Properties

Label 29.3.c.a
Level 29
Weight 3
Character orbit 29.c
Analytic conductor 0.790
Analytic rank 0
Dimension 8
CM No
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 29 \)
Weight: \( k \) = \( 3 \)
Character orbit: \([\chi]\) = 29.c (of order \(4\) and degree \(2\))

Newform invariants

Self dual: No
Analytic conductor: \(0.790192766645\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} + \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \( -\beta_{6} q^{2} \) \( + ( \beta_{2} + \beta_{6} ) q^{3} \) \( + ( \beta_{3} + \beta_{4} - \beta_{5} - \beta_{6} ) q^{4} \) \( + ( -\beta_{3} - \beta_{4} ) q^{5} \) \( + ( -\beta_{1} - \beta_{2} - 2 \beta_{3} - 5 \beta_{4} + \beta_{5} + \beta_{6} ) q^{6} \) \( + ( \beta_{1} - \beta_{2} - \beta_{5} + \beta_{6} + \beta_{7} ) q^{7} \) \( + ( -5 - \beta_{1} + 2 \beta_{3} + 5 \beta_{4} - \beta_{5} - 2 \beta_{7} ) q^{8} \) \( + ( \beta_{1} + \beta_{2} + 3 \beta_{3} + 6 \beta_{4} + \beta_{5} + \beta_{6} ) q^{9} \) \(+O(q^{10})\) \( q\) \( -\beta_{6} q^{2} \) \( + ( \beta_{2} + \beta_{6} ) q^{3} \) \( + ( \beta_{3} + \beta_{4} - \beta_{5} - \beta_{6} ) q^{4} \) \( + ( -\beta_{3} - \beta_{4} ) q^{5} \) \( + ( -\beta_{1} - \beta_{2} - 2 \beta_{3} - 5 \beta_{4} + \beta_{5} + \beta_{6} ) q^{6} \) \( + ( \beta_{1} - \beta_{2} - \beta_{5} + \beta_{6} + \beta_{7} ) q^{7} \) \( + ( -5 - \beta_{1} + 2 \beta_{3} + 5 \beta_{4} - \beta_{5} - 2 \beta_{7} ) q^{8} \) \( + ( \beta_{1} + \beta_{2} + 3 \beta_{3} + 6 \beta_{4} + \beta_{5} + \beta_{6} ) q^{9} \) \( + ( \beta_{1} - \beta_{3} + 3 \beta_{5} + \beta_{7} ) q^{10} \) \( + ( -1 - 4 \beta_{2} - \beta_{3} - \beta_{4} - \beta_{6} - \beta_{7} ) q^{11} \) \( + ( 5 - 2 \beta_{3} - 5 \beta_{4} + 7 \beta_{5} + 2 \beta_{7} ) q^{12} \) \( + ( 2 \beta_{1} + 2 \beta_{2} + 5 \beta_{4} - \beta_{5} - \beta_{6} ) q^{13} \) \( + ( -5 + \beta_{2} + \beta_{3} - 5 \beta_{4} + \beta_{7} ) q^{14} \) \( + ( -2 \beta_{1} - 5 \beta_{5} ) q^{15} \) \( + ( -1 - \beta_{1} + \beta_{2} - 6 \beta_{5} + 6 \beta_{6} - 2 \beta_{7} ) q^{16} \) \( + ( \beta_{2} - 6 \beta_{6} ) q^{17} \) \( + ( 5 - 5 \beta_{1} + \beta_{3} - 5 \beta_{4} - 10 \beta_{5} - \beta_{7} ) q^{18} \) \( + ( -1 - \beta_{2} - 4 \beta_{3} - \beta_{4} + 4 \beta_{6} - 4 \beta_{7} ) q^{19} \) \( + ( 11 + 5 \beta_{5} - 5 \beta_{6} + 2 \beta_{7} ) q^{20} \) \( + ( -5 + 3 \beta_{2} + \beta_{3} - 5 \beta_{4} - 2 \beta_{6} + \beta_{7} ) q^{21} \) \( + ( 5 \beta_{1} + 5 \beta_{2} + 3 \beta_{3} + 5 \beta_{4} + 2 \beta_{5} + 2 \beta_{6} ) q^{22} \) \( + ( -2 \beta_{1} + 2 \beta_{2} + 3 \beta_{7} ) q^{23} \) \( + ( 15 + 6 \beta_{1} - 6 \beta_{2} + 12 \beta_{5} - 12 \beta_{6} + 3 \beta_{7} ) q^{24} \) \( + ( 14 + \beta_{1} - \beta_{2} - 2 \beta_{5} + 2 \beta_{6} ) q^{25} \) \( + ( -5 - 4 \beta_{1} - \beta_{3} + 5 \beta_{4} - 7 \beta_{5} + \beta_{7} ) q^{26} \) \( + ( -15 + 2 \beta_{1} + 3 \beta_{3} + 15 \beta_{4} + 11 \beta_{5} - 3 \beta_{7} ) q^{27} \) \( + ( 2 \beta_{1} + 2 \beta_{2} - 3 \beta_{3} - \beta_{5} - \beta_{6} ) q^{28} \) \( + ( 15 - 5 \beta_{1} + 2 \beta_{2} + 6 \beta_{4} + 7 \beta_{5} + 3 \beta_{6} ) q^{29} \) \( + ( -25 + 2 \beta_{1} - 2 \beta_{2} - 5 \beta_{5} + 5 \beta_{6} - 7 \beta_{7} ) q^{30} \) \( + ( 1 - 7 \beta_{2} + 4 \beta_{3} + \beta_{4} + 9 \beta_{6} + 4 \beta_{7} ) q^{31} \) \( + ( -10 - 4 \beta_{2} - \beta_{3} - 10 \beta_{4} + 13 \beta_{6} - \beta_{7} ) q^{32} \) \( + ( -3 \beta_{1} - 3 \beta_{2} - 6 \beta_{3} - 45 \beta_{4} - 12 \beta_{5} - 12 \beta_{6} ) q^{33} \) \( + ( -\beta_{1} - \beta_{2} + 5 \beta_{3} + 30 \beta_{4} - 6 \beta_{5} - 6 \beta_{6} ) q^{34} \) \( + ( -\beta_{1} - \beta_{2} + 5 \beta_{3} - 10 \beta_{4} + \beta_{5} + \beta_{6} ) q^{35} \) \( + ( -26 - 11 \beta_{5} + 11 \beta_{6} - 5 \beta_{7} ) q^{36} \) \( + ( -10 - 6 \beta_{1} - 8 \beta_{3} + 10 \beta_{4} - 2 \beta_{5} + 8 \beta_{7} ) q^{37} \) \( + ( 5 \beta_{1} + 5 \beta_{2} - 11 \beta_{3} - 20 \beta_{4} + 13 \beta_{5} + 13 \beta_{6} ) q^{38} \) \( + ( -15 + 3 \beta_{1} + 15 \beta_{4} + 15 \beta_{5} ) q^{39} \) \( + ( 25 + 2 \beta_{2} + 3 \beta_{3} + 25 \beta_{4} - 13 \beta_{6} + 3 \beta_{7} ) q^{40} \) \( + ( 1 + 7 \beta_{1} + 5 \beta_{3} - \beta_{4} + 6 \beta_{5} - 5 \beta_{7} ) q^{41} \) \( + ( -4 \beta_{1} - 4 \beta_{2} + \beta_{3} + 10 \beta_{4} + \beta_{5} + \beta_{6} ) q^{42} \) \( + ( -25 + 6 \beta_{2} + \beta_{3} - 25 \beta_{4} - 5 \beta_{6} + \beta_{7} ) q^{43} \) \( + ( 6 + 3 \beta_{1} - 6 \beta_{4} - 3 \beta_{5} ) q^{44} \) \( + ( 36 - 5 \beta_{1} + 5 \beta_{2} + \beta_{5} - \beta_{6} + 3 \beta_{7} ) q^{45} \) \( + ( -7 \beta_{2} + \beta_{3} - 6 \beta_{6} + \beta_{7} ) q^{46} \) \( + ( 10 + 6 \beta_{1} + 2 \beta_{3} - 10 \beta_{4} - 11 \beta_{5} - 2 \beta_{7} ) q^{47} \) \( + ( 40 + 9 \beta_{2} + 13 \beta_{3} + 40 \beta_{4} - 17 \beta_{6} + 13 \beta_{7} ) q^{48} \) \( + ( -9 + 3 \beta_{1} - 3 \beta_{2} - 12 \beta_{7} ) q^{49} \) \( + ( -10 + 2 \beta_{2} - \beta_{3} - 10 \beta_{4} - 10 \beta_{6} - \beta_{7} ) q^{50} \) \( + ( -6 \beta_{1} - 6 \beta_{2} - 11 \beta_{3} - 20 \beta_{4} + 8 \beta_{5} + 8 \beta_{6} ) q^{51} \) \( + ( -15 - 3 \beta_{1} + 3 \beta_{2} - 6 \beta_{5} + 6 \beta_{6} - 9 \beta_{7} ) q^{52} \) \( + ( 35 - 6 \beta_{1} + 6 \beta_{2} - 7 \beta_{5} + 7 \beta_{6} - 10 \beta_{7} ) q^{53} \) \( + ( 55 - 5 \beta_{1} + 5 \beta_{2} - 10 \beta_{5} + 10 \beta_{6} + 7 \beta_{7} ) q^{54} \) \( + ( -11 + 7 \beta_{1} + 3 \beta_{3} + 11 \beta_{4} + 7 \beta_{5} - 3 \beta_{7} ) q^{55} \) \( + ( -25 - 5 \beta_{1} - 8 \beta_{3} + 25 \beta_{4} + 4 \beta_{5} + 8 \beta_{7} ) q^{56} \) \( + ( -\beta_{1} - \beta_{2} + 7 \beta_{3} + 10 \beta_{4} - 23 \beta_{5} - 23 \beta_{6} ) q^{57} \) \( + ( 35 + 3 \beta_{1} - 7 \beta_{2} - 5 \beta_{3} - 15 \beta_{4} + 4 \beta_{5} - 19 \beta_{6} + 2 \beta_{7} ) q^{58} \) \( + ( -10 + 4 \beta_{1} - 4 \beta_{2} + 10 \beta_{5} - 10 \beta_{6} + 17 \beta_{7} ) q^{59} \) \( + ( -25 + 3 \beta_{2} - 10 \beta_{3} - 25 \beta_{4} + 29 \beta_{6} - 10 \beta_{7} ) q^{60} \) \( + ( -24 + 9 \beta_{2} + 2 \beta_{3} - 24 \beta_{4} + 8 \beta_{6} + 2 \beta_{7} ) q^{61} \) \( + ( 3 \beta_{1} + 3 \beta_{2} + 6 \beta_{3} - 45 \beta_{4} ) q^{62} \) \( + ( 3 \beta_{1} + 3 \beta_{2} - 10 \beta_{3} + 20 \beta_{4} - 2 \beta_{5} - 2 \beta_{6} ) q^{63} \) \( + ( \beta_{1} + \beta_{2} - 3 \beta_{3} - 61 \beta_{4} + \beta_{5} + \beta_{6} ) q^{64} \) \( + ( 5 - \beta_{1} + \beta_{2} - \beta_{5} + \beta_{6} + 11 \beta_{7} ) q^{65} \) \( + ( -60 + 12 \beta_{1} + 9 \beta_{3} + 60 \beta_{4} + 33 \beta_{5} - 9 \beta_{7} ) q^{66} \) \( + ( -15 \beta_{1} - 15 \beta_{2} + 6 \beta_{3} + 40 \beta_{4} ) q^{67} \) \( + ( -30 - 7 \beta_{1} + 12 \beta_{3} + 30 \beta_{4} - 28 \beta_{5} - 12 \beta_{7} ) q^{68} \) \( + ( 20 + 3 \beta_{2} + 2 \beta_{3} + 20 \beta_{4} + 20 \beta_{6} + 2 \beta_{7} ) q^{69} \) \( + ( 5 - 3 \beta_{1} + 5 \beta_{3} - 5 \beta_{4} + 2 \beta_{5} - 5 \beta_{7} ) q^{70} \) \( + ( 3 \beta_{1} + 3 \beta_{2} - 4 \beta_{3} + 62 \beta_{4} - 5 \beta_{5} - 5 \beta_{6} ) q^{71} \) \( + ( -75 - 15 \beta_{2} - 12 \beta_{3} - 75 \beta_{4} + 18 \beta_{6} - 