332\)
\(\beta_{9}\)\(=\)\((\)\(-1624725 \nu^{16} + 8295022 \nu^{15} + 22421616 \nu^{14} - 168218710 \nu^{13} - 38246072 \nu^{12} + 1292185188 \nu^{11} - 770532978 \nu^{10} - 4606979421 \nu^{9} + 4723757470 \nu^{8} + 7287976453 \nu^{7} - 9907626526 \nu^{6} - 3380595882 \nu^{5} + 7039828559 \nu^{4} - 765969227 \nu^{3} - 752233888 \nu^{2} + 32793065 \nu - 16113024\)\()/13854332\)
\(\beta_{10}\)\(=\)\((\)\(-2320607 \nu^{16} + 13871476 \nu^{15} + 23761238 \nu^{14} - 276884452 \nu^{13} + 119779992 \nu^{12} + 2072568454 \nu^{11} - 2524139110 \nu^{10} - 7035651475 \nu^{9} + 12362382076 \nu^{8} + 9816264405 \nu^{7} - 24980012280 \nu^{6} - 1752525352 \nu^{5} + 18977545907 \nu^{4} - 3709594629 \nu^{3} - 3249870492 \nu^{2} + 334787557 \nu + 100990532\)\()/13854332\)
\(\beta_{11}\)\(=\)\((\)\(1177751 \nu^{16} - 4876786 \nu^{15} - 19898736 \nu^{14} + 100486682 \nu^{13} + 101420316 \nu^{12} - 791693437 \nu^{11} - 6972284 \nu^{10} + 2952339580 \nu^{9} - 1397185719 \nu^{8} - 5145466896 \nu^{7} + 3888315285 \nu^{6} + 3337708828 \nu^{5} - 3336736615 \nu^{4} - 213355823 \nu^{3} + 708875774 \nu^{2} - 40686922 \nu - 38449256\)\()/6927166\)
\(\beta_{12}\)\(=\)\((\)\(1430016 \nu^{16} - 7943686 \nu^{15} - 16588569 \nu^{14} + 158865051 \nu^{13} - 33046398 \nu^{12} - 1193366371 \nu^{11} + 1225665635 \nu^{10} + 4081521360 \nu^{9} - 6331983519 \nu^{8} - 5820649228 \nu^{7} + 12956859425 \nu^{6} + 1355364934 \nu^{5} - 9771397341 \nu^{4} + 1891397865 \nu^{3} + 1628902717 \nu^{2} - 173708857 \nu - 53459238\)\()/6927166\)
\(\beta_{13}\)\(=\)\((\)\(2212232 \nu^{16} - 10666320 \nu^{15} - 31922034 \nu^{14} + 216378870 \nu^{13} + 78707123 \nu^{12} - 1664310761 \nu^{11} + 862799245 \nu^{10} + 5955878758 \nu^{9} - 5873224414 \nu^{8} - 9524615238 \nu^{7} + 12943391397 \nu^{6} + 4629682174 \nu^{5} - 9935315824 \nu^{4} + 917431766 \nu^{3} + 1598499815 \nu^{2} - 167916467 \nu - 54985350\)\()/6927166\)
\(\beta_{14}\)\(=\)\((\)\(-4464588 \nu^{16} + 22316577 \nu^{15} + 62453009 \nu^{14} - 453309389 \nu^{13} - 117535949 \nu^{12} + 3492081761 \nu^{11} - 2076095689 \nu^{10} - 12518312453 \nu^{9} + 13167816710 \nu^{8} + 20045296036 \nu^{7} - 28679333929 \nu^{6} - 9681635945 \nu^{5} + 22306957823 \nu^{4} - 2117117842 \nu^{3} - 3979676103 \nu^{2} + 424686951 \nu + 161461920\)\()/13854332\)
\(\beta_{15}\)\(=\)\((\)\(-2258564 \nu^{16} + 11405411 \nu^{15} + 30768077 \nu^{14} - 230738128 \nu^{13} - 42311710 \nu^{12} + 1766179249 \nu^{11} - 1184928143 \nu^{10} - 6260382996 \nu^{9} + 7147568361 \nu^{8} + 9783571190 \nu^{7} - 15251807704 \nu^{6} - 4310704213 \nu^{5} + 11481073899 \nu^{4} - 1222783277 \nu^{3} - 1709938046 \nu^{2} + 45368196 \nu + 41232606\)\()/6927166\)
