Properties

Label 722.6.a.r
Level $722$
Weight $6$
Character orbit 722.a
Self dual yes
Analytic conductor $115.797$
Analytic rank $0$
Dimension $15$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 722 = 2 \cdot 19^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 722.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(115.797117905\)
Analytic rank: \(0\)
Dimension: \(15\)
Coefficient field: \(\mathbb{Q}[x]/(x^{15} - \cdots)\)
Defining polynomial: \(x^{15} - 2871 x^{13} - 4674 x^{12} + 3170019 x^{11} + 9081402 x^{10} - 1680307373 x^{9} - 6060225486 x^{8} + 437045334939 x^{7} + 1625551725606 x^{6} - 51993847360191 x^{5} - 166593470925336 x^{4} + 2435236025483701 x^{3} + 5743342864926480 x^{2} - 19853455953893436 x - 34308015556724472\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{6}\cdot 3\cdot 19^{6} \)
Twist minimal: no (minimal twist has level 38)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{14}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 4 q^{2} -\beta_{1} q^{3} + 16 q^{4} + ( 7 + \beta_{5} ) q^{5} -4 \beta_{1} q^{6} + ( 6 - \beta_{1} + \beta_{4} - \beta_{10} ) q^{7} + 64 q^{8} + ( 142 + 2 \beta_{1} - \beta_{2} - \beta_{3} ) q^{9} +O(q^{10})\) \( q + 4 q^{2} -\beta_{1} q^{3} + 16 q^{4} + ( 7 + \beta_{5} ) q^{5} -4 \beta_{1} q^{6} + ( 6 - \beta_{1} + \beta_{4} - \beta_{10} ) q^{7} + 64 q^{8} + ( 142 + 2 \beta_{1} - \beta_{2} - \beta_{3} ) q^{9} + ( 28 + 4 \beta_{5} ) q^{10} + ( 8 + \beta_{3} + \beta_{4} + \beta_{6} - \beta_{13} ) q^{11} -16 \beta_{1} q^{12} + ( -7 - 2 \beta_{1} - \beta_{2} - 4 \beta_{3} + 6 \beta_{4} - \beta_{8} - \beta_{10} - \beta_{11} - \beta_{13} ) q^{13} + ( 24 - 4 \beta_{1} + 4 \beta_{4} - 4 \beta_{10} ) q^{14} + ( 5 - 13 \beta_{1} - \beta_{2} + 3 \beta_{3} + 18 \beta_{4} - 2 \beta_{5} - 3 \beta_{6} - \beta_{7} - \beta_{9} - 5 \beta_{10} - \beta_{11} - 3 \beta_{12} + \beta_{13} ) q^{15} + 256 q^{16} + ( 275 - 2 \beta_{1} + 4 \beta_{3} + \beta_{4} - 3 \beta_{5} + 2 \beta_{6} + \beta_{7} + 2 \beta_{8} + \beta_{9} - 3 \beta_{10} + \beta_{11} + 2 \beta_{12} - \beta_{13} - \beta_{14} ) q^{17} + ( 568 + 8 \beta_{1} - 4 \beta_{2} - 4 \beta_{3} ) q^{18} + ( 112 + 16 \beta_{5} ) q^{20} + ( 224 + 11 \beta_{1} + \beta_{2} - 11 \beta_{3} - 6 \beta_{4} + 22 \beta_{5} + \beta_{6} + 2 \beta_{7} - 3 \beta_{8} + 3 \beta_{9} - 7 \beta_{10} + \beta_{11} + \beta_{12} + 2 \beta_{13} - 2 \beta_{14} ) q^{21} + ( 32 + 4 \beta_{3} + 4 \beta_{4} + 4 \beta_{6} - 4 \beta_{13} ) q^{22} + ( 254 - 23 \beta_{1} - 5 \beta_{2} + 19 \beta_{3} - 35 \beta_{4} + 6 \beta_{5} + 5 \beta_{6} + 4 \beta_{7} - \beta_{8} + 9 \beta_{9} + 3 \beta_{10} - 2 \beta_{11} + \beta_{12} - 4 \beta_{13} + \beta_{14} ) q^{23} -64 \beta_{1} q^{24} + ( 1796 + 15 \beta_{1} - 35 \beta_{3} + 25 \beta_{4} + 20 \beta_{5} - 5 \beta_{7} - 10 \beta_{9} - 5 \beta_{10} + 5 \beta_{11} + 5 \beta_{13} ) q^{25} + ( -28 - 8 \beta_{1} - 4 \beta_{2} - 16 \beta_{3} + 24 \beta_{4} - 4 \beta_{8} - 4 \beta_{10} - 4 \beta_{11} - 4 \beta_{13} ) q^{26} + ( -878 - 167 \beta_{1} + 8 \beta_{2} - 3 \beta_{3} + 3 \beta_{4} + 12 \beta_{5} - 6 \beta_{7} + 7 \beta_{8} + 4 \beta_{9} + 12 \beta_{10} - 3 \beta_{12} + 2 \beta_{13} + 4 \beta_{14} ) q^{27} + ( 96 - 16 \beta_{1} + 16 \beta_{4} - 16 \beta_{10} ) q^{28} + ( 974 - 61 \beta_{1} - 2 \beta_{2} + 7 \beta_{3} + 3 \beta_{4} - 13 \beta_{5} - 2 \beta_{6} + 7 \beta_{7} + \beta_{8} - 8 \beta_{9} + 18 \beta_{10} - 6 \beta_{11} + 8 \beta_{12} + 2 \beta_{13} ) q^{29} + ( 20 - 52 \beta_{1} - 4 \beta_{2} + 12 \beta_{3} + 72 \beta_{4} - 8 \beta_{5} - 12 \beta_{6} - 4 \beta_{7} - 4 \beta_{9} - 20 \beta_{10} - 4 \beta_{11} - 12 \beta_{12} + 4 \beta_{13} ) q^{30} + ( 453 - 14 \beta_{1} + 7 \beta_{2} + 79 \beta_{3} - 56 \beta_{4} - 12 \beta_{6} - 3 \beta_{7} + 5 \beta_{8} - 9 \beta_{9} + 21 \beta_{10} - \beta_{11} - 3 \beta_{12} - 7 \beta_{13} + 7 \beta_{14} ) q^{31} + 1024 q^{32} + ( -250 + 39 \beta_{1} + 5 \beta_{2} - 74 \beta_{3} - 44 \beta_{4} - 44 \beta_{5} - 3 \beta_{6} - 5 \beta_{7} - 20 \beta_{8} - 10 \beta_{9} + 3 \beta_{10} + 9 \beta_{11} + 5 \beta_{12} + 6 \beta_{13} - 3 \beta_{14} ) q^{33} + ( 1100 - 8 \beta_{1} + 16 \beta_{3} + 4 \beta_{4} - 12 \beta_{5} + 8 \beta_{6} + 4 \beta_{7} + 8 \beta_{8} + 4 \beta_{9} - 12 \beta_{10} + 4 \beta_{11} + 8 \beta_{12} - 4 \beta_{13} - 4 \beta_{14} ) q^{34} + ( 871 - 270 \beta_{1} - 4 \beta_{2} + 22 \beta_{3} - 58 \beta_{4} + 14 \beta_{5} - 23 \beta_{6} - 11 \beta_{7} + 8 \beta_{8} - 36 \beta_{9} - 11 \beta_{10} + 7 \beta_{11} - 14 \beta_{12} - 3 \beta_{13} + 6 \beta_{14} ) q^{35} + ( 2272 + 32 \beta_{1} - 16 \beta_{2} - 16 \beta_{3} ) q^{36} + ( -304 - 83 \beta_{1} + 5 \beta_{2} + 55 \beta_{3} - 75 \beta_{4} + 56 \beta_{5} - 2 \beta_{6} + \beta_{7} + 8 \beta_{8} + 16 \beta_{9} + 7 \beta_{10} + 7 \beta_{11} + 2 \beta_{12} - \beta_{13} - 6 \beta_{14} ) q^{37} + ( 291 - 162 \beta_{1} + 11 \beta_{2} + 33 \beta_{3} + 127 \beta_{4} + 81 \beta_{5} + 7 \beta_{6} - \beta_{7} - 7 \beta_{8} + 20 \beta_{9} + 16 \beta_{10} - 9 \beta_{11} + 8 \beta_{12} + 3 \beta_{13} + 6 \beta_{14} ) q^{39} + ( 448 + 64 \beta_{5} ) q^{40} + ( 304 + 30 \beta_{1} + 7 \beta_{2} - 85 \beta_{3} + 46 \beta_{4} - 27 \beta_{5} + 44 \beta_{6} + 8 \beta_{7} + 10 \beta_{8} + 12 \beta_{9} + 38 \beta_{10} - 4 \beta_{11} - 4 \beta_{12} + 12 \beta_{13} + 6 \beta_{14} ) q^{41} + ( 896 + 44 \beta_{1} + 4 \beta_{2} - 44 \beta_{3} - 24 \beta_{4} + 88 \beta_{5} + 4 \beta_{6} + 8 \beta_{7} - 12 \beta_{8} + 12 \beta_{9} - 28 \beta_{10} + 4 \beta_{11} + 4 \beta_{12} + 8 \beta_{13} - 8 \beta_{14} ) q^{42} + ( 6257 - 97 \beta_{1} + 7 \beta_{2} - 9 \beta_{3} - 27 \beta_{4} + 62 \beta_{5} + 21 \beta_{6} - 6 \beta_{7} - 10 \beta_{8} + 24 \beta_{9} - 26 \beta_{10} + 2 \beta_{11} + 3 \beta_{12} + 8 \beta_{13} + 4 \beta_{14} ) q^{43} + ( 128 + 16 \beta_{3} + 16 \beta_{4} + 16 \beta_{6} - 16 \beta_{13} ) q^{44} + ( 2028 - 18 \beta_{1} + 5 \beta_{2} - 256 \beta_{3} - 34 \beta_{4} + 218 \beta_{5} + 25 \beta_{6} + 46 \beta_{7} + 2 \beta_{8} + 46 \beta_{9} + 4 \beta_{10} - 22 \beta_{11} + 49 \beta_{12} + 15 \beta_{13} - 9 \beta_{14} ) q^{45} + ( 1016 - 92 \beta_{1} - 20 \beta_{2} + 76 \beta_{3} - 140 \beta_{4} + 24 \beta_{5} + 20 \beta_{6} + 16 \beta_{7} - 4 \beta_{8} + 36 \beta_{9} + 12 \beta_{10} - 8 \beta_{11} + 4 \beta_{12} - 16 \beta_{13} + 4 \beta_{14} ) q^{46} + ( 69 - 52 \beta_{1} - 10 \beta_{2} + 74 \beta_{3} - 258 \beta_{4} - 21 \beta_{5} - 19 \beta_{6} + 3 \beta_{7} + 25 \beta_{8} - 7 \beta_{9} + \beta_{10} + 7 \beta_{11} - \beta_{12} - 3 \beta_{13} - 16 \beta_{14} ) q^{47} -256 \beta_{1} q^{48} + ( 4238 - 319 \beta_{1} + \beta_{2} + 185 \beta_{3} - 53 \beta_{4} + 56 \beta_{5} + 13 \beta_{6} + 9 \beta_{7} - 24 \beta_{8} - 2 \beta_{9} + 3 \beta_{10} + 27 \beta_{11} - 11 \beta_{12} - 2 \beta_{13} - 7 \beta_{14} ) q^{49} + ( 7184 + 60 \beta_{1} - 140 \beta_{3} + 100 \beta_{4} + 80 \beta_{5} - 20 \beta_{7} - 40 \beta_{9} - 20 \beta_{10} + 20 \beta_{11} + 20 \beta_{13} ) q^{50} + ( 46 - 276 \beta_{1} + 3 \beta_{2} - 133 \beta_{3} - 252 \beta_{4} - 146 \beta_{5} - 20 \beta_{6} - 10 \beta_{7} - 32 \beta_{8} + 5 \beta_{9} - 26 \beta_{10} + 32 \beta_{11} - 3 \beta_{12} + 16 \beta_{13} - 14 \beta_{14} ) q^{51} + ( -112 - 32 \beta_{1} - 16 \beta_{2} - 64 \beta_{3} + 96 \beta_{4} - 16 \beta_{8} - 16 \beta_{10} - 16 \beta_{11} - 16 \beta_{13} ) q^{52} + ( 3205 - 187 \beta_{1} + \beta_{2} + 61 \beta_{3} + 177 \beta_{4} - 105 \beta_{5} - 16 \beta_{6} + 11 \beta_{7} + 51 \beta_{8} + 6 \beta_{9} - 14 \beta_{10} + 4 \beta_{11} - 16 \beta_{12} + 4 \beta_{13} - 10 \beta_{14} ) q^{53} + ( -3512 - 668 \beta_{1} + 32 \beta_{2} - 12 \beta_{3} + 12 \beta_{4} + 48 \beta_{5} - 24 \beta_{7} + 28 \beta_{8} + 16 \beta_{9} + 48 \beta_{10} - 12 \beta_{12} + 8 \beta_{13} + 16 \beta_{14} ) q^{54} + ( 1189 + 624 \beta_{1} + 54 \beta_{2} + 197 \beta_{3} + 180 \beta_{4} - 6 \beta_{5} - 83 \beta_{6} - 48 \beta_{7} + 17 \beta_{8} - 53 \beta_{9} - 46 \beta_{10} + 20 \beta_{11} - 5 \beta_{12} + 12 \beta_{13} + 20 \beta_{14} ) q^{55} + ( 384 - 64 \beta_{1} + 64 \beta_{4} - 64 \beta_{10} ) q^{56} + ( 3896 - 244 \beta_{1} - 8 \beta_{2} + 28 \beta_{3} + 12 \beta_{4} - 52 \beta_{5} - 8 \beta_{6} + 28 \beta_{7} + 4 \beta_{8} - 32 \beta_{9} + 72 \beta_{10} - 24 \beta_{11} + 32 \beta_{12} + 8 \beta_{13} ) q^{58} + ( -574 + 62 \beta_{1} - 35 \beta_{2} + 545 \beta_{3} + 11 \beta_{4} - 105 \beta_{5} - 12 \beta_{6} + 19 \beta_{7} - 27 \beta_{8} + 13 \beta_{9} + 15 \beta_{10} - 31 \beta_{11} + 17 \beta_{12} - 22 \beta_{13} - \beta_{14} ) q^{59} + ( 80 - 208 \beta_{1} - 16 \beta_{2} + 48 \beta_{3} + 288 \beta_{4} - 32 \beta_{5} - 48 \beta_{6} - 16 \beta_{7} - 16 \beta_{9} - 80 \beta_{10} - 16 \beta_{11} - 48 \beta_{12} + 16 \beta_{13} ) q^{60} + ( 7957 + 591 \beta_{1} + 3 \beta_{2} - 173 \beta_{3} + 41 \beta_{4} - 67 \beta_{5} + 34 \beta_{6} - \beta_{7} - 20 \beta_{8} - 30 \beta_{9} + 33 \beta_{10} - 31 \beta_{11} - 22 \beta_{12} + 15 \beta_{13} - 20 \beta_{14} ) q^{61} + ( 1812 - 56 \beta_{1} + 28 \beta_{2} + 316 \beta_{3} - 224 \beta_{4} - 48 \beta_{6} - 12 \beta_{7} + 20 \beta_{8} - 36 \beta_{9} + 84 \beta_{10} - 4 \beta_{11} - 12 \beta_{12} - 28 \beta_{13} + 28 \beta_{14} ) q^{62} + ( -5832 + 316 \beta_{1} + 3 \beta_{2} - 60 \beta_{3} + 606 \beta_{4} + 61 \beta_{5} - 61 \beta_{6} - 9 \beta_{7} + 77 \beta_{8} + 21 \beta_{9} - 43 \beta_{10} - 7 \beta_{11} - 99 \beta_{12} - 3 \beta_{13} + 7 \beta_{14} ) q^{63} + 4096 q^{64} + ( 5386 - 874 \beta_{1} + 36 \beta_{2} + 28 \beta_{3} - 316 \beta_{4} - 117 \beta_{5} + 35 \beta_{6} + 18 \beta_{7} - 17 \beta_{8} - 112 \beta_{9} + 63 \beta_{10} - 47 \beta_{11} + 41 \beta_{12} + 32 \beta_{13} + \beta_{14} ) q^{65} + ( -1000 + 156 \beta_{1} + 20 \beta_{2} - 296 \beta_{3} - 176 \beta_{4} - 176 \beta_{5} - 12 \beta_{6} - 20 \beta_{7} - 80 \beta_{8} - 40 \beta_{9} + 12 \beta_{10} + 36 \beta_{11} + 20 \beta_{12} + 24 \beta_{13} - 12 \beta_{14} ) q^{66} + ( 4283 - 1068 \beta_{1} + 26 \beta_{2} + 219 \beta_{3} - 180 \beta_{4} - 126 \beta_{5} + 4 \beta_{6} + 5 \beta_{7} + 16 \beta_{8} + 80 \beta_{9} - 46 \beta_{10} - 19 \beta_{11} + 16 \beta_{12} - 32 \beta_{13} + 2 \beta_{14} ) q^{67} + ( 4400 - 32 \beta_{1} + 64 \beta_{3} + 16 \beta_{4} - 48 \beta_{5} + 32 \beta_{6} + 16 \beta_{7} + 32 \beta_{8} + 16 \beta_{9} - 48 \beta_{10} + 16 \beta_{11} + 32 \beta_{12} - 16 \beta_{13} - 16 \beta_{14} ) q^{68} + ( 8455 - 955 \beta_{1} - 4 \beta_{2} - 541 \beta_{3} + 1134 \beta_{4} - 177 \beta_{5} - 158 \beta_{6} - 106 \beta_{7} - 8 \beta_{8} - 5 \beta_{9} - 120 \beta_{10} + 68 \beta_{11} - 20 \beta_{12} + 16 \beta_{13} + 3 \beta_{14} ) q^{69} + ( 3484 - 1080 \beta_{1} - 16 \beta_{2} + 88 \beta_{3} - 232 \beta_{4} + 56 \beta_{5} - 92 \beta_{6} - 44 \beta_{7} + 32 \beta_{8} - 144 \beta_{9} - 