Properties

Label 16.4.e.a
Level 16
Weight 4
Character orbit 16.e
Analytic conductor 0.944
Analytic rank 0
Dimension 10
CM No
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 16 = 2^{4} \)
Weight: \( k \) = \( 4 \)
Character orbit: \([\chi]\) = 16.e (of order \(4\) and degree \(2\))

Newform invariants

Self dual: No
Analytic conductor: \(0.944030560092\)
Analytic rank: \(0\)
Dimension: \(10\)
Relative dimension: \(5\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 2^{10} \)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \( -\beta_{4} q^{2} \) \( -\beta_{5} q^{3} \) \( + ( 1 - \beta_{2} + \beta_{5} + \beta_{6} + \beta_{8} ) q^{4} \) \( + ( -1 + \beta_{4} - \beta_{6} - \beta_{7} - \beta_{8} + \beta_{9} ) q^{5} \) \( + ( -3 + \beta_{1} + \beta_{2} + \beta_{3} + \beta_{5} + \beta_{7} - \beta_{8} - \beta_{9} ) q^{6} \) \( + ( -1 - \beta_{1} + 3 \beta_{2} - \beta_{3} + 4 \beta_{4} - \beta_{5} - 2 \beta_{6} + 2 \beta_{7} - \beta_{9} ) q^{7} \) \( + ( -6 - 2 \beta_{1} + 2 \beta_{2} - 3 \beta_{3} + \beta_{4} + \beta_{5} - \beta_{6} - \beta_{7} + \beta_{8} ) q^{8} \) \( + ( 1 + \beta_{1} - 4 \beta_{2} + 2 \beta_{3} + 2 \beta_{5} + 6 \beta_{6} - 2 \beta_{7} - \beta_{9} ) q^{9} \) \(+O(q^{10})\) \( q\) \( -\beta_{4} q^{2} \) \( -\beta_{5} q^{3} \) \( + ( 1 - \beta_{2} + \beta_{5} + \beta_{6} + \beta_{8} ) q^{4} \) \( + ( -1 + \beta_{4} - \beta_{6} - \beta_{7} - \beta_{8} + \beta_{9} ) q^{5} \) \( + ( -3 + \beta_{1} + \beta_{2} + \beta_{3} + \beta_{5} + \beta_{7} - \beta_{8} - \beta_{9} ) q^{6} \) \( + ( -1 - \beta_{1} + 3 \beta_{2} - \beta_{3} + 4 \beta_{4} - \beta_{5} - 2 \beta_{6} + 2 \beta_{7} - \beta_{9} ) q^{7} \) \( + ( -6 - 2 \beta_{1} + 2 \beta_{2} - 3 \beta_{3} + \beta_{4} + \beta_{5} - \beta_{6} - \beta_{7} + \beta_{8} ) q^{8} \) \( + ( 1 + \beta_{1} - 4 \beta_{2} + 2 \beta_{3} + 2 \beta_{5} + 6 \beta_{6} - 2 \beta_{7} - \beta_{9} ) q^{9} \) \( + ( -6 - 8 \beta_{2} + 4 \beta_{3} - \beta_{4} - 2 \beta_{5} - \beta_{6} - 2 \beta_{8} + 2 \beta_{9} ) q^{10} \) \( + ( 2 + 2 \beta_{1} - \beta_{3} - 2 \beta_{4} - 6 \beta_{6} + 2 \beta_{7} + 2 \beta_{8} ) q^{11} \) \( + ( 10 + 2 \beta_{1} + 16 \beta_{2} - 3 \beta_{3} + 3 \beta_{4} - 3 \beta_{5} + \beta_{6} - \beta_{7} + \beta_{8} + 2 \beta_{9} ) q^{12} \) \( + ( 2 - \beta_{1} + \beta_{2} - 9 \beta_{4} + 4 \beta_{5} + 7 \beta_{6} - \beta_{7} + \beta_{8} + 4 \beta_{9} ) q^{13} \) \( + ( 20 + 2 \beta_{1} - 14 \beta_{2} - 2 \beta_{4} - 6 \beta_{5} - 2 \beta_{6} - 2 \beta_{8} - 4 \beta_{9} ) q^{14} \) \( + ( -9 - 5 \beta_{1} - 3 \beta_{2} + \beta_{3} - 14 \beta_{4} - \beta_{5} + 4 \beta_{6} + 2 \beta_{8} - 3 \beta_{9} ) q^{15} \) \( + ( 28 - 6 \beta_{1} - 12 \beta_{2} + 4 \beta_{3} + 4 \beta_{4} - 2 \beta_{5} + 2 \beta_{6} + 2 \beta_{8} - 2 \beta_{9} ) q^{16} \) \( + ( -3 + 3 \beta_{1} - 5 \beta_{2} - 2 \beta_{3} - 6 \beta_{4} + 2 \beta_{5} - 16 \beta_{6} - 2 \beta_{8} - 5 \beta_{9} ) q^{17} \) \( + ( 14 - 2 \beta_{1} + 50 \beta_{2} - 6 \beta_{3} + 6 \beta_{4} + 2 \beta_{5} - \beta_{6} + 2 \beta_{7} - 2 \beta_{8} + 6 \beta_{9} ) q^{18} \) \( + ( 8 \beta_{1} + 10 \beta_{2} + 10 \beta_{4} + 3 \beta_{5} + 18 \beta_{6} + 2 \beta_{7} - 2 \beta_{8} + 2 \beta_{9} ) q^{19} \) \( + ( -22 + 8 \beta_{1} - 48 \beta_{2} + \beta_{3} + 3 \beta_{4} + 3 \beta_{5} - 9 \beta_{6} - \beta_{7} - 5 \beta_{8} + 4 \beta_{9} ) q^{20} \) \( + ( 2 - 4 \beta_{1} + 10 \beta_{2} + 4 \beta_{3} + 20 \beta_{4} - 4 \beta_{6} + 4 \beta_{7} + 4 \beta_{8} + 4 \beta_{9} ) q^{21} \) \( + ( -59 - \beta_{1} + 11 \beta_{2} + \beta_{3} - 4 \beta_{4} + 3 \beta_{5} - 7 \beta_{7} + 5 \beta_{8} - 5 \beta_{9} ) q^{22} \) \( + ( -7 - 7 \beta_{1} - 27 \beta_{2} + 3 \beta_{3} + 12 \beta_{4} + 3 \beta_{5} - 6 \beta_{6} - 10 \beta_{7} + \beta_{9} ) q^{23} \) \( + ( -80 - 2 \beta_{1} + 28 \beta_{2} + 6 \beta_{3} - 6 \beta_{4} + 12 \beta_{6} + 6 \beta_{7} - 4 \beta_{8} - 6 \beta_{9} ) q^{24} \) \( + ( 5 \beta_{2} - 12 \beta_{3} + 16 \beta_{4} - 12 \beta_{5} + 8 \beta_{6} + 8 \beta_{7} - 8 \beta_{9} ) q^{25} \) \( + ( -28 - 6 \beta_{1} - 74 \beta_{2} - 10 \beta_{3} + 5 \beta_{4} + 8 \beta_{5} + 7 \beta_{6} + 2 \beta_{7} + 12 \beta_{8} + 4 \beta_{9} ) q^{26} \) \( + ( 14 + 6 \beta_{1} + 16 \beta_{2} + 4 \beta_{3} - 6 \beta_{4} - 18 \beta_{6} - 10 \beta_{7} - 10 \beta_{8} + 8 \beta_{9} ) q^{27} \) \( + ( 34 + 8 \beta_{1} + 94 \beta_{2} + 10 \beta_{3} - 22 \beta_{4} + 4 \beta_{5} - 8 \beta_{6} + 6 \beta_{7} - 4 \beta_{8} - 4 \beta_{9} ) q^{28} \) \( + ( -16 - 3 \beta_{1} + 13 \beta_{2} - 11 \beta_{4} - 24 \beta_{5} - 11 \beta_{6} + 5 \beta_{7} - 5 \beta_{8} ) q^{29} \) \( + ( 122 - 40 \beta_{2} - 6 \beta_{3} + 10 \beta_{4} + 20 \beta_{5} + 14 \beta_{6} + 2 \beta_{7} + 12 \beta_{8} - 2 \beta_{9} ) q^{30} \) \( + ( 34 + 2 \beta_{1} + 6 \beta_{2} - 4 \beta_{3} + 4 \beta_{4} + 4 \beta_{5} + 8 \beta_{6} - 12 \beta_{8} + 6 \beta_{9} ) q^{31} \) \( + ( 96 + 4 \beta_{1} - 64 \beta_{2} - 6 \beta_{3} - 30 \beta_{4} + 6 \beta_{5} - 14 \beta_{6} - 2 \beta_{7} - 10 \beta_{8} ) q^{32} \) \( + ( 7 - 3 \beta_{1} - 3 \beta_{2} + 14 \beta_{3} - 2 \beta_{4} - 14 \beta_{5} + 10 \beta_{8} - 3 \beta_{9} ) q^{33} \) \( + ( 50 - 2 \beta_{1} + 106 \beta_{2} + 18 \beta_{3} - 12 \beta_{4} - 2 \beta_{5} + 6 \beta_{6} - 14 \beta_{7} + 10 \beta_{8} + 2 \beta_{9} ) q^{34} \) \( + ( 48 - 8 \beta_{1} - 52 \beta_{2} - 12 \beta_{4} + 6 \beta_{5} + 4 \beta_{6} - 12 \beta_{7} + 12 \beta_{8} + 4 \beta_{9} ) q^{35} \) \( + ( -43 - 4 \beta_{1} - 131 \beta_{2} + 3 \beta_{3} - 19 \beta_{4} - 20 \beta_{5} + 36 \beta_{6} + 5 \beta_{7} + 4 \beta_{8} - 4 \beta_{9} ) q^{36} \) \( + ( -5 + 4 \beta_{1} - 10 \beta_{2} - 28 \beta_{3} - 9 \beta_{4} - 7 \beta_{6} + \beta_{7} + \beta_{8} - \beta_{9} ) q^{37} \) \( + ( -127 + \beta_{1} + 97 \beta_{2} - 23 \beta_{3} + 24 \beta_{4} - 19 \beta_{5} - 4 \beta_{6} + 17 \beta_{7} - 5 \beta_{8} + 3 \beta_{9} ) q^{38} \) \( + ( 11 + 11 \beta_{1} + 79 \beta_{2} + 7 \beta_{3} - 12 \beta_{4} + 7 \beta_{5} + 22 \beta_{6} + 10 \beta_{7} - 5 \beta_{9} ) q^{39} \) \( + ( -144 + 2 \beta_{1} + 52 \beta_{2} + 12 \beta_{3} + 16 \beta_{4} - 14 \beta_{5} - 54 \beta_{6} - 8 \beta_{7} - 2 \beta_{8} + 10 \beta_{9} ) q^{40} \) \( + ( 2 + 2 \beta_{1} + 6 \beta_{2} + 24 \beta_{3} - 16 \beta_{4} + 24 \beta_{5} - 12 \beta_{6} + 4 \beta_{7} + 6 \beta_{9} ) q^{41} \) \( + ( -76 + 4 \beta_{1} - 156 \beta_{2} - 12 \beta_{3} - 6 \beta_{4} - 12 \beta_{5} - 18 \beta_{6} - 