\(\beta_{16}\)\(=\)\((\)\(-5588767 \nu^{16} + 28694781 \nu^{15} + 74918371 \nu^{14} - 580575835 \nu^{13} - 79913865 \nu^{12} + 4445515351 \nu^{11} - 3127113413 \nu^{10} - 15773479316 \nu^{9} + 18433074534 \nu^{8} + 24731166679 \nu^{7} - 39180778403 \nu^{6} - 11084633469 \nu^{5} + 29732956670 \nu^{4} - 2992752301 \nu^{3} - 4755103951 \nu^{2} + 276723354 \nu + 155763180\)\()/13854332\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + 3\)
\(\nu^{3}\)\(=\)\(\beta_{3} + 5 \beta_{1}\)
\(\nu^{4}\)\(=\)\(-\beta_{13} - \beta_{10} - \beta_{9} + \beta_{6} + \beta_{5} + \beta_{4} + \beta_{3} + 6 \beta_{2} + 15\)
\(\nu^{5}\)\(=\)\(-\beta_{16} - \beta_{14} - \beta_{13} - \beta_{12} - \beta_{11} + \beta_{9} + \beta_{8} - \beta_{6} + \beta_{5} + 9 \beta_{3} + 28 \beta_{1} + 2\)
\(\nu^{6}\)\(=\)\(-\beta_{16} - 2 \beta_{14} - 11 \beta_{13} - 2 \beta_{11} - 8 \beta_{10} - 8 \beta_{9} + \beta_{8} + \beta_{7} + 10 \beta_{6} + 11 \beta_{5} + 10 \beta_{4} + 10 \beta_{3} + 36 \beta_{2} + \beta_{1} + 86\)
\(\nu^{7}\)\(=\)\(-15 \beta_{16} + \beta_{15} - 14 \beta_{14} - 16 \beta_{13} - 13 \beta_{12} - 14 \beta_{11} - \beta_{10} + 12 \beta_{9} + 13 \beta_{8} - \beta_{7} - 9 \beta_{6} + 14 \beta_{5} + 2 \beta_{4} + 67 \beta_{3} + \beta_{2} + 166 \beta_{1} + 32\)
\(\nu^{8}\)\(=\)\(-17 \beta_{16} - 3 \beta_{15} - 32 \beta_{14} - 96 \beta_{13} - 7 \beta_{12} - 36 \beta_{11} - 53 \beta_{10} - 47 \beta_{9} + 20 \beta_{8} + 12 \beta_{7} + 77 \beta_{6} + 92 \beta_{5} + 79 \beta_{4} + 82 \beta_{3} + 220 \beta_{2} + 17 \beta_{1} + 538\)
\(\nu^{9}\)\(=\)\(-160 \beta_{16} + 13 \beta_{15} - 139 \beta_{14} - 179 \beta_{13} - 126 \beta_{12} - 148 \beta_{11} - 18 \beta_{10} + 106 \beta_{9} + 134 \beta_{8} - 14 \beta_{7} - 58 \beta_{6} + 135 \beta_{5} + 35 \beta_{4} + 476 \beta_{3} + 19 \beta_{2} + 1025 \beta_{1} + 352\)
\(\nu^{10}\)\(=\)\(-202 \beta_{16} - 53 \beta_{15} - 354 \beta_{14} - 785 \beta_{13} - 129 \beta_{12} - 433 \beta_{11} - 341 \beta_{10} - 229 \beta_{9} + 263 \beta_{8} + 101 \beta_{7} + 540 \beta_{6} + 694 \beta_{5} + 582 \beta_{4} + 639 \beta_{3} + 1371 \beta_{2} + 206 \beta_{1} + 3572\)
\(\nu^{11}\)\(=\)\(-1481 \beta_{16} + 105 \beta_{15} - 1229 \beta_{14} - 1730 \beta_{13} - 1102 \beta_{12} - 1406 \beta_{11} - 216 \beta_{10} + 847 \beta_{9} + 1264 \beta_{8} - 144 \beta_{7} - 316 \beta_{6} + 1126 \beta_{5} + 410 \beta_{4} + 3347 \beta_{3} + 242 \beta_{2} + 6530 \beta_{1} + 3364\)
\(\nu^{12}\)\(=\)\(-2070 \beta_{16} - 622 \beta_{15} - 3377 \beta_{14} - 6286 \beta_{13} - 1573 \beta_{12} - 4385 \beta_{11} - 2219 \beta_{10} - 844 \beta_{9} + 2848 \beta_{8} + 720 \beta_{7} + 3632 \beta_{6} + 4999 \beta_{5} + 4194 \beta_{4} + 4908 \beta_{3} + 8716 \beta_{2} + 2160 \beta_{1} + 24725\)
\(\nu^{13}\)\(=\)\(-12736 \beta_{16} + 616 \beta_{15} - 10358 \beta_{14} - 15482 \beta_{13} - 9192 \beta_{12} - 12629 \beta_{11} - 2166 \beta_{10} + 6541 \beta_{9} + 11315 \beta_{8} - 1339 \beta_{7} - 1469 \beta_{6} + 8775 \beta_{5} + 4064 \beta_{4} + 23595 \beta_{3} + 2598 \beta_{2} + 42675 \beta_{1} + 30022\)