44 \beta_{10} + 28 \beta_{11} - 56 \beta_{12} - 12 \beta_{13} + 24 \beta_{14} ) q^{70} + ( -3578 - 1255 \beta_{1} - 19 \beta_{2} - 895 \beta_{3} + 369 \beta_{4} - 120 \beta_{5} - 37 \beta_{6} + 26 \beta_{7} - 33 \beta_{8} + 57 \beta_{9} + 69 \beta_{10} - 14 \beta_{11} - 13 \beta_{12} - 22 \beta_{13} + 25 \beta_{14} ) q^{71} + ( 9088 + 128 \beta_{1} - 64 \beta_{2} - 64 \beta_{3} ) q^{72} + ( -337 - 314 \beta_{1} + 39 \beta_{2} + 105 \beta_{3} - 700 \beta_{4} - 70 \beta_{5} + 73 \beta_{6} - 5 \beta_{7} - 59 \beta_{8} + 15 \beta_{9} - 26 \beta_{10} + 42 \beta_{11} + 49 \beta_{12} + 21 \beta_{13} - 44 \beta_{14} ) q^{73} + ( -1216 - 332 \beta_{1} + 20 \beta_{2} + 220 \beta_{3} - 300 \beta_{4} + 224 \beta_{5} - 8 \beta_{6} + 4 \beta_{7} + 32 \beta_{8} + 64 \beta_{9} + 28 \beta_{10} + 28 \beta_{11} + 8 \beta_{12} - 4 \beta_{13} - 24 \beta_{14} ) q^{74} + ( -5855 - 2901 \beta_{1} + 20 \beta_{2} + 570 \beta_{3} - 1000 \beta_{4} - 125 \beta_{5} + 145 \beta_{6} + 25 \beta_{7} + 45 \beta_{8} - 75 \beta_{9} + 65 \beta_{10} - 65 \beta_{11} - 115 \beta_{12} + 15 \beta_{13} + 20 \beta_{14} ) q^{75} + ( 7983 + 532 \beta_{1} - 16 \beta_{2} + 164 \beta_{3} + 544 \beta_{4} + 114 \beta_{5} + 109 \beta_{6} + 52 \beta_{7} - 95 \beta_{8} + 90 \beta_{9} - 67 \beta_{10} - 33 \beta_{11} + 43 \beta_{12} - 70 \beta_{13} + 17 \beta_{14} ) q^{77} + ( 1164 - 648 \beta_{1} + 44 \beta_{2} + 132 \beta_{3} + 508 \beta_{4} + 324 \beta_{5} + 28 \beta_{6} - 4 \beta_{7} - 28 \beta_{8} + 80 \beta_{9} + 64 \beta_{10} - 36 \beta_{11} + 32 \beta_{12} + 12 \beta_{13} + 24 \beta_{14} ) q^{78} + ( -7850 - 186 \beta_{1} + 35 \beta_{2} + 215 \beta_{3} + 661 \beta_{4} + 191 \beta_{5} + 64 \beta_{6} + 48 \beta_{7} + 25 \beta_{8} + 30 \beta_{9} + 109 \beta_{10} - 56 \beta_{11} + 90 \beta_{12} - 72 \beta_{13} - 10 \beta_{14} ) q^{79} + ( 1792 + 256 \beta_{5} ) q^{80} + ( 32124 + 1703 \beta_{1} - 87 \beta_{2} + 241 \beta_{3} - 724 \beta_{4} - 259 \beta_{5} + 22 \beta_{6} + 93 \beta_{7} - 87 \beta_{8} - 38 \beta_{9} + 128 \beta_{10} - 88 \beta_{11} + 62 \beta_{12} - 4 \beta_{13} - 50 \beta_{14} ) q^{81} + ( 1216 + 120 \beta_{1} + 28 \beta_{2} - 340 \beta_{3} + 184 \beta_{4} - 108 \beta_{5} + 176 \beta_{6} + 32 \beta_{7} + 40 \beta_{8} + 48 \beta_{9} + 152 \beta_{10} - 16 \beta_{11} - 16 \beta_{12} + 48 \beta_{13} + 24 \beta_{14} ) q^{82} + ( 13314 + 1083 \beta_{1} - 24 \beta_{2} - 747 \beta_{3} - 367 \beta_{4} - 145 \beta_{5} + 107 \beta_{6} + 12 \beta_{7} - 63 \beta_{8} - 23 \beta_{9} + 314 \beta_{10} - 84 \beta_{11} - 47 \beta_{12} - 55 \beta_{13} + 24 \beta_{14} ) q^{83} + ( 3584 + 176 \beta_{1} + 16 \beta_{2} - 176 \beta_{3} - 96 \beta_{4} + 352 \beta_{5} + 16 \beta_{6} + 32 \beta_{7} - 48 \beta_{8} + 48 \beta_{9} - 112 \beta_{10} + 16 \beta_{11} + 16 \beta_{12} + 32 \beta_{13} - 32 \beta_{14} ) q^{84} + ( -9946 + 2115 \beta_{1} - 11 \beta_{2} - 1535 \beta_{3} + 860 \beta_{4} + 397 \beta_{5} - 276 \beta_{6} - 104 \beta_{7} + 150 \beta_{8} + 41 \beta_{9} - 144 \beta_{10} + 86 \beta_{11} - 66 \beta_{12} - 46 \beta_{13} + 89 \beta_{14} ) q^{85} + ( 25028 - 388 \beta_{1} + 28 \beta_{2} - 36 \beta_{3} - 108 \beta_{4} + 248 \beta_{5} + 84 \beta_{6} - 24 \beta_{7} - 40 \beta_{8} + 96 \beta_{9} - 104 \beta_{10} + 8 \beta_{11} + 12 \beta_{12} + 32 \beta_{13} + 16 \beta_{14} ) q^{86} + ( 25074 - 1799 \beta_{1} + 18 \beta_{2} + 291 \beta_{3} + 1459 \beta_{4} - 207 \beta_{5} + 22 \beta_{6} - 143 \beta_{7} + 16 \beta_{8} - 66 \beta_{9} + 270 \beta_{10} - 51 \beta_{11} - 82 \beta_{12} + 35 \beta_{13} + 48 \beta_{14} ) q^{87} + ( 512 + 64 \beta_{3} + 64 \beta_{4} + 64 \beta_{6} - 64 \beta_{13} ) q^{88} + ( -6734 - 309 \beta_{1} - 15 \beta_{2} + 499 \beta_{3} - 946 \beta_{4} + 82 \beta_{5} - 50 \beta_{6} - 46 \beta_{7} + 148 \beta_{8} + 93 \beta_{9} - 108 \beta_{10} + 56 \beta_{11} + 26 \beta_{12} - 68 \beta_{13} - 41 \beta_{14} ) q^{89} + ( 8112 - 72 \beta_{1} + 20 \beta_{2} - 1024 \beta_{3} - 136 \beta_{4} + 872 \beta_{5} + 100 \beta_{6} + 184 \beta_{7} + 8 \beta_{8} + 184 \beta_{9} + 16 \beta_{10} - 88 \beta_{11} + 196 \beta_{12} + 60 \beta_{13} - 36 \beta_{14} ) q^{90} + ( 27458 + 362 \beta_{1} + 71 \beta_{2} + 511 \beta_{3} - 63 \beta_{4} + 187 \beta_{5} + 208 \beta_{6} - 72 \beta_{7} - 97 \beta_{8} - 42 \beta_{9} + 285 \beta_{10} + 32 \beta_{11} - 132 \beta_{12} - 10 \beta_{13} + 30 \beta_{14} ) q^{91} + ( 4064 - 368 \beta_{1} - 80 \beta_{2} + 304 \beta_{3} - 560 \beta_{4} + 96 \beta_{5} + 80 \beta_{6} + 64 \beta_{7} - 16 \beta_{8} + 144 \beta_{9} + 48 \beta_{10} - 32 \beta_{11} + 16 \beta_{12} - 64 \beta_{13} + 16 \beta_{14} ) q^{92} + ( 11357 + 2002 \beta_{1} - 161 \beta_{2} + 1538 \beta_{3} - 1172 \beta_{4} - 642 \beta_{5} - 230 \beta_{6} - 48 \beta_{7} - 207 \beta_{8} - 198 \beta_{9} + 139 \beta_{10} + 65 \beta_{11} + 56 \beta_{12} - 37 \beta_{13} - 46 \beta_{14} ) q^{93} + ( 276 - 208 \beta_{1} - 40 \beta_{2} + 296 \beta_{3} - 1032 \beta_{4} - 84 \beta_{5} - 76 \beta_{6} + 12 \beta_{7} + 100 \beta_{8} - 28 \beta_{9} + 4 \beta_{10} + 28 \beta_{11} - 4 \beta_{12} - 12 \beta_{13} - 64 \beta_{14} ) q^{94} -1024 \beta_{1} q^{96} + ( 19998 + 306 \beta_{1} + 38 \beta_{2} - 773 \beta_{3} + 1909 \beta_{4} - 422 \beta_{5} - 65 \beta_{6} + \beta_{7} - 85 \beta_{8} - 205 \beta_{9} - 28 \beta_{10} + 166 \beta_{11} + 99 \beta_{12} + 33 \beta_{13} - 36 \beta_{14} ) q^{97} + ( 16952 - 1276 \beta_{1} + 4 \beta_{2} + 740 \beta_{3} - 212 \beta_{4} + 224 \beta_{5} + 52 \beta_{6} + 36 \beta_{7} - 96 \beta_{8} - 8 \beta_{9} + 12 \beta_{10} + 108 \beta_{11} - 44 \beta_{12} - 8 \beta_{13} - 28 \beta_{14} ) q^{98} + ( -10062 + 1048 \beta_{1} + 44 \beta_{2} + 2803 \beta_{3} - 1829 \beta_{4} - 756 \beta_{5} + 139 \beta_{6} + 80 \beta_{7} + 348 \beta_{8} + 24 \beta_{9} + 188 \beta_{10} - 18 \beta_{11} - 71 \beta_{12} - 84 \beta_{13} + 173 \beta_{14} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 15q + 60q^{2} + 240q^{4} + 108q^{5} + 84q^{7} + 960q^{8} + 2127q^{9} + O(q^{10}) \) \( 15q + 60q^{2} + 240q^{4} + 108q^{5} + 84q^{7} + 960q^{8} + 2127q^{9} + 432q^{10} + 126q^{11} - 114q^{13} + 336q^{14} + 3840q^{16} + 4119q^{17} + 8508q^{18} + 1728q^{20} + 3408q^{21} + 504q^{22} + 3936q^{23} + 26895q^{25} - 456q^{26} - 13017q^{27} + 1344q^{28} + 14658q^{29} + 6840q^{31} + 15360q^{32} - 3945q^{33} + 16476q^{34} + 12636q^{35} + 34032q^{36} - 4278q^{37} + 4956q^{39} + 6912q^{40} + 5112q^{41} + 13632q^{42} + 94191q^{43} + 2016q^{44} + 31770q^{45} + 15744q^{46} + 702q^{47} + 63777q^{49} + 107580q^{50} - 108q^{51} - 1824q^{52} + 47544q^{53} - 52068q^{54} + 16848q^{55} + 5376q^{56} + 58632q^{58} - 8832q^{59} + 119196q^{61} + 27360q^{62} - 88068q^{63} + 61440q^{64} + 80646q^{65} - 15780q^{66} + 64248q^{67} + 65904q^{68} + 124224q^{69} + 50544q^{70} - 53364q^{71} + 136128q^{72} - 4908q^{73} - 17112q^{74} - 87480q^{75} + 121218q^{77} + 19824q^{78} - 115500q^{79} + 27648q^{80} + 481659q^{81} + 20448q^{82} + 201630q^{83} + 54528q^{84} - 150282q^{85} + 376764q^{86} + 376512q^{87} + 8064q^{88} - 101505q^{89} + 127080q^{90} + 414918q^{91} + 62976q^{92} + 165960q^{93} + 2808q^{94} + 297114q^{97} + 255108q^{98} - 149895q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{15} - 2871 x^{13} - 4674 x^{12} + 3170019 x^{11} + 9081402 x^{10} - 1680307373 x^{9} - 6060225486 x^{8} + 437045334939 x^{7} + 1625551725606 x^{6} - 51993847360191 x^{5} - 166593470925336 x^{4} + 2435236025483701 x^{3} + 5743342864926480 x^{2} - 19853455953893436 x - 34308015556724472\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\(\)\(40\!\cdots\!68\)\( \nu^{14} + \)\(11\!\cdots\!63\)\( \nu^{13} - \)\(11\!\cdots\!02\)\( \nu^{12} - \)\(51\!\cdots\!33\)\( \nu^{11} + \)\(11\!\cdots\!24\)\( \nu^{10} + \)\(67\!\cdots\!97\)\( \nu^{9} - \)\(54\!\cdots\!70\)\( \nu^{8} - \)\(36\!\cdots\!79\)\( \nu^{7} + \)\(11\!\cdots\!22\)\( \nu^{6} + \)\(73\!\cdots\!69\)\( \nu^{5} - \)\(74\!\cdots\!72\)\( \nu^{4} - \)\(31\!\cdots\!81\)\( \nu^{3} + \)\(74\!\cdots\!46\)\( \nu^{2} + \)\(17\!\cdots\!21\)\( \nu - \)\(12\!\cdots\!60\)\(\)\()/ \)\(31\!\cdots\!06\)\( \)
\(\beta_{2}\)\(=\)\((\)\(\)\(46\!\cdots\!99\)\( \nu^{14} - \)\(68\!\cdots\!78\)\( \nu^{13} - \)\(15\!\cdots\!05\)\( \nu^{12} + \)\(18\!\cdots\!84\)\( \nu^{11} + \)\(23\!\cdots\!77\)\( \nu^{10} - \)\(19\!\cdots\!88\)\( \nu^{9} - \)\(19\!\cdots\!31\)\( \nu^{8} + \)\(89\!\cdots\!00\)\( \nu^{7} + \)\(86\!\cdots\!61\)\( \nu^{6} - \)\(18\!\cdots\!36\)\( \nu^{5} - \)\(16\!\cdots\!49\)\( \nu^{4} + \)\(11\!\cdots\!94\)\( \nu^{3} + \)\(94\!\cdots\!93\)\( \nu^{2} - \)\(19\!\cdots\!58\)\( \nu + \)\(34\!\cdots\!66\)\(\)\()/ \)\(10\!\cdots\!02\)\( \)
\(\beta_{3}\)\(=\)\((\)\(-\)\(33\!\cdots\!43\)\( \nu^{14} + \)\(13\!\cdots\!12\)\( \nu^{13} + \)\(98\!\cdots\!57\)\( \nu^{12} - \)\(36\!\cdots\!98\)\( \nu^{11} - \)\(12\!\cdots\!45\)\( \nu^{10} + \)\(36\!\cdots\!30\)\( \nu^{9} + \)\(80\!\cdots\!35\)\( \nu^{8} - \)\(16\!\cdots\!22\)\( \nu^{7} - \)\(28\!\cdots\!29\)\( \nu^{6} + \)\(32\!\cdots\!46\)\( \nu^{5} + \)\(46\!\cdots\!41\)\( \nu^{4} - \)\(19\!\cdots\!00\)\( \nu^{3} - \)\(26\!\cdots\!27\)\( \nu^{2} - \)\(44\!\cdots\!40\)\( \nu + \)\(16\!\cdots\!32\)\(\)\()/ \)\(33\!\cdots\!48\)\( \)
\(\beta_{4}\)\(=\)\((\)\(\)\(40\!\cdots\!68\)\( \nu^{14} + \)\(11\!\cdots\!63\)\( \nu^{13} - \)\(11\!\cdots\!02\)\( \nu^{12} - \)\(51\!\cdots\!33\)\( \nu^{11} + \)\(11\!\cdots\!24\)\( \nu^{10} + \)\(67\!\cdots\!97\)\( \nu^{9} - \)\(54\!\cdots\!70\)\( \nu^{8} - \)\(36\!\cdots\!79\)\( \nu^{7} + \)\(11\!\cdots\!22\)\( \nu^{6} + \)\(73\!\cdots\!69\)\( \nu^{5} - \)\(74\!\cdots\!72\)\( \nu^{4} - \)\(31\!\cdots\!81\)\( \nu^{3} + \)\(74\!\cdots\!46\)\( \nu^{2} - \)\(13\!\cdots\!85\)\( \nu - \)\(12\!\cdots\!60\)\(\)\()/ \)\(16\!\cdots\!74\)\( \)
\(\beta_{5}\)\(=\)\((\)\(\)\(12\!\cdots\!29\)\( \nu^{14} + \)\(27\!\cdots\!06\)\( \nu^{13} - \)\(36\!\cdots\!55\)\( \nu^{12} - \)\(80\!\cdots\!86\)\( \nu^{11} + \)\(39\!\cdots\!64\)\( \nu^{10} + \)\(87\!\cdots\!30\)\( \nu^{9} - \)\(19\!\cdots\!53\)\( \nu^{8} - \)\(43\!\cdots\!64\)\( \nu^{7} + \)\(43\!\cdots\!84\)\( \nu^{6} + \)\(93\!\cdots\!46\)\( \nu^{5} - \)\(36\!\cdots\!47\)\( \nu^{4} - \)\(68\!\cdots\!90\)\( \nu^{3} + \)\(12\!\cdots\!20\)\( \nu^{2} + \)\(80\!\cdots\!76\)\( \nu - \)\(15\!\cdots\!75\)\(\)\()/ \)\(30\!\cdots\!67\)\( \)
\(\beta_{6}\)\(=\)\((\)\(\)\(67\!\cdots\!99\)\( \nu^{14} + \)\(11\!\cdots\!56\)\( \nu^{13} - \)\(15\!\cdots\!41\)\( \nu^{12} - \)\(39\!\cdots\!86\)\( \nu^{11} + \)\(11\!\cdots\!09\)\( \nu^{10} + \)\(47\!\cdots\!06\)\( \nu^{9} - \)\(12\!\cdots\!99\)\( \nu^{8} - \)\(24\!\cdots\!90\)\( \nu^{7} - \)\(17\!\cdots\!99\)\( \nu^{6} + \)\(55\!\cdots\!18\)\( \nu^{5} + \)\(58\!\cdots\!67\)\( \nu^{4} - \)\(41\!\cdots\!08\)\( \nu^{3} - \)\(43\!\cdots\!17\)\( \nu^{2} + \)\(62\!\cdots\!12\)\( \nu + \)\(12\!\cdots\!64\)\(\)\()/ \)\(12\!\cdots\!68\)\( \)
