Properties

Label 29.2.e.a
Level 29
Weight 2
Character orbit 29.e
Analytic conductor 0.232
Analytic rank 0
Dimension 12
CM No
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 29 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 29.e (of order \(14\) and degree \(6\))

Newform invariants

Self dual: No
Analytic conductor: \(0.231566165862\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(2\) over \(\Q(\zeta_{14})\)
Coefficient field: 12.0.7877952219361.1
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Sato-Tate group: $\mathrm{SU}(2)[C_{14}]$

$q$-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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{12}\mathstrut -\mathstrut \) \(3\) \(x^{11}\mathstrut +\mathstrut \) \(13\) \(x^{9}\mathstrut -\mathstrut \) \(18\) \(x^{8}\mathstrut -\mathstrut \) \(14\) \(x^{7}\mathstrut +\mathstrut \) \(57\) \(x^{6}\mathstrut -\mathstrut \) \(28\) \(x^{5}\mathstrut -\mathstrut \) \(72\) \(x^{4}\mathstrut +\mathstrut \) \(104\) \(x^{3}\mathstrut -\mathstrut \) \(96\) \(x\mathstrut +\mathstrut \) \(64\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{11} + 7 \nu^{10} - 10 \nu^{9} - 7 \nu^{8} + 40 \nu^{7} - 14 \nu^{6} - 83 \nu^{5} + 102 \nu^{4} + 68 \nu^{3} - 176 \nu^{2} + 32 \nu + 96 \)\()/128\)
\(\beta_{2}\)\(=\)\((\)\( -\nu^{11} + 5 \nu^{10} - 10 \nu^{9} + 7 \nu^{8} + 4 \nu^{7} - 58 \nu^{6} + 59 \nu^{5} + 38 \nu^{4} - 196 \nu^{3} + 96 \nu^{2} + 256 \nu - 288 \)\()/128\)
\(\beta_{3}\)\(=\)\((\)\( -\nu^{11} + 5 \nu^{10} - 10 \nu^{9} + 7 \nu^{8} + 36 \nu^{7} - 58 \nu^{6} - 5 \nu^{5} + 134 \nu^{4} - 100 \nu^{3} - 128 \nu^{2} + 192 \nu - 32 \)\()/128\)
\(\beta_{4}\)\(=\)\((\)\( \nu^{11} - 3 \nu^{10} + 4 \nu^{9} + \nu^{8} - 18 \nu^{7} + 22 \nu^{6} + 17 \nu^{5} - 52 \nu^{4} + 44 \nu^{3} + 40 \nu^{2} - 80 \nu + 64 \)\()/64\)
\(\beta_{5}\)\(=\)\((\)\( -\nu^{11} + 5 \nu^{10} - 10 \nu^{9} - 25 \nu^{8} + 68 \nu^{7} + 6 \nu^{6} - 165 \nu^{5} + 134 \nu^{4} + 220 \nu^{3} - 288 \nu^{2} + 224 \)\()/128\)
\(\beta_{6}\)\(=\)\((\)\( 3 \nu^{11} + \nu^{10} - 18 \nu^{9} + 11 \nu^{8} + 36 \nu^{7} - 50 \nu^{6} - 49 \nu^{5} + 94 \nu^{4} + 12 \nu^{3} - 128 \nu^{2} + 96 \)\()/128\)
\(\beta_{7}\)\(=\)\((\)\( -\nu^{10} + 3 \nu^{9} - 9 \nu^{7} + 10 \nu^{6} + 6 \nu^{5} - 29 \nu^{4} + 16 \nu^{3} + 20 \nu^{2} - 40 \nu + 16 \)\()/16\)
