Properties

Label 17.4.d.a
Level 17
Weight 4
Character orbit 17.d
Analytic conductor 1.003
Analytic rank 0
Dimension 12
CM No
Inner twists 2

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) = \( 17 \)
Weight: \( k \) = \( 4 \)
Character orbit: \([\chi]\) = 17.d (of order \(8\) and degree \(4\))

Newform invariants

Self dual: No
Analytic conductor: \(1.0030324701\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(3\) over \(\Q(\zeta_{8})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Sato-Tate group: $\mathrm{SU}(2)[C_{8}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( -\beta_{3} - \beta_{10} ) q^{2} + \beta_{8} q^{3} + ( -\beta_{1} + \beta_{2} + \beta_{3} + \beta_{5} - \beta_{6} - \beta_{9} ) q^{4} + ( -2 - 2 \beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} + \beta_{6} - \beta_{7} + \beta_{9} - \beta_{11} ) q^{5} + ( 1 + \beta_{1} + \beta_{2} + \beta_{3} + 2 \beta_{4} - \beta_{6} + \beta_{7} - \beta_{8} + \beta_{9} + 2 \beta_{10} + \beta_{11} ) q^{6} + ( -2 + \beta_{1} + 2 \beta_{3} - 2 \beta_{4} + 2 \beta_{6} - 2 \beta_{8} + 2 \beta_{10} - \beta_{11} ) q^{7} + ( 2 + 2 \beta_{1} - 3 \beta_{2} + 2 \beta_{4} - 3 \beta_{5} + \beta_{6} + \beta_{7} - 3 \beta_{8} + 2 \beta_{9} + 3 \beta_{11} ) q^{8} + ( -4 + 2 \beta_{1} + 5 \beta_{2} + 2 \beta_{4} + \beta_{5} + 2 \beta_{6} + 2 \beta_{7} + \beta_{8} - 4 \beta_{9} ) q^{9} +O(q^{10})\) \( q + ( -\beta_{3} - \beta_{10} ) q^{2} + \beta_{8} q^{3} + ( -\beta_{1} + \beta_{2} + \beta_{3} + \beta_{5} - \beta_{6} - \beta_{9} ) q^{4} + ( -2 - 2 \beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} + \beta_{6} - \beta_{7} + \beta_{9} - \beta_{11} ) q^{5} + ( 1 + \beta_{1} + \beta_{2} + \beta_{3} + 2 \beta_{4} - \beta_{6} + \beta_{7} - \beta_{8} + \beta_{9} + 2 \beta_{10} + \beta_{11} ) q^{6} + ( -2 + \beta_{1} + 2 \beta_{3} - 2 \beta_{4} + 2 \beta_{6} - 2 \beta_{8} + 2 \beta_{10} - \beta_{11} ) q^{7} + ( 2 + 2 \beta_{1} - 3 \beta_{2} + 2 \beta_{4} - 3 \beta_{5} + \beta_{6} + \beta_{7} - 3 \beta_{8} + 2 \beta_{9} + 3 \beta_{11} ) q^{8} + ( -4 + 2 \beta_{1} + 5 \beta_{2} + 2 \beta_{4} + \beta_{5} + 2 \beta_{6} + 2 \beta_{7} + \beta_{8} - 4 \beta_{9} ) q^{9} + ( -10 - 4 \beta_{1} + 3 \beta_{2} + 10 \beta_{3} - 4 \beta_{4} + 3 \beta_{5} - 3 \beta_{6} + 3 \beta_{8} - 3 \beta_{9} + 4 \beta_{10} + 4 \beta_{11} ) q^{10} + ( 2 - 7 \beta_{1} - 8 \beta_{2} + 2 \beta_{3} + \beta_{4} - \beta_{6} - 5 \beta_{7} - \beta_{8} - 8 \beta_{9} + \beta_{10} - 7 \beta_{11} ) q^{11} + ( 5 + 4 \beta_{1} + 5 \beta_{2} - 19 \beta_{3} - \beta_{4} - 3 \beta_{5} - \beta_{6} - 3 \beta_{7} + 19 \beta_{9} - 4 \beta_{10} - \beta_{11} ) q^{12} + ( 6 \beta_{1} - 11 \beta_{2} - 11 \beta_{3} + 3 \beta_{5} - 3 \beta_{6} + 4 \beta_{7} + 4 \beta_{8} + 2 \beta_{9} - 6 \beta_{10} - 6 \beta_{11} ) q^{13} + ( -7 - 4 \beta_{1} + 7 \beta_{2} + 11 \beta_{3} - 2 \beta_{4} + \beta_{5} - \beta_{7} + 6 \beta_{8} + 11 \beta_{9} - 4 \beta_{10} + 2 \beta_{11} ) q^{14} + ( 20 - 5 \beta_{1} + 12 \beta_{3} + 5 \beta_{4} - 2 \beta_{6} + 2 \beta_{7} - 20 \beta_{9} + 2 \beta_{10} ) q^{15} + ( 21 + 9 \beta_{2} - 9 \beta_{3} - 5 \beta_{4} + 5 \beta_{5} + 5 \beta_{6} + 2 \beta_{7} - 2 \beta_{8} - 8 \beta_{10} + 8 \beta_{11} ) q^{16} + ( 12 - 4 \beta_{1} - 6 \beta_{2} - 10 \beta_{3} + \beta_{4} - 8 \beta_{5} + 4 \beta_{6} - 2 \beta_{7} + \beta_{8} - 31 \beta_{9} - 15 \beta_{10} + 8 \beta_{11} ) q^{17} + ( 7 + 16 \beta_{2} - 16 \beta_{3} + 13 \beta_{4} + \beta_{5} + \beta_{6} - 2 \beta_{7} + 2 \beta_{8} - \beta_{10} + \beta_{11} ) q^{18} + ( -14 + 19 \beta_{1} - 16 \beta_{3} - 19 \beta_{4} + 3 \beta_{5} - 3 \beta_{6} + 3 \beta_{7} - 3 \beta_{8} + 14 \beta_{9} + 14 \beta_{10} ) q^{19} + ( 38 + 8 \beta_{1} - 38 \beta_{2} + 29 \beta_{3} + 12 \beta_{4} - 6 \beta_{5} + 6 \beta_{7} - 9 \beta_{8} + 29 \beta_{9} + 8 \beta_{10} - 12 \beta_{11} ) q^{20} + ( -16 \beta_{1} + 40 \beta_{2} + 40 \beta_{3} + \beta_{5} - \beta_{6} - 5 \beta_{7} - 5 \beta_{8} + 10 \beta_{9} - 5 \beta_{10} - 5 \beta_{11} ) q^{21} + ( -65 - 7 \beta_{1} - 65 \beta_{2} - 13 \beta_{3} - 16 \beta_{4} - 4 \beta_{5} - \beta_{6} - 4 \beta_{7} + 13 \beta_{9} + 7 \beta_{10} - 16 \beta_{11} ) q^{22} + ( -20 + 9 \beta_{1} + 24 \beta_{2} - 20 \beta_{3} + 24 \beta_{4} + 24 \beta_{9} + 24 \beta_{10} + 9 \beta_{11} ) q^{23} + ( -21 - 4 \beta_{1} + 47 \beta_{2} + 21 \beta_{3} - 13 \beta_{4} + 5 \beta_{5} + \beta_{6} - \beta_{8} - 47 \beta_{9} + 13 \beta_{10} + 4 \beta_{11} ) q^{24} + ( -27 - 7 \beta_{1} - 47 \beta_{2} - 7 \beta_{4} + 10 \beta_{5} - 3 \beta_{6} - 3 \beta_{7} + 10 \beta_{8} - 27 \beta_{9} + 22 \beta_{11} ) q^{25} + ( -57 + 4 \beta_{1} + 64 \beta_{2} + 4 \beta_{4} + 2 \beta_{5} - 3 \beta_{6} - 3 \beta_{7} + 2 \beta_{8} - 57 \beta_{9} - 2 \beta_{11} ) q^{26} + ( -60 - 4 \beta_{1} - 26 \beta_{2} + 60 \beta_{3} - 18 \beta_{5} + 11 \beta_{6} - 11 \beta_{8} + 26 \beta_{9} + 4 \beta_{11} ) q^{27} + ( 35 - 2 \beta_{1} - 31 \beta_{2} + 35 \beta_{3} + 12 \beta_{4} + 3 \beta_{6} + 12 \beta_{7} + 3 \beta_{8} - 31 \beta_{9} + 12 \beta_{10} - 2 \beta_{11} ) q^{28} + ( 51 + 10 \beta_{1} + 51 \beta_{2} - 88 \beta_{3} - 3 \beta_{4} + 14 \beta_{5} - 9 \beta_{6} + 14 \beta_{7} + 88 \beta_{9} - 10 \beta_{10} - 3 \beta_{11} ) q^{29} + ( 20 \beta_{1} - 62 \beta_{2} - 62 \beta_{3} - 11 \beta_{5} + 11 \beta_{6} - 7 \beta_{7} - 7 \beta_{8} + 42 \beta_{9} - 22 \beta_{10} - 22 \beta_{11} ) q^{30} + ( 6 + 3 \beta_{1} - 6 \beta_{2} + 74 \beta_{3} - 4 \beta_{4} - 6 \beta_{5} + 6 \beta_{7} - 14 \beta_{8} + 74 \beta_{9} + 3 \beta_{10} + 4 \beta_{11} ) q^{31} + ( 72 - 6 \beta_{1} - 3 \beta_{3} + 6 \beta_{4} - 2 \beta_{5} + 10 \beta_{6} - 10 \beta_{7} + 2 \beta_{8} - 72 \beta_{9} - 15 \beta_{10} ) q^{32} + ( 130 + 56 \beta_{2} - 56 \beta_{3} + 6 \beta_{4} - 11 \beta_{5} - 11 \beta_{6} - 16 \beta_{7} + 16 \beta_{8} - 12 \beta_{10} + 12 \beta_{11} ) q^{33} + ( 56 - 13 \beta_{1} - 45 \beta_{2} + 10 \beta_{3} - 18 \beta_{4} + 25 \beta_{5} - 21 \beta_{6} + 2 \beta_{7} - \beta_{8} - 122 \beta_{9} - 2 \beta_{10} - 8 \beta_{11} ) q^{34} + ( -44 + 48 \beta_{2} - 48 \beta_{3} - 12 \beta_{4} + 6 \beta_{5} + 6 \beta_{6} - 2 \beta_{7} + 2 \beta_{8} - 11 \beta_{10} + 11 \beta_{11} ) q^{35} + ( 45 - 5 \beta_{1} - 67 \beta_{3} + 5 \beta_{4} - 13 \beta_{5} - \beta_{6} + \beta_{7} + 13 \beta_{8} - 45 \beta_{9} - \beta_{10} ) q^{36} + ( 98 - 26 \beta_{1} - 98 \beta_{2} + 73 \beta_{3} + 15 \beta_{4} + 12 \beta_{5} - 12 \beta_{7} + 9 \beta_{8} + 73 \beta_{9} - 26 \beta_{10} - 15 \beta_{11} ) q^{37} + ( -10 \beta_{1} + 146 \beta_{2} + 146 \beta_{3} - 4 \beta_{5} + 4 \beta_{6} + 16 \beta_{7} + 16 \beta_{8} + 58 \beta_{9} + 2 \beta_{10} + 2 \beta_{11} ) q^{38} + ( -86 - 4 \beta_{1} - 86 \beta_{2} - 32 \beta_{3} + 14 \beta_{4} + 26 \beta_{5} - 4 \beta_{6} + 26 \beta_{7} + 32 \beta_{9} + 4 \beta_{10} + 14 \beta_{11} ) q^{39} + ( -139 + 6 \beta_{1} + 20 \beta_{2} - 139 \beta_{3} - 30 \beta_{4} + 11 \beta_{6} - 21 \beta_{7} + 11 \beta_{8} + 20 \beta_{9} - 30 \beta_{10} + 6 \beta_{11} ) q^{40} + ( -1 + 21 \beta_{1} + 44 \beta_{2} + \beta_{3} - 7 \beta_{4} - 12 \beta_{5} - 23 \beta_{6} + 23 \beta_{8} - 44 \beta_{9} + 7 \beta_{10} - 21 \beta_{11} ) q^{41} + ( 6 + 26 \beta_{1} - 120 \beta_{2} + 26 \beta_{4} - 8 \beta_{5} - 4 \beta_{6} - 4 \beta_{7} - 8 \beta_{8} + 6 \beta_{9} - 4 \beta_{11} ) q^{42} + ( -136 + 6 \beta_{1} + 116 \beta_{2} + 6 \beta_{4} - 28 \beta_{5} - \beta_{6} - \beta_{7} - 28 \beta_{8} - 136 \beta_{9} - 18 \beta_{11} ) q^{43} + ( -115 - 7 \beta_{1} - 51 \beta_{2} + 115 \beta_{3} - 22 \beta_{4} + 33 \beta_{5} - 4 \beta_{6} + 4 \beta_{8} + 51 \beta_{9} + 22 \beta_{10} + 7 \beta_{11} ) q^{44} + ( -11 - 10 \beta_{1} - 54 \beta_{2} - 11 \beta_{3} - 13 \beta_{4} - 3 \beta_{6} + 9 \beta_{7} - 3 \beta_{8} - 54 \beta_{9} - 13 \beta_{10} - 10 \beta_{11} ) q^{45} + ( 63 - 20 \beta_{1} + 63 \beta_{2} - 187 \beta_{3} + 24 \beta_{4} - 15 \beta_{5} + 48 \beta_{6} - 15 \beta_{7} + 187 \beta_{9} + 20 \beta_{10} + 24 \beta_{11} ) q^{46} + ( -30 \beta_{1} + 6 \beta_{5} - 6 \beta_{6} - 22 \beta_{7} - 22 \beta_{8} - 112 \beta_{9} + 52 \beta_{10} + 52 \beta_{11} ) q^{47} + ( -51 + 5 \beta_{1} + 51 \beta_{2} + 149 \beta_{3} + 24 \beta_{4} + 3 \beta_{5} - 3 \beta_{7} - 19 \beta_{8} + 149 \beta_{9} + 5 \beta_{10} - 24 \beta_{11} ) q^{48} + ( 58 + 43 \beta_{1} + 125 \beta_{3} - 43 \beta_{4} - \beta_{5} - 5 \beta_{6} + 5 \beta_{7} + \beta_{8} - 58 \beta_{9} + 34 \beta_{10} ) q^{49} + ( 157 - 111 \beta_{2} + 111 \beta_{3} - 13 \beta_{4} - 17 \beta_{5} - 17 \beta_{6} + 52 \beta_{7} - 52 \beta_{8} + 46 \beta_{10} - 46 \beta_{11} ) q^{50} + ( 34 + 34 \beta_{1} - 34 \beta_{2} + 68 \beta_{3} + 34 \beta_{4} + 34 \beta_{6} + 17 \beta_{7} - 136 \beta_{9} + 51 \beta_{10} + 17 \beta_{11} ) q^{51} + ( 22 + 59 \beta_{2} - 59 \beta_{3} + 18 \beta_{4} - 30 \beta_{5} - 30 \beta_{6} + 25 \beta_{7} - 25 \beta_{8} + 20 \beta_{10} - 20 \beta_{11} ) q^{52} + ( 29 - 39 \beta_{1} - 236 \beta_{3} + 39 \beta_{4} + 13 \beta_{5} + 40 \beta_{6} - 40 \beta_{7} - 13 \beta_{8} - 29 \beta_{9} - 58 \beta_{10} ) q^{53} + ( -8 + 45 \beta_{1} + 8 \beta_{2} + 60 \beta_{3} - 73 \beta_{4} + 25 \beta_{5} - 25 \beta_{7} + 32 \beta_{8} + 60 \beta_{9} + 45 \beta_{10} + 73 \beta_{11} ) q^{54} + ( 66 \beta_{1} + 122 \beta_{2} + 122 \beta_{3} - 22 \beta_{5} + 22 \beta_{6} - 6 \beta_{7} - 6 \beta_{8} + 132 \beta_{9} + 15 \beta_{10} + 15 \beta_{11} ) q^{55} + ( -127 + 40 \beta_{1} - 127 \beta_{2} - 101 \beta_{3} + 22 \beta_{4} - 15 \beta_{5} - 18 \beta_{6} - 15 \beta_{7} + 101 \beta_{9} - 40 \beta_{10} + 22 \beta_{11} ) q^{56} + ( -56 - 71 \beta_{1} + 14 \beta_{2} - 56 \beta_{3} - 21 \beta_{4} - 6 \beta_{6} + 48 \beta_{7} - 6 \beta_{8} + 14 \beta_{9} - 21 \beta_{10} - 71 \beta_{11} ) q^{57} + ( 39 - 56 \beta_{1} + 172 \beta_{2} - 39 \beta_{3} + 84 \beta_{4} - 17 \beta_{5} + 10 \beta_{6} - 10 \beta_{8} - 172 \beta_{9} - 84 \beta_{10} + 56 \beta_{11} ) q^{58} + ( -94 - 54 \beta_{1} - 356 \beta_{2} - 54 \beta_{4} + 14 \beta_{5} + 15 \beta_{6} + 15 \beta_{7} + 14 \beta_{8} - 94 \beta_{9} - 110 \beta_{11} ) q^{59} + ( -112 - 54 \beta_{1} + 158 \beta_{2} - 54 \beta_{4} + 55 \beta_{5} - 33 \beta_{6} - 33 \beta_{7} + 55 \beta_{8} - 112 \beta_{9} + 56 \beta_{11} ) q^{60} + ( -54 + 68 \beta_{1} - 51 \beta_{2} + 54 \beta_{3} + 85 \beta_{4} - \beta_{5} - 29 \beta_{6} + 29 \beta_{8} + 51 \beta_{9} - 85 \beta_{10} - 68 \beta_{11} ) q^{61} + ( 13 + 78 \beta_{1} + 89 \beta_{2} + 13 \beta_{3} - 40 \beta_{4} + 19 \beta_{6} - 18 \beta_{7} + 19 \beta_{8} + 89 \beta_{9} - 40 \beta_{10} + 78 \beta_{11} ) q^{62} + ( 106 - 8 \beta_{1} + 106 \beta_{2} - 54 \beta_{3} - 35 \beta_{4} - 18 \beta_{5} - 42 \beta_{6} - 18 \beta_{7} + 54 \beta_{9} + 8 \beta_{10} - 35 \beta_{11} ) q^{63} + ( -79 \beta_{1} - 29 \beta_{2} - 29 \beta_{3} + 15 \beta_{5} - 15 \beta_{6} + 44 \beta_{7} + 44 \beta_{8} + 41 \beta_{9} + 8 \beta_{10} + 8 \beta_{11} ) q^{64} + ( 69 - 12 \beta_{1} - 69 \beta_{2} + 23 \beta_{3} - 36 \beta_{4} + 23 \beta_{5} - 23 \beta_{7} + 48 \beta_{8} + 23 \beta_{9} - 12 \beta_{10} + 36 \beta_{11} ) q^{65} + ( 124 - 39 \beta_{1} - 100 \beta_{3} + 39 \beta_{4} + 28 \beta_{5} - 27 \beta_{6} + 27 \beta_{7} - 28 \beta_{8} - 124 \beta_{9} - 44 \beta_{10} ) q^{66} + ( 144 + 128 \beta_{2} - 128 \beta_{3} + 50 \beta_{4} + 25 \beta_{5} + 25 \beta_{6} - 66 \beta_{7} + 66 \beta_{8} + 35 \beta_{10} - 35 \beta_{11} ) q^{67} + ( 175 + 38 \beta_{1} - 181 \beta_{2} + 27 \beta_{3} + 16 \beta_{4} - 43 \beta_{5} - 4 \beta_{6} - 32 \beta_{7} - 18 \beta_{8} + 82 \beta_{9} - 2 \beta_{10} - 93 \beta_{11} ) q^{68} + ( -66 - 18 \beta_{2} + 18 \beta_{3} - 114 \beta_{4} + 9 \beta_{5} + 9 \beta_{6} + 5 \beta_{7} - 5 \beta_{8} + 3 \beta_{10} - 3 \beta_{11} ) q^{69} + ( 160 - 68 \beta_{1} + 154 \beta_{3} + 68 \beta_{4} + 13 \beta_{5} - 33 \beta_{6} + 33 \beta_{7} - 13 \beta_{8} - 160 \beta_{9} + 40 \beta_{10} ) q^{70} + ( -16 - 43 \beta_{1} + 16 \beta_{2} + 50 \beta_{3} - 44 \beta_{4} - 42 \beta_{5} + 42 \beta_{7} - 14 \beta_{8} + 50 \beta_{9} - 43 \beta_{10} + 44 \beta_{11} ) q^{71} + ( 67 \beta_{1} + 70 \beta_{2} + 70 \beta_{3} + 23 \beta_{5} - 23 \beta_{6} + 2 \beta_{7} + 2 \beta_{8} + 