\(\beta_{7}\)\(=\)\((\)\(\)\(14\!\cdots\!80\)\( \nu^{14} - \)\(59\!\cdots\!57\)\( \nu^{13} - \)\(44\!\cdots\!94\)\( \nu^{12} + \)\(16\!\cdots\!75\)\( \nu^{11} + \)\(54\!\cdots\!46\)\( \nu^{10} - \)\(16\!\cdots\!89\)\( \nu^{9} - \)\(34\!\cdots\!04\)\( \nu^{8} + \)\(81\!\cdots\!51\)\( \nu^{7} + \)\(11\!\cdots\!76\)\( \nu^{6} - \)\(18\!\cdots\!11\)\( \nu^{5} - \)\(19\!\cdots\!96\)\( \nu^{4} + \)\(17\!\cdots\!95\)\( \nu^{3} + \)\(12\!\cdots\!04\)\( \nu^{2} - \)\(57\!\cdots\!27\)\( \nu - \)\(81\!\cdots\!22\)\(\)\()/ \)\(20\!\cdots\!78\)\( \)
\(\beta_{8}\)\(=\)\((\)\(-\)\(21\!\cdots\!75\)\( \nu^{14} - \)\(52\!\cdots\!90\)\( \nu^{13} + \)\(61\!\cdots\!21\)\( \nu^{12} + \)\(23\!\cdots\!48\)\( \nu^{11} - \)\(67\!\cdots\!05\)\( \nu^{10} - \)\(32\!\cdots\!12\)\( \nu^{9} + \)\(35\!\cdots\!71\)\( \nu^{8} + \)\(18\!\cdots\!40\)\( \nu^{7} - \)\(92\!\cdots\!29\)\( \nu^{6} - \)\(43\!\cdots\!52\)\( \nu^{5} + \)\(11\!\cdots\!53\)\( \nu^{4} + \)\(35\!\cdots\!86\)\( \nu^{3} - \)\(60\!\cdots\!59\)\( \nu^{2} - \)\(46\!\cdots\!14\)\( \nu + \)\(48\!\cdots\!08\)\(\)\()/ \)\(12\!\cdots\!68\)\( \)
\(\beta_{9}\)\(=\)\((\)\(\)\(92\!\cdots\!91\)\( \nu^{14} - \)\(62\!\cdots\!54\)\( \nu^{13} - \)\(25\!\cdots\!29\)\( \nu^{12} - \)\(19\!\cdots\!56\)\( \nu^{11} + \)\(27\!\cdots\!25\)\( \nu^{10} + \)\(48\!\cdots\!64\)\( \nu^{9} - \)\(13\!\cdots\!63\)\( \nu^{8} - \)\(30\!\cdots\!72\)\( \nu^{7} + \)\(28\!\cdots\!13\)\( \nu^{6} + \)\(62\!\cdots\!52\)\( \nu^{5} - \)\(23\!\cdots\!45\)\( \nu^{4} - \)\(30\!\cdots\!90\)\( \nu^{3} + \)\(46\!\cdots\!99\)\( \nu^{2} + \)\(10\!\cdots\!34\)\( \nu - \)\(23\!\cdots\!28\)\(\)\()/ \)\(41\!\cdots\!56\)\( \)
\(\beta_{10}\)\(=\)\((\)\(\)\(17\!\cdots\!83\)\( \nu^{14} + \)\(23\!\cdots\!60\)\( \nu^{13} - \)\(48\!\cdots\!97\)\( \nu^{12} - \)\(14\!\cdots\!78\)\( \nu^{11} + \)\(51\!\cdots\!89\)\( \nu^{10} + \)\(21\!\cdots\!58\)\( \nu^{9} - \)\(25\!\cdots\!31\)\( \nu^{8} - \)\(12\!\cdots\!82\)\( \nu^{7} + \)\(59\!\cdots\!85\)\( \nu^{6} + \)\(27\!\cdots\!76\)\( \nu^{5} - \)\(56\!\cdots\!73\)\( \nu^{4} - \)\(15\!\cdots\!76\)\( \nu^{3} + \)\(18\!\cdots\!21\)\( \nu^{2} - \)\(57\!\cdots\!76\)\( \nu - \)\(10\!\cdots\!72\)\(\)\()/ \)\(61\!\cdots\!34\)\( \)
\(\beta_{11}\)\(=\)\((\)\(\)\(76\!\cdots\!06\)\( \nu^{14} + \)\(20\!\cdots\!89\)\( \nu^{13} - \)\(20\!\cdots\!10\)\( \nu^{12} - \)\(94\!\cdots\!41\)\( \nu^{11} + \)\(21\!\cdots\!22\)\( \nu^{10} + \)\(12\!\cdots\!77\)\( \nu^{9} - \)\(10\!\cdots\!72\)\( \nu^{8} - \)\(68\!\cdots\!89\)\( \nu^{7} + \)\(21\!\cdots\!24\)\( \nu^{6} + \)\(13\!\cdots\!53\)\( \nu^{5} - \)\(15\!\cdots\!74\)\( \nu^{4} - \)\(59\!\cdots\!85\)\( \nu^{3} + \)\(34\!\cdots\!28\)\( \nu^{2} - \)\(23\!\cdots\!13\)\( \nu - \)\(82\!\cdots\!32\)\(\)\()/ \)\(20\!\cdots\!78\)\( \)
\(\beta_{12}\)\(=\)\((\)\(-\)\(17\!\cdots\!99\)\( \nu^{14} - \)\(45\!\cdots\!91\)\( \nu^{13} + \)\(50\!\cdots\!37\)\( \nu^{12} + \)\(20\!\cdots\!94\)\( \nu^{11} - \)\(54\!\cdots\!06\)\( \nu^{10} - \)\(27\!\cdots\!65\)\( \nu^{9} + \)\(27\!\cdots\!21\)\( \nu^{8} + \)\(15\!\cdots\!81\)\( \nu^{7} - \)\(66\!\cdots\!19\)\( \nu^{6} - \)\(33\!\cdots\!28\)\( \nu^{5} + \)\(69\!\cdots\!06\)\( \nu^{4} + \)\(22\!\cdots\!84\)\( \nu^{3} - \)\(28\!\cdots\!80\)\( \nu^{2} - \)\(13\!\cdots\!66\)\( \nu + \)\(17\!\cdots\!37\)\(\)\()/ \)\(30\!\cdots\!67\)\( \)
\(\beta_{13}\)\(=\)\((\)\(-\)\(78\!\cdots\!03\)\( \nu^{14} - \)\(99\!\cdots\!12\)\( \nu^{13} + \)\(22\!\cdots\!49\)\( \nu^{12} + \)\(64\!\cdots\!46\)\( \nu^{11} - \)\(24\!\cdots\!81\)\( \nu^{10} - \)\(10\!\cdots\!66\)\( \nu^{9} + \)\(12\!\cdots\!79\)\( \nu^{8} + \)\(60\!\cdots\!86\)\( \nu^{7} - \)\(31\!\cdots\!49\)\( \nu^{6} - \)\(15\!\cdots\!02\)\( \nu^{5} + \)\(34\!\cdots\!61\)\( \nu^{4} + \)\(13\!\cdots\!96\)\( \nu^{3} - \)\(13\!\cdots\!07\)\( \nu^{2} - \)\(26\!\cdots\!08\)\( \nu + \)\(51\!\cdots\!60\)\(\)\()/ \)\(12\!\cdots\!68\)\( \)
\(\beta_{14}\)\(=\)\((\)\(-\)\(16\!\cdots\!47\)\( \nu^{14} - \)\(12\!\cdots\!44\)\( \nu^{13} + \)\(45\!\cdots\!87\)\( \nu^{12} + \)\(42\!\cdots\!14\)\( \nu^{11} - \)\(47\!\cdots\!75\)\( \nu^{10} - \)\(50\!\cdots\!66\)\( \nu^{9} + \)\(22\!\cdots\!03\)\( \nu^{8} + \)\(26\!\cdots\!32\)\( \nu^{7} - \)\(43\!\cdots\!61\)\( \nu^{6} - \)\(56\!\cdots\!32\)\( \nu^{5} + \)\(23\!\cdots\!01\)\( \nu^{4} + \)\(40\!\cdots\!38\)\( \nu^{3} + \)\(53\!\cdots\!31\)\( \nu^{2} - \)\(53\!\cdots\!32\)\( \nu - \)\(16\!\cdots\!00\)\(\)\()/ \)\(20\!\cdots\!78\)\( \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-\beta_{4} + 19 \beta_{1}\)\()/19\)
\(\nu^{2}\)\(=\)\((\)\(2 \beta_{12} - 2 \beta_{11} + 2 \beta_{10} + 2 \beta_{9} - 2 \beta_{8} + 2 \beta_{7} + 2 \beta_{6} + 4 \beta_{5} - 2 \beta_{4} - 20 \beta_{3} - 19 \beta_{2} + 40 \beta_{1} + 7273\)\()/19\)
\(\nu^{3}\)\(=\)\((\)\(-97 \beta_{14} - 38 \beta_{13} + 96 \beta_{12} + 30 \beta_{11} - 186 \beta_{10} - 85 \beta_{9} - 148 \beta_{8} + 162 \beta_{7} + 15 \beta_{6} - 183 \beta_{5} - 1308 \beta_{4} + 66 \beta_{3} - 173 \beta_{2} + 12479 \beta_{1} + 17922\)\()/19\)
\(\nu^{4}\)\(=\)\((\)\(-1360 \beta_{14} + 190 \beta_{13} + 4214 \beta_{12} - 3954 \beta_{11} + 5642 \beta_{10} + 1476 \beta_{9} - 4063 \beta_{8} + 4459 \beta_{7} + 2934 \beta_{6} + 4509 \beta_{5} - 20395 \beta_{4} - 11475 \beta_{3} - 16028 \beta_{2} + 67189 \beta_{1} + 4817020\)\()/19\)
