Properties

Label 16.5.f.a
Level 16
Weight 5
Character orbit 16.f
Analytic conductor 1.654
Analytic rank 0
Dimension 14
CM No
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 16 = 2^{4} \)
Weight: \( k \) = \( 5 \)
Character orbit: \([\chi]\) = 16.f (of order \(4\) and degree \(2\))

Newform invariants

Self dual: No
Analytic conductor: \(1.65391940934\)
Analytic rank: \(0\)
Dimension: \(14\)
Relative dimension: \(7\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 2^{21} \)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{13}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \( + \beta_{4} q^{2} \) \( -\beta_{6} q^{3} \) \( + ( -1 + 2 \beta_{3} - \beta_{11} ) q^{4} \) \( + ( \beta_{2} + \beta_{3} - \beta_{7} ) q^{5} \) \( + ( 5 - \beta_{2} - \beta_{3} - \beta_{10} + \beta_{12} ) q^{6} \) \( + ( -1 - 2 \beta_{1} + \beta_{4} + \beta_{5} + \beta_{7} + \beta_{11} - \beta_{12} - \beta_{13} ) q^{7} \) \( + ( -6 + 3 \beta_{1} - \beta_{2} + 2 \beta_{3} - \beta_{4} - \beta_{5} + 3 \beta_{6} + \beta_{7} + \beta_{8} - \beta_{9} + \beta_{10} + \beta_{11} - \beta_{12} + \beta_{13} ) q^{8} \) \( + ( \beta_{1} + \beta_{2} - 17 \beta_{3} - 7 \beta_{4} - \beta_{5} + \beta_{7} - \beta_{8} + \beta_{9} + 2 \beta_{10} + 3 \beta_{11} + 2 \beta_{13} ) q^{9} \) \(+O(q^{10})\) \( q\) \( + \beta_{4} q^{2} \) \( -\beta_{6} q^{3} \) \( + ( -1 + 2 \beta_{3} - \beta_{11} ) q^{4} \) \( + ( \beta_{2} + \beta_{3} - \beta_{7} ) q^{5} \) \( + ( 5 - \beta_{2} - \beta_{3} - \beta_{10} + \beta_{12} ) q^{6} \) \( + ( -1 - 2 \beta_{1} + \beta_{4} + \beta_{5} + \beta_{7} + \beta_{11} - \beta_{12} - \beta_{13} ) q^{7} \) \( + ( -6 + 3 \beta_{1} - \beta_{2} + 2 \beta_{3} - \beta_{4} - \beta_{5} + 3 \beta_{6} + \beta_{7} + \beta_{8} - \beta_{9} + \beta_{10} + \beta_{11} - \beta_{12} + \beta_{13} ) q^{8} \) \( + ( \beta_{1} + \beta_{2} - 17 \beta_{3} - 7 \beta_{4} - \beta_{5} + \beta_{7} - \beta_{8} + \beta_{9} + 2 \beta_{10} + 3 \beta_{11} + 2 \beta_{13} ) q^{9} \) \( + ( -10 - 3 \beta_{1} - \beta_{2} - 10 \beta_{3} + 3 \beta_{4} + 5 \beta_{6} + 2 \beta_{7} - 2 \beta_{8} + 3 \beta_{9} + 3 \beta_{10} - 2 \beta_{11} + \beta_{12} - \beta_{13} ) q^{10} \) \( + ( 11 + 7 \beta_{1} - 2 \beta_{2} + 13 \beta_{3} - 11 \beta_{4} - \beta_{5} - 2 \beta_{6} - 4 \beta_{7} + \beta_{8} - 2 \beta_{9} - \beta_{10} + 2 \beta_{11} - 2 \beta_{13} ) q^{11} \) \( + ( -23 - 2 \beta_{1} + 2 \beta_{2} - \beta_{3} + 2 \beta_{5} + 4 \beta_{6} - \beta_{7} - 4 \beta_{9} - 10 \beta_{10} + \beta_{11} - 2 \beta_{12} ) q^{12} \) \( + ( -11 - 18 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} - 8 \beta_{4} + 2 \beta_{6} + 4 \beta_{7} + 2 \beta_{8} + \beta_{9} - 2 \beta_{10} - 3 \beta_{11} + 2 \beta_{12} + 2 \beta_{13} ) q^{13} \) \( + ( 2 + \beta_{2} + 44 \beta_{3} + 4 \beta_{4} + 2 \beta_{5} - 7 \beta_{6} - 4 \beta_{7} + 2 \beta_{8} + 3 \beta_{9} + 13 \beta_{10} + 3 \beta_{12} - \beta_{13} ) q^{14} \) \( + ( 10 + 27 \beta_{1} - 4 \beta_{2} + 21 \beta_{3} + 20 \beta_{4} + 6 \beta_{5} - 2 \beta_{6} - \beta_{7} - \beta_{8} - 2 \beta_{10} - 7 \beta_{11} - \beta_{12} - 3 \beta_{13} ) q^{15} \) \( + ( -6 + 10 \beta_{1} + 4 \beta_{2} - 54 \beta_{3} - 12 \beta_{4} - 6 \beta_{5} - 14 \beta_{6} - 2 \beta_{7} + 4 \beta_{8} - 2 \beta_{9} - 8 \beta_{10} - 4 \beta_{12} + 2 \beta_{13} ) q^{16} \) \( + ( -8 - 25 \beta_{1} + 3 \beta_{2} - 14 \beta_{3} + 29 \beta_{4} - 5 \beta_{5} + 6 \beta_{6} - 3 \beta_{7} - 7 \beta_{8} + \beta_{9} + 4 \beta_{10} - 7 \beta_{11} + 2 \beta_{12} + 4 \beta_{13} ) q^{17} \) \( + ( 104 - 13 \beta_{1} + 8 \beta_{3} - 4 \beta_{4} - 2 \beta_{5} - 22 \beta_{6} - 10 \beta_{8} + 2 \beta_{9} - 4 \beta_{10} + 4 \beta_{11} + 4 \beta_{12} - 2 \beta_{13} ) q^{18} \) \( + ( -47 - 9 \beta_{1} - 6 \beta_{2} + 21 \beta_{3} + 47 \beta_{4} - 7 \beta_{5} + 3 \beta_{6} + 10 \beta_{7} + 5 \beta_{8} - 2 \beta_{9} - 2 \beta_{10} - 4 \beta_{11} - 2 \beta_{12} - 4 \beta_{13} ) q^{19} \) \( + ( 129 - 2 \beta_{1} + 2 \beta_{2} + 39 \beta_{3} - 2 \beta_{4} + 10 \beta_{5} - 10 \beta_{6} + \beta_{7} - 2 \beta_{8} - 2 \beta_{9} + 22 \beta_{10} + 5 \beta_{11} - 6 \beta_{12} + 2 \beta_{13} ) q^{20} \) \( + ( 18 + 76 \beta_{1} - 2 \beta_{2} + 4 \beta_{3} + 30 \beta_{4} + 2 \beta_{5} + 10 \beta_{6} - 4 \beta_{7} + 12 \beta_{8} + 2 \beta_{9} + 2 \beta_{10} + 4 \beta_{11} + 2 \beta_{12} + 6 \beta_{13} ) q^{21} \) \( + ( 65 + 8 \beta_{1} + 2 \beta_{2} - 147 \beta_{3} + 16 \beta_{4} + 8 \beta_{5} + 11 \beta_{6} + 4 \beta_{7} + 12 \beta_{8} + 5 \beta_{9} - 22 \beta_{10} + 4 \beta_{11} - 2 \beta_{12} - 3 \beta_{13} ) q^{22} \) \( + ( 47 - 80 \beta_{1} - 8 \beta_{2} - 2 \beta_{3} - 23 \beta_{4} + 13 \beta_{5} - 6 \beta_{6} - \beta_{7} - 6 \beta_{8} - 4 \beta_{9} - 2 \beta_{10} + 7 \beta_{11} + 5 \beta_{12} + \beta_{13} ) q^{23} \) \( + ( -140 - 10 \beta_{1} + 8 \beta_{2} + 244 \beta_{3} - 32 \beta_{4} - 14 \beta_{5} + 26 \beta_{6} - 4 \beta_{7} + 4 \beta_{8} - 2 \beta_{9} + 4 \beta_{10} - 6 \beta_{11} - 6 \beta_{13} ) q^{24} \) \( + ( 2 + 10 \beta_{1} + \beta_{3} - 98 \beta_{4} - 6 \beta_{5} + 12 \beta_{6} - 18 \beta_{8} - 4 \beta_{9} - 16 \beta_{10} - 8 \beta_{11} + 4 \beta_{12} - 8 \beta_{13} ) q^{25} \) \( + ( -238 - 13 \beta_{1} + 3 \beta_{2} - 146 \beta_{3} - 21 \beta_{4} - 10 \beta_{5} + 15 \beta_{6} - 18 \beta_{7} - 16 \beta_{8} - 11 \beta_{9} + 11 \beta_{10} + 14 \beta_{11} + \beta_{12} + 5 \beta_{13} ) q^{26} \) \( + ( -123 + 49 \beta_{1} + 6 \beta_{2} - 73 \beta_{3} - 113 \beta_{4} - 19 \beta_{5} + 6 \beta_{6} + 12 \beta_{7} + 7 \beta_{8} + 14 \beta_{9} + 4 \beta_{10} - 10 \beta_{11} - 4 \beta_{12} + 6 \beta_{13} ) q^{27} \) \( + ( -274 + 22 \beta_{1} - 10 \beta_{2} - 