12 \beta_{7} ) q^{72} \) \( + ( -20 - 6 \beta_{1} + 4 \beta_{3} + 20 \beta_{4} - 14 \beta_{5} - 4 \beta_{7} ) q^{73} \) \( + ( -10 + 14 \beta_{1} - 14 \beta_{2} + 4 \beta_{5} - 4 \beta_{6} + 8 \beta_{7} ) q^{74} \) \( + ( 18 \beta_{2} + 3 \beta_{3} + 6 \beta_{6} + 3 \beta_{7} ) q^{75} \) \( + ( 61 + 5 \beta_{1} - 13 \beta_{3} - 61 \beta_{4} + 52 \beta_{5} + 13 \beta_{7} ) q^{76} \) \( + ( 25 - 17 \beta_{2} - 2 \beta_{3} + 25 \beta_{4} + 10 \beta_{6} - 2 \beta_{7} ) q^{77} \) \( + ( 75 - 3 \beta_{1} + 3 \beta_{2} + 18 \beta_{7} ) q^{78} \) \( + ( -46 - 14 \beta_{2} - 6 \beta_{3} - 46 \beta_{4} + 7 \beta_{6} - 6 \beta_{7} ) q^{79} \) \( + ( -5 \beta_{1} - 5 \beta_{2} + 9 \beta_{3} + 21 \beta_{4} - 24 \beta_{5} - 24 \beta_{6} ) q^{80} \) \( + ( -21 + 20 \beta_{1} - 20 \beta_{2} + 11 \beta_{5} - 11 \beta_{6} + 3 \beta_{7} ) q^{81} \) \( + ( 30 - 12 \beta_{1} + 12 \beta_{2} - 3 \beta_{5} + 3 \beta_{6} + 3 \beta_{7} ) q^{82} \) \( + ( 30 + 7 \beta_{1} - 7 \beta_{2} + 10 \beta_{5} - 10 \beta_{6} - 21 \beta_{7} ) q^{83} \) \( + ( -15 - 5 \beta_{1} + 15 \beta_{4} - 2 \beta_{5} ) q^{84} \) \( + ( 5 \beta_{1} - 7 \beta_{3} + 16 \beta_{5} + 7 \beta_{7} ) q^{85} \) \( + ( -7 \beta_{1} - 7 \beta_{2} + \beta_{3} + 25 \beta_{4} + 18 \beta_{5} + 18 \beta_{6} ) q^{86} \) \( + ( 15 + 16 \beta_{1} + 11 \beta_{2} + 8 \beta_{3} + 35 \beta_{4} - 10 \beta_{5} + 33 \beta_{6} - 9 \beta_{7} ) q^{87} \) \( + ( 5 - 23 \beta_{1} + 23 \beta_{2} - 5 \beta_{5} + 5 \beta_{6} + 12 \beta_{7} ) q^{88} \) \( + ( -11 + 31 \beta_{2} - 8 \beta_{3} - 11 \beta_{4} - 12 \beta_{6} - 8 \beta_{7} ) q^{89} \) \( + ( 5 - 13 \beta_{2} - \beta_{3} + 5 \beta_{4} - 44 \beta_{6} - \beta_{7} ) q^{90} \) \( + ( 4 \beta_{1} + 4 \beta_{2} + 15 \beta_{3} - 50 \beta_{4} + \beta_{5} + \beta_{6} ) q^{91} \) \( + ( -2 \beta_{1} - 2 \beta_{2} + 3 \beta_{3} + 30 \beta_{4} - 8 \beta_{5} - 8 \beta_{6} ) q^{92} \) \( + ( 14 \beta_{1} + 14 \beta_{2} + 11 \beta_{3} - 25 \beta_{4} - 6 \beta_{5} - 6 \beta_{6} ) q^{93} \) \( + ( -55 - 8 \beta_{1} + 8 \beta_{2} - 5 \beta_{5} + 5 \beta_{6} - 9 \beta_{7} ) q^{94} \) \( + ( -41 + 5 \beta_{1} + 2 \beta_{3} + 41 \beta_{4} - 26 \beta_{5} - 2 \beta_{7} ) q^{95} \) \( + ( 2 \beta_{1} + 2 \beta_{2} + 22 \beta_{3} + 25 \beta_{4} - 35 \beta_{5} - 35 \beta_{6} ) q^{96} \) \( + ( -5 - 7 \beta_{1} - 16 \beta_{3} + 5 \beta_{4} - 