Properties

Label 22.3.d.a
Level 22
Weight 3
Character orbit 22.d
Analytic conductor 0.599
Analytic rank 0
Dimension 8
CM No
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 22 = 2 \cdot 11 \)
Weight: \( k \) = \( 3 \)
Character orbit: \([\chi]\) = 22.d (of order \(10\) and degree \(4\))

Newform invariants

Self dual: No
Analytic conductor: \(0.599456581593\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{10})\)
Coefficient field: 8.0.64000000.1
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

$q$-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.

\(f(q)\) \(=\) \( q\) \( + \beta_{1} q^{2} \) \( + ( 1 - \beta_{1} - 2 \beta_{2} + \beta_{3} + 2 \beta_{4} - \beta_{5} - \beta_{6} + 2 \beta_{7} ) q^{3} \) \( + 2 \beta_{2} q^{4} \) \( + ( -2 \beta_{1} - \beta_{2} - 3 \beta_{4} - \beta_{6} - 2 \beta_{7} ) q^{5} \) \( + ( -4 + \beta_{1} + 2 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} + 2 \beta_{6} - \beta_{7} ) q^{6} \) \( + ( -5 + \beta_{1} + 2 \beta_{2} - \beta_{3} + \beta_{4} + 4 \beta_{6} - 2 \beta_{7} ) q^{7} \) \( + 2 \beta_{3} q^{8} \) \( + ( -1 + 2 \beta_{1} + \beta_{2} - 4 \beta_{3} + 4 \beta_{5} + \beta_{6} ) q^{9} \) \(+O(q^{10})\) \( q\) \( + \beta_{1} q^{2} \) \( + ( 1 - \beta_{1} - 2 \beta_{2} + \beta_{3} + 2 \beta_{4} - \beta_{5} - \beta_{6} + 2 \beta_{7} ) q^{3} \) \( + 2 \beta_{2} q^{4} \) \( + ( -2 \beta_{1} - \beta_{2} - 3 \beta_{4} - \beta_{6} - 2 \beta_{7} ) q^{5} \) \( + ( -4 + \beta_{1} + 2 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} + 2 \beta_{6} - \beta_{7} ) q^{6} \) \( + ( -5 + \beta_{1} + 2 \beta_{2} - \beta_{3} + \beta_{4} + 4 \beta_{6} - 2 \beta_{7} ) q^{7} \) \( + 2 \beta_{3} q^{8} \) \( + ( -1 + 2 \beta_{1} + \beta_{2} - 4 \beta_{3} + 4 \beta_{5} + \beta_{6} ) q^{9} \) \( + ( 4 - 8 \beta_{2} - \beta_{3} + 4 \beta_{4} - 3 \beta_{5} - 4 \beta_{6} - \beta_{7} ) q^{10} \) \( + ( -1 + 2 \beta_{1} + 4 \beta_{2} + 3 \beta_{3} - 2 \beta_{4} + 4 \beta_{5} - 4 \beta_{6} - 3 \beta_{7} ) q^{11} \) \( + ( 2 - 4 \beta_{1} + 2 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} + 2 \beta_{6} + 2 \beta_{7} ) q^{12} \) \( + ( 1 - 4 \beta_{1} + 5 \beta_{2} - 4 \beta_{4} - 10 \beta_{5} + 2 \beta_{6} + 10 \beta_{7} ) q^{13} \) \( + ( 4 - 5 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} + 2 \beta_{4} + \beta_{5} - 4 \beta_{6} + 4 \beta_{7} ) q^{14} \) \( + ( 10 + 3 \beta_{1} - 9 \beta_{2} + 9 \beta_{3} + 10 \beta_{4} - 6 \beta_{5} + 3 \beta_{7} ) q^{15} \) \( + 4 \beta_{4} q^{16} \) \( + ( 6 + 8 \beta_{1} - \beta_{2} - 