196 \beta_{3} + 14 \beta_{4} + 18 \beta_{5} + 2 \beta_{6} + 6 \beta_{7} - 10 \beta_{8} + 18 \beta_{9} - 6 \beta_{10} - 4 \beta_{11} + 6 \beta_{12} + 6 \beta_{13} ) q^{28} \) \( + ( 15 - 72 \beta_{1} - 4 \beta_{2} + 60 \beta_{3} - 28 \beta_{4} + 12 \beta_{5} - 4 \beta_{6} - 8 \beta_{7} + 24 \beta_{8} - 3 \beta_{9} + 28 \beta_{10} + 15 \beta_{11} - 12 \beta_{12} - 4 \beta_{13} ) q^{29} \) \( + ( -268 + 36 \beta_{1} - 9 \beta_{2} + 454 \beta_{3} + 8 \beta_{4} + 6 \beta_{5} + 13 \beta_{6} + 32 \beta_{7} + 26 \beta_{8} - 9 \beta_{9} + 3 \beta_{10} - 4 \beta_{11} - 11 \beta_{12} + 3 \beta_{13} ) q^{30} \) \( + ( 58 + 96 \beta_{1} + 20 \beta_{2} - 8 \beta_{3} + 38 \beta_{4} + 14 \beta_{5} + 2 \beta_{6} - 6 \beta_{7} - 12 \beta_{8} - 8 \beta_{9} + 10 \beta_{10} + 30 \beta_{11} + 2 \beta_{12} + 10 \beta_{13} ) q^{31} \) \( + ( 224 - 50 \beta_{1} - 22 \beta_{2} - 500 \beta_{3} + 14 \beta_{4} - 18 \beta_{5} + 2 \beta_{6} + 10 \beta_{7} + 2 \beta_{8} + 2 \beta_{9} - 2 \beta_{10} - 2 \beta_{11} + 18 \beta_{12} - 10 \beta_{13} ) q^{32} \) \( + ( 14 - 69 \beta_{1} - 7 \beta_{2} - 52 \beta_{3} + 125 \beta_{4} + 3 \beta_{5} - 38 \beta_{6} + 11 \beta_{7} - 19 \beta_{8} - \beta_{9} - 32 \beta_{10} + 23 \beta_{11} - 10 \beta_{12} - 16 \beta_{13} ) q^{33} \) \( + ( 520 + 4 \beta_{1} - 6 \beta_{2} + 272 \beta_{3} - 18 \beta_{4} - 14 \beta_{5} + 24 \beta_{6} - 4 \beta_{7} - 10 \beta_{8} - 4 \beta_{9} - 10 \beta_{10} - 32 \beta_{11} - 14 \beta_{12} + 8 \beta_{13} ) q^{34} \) \( + ( 90 - 74 \beta_{1} + 32 \beta_{2} - 122 \beta_{3} + 74 \beta_{4} - 26 \beta_{5} + 2 \beta_{6} - 44 \beta_{7} + 2 \beta_{8} + 8 \beta_{9} + 8 \beta_{10} + 16 \beta_{11} + 8 \beta_{12} + 12 \beta_{13} ) q^{35} \) \( + ( 846 + 4 \beta_{1} - 12 \beta_{2} + 317 \beta_{3} + 104 \beta_{4} + 20 \beta_{5} + 16 \beta_{6} - 5 \beta_{7} - 16 \beta_{8} - 20 \beta_{10} + 28 \beta_{12} - 8 \beta_{13} ) q^{36} \) \( + ( -62 + 136 \beta_{1} - 7 \beta_{2} + 111 \beta_{3} + 58 \beta_{4} + 22 \beta_{5} - 70 \beta_{6} + 33 \beta_{7} + 16 \beta_{8} - 6 \beta_{9} - 6 \beta_{10} - 12 \beta_{11} - 6 \beta_{12} - 26 \beta_{13} ) q^{37} \) \( + ( 245 + 40 \beta_{1} + 13 \beta_{2} - 665 \beta_{3} - 32 \beta_{4} - 4 \beta_{5} - 6 \beta_{6} - 28 \beta_{7} + 24 \beta_{8} - 18 \beta_{9} + 29 \beta_{10} - 36 \beta_{11} - 5 \beta_{12} + 14 \beta_{13} ) q^{38} \) \( + ( 91 - 130 \beta_{1} + 40 \beta_{2} + 32 \beta_{3} - 35 \beta_{4} + 13 \beta_{5} + 20 \beta_{6} - 19 \beta_{7} - 8 \beta_{8} + 24 \beta_{9} + 4 \beta_{10} - 51 \beta_{11} - 5 \beta_{12} + 11 \beta_{13} ) q^{39} \) \( + ( -328 + 52 \beta_{1} - 14 \beta_{2} + 804 \beta_{3} + 110 \beta_{4} - 16 \beta_{5} - 68 \beta_{6} - 6 \beta_{7} + 6 \beta_{8} + 32 \beta_{9} + 2 \beta_{10} + 4 \beta_{11} + 18 \beta_{12} + 16 \beta_{13} ) q^{40} \) \( + ( 10 + 76 \beta_{1} - 14 \beta_{2} + 34 \beta_{3} - 72 \beta_{4} - 76 \beta_{6} - 22 \beta_{7} - 4 \beta_{8} - 2 \beta_{9} + 60 \beta_{10} - 10 \beta_{11} - 20 \beta_{12} + 4 \beta_{13} ) q^{41} \) \( + ( -1220 - 6 \beta_{1} + 4 \beta_{2} - 452 \beta_{3} - 2 \beta_{4} - 40 \beta_{6} + 64 \beta_{7} - 8 \beta_{8} - 8 \beta_{9} - 60 \beta_{10} - 32 \beta_{11} - 20 \beta_{12} - 8 \beta_{13} ) q^{42} \) \( + ( 140 - 4 \beta_{1} + 8 \beta_{2} + 148 \beta_{3} - 124 \beta_{4} - 20 \beta_{5} + 8 \beta_{6} + 16 \beta_{7} + 4 \beta_{8} - 40 \beta_{9} - \beta_{10} + 24 \beta_{11} + 16 \beta_{12} + 8 \beta_{13} ) q^{43} \) \( + ( -1103 - 180 \beta_{1} + 8 \beta_{2} - 303 \beta_{3} + 102 \beta_{4} + 20 \beta_{5} - 18 \beta_{6} - 5 \beta_{7} - 10 \beta_{8} - 10 \beta_{9} + 80 \beta_{10} - 5 \beta_{11} + 4 \beta_{12} - 34 \beta_{13} ) q^{44} \) \( + ( 111 - 34 \beta_{1} - 18 \beta_{2} + 130 \beta_{3} - 28 \beta_{4} + 4 \beta_{5} - 18 \beta_{6} - 36 \beta_{7} + 2 \beta_{8} - 5 \beta_{9} - 134 \beta_{10} - 17 \beta_{11} + 22 \beta_{12} - 18 \beta_{13} ) q^{45} \) \( + ( -358 - 8 \beta_{1} + 17 \beta_{2} + 1316 \beta_{3} - 36 \beta_{4} + 6 \beta_{5} + 5 \beta_{6} - 92 \beta_{7} + 6 \beta_{8} - 17 \beta_{9} - 107 \beta_{10} + 32 \beta_{11} - 5 \beta_{12} + 3 \beta_{13} ) q^{46} \) \( + ( -34 - 12 \beta_{1} - 28 \beta_{2} - 468 \beta_{3} + 50 \beta_{4} + 2 \beta_{5} + 26 \beta_{6} + 58 \beta_{7} + 48 \beta_{9} - 6 \beta_{10} - 18 \beta_{11} + 10 \beta_{12} + 10 \beta_{13} ) q^{47} \) \( + ( 528 + 202 \beta_{1} + 26 \beta_{2} - 1452 \beta_{3} - 66 \beta_{4} + 2 \beta_{5} + 86 \beta_{6} + 2 \beta_{7} - 6 \beta_{8} + 30 \beta_{9} + 102 \beta_{10} + 10 \beta_{11} - 14 \beta_{12} + 10 \beta_{13} ) q^{48} \) \( + ( 79 + 102 \beta_{1} - 26 \beta_{2} - 4 \beta_{3} - 22 \beta_{4} - 10 \beta_{5} + 100 \beta_{6} + 2 \beta_{7} + 10 \beta_{8} - 14 \beta_{9} + 112 \beta_{10} + 26 \beta_{11} + 12 \beta_{12} ) q^{49} \) \( + ( 1448 - \beta_{1} + 28 \beta_{2} + 336 \beta_{3} + 8 \beta_{4} + 8 \beta_{5} + 124 \beta_{6} + 32 \beta_{7} + 8 \beta_{8} - 12 \beta_{9} + 44 \beta_{10} + 96 \beta_{11} - 12 \beta_{12} + 4 \beta_{13} ) q^{50} \) \( + ( -231 + 31 \beta_{1} - 50 \beta_{2} + 217 \beta_{3} - 93 \beta_{4} + 5 \beta_{5} - 10 \beta_{6} + 26 \beta_{7} - 3 \beta_{8} + 2 \beta_{9} + 2 \beta_{10} + 4 \beta_{11} + 2 \beta_{12} + 24 \beta_{13} ) q^{51} \) \( + ( 1345 - 142 \beta_{1} + 30 \beta_{2} + 641 \beta_{3} - 282 \beta_{4} - 18 \beta_{5} + 30 \beta_{6} - \beta_{7} - 2 \beta_{8} + 38 \beta_{9} - 118 \beta_{10} - 35 \beta_{11} - 18 \beta_{12} + 2 \beta_{13} ) q^{52} \) \( + ( -134 - 156 \beta_{1} + 13 \beta_{2} + 75 \beta_{3} - 42 \beta_{4} - 22 \beta_{5} + 202 \beta_{6} - 19 \beta_{7} - 12 \beta_{8} - 14 \beta_{9} - 14 \beta_{10} - 28 \beta_{11} - 14 \beta_{12} + 6 \beta_{13} ) q^{53} \) \( + ( 824 - 16 \beta_{1} - 