2 \beta_{5} + 16 \beta_{7} ) q^{97} \) \( + ( 18 \beta_{2} - 9 \beta_{3} + 33 \beta_{6} - 9 \beta_{7} ) q^{98} \) \( + ( 81 - 39 \beta_{1} - 18 \beta_{3} - 81 \beta_{4} - 48 \beta_{5} + 18 \beta_{7} ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(8q \) \(\mathstrut +\mathstrut 2q^{2} \) \(\mathstrut -\mathstrut 2q^{3} \) \(\mathstrut -\mathstrut 4q^{7} \) \(\mathstrut -\mathstrut 42q^{8} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(8q \) \(\mathstrut +\mathstrut 2q^{2} \) \(\mathstrut -\mathstrut 2q^{3} \) \(\mathstrut -\mathstrut 4q^{7} \) \(\mathstrut -\mathstrut 42q^{8} \) \(\mathstrut +\mathstrut 6q^{10} \) \(\mathstrut -\mathstrut 6q^{11} \) \(\mathstrut +\mathstrut 54q^{12} \) \(\mathstrut -\mathstrut 40q^{14} \) \(\mathstrut -\mathstrut 10q^{15} \) \(\mathstrut -\mathstrut 32q^{16} \) \(\mathstrut +\mathstrut 12q^{17} \) \(\mathstrut +\mathstrut 20q^{18} \) \(\mathstrut -\mathstrut 16q^{19} \) \(\mathstrut +\mathstrut 108q^{20} \) \(\mathstrut -\mathstrut 36q^{21} \) \(\mathstrut +\mathstrut 168q^{24} \) \(\mathstrut +\mathstrut 104q^{25} \) \(\mathstrut -\mathstrut 54q^{26} \) \(\mathstrut -\mathstrut 98q^{27} \) \(\mathstrut +\mathstrut 128q^{29} \) \(\mathstrut -\mathstrut 220q^{30} \) \(\mathstrut -\mathstrut 10q^{31} \) \(\mathstrut -\mathstrut 106q^{32} \) \(\mathstrut -\mathstrut 252q^{36} \) \(\mathstrut -\mathstrut 84q^{37} \) \(\mathstrut -\mathstrut 90q^{39} \) \(\mathstrut +\mathstrut 226q^{40} \) \(\mathstrut +\mathstrut 20q^{41} \) \(\mathstrut -\mathstrut 190q^{43} \) \(\mathstrut +\mathstrut 42q^{44} \) \(\mathstrut +\mathstrut 292q^{45} \) \(\mathstrut +\mathstrut 12q^{46} \) \(\mathstrut +\mathstrut 58q^{47} \) \(\mathstrut +\mathstrut 354q^{48} \) \(\mathstrut -\mathstrut 72q^{49} \) \(\mathstrut -\mathstrut 60q^{50} \) \(\mathstrut -\mathstrut 144q^{52} \) \(\mathstrut +\mathstrut 252q^{53} \) \(\mathstrut +\mathstrut 400q^{54} \) \(\mathstrut -\mathstrut 74q^{55} \) \(\mathstrut -\mathstrut 192q^{56} \) \(\mathstrut +\mathstrut 326q^{58} \) \(\mathstrut -\mathstrut 40q^{59} \) \(\mathstrut -\mathstrut 258q^{60} \) \(\mathstrut -\mathstrut 208q^{61} \) \(\mathstrut +\mathstrut 36q^{65} \) \(\mathstrut -\mathstrut 414q^{66} \) \(\mathstrut -\mathstrut 296q^{68} \) \(\mathstrut +\mathstrut 120q^{69} \) \(\mathstrut +\mathstrut 44q^{70} \) \(\mathstrut -\mathstrut 636q^{72} \) \(\mathstrut -\mathstrut 188q^{73} \) \(\mathstrut -\mathstrut 64q^{74} \) \(\mathstrut -\mathstrut 12q^{75} \) \(\mathstrut +\mathstrut 592q^{76} \) \(\mathstrut +\mathstrut 180q^{77} \) \(\mathstrut +\mathstrut 600q^{78} \) \(\mathstrut -\mathstrut 382q^{79} \) \(\mathstrut -\mathstrut 124q^{81} \) \(\mathstrut +\mathstrut 228q^{82} \) \(\mathstrut +\mathstrut 280q^{83} \) \(\mathstrut -\mathstrut 124q^{84} \) \(\mathstrut +\mathstrut 32q^{85} \) \(\mathstrut +\mathstrut 34q^{87} \) \(\mathstrut +\mathstrut 20q^{88} \) \(\mathstrut -\mathstrut 64q^{89} \) \(\mathstrut +\mathstrut 128q^{90} \) \(\mathstrut -\mathstrut 460q^{94} \) \(\mathstrut -\mathstrut 380q^{95} \) \(\mathstrut -\mathstrut 44q^{97} \) \(\mathstrut -\mathstrut 66q^{98} \) \(\mathstrut +\mathstrut 552q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8}\mathstrut +\mathstrut \) \(18\) \(x^{6}\mathstrut +\mathstrut \) \(91\) \(x^{4}\mathstrut +\mathstrut \) \(126\) \(x^{2}\mathstrut +\mathstrut \) \(25\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{4} + 9 \nu^{2} + 6 \nu + 5 \)\()/6\)
\(\beta_{2}\)\(=\)\((\)\( -\nu^{4} - 9 \nu^{2} + 6 \nu - 5 \)\()/6\)
\(\beta_{3}\)\(=\)\((\)\( -\nu^{5} - 15 \nu^{3} - 47 \nu \)\()/6\)
\(\beta_{4}\)\(=\)\((\)\( -\nu^{7} - 18 \nu^{5} - 86 \nu^{3} - 81 \nu \)\()/30\)
\(\beta_{5}\)\(=\)\((\)\( \nu^{7} + 17 \nu^{5} + \nu^{4} + 77 \nu^{3} + 15 \nu^{2} + 82 \nu + 35 \)\()/12\)
\(\beta_{6}\)\(=\)\((\)\( \nu^{7} + 17 \nu^{5} - \nu^{4} + 77 \nu^{3} - 15 \nu^{2} + 82 \nu - 35 \)\()/12\)
\(\beta_{7}\)\(=\)\((\)\( \nu^{6} + 16 \nu^{4} + 62 \nu^{2} + 35 \)\()/6\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{2}\mathstrut +\mathstrut \) \(\beta_{1}\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(-\)\(2\) \(\beta_{6}\mathstrut +\mathstrut \) \(2\) \(\beta_{5}\mathstrut +\mathstrut \) \(\beta_{2}\mathstrut -\mathstrut \) \(\beta_{1}\mathstrut -\mathstrut \) \(10\)\()/2\)
\(\nu^{3}\)\(=\)\(\beta_{6}\mathstrut +\mathstrut \) \(\beta_{5}\mathstrut +\mathstrut \) \(5\) \(\beta_{4}\mathstrut -\mathstrut \) \(\beta_{3}\mathstrut -\mathstrut \) \(4\) \(\beta_{2}\mathstrut -\mathstrut \) \(4\) \(\beta_{1}\)
\(\nu^{4}\)\(=\)\((\)\(18\) \(\beta_{6}\mathstrut -\mathstrut \) \(18\) \(\beta_{5}\mathstrut -\mathstrut \) \(15\) \(\beta_{2}\mathstrut +\mathstrut \) \(15\) \(\beta_{1}\mathstrut +\mathstrut \) \(80\)\()/2\)