12 \beta_{7} - 20 \beta_{8} - 4 \beta_{9} ) q^{42} \) \( + ( -72 - 8 \beta_{1} - 80 \beta_{2} + 3 \beta_{3} - 8 \beta_{4} + 40 \beta_{6} + 8 \beta_{7} + 8 \beta_{8} - 16 \beta_{9} ) q^{43} \) \( + ( 72 - 14 \beta_{1} + 134 \beta_{2} + 5 \beta_{3} + 63 \beta_{4} + 19 \beta_{5} + 27 \beta_{6} - 9 \beta_{7} - \beta_{8} + 6 \beta_{9} ) q^{44} \) \( + ( 2 + 5 \beta_{1} - 9 \beta_{2} + 37 \beta_{4} + 44 \beta_{5} - 11 \beta_{6} - 3 \beta_{7} + 3 \beta_{8} - 12 \beta_{9} ) q^{45} \) \( + ( 104 - 10 \beta_{1} - 34 \beta_{2} + 20 \beta_{3} - 6 \beta_{4} - 2 \beta_{5} - 30 \beta_{6} - 12 \beta_{7} - 22 \beta_{8} + 16 \beta_{9} ) q^{46} \) \( + ( -102 + 10 \beta_{1} - 2 \beta_{2} - 4 \beta_{3} + 36 \beta_{4} + 4 \beta_{5} - 24 \beta_{6} + 20 \beta_{8} - 2 \beta_{9} ) q^{47} \) \( + ( 168 + 16 \beta_{1} - 16 \beta_{2} - 26 \beta_{3} + 86 \beta_{4} - 2 \beta_{5} + 34 \beta_{6} + 10 \beta_{7} + 6 \beta_{8} + 4 \beta_{9} ) q^{48} \) \( + ( 1 - 10 \beta_{1} + 22 \beta_{2} - 36 \beta_{3} + 20 \beta_{4} + 36 \beta_{5} + 64 \beta_{6} - 4 \beta_{8} + 22 \beta_{9} ) q^{49} \) \( + ( 80 + 8 \beta_{1} + 120 \beta_{2} + 8 \beta_{4} - 24 \beta_{5} - 11 \beta_{6} + 32 \beta_{7} - 8 \beta_{8} - 32 \beta_{9} ) q^{50} \) \( + ( -160 - 16 \beta_{1} + 130 \beta_{2} - 22 \beta_{4} - 22 \beta_{5} - 78 \beta_{6} + 18 \beta_{7} - 18 \beta_{8} - 14 \beta_{9} ) q^{51} \) \( + ( -56 - 28 \beta_{1} - 122 \beta_{2} - 21 \beta_{3} + 41 \beta_{4} - \beta_{5} - 85 \beta_{6} - 3 \beta_{7} + 15 \beta_{8} - 16 \beta_{9} ) q^{52} \) \( + ( -5 + 12 \beta_{1} - 38 \beta_{2} + 68 \beta_{3} - 85 \beta_{4} + 37 \beta_{6} - 19 \beta_{7} - 19 \beta_{8} - 21 \beta_{9} ) q^{53} \) \( + ( -120 + 4 \beta_{1} + 12 \beta_{2} + 44 \beta_{3} - 36 \beta_{4} + 24 \beta_{6} - 12 \beta_{7} - 8 \beta_{8} + 24 \beta_{9} ) q^{54} \) \( + ( 15 + 15 \beta_{1} - 157 \beta_{2} - 25 \beta_{3} - 60 \beta_{4} - 25 \beta_{5} - 18 \beta_{6} + 18 \beta_{7} + 15 \beta_{9} ) q^{55} \) \( + ( -20 + 16 \beta_{1} + 60 \beta_{2} - 14 \beta_{3} - 38 \beta_{4} + 14 \beta_{5} + 114 \beta_{6} - 18 \beta_{7} + 22 \beta_{8} + 12 \beta_{9} ) q^{56} \) \( + ( -9 - 9 \beta_{1} - 21 \beta_{2} - 18 \beta_{3} - 64 \beta_{4} - 18 \beta_{5} - 54 \beta_{6} - 46 \beta_{7} + 41 \beta_{9} ) q^{57} \) \( + ( 16 + 34 \beta_{1} - 106 \beta_{2} + 30 \beta_{3} - 3 \beta_{4} + 20 \beta_{5} + 19 \beta_{6} + 18 \beta_{7} - 8 \beta_{8} - 24 \beta_{9} ) q^{58} \) \( + ( 164 - 36 \beta_{1} + 192 \beta_{2} - 29 \beta_{3} + 84 \beta_{4} + 60 \beta_{6} + 28 \beta_{7} + 28 \beta_{8} - 8 \beta_{9} ) q^{59} \) \( + ( -22 - 36 \beta_{1} + 58 \beta_{2} - 44 \beta_{3} - 96 \beta_{4} - 14 \beta_{5} - 50 \beta_{6} - 12 \beta_{7} + 18 \beta_{8} ) q^{60} \) \( + ( 94 + 19 \beta_{1} - 79 \beta_{2} + 67 \beta_{4} - 12 \beta_{5} + 51 \beta_{6} - 21 \beta_{7} + 21 \beta_{8} - 4 \beta_{9} ) q^{61} \) \( + ( 12 + 4 \beta_{1} - 68 \beta_{2} + 4 \beta_{3} - 36 \beta_{4} - 36 \beta_{5} - 4 \beta_{6} + 20 \beta_{7} + 4 \beta_{8} + 12 \beta_{9} ) q^{62} \) \( + ( 267 + 15 \beta_{1} - 7 \beta_{2} + 33 \beta_{3} + 26 \beta_{4} - 33 \beta_{5} - 44 \beta_{6} + 10 \beta_{8} - 7 \beta_{9} ) q^{63} \) \( + ( 80 - 4 \beta_{1} + 32 \beta_{2} + 28 \beta_{3} - 116 \beta_{4} - 32 \beta_{6} - 4 \beta_{7} + 40 \beta_{8} + 12 \beta_{9} ) q^{64} \) \( + ( -44 + 18 \beta_{1} + 26 \beta_{2} + 40 \beta_{3} + 36 \beta_{4} - 40 \beta_{5} + 16 \beta_{6} - 52 \beta_{8} + 26 \beta_{9} ) q^{65} \) \( + ( -58 + 18 \beta_{1} - 10 \beta_{2} - 10 \beta_{3} + 50 \beta_{5} + 2 \beta_{6} - 10 \beta_{7} - 26 \beta_{8} - 10 \beta_{9} ) q^{66} \) \( + ( 176 - 202 \beta_{2} + 30 \beta_{4} - 13 \beta_{5} - 74 \beta_{6} + 22 \beta_{7} - 22 \beta_{8} - 26 \beta_{9} ) q^{67} \) \( + ( -78 - 4 \beta_{1} + 38 \beta_{2} - 4 \beta_{3} - 36 \beta_{4} + 66 \beta_{5} + 130 \beta_{6} - 20 \beta_{7} - 22 \beta_{8} + 20 \beta_{9} ) q^{68} \) \( + ( 58 - 12 \beta_{1} + 54 \beta_{2} - 52 \beta_{3} + 48 \beta_{6} + 8 \beta_{7} + 8 \beta_{8} - 16 \beta_{9} ) q^{69} \) \( + ( 2 - 30 \beta_{1} - 62 \beta_{2} + 2 \beta_{3} - 40 \beta_{4} + 42 \beta_{5} - 32 \beta_{6} - 14 \beta_{7} + 6 \beta_{8} + 6 \beta_{9} ) q^{70} \) \( + ( 1 + \beta_{1} + 333 \beta_{2} - 21 \beta_{3} - 20 \beta_{4} - 21 \beta_{5} + 10 \beta_{6} - 26 \beta_{7} + 9 \beta_{9} ) q^{71} \) \( + ( 70 + 24 \beta_{1} - 14 \beta_{2} - 33 \beta_{3} + 79 \beta_{4} + 45 \beta_{5} - 113 \beta_{6} + 49 \beta_{7} + \beta_{8} - 26 \beta_{9} ) q^{72} \) \( + ( -29 - 29 \beta_{1} + 37 \beta_{2} + 14 \beta_{3} + 64 \beta_{4} + 14 \beta_{5} - 62 \beta_{6} + 10 \beta_{7} - 3 \beta_{9} ) q^{73} \) \( + ( -30 - 28 \beta_{1} + 140 \beta_{2} + 32 \beta_{3} + 3 \beta_{4} - 18 \beta_{5} - \beta_{6} + 20 \beta_{7} + 38 \beta_{8} - 30 \beta_{9} ) q^{74} \) \( + ( -312 - 288 \beta_{2} + 13 \beta_{3} + 16 \beta_{4} - 16 \beta_{6} - 32 \beta_{7} - 32 \beta_{8} + 24 \beta_{9} ) q^{75} \) \( + ( -110 + 18 \beta_{1} - 104 \beta_{2} + 17 \beta_{3} + 119 \beta_{4} - 55 \beta_{5} + 53 \beta_{6} + 43 \beta_{7} - 3 \beta_{8} - 54 \beta_{9} ) q^{76} \) \( + ( -42 - 8 \beta_{1} + 14 \beta_{2} - 8 \beta_{4} - 12 \beta_{5} - 88 \beta_{6} + 32 \beta_{7} - 32 \beta_{8} - 20 \beta_{9} ) q^{77} \) \( + ( -68 + 10 \beta_{1} + 138 \beta_{2} - 56 \beta_{3} + 22 \beta_{4} + 2 \beta_{5} + 86 \beta_{6} + 8 \beta_{7} + 22 \beta_{8} + 4 \beta_{9} ) q^{78} \) \( + ( -452 + 12 \beta_{1} + 36 \beta_{2} - 8 \beta_{3} + 40 \beta_{4} + 8 \beta_{5} + 48 \beta_{6} - 56 \beta_{8} + 36 \beta_{9} ) q^{79} \) \( + ( -280 + 20 \beta_{1} - 112 \beta_{2} + 74 \beta_{3} + 90 \beta_{4} + 14 \beta_{5} + 34 \beta_{6} - 50 \beta_{7} - 50 \beta_{8} + 8 \beta_{9} ) q^{80} \) \( + ( 50 - 15 \beta_{1} - 15 \beta_{2} - 18 \beta_{3} + 6 \beta_{4} + 18 \beta_{5} + 66 \beta_{8} - 15 \beta_{9} ) q^{81} \) \( + ( -84 + 4 \beta_{1} - 228 \beta_{2} - 44 \beta_{3} - 12 \beta_{4} + 12 \beta_{5} + 12 \beta_{6} - 60 \beta_{7} + 20 \beta_{8} + 44 \beta_{9} ) q^{82} \) \( + ( -240 - 24 \beta_{1} + 256 \beta_{2} - 56 \beta_{4} + 55 \beta_{5} + 104 \beta_{6} - 72 \beta_{7} + 72 \beta_{8} + 40 \beta_{9} ) q^{83} \) \( + ( 188 + 8 \beta_{1} + 232 \beta_{2} + 86 \beta_{3} + 42 \beta_{4} - 22 \beta_{5} - 134 \beta_{6} + 26 \beta_{7} - 6 \beta_{8} + 8 \beta_{9} ) q^{84} \) \( + ( -22 + 20 \beta_{1} - 18 \beta_{2} - 28 \beta_{3} + 32 \beta_{4} - 112 \beta_{6} + 24 \beta_{7} + 24 \beta_{8} + 24 \beta_{9} ) q^{85} \) \( + ( 333 + 3 \beta_{1} + 87 \beta_{2} - 67 \beta_{3} + 112 \beta_{4} + 19 \beta_{5} - 64 \beta_{6} + 29 \beta_{7} + 13 \beta_{8} - 13 \beta_{9} ) q^{86} \) \( + ( 17 + 17 \beta_{1} - 547 \beta_{2} + 83 \beta_{3} + 44 \beta_{4} + 83 \beta_{5} + 106 \beta_{6} + 6 \beta_{7} - 39 \beta_{9} ) q^{87} \) \( + ( 172 - 22 \beta_{1} - 176 \beta_{2} - 32 \beta_{3} - 60 \beta_{4} - 70 \beta_{5} + 74 \beta_{6} + 4 \beta_{7} - 58 \beta_{8} + 34 \beta_{9} ) q^{88} \) \( + ( 53 + 53 \beta_{1} + 35 \beta_{2} - 14 \beta_{3} - 14 \beta_{5} + 110 \beta_{6} + 102 \beta_{7} - 53 \beta_{9} ) q^{89} \) \( + ( 172 - 50 \beta_{1} + 290 \beta_{2} - 30 \beta_{3} - 5 \beta_{4} - 72 \beta_{5} - 31 \beta_{6} - 58 \beta_{7} + 4 \beta_{8} + 44 \beta_{9} ) q^{90} \) \( + ( 392 + 64 \beta_{1} + 288 \beta_{2} + 70 \beta_{3} - 240 \beta_{4} - 16 \beta_{6} - 32 \beta_{7} - 32 \beta_{8} - 40 \beta_{9} ) q^{91} \) \( + ( 98 + 32 \beta_{1} - 434 \beta_{2} + 58 \beta_{3} - 166 \beta_{4} - 4 \beta_{5} - 32 \beta_{6} - 26 \beta_{7} - 28 \beta_{8} + 52 \beta_{9} ) q^{92} \) \( + ( -212 - 36 \beta_{1} + 184 \beta_{2} - 100 \beta_{4} - 40 \beta_{5} - 68 \beta_{6} + 12 \beta_{7} - 12 \beta_{8} + 8 \beta_{9} ) q^{93} \) \( + ( -372 - 28 \beta_{1} + 252 \beta_{2} + 4 \beta_{3} + 108 \beta_{4} - 4 \beta_{5} - 36 \beta_{6} - 44 \beta_{7} - 28 \beta_{8} - 20 \beta_{9} ) q^{94} \) \( + ( 673 - 43 \beta_{1} - 21 \beta_{2} - 107 \beta_{3} - 74 \beta_{4} + 107 \beta_{5} + 44 \beta_{6} + 54 \beta_{8} - 21 \beta_{9} ) q^{95} \) \( + ( -432 - 20 \beta_{1} + 304 \beta_{2} - 32 \beta_{3} - 112 \beta_{4} - 108 \beta_{5} - 116 \beta_{6} + 56 \beta_{7} - 52 \beta_{8} - 44 \beta_{9} ) q^{96} \) \( + ( 25 - 49 \beta_{1} - 73 \beta_{2} - 10 \beta_{3} - 158 \beta_{4} + 10 \beta_{5} - 48 \beta_{6} + 86 \beta_{8} - 73 \beta_{9} ) q^{97} \) \( + ( 36 - 68 \beta_{1} - 364 \beta_{2} - 60 \beta_{3} + 49 \beta_{4} - 100 \beta_{5} - 20 \beta_{6} + 68 \beta_{7} + 52 \beta_{8} + 4 \beta_{9} ) q^{98} \) \( + ( 544 + 56 \beta_{1} - 436 \beta_{2} - 12 \beta_{4} + 23 \beta_{5} + 196 \beta_{6} + 20 \beta_{7} - 20 \beta_{8} + 52 \beta_{9} ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(10q \) \(\mathstrut -\mathstrut 2q^{2} \) \(\mathstrut -\mathstrut 2q^{3} \) \(\mathstrut +\mathstrut 8q^{4} \) \(\mathstrut -\mathstrut 2q^{5} \) \(\mathstrut -\mathstrut 32q^{6} \) \(\mathstrut -\mathstrut 44q^{8} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(10q \) \(\mathstrut -\mathstrut 2q^{2} \) \(\mathstrut -\mathstrut 2q^{3} \) \(\mathstrut +\mathstrut 8q^{4} \) \(\mathstrut -\mathstrut 2q^{5} \) \(\mathstrut -\mathstrut 32q^{6} \) \(\mathstrut -\mathstrut 44q^{8} \) \(\mathstrut -\mathstrut 68q^{10} \) \(\mathstrut +\mathstrut 18q^{11} \) \(\mathstrut +\mathstrut 100q^{12} \) \(\mathstrut -\mathstrut 2q^{13} \) \(\mathstrut +\mathstrut 188q^{14} \) \(\mathstrut -\mathstrut 124q^{15} \) \(\mathstrut +\mathstrut 280q^{16} \) \(\mathstrut -\mathstrut 4q^{17} \) \(\mathstrut +\mathstrut 174q^{18} \) \(\mathstrut -\mathstrut 26q^{19} \) \(\mathstrut -\mathstrut 196q^{20} \) \(\mathstrut +\mathstrut 52q^{21} \) \(\mathstrut -\mathstrut 588q^{22} \) \(\mathstrut -\mathstrut 848q^{24} \) \(\mathstrut -\mathstrut 