34 \beta_{2} - 1920 \beta_{3} - 40 \beta_{4} - 8 \beta_{5} - 98 \beta_{6} + 60 \beta_{7} - 28 \beta_{8} - 14 \beta_{9} + 118 \beta_{10} + 124 \beta_{11} + 2 \beta_{12} - 6 \beta_{13} ) q^{54} \) \( + ( -701 + 210 \beta_{1} - 40 \beta_{2} - 68 \beta_{3} + 101 \beta_{4} - 35 \beta_{5} + 32 \beta_{6} + 65 \beta_{7} + 12 \beta_{8} - 48 \beta_{9} + 24 \beta_{10} + 49 \beta_{11} - 17 \beta_{12} - 9 \beta_{13} ) q^{55} \) \( + ( -552 - 126 \beta_{1} - 26 \beta_{2} + 1784 \beta_{3} - 322 \beta_{4} + 42 \beta_{5} - 118 \beta_{6} + 38 \beta_{7} - 14 \beta_{8} - 54 \beta_{9} - 30 \beta_{10} + 42 \beta_{11} - 26 \beta_{12} - 26 \beta_{13} ) q^{56} \) \( + ( -16 - 65 \beta_{1} - \beta_{2} - 98 \beta_{3} + 279 \beta_{4} + 17 \beta_{5} + 176 \beta_{6} + 47 \beta_{7} + 33 \beta_{8} + 31 \beta_{9} - 146 \beta_{10} + 13 \beta_{11} + 16 \beta_{12} + 14 \beta_{13} ) q^{57} \) \( + ( -1454 + 43 \beta_{1} - 11 \beta_{2} - 658 \beta_{3} + 59 \beta_{4} + 18 \beta_{5} - 131 \beta_{6} - 110 \beta_{7} + 48 \beta_{8} + 63 \beta_{9} - 3 \beta_{10} + 18 \beta_{11} + 7 \beta_{12} - \beta_{13} ) q^{58} \) \( + ( -242 - 114 \beta_{1} - 32 \beta_{2} - 330 \beta_{3} + 406 \beta_{4} + 66 \beta_{5} - 32 \beta_{6} - 64 \beta_{7} - 30 \beta_{8} + 56 \beta_{9} - 13 \beta_{10} - 68 \beta_{11} + 12 \beta_{12} - 32 \beta_{13} ) q^{59} \) \( + ( -1932 + 482 \beta_{1} + 34 \beta_{2} - 870 \beta_{3} - 322 \beta_{4} - 74 \beta_{5} - 94 \beta_{6} - 36 \beta_{7} + 38 \beta_{8} - 46 \beta_{9} + 46 \beta_{10} + 34 \beta_{11} - 6 \beta_{12} + 46 \beta_{13} ) q^{60} \) \( + ( -199 + 202 \beta_{1} + 34 \beta_{2} - 362 \beta_{3} + 180 \beta_{4} - 28 \beta_{5} + 34 \beta_{6} + 68 \beta_{7} - 74 \beta_{8} + 21 \beta_{9} + 294 \beta_{10} - 31 \beta_{11} + 10 \beta_{12} + 34 \beta_{13} ) q^{61} \) \( + ( -856 - 104 \beta_{1} + 36 \beta_{2} + 1624 \beta_{3} - 24 \beta_{4} - 36 \beta_{5} + 8 \beta_{6} + 96 \beta_{7} - 92 \beta_{8} + 64 \beta_{9} - 36 \beta_{10} - 88 \beta_{11} + 52 \beta_{12} - 8 \beta_{13} ) q^{62} \) \( + ( -114 - 317 \beta_{1} + 4 \beta_{2} + 1501 \beta_{3} - 160 \beta_{4} - 46 \beta_{5} - 22 \beta_{6} - 101 \beta_{7} + 47 \beta_{8} - 80 \beta_{9} - 54 \beta_{10} - 51 \beta_{11} - 21 \beta_{12} - 55 \beta_{13} ) q^{63} \) \( + ( 924 - 512 \beta_{1} + 52 \beta_{2} - 1636 \beta_{3} + 212 \beta_{4} + 64 \beta_{5} + 64 \beta_{6} - 80 \beta_{7} + 12 \beta_{8} - 56 \beta_{9} - 92 \beta_{10} + 4 \beta_{11} - 28 \beta_{12} + 24 \beta_{13} ) q^{64} \) \( + ( -204 + 104 \beta_{1} + 62 \beta_{2} + 214 \beta_{3} - 484 \beta_{4} + 12 \beta_{5} - 180 \beta_{6} - 26 \beta_{7} + 64 \beta_{8} + 14 \beta_{9} - 228 \beta_{10} - 122 \beta_{11} + 12 \beta_{12} + 60 \beta_{13} ) q^{65} \) \( + ( 2120 - 12 \beta_{1} - 6 \beta_{2} + 1480 \beta_{3} + 70 \beta_{4} + 54 \beta_{5} - 76 \beta_{6} - 92 \beta_{7} + 34 \beta_{8} + 8 \beta_{9} + 78 \beta_{10} - 128 \beta_{11} + 74 \beta_{12} - 52 \beta_{13} ) q^{66} \) \( + ( 643 + 341 \beta_{1} + 10 \beta_{2} - 365 \beta_{3} - 223 \beta_{4} + 119 \beta_{5} + 7 \beta_{6} + 78 \beta_{7} - \beta_{8} - 42 \beta_{9} - 42 \beta_{10} - 84 \beta_{11} - 42 \beta_{12} - 88 \beta_{13} ) q^{67} \) \( + ( 1554 + 448 \beta_{1} - 40 \beta_{2} + 624 \beta_{3} + 516 \beta_{4} - 72 \beta_{5} - 20 \beta_{6} + 40 \beta_{7} + 92 \beta_{8} - 52 \beta_{9} - 8 \beta_{10} + 34 \beta_{11} - 80 \beta_{12} + 12 \beta_{13} ) q^{68} \) \( + ( 494 - 604 \beta_{1} + 14 \beta_{2} - 696 \beta_{3} - 278 \beta_{4} - 74 \beta_{5} - 338 \beta_{6} - 112 \beta_{7} - 92 \beta_{8} + 54 \beta_{9} + 54 \beta_{10} + 108 \beta_{11} + 54 \beta_{12} + 98 \beta_{13} ) q^{69} \) \( + ( 946 - 232 \beta_{1} - 38 \beta_{2} - 1674 \beta_{3} + 200 \beta_{4} + 24 \beta_{5} + 76 \beta_{6} - 8 \beta_{7} - 112 \beta_{8} + 100 \beta_{9} - 38 \beta_{10} - 184 \beta_{11} + 22 \beta_{12} - 60 \beta_{13} ) q^{70} \) \( + ( 1807 + 608 \beta_{1} - 72 \beta_{2} + 14 \beta_{3} + 73 \beta_{4} - 51 \beta_{5} - 150 \beta_{6} - 81 \beta_{7} + 42 \beta_{8} + 28 \beta_{9} - 50 \beta_{10} + 119 \beta_{11} + 53 \beta_{12} - 47 \beta_{13} ) q^{71} \) \( + ( -1118 + 213 \beta_{1} + 75 \beta_{2} + 1886 \beta_{3} + 727 \beta_{4} + 69 \beta_{5} + 205 \beta_{6} + 13 \beta_{7} - 35 \beta_{8} - 51 \beta_{9} - 127 \beta_{10} - 77 \beta_{11} - 37 \beta_{12} + 3 \beta_{13} ) q^{72} \) \( + ( -52 - 409 \beta_{1} + 83 \beta_{2} + 68 \beta_{3} + 367 \beta_{4} - 7 \beta_{5} - 208 \beta_{6} + 27 \beta_{7} + 57 \beta_{8} - 37 \beta_{9} + 254 \beta_{10} + 65 \beta_{11} + 64 \beta_{12} - 18 \beta_{13} ) q^{73} \) \( + ( -1714 + 37 \beta_{1} - 23 \beta_{2} - 498 \beta_{3} - 13 \beta_{4} + 16 \beta_{5} + 79 \beta_{6} + 86 \beta_{7} + 50 \beta_{8} - 7 \beta_{9} + 181 \beta_{10} + 10 \beta_{11} + 103 \beta_{12} + 45 \beta_{13} ) q^{74} \) \( + ( 1300 - 52 \beta_{1} - 4 \beta_{2} + 1024 \beta_{3} + 720 \beta_{4} + 112 \beta_{5} - 4 \beta_{6} - 8 \beta_{7} - 12 \beta_{8} - 12 \beta_{9} + 21 \beta_{10} + 128 \beta_{11} - 116 \beta_{12} - 4 \beta_{13} ) q^{75} \) \( + ( -2015 - 694 \beta_{1} - 58 \beta_{2} - 1153 \beta_{3} + 204 \beta_{4} - 122 \beta_{5} + 176 \beta_{6} + 75 \beta_{7} + 44 \beta_{8} + 24 \beta_{9} - 126 \beta_{10} + 5 \beta_{11} - 46 \beta_{12} + 68 \beta_{13} ) q^{76} \) \( + ( -384 + 530 \beta_{1} + 66 \beta_{2} - 570 \beta_{3} + 28 \beta_{4} - 68 \beta_{5} + 66 \beta_{6} + 132 \beta_{7} - 50 \beta_{8} - 2 \beta_{9} - 330 \beta_{10} + 104 \beta_{11} - 102 \beta_{12} + 66 \beta_{13} ) q^{77} \) \( + ( -558 + 32 \beta_{1} - 119 \beta_{2} + 1164 \beta_{3} + 212 \beta_{4} - 38 \beta_{5} + 105 \beta_{6} + 12 \beta_{7} - 70 \beta_{8} + 35 \beta_{9} + 293 \beta_{10} + 64 \beta_{11} - 21 \beta_{12} - 