\(\nu^{5}\)\(=\)\((\)\(-108532 \beta_{14} - 34010 \beta_{13} + 131916 \beta_{12} + 27512 \beta_{11} - 132077 \beta_{10} - 52170 \beta_{9} - 171388 \beta_{8} + 195628 \beta_{7} + 42390 \beta_{6} - 96886 \beta_{5} - 1942502 \beta_{4} + 40390 \beta_{3} - 209555 \beta_{2} + 9383830 \beta_{1} + 27567731\)\()/19\)
\(\nu^{6}\)\(=\)\((\)\(-1819163 \beta_{14} + 348536 \beta_{13} + 5433359 \beta_{12} - 4224877 \beta_{11} + 6529751 \beta_{10} + 1706188 \beta_{9} - 5207847 \beta_{8} + 5620872 \beta_{7} + 3833053 \beta_{6} + 7838294 \beta_{5} - 30702514 \beta_{4} - 9942074 \beta_{3} - 13152321 \beta_{2} + 82086760 \beta_{1} + 3629933860\)\()/19\)
\(\nu^{7}\)\(=\)\((\)\(-106782040 \beta_{14} - 25659101 \beta_{13} + 160647482 \beta_{12} + 20459193 \beta_{11} - 57171900 \beta_{10} - 25258339 \beta_{9} - 176473923 \beta_{8} + 205955454 \beta_{7} + 59910555 \beta_{6} + 21055603 \beta_{5} - 2227675502 \beta_{4} - 27044834 \beta_{3} - 215067524 \beta_{2} + 7528582711 \beta_{1} + 32796875225\)\()/19\)
\(\nu^{8}\)\(=\)\((\)\(-2101820106 \beta_{14} + 486960082 \beta_{13} + 6208915457 \beta_{12} - 3915486941 \beta_{11} + 6599883213 \beta_{10} + 1819989763 \beta_{9} - 5746085449 \beta_{8} + 6178946184 \beta_{7} + 4259603655 \beta_{6} + 10804243073 \beta_{5} - 37508951792 \beta_{4} - 11537606425 \beta_{3} - 11022118554 \beta_{2} + 90288520217 \beta_{1} + 2910272012244\)\()/19\)
\(\nu^{9}\)\(=\)\((\)\(-103318575512 \beta_{14} - 17505847961 \beta_{13} + 184253758157 \beta_{12} + 12758261518 \beta_{11} + 3503748183 \beta_{10} - 6593912618 \beta_{9} - 180541496880 \beta_{8} + 208749564543 \beta_{7} + 71943125046 \beta_{6} + 127662346885 \beta_{5} - 2312807532584 \beta_{4} - 115832538463 \beta_{3} - 211089624822 \beta_{2} + 6307250811039 \beta_{1} + 35585553569167\)\()/19\)
\(\nu^{10}\)\(=\)\((\)\(-2314506788037 \beta_{14} + 611868793642 \beta_{13} + 6718510062981 \beta_{12} - 3436204036483 \beta_{11} + 6446103049979 \beta_{10} + 1794990811936 \beta_{9} - 6018706544711 \beta_{8} + 6423096617722 \beta_{7} + 4373689048329 \beta_{6} + 12942984854783 \beta_{5} - 42476925886514 \beta_{4} - 13939738796066 \beta_{3} - 9478761805354 \beta_{2} + 94777178869950 \beta_{1} + 2433686711301360\)\()/19\)
\(\nu^{11}\)\(=\)\((\)\(-100447494063224 \beta_{14} - 9761809753832 \beta_{13} + 202526285762246 \beta_{12} + 5854516393761 \beta_{11} + 48292593257482 \beta_{10} + 6192246770573 \beta_{9} - 185176095272694 \beta_{8} + 208414750784623 \beta_{7} + 79989780455556 \beta_{6} + 215034748629711 \beta_{5} - 2303746850761660 \beta_{4} - 209771977820271 \beta_{3} - 204251380052439 \beta_{2} + 5463395907258479 \beta_{1} + 37022077134045267\)\()/19\)
\(\nu^{12}\)\(=\)\((\)\(-2480676302833615 \beta_{14} + 721423171444289 \beta_{13} + 7061018188206147 \beta_{12} - 2944079019030926 \beta_{11} + 6247420570537850 \beta_{10} + 1696443005193974 \beta_{9} - 6181145737016242 \beta_{8} + 6499470211523256 \beta_{7} + 4322144777428537 \beta_{6} + 14364399165839275 \beta_{5} - 46187795514399380 \beta_{4} - 16267017509263180 \beta_{3} - 8352370814518331 \beta_{2} + 97167083931194694 \beta_{1} + 2103225877621107699\)\()/19\)
\(\nu^{13}\)\(=\)\((\)\(-98467910169687803 \beta_{14} - 2545495341263264 \beta_{13} + 216073512961131523 \beta_{12} + 214307783380624 \beta_{11} + 80446717560439182 \beta_{10} + 14795771672549923 \beta_{9} - 189883356478368497 \beta_{8} + 206611485280131540 \beta_{7} + 85117393281833262 \beta_{6} + 284114537848223858 \beta_{5} - 2254314782326591525 \beta_{4} - 300068718204594516 \beta_{3} - 196962787911948147 \beta_{2} + 4861924279296478217 \beta_{1} + 37706446110039766494\)\()/19\)
\(\nu^{14}\)\(=\)\((\)\(-2610305957155644383 \beta_{14} + 813992601822006193 \beta_{13} + 7295137110306615336 \beta_{12} - 2497387721195616171 \beta_{11} + 6057753222956392545 \beta_{10} + 1571285778243187631 \beta_{9} - 6293720050026748718 \beta_{8} + 6486906170602019745 \beta_{7} + 4191419996174775486 \beta_{6} + 15265012098058714432 \beta_{5} - 48935932543249543131 \beta_{4} - 18251765048210922114 \beta_{3} - 7520570957827058099 \beta_{2} + 98353729958252121694 \beta_{1} + 1867133865590550389336\)\()/19\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
31.5298
28.1551
25.8045
18.3407
10.3629
9.74479
3.01062
−1.51331
−3.66933
−10.7498
−12.5784
−20.7673
−23.5180
−26.2549
−27.8974
4.00000 −29.6504 16.0000 89.4742 −118.602 −137.413 64.0000 636.149 357.897
1.2 4.00000 −28.5024 16.0000 −35.3418 −114.010 85.8987 64.0000 569.386 −141.367
1.3 4.00000 −27.3366 16.0000 −91.1494 −109.346 45.5647 64.0000 504.290 −364.597
1.4 4.00000 −19.8728 16.0000 105.633 −79.4911 36.4788 64.0000 151.927 422.533
1.5 4.00000 −10.7102 16.0000 59.4057 −42.8408 −230.434 64.0000 −128.291 237.623
1.6 4.00000 −7.86541 16.0000 −51.1536 −31.4616 18.3720 64.0000 −181.135 −204.615
1.7 4.00000 −1.13124 16.0000 11.4222 −4.52494 15.4828 64.0000 −241.720 45.6887
1.8 4.00000 1.16601 16.0000 −65.1319 4.66404 94.5243 64.0000 −241.640 −260.528
1.9 4.00000 2.13724 16.0000 −10.7945 8.54896 −195.108 64.0000 −238.432 −43.1779
1.10 4.00000 10.4025 16.0000 95.5276 41.6100 232.219 64.0000 −134.788 382.111
1.11 4.00000 11.0464 16.0000 −52.2115 44.1854 215.645 64.0000 −120.978 −208.846
1.12 4.00000 22.6466 16.0000 79.6824 90.5866 191.779 64.0000 269.871 318.730
1.13 4.00000 25.3974 16.0000 −93.4251 101.589 −182.176 64.0000 402.026 −373.700
1.14 4.00000 25.9076 16.0000 −18.4597 103.630 −122.093 64.0000 428.205 −73.8387