19 \beta_{9} + \beta_{10} + \beta_{11} ) q^{72} + ( 139 - 5 \beta_{1} + 139 \beta_{2} - 42 \beta_{3} + 73 \beta_{4} - 41 \beta_{5} + 70 \beta_{6} - 41 \beta_{7} + 42 \beta_{9} + 5 \beta_{10} + 73 \beta_{11} ) q^{73} + ( -198 + 46 \beta_{1} - 109 \beta_{2} - 198 \beta_{3} - 68 \beta_{4} - 32 \beta_{6} + 3 \beta_{7} - 32 \beta_{8} - 109 \beta_{9} - 68 \beta_{10} + 46 \beta_{11} ) q^{74} + ( 30 + 77 \beta_{1} - 190 \beta_{2} - 30 \beta_{3} + 13 \beta_{4} + 41 \beta_{5} + 50 \beta_{6} - 50 \beta_{8} + 190 \beta_{9} - 13 \beta_{10} - 77 \beta_{11} ) q^{75} + ( 92 + 38 \beta_{1} + 142 \beta_{2} + 38 \beta_{4} - 44 \beta_{5} + 22 \beta_{6} + 22 \beta_{7} - 44 \beta_{8} + 92 \beta_{9} - 6 \beta_{11} ) q^{76} + ( -88 + 35 \beta_{1} + 14 \beta_{2} + 35 \beta_{4} - 9 \beta_{5} + 67 \beta_{6} + 67 \beta_{7} - 9 \beta_{8} - 88 \beta_{9} + 32 \beta_{11} ) q^{77} + ( 64 + 2 \beta_{1} - 22 \beta_{2} - 64 \beta_{3} - 12 \beta_{4} - 64 \beta_{5} + 48 \beta_{6} - 48 \beta_{8} + 22 \beta_{9} + 12 \beta_{10} - 2 \beta_{11} ) q^{78} + ( -42 + 27 \beta_{1} + 6 \beta_{2} - 42 \beta_{3} + 6 \beta_{4} - 64 \beta_{6} - 24 \beta_{7} - 64 \beta_{8} + 6 \beta_{9} + 6 \beta_{10} + 27 \beta_{11} ) q^{79} + ( -139 - 96 \beta_{1} - 139 \beta_{2} + 132 \beta_{3} - 32 \beta_{4} + 20 \beta_{5} - 31 \beta_{6} + 20 \beta_{7} - 132 \beta_{9} + 96 \beta_{10} - 32 \beta_{11} ) q^{80} + ( -28 \beta_{1} + 62 \beta_{2} + 62 \beta_{3} - 12 \beta_{5} + 12 \beta_{6} - 7 \beta_{7} - 7 \beta_{8} - 165 \beta_{9} + 86 \beta_{10} + 86 \beta_{11} ) q^{81} + ( -150 + 82 \beta_{1} + 150 \beta_{2} + 87 \beta_{3} + 43 \beta_{4} + 3 \beta_{5} - 3 \beta_{7} + 8 \beta_{8} + 87 \beta_{9} + 82 \beta_{10} - 43 \beta_{11} ) q^{82} + ( -118 - 16 \beta_{1} + 272 \beta_{3} + 16 \beta_{4} - 19 \beta_{5} - 12 \beta_{6} + 12 \beta_{7} + 19 \beta_{8} + 118 \beta_{9} + 94 \beta_{10} ) q^{83} + ( -308 - 110 \beta_{2} + 110 \beta_{3} - 32 \beta_{4} + 74 \beta_{5} + 74 \beta_{6} - 34 \beta_{7} + 34 \beta_{8} - 42 \beta_{10} + 42 \beta_{11} ) q^{84} + ( -279 - 60 \beta_{1} + 165 \beta_{2} + 122 \beta_{3} - 53 \beta_{4} - \beta_{5} - 42 \beta_{6} + 21 \beta_{7} + 15 \beta_{8} - 108 \beta_{9} + 98 \beta_{10} + 69 \beta_{11} ) q^{85} + ( -98 - 72 \beta_{2} + 72 \beta_{3} + 2 \beta_{4} + 34 \beta_{5} + 34 \beta_{6} - 79 \beta_{7} + 79 \beta_{8} + 111 \beta_{10} - 111 \beta_{11} ) q^{86} + ( -266 - \beta_{1} - 72 \beta_{3} + \beta_{4} - 20 \beta_{5} - 96 \beta_{6} + 96 \beta_{7} + 20 \beta_{8} + 266 \beta_{9} - 22 \beta_{10} ) q^{87} + ( -37 - 34 \beta_{1} + 37 \beta_{2} - 269 \beta_{3} + 75 \beta_{4} - 34 \beta_{5} + 34 \beta_{7} - 63 \beta_{8} - 269 \beta_{9} - 34 \beta_{10} - 75 \beta_{11} ) q^{88} + ( 96 \beta_{1} - 249 \beta_{2} - 249 \beta_{3} + 21 \beta_{5} - 21 \beta_{6} - 53 \beta_{7} - 53 \beta_{8} + 194 \beta_{9} + 7 \beta_{10} + 7 \beta_{11} ) q^{89} + ( -108 + 10 \beta_{1} - 108 \beta_{2} + 109 \beta_{3} - 54 \beta_{4} - 9 \beta_{5} - 11 \beta_{6} - 9 \beta_{7} - 109 \beta_{9} - 10 \beta_{10} - 54 \beta_{11} ) q^{90} + ( 