264q^{26} \) \(\mathstrut +\mathstrut 184q^{27} \) \(\mathstrut +\mathstrut 280q^{28} \) \(\mathstrut -\mathstrut 202q^{29} \) \(\mathstrut +\mathstrut 1236q^{30} \) \(\mathstrut +\mathstrut 368q^{31} \) \(\mathstrut +\mathstrut 968q^{32} \) \(\mathstrut -\mathstrut 4q^{33} \) \(\mathstrut +\mathstrut 436q^{34} \) \(\mathstrut +\mathstrut 476q^{35} \) \(\mathstrut -\mathstrut 596q^{36} \) \(\mathstrut -\mathstrut 10q^{37} \) \(\mathstrut -\mathstrut 1232q^{38} \) \(\mathstrut -\mathstrut 1336q^{40} \) \(\mathstrut -\mathstrut 680q^{42} \) \(\mathstrut -\mathstrut 838q^{43} \) \(\mathstrut +\mathstrut 868q^{44} \) \(\mathstrut +\mathstrut 194q^{45} \) \(\mathstrut +\mathstrut 1132q^{46} \) \(\mathstrut -\mathstrut 944q^{47} \) \(\mathstrut +\mathstrut 1768q^{48} \) \(\mathstrut +\mathstrut 94q^{49} \) \(\mathstrut +\mathstrut 726q^{50} \) \(\mathstrut -\mathstrut 1500q^{51} \) \(\mathstrut -\mathstrut 236q^{52} \) \(\mathstrut -\mathstrut 378q^{53} \) \(\mathstrut -\mathstrut 1376q^{54} \) \(\mathstrut -\mathstrut 488q^{56} \) \(\mathstrut +\mathstrut 8q^{58} \) \(\mathstrut +\mathstrut 1706q^{59} \) \(\mathstrut -\mathstrut 192q^{60} \) \(\mathstrut +\mathstrut 910q^{61} \) \(\mathstrut -\mathstrut 80q^{62} \) \(\mathstrut +\mathstrut 2628q^{63} \) \(\mathstrut +\mathstrut 512q^{64} \) \(\mathstrut -\mathstrut 492q^{65} \) \(\mathstrut -\mathstrut 428q^{66} \) \(\mathstrut +\mathstrut 1942q^{67} \) \(\mathstrut -\mathstrut 880q^{68} \) \(\mathstrut +\mathstrut 580q^{69} \) \(\mathstrut +\mathstrut 160q^{70} \) \(\mathstrut +\mathstrut 1092q^{72} \) \(\mathstrut -\mathstrut 452q^{74} \) \(\mathstrut -\mathstrut 2954q^{75} \) \(\mathstrut -\mathstrut 1228q^{76} \) \(\mathstrut -\mathstrut 268q^{77} \) \(\mathstrut -\mathstrut 772q^{78} \) \(\mathstrut -\mathstrut 4416q^{79} \) \(\mathstrut -\mathstrut 2648q^{80} \) \(\mathstrut +\mathstrut 482q^{81} \) \(\mathstrut -\mathstrut 704q^{82} \) \(\mathstrut -\mathstrut 2562q^{83} \) \(\mathstrut +\mathstrut 1960q^{84} \) \(\mathstrut -\mathstrut 12q^{85} \) \(\mathstrut +\mathstrut 3764q^{86} \) \(\mathstrut +\mathstrut 1528q^{88} \) \(\mathstrut +\mathstrut 1896q^{90} \) \(\mathstrut +\mathstrut 3332q^{91} \) \(\mathstrut +\mathstrut 632q^{92} \) \(\mathstrut -\mathstrut 2192q^{93} \) \(\mathstrut -\mathstrut 3248q^{94} \) \(\mathstrut +\mathstrut 6900q^{95} \) \(\mathstrut -\mathstrut 4432q^{96} \) \(\mathstrut -\mathstrut 4q^{97} \) \(\mathstrut +\mathstrut 314q^{98} \) \(\mathstrut +\mathstrut 4958q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{10}\mathstrut -\mathstrut \) \(2\) \(x^{9}\mathstrut -\mathstrut \) \(x^{8}\mathstrut +\mathstrut \) \(6\) \(x^{7}\mathstrut +\mathstrut \) \(14\) \(x^{6}\mathstrut -\mathstrut \) \(80\) \(x^{5}\mathstrut +\mathstrut \) \(56\) \(x^{4}\mathstrut +\mathstrut \) \(96\) \(x^{3}\mathstrut -\mathstrut \) \(64\) \(x^{2}\mathstrut -\mathstrut \) \(512\) \(x\mathstrut +\mathstrut \) \(1024\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 4 \nu - 1 \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{9} - 14 \nu^{8} + 7 \nu^{7} + 82 \nu^{6} - 170 \nu^{5} - 120 \nu^{4} + 536 \nu^{3} + 384 \nu^{2} - 2752 \nu + 3072 \)\()/1280\)