17 \beta_{13} ) q^{78} \) \( + ( -184 - 348 \beta_{1} + 8 \beta_{2} - 1948 \beta_{3} - 648 \beta_{4} - 112 \beta_{5} - 148 \beta_{6} - 76 \beta_{7} + 4 \beta_{8} - 40 \beta_{9} + 116 \beta_{10} + 12 \beta_{11} - 36 \beta_{12} + 4 \beta_{13} ) q^{79} \) \( + ( 476 + 882 \beta_{1} - 114 \beta_{2} - 1232 \beta_{3} - 318 \beta_{4} + 90 \beta_{5} - 194 \beta_{6} + 66 \beta_{7} - 58 \beta_{8} - 66 \beta_{9} - 230 \beta_{10} - 90 \beta_{11} + 22 \beta_{12} - 54 \beta_{13} ) q^{80} \) \( + ( -135 + 35 \beta_{1} + 93 \beta_{2} + 280 \beta_{3} - 411 \beta_{4} + 91 \beta_{5} + 298 \beta_{6} - 41 \beta_{7} + 69 \beta_{8} + 83 \beta_{9} + 288 \beta_{10} - 61 \beta_{11} - 42 \beta_{12} - 32 \beta_{13} ) q^{81} \) \( + ( 1240 + 216 \beta_{1} - 148 \beta_{2} + 640 \beta_{3} + 32 \beta_{4} - 12 \beta_{5} - 208 \beta_{6} + 96 \beta_{7} + 132 \beta_{8} + 56 \beta_{9} - 172 \beta_{10} + 72 \beta_{11} + 12 \beta_{12} + 32 \beta_{13} ) q^{82} \) \( + ( -1300 - 108 \beta_{1} + 20 \beta_{2} + 1472 \beta_{3} - 424 \beta_{4} + 104 \beta_{5} - 21 \beta_{6} + 40 \beta_{7} - 100 \beta_{8} + 36 \beta_{9} + 36 \beta_{10} + 72 \beta_{11} + 36 \beta_{12} - 60 \beta_{13} ) q^{83} \) \( + ( 1170 - 360 \beta_{1} - 24 \beta_{2} + 630 \beta_{3} - 1216 \beta_{4} - 120 \beta_{5} + 48 \beta_{6} - 34 \beta_{7} + 32 \beta_{8} - 112 \beta_{9} + 312 \beta_{10} + 86 \beta_{11} + 88 \beta_{12} + 32 \beta_{13} ) q^{84} \) \( + ( 434 - 544 \beta_{1} - 14 \beta_{2} - 520 \beta_{3} - 382 \beta_{4} + 46 \beta_{5} + 394 \beta_{6} + 88 \beta_{7} - 136 \beta_{8} + 26 \beta_{9} + 26 \beta_{10} + 52 \beta_{11} + 26 \beta_{12} - 74 \beta_{13} ) q^{85} \) \( + ( -271 - 32 \beta_{1} + 152 \beta_{2} - 1579 \beta_{3} - 64 \beta_{4} - 128 \beta_{5} + 265 \beta_{6} - 80 \beta_{7} - 80 \beta_{8} - 9 \beta_{9} - 200 \beta_{10} + 48 \beta_{11} + 40 \beta_{12} + 87 \beta_{13} ) q^{86} \) \( + ( -3353 + 480 \beta_{1} + 88 \beta_{2} + 94 \beta_{3} - 47 \beta_{4} - 75 \beta_{5} - 62 \beta_{6} + 7 \beta_{7} + 58 \beta_{8} + 28 \beta_{9} - 122 \beta_{10} - 113 \beta_{11} - 35 \beta_{12} + 25 \beta_{13} ) q^{87} \) \( + ( -44 - 472 \beta_{1} + 34 \beta_{2} + 556 \beta_{3} - 1014 \beta_{4} + 68 \beta_{5} + 196 \beta_{6} - 162 \beta_{7} + 18 \beta_{8} + 160 \beta_{9} + 202 \beta_{10} - 100 \beta_{11} + 58 \beta_{12} + 104 \beta_{13} ) q^{88} \) \( + ( 92 + 193 \beta_{1} + 13 \beta_{2} + 420 \beta_{3} + 457 \beta_{4} + 47 \beta_{5} + 256 \beta_{6} - 187 \beta_{7} + 127 \beta_{8} - 75 \beta_{9} - 318 \beta_{10} + 63 \beta_{11} - 112 \beta_{12} + 50 \beta_{13} ) q^{89} \) \( + ( -346 + 37 \beta_{1} + \beta_{2} + 218 \beta_{3} + 109 \beta_{4} + 98 \beta_{5} + 405 \beta_{6} - 6 \beta_{7} + 64 \beta_{8} - 137 \beta_{9} - 199 \beta_{10} + 26 \beta_{11} - 53 \beta_{12} - 121 \beta_{13} ) q^{90} \) \( + ( -2124 - 468 \beta_{1} + 28 \beta_{2} - 2176 \beta_{3} + 144 \beta_{4} + 48 \beta_{5} + 28 \beta_{6} + 56 \beta_{7} + 20 \beta_{8} - 108 \beta_{9} - 30 \beta_{10} + 64 \beta_{11} + 44 \beta_{12} + 28 \beta_{13} ) q^{91} \) \( + ( -602 + 1198 \beta_{1} - 34 \beta_{2} + 1108 \beta_{3} - 306 \beta_{4} + 42 \beta_{5} + 226 \beta_{6} + 46 \beta_{7} + 38 \beta_{8} + 50 \beta_{9} - 270 \beta_{10} - 116 \beta_{11} - 34 \beta_{12} - 202 \beta_{13} ) q^{92} \) \( + ( 796 + 268 \beta_{1} - 100 \beta_{2} + 716 \beta_{3} - 136 \beta_{4} - 40 \beta_{5} - 100 \beta_{6} - 200 \beta_{7} - 76 \beta_{8} - 32 \beta_{9} + 308 \beta_{10} - 108 \beta_{11} + 140 \beta_{12} - 100 \beta_{13} ) q^{93} \) \( + ( 24 - 264 \beta_{1} - 20 \beta_{2} - 8 \beta_{3} - 40 \beta_{4} + 76 \beta_{5} - 368 \beta_{6} - 32 \beta_{7} - 44 \beta_{8} - 168 \beta_{9} + 132 \beta_{10} + 104 \beta_{11} - 52 \beta_{12} - 16 \beta_{13} ) q^{94} \) \( + ( -216 - 345 \beta_{1} - 64 \beta_{2} + 2697 \beta_{3} + 278 \beta_{4} - 24 \beta_{5} + 96 \beta_{6} + 289 \beta_{7} + 23 \beta_{8} + 208 \beta_{9} + 48 \beta_{10} - \beta_{11} + 81 \beta_{12} + 63 \beta_{13} ) q^{95} \) \( + ( -116 - 1396 \beta_{1} - 396 \beta_{3} + 680 \beta_{4} + 28 \beta_{5} - 348 \beta_{6} + 252 \beta_{7} - 88 \beta_{8} + 156 \beta_{9} + 344 \beta_{10} + 64 \beta_{11} - 32 \beta_{12} + 20 \beta_{13} ) q^{96} \) \( + ( 608 + 819 \beta_{1} - 201 \beta_{2} - 78 \beta_{3} + 25 \beta_{4} - 81 \beta_{5} - 330 \beta_{6} + \beta_{7} + 61 \beta_{8} - 83 \beta_{9} - 212 \beta_{10} + 237 \beta_{11} + 82 \beta_{12} - 36 \beta_{13} ) q^{97} \) \( + ( -976 - 24 \beta_{1} + 180 \beta_{2} - 1600 \beta_{3} - 37 \beta_{4} + 20 \beta_{5} - 128 \beta_{6} + 24 \beta_{7} + 28 \beta_{8} + 72 \beta_{9} - 180 \beta_{10} - 124 \beta_{12} + 128 \beta_{13} ) q^{98} \) \( + ( 3370 - 106 \beta_{1} - 88 \beta_{2} - 3506 \beta_{3} - 334 \beta_{4} - 98 \beta_{5} - 5 \beta_{6} - 156 \beta_{7} + 2 \beta_{8} + 32 \beta_{9} + 32 \beta_{10} + 64 \beta_{11} + 32 \beta_{12} + 244 \beta_{13} ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(14q \) \(\mathstrut -\mathstrut 2q^{2} \) \(\mathstrut -\mathstrut 2q^{3} \) \(\mathstrut -\mathstrut 8q^{4} \) \(\mathstrut -\mathstrut 2q^{5} \) \(\mathstrut +\mathstrut 64q^{6} \) \(\mathstrut -\mathstrut 4q^{7} \) \(\mathstrut -\mathstrut 92q^{8} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(14q \) \(\mathstrut -\mathstrut 2q^{2} \) \(\mathstrut -\mathstrut 2q^{3} \) \(\mathstrut -\mathstrut 8q^{4} \) \(\mathstrut -\mathstrut 2q^{5} \) \(\mathstrut +\mathstrut 64q^{6} \) \(\mathstrut -\mathstrut 4q^{7} \) \(\mathstrut -\mathstrut 92q^{8} \) \(\mathstrut -\mathstrut 100q^{10} \) \(\mathstrut +\mathstrut 94q^{11} \) \(\mathstrut -\mathstrut 332q^{12} \) \(\mathstrut -\mathstrut 2q^{13} \) \(\mathstrut +\mathstrut 44q^{14} \) \(\mathstrut -\mathstrut 168q^{16} \) \(\mathstrut -\mathstrut 4q^{17} \) \(\mathstrut +\mathstrut 1390q^{18} \) \(\mathstrut -\mathstrut 706q^{19} \) \(\mathstrut +\mathstrut 1900q^{20} \) \(\mathstrut -\mathstrut 164q^{21} \) \(\mathstrut +\mathstrut 900q^{22} \) \(\mathstrut +\mathstrut 1148q^{23} \) \(\mathstrut -\mathstrut 1872q^{24} \) \(\mathstrut -\mathstrut 3416q^{26} \) \(\mathstrut -\mathstrut 1664q^{27} \) \(\mathstrut -\mathstrut 3784q^{28} \) \(\mathstrut +\mathstrut 862q^{29} \) \(\mathstrut -\mathstrut 3740q^{30} \) \(\mathstrut +\mathstrut 3208q^{32} \) \(\mathstrut -\mathstrut 4q^{33} \) \(\mathstrut +\mathstrut 7508q^{34} \) \(\mathstrut +\mathstrut 1340q^{35} \) \(\mathstrut +\mathstrut 11468q^{36} \) \(\mathstrut -\mathstrut 1826q^{37} \) \(\mathstrut +\mathstrut 3568q^{38} \) \(\mathstrut +\mathstrut 2684q^{39} \) \(\mathstrut -\mathstrut 5144q^{40} \) \(\mathstrut -\mathstrut 17064q^{42} \) \(\mathstrut +\mathstrut 1694q^{43} \) \(\mathstrut -\mathstrut 14636q^{44} \) \(\mathstrut +\mathstrut 1410q^{45} \) \(\mathstrut -\mathstrut 5316q^{46} \) \(\mathstrut +\mathstrut 6888q^{48} \) \(\mathstrut +\mathstrut 682q^{49} \) \(\mathstrut +\mathstrut 20070q^{50} \) \(\mathstrut -\mathstrut 3012q^{51} \) \(\mathstrut +\mathstrut 20452q^{52} \) \(\mathstrut -\mathstrut 482q^{53} \) \(\mathstrut +\mathstrut 10784q^{54} \) \(\mathstrut -\mathstrut 11780q^{55} \) \(\mathstrut -\mathstrut 6952q^{56} \) \(\mathstrut -\mathstrut 20456q^{58} \) \(\mathstrut -\mathstrut 2786q^{59} \) \(\mathstrut -\mathstrut 29920q^{60} \) \(\mathstrut -\mathstrut 3778q^{61} \) \(\mathstrut -\mathstrut 11472q^{62} \) \(\mathstrut +\mathstrut 15808q^{64} \) \(\mathstrut -\mathstrut 2020q^{65} \) \(\mathstrut +\mathstrut 30148q^{66} \) \(\mathstrut +\mathstrut 7998q^{67} \) \(\mathstrut +\mathstrut 18032q^{68} \) \(\mathstrut +\mathstrut 9628q^{69} \) \(\mathstrut +\mathstrut 15296q^{70} \) \(\mathstrut +\mathstrut 19964q^{71} \) \(\mathstrut -\mathstrut 17708q^{72} \) \(\mathstrut -\mathstrut 23780q^{74} \) \(\mathstrut +\mathstrut 17570q^{75} \) \(\mathstrut -\mathstrut 23996q^{76} \) \(\mathstrut -\mathstrut 9508q^{77} \) \(\mathstrut -\mathstrut 8052q^{78} \) \(\mathstrut +\mathstrut 1384q^{80} \) \(\mathstrut +\mathstrut 1454q^{81} \) \(\mathstrut +\mathstrut 16016q^{82} \) \(\mathstrut -\mathstrut 17282q^{83} \) \(\mathstrut +\mathstrut 19624q^{84} \) \(\mathstrut +\mathstrut 9948q^{85} \) \(\mathstrut -\mathstrut 4796q^{86} \) \(\mathstrut -\mathstrut 49284q^{87} \) \(\mathstrut +\mathstrut 7288q^{88} \) \(\mathstrut -\mathstrut 5416q^{90} \) \(\mathstrut -\mathstrut 28036q^{91} \) \(\mathstrut -\mathstrut 14632q^{92} \) \(\mathstrut +\mathstrut 8896q^{93} \) \(\mathstrut +\mathstrut 432q^{94} \) \(\mathstrut +\mathstrut 6064q^{96} \) \(\mathstrut -\mathstrut 4q^{97} \) \(\mathstrut -\mathstrut 12246q^{98} \) \(\mathstrut +\mathstrut 49214q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{14}\mathstrut -\mathstrut \) \(4\) \(x^{13}\mathstrut +\mathstrut \) \(15\) \(x^{12}\mathstrut -\mathstrut \) \(34\) \(x^{11}\mathstrut +\mathstrut \) \(62\) \(x^{10}\mathstrut -\mathstrut \) \(312\) \(x^{9}\mathstrut +\mathstrut \) \(1432\) \(x^{8}\mathstrut -\mathstrut \) \(4960\) \(x^{7}\mathstrut +\mathstrut \) \(11456\) \(x^{6}\mathstrut -\mathstrut \) \(19968\) \(x^{5}\mathstrut +\mathstrut \) \(31744\) \(x^{4}\mathstrut -\mathstrut \) \(139264\) \(x^{3}\mathstrut +\mathstrut \) \(491520\) \(x^{2}\mathstrut -\mathstrut \) \(1048576\) \(x\mathstrut +\mathstrut \) \(2097152\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\(-\)\(2291\) \(\nu^{13}\mathstrut -\mathstrut \) \(13836\) \(\nu^{12}\mathstrut +\mathstrut \) \(73187\) \(\nu^{11}\mathstrut -\mathstrut \) \(134754\) \(\nu^{10}\mathstrut -\mathstrut \) \(13546\) \(\nu^{9}\mathstrut +\mathstrut \) \(515160\) \(\nu^{8}\mathstrut +\mathstrut \) \(2711160\) \(\nu^{7}\mathstrut -\mathstrut \) \(25589280\) \(\nu^{6}\mathstrut +\mathstrut \) \(58745024\) \(\nu^{5}\mathstrut -\mathstrut \) \(46252032\) \(\nu^{4}\mathstrut -\mathstrut \) \(25039872\) \(\nu^{3}\mathstrut -\mathstrut \) \(96387072\) \(\nu^{2}\mathstrut +\mathstrut \) \(2307620864\) \(\nu\mathstrut -\mathstrut \) \(8127774720\)\()/\)\(678952960\)
\(\beta_{2}\)\(=\)\((\)\(2559\) \(\nu^{13}\mathstrut +\mathstrut \) \(209196\) \(\nu^{12}\mathstrut -\mathstrut \) \(627631\) \(\nu^{11}\mathstrut +\mathstrut \) \(1042042\) \(\nu^{10}\mathstrut -\mathstrut \) \(73358\) \(\nu^{9}\mathstrut +\mathstrut \) \(4773352\) \(\nu^{8}\mathstrut -\mathstrut \) \(52296280\) \(\nu^{7}\mathstrut +\mathstrut \) \(214229280\) \(\nu^{6}\mathstrut -\mathstrut \) \(430709696\) \(\nu^{5}\mathstrut +\mathstrut \) \(72770560\) \(\nu^{4}\mathstrut -\mathstrut \) \(10267648\) \(\nu^{3}\mathstrut +\mathstrut \) \(3766566912\) \(\nu^{2}\mathstrut -\mathstrut \) \(21547548672\) \(\nu\mathstrut +\mathstrut \) \(48311304192\)\()/\)\(678952960\)
\(\beta_{3}\)\(=\)\((\)\(2875\) \(\nu^{13}\mathstrut -\mathstrut \) \(13444\) \(\nu^{12}\mathstrut +\mathstrut \) \(26581\) \(\nu^{11}\mathstrut -\mathstrut \) \(16062\) \(\nu^{10}\mathstrut +\mathstrut \) \(24954\) \(\nu^{9}\mathstrut -\mathstrut \) \(748984\) \(\nu^{8}\mathstrut +\mathstrut \) \(4619080\) \(\nu^{7}\mathstrut -\mathstrut \) \(10623840\) \(\nu^{6}\mathstrut +\mathstrut \) \(11499840\) \(\nu^{5}\mathstrut -\mathstrut \) \(5960704\) \(\nu^{4}\mathstrut +\mathstrut \) \(51930112\) \(\nu^{3}\mathstrut -\mathstrut \) \(429211648\) \(\nu^{2}\mathstrut +\mathstrut \) \(1316257792\) \(\nu\mathstrut -\mathstrut \) \(1279524864\)\()/\)\(678952960\)
