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.

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

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

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

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 \(S_{3}^{\mathrm{new}}(28, [\chi])\).