1.15 4.00000 26.3653 16.0000 84.5221 105.461 15.2607 64.0000 452.132 338.089
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.15
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(19\) \(1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 722.6.a.r 15
19.b odd 2 1 722.6.a.q 15
19.e even 9 2 38.6.e.b 30
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
38.6.e.b 30 19.e even 9 2
722.6.a.q 15 19.b odd 2 1
722.6.a.r 15 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(16\!\cdots\!43\)\( T_{3}^{6} - \)\(46\!\cdots\!52\)\( T_{3}^{5} + \)\(15\!\cdots\!73\)\( T_{3}^{4} + \)\(16\!\cdots\!42\)\( T_{3}^{3} - \)\(39\!\cdots\!41\)\( T_{3}^{2} - \)\(20\!\cdots\!61\)\( T_{3} + \)\(49\!\cdots\!71\)\( \)">\(T_{3}^{15} - \cdots\) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(722))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( -4 + T )^{15} \)
$3$ \( 4922141548222971 - 2028046240465461 T - 3989649082088241 T^{2} + 1690978504212342 T^{3} + 156214328638473 T^{4} - 46284758138952 T^{5} - 1645531637743 T^{6} + 434202352248 T^{7} + 6021674247 T^{8} - 1709279884 T^{9} - 8713875 T^{10} + 3213570 T^{11} + 4339 T^{12} - 2886 T^{13} + T^{15} \)
$5$ \( -\)\(43\!\cdots\!16\)\( - \)\(39\!\cdots\!32\)\( T + \)\(29\!\cdots\!98\)\( T^{2} + \)\(33\!\cdots\!33\)\( T^{3} + \)\(33\!\cdots\!08\)\( T^{4} - 30026575789086981261 T^{5} - 461339133564115250 T^{6} + 12192907694053605 T^{7} + 186633454199520 T^{8} - 2850496741725 T^{9} - 35032443324 T^{10} + 400875657 T^{11} + 3136722 T^{12} - 31053 T^{13} - 108 T^{14} + T^{15} \)
$7$ \( \)\(77\!\cdots\!12\)\( - \)\(18\!\cdots\!48\)\( T + \)\(17\!\cdots\!62\)\( T^{2} - \)\(74\!\cdots\!53\)\( T^{3} + \)\(14\!\cdots\!00\)\( T^{4} - \)\(15\!\cdots\!48\)\( T^{5} - \)\(29\!\cdots\!24\)\( T^{6} + 2279819863211981670 T^{7} + 21541507819697910 T^{8} - 224626159105871 T^{9} - 752836226922 T^{10} + 8820593586 T^{11} + 12796586 T^{12} - 154413 T^{13} - 84 T^{14} + T^{15} \)
$11$ \( \)\(66\!\cdots\!17\)\( + \)\(61\!\cdots\!27\)\( T + \)\(14\!\cdots\!24\)\( T^{2} + \)\(10\!\cdots\!07\)\( T^{3} - \)\(27\!\cdots\!58\)\( T^{4} - \)\(32\!\cdots\!94\)\( T^{5} + \)\(12\!\cdots\!63\)\( T^{6} + \)\(39\!\cdots\!81\)\( T^{7} + 5290603569275748144 T^{8} - 247182173496103245 T^{9} - 58629534368133 T^{10} + 836972776335 T^{11} + 157590693 T^{12} - 1456764 T^{13} - 126 T^{14} + T^{15} \)
$13$ \( \)\(17\!\cdots\!96\)\( - \)\(17\!\cdots\!52\)\( T - \)\(68\!\cdots\!50\)\( T^{2} + \)\(64\!\cdots\!33\)\( T^{3} + \)\(96\!\cdots\!72\)\( T^{4} - \)\(86\!\cdots\!94\)\( T^{5} - \)\(58\!\cdots\!14\)\( T^{6} + \)\(56\!\cdots\!49\)\( T^{7} + \)\(14\!\cdots\!44\)\( T^{8} - 1923671324247005906 T^{9} - 104585960471160 T^{10} + 3444994413402 T^{11} - 110416072 T^{12} - 3007272 T^{13} + 114 T^{14} + T^{15} \)
$17$ \( \)\(45\!\cdots\!11\)\( + \)\(53\!\cdots\!35\)\( T - \)\(48\!\cdots\!89\)\( T^{2} - \)\(29\!\cdots\!93\)\( T^{3} + \)\(44\!\cdots\!05\)\( T^{4} + \)\(24\!\cdots\!13\)\( T^{5} - \)\(11\!\cdots\!57\)\( T^{6} - \)\(50\!\cdots\!89\)\( T^{7} + \)\(15\!\cdots\!32\)\( T^{8} + 35969928330142381536 T^{9} - 102185741822630058 T^{10} - 3508305852762 T^{11} + 33691850511 T^{12} - 4416669 T^{13} - 4119 T^{14} + T^{15} \)
$19$ \( T^{15} \)
$23$ \( -\)\(50\!\cdots\!28\)\( + \)\(42\!\cdots\!08\)\( T - \)\(38\!\cdots\!28\)\( T^{2} - \)\(37\!\cdots\!28\)\( T^{3} + \)\(53\!\cdots\!16\)\( T^{4} + \)\(10\!\cdots\!68\)\( T^{5} - \)\(12\!\cdots\!28\)\( T^{6} - \)\(61\!\cdots\!72\)\( T^{7} + \)\(94\!\cdots\!68\)\( T^{8} + 33004470979913530881 T^{9} - 2254100471335670856 T^{10} + 362434278880671 T^{11} + 171292390182 T^{12} - 37758720 T^{13} - 3936 T^{14} + T^{15} \)
$29$ \( \)\(53\!\cdots\!84\)\( - \)\(37\!\cdots\!68\)\( T - \)\(14\!\cdots\!44\)\( T^{2} - \)\(64\!\cdots\!11\)\( T^{3} + \)\(11\!\cdots\!54\)\( T^{4} + \)\(17\!\cdots\!07\)\( T^{5} - \)\(21\!\cdots\!60\)\( T^{6} - \)\(23\!\cdots\!59\)\( T^{7} + \)\(18\!\cdots\!34\)\( T^{8} + \)\(94\!\cdots\!79\)\( T^{9} - 83365764209575333338 T^{10} + 149965848401109 T^{11} + 1812599797788 T^{12} - 75710937 T^{13} - 14658 T^{14} + T^{15} \)
$31$ \( \)\(42\!\cdots\!72\)\( + \)\(23\!\cdots\!12\)\( T - \)\(59\!\cdots\!90\)\( T^{2} - \)\(15\!\cdots\!97\)\( T^{3} + \)\(23\!\cdots\!62\)\( T^{4} - \)\(12\!\cdots\!49\)\( T^{5} - \)\(21\!\cdots\!16\)\( T^{6} + \)\(23\!\cdots\!24\)\( T^{7} + \)\(84\!\cdots\!38\)\( T^{8} - \)\(10\!\cdots\!19\)\( T^{9} - \)\(16\!\cdots\!66\)\( T^{10} + 23779906369621608 T^{11} + 1708645160600 T^{12} - 247440792 T^{13} - 6840 T^{14} + T^{15} \)
$37$ \( -\)\(71\!\cdots\!96\)\( - \)\(19\!\cdots\!00\)\( T + \)\(20\!\cdots\!92\)\( T^{2} + \)\(50\!\cdots\!84\)\( T^{3} - \)\(55\!\cdots\!76\)\( T^{4} - \)\(76\!\cdots\!36\)\( T^{5} + \)\(54\!\cdots\!24\)\( T^{6} + \)\(67\!\cdots\!72\)\( T^{7} - \)\(23\!\cdots\!22\)\( T^{8} - \)\(30\!\cdots\!87\)\( T^{9} + \)\(41\!\cdots\!44\)\( T^{10} + 59454972081163797 T^{11} - 2453924633698 T^{12} - 426700914 T^{13} + 4278 T^{14} + T^{15} \)
$41$ \( \)\(92\!\cdots\!87\)\( - \)\(11\!\cdots\!95\)\( T - \)\(76\!\cdots\!78\)\( T^{2} + \)\(55\!\cdots\!83\)\( T^{3} + \)\(13\!\cdots\!11\)\( T^{4} - \)\(12\!\cdots\!09\)\( T^{5} - \)\(49\!\cdots\!14\)\( T^{6} + \)\(50\!\cdots\!23\)\( T^{7} + \)\(62\!\cdots\!86\)\( T^{8} - \)\(70\!\cdots\!92\)\( T^{9} - \)\(31\!\cdots\!40\)\( T^{10} + 412802742063536010 T^{11} + 6812382897195 T^{12} - 1064200959 T^{13} - 5112 T^{14} + T^{15} \)