\(\beta_{4}\)\(=\)\((\)\(7471\) \(\nu^{13}\mathstrut -\mathstrut \) \(6884\) \(\nu^{12}\mathstrut +\mathstrut \) \(4513\) \(\nu^{11}\mathstrut -\mathstrut \) \(41366\) \(\nu^{10}\mathstrut +\mathstrut \) \(334706\) \(\nu^{9}\mathstrut -\mathstrut \) \(2131320\) \(\nu^{8}\mathstrut +\mathstrut \) \(4706600\) \(\nu^{7}\mathstrut -\mathstrut \) \(103520\) \(\nu^{6}\mathstrut +\mathstrut \) \(597056\) \(\nu^{5}\mathstrut -\mathstrut \) \(57182208\) \(\nu^{4}\mathstrut +\mathstrut \) \(189473792\) \(\nu^{3}\mathstrut -\mathstrut \) \(625000448\) \(\nu^{2}\mathstrut +\mathstrut \) \(238452736\) \(\nu\mathstrut +\mathstrut \) \(2696151040\)\()/\)\(678952960\)
\(\beta_{5}\)\(=\)\((\)\(-\)\(739\) \(\nu^{13}\mathstrut +\mathstrut \) \(1452\) \(\nu^{12}\mathstrut -\mathstrut \) \(2221\) \(\nu^{11}\mathstrut +\mathstrut \) \(4102\) \(\nu^{10}\mathstrut -\mathstrut \) \(25690\) \(\nu^{9}\mathstrut +\mathstrut \) \(205736\) \(\nu^{8}\mathstrut -\mathstrut \) \(666120\) \(\nu^{7}\mathstrut +\mathstrut \) \(766240\) \(\nu^{6}\mathstrut -\mathstrut \) \(864064\) \(\nu^{5}\mathstrut +\mathstrut \) \(4510208\) \(\nu^{4}\mathstrut -\mathstrut \) \(17243136\) \(\nu^{3}\mathstrut +\mathstrut \) \(75300864\) \(\nu^{2}\mathstrut +\mathstrut \) \(276922368\) \(\nu\mathstrut -\mathstrut \) \(198180864\)\()/48496640\)
\(\beta_{6}\)\(=\)\((\)\(-\)\(11609\) \(\nu^{13}\mathstrut +\mathstrut \) \(81068\) \(\nu^{12}\mathstrut -\mathstrut \) \(216535\) \(\nu^{11}\mathstrut +\mathstrut \) \(92170\) \(\nu^{10}\mathstrut -\mathstrut \) \(445086\) \(\nu^{9}\mathstrut +\mathstrut \) \(4629992\) \(\nu^{8}\mathstrut -\mathstrut \) \(22287000\) \(\nu^{7}\mathstrut +\mathstrut \) \(63665440\) \(\nu^{6}\mathstrut -\mathstrut \) \(81223104\) \(\nu^{5}\mathstrut +\mathstrut \) \(58993664\) \(\nu^{4}\mathstrut -\mathstrut \) \(414198784\) \(\nu^{3}\mathstrut +\mathstrut \) \(2235252736\) \(\nu^{2}\mathstrut -\mathstrut \) \(5795840000\) \(\nu\mathstrut +\mathstrut \) \(7904428032\)\()/\)\(678952960\)
\(\beta_{7}\)\(=\)\((\)\(-\)\(17845\) \(\nu^{13}\mathstrut +\mathstrut \) \(27996\) \(\nu^{12}\mathstrut -\mathstrut \) \(118459\) \(\nu^{11}\mathstrut +\mathstrut \) \(66818\) \(\nu^{10}\mathstrut +\mathstrut \) \(149274\) \(\nu^{9}\mathstrut +\mathstrut \) \(3146376\) \(\nu^{8}\mathstrut -\mathstrut \) \(12122680\) \(\nu^{7}\mathstrut +\mathstrut \) \(32805280\) \(\nu^{6}\mathstrut -\mathstrut \) \(20326080\) \(\nu^{5}\mathstrut -\mathstrut \) \(66960384\) \(\nu^{4}\mathstrut +\mathstrut \) \(221668352\) \(\nu^{3}\mathstrut +\mathstrut \) \(1140867072\) \(\nu^{2}\mathstrut -\mathstrut \) \(3096936448\) \(\nu\mathstrut +\mathstrut \) \(1436286976\)\()/\)\(678952960\)
\(\beta_{8}\)\(=\)\((\)\(-\)\(19011\) \(\nu^{13}\mathstrut -\mathstrut \) \(91628\) \(\nu^{12}\mathstrut +\mathstrut \) \(527155\) \(\nu^{11}\mathstrut -\mathstrut \) \(1059490\) \(\nu^{10}\mathstrut +\mathstrut \) \(1147766\) \(\nu^{9}\mathstrut +\mathstrut \) \(4999448\) \(\nu^{8}\mathstrut +\mathstrut \) \(17139960\) \(\nu^{7}\mathstrut -\mathstrut \) \(124652320\) \(\nu^{6}\mathstrut +\mathstrut \) \(329833664\) \(\nu^{5}\mathstrut -\mathstrut \) \(256497664\) \(\nu^{4}\mathstrut -\mathstrut \) \(309441536\) \(\nu^{3}\mathstrut +\mathstrut \) \(179912704\) \(\nu^{2}\mathstrut +\mathstrut \) \(8940912640\) \(\nu\mathstrut -\mathstrut \) \(36868718592\)\()/\)\(678952960\)
\(\beta_{9}\)\(=\)\((\)\(28401\) \(\nu^{13}\mathstrut -\mathstrut \) \(32588\) \(\nu^{12}\mathstrut -\mathstrut \) \(184641\) \(\nu^{11}\mathstrut +\mathstrut \) \(518662\) \(\nu^{10}\mathstrut -\mathstrut \) \(277010\) \(\nu^{9}\mathstrut -\mathstrut \) \(2018984\) \(\nu^{8}\mathstrut -\mathstrut \) \(431400\) \(\nu^{7}\mathstrut +\mathstrut \) \(42427360\) \(\nu^{6}\mathstrut -\mathstrut \) \(178329664\) \(\nu^{5}\mathstrut +\mathstrut \) \(148231168\) \(\nu^{4}\mathstrut -\mathstrut \) \(371536896\) \(\nu^{3}\mathstrut -\mathstrut \) \(414662656\) \(\nu^{2}\mathstrut -\mathstrut \) \(5578063872\) \(\nu\mathstrut +\mathstrut \) \(19838795776\)\()/\)\(678952960\)
\(\beta_{10}\)\(=\)\((\)\(30299\) \(\nu^{13}\mathstrut -\mathstrut \) \(175428\) \(\nu^{12}\mathstrut +\mathstrut \) \(396085\) \(\nu^{11}\mathstrut -\mathstrut \) \(537790\) \(\nu^{10}\mathstrut +\mathstrut \) \(1159226\) \(\nu^{9}\mathstrut -\mathstrut \) \(9651512\) \(\nu^{8}\mathstrut +\mathstrut \) \(53625160\) \(\nu^{7}\mathstrut -\mathstrut \) \(150852960\) \(\nu^{6}\mathstrut +\mathstrut \) \(203316544\) \(\nu^{5}\mathstrut -\mathstrut \) \(134759424\) \(\nu^{4}\mathstrut +\mathstrut \) \(982549504\) \(\nu^{3}\mathstrut -\mathstrut \) \(4972855296\) \(\nu^{2}\mathstrut +\mathstrut \) \(16046653440\) \(\nu\mathstrut -\mathstrut \) \(20694433792\)\()/\)\(678952960\)
\(\beta_{11}\)\(=\)\((\)\(-\)\(8885\) \(\nu^{13}\mathstrut +\mathstrut \) \(54092\) \(\nu^{12}\mathstrut -\mathstrut \) \(127643\) \(\nu^{11}\mathstrut +\mathstrut \) \(205586\) \(\nu^{10}\mathstrut -\mathstrut \) \(440742\) \(\nu^{9}\mathstrut +\mathstrut \) \(3542632\) \(\nu^{8}\mathstrut -\mathstrut \) \(17765240\) \(\nu^{7}\mathstrut +\mathstrut \) \(49878560\) \(\nu^{6}\mathstrut -\mathstrut \) \(61533120\) \(\nu^{5}\mathstrut +\mathstrut \) \(86843392\) \(\nu^{4}\mathstrut -\mathstrut \) \(323572736\) \(\nu^{3}\mathstrut +\mathstrut \) \(1651974144\) \(\nu^{2}\mathstrut -\mathstrut \) \(5472813056\) \(\nu\mathstrut +\mathstrut \) \(5876088832\)\()/\)\(169738240\)
