Properties

Label 28.3.g.a
Level 28
Weight 3
Character orbit 28.g
Analytic conductor 0.763
Analytic rank 0
Dimension 12
CM No
Inner twists 4

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Newspace parameters

Level: \( N \) = \( 28 = 2^{2} \cdot 7 \)
Weight: \( k \) = \( 3 \)
Character orbit: \([\chi]\) = 28.g (of order \(6\) and degree \(2\))

Newform invariants

Self dual: No
Analytic conductor: \(0.762944740209\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{8} \)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{6} q^{2} + \beta_{4} q^{3} + ( -\beta_{1} + \beta_{5} + \beta_{9} ) q^{4} + ( -1 - \beta_{5} + \beta_{7} - \beta_{9} ) q^{5} + ( \beta_{1} + \beta_{2} - \beta_{3} + \beta_{6} + 2 \beta_{10} + \beta_{11} ) q^{6} + ( -1 + \beta_{1} - 2 \beta_{4} - \beta_{5} - 2 \beta_{6} - \beta_{7} - \beta_{8} - \beta_{9} - \beta_{10} - \beta_{11} ) q^{7} + ( -1 - 2 \beta_{2} + \beta_{3} - 2 \beta_{6} + 2 \beta_{8} - 2 \beta_{10} - \beta_{11} ) q^{8} + ( 1 + \beta_{5} - 2 \beta_{6} - \beta_{7} + \beta_{9} + \beta_{11} ) q^{9} +O(q^{10})\) \( q + \beta_{6} q^{2} + \beta_{4} q^{3} + ( -\beta_{1} + \beta_{5} + \beta_{9} ) q^{4} + ( -1 - \beta_{5} + \beta_{7} - \beta_{9} ) q^{5} + ( \beta_{1} + \beta_{2} - \beta_{3} + \beta_{6} + 2 \beta_{10} + \beta_{11} ) q^{6} + ( -1 + \beta_{1} - 2 \beta_{4} - \beta_{5} - 2 \beta_{6} - \beta_{7} - \beta_{8} - \beta_{9} - \beta_{10} - \beta_{11} ) q^{7} + ( -1 - 2 \beta_{2} + \beta_{3} - 2 \beta_{6} + 2 \beta_{8} - 2 \beta_{10} - \beta_{11} ) q^{8} + ( 1 + \beta_{5} - 2 \beta_{6} - \beta_{7} + \beta_{9} + \beta_{11} ) q^{9} + ( \beta_{2} + \beta_{3} - 2 \beta_{4} + \beta_{5} - 2 \beta_{7} - 2 \beta_{8} ) q^{10} + ( -\beta_{1} - 2 \beta_{2} + \beta_{3} + \beta_{4} + \beta_{5} + \beta_{7} + \beta_{8} + \beta_{9} ) q^{11} + ( -6 - 2 \beta_{4} - 6 \beta_{5} - 2 \beta_{6} + 2 \beta_{7} - \beta_{9} - 2 \beta_{10} - \beta_{11} ) q^{12} + ( -1 + \beta_{1} + 2 \beta_{2} + \beta_{3} + 2 \beta_{6} + \beta_{8} - \beta_{11} ) q^{13} + ( 3 - \beta_{3} + 2 \beta_{4} + 6 \beta_{5} + \beta_{6} - 2 \beta_{8} - \beta_{9} + 2 \beta_{11} ) q^{14} + ( 2 - \beta_{1} + 4 \beta_{2} - 2 \beta_{3} + 4 \beta_{6} + \beta_{8} + 3 \beta_{10} + 2 \beta_{11} ) q^{15} + ( 2 + 4 \beta_{4} + 2 \beta_{5} + 4 \beta_{10} - 2 \beta_{11} ) q^{16} + ( -\beta_{1} - 6 \beta_{2} - 3 \beta_{3} + 2 \beta_{5} - \beta_{7} - \beta_{8} + \beta_{9} ) q^{17} + ( 2 \beta_{1} - \beta_{3} + 2 \beta_{4} - 11 \beta_{5} + 2 \beta_{7} + 2 \beta_{8} - 2 \beta_{9} ) q^{18} + ( 1 + \beta_{4} + \beta_{5} + 2 \beta_{6} + 3 \beta_{7} + 3 \beta_{9} + \beta_{10} + \beta_{11} ) q^{19} + ( 13 - \beta_{1} - 2 \beta_{3} - 4 \beta_{10} + 2 \beta_{11} ) q^{20} + ( -7 - 3 \beta_{1} + 4 \beta_{2} + 2 \beta_{3} + 6 \beta_{6} - 2 \beta_{7} - 3 \beta_{8} + 2 \beta_{9} - 3 \beta_{11} ) q^{21} + ( 3 - \beta_{1} - \beta_{2} + 2 \beta_{3} - \beta_{6} + 2 \beta_{8} - 2 \beta_{11} ) q^{22} + ( 3 - 3 \beta_{4} + 3 \beta_{5} + 6 \beta_{6} - 2 \beta_{7} - 2 \beta_{9} - 3 \beta_{10} + 3 \beta_{11} ) q^{23} + ( 2 \beta_{2} + 5 \beta_{3} + 2 \beta_{4} + 9 \beta_{5} - 2 \beta_{7} - 2 \beta_{8} ) q^{24} + ( 2 \beta_{1} + 4 \beta_{2} + 2 \beta_{3} - 2 \beta_{5} + 2 \beta_{7} + 2 \beta_{8} - 2 \beta_{9} ) q^{25} + ( 9 + 2 \beta_{4} + 9 \beta_{5} + 2 \beta_{7} + 2 \beta_{9} + 2 \beta_{10} - \beta_{11} ) q^{26} + ( -5 - \beta_{1} - 10 \beta_{2} + 5 \beta_{3} - 10 \beta_{6} + \beta_{8} - \beta_{10} - 5 \beta_{11} ) q^{27} + ( -4 + \beta_{1} - 2 \beta_{2} - 3 \beta_{3} - 2 \beta_{4} - 12 \beta_{5} - 2 \beta_{7} - 2 \beta_{8} + \beta_{9} + 4 \beta_{10} + 6 \beta_{11} ) q^{28} + ( 7 + 3 \beta_{1} - 6 \beta_{2} - 3 \beta_{3} - 6 \beta_{6} + 3 \beta_{8} + 3 \beta_{11} ) q^{29} + ( -15 - 8 \beta_{4} - 15 \beta_{5} - 5 \beta_{6} - 2 \beta_{7} + \beta_{9} - 8 \beta_{10} - 6 \beta_{11} ) q^{30} + ( 3 \beta_{1} + 16 \beta_{2} - 8 \beta_{3} - 3 \beta_{4} - 8 \beta_{5} - 3 \beta_{7} - 3 \beta_{8} - 3 \beta_{9} ) q^{31} + ( 4 \beta_{1} - 4 \beta_{3} - 8 \beta_{4} + 16 \beta_{5} - 4 \beta_{9} ) q^{32} + ( -3 - 3 \beta_{5} - 4 \beta_{6} + 2 \beta_{11} ) q^{33} + ( -29 - 6 \beta_{1} - 5 \beta_{2} - \beta_{3} - 5 \beta_{6} + 2 \beta_{8} - 2 \beta_{10} + \beta_{11} ) q^{34} + ( -4 - \beta_{1} - 10 \beta_{2} + 5 \beta_{3} + 11 \beta_{4} + \beta_{5} - 8 \beta_{6} + 3 \beta_{7} + \beta_{8} + 3 \beta_{9} + 5 \beta_{10} - 4 \beta_{11} ) q^{35} + ( -10 + 2 \beta_{1} + 12 \beta_{2} + 12 \beta_{6} - 4 \beta_{8} + 8 \beta_{10} ) q^{36} + ( 10 + 10 \beta_{5} - 10 \beta_{6} + 4 \beta_{7} - 4 \beta_{9} + 5 \beta_{11} ) q^{37} + ( -\beta_{1} - 5 \beta_{2} + 8 \beta_{3} + 4 \beta_{4} + 3 \beta_{5} + 6 \beta_{7} + 6 \beta_{8} + \beta_{9} ) q^{38} + ( 3 \beta_{1} + 2 \beta_{2} - \beta_{3} - 6 \beta_{4} - \beta_{5} - 3 \beta_{7} - 3 \beta_{8} - 3 \beta_{9} ) q^{39} + ( -17 + 6 \beta_{4} - 17 \beta_{5} + 14 \beta_{6} - 2 \beta_{7} + 4 \beta_{9} + 6 \beta_{10} + 3 \beta_{11} ) q^{40} + ( -1 - 5 \beta_{1} + 10 \beta_{2} + 5 \beta_{3} + 10 \beta_{6} - 5 \beta_{8} - 5 \beta_{11} ) q^{41} + ( 