\(\beta_{8}\)\(=\)\((\)\( \nu^{11} - 5 \nu^{10} + 10 \nu^{9} + 9 \nu^{8} - 36 \nu^{7} + 26 \nu^{6} + 53 \nu^{5} - 86 \nu^{4} - 12 \nu^{3} + 96 \nu^{2} - 64 \nu + 32 \)\()/64\)
\(\beta_{9}\)\(=\)\((\)\( \nu^{11} - 5 \nu^{9} + 5 \nu^{8} + 9 \nu^{7} - 16 \nu^{6} - 5 \nu^{5} + 31 \nu^{4} - 12 \nu^{2} + 8 \nu + 16 \)\()/32\)
\(\beta_{10}\)\(=\)\((\)\( 3 \nu^{11} - 11 \nu^{10} - 14 \nu^{9} + 59 \nu^{8} - 40 \nu^{7} - 122 \nu^{6} + 199 \nu^{5} + 82 \nu^{4} - 420 \nu^{3} + 176 \nu^{2} + 352 \nu - 352 \)\()/128\)
\(\beta_{11}\)\(=\)\((\)\( -15 \nu^{11} + 31 \nu^{10} + 30 \nu^{9} - 151 \nu^{8} + 80 \nu^{7} + 290 \nu^{6} - 435 \nu^{5} - 146 \nu^{4} + 740 \nu^{3} - 368 \nu^{2} - 416 \nu + 480 \)\()/128\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{8}\mathstrut +\mathstrut \) \(\beta_{5}\mathstrut +\mathstrut \) \(\beta_{2}\)
\(\nu^{2}\)\(=\)\(\beta_{9}\mathstrut -\mathstrut \) \(\beta_{6}\mathstrut -\mathstrut \) \(\beta_{4}\mathstrut -\mathstrut \) \(\beta_{3}\mathstrut +\mathstrut \) \(1\)
\(\nu^{3}\)\(=\)\(\beta_{9}\mathstrut +\mathstrut \) \(\beta_{7}\mathstrut -\mathstrut \) \(\beta_{6}\mathstrut +\mathstrut \) \(\beta_{5}\mathstrut +\mathstrut \) \(\beta_{4}\mathstrut +\mathstrut \) \(\beta_{3}\mathstrut +\mathstrut \) \(\beta_{2}\mathstrut -\mathstrut \) \(1\)
\(\nu^{4}\)\(=\)\(\beta_{10}\mathstrut +\mathstrut \) \(\beta_{9}\mathstrut -\mathstrut \) \(2\) \(\beta_{8}\mathstrut -\mathstrut \) \(3\) \(\beta_{6}\mathstrut -\mathstrut \) \(\beta_{5}\mathstrut -\mathstrut \) \(\beta_{3}\mathstrut -\mathstrut \) \(2\) \(\beta_{2}\mathstrut +\mathstrut \) \(2\) \(\beta_{1}\mathstrut +\mathstrut \) \(1\)
\(\nu^{5}\)\(=\)\(-\)\(\beta_{10}\mathstrut -\mathstrut \) \(\beta_{8}\mathstrut -\mathstrut \) \(\beta_{7}\mathstrut -\mathstrut \) \(\beta_{5}\mathstrut +\mathstrut \) \(5\) \(\beta_{4}\mathstrut +\mathstrut \) \(4\) \(\beta_{3}\mathstrut -\mathstrut \) \(2\) \(\beta_{1}\mathstrut -\mathstrut \) \(2\)
\(\nu^{6}\)\(=\)\(2\) \(\beta_{11}\mathstrut +\mathstrut \) \(2\) \(\beta_{10}\mathstrut +\mathstrut \) \(2\) \(\beta_{9}\mathstrut +\mathstrut \) \(3\) \(\beta_{8}\mathstrut -\mathstrut \) \(3\) \(\beta_{7}\mathstrut +\mathstrut \) \(2\) \(\beta_{6}\mathstrut -\mathstrut \) \(2\) \(\beta_{5}\mathstrut +\mathstrut \) \(2\) \(\beta_{4}\mathstrut +\mathstrut \) \(2\) \(\beta_{3}\mathstrut -\mathstrut \) \(2\) \(\beta_{2}\mathstrut -\mathstrut \) \(2\) \(\beta_{1}\mathstrut -\mathstrut \) \(4\)