6 \beta_{3} + 3 \beta_{4} + 6 \beta_{5} - 5 \beta_{6} - 8 \beta_{7} ) q^{17} \) \( + ( -\beta_{1} + 4 \beta_{2} + \beta_{3} - 8 \beta_{4} + 8 \beta_{6} + \beta_{7} ) q^{18} \) \( + ( -3 + 11 \beta_{1} - 4 \beta_{2} - 5 \beta_{3} - 4 \beta_{4} + 11 \beta_{5} - 3 \beta_{6} ) q^{19} \) \( + ( 2 + 4 \beta_{1} - 2 \beta_{2} - 8 \beta_{3} + 4 \beta_{5} - 8 \beta_{6} - 4 \beta_{7} ) q^{20} \) \( + ( -6 + 12 \beta_{2} - 8 \beta_{3} - 3 \beta_{4} - 2 \beta_{5} + 9 \beta_{6} - 8 \beta_{7} ) q^{21} \) \( + ( 6 - \beta_{1} - 2 \beta_{2} + 4 \beta_{3} + 12 \beta_{4} - 2 \beta_{5} + 2 \beta_{6} - 4 \beta_{7} ) q^{22} \) \( + ( -16 - 16 \beta_{1} + 12 \beta_{3} - 6 \beta_{4} - 8 \beta_{5} + 6 \beta_{6} + 4 \beta_{7} ) q^{23} \) \( + ( -4 + 2 \beta_{1} - 4 \beta_{2} - 2 \beta_{5} + 2 \beta_{7} ) q^{24} \) \( + ( -8 - 16 \beta_{1} + 9 \beta_{2} + 4 \beta_{3} - 9 \beta_{4} + 8 \beta_{5} + 8 \beta_{6} + 8 \beta_{7} ) q^{25} \) \( + ( -20 + \beta_{1} + 12 \beta_{2} + 5 \beta_{3} - 20 \beta_{4} - 4 \beta_{5} + 2 \beta_{7} ) q^{26} \) \( + ( -7 \beta_{1} - 4 \beta_{2} - 6 \beta_{3} + 5 \beta_{4} - 6 \beta_{5} - 4 \beta_{6} - 7 \beta_{7} ) q^{27} \) \( + ( -8 + 4 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} - 4 \beta_{4} + 2 \beta_{5} + 10 \beta_{6} - 4 \beta_{7} ) q^{28} \) \( + ( 17 + 6 \beta_{1} - 18 \beta_{2} - 6 \beta_{3} + 19 \beta_{4} - 36 \beta_{6} - 4 \beta_{7} ) q^{29} \) \( + ( -6 + 10 \beta_{1} + 12 \beta_{2} - 9 \beta_{3} + 12 \beta_{4} + 10 \beta_{5} - 6 \beta_{6} ) q^{30} \) \( + ( 3 + \beta_{1} - 3 \beta_{2} - 2 \beta_{3} + 4 \beta_{5} + 14 \beta_{6} + 2 \beta_{7} ) q^{31} \) \( + 4 \beta_{5} q^{32} \) \( + ( -9 - 4 \beta_{1} - 8 \beta_{2} + 16 \beta_{3} - 18 \beta_{4} - 8 \beta_{5} + 19 \beta_{6} + 6 \beta_{7} ) q^{33} \) \( + ( 16 + 6 \beta_{1} - \beta_{3} + 4 \beta_{4} + 3 \beta_{5} - 4 \beta_{6} - 5 \beta_{7} ) q^{34} \) \( + ( 14 + 7 \beta_{1} + 14 \beta_{4} - 3 \beta_{5} - 7 \beta_{6} + 3 \beta_{7} ) q^{35} \) \( + ( -2 + 4 \beta_{3} - 8 \beta_{5} + 2 \beta_{6} + 8 \beta_{7} ) q^{36} \) \( + ( 3 + 2 \beta_{1} - 6 \beta_{2} + 6 \beta_{3} + 3 \beta_{4} - 4 \beta_{5} + 2 \beta_{7} ) q^{37} \) \( + ( -3 \beta_{1} + 22 \beta_{2} - 4 \beta_{3} - 10 \beta_{4} - 4 \beta_{5} + 22 \beta_{6} - 3 \beta_{7} ) q^{38} \) \( + ( 26 - \beta_{1} - 14 \beta_{3} + 13 \beta_{4} + 14 \beta_{5} - 26 \beta_{6} + \beta_{7} ) q^{39} \) \( + ( 8 + 2 \beta_{1} - 2 \beta_{3} - 8 \beta_{4} - 8 \beta_{7} ) q^{40} \) \( + ( 25 - 8 \beta_{1} - 12 \beta_{2} + 8 \beta_{3} - 12 \beta_{4} - 