\(\beta_{12}\)\(=\)\((\)\(-\)\(73363\) \(\nu^{13}\mathstrut +\mathstrut \) \(121444\) \(\nu^{12}\mathstrut +\mathstrut \) \(210563\) \(\nu^{11}\mathstrut -\mathstrut \) \(703506\) \(\nu^{10}\mathstrut -\mathstrut \) \(613450\) \(\nu^{9}\mathstrut +\mathstrut \) \(15576312\) \(\nu^{8}\mathstrut -\mathstrut \) \(36025480\) \(\nu^{7}\mathstrut -\mathstrut \) \(13428640\) \(\nu^{6}\mathstrut +\mathstrut \) \(250450112\) \(\nu^{5}\mathstrut -\mathstrut \) \(165837824\) \(\nu^{4}\mathstrut -\mathstrut \) \(888906752\) \(\nu^{3}\mathstrut +\mathstrut \) \(2383659008\) \(\nu^{2}\mathstrut -\mathstrut \) \(142639104\) \(\nu\mathstrut -\mathstrut \) \(37401919488\)\()/\)\(678952960\)
\(\beta_{13}\)\(=\)\((\)\(-\)\(74425\) \(\nu^{13}\mathstrut +\mathstrut \) \(92076\) \(\nu^{12}\mathstrut +\mathstrut \) \(124041\) \(\nu^{11}\mathstrut -\mathstrut \) \(284342\) \(\nu^{10}\mathstrut +\mathstrut \) \(496674\) \(\nu^{9}\mathstrut +\mathstrut \) \(12413416\) \(\nu^{8}\mathstrut -\mathstrut \) \(28455320\) \(\nu^{7}\mathstrut -\mathstrut \) \(40588000\) \(\nu^{6}\mathstrut +\mathstrut \) \(293167680\) \(\nu^{5}\mathstrut -\mathstrut \) \(121549824\) \(\nu^{4}\mathstrut -\mathstrut \) \(725675008\) \(\nu^{3}\mathstrut +\mathstrut \) \(2751741952\) \(\nu^{2}\mathstrut +\mathstrut \) \(4283072512\) \(\nu\mathstrut -\mathstrut \) \(48499523584\)\()/\)\(678952960\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{5}\mathstrut +\mathstrut \) \(\beta_{4}\mathstrut +\mathstrut \) \(\beta_{3}\mathstrut +\mathstrut \) \(2\)\()/8\)
\(\nu^{2}\)\(=\)\((\)\(-\)\(2\) \(\beta_{12}\mathstrut -\mathstrut \) \(2\) \(\beta_{10}\mathstrut -\mathstrut \) \(2\) \(\beta_{9}\mathstrut +\mathstrut \) \(\beta_{8}\mathstrut -\mathstrut \) \(2\) \(\beta_{4}\mathstrut +\mathstrut \) \(4\) \(\beta_{3}\mathstrut +\mathstrut \) \(3\) \(\beta_{1}\mathstrut -\mathstrut \) \(7\)\()/8\)
\(\nu^{3}\)\(=\)\((\)\(-\)\(2\) \(\beta_{13}\mathstrut +\mathstrut \) \(4\) \(\beta_{11}\mathstrut +\mathstrut \) \(8\) \(\beta_{10}\mathstrut -\mathstrut \) \(4\) \(\beta_{9}\mathstrut -\mathstrut \) \(\beta_{8}\mathstrut +\mathstrut \) \(4\) \(\beta_{7}\mathstrut -\mathstrut \) \(2\) \(\beta_{6}\mathstrut -\mathstrut \) \(6\) \(\beta_{4}\mathstrut -\mathstrut \) \(16\) \(\beta_{3}\mathstrut +\mathstrut \) \(2\) \(\beta_{2}\mathstrut +\mathstrut \) \(9\) \(\beta_{1}\mathstrut -\mathstrut \) \(1\)\()/8\)
\(\nu^{4}\)\(=\)\((\)\(2\) \(\beta_{13}\mathstrut +\mathstrut \) \(12\) \(\beta_{11}\mathstrut +\mathstrut \) \(32\) \(\beta_{10}\mathstrut -\mathstrut \) \(4\) \(\beta_{9}\mathstrut -\mathstrut \) \(\beta_{8}\mathstrut -\mathstrut \) \(12\) \(\beta_{7}\mathstrut +\mathstrut \) \(10\) \(\beta_{6}\mathstrut -\mathstrut \) \(26\) \(\beta_{4}\mathstrut -\mathstrut \) \(116\) \(\beta_{3}\mathstrut -\mathstrut \) \(2\) \(\beta_{2}\mathstrut -\mathstrut \) \(59\) \(\beta_{1}\mathstrut -\mathstrut \) \(5\)\()/8\)
\(\nu^{5}\)\(=\)\((\)\(10\) \(\beta_{13}\mathstrut -\mathstrut \) \(8\) \(\beta_{12}\mathstrut +\mathstrut \) \(52\) \(\beta_{11}\mathstrut +\mathstrut \) \(24\) \(\beta_{10}\mathstrut -\mathstrut \) \(4\) \(\beta_{9}\mathstrut -\mathstrut \) \(7\) \(\beta_{8}\mathstrut +\mathstrut \) \(12\) \(\beta_{7}\mathstrut +\mathstrut \) \(2\) \(\beta_{6}\mathstrut -\mathstrut \) \(6\) \(\beta_{5}\mathstrut +\mathstrut \) \(160\) \(\beta_{4}\mathstrut +\mathstrut \) \(162\) \(\beta_{3}\mathstrut -\mathstrut \) \(18\) \(\beta_{2}\mathstrut +\mathstrut \) \(79\) \(\beta_{1}\mathstrut +\mathstrut \) \(765\)\()/8\)
\(\nu^{6}\)\(=\)\((\)\(-\)\(26\) \(\beta_{13}\mathstrut +\mathstrut \) \(12\) \(\beta_{12}\mathstrut +\mathstrut \) \(84\) \(\beta_{11}\mathstrut -\mathstrut \) \(52\) \(\beta_{10}\mathstrut -\mathstrut \) \(16\) \(\beta_{9}\mathstrut +\mathstrut \) \(35\) \(\beta_{8}\mathstrut +\mathstrut \) \(76\) \(\beta_{7}\mathstrut -\mathstrut \) \(66\) \(\beta_{6}\mathstrut +\mathstrut \) \(104\) \(\beta_{5}\mathstrut +\mathstrut \) \(454\) \(\beta_{4}\mathstrut +\mathstrut \) \(860\) \(\beta_{3}\mathstrut -\mathstrut \) \(54\) \(\beta_{2}\mathstrut -\mathstrut \) \(247\) \(\beta_{1}\mathstrut -\mathstrut \) \(1585\)\()/8\)
\(\nu^{7}\)\(=\)\((\)\(34\) \(\beta_{13}\mathstrut -\mathstrut \) \(136\) \(\beta_{12}\mathstrut -\mathstrut \) \(28\) \(\beta_{11}\mathstrut -\mathstrut \) \(344\) \(\beta_{10}\mathstrut -\mathstrut \) \(260\) \(\beta_{9}\mathstrut +\mathstrut \) \(185\) \(\beta_{8}\mathstrut +\mathstrut \) \(44\) \(\beta_{7}\mathstrut +\mathstrut \) \(330\) \(\beta_{6}\mathstrut -\mathstrut \) \(270\) \(\beta_{5}\mathstrut -\mathstrut \) \(416\) \(\beta_{4}\mathstrut +\mathstrut \) \(5506\) \(\beta_{3}\mathstrut +\mathstrut \) \(22\) \(\beta_{2}\mathstrut -\mathstrut \) \(841\) \(\beta_{1}\mathstrut -\mathstrut \) \(1579\)\()/8\)
\(\nu^{8}\)\(=\)\((\)\(-\)\(74\) \(\beta_{13}\mathstrut +\mathstrut \) \(268\) \(\beta_{12}\mathstrut +\mathstrut \) \(516\) \(\beta_{11}\mathstrut +\mathstrut \) \(1100\) \(\beta_{10}\mathstrut +\mathstrut \) \(224\) \(\beta_{9}\mathstrut +\mathstrut \) \(343\) \(\beta_{8}\mathstrut +\mathstrut \) \(412\) \(\beta_{7}\mathstrut +\mathstrut \) \(814\) \(\beta_{6}\mathstrut -\mathstrut \) \(60\) \(\beta_{5}\mathstrut -\mathstrut \) \(1102\) \(\beta_{4}\mathstrut +\mathstrut \) \(8000\) \(\beta_{3}\mathstrut +\mathstrut \) \(586\) \(\beta_{2}\mathstrut +\mathstrut \) \(333\) \(\beta_{1}\mathstrut +\mathstrut \) \(8371\)\()/8\)
\(\nu^{9}\)\(=\)\((\)\(1946\) \(\beta_{13}\mathstrut -\mathstrut \) \(528\) \(\beta_{12}\mathstrut -\mathstrut \) \(380\) \(\beta_{11}\mathstrut +\mathstrut \) \(2848\) \(\beta_{10}\mathstrut +\mathstrut \) \(1364\) \(\beta_{9}\mathstrut +\mathstrut \) \(1437\) \(\beta_{8}\mathstrut -\mathstrut \) \(1524\) \(\beta_{7}\mathstrut +\mathstrut \) \(1154\) \(\beta_{6}\mathstrut +\mathstrut \) \(614\) \(\beta_{5}\mathstrut +\mathstrut \) \(532\) \(\beta_{4}\mathstrut -\mathstrut \) \(15042\) \(\beta_{3}\mathstrut +\mathstrut \) \(1134\) \(\beta_{2}\mathstrut -\mathstrut \) \(10389\) \(\beta_{1}\mathstrut +\mathstrut \) \(4841\)\()/8\)