\(\nu^{5}\)\(=\)\((\)\(-\)\(30\) \(\beta_{6}\mathstrut -\mathstrut \) \(30\) \(\beta_{5}\mathstrut -\mathstrut \) \(150\) \(\beta_{4}\mathstrut +\mathstrut \) \(18\) \(\beta_{3}\mathstrut +\mathstrut \) \(73\) \(\beta_{2}\mathstrut +\mathstrut \) \(73\) \(\beta_{1}\)\()/2\)
\(\nu^{6}\)\(=\)\(6\) \(\beta_{7}\mathstrut -\mathstrut \) \(82\) \(\beta_{6}\mathstrut +\mathstrut \) \(82\) \(\beta_{5}\mathstrut +\mathstrut \) \(89\) \(\beta_{2}\mathstrut -\mathstrut \) \(89\) \(\beta_{1}\mathstrut -\mathstrut \) \(365\)
\(\nu^{7}\)\(=\)\((\)\(368\) \(\beta_{6}\mathstrut +\mathstrut \) \(368\) \(\beta_{5}\mathstrut +\mathstrut \) \(1780\) \(\beta_{4}\mathstrut -\mathstrut \) \(152\) \(\beta_{3}\mathstrut -\mathstrut \) \(707\) \(\beta_{2}\mathstrut -\mathstrut \) \(707\) \(\beta_{1}\)\()/2\)

Character Values

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

\(n\) \(2\)
\(\chi(n)\) \(\beta_{4}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
12.1
2.35663i
3.22189i
1.35225i
0.486981i
2.35663i
3.22189i
1.35225i
0.486981i
−1.45515 1.45515i 3.81178 + 3.81178i 0.234947i 3.14526i 11.0935i 0.342313 −5.47873 + 5.47873i 20.0593i −4.57683 + 4.57683i
12.2 −1.07935 1.07935i −2.14254 2.14254i 1.67001i 0.488689i 4.62511i 8.09117 −6.11992 + 6.11992i 0.180982i −0.527467 + 0.527467i
12.3 0.909588 + 0.909588i 0.442660 + 0.442660i 2.34530i 4.16447i 0.805276i −9.68815 5.77161 5.77161i 8.60810i −3.78796 + 3.78796i
12.4 2.62492 + 2.62492i −3.11190 3.11190i 9.78036i 4.53053i 16.3369i −0.745339 −15.1730 + 15.1730i 10.3678i 11.8923 11.8923i
17.1 −1.45515 + 1.45515i 3.81178 3.81178i 0.234947i 3.14526i 11.0935i 0.342313 −5.47873 5.47873i 20.0593i −4.57683 4.57683i
17.2 −1.07935 + 1.07935i −2.14254 + 2.14254i 1.67001i 0.488689i 4.62511i 8.09117 −6.11992 6.11992i 0.180982i −0.527467 0.527467i
17.3 0.909588 0.909588i 0.442660 0.442660i 2.34530i 4.16447i 0.805276i −9.68815 5.77161 + 5.77161i 8.60810i −3.78796 3.78796i
17.4 2.62492 2.62492i −3.11190 + 3.11190i 9.78036i 4.53053i 16.3369i −0.745339 −15.1730 15.1730i 10.3678i 11.8923 + 11.8923i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 17.4
Significant digits:
Format:

Inner twists

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

Hecke kernels

There are no other newforms in \(S_{3}^{\mathrm{new}}(29, [\chi])\).