\(\nu^{14}\)\(=\)\(-19563 \beta_{16} - 6149 \beta_{15} - 29869 \beta_{14} - 50019 \beta_{13} - 16060 \beta_{12} - 40482 \beta_{11} - 14795 \beta_{10} - 677 \beta_{9} + 27643 \beta_{8} + 4547 \beta_{7} + 23977 \beta_{6} + 35301 \beta_{5} + 30136 \beta_{4} + 37611 \beta_{3} + 56538 \beta_{2} + 20845 \beta_{1} + 176300\)
\(\nu^{15}\)\(=\)\(-105024 \beta_{16} + 2128 \beta_{15} - 85333 \beta_{14} - 132384 \beta_{13} - 74733 \beta_{12} - 109542 \beta_{11} - 19696 \beta_{10} + 50118 \beta_{9} + 97739 \beta_{8} - 11906 \beta_{7} - 5116 \beta_{6} + 66169 \beta_{5} + 36934 \beta_{4} + 167666 \beta_{3} + 25381 \beta_{2} + 285081 \beta_{1} + 257719\)
\(\nu^{16}\)\(=\)\(-175725 \beta_{16} - 55584 \beta_{15} - 252927 \beta_{14} - 397350 \beta_{13} - 149019 \beta_{12} - 353544 \beta_{11} - 101394 \beta_{10} + 30387 \beta_{9} + 250809 \beta_{8} + 25227 \beta_{7} + 157342 \beta_{6} + 247838 \beta_{5} + 217784 \beta_{4} + 288865 \beta_{3} + 374172 \beta_{2} + 190567 \beta_{1} + 1284423\)

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
−2.51266
−2.36348
−1.99653
−1.82251
−1.09071
−0.358260
−0.207529
−0.107621
0.226775
0.695551
1.36177
1.43824
1.93329
2.18773
2.36706
2.45146
2.79743
−2.51266 2.89863 4.31346 2.77794 −7.28327 −0.223033 −5.81294 5.40206 −6.98003
1.2 −2.36348 −2.15844 3.58604 −0.000712583 0 5.10144 −2.22015 −3.74859 1.65887 0.00168418
1.3 −1.99653 1.39055 1.98612 −2.55061 −2.77626 −4.72921 0.0277164 −1.06638 5.09235
1.4 −1.82251 −0.490735 1.32154 3.61587 0.894370 2.81440 1.23649 −2.75918 −6.58995
1.5 −1.09071 2.75693 −0.810342 1.35282 −3.00702 0.0387215 3.06528 4.60064 −1.47554
1.6 −0.358260 −1.20564 −1.87165 −1.57855 0.431934 1.55949 1.38706 −1.54642 0.565531
1.7 −0.207529 2.64177 −1.95693 −1.74345 −0.548246 3.85565 0.821180 3.97897 0.361817
1.8 −0.107621 −2.82440 −1.98842 −3.72669 0.303965 −3.10404 0.429238 4.97724 0.401070
1.9 0.226775 −1.51708 −1.94857 2.45530 −0.344035 0.487144 −0.895438 −0.698475 0.556801
1.10 0.695551 1.81098 −1.51621 3.87525 1.25963 2.52014 −2.44570 0.279648 2.69543
1.11 1.36177 1.04922 −0.145589 0.274763 1.42880 2.68858 −2.92179 −1.89913 0.374163
1.12 1.43824 3.12218 0.0685351 1.24047 4.49045 −3.90458 −2.77791 6.74802 1.78410
1.13 1.93329 −3.25199 1.73762 0.302138 −6.28704 2.99039 −0.507260 7.57541 0.584120
1.14 2.18773 1.16554 2.78617 0.568409 2.54988 0.610781 1.71993 −1.64153 1.24353
1.15 2.36706 2.46697 3.60297 −3.97349 5.83947 −0.131055 3.79433 3.08596 −9.40548
1.16 2.45146 −0.386523 4.00965 3.21890 −0.947545 −4.43315 4.92658 −2.85060 7.89101
1.17 2.79743 −1.46796 5.82560 −1.10838 −4.10651 2.17992 10.7018 −0.845091 −3.10061
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.17
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 \(T_{2}^{17} - \cdots\) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(349))\).