8 \beta_{5} + 25 \beta_{6} ) q^{41} \) \( + ( 16 - 6 \beta_{1} - 16 \beta_{2} + 12 \beta_{3} - 3 \beta_{5} - 20 \beta_{6} + 9 \beta_{7} ) q^{42} \) \( + ( -14 + 28 \beta_{2} + 4 \beta_{3} - 24 \beta_{4} - 10 \beta_{5} + 4 \beta_{6} + 4 \beta_{7} ) q^{43} \) \( + ( 8 + 6 \beta_{1} - 10 \beta_{2} - 2 \beta_{3} + 16 \beta_{4} + 12 \beta_{5} - 12 \beta_{6} + 2 \beta_{7} ) q^{44} \) \( + ( -3 + 16 \beta_{1} - 12 \beta_{3} + 28 \beta_{4} + 8 \beta_{5} - 28 \beta_{6} - 4 \beta_{7} ) q^{45} \) \( + ( -8 - 16 \beta_{1} - 24 \beta_{2} + 16 \beta_{4} - 6 \beta_{5} - 8 \beta_{6} + 6 \beta_{7} ) q^{46} \) \( + ( -13 + 9 \beta_{1} + 6 \beta_{2} + \beta_{3} - 6 \beta_{4} - 11 \beta_{5} + 13 \beta_{6} + 2 \beta_{7} ) q^{47} \) \( + ( -4 - 4 \beta_{1} + 8 \beta_{2} - 4 \beta_{3} - 4 \beta_{4} ) q^{48} \) \( + ( 10 \beta_{1} - 25 \beta_{2} + 14 \beta_{3} + 22 \beta_{4} + 14 \beta_{5} - 25 \beta_{6} + 10 \beta_{7} ) q^{49} \) \( + ( -16 - 8 \beta_{1} - 16 \beta_{2} + 9 \beta_{3} - 8 \beta_{4} - 9 \beta_{5} + 32 \beta_{6} + 8 \beta_{7} ) q^{50} \) \( + ( -18 - 11 \beta_{1} + 15 \beta_{2} + 11 \beta_{3} - 12 \beta_{4} + 30 \beta_{6} + 5 \beta_{7} ) q^{51} \) \( + ( -4 - 20 \beta_{1} + 6 \beta_{2} + 12 \beta_{3} + 6 \beta_{4} - 20 \beta_{5} - 4 \beta_{6} ) q^{52} \) \( + ( -57 + 6 \beta_{1} + 57 \beta_{2} - 12 \beta_{3} + 34 \beta_{6} - 12 \beta_{7} ) q^{53} \) \( + ( 14 - 28 \beta_{2} - 4 \beta_{3} + 2 \beta_{4} + 5 \beta_{5} - 26 \beta_{6} - 4 \beta_{7} ) q^{54} \) \( + ( -12 + 13 \beta_{1} + 15 \beta_{2} - 41 \beta_{3} - 35 \beta_{4} + 15 \beta_{5} - 15 \beta_{6} - 25 \beta_{7} ) q^{55} \) \( + ( 8 - 8 \beta_{1} - 2 \beta_{3} + 4 \beta_{4} - 4 \beta_{5} - 4 \beta_{6} + 10 \beta_{7} ) q^{56} \) \( + ( -37 + 10 \beta_{1} + 11 \beta_{2} - 48 \beta_{4} + 26 \beta_{5} + 24 \beta_{6} - 26 \beta_{7} ) q^{57} \) \( + ( 8 + 17 \beta_{1} + 4 \beta_{2} - 18 \beta_{3} - 4 \beta_{4} + 19 \beta_{5} - 8 \beta_{6} - 36 \beta_{7} ) q^{58} \) \( + ( 15 - 13 \beta_{1} - 20 \beta_{2} - 33 \beta_{3} + 15 \beta_{4} + 20 \beta_{5} - 10 \beta_{7} ) q^{59} \) \( + ( -6 \beta_{1} + 20 \beta_{2} + 12 \beta_{3} - 18 \beta_{4} + 12 \beta_{5} + 20 \beta_{6} - 6 \beta_{7} ) q^{60} \) \( + ( 10 - 6 \beta_{1} + 9 \beta_{2} + 34 \beta_{3} + 5 \beta_{4} - 34 \beta_{5} - 19 \beta_{6} + 6 \beta_{7} ) q^{61} \) \( + ( -4 + 3 \beta_{1} + 6 \beta_{2} - 3 \beta_{3} - 8 \beta_{4} + 12 \beta_{6} + 14 \beta_{7} ) q^{62} \) \( + ( -2 - 12 \beta_{1} - \beta_{2} + 13 \beta_{3} - \beta_{4} - 12 \beta_{5} - 2 \beta_{6} ) q^{63} \) \( + 8 \beta_{6} q^{64} \) \( + ( 21 - 42 \beta_{2} + 18 \beta_{3} + 51 \beta_{4} + 10 \beta_{5} + 9 \beta_{6} + 18 \beta_{7} ) q^{65} \) \( + ( -12 - 9 \beta_{1} + 4 \beta_{2} - 8 \beta_{3} + 20 \beta_{4} - 18 \beta_{5} - 4 \beta_{6} + 19 \beta_{7} ) q^{66} \) \( + ( 12 + 4 \beta_{1} - 14 \beta_{3} - 4 \beta_{4} + 2 \beta_{5} + 4 \beta_{6} + 10 \beta_{7} ) q^{67} \) \( + ( 10 + 16 \beta_{1} + 2 \beta_{2} + 8 \beta_{4} + 4 \beta_{5} - 4 \beta_{6} - 4 \beta_{7} ) q^{68} \) \( + ( 30 + 26 \beta_{1} - 26 \beta_{2} - 4 \beta_{3} + 26 \beta_{4} - 18 \beta_{5} - 30 \beta_{6} - 8 \beta_{7} ) q^{69} \) \( + ( -6 + 14 \beta_{1} + 20 \beta_{2} - 6 \beta_{4} + 14 \beta_{5} - 7 \beta_{7} ) q^{70} \) \( + ( 19 \beta_{1} - 3 \beta_{2} - 7 \beta_{3} - 33 \beta_{4} - 7 \beta_{5} - 3 \beta_{6} + 19 \beta_{7} ) q^{71} \) \( + ( -16 - 2 \beta_{1} + 16 \beta_{2} - 8 \beta_{4} + 2 \beta_{7} ) q^{72} \) \( + ( -9 - 22 \beta_{1} + 2 \beta_{2} + 22 \beta_{3} + 5 \beta_{4} + 4 \beta_{6} + 26 \beta_{7} ) q^{73} \) \( + ( -4 + 3 \beta_{1} + 8 \beta_{2} - 6 \beta_{3} + 8 \beta_{4} + 3 \beta_{5} - 4 \beta_{6} ) q^{74} \) \( + ( 66 - 21 \beta_{1} - 66 \beta_{2} + 42 \beta_{3} - 37 \beta_{5} - 39 \beta_{6} + 5 \beta_{7} ) q^{75} \) \( + ( 6 - 12 \beta_{2} + 22 \beta_{3} - 2 \beta_{4} - 10 \beta_{5} - 14 \beta_{6} + 22 \beta_{7} ) q^{76} \) \( + ( 39 - 34 \beta_{1} - 2 \beta_{2} + 4 \beta_{3} + 23 \beta_{4} - 24 \beta_{5} + 2 \beta_{6} + 40 \beta_{7} ) q^{77} \) \( + ( -2 + 26 \beta_{1} - 30 \beta_{4} + 13 \beta_{5} + 30 \beta_{6} - 26 \beta_{7} ) q^{78} \) \( + ( 59 - 17 \beta_{1} + 15 \beta_{2} + 44 \beta_{4} + 24 \beta_{5} - 22 \beta_{6} - 24 \beta_{7} ) q^{79} \) \( + ( 16 + 8 \beta_{1} - 12 \beta_{2} + 12 \beta_{4} - 8 \beta_{5} - 16 \beta_{6} ) q^{80} \) \( + ( 22 - 4 \beta_{1} + 24 \beta_{2} - 32 \beta_{3} + 22 \beta_{4} + 28 \beta_{5} - 14 \beta_{7} ) q^{81} \) \( + ( 25 \beta_{1} - 16 \beta_{2} - 12 \beta_{3} + 16 \beta_{4} - 12 \beta_{5} - 16 \beta_{6} + 25 \beta_{7} ) q^{82} \) \( + ( -30 - 39 \beta_{1} - \beta_{2} + 33 \beta_{3} - 15 \beta_{4} - 33 \beta_{5} + 31 \beta_{6} + 39 \beta_{7} ) q^{83} \) \( + ( -18 + 16 \beta_{1} + 6 \beta_{2} - 16 \beta_{3} + 6 \beta_{4} + 12 \beta_{6} - 20 \beta_{7} ) q^{84} \) \( + ( -33 - 22 \beta_{1} - 22 \beta_{2} - 26 \beta_{3} - 22 \beta_{4} - 22 \beta_{5} - 33 \beta_{6} ) q^{85} \) \( + ( -8 - 14 \beta_{1} + 8 \beta_{2} + 28 \beta_{3} - 24 \beta_{5} - 12 \beta_{6} + 4 \beta_{7} ) q^{86} \) \( + ( -32 + 64 \beta_{2} + 8 \beta_{3} - 43 \beta_{4} + 47 \beta_{5} + 