\(\beta_{3}\)\(=\)\((\)\( 3 \nu^{9} - 2 \nu^{8} + 101 \nu^{7} - 114 \nu^{6} + 210 \nu^{5} - 120 \nu^{4} + 8 \nu^{3} - 3008 \nu^{2} + 3264 \nu - 7424 \)\()/1280\)
\(\beta_{4}\)\(=\)\((\)\( 7 \nu^{9} - 18 \nu^{8} + 49 \nu^{7} + 14 \nu^{6} - 230 \nu^{5} + 120 \nu^{4} + 872 \nu^{3} - 1472 \nu^{2} - 1984 \nu + 7424 \)\()/1280\)
\(\beta_{5}\)\(=\)\((\)\( -17 \nu^{9} + 38 \nu^{8} - 39 \nu^{7} + 86 \nu^{6} + 90 \nu^{5} + 520 \nu^{4} - 792 \nu^{3} + 2752 \nu^{2} - 1856 \nu - 4864 \)\()/1280\)
\(\beta_{6}\)\(=\)\((\)\( 17 \nu^{9} - 38 \nu^{8} + 39 \nu^{7} + 74 \nu^{6} - 90 \nu^{5} - 680 \nu^{4} + 1432 \nu^{3} - 512 \nu^{2} - 2624 \nu + 2304 \)\()/1280\)
\(\beta_{7}\)\(=\)\((\)\( -5 \nu^{9} + 6 \nu^{8} + 29 \nu^{7} - 58 \nu^{6} + 18 \nu^{5} + 184 \nu^{4} + 136 \nu^{3} - 1216 \nu^{2} + 1088 \nu - 1280 \)\()/256\)
\(\beta_{8}\)\(=\)\((\)\( -37 \nu^{9} + 158 \nu^{8} - 179 \nu^{7} + 46 \nu^{6} - 350 \nu^{5} + 1800 \nu^{4} - 4472 \nu^{3} + 3712 \nu^{2} - 3776 \nu + 5376 \)\()/1280\)
\(\beta_{9}\)\(=\)\((\)\( -49 \nu^{9} + 46 \nu^{8} + 297 \nu^{7} - 818 \nu^{6} + 10 \nu^{5} + 2040 \nu^{4} + 616 \nu^{3} - 11136 \nu^{2} + 17088 \nu - 7168 \)\()/1280\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{1}\mathstrut +\mathstrut \) \(1\)\()/4\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{9}\mathstrut -\mathstrut \) \(2\) \(\beta_{7}\mathstrut +\mathstrut \) \(2\) \(\beta_{6}\mathstrut +\mathstrut \) \(2\) \(\beta_{5}\mathstrut -\mathstrut \) \(\beta_{2}\mathstrut +\mathstrut \) \(2\)\()/4\)
\(\nu^{3}\)\(=\)\((\)\(2\) \(\beta_{7}\mathstrut +\mathstrut \) \(6\) \(\beta_{6}\mathstrut +\mathstrut \) \(2\) \(\beta_{5}\mathstrut -\mathstrut \) \(4\) \(\beta_{3}\mathstrut -\mathstrut \) \(6\) \(\beta_{2}\mathstrut +\mathstrut \) \(\beta_{1}\mathstrut -\mathstrut \) \(1\)\()/4\)
\(\nu^{4}\)\(=\)\((\)\(-\)\(\beta_{9}\mathstrut -\mathstrut \) \(2\) \(\beta_{8}\mathstrut -\mathstrut \) \(8\) \(\beta_{6}\mathstrut +\mathstrut \) \(4\) \(\beta_{5}\mathstrut +\mathstrut \) \(14\) \(\beta_{4}\mathstrut -\mathstrut \) \(2\) \(\beta_{3}\mathstrut -\mathstrut \) \(11\) \(\beta_{2}\mathstrut -\mathstrut \) \(34\)\()/4\)
\(\nu^{5}\)\(=\)\((\)\(-\)\(4\) \(\beta_{9}\mathstrut -\mathstrut \) \(4\) \(\beta_{8}\mathstrut +\mathstrut \) \(6\) \(\beta_{7}\mathstrut -\mathstrut \) \(6\) \(\beta_{6}\mathstrut +\mathstrut \) \(6\) \(\beta_{5}\mathstrut +\mathstrut \) \(4\) \(\beta_{4}\mathstrut -\mathstrut \) \(18\) \(\beta_{2}\mathstrut -\mathstrut \) \(5\) \(\beta_{1}\mathstrut +\mathstrut \) \(73\)\()/4\)
\(\nu^{6}\)\(=\)\((\)\(-\)\(15\) \(\beta_{9}\mathstrut -\mathstrut \) \(2\) \(\beta_{8}\mathstrut +\mathstrut \) \(20\) \(\beta_{7}\mathstrut -\mathstrut \) \(28\) \(\beta_{6}\mathstrut +\mathstrut \) \(14\) \(\beta_{4}\mathstrut +\mathstrut \) \(14\) \(\beta_{3}\mathstrut +\mathstrut \) \(27\) \(\beta_{2}\mathstrut +\mathstrut \) \(24\) \(\beta_{1}\mathstrut +\mathstrut \) \(34\)\()/4\)
\(\nu^{7}\)\(=\)\((\)\(20\) \(\beta_{9}\mathstrut +\mathstrut \) \(4\) \(\beta_{8}\mathstrut -\mathstrut \) \(46\) \(\beta_{7}\mathstrut +\mathstrut \) \(30\) \(\beta_{6}\mathstrut +\mathstrut \) \(50\) \(\beta_{5}\mathstrut +\mathstrut \) \(12\) \(\beta_{4}\mathstrut +\mathstrut \) \(64\) \(\beta_{3}\mathstrut +\mathstrut \) \(42\) \(\beta_{2}\mathstrut +\mathstrut \) \(5\) \(\beta_{1}\mathstrut +\mathstrut \) \(207\)\()/4\)