23 - 2 \beta_{1} - \beta_{2} - 2 \beta_{3} - 2 \beta_{4} + 31 \beta_{5} - 5 \beta_{6} - 2 \beta_{7} + 4 \beta_{8} + 6 \beta_{9} - 6 \beta_{10} + 3 \beta_{11} ) q^{42} + ( -2 - 4 \beta_{2} + 2 \beta_{3} - 4 \beta_{6} - 16 \beta_{10} - 2 \beta_{11} ) q^{43} + ( 10 - 2 \beta_{4} + 10 \beta_{5} + 2 \beta_{6} - 2 \beta_{7} - \beta_{9} - 2 \beta_{10} - 5 \beta_{11} ) q^{44} + ( \beta_{1} - 10 \beta_{2} - 5 \beta_{3} - 25 \beta_{5} + \beta_{7} + \beta_{8} - \beta_{9} ) q^{45} + ( -9 \beta_{1} + \beta_{2} - 3 \beta_{3} + 2 \beta_{4} - 24 \beta_{5} - 4 \beta_{7} - 4 \beta_{8} + 9 \beta_{9} ) q^{46} + ( -3 \beta_{4} - 7 \beta_{7} - 7 \beta_{9} - 3 \beta_{10} ) q^{47} + ( 46 + 4 \beta_{1} - 6 \beta_{3} + 4 \beta_{10} + 6 \beta_{11} ) q^{48} + ( 14 + 9 \beta_{1} + 2 \beta_{2} + \beta_{3} + 10 \beta_{6} + 6 \beta_{7} + 9 \beta_{8} - 6 \beta_{9} - 5 \beta_{11} ) q^{49} + ( 18 + 4 \beta_{1} + 4 \beta_{2} + 2 \beta_{3} + 4 \beta_{6} - 4 \beta_{8} + 4 \beta_{10} - 2 \beta_{11} ) q^{50} + ( 7 + 17 \beta_{4} + 7 \beta_{5} + 14 \beta_{6} + 7 \beta_{7} + 7 \beta_{9} + 17 \beta_{10} + 7 \beta_{11} ) q^{51} + ( 2 \beta_{1} - 12 \beta_{2} + 4 \beta_{3} + 14 \beta_{5} + 4 \beta_{7} + 4 \beta_{8} - 2 \beta_{9} ) q^{52} + ( -6 \beta_{1} + 2 \beta_{2} + \beta_{3} + 16 \beta_{5} - 6 \beta_{7} - 6 \beta_{8} + 6 \beta_{9} ) q^{53} + ( 27 + 27 \beta_{5} - \beta_{6} - 2 \beta_{7} - 9 \beta_{9} - 2 \beta_{11} ) q^{54} + ( 3 + 2 \beta_{1} + 6 \beta_{2} - 3 \beta_{3} + 6 \beta_{6} - 2 \beta_{8} + 3 \beta_{10} + 3 \beta_{11} ) q^{55} + ( -21 - 4 \beta_{1} + 14 \beta_{2} - \beta_{3} - 4 \beta_{4} - 46 \beta_{5} + 2 \beta_{6} + 4 \beta_{7} + 2 \beta_{8} - 4 \beta_{9} - 10 \beta_{10} - \beta_{11} ) q^{56} + ( -23 - 6 \beta_{1} - 4 \beta_{2} - 2 \beta_{3} - 4 \beta_{6} - 6 \beta_{8} + 2 \beta_{11} ) q^{57} + ( -33 + 6 \beta_{4} - 33 \beta_{5} + 4 \beta_{6} + 6 \beta_{7} - 6 \beta_{9} + 6 \beta_{10} - 3 \beta_{11} ) q^{58} + ( -8 \beta_{1} + 4 \beta_{2} - 2 \beta_{3} - 7 \beta_{4} - 2 \beta_{5} + 8 \beta_{7} + 8 \beta_{8} + 8 \beta_{9} ) q^{59} + ( -3 \beta_{1} + 2 \beta_{2} + 5 \beta_{3} + 18 \beta_{4} + 40 \beta_{5} + 2 \beta_{7} + 2 \beta_{8} + 3 \beta_{9} ) q^{60} + ( 20 + 20 \beta_{5} + 2 \beta_{6} - 8 \beta_{7} + 8 \beta_{9} - \beta_{11} ) q^{61} + ( -39 + 13 \beta_{1} + 3 \beta_{2} - 6 \beta_{3} + 3 \beta_{6} - 6 \beta_{8} + 6 \beta_{11} ) q^{62} + ( -1 - 4 \beta_{1} + 4 \beta_{2} - 2 \beta_{3} - 18 \beta_{4} - 3 \beta_{5} - 2 \beta_{6} - \beta_{7} + 4 \beta_{8} - \beta_{9} - 