21 \beta_{6} + 8 \beta_{7} ) q^{87} \) \( + ( -4 + 8 \beta_{1} + 16 \beta_{2} - 10 \beta_{3} - 8 \beta_{4} + 16 \beta_{5} + 28 \beta_{6} - 12 \beta_{7} ) q^{88} \) \( + ( -30 + 48 \beta_{1} - 6 \beta_{3} - 66 \beta_{4} + 24 \beta_{5} + 66 \beta_{6} - 42 \beta_{7} ) q^{89} \) \( + ( 8 - 3 \beta_{1} + 24 \beta_{2} - 16 \beta_{4} + 28 \beta_{5} + 8 \beta_{6} - 28 \beta_{7} ) q^{90} \) \( + ( -47 + 45 \beta_{1} - 3 \beta_{2} - 42 \beta_{3} + 3 \beta_{4} + 39 \beta_{5} + 47 \beta_{6} - 84 \beta_{7} ) q^{91} \) \( + ( -12 - 8 \beta_{1} - 20 \beta_{2} - 24 \beta_{3} - 12 \beta_{4} + 16 \beta_{5} - 8 \beta_{7} ) q^{92} \) \( + ( -2 \beta_{1} - 22 \beta_{2} - 10 \beta_{3} + 23 \beta_{4} - 10 \beta_{5} - 22 \beta_{6} - 2 \beta_{7} ) q^{93} \) \( + ( -4 - 13 \beta_{1} + 22 \beta_{2} + 6 \beta_{3} - 2 \beta_{4} - 6 \beta_{5} - 18 \beta_{6} + 13 \beta_{7} ) q^{94} \) \( + ( -8 + 29 \beta_{1} - 25 \beta_{2} - 29 \beta_{3} + 58 \beta_{4} - 50 \beta_{6} - 15 \beta_{7} ) q^{95} \) \( + ( -4 \beta_{1} - 8 \beta_{2} + 8 \beta_{3} - 8 \beta_{4} - 4 \beta_{5} ) q^{96} \) \( + ( 1 - 12 \beta_{1} - \beta_{2} + 24 \beta_{3} - 8 \beta_{5} - 12 \beta_{6} + 16 \beta_{7} ) q^{97} \) \( + ( -20 + 40 \beta_{2} - 25 \beta_{3} + 8 \beta_{4} + 22 \beta_{5} + 48 \beta_{6} - 25 \beta_{7} ) q^{98} \) \( + ( -33 + 11 \beta_{2} + 22 \beta_{3} + 22 \beta_{4} - 33 \beta_{5} - 11 \beta_{6} + 33 \beta_{7} ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(8q \) \(\mathstrut -\mathstrut 2q^{3} \) \(\mathstrut +\mathstrut 4q^{4} \) \(\mathstrut +\mathstrut 2q^{5} \) \(\mathstrut -\mathstrut 20q^{6} \) \(\mathstrut -\mathstrut 30q^{7} \) \(\mathstrut -\mathstrut 4q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(8q \) \(\mathstrut -\mathstrut 2q^{3} \) \(\mathstrut +\mathstrut 4q^{4} \) \(\mathstrut +\mathstrut 2q^{5} \) \(\mathstrut -\mathstrut 20q^{6} \) \(\mathstrut -\mathstrut 30q^{7} \) \(\mathstrut -\mathstrut 4q^{9} \) \(\mathstrut -\mathstrut 4q^{11} \) \(\mathstrut +\mathstrut 24q^{12} \) \(\mathstrut +\mathstrut 30q^{13} \) \(\mathstrut +\mathstrut 16q^{14} \) \(\mathstrut +\mathstrut 42q^{15} \) \(\mathstrut -\mathstrut 8q^{16} \) \(\mathstrut +\mathstrut 30q^{17} \) \(\mathstrut +\mathstrut 40q^{18} \) \(\mathstrut -\mathstrut 30q^{19} \) \(\mathstrut -\mathstrut 4q^{20} \) \(\mathstrut +\mathstrut 24q^{22} \) \(\mathstrut -\mathstrut 104q^{23} \) \(\mathstrut -\mathstrut 40q^{24} \) \(\mathstrut -\mathstrut 12q^{25} \) \(\mathstrut -\mathstrut 96q^{26} \) \(\mathstrut -\mathstrut 