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 \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.65391940934\)
Analytic rank: \(0\)
Dimension: \(14\)
Relative dimension: \(7\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} - \cdots)\)
Defining polynomial: \(x^{14} - 4 x^{13} + 15 x^{12} - 34 x^{11} + 62 x^{10} - 312 x^{9} + 1432 x^{8} - 4960 x^{7} + 11456 x^{6} - 19968 x^{5} + 31744 x^{4} - 139264 x^{3} + 491520 x^{2} - 1048576 x + 2097152\)
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 2^{21} \)
Twist minimal: yes
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})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14q - 2q^{2} - 2q^{3} - 8q^{4} - 2q^{5} + 64q^{6} - 4q^{7} - 92q^{8} + O(q^{10}) \) \( 14q - 2q^{2} - 2q^{3} - 8q^{4} - 2q^{5} + 64q^{6} - 4q^{7} - 92q^{8} - 100q^{10} + 94q^{11} - 332q^{12} - 2q^{13} + 44q^{14} - 168q^{16} - 4q^{17} + 1390q^{18} - 706q^{19} + 1900q^{20} - 164q^{21} + 900q^{22} + 1148q^{23} - 1872q^{24} - 3416q^{26} - 1664q^{27} - 3784q^{28} + 862q^{29} - 3740q^{30} + 3208q^{32} - 4q^{33} + 7508q^{34} + 1340q^{35} + 11468q^{36} - 1826q^{37} + 3568q^{38} + 2684q^{39} - 5144q^{40} - 17064q^{42} + 1694q^{43} - 14636q^{44} + 1410q^{45} - 5316q^{46} + 6888q^{48} + 682q^{49} + 20070q^{50} - 3012q^{51} + 20452q^{52} - 482q^{53} + 10784q^{54} - 11780q^{55} - 6952q^{56} - 20456q^{58} - 2786q^{59} - 29920q^{60} - 3778q^{61} - 11472q^{62} + 15808q^{64} - 2020q^{65} + 30148q^{66} + 7998q^{67} + 18032q^{68} + 9628q^{69} + 15296q^{70} + 19964q^{71} - 17708q^{72} - 23780q^{74} + 17570q^{75} - 23996q^{76} - 9508q^{77} - 8052q^{78} + 1384q^{80} + 1454q^{81} + 16016q^{82} - 17282q^{83} + 19624q^{84} + 9948q^{85} - 4796q^{86} - 49284q^{87} + 7288q^{88} - 5416q^{90} - 28036q^{91} - 14632q^{92} + 8896q^{93} + 432q^{94} + 6064q^{96} - 4q^{97} - 12246q^{98} + 49214q^{99} + O(q^{100}) \)

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

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

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 Parity Ord Mult Type
1.a even 1 1 trivial
16.f odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 16.5.f.a 14
3.b odd 2 1 144.5.m.a 14
4.b odd 2 1 64.5.f.a 14
8.b even 2 1 128.5.f.b 14
8.d odd 2 1 128.5.f.a 14
12.b even 2 1 576.5.m.a 14
16.e even 4 1 64.5.f.a 14
16.e even 4 1 128.5.f.a 14
16.f odd 4 1 inner 16.5.f.a 14
16.f odd 4 1 128.5.f.b 14
48.i odd 4 1 576.5.m.a 14
48.k even 4 1 144.5.m.a 14
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