6 \beta_{10} - \beta_{11} ) q^{63} + ( -28 - 8 \beta_{1} - 24 \beta_{2} + 4 \beta_{3} - 24 \beta_{6} - 8 \beta_{8} - 8 \beta_{10} - 4 \beta_{11} ) q^{64} + ( -19 - 19 \beta_{5} - 2 \beta_{6} - 7 \beta_{7} + 7 \beta_{9} + \beta_{11} ) q^{65} + ( 4 \beta_{1} + 5 \beta_{2} - 20 \beta_{5} - 4 \beta_{9} ) q^{66} + ( -2 \beta_{1} - 16 \beta_{2} + 8 \beta_{3} + 9 \beta_{4} + 8 \beta_{5} + 2 \beta_{7} + 2 \beta_{8} + 2 \beta_{9} ) q^{67} + ( -23 - 8 \beta_{4} - 23 \beta_{5} - 36 \beta_{6} - 12 \beta_{7} - 3 \beta_{9} - 8 \beta_{10} - 8 \beta_{11} ) q^{68} + ( -19 + 9 \beta_{1} + 9 \beta_{8} ) q^{69} + ( 27 + 9 \beta_{1} + 5 \beta_{2} - 2 \beta_{3} - 6 \beta_{4} + 30 \beta_{5} + 4 \beta_{6} + 4 \beta_{7} + 6 \beta_{8} - 13 \beta_{9} + 10 \beta_{10} + 3 \beta_{11} ) q^{70} + ( 4 + 6 \beta_{1} + 8 \beta_{2} - 4 \beta_{3} + 8 \beta_{6} - 6 \beta_{8} + 24 \beta_{10} + 4 \beta_{11} ) q^{71} + ( 22 - 12 \beta_{4} + 22 \beta_{5} - 12 \beta_{6} + 4 \beta_{7} + 4 \beta_{9} - 12 \beta_{10} + 2 \beta_{11} ) q^{72} + ( 2 \beta_{1} + 4 \beta_{2} + 2 \beta_{3} + 37 \beta_{5} + 2 \beta_{7} + 2 \beta_{8} - 2 \beta_{9} ) q^{73} + ( 10 \beta_{1} - 5 \beta_{2} + 4 \beta_{3} - 8 \beta_{4} - 46 \beta_{5} - 8 \beta_{7} - 8 \beta_{8} - 10 \beta_{9} ) q^{74} + ( -2 - 8 \beta_{4} - 2 \beta_{5} - 4 \beta_{6} - 6 \beta_{7} - 6 \beta_{9} - 8 \beta_{10} - 2 \beta_{11} ) q^{75} + ( 46 - \beta_{1} + 2 \beta_{2} + 9 \beta_{3} + 2 \beta_{6} + 2 \beta_{8} + 6 \beta_{10} - 9 \beta_{11} ) q^{76} + ( -35 - 2 \beta_{1} - 2 \beta_{2} - \beta_{3} - 21 \beta_{5} - 10 \beta_{6} + \beta_{7} - 2 \beta_{8} - \beta_{9} + 5 \beta_{11} ) q^{77} + ( 3 - 4 \beta_{1} - 3 \beta_{3} - 6 \beta_{8} - 6 \beta_{10} + 3 \beta_{11} ) q^{78} + ( -7 - 27 \beta_{4} - 7 \beta_{5} - 14 \beta_{6} - 2 \beta_{7} - 2 \beta_{9} - 27 \beta_{10} - 7 \beta_{11} ) q^{79} + ( -8 \beta_{1} + 24 \beta_{2} - 6 \beta_{3} - 4 \beta_{4} - 10 \beta_{5} + 8 \beta_{7} + 8 \beta_{8} + 8 \beta_{9} ) q^{80} + ( \beta_{1} - 6 \beta_{2} - 3 \beta_{3} - 24 \beta_{5} + \beta_{7} + \beta_{8} - \beta_{9} ) q^{81} + ( 55 - 10 \beta_{4} + 55 \beta_{5} + 4 \beta_{6} - 10 \beta_{7} + 10 \beta_{9} - 10 \beta_{10} + 5 \beta_{11} ) q^{82} + ( 2 + 6 \beta_{1} + 4 \beta_{2} - 2 \beta_{3} + 4 \beta_{6} - 6 \beta_{8} + 8 \beta_{10} + 2 \beta_{11} ) q^{83} + ( -27 + 2 \beta_{1} - 36 \beta_{2} + 4 \beta_{3} + 20 \beta_{4} - 37 \beta_{5} - 12 \beta_{6} + 8 \beta_{7} + 12 \beta_{8} + \beta_{9} + 4 \beta_{10} - 6 \beta_{11} ) q^{84} + ( 18 - 10 \beta_{1} + 