26q^{27} \) \(\mathstrut -\mathstrut 40q^{28} \) \(\mathstrut -\mathstrut 10q^{29} \) \(\mathstrut -\mathstrut 60q^{30} \) \(\mathstrut +\mathstrut 46q^{31} \) \(\mathstrut -\mathstrut 14q^{33} \) \(\mathstrut +\mathstrut 112q^{34} \) \(\mathstrut +\mathstrut 70q^{35} \) \(\mathstrut -\mathstrut 12q^{36} \) \(\mathstrut +\mathstrut 6q^{37} \) \(\mathstrut +\mathstrut 108q^{38} \) \(\mathstrut +\mathstrut 130q^{39} \) \(\mathstrut +\mathstrut 80q^{40} \) \(\mathstrut +\mathstrut 250q^{41} \) \(\mathstrut +\mathstrut 56q^{42} \) \(\mathstrut -\mathstrut 12q^{44} \) \(\mathstrut -\mathstrut 136q^{45} \) \(\mathstrut -\mathstrut 160q^{46} \) \(\mathstrut -\mathstrut 54q^{47} \) \(\mathstrut -\mathstrut 8q^{48} \) \(\mathstrut -\mathstrut 144q^{49} \) \(\mathstrut -\mathstrut 80q^{50} \) \(\mathstrut -\mathstrut 30q^{51} \) \(\mathstrut -\mathstrut 40q^{52} \) \(\mathstrut -\mathstrut 274q^{53} \) \(\mathstrut -\mathstrut 26q^{55} \) \(\mathstrut +\mathstrut 48q^{56} \) \(\mathstrut -\mathstrut 130q^{57} \) \(\mathstrut +\mathstrut 64q^{58} \) \(\mathstrut +\mathstrut 50q^{59} \) \(\mathstrut +\mathstrut 116q^{60} \) \(\mathstrut +\mathstrut 50q^{61} \) \(\mathstrut +\mathstrut 20q^{62} \) \(\mathstrut -\mathstrut 20q^{63} \) \(\mathstrut +\mathstrut 16q^{64} \) \(\mathstrut -\mathstrut 136q^{66} \) \(\mathstrut +\mathstrut 112q^{67} \) \(\mathstrut +\mathstrut 60q^{68} \) \(\mathstrut +\mathstrut 76q^{69} \) \(\mathstrut +\mathstrut 4q^{70} \) \(\mathstrut +\mathstrut 54q^{71} \) \(\mathstrut -\mathstrut 80q^{72} \) \(\mathstrut -\mathstrut 70q^{73} \) \(\mathstrut -\mathstrut 40q^{74} \) \(\mathstrut +\mathstrut 318q^{75} \) \(\mathstrut +\mathstrut 266q^{77} \) \(\mathstrut +\mathstrut 104q^{78} \) \(\mathstrut +\mathstrut 370q^{79} \) \(\mathstrut +\mathstrut 48q^{80} \) \(\mathstrut +\mathstrut 180q^{81} \) \(\mathstrut -\mathstrut 96q^{82} \) \(\mathstrut -\mathstrut 150q^{83} \) \(\mathstrut -\mathstrut 120q^{84} \) \(\mathstrut -\mathstrut 330q^{85} \) \(\mathstrut -\mathstrut 72q^{86} \) \(\mathstrut +\mathstrut 72q^{88} \) \(\mathstrut +\mathstrut 24q^{89} \) \(\mathstrut +\mathstrut 160q^{90} \) \(\mathstrut -\mathstrut 294q^{91} \) \(\mathstrut -\mathstrut 112q^{92} \) \(\mathstrut -\mathstrut 134q^{93} \) \(\mathstrut -\mathstrut 20q^{94} \) \(\mathstrut -\mathstrut 330q^{95} \) \(\mathstrut -\mathstrut 18q^{97} \) \(\mathstrut -\mathstrut 308q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8}\mathstrut -\mathstrut \) \(2\) \(x^{6}\mathstrut +\mathstrut \) \(4\) \(x^{4}\mathstrut -\mathstrut \) \(8\) \(x^{2}\mathstrut +\mathstrut \) \(16\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} \)\(/2\)