$43$ \( \)\(60\!\cdots\!87\)\( + \)\(22\!\cdots\!13\)\( T - \)\(25\!\cdots\!03\)\( T^{2} + \)\(28\!\cdots\!63\)\( T^{3} - \)\(12\!\cdots\!97\)\( T^{4} + \)\(27\!\cdots\!87\)\( T^{5} - \)\(69\!\cdots\!91\)\( T^{6} - \)\(93\!\cdots\!43\)\( T^{7} + \)\(22\!\cdots\!36\)\( T^{8} - \)\(24\!\cdots\!10\)\( T^{9} + \)\(11\!\cdots\!10\)\( T^{10} + 235453134357446178 T^{11} - 62607621008431 T^{12} + 3558934143 T^{13} - 94191 T^{14} + T^{15} \)
$47$ \( \)\(33\!\cdots\!08\)\( - \)\(29\!\cdots\!20\)\( T - \)\(73\!\cdots\!10\)\( T^{2} + \)\(78\!\cdots\!03\)\( T^{3} + \)\(73\!\cdots\!30\)\( T^{4} - \)\(51\!\cdots\!12\)\( T^{5} - \)\(22\!\cdots\!86\)\( T^{6} + \)\(13\!\cdots\!81\)\( T^{7} + \)\(14\!\cdots\!02\)\( T^{8} - \)\(13\!\cdots\!56\)\( T^{9} - 60084286586645313246 T^{10} + 639982979498803002 T^{11} - 732933450246 T^{12} - 1368284232 T^{13} - 702 T^{14} + T^{15} \)
$53$ \( \)\(31\!\cdots\!72\)\( - \)\(90\!\cdots\!76\)\( T + \)\(75\!\cdots\!40\)\( T^{2} - \)\(14\!\cdots\!15\)\( T^{3} - \)\(70\!\cdots\!48\)\( T^{4} + \)\(28\!\cdots\!18\)\( T^{5} - \)\(25\!\cdots\!60\)\( T^{6} - \)\(92\!\cdots\!55\)\( T^{7} + \)\(65\!\cdots\!24\)\( T^{8} + \)\(79\!\cdots\!18\)\( T^{9} - \)\(73\!\cdots\!68\)\( T^{10} + 13019822665049070 T^{11} + 117418231167264 T^{12} - 1828343088 T^{13} - 47544 T^{14} + T^{15} \)
$59$ \( \)\(11\!\cdots\!77\)\( - \)\(19\!\cdots\!65\)\( T - \)\(58\!\cdots\!08\)\( T^{2} + \)\(16\!\cdots\!77\)\( T^{3} + \)\(44\!\cdots\!24\)\( T^{4} - \)\(18\!\cdots\!16\)\( T^{5} + \)\(56\!\cdots\!85\)\( T^{6} + \)\(27\!\cdots\!55\)\( T^{7} - \)\(15\!\cdots\!62\)\( T^{8} - \)\(16\!\cdots\!69\)\( T^{9} + \)\(70\!\cdots\!67\)\( T^{10} + 4693292038628757783 T^{11} - 56724593345247 T^{12} - 4085385390 T^{13} + 8832 T^{14} + T^{15} \)
$61$ \( \)\(15\!\cdots\!32\)\( - \)\(23\!\cdots\!88\)\( T + \)\(66\!\cdots\!90\)\( T^{2} + \)\(11\!\cdots\!77\)\( T^{3} - \)\(30\!\cdots\!64\)\( T^{4} - \)\(43\!\cdots\!72\)\( T^{5} + \)\(39\!\cdots\!42\)\( T^{6} - \)\(15\!\cdots\!80\)\( T^{7} - \)\(12\!\cdots\!84\)\( T^{8} + \)\(88\!\cdots\!27\)\( T^{9} - \)\(14\!\cdots\!20\)\( T^{10} - 12084544117870210638 T^{11} + 261389296864184 T^{12} + 2112402261 T^{13} - 119196 T^{14} + T^{15} \)
$67$ \( \)\(13\!\cdots\!88\)\( - \)\(56\!\cdots\!20\)\( T + \)\(21\!\cdots\!68\)\( T^{2} + \)\(16\!\cdots\!80\)\( T^{3} + \)\(18\!\cdots\!96\)\( T^{4} - \)\(10\!\cdots\!28\)\( T^{5} - \)\(16\!\cdots\!03\)\( T^{6} + \)\(25\!\cdots\!75\)\( T^{7} + \)\(55\!\cdots\!51\)\( T^{8} - \)\(58\!\cdots\!68\)\( T^{9} - \)\(79\!\cdots\!02\)\( T^{10} + 10109894630730413319 T^{11} + 421992839301791 T^{12} - 6199228029 T^{13} - 64248 T^{14} + T^{15} \)
$71$ \( -\)\(19\!\cdots\!12\)\( - \)\(69\!\cdots\!04\)\( T + \)\(40\!\cdots\!40\)\( T^{2} + \)\(51\!\cdots\!12\)\( T^{3} - \)\(19\!\cdots\!80\)\( T^{4} - \)\(14\!\cdots\!32\)\( T^{5} + \)\(41\!\cdots\!16\)\( T^{6} + \)\(21\!\cdots\!20\)\( T^{7} - \)\(42\!\cdots\!76\)\( T^{8} - \)\(16\!\cdots\!11\)\( T^{9} + \)\(23\!\cdots\!40\)\( T^{10} + 64107892093091700363 T^{11} - 580699062426138 T^{12} - 12798229332 T^{13} + 53364 T^{14} + T^{15} \)
$73$ \( \)\(33\!\cdots\!24\)\( - \)\(16\!\cdots\!92\)\( T - \)\(92\!\cdots\!28\)\( T^{2} + \)\(47\!\cdots\!00\)\( T^{3} + \)\(14\!\cdots\!72\)\( T^{4} - \)\(27\!\cdots\!96\)\( T^{5} - \)\(40\!\cdots\!91\)\( T^{6} + \)\(63\!\cdots\!67\)\( T^{7} + \)\(35\!\cdots\!79\)\( T^{8} - \)\(68\!\cdots\!80\)\( T^{9} - \)\(90\!\cdots\!30\)\( T^{10} + 37010331202980053727 T^{11} - 10278533867305 T^{12} - 9769958337 T^{13} + 4908 T^{14} + T^{15} \)
$79$ \( \)\(34\!\cdots\!44\)\( - \)\(57\!\cdots\!64\)\( T - \)\(52\!\cdots\!42\)\( T^{2} + \)\(96\!\cdots\!55\)\( T^{3} + \)\(10\!\cdots\!20\)\( T^{4} - \)\(44\!\cdots\!51\)\( T^{5} + \)\(67\!\cdots\!46\)\( T^{6} + \)\(58\!\cdots\!83\)\( T^{7} - \)\(13\!\cdots\!82\)\( T^{8} - \)\(29\!\cdots\!91\)\( T^{9} + \)\(73\!\cdots\!28\)\( T^{10} + 82495255868171481963 T^{11} - 1595816724075322 T^{12} - 14149907055 T^{13} + 115500 T^{14} + T^{15} \)
$83$ \( -\)\(54\!\cdots\!23\)\( + \)\(18\!\cdots\!07\)\( T + \)\(21\!\cdots\!00\)\( T^{2} - \)\(85\!\cdots\!25\)\( T^{3} - \)\(24\!\cdots\!34\)\( T^{4} + \)\(13\!\cdots\!10\)\( T^{5} + \)\(57\!\cdots\!47\)\( T^{6} - \)\(88\!\cdots\!59\)\( T^{7} + \)\(58\!\cdots\!60\)\( T^{8} + \)\(24\!\cdots\!15\)\( T^{9} - \)\(33\!\cdots\!09\)\( T^{10} - \)\(15\!\cdots\!45\)\( T^{11} + 4779745651687341 T^{12} - 10949169384 T^{13} - 201630 T^{14} + T^{15} \)
$89$ \( -\)\(81\!\cdots\!39\)\( - \)\(23\!\cdots\!51\)\( T + \)\(93\!\cdots\!20\)\( T^{2} - \)\(74\!\cdots\!16\)\( T^{3} - \)\(17\!\cdots\!44\)\( T^{4} + \)\(52\!\cdots\!56\)\( T^{5} + \)\(13\!\cdots\!48\)\( T^{6} + \)\(56\!\cdots\!74\)\( T^{7} - \)\(41\!\cdots\!66\)\( T^{8} - \)\(29\!\cdots\!52\)\( T^{9} + \)\(59\!\cdots\!24\)\( T^{10} + \)\(49\!\cdots\!52\)\( T^{11} - 3963231129869220 T^{12} - 36837954108 T^{13} + 101505 T^{14} + T^{15} \)
$97$ \( \)\(55\!\cdots\!19\)\( + \)\(28\!\cdots\!81\)\( T - \)\(94\!\cdots\!80\)\( T^{2} - \)\(32\!\cdots\!17\)\( T^{3} + \)\(62\!\cdots\!74\)\( T^{4} + \)\(12\!\cdots\!62\)\( T^{5} - \)\(21\!\cdots\!39\)\( T^{6} - \)\(20\!\cdots\!21\)\( T^{7} + \)\(37\!\cdots\!96\)\( T^{8} + \)\(13\!\cdots\!59\)\( T^{9} - \)\(35\!\cdots\!39\)\( T^{10} - 41683756452765739275 T^{11} + 16447354218645623 T^{12} - 33088849674 T^{13} - 297114 T^{14} + T^{15} \)
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