14 \beta_{2} + 7 \beta_{3} + 14 \beta_{6} - 10 \beta_{8} - 7 \beta_{11} ) q^{85} + ( 12 + 32 \beta_{4} + 12 \beta_{5} + 16 \beta_{6} + 12 \beta_{9} + 32 \beta_{10} + 16 \beta_{11} ) q^{86} + ( -3 \beta_{1} - 30 \beta_{2} + 15 \beta_{3} + 34 \beta_{4} + 15 \beta_{5} + 3 \beta_{7} + 3 \beta_{8} + 3 \beta_{9} ) q^{87} + ( -4 \beta_{1} - 14 \beta_{2} - 3 \beta_{3} + 2 \beta_{4} + 37 \beta_{5} - 2 \beta_{7} - 2 \beta_{8} + 4 \beta_{9} ) q^{88} + ( -1 - \beta_{5} + 4 \beta_{6} + 4 \beta_{7} - 4 \beta_{9} - 2 \beta_{11} ) q^{89} + ( -51 - 10 \beta_{1} + 20 \beta_{2} + \beta_{3} + 20 \beta_{6} - 2 \beta_{8} + 2 \beta_{10} - \beta_{11} ) q^{90} + ( 3 - \beta_{1} + 18 \beta_{2} - 9 \beta_{3} + 4 \beta_{4} - 6 \beta_{5} + 6 \beta_{6} - 4 \beta_{7} + \beta_{8} - 4 \beta_{9} - 2 \beta_{10} + 3 \beta_{11} ) q^{91} + ( -24 + 3 \beta_{1} + 18 \beta_{2} - \beta_{3} + 18 \beta_{6} + 18 \beta_{8} - 14 \beta_{10} + \beta_{11} ) q^{92} + ( 24 + 24 \beta_{5} + 22 \beta_{6} + 10 \beta_{7} - 10 \beta_{9} - 11 \beta_{11} ) q^{93} + ( -3 \beta_{1} + 11 \beta_{2} - 18 \beta_{3} - 8 \beta_{4} - 21 \beta_{5} - 14 \beta_{7} - 14 \beta_{8} + 3 \beta_{9} ) q^{94} + ( 14 \beta_{2} - 7 \beta_{3} + 13 \beta_{4} - 7 \beta_{5} ) q^{95} + ( -44 - 44 \beta_{5} + 40 \beta_{6} + 8 \beta_{7} - 4 \beta_{9} ) q^{96} + ( 55 - 3 \beta_{1} - 30 \beta_{2} - 15 \beta_{3} - 30 \beta_{6} - 3 \beta_{8} + 15 \beta_{11} ) q^{97} + ( 1 - 8 \beta_{1} - 4 \beta_{2} + 6 \beta_{3} + 6 \beta_{4} + 47 \beta_{5} + 15 \beta_{6} + 6 \beta_{7} - 12 \beta_{8} + 10 \beta_{9} + 18 \beta_{10} - 9 \beta_{11} ) q^{98} + ( 3 + 5 \beta_{1} + 6 \beta_{2} - 3 \beta_{3} + 6 \beta_{6} - 5 \beta_{8} - 2 \beta_{10} + 3 \beta_{11} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12q - 2q^{2} - 4q^{4} - 2q^{5} - 12q^{6} - 8q^{8} + 4q^{9} + O(q^{10}) \) \( 12q - 2q^{2} - 4q^{4} - 2q^{5} - 12q^{6} - 8q^{8} + 4q^{9} - 2q^{10} - 24q^{12} - 24q^{13} + 2q^{14} + 16q^{16} - 2q^{17} + 56q^{18} + 152q^{20} - 78q^{21} + 44q^{22} - 44q^{24} + 56q^{26} + 8q^{28} + 72q^{29} - 74q^{30} - 112q^{32} - 14q^{33} - 316q^{34} - 160q^{36} + 86q^{37} - 2q^{38} - 148q^{40} + 8q^{41} + 68q^{42} + 64q^{44} + 156q^{45} + 162q^{46} + 512q^{48} + 108q^{49} + 208q^{50} - 64q^{52} - 74q^{53} + 182q^{54} + 16q^{56} - 220q^{57} - 176q^{58} - 232q^{60} + 86q^{61} - 532q^{62} - 160q^{64} - 140q^{65} + 102q^{66} - 68q^{68} - 300q^{69} + 90q^{70} + 152q^{72} - 234q^{73} + 290q^{74} + 576q^{76} - 262q^{77} + 64q^{78} + 146q^{81} + 272q^{82} - 28q^{84} + 