\(\beta_{3}\)\(=\)\( \nu^{3} \)\(/2\)
\(\beta_{4}\)\(=\)\( \nu^{4} \)\(/4\)
\(\beta_{5}\)\(=\)\( \nu^{5} \)\(/4\)
\(\beta_{6}\)\(=\)\( \nu^{6} \)\(/8\)
\(\beta_{7}\)\(=\)\( \nu^{7} \)\(/8\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(2\) \(\beta_{2}\)
\(\nu^{3}\)\(=\)\(2\) \(\beta_{3}\)
\(\nu^{4}\)\(=\)\(4\) \(\beta_{4}\)
\(\nu^{5}\)\(=\)\(4\) \(\beta_{5}\)
\(\nu^{6}\)\(=\)\(8\) \(\beta_{6}\)
\(\nu^{7}\)\(=\)\(8\) \(\beta_{7}\)

Character Values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/22\mathbb{Z}\right)^\times\).

\(n\) \(13\)
\(\chi(n)\) \(\beta_{2}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
7.1
−0.831254 + 1.14412i
0.831254 1.14412i
−1.34500 0.437016i
1.34500 + 0.437016i
−1.34500 + 0.437016i
1.34500 0.437016i
−0.831254 1.14412i
0.831254 + 1.14412i
−0.831254 + 1.14412i −1.32276 + 4.07104i −0.618034 1.90211i 6.27955 4.56236i −3.55822 4.89747i −2.67724 + 0.869888i 2.68999 + 0.874032i −7.54250 5.47994i 10.9771i
7.2 0.831254 1.14412i −0.295274 + 0.908759i −0.618034 1.90211i −2.42545 + 1.76219i 0.794285 + 1.09324i −3.70473 + 1.20374i −2.68999 0.874032i 6.54250 + 4.75340i 4.23984i
13.1 −1.34500 0.437016i 2.48527 1.80565i 1.61803 + 1.17557i −0.399565 1.22973i −4.13178 + 1.34250i −6.48527 + 8.92621i −1.66251 2.28825i 0.135021 0.415553i 1.82860i
13.2 1.34500 + 0.437016i −1.86723 + 1.35662i 1.61803 + 1.17557i −2.45454 7.55429i −3.10429 + 1.00865i −2.13277 + 2.93550i 1.66251 + 2.28825i −1.13502 + 3.49324i 11.2332i
17.1 −1.34500 + 0.437016i 2.48527 + 1.80565i 1.61803 1.17557i −0.399565 + 1.22973i −4.13178 1.34250i −6.48527 8.92621i −1.66251 + 2.28825i 0.135021 + 0.415553i 1.82860i
17.2 1.34500 0.437016i −1.86723 1.35662i 1.61803 1.17557i −2.45454 + 7.55429i −3.10429 1.00865i −2.13277 2.93550i 1.66251 2.28825i −1.13502 3.49324i 11.2332i
19.1 −0.831254 1.14412i −1.32276 4.07104i −0.618034 + 1.90211i 6.27955 + 4.56236i −3.55822 + 4.89747i −2.67724 0.869888i 2.68999 0.874032i −7.54250 + 5.47994i 10.9771i
19.2 0.831254 + 1.14412i −0.295274 0.908759i −0.618034 + 1.90211i −2.42545 1.76219i 0.794285 1.09324i −3.70473 1.20374i −2.68999 + 0.874032i 6.54250 4.75340i 4.23984i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 19.2
Significant digits:
Format:

Inner twists

Char. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
11.d Odd 1 yes

Hecke kernels

There are no other newforms in \(S_{3}^{\mathrm{new}}(22, [\chi])\).