\(\nu^{8}\)\(=\)\((\)\(7\) \(\beta_{9}\mathstrut +\mathstrut \) \(58\) \(\beta_{8}\mathstrut +\mathstrut \) \(36\) \(\beta_{7}\mathstrut +\mathstrut \) \(292\) \(\beta_{6}\mathstrut +\mathstrut \) \(56\) \(\beta_{5}\mathstrut -\mathstrut \) \(54\) \(\beta_{4}\mathstrut -\mathstrut \) \(22\) \(\beta_{3}\mathstrut -\mathstrut \) \(179\) \(\beta_{2}\mathstrut +\mathstrut \) \(48\) \(\beta_{1}\mathstrut +\mathstrut \) \(326\)\()/4\)
\(\nu^{9}\)\(=\)\((\)\(4\) \(\beta_{9}\mathstrut +\mathstrut \) \(28\) \(\beta_{8}\mathstrut -\mathstrut \) \(98\) \(\beta_{7}\mathstrut +\mathstrut \) \(210\) \(\beta_{6}\mathstrut +\mathstrut \) \(94\) \(\beta_{5}\mathstrut +\mathstrut \) \(372\) \(\beta_{4}\mathstrut -\mathstrut \) \(674\) \(\beta_{2}\mathstrut +\mathstrut \) \(35\) \(\beta_{1}\mathstrut -\mathstrut \) \(1111\)\()/4\)

Character Values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/16\mathbb{Z}\right)^\times\).

\(n\) \(5\) \(15\)
\(\chi(n)\) \(\beta_{2}\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
5.1
1.28199 1.53509i
1.97476 0.316760i
−1.62580 1.16481i
0.932438 + 1.76934i
−1.56339 + 1.24732i
1.28199 + 1.53509i
1.97476 + 0.316760i
−1.62580 + 1.16481i
0.932438 1.76934i
−1.56339 1.24732i
−2.81708 0.253099i 5.49618 5.49618i 7.87188 + 1.42600i −4.66372 4.66372i −16.8743 + 14.0921i 24.8965i −21.8148 6.00953i 33.4160i 11.9577 + 14.3185i
5.2 −2.29152 + 1.65800i −5.96513 + 5.96513i 2.50210 7.59865i 8.67959 + 8.67959i 3.77903 23.5594i 1.63924i 6.86495 + 21.5609i 44.1656i −34.2802 5.49869i
5.3 0.460984 2.79061i 0.756776 0.756776i −7.57499 2.57285i 8.22587 + 8.22587i −1.76300 2.46073i 2.67171i −10.6718 + 19.9528i 25.8546i 26.7472 19.1632i
5.4 0.836901 + 2.70178i 1.98356 1.98356i −6.59919 + 4.52224i −0.596848 0.596848i 7.01918 + 3.69910i 29.0828i −17.7410 14.0449i 19.1310i 1.11305 2.11205i
5.5 2.81071 0.316066i −3.27139 + 3.27139i 7.80020 1.77674i −12.6449 12.6449i −8.16095 + 10.2289i 13.8754i 21.3626 7.45928i 5.59607i −39.5378 31.5445i
13.1 −2.81708 + 0.253099i 5.49618 + 5.49618i 7.87188 1.42600i −4.66372 + 4.66372i −16.8743 14.0921i 24.8965i −21.8148 + 6.00953i 33.4160i 11.9577 14.3185i
13.2 −2.29152 1.65800i −5.96513 5.96513i 2.50210 + 7.59865i 8.67959 8.67959i 3.77903 + 23.5594i 1.63924i 6.86495 21.5609i 44.1656i −34.2802 + 5.49869i
13.3 0.460984 + 2.79061i 0.756776 + 0.756776i −7.57499 + 2.57285i 8.22587 8.22587i −1.76300 + 2.46073i 2.67171i −10.6718 19.9528i 25.8546i 26.7472 + 19.1632i
13.4 0.836901 2.70178i 1.98356 + 1.98356i −6.59919 4.52224i −0.596848 + 0.596848i 7.01918 3.69910i 29.0828i −17.7410 + 14.0449i 19.1310i 1.11305 + 2.11205i
13.5 2.81071 + 0.316066i −3.27139 3.27139i 7.80020 + 1.77674i −12.6449 + 12.6449i −8.16095 10.2289i 13.8754i 21.3626 + 7.45928i 5.59607i −39.5378 + 31.5445i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 13.5
Significant digits:
Format:

Inner twists

Char. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
16.e Even 1 yes

Hecke kernels

There are no other newforms in \(S_{4}^{\mathrm{new}}(16, [\chi])\).