16.5.f.a 14 1.a even 1 1 trivial
16.5.f.a 14 16.f odd 4 1 inner
64.5.f.a 14 4.b odd 2 1
64.5.f.a 14 16.e even 4 1
128.5.f.a 14 8.d odd 2 1
128.5.f.a 14 16.e even 4 1
128.5.f.b 14 8.b even 2 1
128.5.f.b 14 16.f odd 4 1
144.5.m.a 14 3.b odd 2 1
144.5.m.a 14 48.k even 4 1
576.5.m.a 14 12.b even 2 1
576.5.m.a 14 48.i odd 4 1

Hecke kernels

This newform subspace is the entire newspace \(S_{5}^{\mathrm{new}}(16, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 268435456 + 33554432 T + 6291456 T^{2} + 2621440 T^{3} + 491520 T^{4} - 90112 T^{5} - 46080 T^{6} + 2560 T^{7} - 2880 T^{8} - 352 T^{9} + 120 T^{10} + 40 T^{11} + 6 T^{12} + 2 T^{13} + T^{14} \)
$3$ \( 2016379008 + 43496175744 T + 469137324096 T^{2} - 34747646976 T^{3} + 1358008416 T^{4} + 786717792 T^{5} + 409777200 T^{6} - 14203392 T^{7} + 551960 T^{8} + 288664 T^{9} + 62860 T^{10} + 448 T^{11} + 2 T^{12} + 2 T^{13} + T^{14} \)
$5$ \( 95385721607700608 + 27076575211548800 T + 3843032860840000 T^{2} - 208330635859968 T^{3} + 72061290078304 T^{4} + 14025915627616 T^{5} + 1305653282864 T^{6} + 19416658944 T^{7} + 3101720 T^{8} + 2850584 T^{9} + 3106060 T^{10} + 2688 T^{11} + 2 T^{12} + 2 T^{13} + T^{14} \)
$7$ \( ( -82884464768 - 13846485056 T - 67418144 T^{2} + 20706288 T^{3} + 35592 T^{4} - 8572 T^{5} + 2 T^{6} + T^{7} )^{2} \)
$11$ \( \)\(47\!\cdots\!12\)\( + \)\(30\!\cdots\!92\)\( T + \)\(95\!\cdots\!36\)\( T^{2} + \)\(65\!\cdots\!84\)\( T^{3} + \)\(20\!\cdots\!48\)\( T^{4} + 2830209232291459680 T^{5} + 219305901738719280 T^{6} + 1162127446795776 T^{7} + 2291939461144 T^{8} - 32065636328 T^{9} + 784133132 T^{10} + 2341824 T^{11} + 4418 T^{12} - 94 T^{13} + T^{14} \)
$13$ \( \)\(24\!\cdots\!52\)\( - \)\(17\!\cdots\!12\)\( T + \)\(62\!\cdots\!36\)\( T^{2} - \)\(78\!\cdots\!80\)\( T^{3} + \)\(49\!\cdots\!08\)\( T^{4} - 13395006265199179680 T^{5} + 938835469167525936 T^{6} - 8456611120356352 T^{7} + 24609579825688 T^{8} + 662386022680 T^{9} + 6547368972 T^{10} + 6826112 T^{11} + 2 T^{12} + 2 T^{13} + T^{14} \)
$17$ \( ( -12009518203797632 - 119000879480896 T + 1432677679200 T^{2} + 14701879344 T^{3} - 13778968 T^{4} - 250892 T^{5} + 2 T^{6} + T^{7} )^{2} \)
$19$ \( \)\(37\!\cdots\!48\)\( - \)\(19\!\cdots\!84\)\( T + \)\(52\!\cdots\!36\)\( T^{2} + \)\(10\!\cdots\!20\)\( T^{3} + \)\(87\!\cdots\!60\)\( T^{4} + \)\(29\!\cdots\!48\)\( T^{5} + \)\(54\!\cdots\!96\)\( T^{6} + 1364277329112460800 T^{7} + 15308192398571800 T^{8} + 46983788219800 T^{9} + 71849973004 T^{10} + 9374912 T^{11} + 249218 T^{12} + 706 T^{13} + T^{14} \)