268q^{85} - 16q^{86} - 188q^{88} + 6q^{89} - 640q^{90} - 448q^{92} + 162q^{93} + 102q^{94} - 320q^{96} + 744q^{97} - 190q^{98} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{12} - 3 x^{11} - 4 x^{10} + 3 x^{9} + 86 x^{8} - 163 x^{7} + 155 x^{6} - 166 x^{5} + 164 x^{4} - 116 x^{3} + 60 x^{2} - 20 x + 4\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\(548052 \nu^{11} + 1422116 \nu^{10} - 6191612 \nu^{9} - 21007396 \nu^{8} + 23867696 \nu^{7} + 160225554 \nu^{6} + 25497908 \nu^{5} - 20918128 \nu^{4} - 126880702 \nu^{3} - 5442024 \nu^{2} + 1537456 \nu - 9243685\)\()/5128417\)
\(\beta_{2}\)\(=\)\((\)\(-2016555 \nu^{11} + 4608465 \nu^{10} + 11867940 \nu^{9} + 1459025 \nu^{8} - 175954938 \nu^{7} + 201471321 \nu^{6} - 122846749 \nu^{5} + 215038026 \nu^{4} - 177078348 \nu^{3} + 48635510 \nu^{2} - 46482276 \nu + 13204764\)\()/10256834\)
\(\beta_{3}\)\(=\)\((\)\(6080681 \nu^{11} + 174798 \nu^{10} - 64448111 \nu^{9} - 87898803 \nu^{8} + 490503169 \nu^{7} + 562380793 \nu^{6} - 787680086 \nu^{5} + 461315491 \nu^{4} - 855472904 \nu^{3} + 799718192 \nu^{2} - 519820470 \nu + 199145960\)\()/20513668\)
\(\beta_{4}\)\(=\)\((\)\(-8092731 \nu^{11} + 13062376 \nu^{10} + 58450763 \nu^{9} + 38422753 \nu^{8} - 688812905 \nu^{7} + 358968697 \nu^{6} - 67350324 \nu^{5} + 436258613 \nu^{4} - 177419776 \nu^{3} - 102428648 \nu^{2} + 66312934 \nu - 38518016\)\()/20513668\)
\(\beta_{5}\)\(=\)\((\)\(8864783 \nu^{11} - 20455402 \nu^{10} - 48834261 \nu^{9} - 8785933 \nu^{8} + 751559539 \nu^{7} - 927115697 \nu^{6} + 797312818 \nu^{5} - 981317799 \nu^{4} + 836256216 \nu^{3} - 526622432 \nu^{2} + 236999870 \nu - 70375800\)\()/20513668\)
\(\beta_{6}\)\(=\)\((\)\(-2781796 \nu^{11} + 6284192 \nu^{10} + 15510516 \nu^{9} + 3410783 \nu^{8} - 234558069 \nu^{7} + 282488251 \nu^{6} - 243549767 \nu^{5} + 278083421 \nu^{4} - 254949459 \nu^{3} + 160381312 \nu^{2} - 43781960 \nu + 15160294\)\()/5128417\)
\(\beta_{7}\)\(=\)\((\)\(11588911 \nu^{11} - 2778978 \nu^{10} - 108332469 \nu^{9} - 167711837 \nu^{8} + 897253275 \nu^{7} + 805776575 \nu^{6} - 546730846 \nu^{5} - 195583759 \nu^{4} - 691987512 \nu^{3} + 476417312 \nu^{2} - 253397978 \nu + 51100076\)\()/20513668\)
\(\beta_{8}\)\(=\)\((\)\(7577433 \nu^{11} - 23003531 \nu^{10} - 36322258 \nu^{9} + 36406939 \nu^{8} + 694698148 \nu^{7} - 1232772201 \nu^{6} + 650456281 \nu^{5} - 847212044 \nu^{4} + 921404444 \nu^{3} - 415827090 \nu^{2} + 151851740 \nu - 10748502\)\()/10256834\)