242 - 34 \beta_{1} + 140 \beta_{2} + 242 \beta_{3} - 26 \beta_{4} + 22 \beta_{6} - 102 \beta_{7} + 22 \beta_{8} + 140 \beta_{9} - 26 \beta_{10} - 34 \beta_{11} ) q^{91} + ( 77 - 106 \beta_{1} + 191 \beta_{2} - 77 \beta_{3} - 6 \beta_{4} + 38 \beta_{5} - 19 \beta_{6} + 19 \beta_{8} - 191 \beta_{9} + 6 \beta_{10} + 106 \beta_{11} ) q^{92} + ( -82 - 33 \beta_{1} + 246 \beta_{2} - 33 \beta_{4} - 3 \beta_{5} - 77 \beta_{6} - 77 \beta_{7} - 3 \beta_{8} - 82 \beta_{9} + 2 \beta_{11} ) q^{93} + ( 402 - 60 \beta_{1} - 512 \beta_{2} - 60 \beta_{4} - 72 \beta_{5} + 58 \beta_{6} + 58 \beta_{7} - 72 \beta_{8} + 402 \beta_{9} - 180 \beta_{11} ) q^{94} + ( 244 - 55 \beta_{1} + 64 \beta_{2} - 244 \beta_{3} + 27 \beta_{4} + 80 \beta_{5} - 28 \beta_{6} + 28 \beta_{8} - 64 \beta_{9} - 27 \beta_{10} + 55 \beta_{11} ) q^{95} + ( 187 + 17 \beta_{1} + 15 \beta_{2} + 187 \beta_{3} + 48 \beta_{4} + 53 \beta_{6} - 21 \beta_{7} + 53 \beta_{8} + 15 \beta_{9} + 48 \beta_{10} + 17 \beta_{11} ) q^{96} + ( 22 + 104 \beta_{1} + 22 \beta_{2} + 331 \beta_{3} + 108 \beta_{4} + 75 \beta_{5} - 28 \beta_{6} + 75 \beta_{7} - 331 \beta_{9} - 104 \beta_{10} + 108 \beta_{11} ) q^{97} + ( 147 \beta_{1} + 254 \beta_{2} + 254 \beta_{3} + 38 \beta_{7} + 38 \beta_{8} + 323 \beta_{9} - 60 \beta_{10} - 60 \beta_{11} ) q^{98} + ( 272 - 93 \beta_{1} - 272 \beta_{2} - 536 \beta_{3} - 17 \beta_{4} - 72 \beta_{5} + 72 \beta_{7} + 47 \beta_{8} - 536 \beta_{9} - 93 \beta_{10} + 17 \beta_{11} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12q - 4q^{2} - 4q^{3} - 20q^{5} + 20q^{6} - 4q^{7} + 28q^{8} - 64q^{9} + O(q^{10}) \) \( 12q - 4q^{2} - 4q^{3} - 20q^{5} + 20q^{6} - 4q^{7} + 28q^{8} - 64q^{9} - 116q^{10} + 40q^{11} + 56q^{12} - 132q^{14} + 244q^{15} + 184q^{16} + 52q^{17} - 12q^{19} + 572q^{20} - 620q^{22} - 276q^{23} - 184q^{24} - 464q^{25} - 708q^{26} - 664q^{27} + 452q^{28} + 632q^{29} + 188q^{31} + 700q^{32} + 1400q^{33} + 764q^{34} - 632q^{35} + 524q^{36} + 940q^{37} - 1112q^{39} - 1864q^{40} + 176q^{41} + 48q^{42} - 1360q^{43} - 1364q^{44} - 32q^{45} + 452q^{46} - 540q^{48} + 1044q^{49} + 2856q^{50} + 340q^{51} + 792q^{52} - 360q^{53} - 244q^{54} - 1788q^{56} - 148q^{57} - 360q^{58} - 584q^{59} - 1792q^{60} - 1052q^{61} - 380q^{62} + 1752q^{63} + 404q^{65} + 1372q^{66} + 1080q^{67} + 2532q^{68} - 344q^{69} + 2072q^{70} + 28q^{71} + 824q^{73} - 2292q^{74} + 400q^{75} + 1328q^{76} - 1252q^{77} + 1128q^{78} - 196q^{79} - 904q^{80} - 1528q^{82} - 1008q^{83} - 4768q^{84} - 2824q^{85} - 1200q^{86} - 2516q^{87} - 56q^{88} - 860q^{90} + 2456q^{91} + 396q^{92} - 836q^{93} + 6360q^{94} + 2172q^{95} + 1668q^{96} - 904q^{97} + 3280q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{12} + 54 x^{10} + 1085 x^{8} + 9836 x^{6} + 38276 x^{4} + 49664 x^{2} + 16384\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( -\nu^{11} - 408 \nu^{10} + 10 \nu^{9} - 17680 \nu^{8} - 1725 \nu^{7} - 268600 \nu^{6} - 73516 \nu^{5} - 1728288 \nu^{4} - 729732 \nu^{3} - 4623456 \nu^{2} - 1741056 \nu - 2889728 \)\()/1392640\)