$23$ \( ( 54911897542109056 - 1959672332215360 T - 7489328380960 T^{2} + 215281147888 T^{3} + 327850376 T^{4} - 934844 T^{5} - 574 T^{6} + T^{7} )^{2} \)
$29$ \( \)\(27\!\cdots\!28\)\( - \)\(58\!\cdots\!72\)\( T + \)\(63\!\cdots\!64\)\( T^{2} - \)\(36\!\cdots\!92\)\( T^{3} + \)\(12\!\cdots\!92\)\( T^{4} - \)\(25\!\cdots\!48\)\( T^{5} + \)\(35\!\cdots\!00\)\( T^{6} - \)\(63\!\cdots\!52\)\( T^{7} + 2138874201321723160 T^{8} - 4180270176993128 T^{9} + 4196992328204 T^{10} - 435330432 T^{11} + 371522 T^{12} - 862 T^{13} + T^{14} \)
$31$ \( \)\(71\!\cdots\!96\)\( + \)\(40\!\cdots\!52\)\( T^{2} + \)\(76\!\cdots\!80\)\( T^{4} + \)\(57\!\cdots\!84\)\( T^{6} + 13297300404014415872 T^{8} + 13339923865600 T^{10} + 6024960 T^{12} + T^{14} \)
$37$ \( \)\(51\!\cdots\!68\)\( - \)\(19\!\cdots\!88\)\( T + \)\(35\!\cdots\!04\)\( T^{2} + \)\(35\!\cdots\!92\)\( T^{3} + \)\(15\!\cdots\!28\)\( T^{4} + \)\(12\!\cdots\!88\)\( T^{5} + \)\(64\!\cdots\!32\)\( T^{6} + \)\(10\!\cdots\!64\)\( T^{7} + 37374659035443657496 T^{8} + 35578094339272856 T^{9} + 17274822237964 T^{10} + 1554716288 T^{11} + 1667138 T^{12} + 1826 T^{13} + T^{14} \)
$41$ \( \)\(24\!\cdots\!04\)\( + \)\(51\!\cdots\!76\)\( T^{2} + \)\(20\!\cdots\!12\)\( T^{4} + \)\(31\!\cdots\!48\)\( T^{6} + \)\(23\!\cdots\!56\)\( T^{8} + 85694612004864 T^{10} + 15036672 T^{12} + T^{14} \)
$43$ \( \)\(11\!\cdots\!68\)\( - \)\(68\!\cdots\!60\)\( T + \)\(20\!\cdots\!00\)\( T^{2} - \)\(88\!\cdots\!68\)\( T^{3} + \)\(18\!\cdots\!68\)\( T^{4} - \)\(15\!\cdots\!16\)\( T^{5} + \)\(42\!\cdots\!04\)\( T^{6} - \)\(13\!\cdots\!48\)\( T^{7} + \)\(19\!\cdots\!00\)\( T^{8} - 142596531803367912 T^{9} + 57605028643980 T^{10} - 8486946368 T^{11} + 1434818 T^{12} - 1694 T^{13} + T^{14} \)
$47$ \( \)\(12\!\cdots\!76\)\( + \)\(10\!\cdots\!80\)\( T^{2} + \)\(38\!\cdots\!80\)\( T^{4} + \)\(52\!\cdots\!84\)\( T^{6} + \)\(32\!\cdots\!52\)\( T^{8} + 104964818558976 T^{10} + 16427776 T^{12} + T^{14} \)
$53$ \( \)\(29\!\cdots\!88\)\( - \)\(53\!\cdots\!48\)\( T + \)\(48\!\cdots\!04\)\( T^{2} - \)\(13\!\cdots\!72\)\( T^{3} + \)\(16\!\cdots\!92\)\( T^{4} - \)\(29\!\cdots\!76\)\( T^{5} + \)\(28\!\cdots\!44\)\( T^{6} - \)\(75\!\cdots\!52\)\( T^{7} + 30136282404129217816 T^{8} + 411910984078744 T^{9} + 119500824744716 T^{10} - 9361537920 T^{11} + 116162 T^{12} + 482 T^{13} + T^{14} \)