\(\beta_{9}\)\(=\)\((\)\(-18854063 \nu^{11} + 54731322 \nu^{10} + 82312181 \nu^{9} - 56986907 \nu^{8} - 1629791971 \nu^{7} + 2949221097 \nu^{6} - 2465476938 \nu^{5} + 2338940247 \nu^{4} - 2643323856 \nu^{3} + 1684290272 \nu^{2} - 641927134 \nu + 162375652\)\()/20513668\)
\(\beta_{10}\)\(=\)\((\)\(13906733 \nu^{11} - 32333536 \nu^{10} - 77826224 \nu^{9} - 8906454 \nu^{8} + 1190477049 \nu^{7} - 1470041616 \nu^{6} + 1124752022 \nu^{5} - 1425602332 \nu^{4} + 1248314483 \nu^{3} - 684256636 \nu^{2} + 248435884 \nu - 51147512\)\()/10256834\)
\(\beta_{11}\)\(=\)\((\)\(-6254493 \nu^{11} + 16328866 \nu^{10} + 30974575 \nu^{9} - 6021165 \nu^{8} - 537499493 \nu^{7} + 811952879 \nu^{6} - 686835394 \nu^{5} + 792739733 \nu^{4} - 729655908 \nu^{3} + 462671520 \nu^{2} - 200722666 \nu + 44275732\)\()/2930524\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{11} + 2 \beta_{10} + 2 \beta_{6} + 3 \beta_{5} + 2 \beta_{4} + 3\)\()/4\)
\(\nu^{2}\)\(=\)\((\)\(-2 \beta_{9} + 2 \beta_{8} + 2 \beta_{7} - 11 \beta_{5} + 2 \beta_{4} - \beta_{3} - 6 \beta_{2} + 2 \beta_{1}\)\()/4\)
\(\nu^{3}\)\(=\)\((\)\(9 \beta_{11} + 18 \beta_{10} + 2 \beta_{8} + 6 \beta_{6} - 9 \beta_{3} + 6 \beta_{2} + 4 \beta_{1} + 25\)\()/4\)
\(\nu^{4}\)\(=\)\((\)\(-\beta_{11} + 38 \beta_{10} - 14 \beta_{9} + 18 \beta_{7} + 54 \beta_{6} - 75 \beta_{5} + 38 \beta_{4} - 75\)\()/4\)
\(\nu^{5}\)\(=\)\((\)\(-52 \beta_{9} + 38 \beta_{8} + 38 \beta_{7} - 293 \beta_{5} - 122 \beta_{4} - 81 \beta_{3} + 2 \beta_{2} + 52 \beta_{1}\)\()/4\)
\(\nu^{6}\)\(=\)\((\)\(71 \beta_{11} + 462 \beta_{10} - 122 \beta_{8} + 482 \beta_{6} - 71 \beta_{3} + 482 \beta_{2} - 70 \beta_{1} - 379\)\()/4\)
\(\nu^{7}\)\(=\)\((\)\(-651 \beta_{11} - 622 \beta_{10} - 532 \beta_{9} + 462 \beta_{7} + 462 \beta_{6} - 2987 \beta_{5} - 622 \beta_{4} - 2987\)\()/4\)
\(\nu^{8}\)\(=\)\((\)\(90 \beta_{9} - 622 \beta_{8} - 622 \beta_{7} + 391 \beta_{5} - 4734 \beta_{4} - 1283 \beta_{3} + 3830 \beta_{2} - 90 \beta_{1}\)\()/4\)
\(\nu^{9}\)\(=\)\((\)\(-4513 \beta_{11} - 802 \beta_{10} - 4734 \beta_{8} + 7942 \beta_{6} + 4513 \beta_{3} + 7942 \beta_{2} - 4824 \beta_{1} - 26985\)\()/4\)
\(\nu^{10}\)\(=\)\((\)\(-15935 \beta_{11} - 42942 \beta_{10} - 4022 \beta_{9} - 802 \beta_{7} - 26154 \beta_{6} - 23505 \beta_{5} - 42942 \beta_{4} - 23505\)\()/4\)
\(\nu^{11}\)\(=\)\((\)\(38920 \beta_{9} - 42942 \beta_{8} - 42942 \beta_{7} + 216715 \beta_{5} - 35802 \beta_{4} + 24239 \beta_{3} + 96854 \beta_{2} - 38920 \beta_{1}\)\()/4\)

Character Values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/28\mathbb{Z}\right)^\times\).