\(\nu^{10}\)\(=\)\((\)\(4046\) \(\beta_{13}\mathstrut -\mathstrut \) \(3372\) \(\beta_{12}\mathstrut +\mathstrut \) \(3460\) \(\beta_{11}\mathstrut -\mathstrut \) \(5708\) \(\beta_{10}\mathstrut -\mathstrut \) \(248\) \(\beta_{9}\mathstrut -\mathstrut \) \(157\) \(\beta_{8}\mathstrut +\mathstrut \) \(700\) \(\beta_{7}\mathstrut -\mathstrut \) \(13850\) \(\beta_{6}\mathstrut -\mathstrut \) \(320\) \(\beta_{5}\mathstrut +\mathstrut \) \(2910\) \(\beta_{4}\mathstrut +\mathstrut \) \(73348\) \(\beta_{3}\mathstrut +\mathstrut \) \(2914\) \(\beta_{2}\mathstrut +\mathstrut \) \(16569\) \(\beta_{1}\mathstrut +\mathstrut \) \(84351\)\()/8\)
\(\nu^{11}\)\(=\)\((\)\(-\)\(7230\) \(\beta_{13}\mathstrut +\mathstrut \) \(3624\) \(\beta_{12}\mathstrut +\mathstrut \) \(4260\) \(\beta_{11}\mathstrut -\mathstrut \) \(13896\) \(\beta_{10}\mathstrut -\mathstrut \) \(10932\) \(\beta_{9}\mathstrut +\mathstrut \) \(9033\) \(\beta_{8}\mathstrut -\mathstrut \) \(12724\) \(\beta_{7}\mathstrut -\mathstrut \) \(49206\) \(\beta_{6}\mathstrut +\mathstrut \) \(13506\) \(\beta_{5}\mathstrut +\mathstrut \) \(1984\) \(\beta_{4}\mathstrut +\mathstrut \) \(10978\) \(\beta_{3}\mathstrut +\mathstrut \) \(5686\) \(\beta_{2}\mathstrut -\mathstrut \) \(27465\) \(\beta_{1}\mathstrut -\mathstrut \) \(143499\)\()/8\)
\(\nu^{12}\)\(=\)\((\)\(15734\) \(\beta_{13}\mathstrut -\mathstrut \) \(5300\) \(\beta_{12}\mathstrut -\mathstrut \) \(8476\) \(\beta_{11}\mathstrut +\mathstrut \) \(1484\) \(\beta_{10}\mathstrut -\mathstrut \) \(56128\) \(\beta_{9}\mathstrut -\mathstrut \) \(1297\) \(\beta_{8}\mathstrut -\mathstrut \) \(88068\) \(\beta_{7}\mathstrut +\mathstrut \) \(52590\) \(\beta_{6}\mathstrut -\mathstrut \) \(40244\) \(\beta_{5}\mathstrut -\mathstrut \) \(79110\) \(\beta_{4}\mathstrut +\mathstrut \) \(378904\) \(\beta_{3}\mathstrut +\mathstrut \) \(39018\) \(\beta_{2}\mathstrut -\mathstrut \) \(13019\) \(\beta_{1}\mathstrut +\mathstrut \) \(245835\)\()/8\)
\(\nu^{13}\)\(=\)\((\)\(-\)\(23574\) \(\beta_{13}\mathstrut -\mathstrut \) \(6464\) \(\beta_{12}\mathstrut +\mathstrut \) \(178052\) \(\beta_{11}\mathstrut +\mathstrut \) \(253168\) \(\beta_{10}\mathstrut +\mathstrut \) \(23716\) \(\beta_{9}\mathstrut +\mathstrut \) \(11813\) \(\beta_{8}\mathstrut -\mathstrut \) \(104692\) \(\beta_{7}\mathstrut +\mathstrut \) \(100306\) \(\beta_{6}\mathstrut +\mathstrut \) \(48462\) \(\beta_{5}\mathstrut +\mathstrut \) \(340348\) \(\beta_{4}\mathstrut +\mathstrut \) \(55670\) \(\beta_{3}\mathstrut +\mathstrut \) \(85822\) \(\beta_{2}\mathstrut +\mathstrut \) \(747987\) \(\beta_{1}\mathstrut +\mathstrut \) \(313681\)\()/8\)

Character Values

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

\(n\) \(5\) \(15\)
\(\chi(n)\) \(-\beta_{3}\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3.1
2.24452 1.72109i
0.153862 2.82424i
2.79265 + 0.448449i
−2.15805 1.82834i
1.03712 + 2.63142i
0.336831 + 2.80830i
−2.40693 + 1.48549i
2.24452 + 1.72109i
0.153862 + 2.82424i
2.79265 0.448449i
−2.15805 + 1.82834i
1.03712 2.63142i
0.336831 2.80830i
−2.40693 1.48549i
−3.96560 + 0.523430i 5.54016 5.54016i 15.4520 4.15143i 21.7374 21.7374i −19.0702 + 24.8700i −6.62054 −59.1037 + 24.5510i 19.6133i −74.8239 + 97.5799i
3.2 −2.97810 2.67038i −9.42589 + 9.42589i 1.73818 + 15.9053i −2.84710 + 2.84710i 53.2419 2.90058i −76.7794 37.2967 52.0092i 96.6949i 16.0818 0.876123i
3.3 −2.34420 + 3.24110i −4.63552 + 4.63552i −5.00945 15.1956i −29.2002 + 29.2002i −4.15759 25.8908i 59.6196 60.9935 + 19.3854i 38.0239i −26.1896 163.092i
3.4 0.329715 3.98639i 3.91498 3.91498i −15.7826 2.62875i 4.72348 4.72348i −14.3158 16.8975i 45.3712 −15.6830 + 62.0487i 50.3458i −17.2722 20.3870i
3.5 1.59430 + 3.66854i 11.5209 11.5209i −10.9164 + 11.6975i −14.6016 + 14.6016i 60.6325 + 23.8971i −24.0210 −60.3169 21.3980i 184.461i −76.8459 30.2872i
3.6 2.47147 + 3.14513i −7.86839 + 7.86839i −3.78368 + 15.5462i 27.2309 27.2309i −44.1936 5.30063i 50.3097 −58.2460 + 26.5217i 42.8233i 152.945 + 18.3444i
3.7 3.89242 0.921438i −0.0461995 + 0.0461995i 14.3019 7.17325i −8.04297 + 8.04297i −0.137258 + 0.222398i −49.8797 49.0594 41.0996i 80.9957i −23.8955 + 38.7177i
11.1 −3.96560 0.523430i 5.54016 + 5.54016i 15.4520 + 4.15143i 21.7374 + 21.7374i −19.0702 24.8700i −6.62054 −59.1037 24.5510i 19.6133i −74.8239 97.5799i
11.2 −2.97810 + 2.67038i −9.42589 9.42589i 1.73818 15.9053i −2.84710 2.84710i 53.2419 + 2.90058i −76.7794 37.2967 + 52.0092i 96.6949i 16.0818 + 0.876123i
11.3 −2.34420 3.24110i −4.63552 4.63552i −5.00945 + 15.1956i −29.2002 29.2002i −4.15759 + 25.8908i 59.6196 60.9935 19.3854i 38.0239i −26.1896 + 163.092i
11.4 0.329715 + 3.98639i 3.91498 + 3.91498i −15.7826 + 2.62875i 4.72348 + 4.72348i −14.3158 + 16.8975i 45.3712 −15.6830 62.0487i 50.3458i −17.2722 + 20.3870i
11.5 1.59430 3.66854i 11.5209 + 11.5209i −10.9164 11.6975i −14.6016 14.6016i 60.6325 23.8971i −24.0210 −60.3169 + 21.3980i 184.461i −76.8459 + 30.2872i
11.6 2.47147 3.14513i −7.86839 7.86839i −3.78368 15.5462i 27.2309 + 27.2309i −44.1936 + 5.30063i 50.3097 −58.2460 26.5217i 42.8233i 152.945 18.3444i
11.7 3.89242 + 0.921438i −0.0461995 0.0461995i 14.3019 + 7.17325i −8.04297 8.04297i −0.137258 0.222398i −49.8797 49.0594 + 41.0996i 80.9957i −23.8955 38.7177i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 11.7
Significant digits:
Format:

Inner twists

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

Hecke kernels

There are no other newforms in \(S_{5}^{\mathrm{new}}(16, [\chi])\).