\(\beta_{3}\)\(=\)\((\)\( -\nu^{11} + 408 \nu^{10} + 10 \nu^{9} + 17680 \nu^{8} - 1725 \nu^{7} + 268600 \nu^{6} - 73516 \nu^{5} + 1728288 \nu^{4} - 729732 \nu^{3} + 4623456 \nu^{2} - 1741056 \nu + 2889728 \)\()/1392640\)
\(\beta_{4}\)\(=\)\((\)\( \nu^{10} + 41 \nu^{8} + 569 \nu^{6} + 3051 \nu^{4} + 5498 \nu^{2} + 2432 \)\()/544\)
\(\beta_{5}\)\(=\)\((\)\( 241 \nu^{11} - 280 \nu^{10} + 19350 \nu^{9} + 2800 \nu^{8} + 502765 \nu^{7} + 387400 \nu^{6} + 5292396 \nu^{5} + 5701600 \nu^{4} + 20629572 \nu^{3} + 23023520 \nu^{2} + 17121536 \nu + 12462080 \)\()/1392640\)
\(\beta_{6}\)\(=\)\((\)\( -241 \nu^{11} - 280 \nu^{10} - 19350 \nu^{9} + 2800 \nu^{8} - 502765 \nu^{7} + 387400 \nu^{6} - 5292396 \nu^{5} + 5701600 \nu^{4} - 20629572 \nu^{3} + 23023520 \nu^{2} - 17121536 \nu + 12462080 \)\()/1392640\)
\(\beta_{7}\)\(=\)\((\)\( -325 \nu^{11} + 872 \nu^{10} - 18510 \nu^{9} + 34800 \nu^{8} - 386545 \nu^{7} + 459720 \nu^{6} - 3460060 \nu^{5} + 2176992 \nu^{4} - 10989460 \nu^{3} + 1717664 \nu^{2} - 1301760 \nu - 5347328 \)\()/1392640\)
\(\beta_{8}\)\(=\)\((\)\(-325 \nu^{11} - 872 \nu^{10} - 18510 \nu^{9} - 34800 \nu^{8} - 386545 \nu^{7} - 459720 \nu^{6} - 3460060 \nu^{5} - 2176992 \nu^{4} - 10989460 \nu^{3} - 1717664 \nu^{2} - 1301760 \nu + 5347328\)\()/1392640\)
\(\beta_{9}\)\(=\)\((\)\( 19 \nu^{11} + 898 \nu^{9} + 15367 \nu^{7} + 114052 \nu^{5} + 336716 \nu^{3} + 239872 \nu \)\()/69632\)
\(\beta_{10}\)\(=\)\((\)\( -51 \nu^{11} + 8 \nu^{10} - 2210 \nu^{9} - 80 \nu^{8} - 33575 \nu^{7} - 7960 \nu^{6} - 216036 \nu^{5} - 86432 \nu^{4} - 577932 \nu^{3} - 211424 \nu^{2} - 361216 \nu + 2048 \)\()/174080\)
\(\beta_{11}\)\(=\)\((\)\( -51 \nu^{11} - 8 \nu^{10} - 2210 \nu^{9} + 80 \nu^{8} - 33575 \nu^{7} + 7960 \nu^{6} - 216036 \nu^{5} + 86432 \nu^{4} - 577932 \nu^{3} + 211424 \nu^{2} - 361216 \nu - 2048 \)\()/174080\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{8} - \beta_{7} + \beta_{4} - \beta_{3} + \beta_{2} - 8\)
\(\nu^{3}\)\(=\)\(2 \beta_{11} + 2 \beta_{10} + 6 \beta_{9} + \beta_{8} + \beta_{7} - \beta_{3} - \beta_{2} - 13 \beta_{1}\)
\(\nu^{4}\)\(=\)\(-16 \beta_{8} + 16 \beta_{7} - \beta_{6} - \beta_{5} - 21 \beta_{4} + 31 \beta_{3} - 31 \beta_{2} + 106\)
\(\nu^{5}\)\(=\)\(-50 \beta_{11} - 50 \beta_{10} - 138 \beta_{9} - 15 \beta_{8} - 15 \beta_{7} - 4 \beta_{6} + 4 \beta_{5} + 19 \beta_{3} + 19 \beta_{2} + 189 \beta_{1}\)
\(\nu^{6}\)\(=\)\(-28 \beta_{11} + 28 \beta_{10} + 250 \beta_{8} - 250 \beta_{7} + 31 \beta_{6} + 31 \beta_{5} + 373 \beta_{4} - 619 \beta_{3} + 619 \beta_{2} - 1574\)
\(\nu^{7}\)\(=\)\(990 \beta_{11} + 990 \beta_{10} + 2602 \beta_{9} + 191 \beta_{8} + 191 \beta_{7} + 120 \beta_{6} - 120 \beta_{5} - 535 \beta_{3} - 535 \beta_{2} - 2885 \beta_{1}\)
\(\nu^{8}\)\(=\)\(1072 \beta_{11} - 1072 \beta_{10} - 3946 \beta_{8} + 3946 \beta_{7} - 679 \beta_{6} - 679 \beta_{5} - 6349 \beta_{4} + 11187 \beta_{3} - 11187 \beta_{2} + 24438\)
\(\nu^{9}\)\(=\)\(-18242 \beta_{11} - 18242 \beta_{10} - 46402 \beta_{9} - 2195 \beta_{8} - 2195 \beta_{7} - 2796 \beta_{6} + 2796 \beta_{5} + 13567 \beta_{3} + 13567 \beta_{2} + 45213 \beta_{1}\)