\(n\) \(15\) \(17\)
\(\chi(n)\) \(-1\) \(-1 - \beta_{5}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
11.1
−0.407369 + 0.812545i
−2.29733 + 1.90372i
2.79733 1.03769i
0.121721 + 0.507075i
0.907369 + 0.0534805i
0.378279 + 0.358951i
−0.407369 0.812545i
−2.29733 1.90372i
2.79733 + 1.03769i
0.121721 0.507075i
0.907369 0.0534805i
0.378279 0.358951i
−1.98615 + 0.234945i 1.86796 + 1.07847i 3.88960 0.933271i 3.25304 + 5.63443i −3.96343 1.70313i −2.39669 6.57692i −7.50608 + 2.76746i −2.17382 3.76517i −7.78481 10.4265i
11.2 −1.51615 1.30434i −3.95004 2.28056i 0.597396 + 3.95514i −2.62655 4.54932i 3.01422 + 8.60985i 5.86799 3.81663i 4.25310 6.77577i 5.90188 + 10.2224i −1.95163 + 10.3234i
11.3 −0.371518 1.96519i 3.95004 + 2.28056i −3.72395 + 1.46021i −2.62655 4.54932i 3.01422 8.60985i −5.86799 + 3.81663i 4.25310 + 6.77577i 5.90188 + 10.2224i −7.96447 + 6.85183i
11.4 −0.104798 + 1.99725i 1.63031 + 0.941260i −3.97803 0.418616i −1.12649 1.95113i −2.05079 + 3.15750i 6.84270 + 1.47562i 1.25297 7.90127i −2.72806 4.72514i 4.01495 2.04540i
11.5 1.19654 1.60259i −1.86796 1.07847i −1.13656 3.83513i 3.25304 + 5.63443i −3.96343 + 1.70313i 2.39669 + 6.57692i −7.50608 2.76746i −2.17382 3.76517i 12.9221 + 1.52857i
11.6 1.78207 + 0.907869i −1.63031 0.941260i 2.35155 + 3.23577i −1.12649 1.95113i −2.05079 3.15750i −6.84270 1.47562i 1.25297 + 7.90127i −2.72806 4.72514i −0.236107 4.49975i
23.1 −1.98615 0.234945i 1.86796 1.07847i 3.88960 + 0.933271i 3.25304 5.63443i −3.96343 + 1.70313i −2.39669 + 6.57692i −7.50608 2.76746i −2.17382 + 3.76517i −7.78481 + 10.4265i
23.2 −1.51615 + 1.30434i −3.95004 + 2.28056i 0.597396 3.95514i −2.62655 + 4.54932i 3.01422 8.60985i 5.86799 + 3.81663i 4.25310 + 6.77577i 5.90188 10.2224i −1.95163 10.3234i
23.3 −0.371518 + 1.96519i 3.95004 2.28056i −3.72395 1.46021i −2.62655 + 4.54932i 3.01422 + 8.60985i −5.86799 3.81663i 4.25310 6.77577i 5.90188 10.2224i −7.96447 6.85183i
23.4 −0.104798 1.99725i 1.63031 0.941260i −3.97803 + 0.418616i −1.12649 + 1.95113i −2.05079 3.15750i 6.84270 1.47562i 1.25297 + 7.90127i −2.72806 + 4.72514i 4.01495 + 2.04540i
23.5 1.19654 + 1.60259i −1.86796 + 1.07847i −1.13656 + 3.83513i 3.25304 5.63443i −3.96343 1.70313i 2.39669 6.57692i −7.50608 + 2.76746i −2.17382 + 3.76517i 12.9221 1.52857i
23.6 1.78207 0.907869i −1.63031 + 0.941260i 2.35155 3.23577i −1.12649 + 1.95113i −2.05079 + 3.15750i −6.84270 + 1.47562i 1.25297 7.90127i −2.72806 + 4.72514i −0.236107 + 4.49975i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 23.6
Significant digits:
Format:

Inner twists

Char. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
4.b Odd 1 yes
7.c Even 1 yes
28.g Odd 1 yes

Hecke kernels

There are no other newforms in \(S_{3}^{\mathrm{new}}(28, [\chi])\).