$59$ \( \)\(35\!\cdots\!72\)\( + \)\(10\!\cdots\!16\)\( T + \)\(15\!\cdots\!24\)\( T^{2} + \)\(13\!\cdots\!56\)\( T^{3} + \)\(80\!\cdots\!12\)\( T^{4} + \)\(31\!\cdots\!96\)\( T^{5} + \)\(82\!\cdots\!28\)\( T^{6} + \)\(18\!\cdots\!16\)\( T^{7} + \)\(70\!\cdots\!88\)\( T^{8} + 2819291680457879576 T^{9} + 687445565640076 T^{10} + 61476408000 T^{11} + 3880898 T^{12} + 2786 T^{13} + T^{14} \)
$61$ \( \)\(12\!\cdots\!92\)\( + \)\(21\!\cdots\!00\)\( T + \)\(19\!\cdots\!00\)\( T^{2} - \)\(29\!\cdots\!64\)\( T^{3} + \)\(57\!\cdots\!88\)\( T^{4} + \)\(39\!\cdots\!88\)\( T^{5} + \)\(13\!\cdots\!44\)\( T^{6} - \)\(24\!\cdots\!80\)\( T^{7} + \)\(39\!\cdots\!76\)\( T^{8} + 2978236711893026328 T^{9} + 1123719102824460 T^{10} - 38724599168 T^{11} + 7136642 T^{12} + 3778 T^{13} + T^{14} \)
$67$ \( \)\(40\!\cdots\!08\)\( + \)\(14\!\cdots\!84\)\( T + \)\(26\!\cdots\!16\)\( T^{2} + \)\(13\!\cdots\!68\)\( T^{3} + \)\(29\!\cdots\!56\)\( T^{4} - \)\(34\!\cdots\!52\)\( T^{5} + \)\(31\!\cdots\!56\)\( T^{6} + \)\(15\!\cdots\!00\)\( T^{7} + \)\(38\!\cdots\!48\)\( T^{8} - 8669080375183272552 T^{9} + 1177758166491660 T^{10} + 113497816512 T^{11} + 31984002 T^{12} - 7998 T^{13} + T^{14} \)
$71$ \( ( -\)\(38\!\cdots\!80\)\( - \)\(73\!\cdots\!68\)\( T - 3205242620439887904 T^{2} + 628961169111024 T^{3} + 356646354056 T^{4} - 32864444 T^{5} - 9982 T^{6} + T^{7} )^{2} \)
$73$ \( \)\(67\!\cdots\!16\)\( + \)\(23\!\cdots\!48\)\( T^{2} + \)\(23\!\cdots\!60\)\( T^{4} + \)\(59\!\cdots\!72\)\( T^{6} + \)\(55\!\cdots\!72\)\( T^{8} + 18004735458743808 T^{10} + 229001536 T^{12} + T^{14} \)
$79$ \( \)\(80\!\cdots\!76\)\( + \)\(17\!\cdots\!96\)\( T^{2} + \)\(52\!\cdots\!92\)\( T^{4} + \)\(64\!\cdots\!36\)\( T^{6} + \)\(39\!\cdots\!92\)\( T^{8} + 12122330541457408 T^{10} + 181267456 T^{12} + T^{14} \)
$83$ \( \)\(30\!\cdots\!48\)\( + \)\(32\!\cdots\!96\)\( T + \)\(17\!\cdots\!96\)\( T^{2} + \)\(46\!\cdots\!44\)\( T^{3} + \)\(61\!\cdots\!28\)\( T^{4} + \)\(60\!\cdots\!12\)\( T^{5} + \)\(87\!\cdots\!84\)\( T^{6} + \)\(41\!\cdots\!36\)\( T^{7} + \)\(88\!\cdots\!24\)\( T^{8} - 461127974893871208 T^{9} + 129090565299084 T^{10} + 268544938176 T^{11} + 149333762 T^{12} + 17282 T^{13} + T^{14} \)
$89$ \( \)\(10\!\cdots\!96\)\( + \)\(14\!\cdots\!80\)\( T^{2} + \)\(54\!\cdots\!80\)\( T^{4} + \)\(52\!\cdots\!20\)\( T^{6} + \)\(20\!\cdots\!84\)\( T^{8} + 39327084461872640 T^{10} + 329862464 T^{12} + T^{14} \)
$97$ \( ( -\)\(50\!\cdots\!96\)\( - \)\(13\!\cdots\!60\)\( T - 15741851025018318752 T^{2} + 13158120305248304 T^{3} + 250697829864 T^{4} - 231854348 T^{5} + 2 T^{6} + T^{7} )^{2} \)
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