\(\nu^{7}\)\(=\)\(-\)\(5\) \(\beta_{10}\mathstrut +\mathstrut \) \(\beta_{9}\mathstrut +\mathstrut \) \(6\) \(\beta_{8}\mathstrut -\mathstrut \) \(5\) \(\beta_{7}\mathstrut +\mathstrut \) \(5\) \(\beta_{6}\mathstrut +\mathstrut \) \(5\) \(\beta_{3}\mathstrut +\mathstrut \) \(\beta_{2}\mathstrut -\mathstrut \) \(10\) \(\beta_{1}\mathstrut -\mathstrut \) \(5\)
\(\nu^{8}\)\(=\)\(4\) \(\beta_{11}\mathstrut +\mathstrut \) \(4\) \(\beta_{10}\mathstrut +\mathstrut \) \(10\) \(\beta_{9}\mathstrut +\mathstrut \) \(11\) \(\beta_{8}\mathstrut +\mathstrut \) \(4\) \(\beta_{7}\mathstrut +\mathstrut \) \(4\) \(\beta_{6}\mathstrut +\mathstrut \) \(\beta_{5}\mathstrut -\mathstrut \) \(6\) \(\beta_{4}\mathstrut +\mathstrut \) \(8\) \(\beta_{3}\mathstrut +\mathstrut \) \(\beta_{2}\mathstrut -\mathstrut \) \(4\) \(\beta_{1}\mathstrut -\mathstrut \) \(10\)
\(\nu^{9}\)\(=\)\(-\)\(6\) \(\beta_{11}\mathstrut -\mathstrut \) \(10\) \(\beta_{10}\mathstrut -\mathstrut \) \(3\) \(\beta_{9}\mathstrut +\mathstrut \) \(4\) \(\beta_{8}\mathstrut -\mathstrut \) \(11\) \(\beta_{6}\mathstrut +\mathstrut \) \(4\) \(\beta_{5}\mathstrut -\mathstrut \) \(15\) \(\beta_{4}\mathstrut -\mathstrut \) \(5\) \(\beta_{3}\mathstrut +\mathstrut \) \(6\) \(\beta_{1}\mathstrut +\mathstrut \) \(5\)
\(\nu^{10}\)\(=\)\(2\) \(\beta_{11}\mathstrut +\mathstrut \) \(9\) \(\beta_{9}\mathstrut +\mathstrut \) \(9\) \(\beta_{7}\mathstrut -\mathstrut \) \(7\) \(\beta_{6}\mathstrut -\mathstrut \) \(9\) \(\beta_{5}\mathstrut +\mathstrut \) \(\beta_{4}\mathstrut +\mathstrut \) \(9\) \(\beta_{3}\mathstrut +\mathstrut \) \(5\) \(\beta_{2}\mathstrut +\mathstrut \) \(18\) \(\beta_{1}\mathstrut -\mathstrut \) \(1\)
\(\nu^{11}\)\(=\)\(-\)\(18\) \(\beta_{11}\mathstrut -\mathstrut \) \(29\) \(\beta_{10}\mathstrut -\mathstrut \) \(29\) \(\beta_{9}\mathstrut +\mathstrut \) \(8\) \(\beta_{8}\mathstrut -\mathstrut \) \(28\) \(\beta_{7}\mathstrut -\mathstrut \) \(7\) \(\beta_{6}\mathstrut +\mathstrut \) \(\beta_{5}\mathstrut -\mathstrut \) \(39\) \(\beta_{3}\mathstrut +\mathstrut \) \(8\) \(\beta_{2}\mathstrut +\mathstrut \) \(36\) \(\beta_{1}\mathstrut +\mathstrut \) \(11\)