\(\nu^{10}\)\(=\)\(-28020 \beta_{11} + 28020 \beta_{10} + 62854 \beta_{8} - 62854 \beta_{7} + 13251 \beta_{6} + 13251 \beta_{5} + 107189 \beta_{4} - 195539 \beta_{3} + 195539 \beta_{2} - 388206\)
\(\nu^{11}\)\(=\)\(326166 \beta_{11} + 326166 \beta_{10} + 814346 \beta_{9} + 21583 \beta_{8} + 21583 \beta_{7} + 59104 \beta_{6} - 59104 \beta_{5} - 304847 \beta_{3} - 304847 \beta_{2} - 720309 \beta_{1}\)

Character Values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/17\mathbb{Z}\right)^\times\).

\(n\) \(3\)
\(\chi(n)\) \(-\beta_{2}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2.1
3.86166i
0.705468i
4.15292i
3.68604i
1.22788i
2.49971i
3.86166i
0.705468i
4.15292i
3.68604i
1.22788i
2.49971i
−3.43772 3.43772i −4.67995 + 1.93850i 15.6358i −7.10390 17.1503i 22.7523 + 9.42432i 5.36561 12.9537i 26.2496 26.2496i −0.947753 + 0.947753i −34.5367 + 83.3791i
2.2 −1.20595 1.20595i 4.10553 1.70057i 5.09138i 2.60601 + 6.29147i −7.00185 2.90026i −5.31013 + 12.8198i −15.7875 + 15.7875i −5.12843 + 5.12843i 4.44447 10.7299i
2.3 2.22945 + 2.22945i −1.83980 + 0.762069i 1.94089i −1.91633 4.62643i −5.80073 2.40274i 1.06584 2.57316i 13.5085 13.5085i −16.2878 + 16.2878i 6.04203 14.5867i
8.1 −1.89932 + 1.89932i −1.65755 + 4.00167i 0.785167i 1.92782 + 0.798529i −4.45224 10.7487i 23.0956 9.56650i −16.6858 16.6858i 5.82599 + 5.82599i −5.17821 + 2.14488i
8.2 −0.161134 + 0.161134i 3.15299 7.61199i 7.94807i 2.54200 + 1.05293i 0.718496 + 1.73460i −19.8837 + 8.23610i −2.56978 2.56978i −28.9092 28.9092i −0.579266 + 0.239940i
8.3 2.47467 2.47467i −1.08123 + 2.61032i 4.24796i −8.05561 3.33674i 3.78400 + 9.13537i −6.33320 + 2.62330i 9.28506 + 9.28506i 13.4472 + 13.4472i −28.1923 + 11.6776i
9.1 −3.43772 + 3.43772i −4.67995 1.93850i 15.6358i −7.10390 + 17.1503i 22.7523 9.42432i 5.36561 + 12.9537i 26.2496 + 26.2496i −0.947753 0.947753i −34.5367 83.3791i
9.2 −1.20595 + 1.20595i 4.10553 + 1.70057i 5.09138i 2.60601 6.29147i −7.00185 + 2.90026i −5.31013 12.8198i −15.7875 15.7875i −5.12843 5.12843i 4.44447 + 10.7299i
9.3 2.22945 2.22945i −1.83980 0.762069i 1.94089i −1.91633 + 4.62643i −5.80073 + 2.40274i 1.06584 + 2.57316i 13.5085 + 13.5085i −16.2878 16.2878i 6.04203 + 14.5867i
15.1 −1.89932 1.89932i −1.65755 4.00167i 0.785167i 1.92782 0.798529i −4.45224 + 10.7487i 23.0956 + 9.56650i −16.6858 + 16.6858i 5.82599 5.82599i −5.17821 2.14488i
15.2 −0.161134 0.161134i 3.15299 + 7.61199i 7.94807i 2.54200 1.05293i 0.718496 1.73460i −19.8837 8.23610i −2.56978 + 2.56978i −28.9092 + 28.9092i −0.579266 0.239940i
15.3 2.47467 + 2.47467i −1.08123 2.61032i 4.24796i −8.05561 + 3.33674i 3.78400 9.13537i −6.33320 2.62330i 9.28506 9.28506i 13.4472 13.4472i −28.1923 11.6776i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 15.3
Significant digits:
Format:

Inner twists

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

Hecke kernels

There are no other newforms in \(S_{4}^{\mathrm{new}}(17, [\chi])\).