Character Values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/29\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(\beta_{11}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
4.1
0.639551 + 1.26134i
1.38491 0.286410i
1.23295 + 0.692694i
−1.41140 + 0.0891373i
1.23295 0.692694i
−1.41140 0.0891373i
−1.25719 0.647667i
0.911180 + 1.08155i
−1.25719 + 0.647667i
0.911180 1.08155i
0.639551 1.26134i
1.38491 + 0.286410i
−2.21089 0.504621i −1.23248 2.55926i 2.83146 + 1.36356i 0.0128801 0.0564316i 1.43341 + 6.28018i 1.40728 0.677709i −2.02596 1.61565i −3.16036 + 3.96297i −0.0569531 + 0.118264i
4.2 −0.536089 0.122359i 0.855966 + 1.77743i −1.52952 0.736577i 0.610610 2.67526i −0.241390 1.05760i −4.03077 + 1.94112i 1.58965 + 1.26771i −0.556117 + 0.697349i −0.654683 + 1.35946i
5.1 −0.909335 + 0.725171i 0.960118 0.219141i −0.144024 + 0.631009i −1.18424 1.48499i −0.714155 + 0.895521i −0.339509 1.48749i −1.33591 2.77404i −1.82910 + 0.880850i 2.15374 + 0.491577i
5.2 1.21127 0.965958i −2.86109 + 0.653024i 0.0890656 0.390222i 0.283269 + 0.355208i −2.83476 + 3.55468i −0.759522 3.32768i 1.07536 + 2.23300i 5.05647 2.43507i 0.686232 + 0.156628i
6.1 −0.909335 0.725171i 0.960118 + 0.219141i −0.144024 0.631009i −1.18424 + 1.48499i −0.714155 0.895521i −0.339509 + 1.48749i −1.33591 + 2.77404i −1.82910 0.880850i 2.15374 0.491577i
6.2 1.21127 + 0.965958i −2.86109 0.653024i 0.0890656 + 0.390222i 0.283269 0.355208i −2.83476 3.55468i −0.759522 + 3.32768i 1.07536 2.23300i 5.05647 + 2.43507i 0.686232 0.156628i
9.1 −1.12916 2.34472i −0.343489 + 0.273923i −2.97573 + 3.73144i 2.32488 1.11960i 1.03013 + 0.496082i 0.0468435 + 0.0587399i 7.03485 + 1.60566i −0.624612 + 2.73660i −5.25031 4.18698i
9.2 0.0741982 + 0.154074i −0.879032 + 0.701005i 1.22875 1.54080i −2.54740 + 1.22676i −0.173229 0.0834229i −1.82432 2.28763i 0.662012 + 0.151100i −0.386273 + 1.69237i −0.378025 0.301465i
13.1 −1.12916 + 2.34472i −0.343489 0.273923i −2.97573 3.73144i 2.32488 + 1.11960i 1.03013 0.496082i 0.0468435 0.0587399i 7.03485 1.60566i −0.624612 2.73660i −5.25031 + 4.18698i
13.2 0.0741982 0.154074i −0.879032 0.701005i 1.22875 + 1.54080i −2.54740 1.22676i −0.173229 + 0.0834229i −1.82432 + 2.28763i 0.662012 0.151100i −0.386273 1.69237i −0.378025 + 0.301465i
22.1 −2.21089 + 0.504621i −1.23248 + 2.55926i 2.83146 1.36356i 0.0128801 + 0.0564316i 1.43341 6.28018i 1.40728 + 0.677709i −2.02596 + 1.61565i −3.16036 3.96297i −0.0569531 0.118264i
22.2 −0.536089 + 0.122359i 0.855966 1.77743i −1.52952 + 0.736577i 0.610610 + 2.67526i −0.241390 + 1.05760i −4.03077 1.94112i 1.58965 1.26771i −0.556117 0.697349i −0.654683 1.35946i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 22.2
Significant digits:
Format:

Inner twists

Char. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
29.e Even 1 yes

Hecke kernels

There are no other newforms in \(S_{2}^{\mathrm{new}}(29, [\chi])\).