Properties

Label 13.5.f.a
Level 13
Weight 5
Character orbit 13.f
Analytic conductor 1.344
Analytic rank 0
Dimension 16
CM No
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 13 \)
Weight: \( k \) = \( 5 \)
Character orbit: \([\chi]\) = 13.f (of order \(12\) and degree \(4\))

Newform invariants

Self dual: No
Analytic conductor: \(1.34380952009\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(4\) over \(\Q(\zeta_{12})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 2^{4}\cdot 3^{2}\cdot 13^{2} \)
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \( -\beta_{12} q^{2} \) \( + ( -1 - \beta_{8} + \beta_{9} + \beta_{12} - \beta_{14} ) q^{3} \) \( + ( 8 \beta_{3} + \beta_{4} - \beta_{5} - \beta_{11} + \beta_{12} ) q^{4} \) \( + ( 4 - 8 \beta_{1} - 5 \beta_{2} - 2 \beta_{3} + \beta_{5} + \beta_{8} - \beta_{9} - \beta_{10} - 2 \beta_{12} - \beta_{13} - \beta_{14} + \beta_{15} ) q^{5} \) \( + ( -9 + 14 \beta_{1} + 9 \beta_{2} - 13 \beta_{3} - \beta_{4} + \beta_{6} - \beta_{9} + \beta_{10} + 2 \beta_{11} + \beta_{13} + \beta_{14} - 2 \beta_{15} ) q^{6} \) \( + ( 8 - 7 \beta_{1} + 7 \beta_{2} - 6 \beta_{3} - 2 \beta_{6} - \beta_{7} + \beta_{8} + \beta_{11} + 2 \beta_{13} + 2 \beta_{14} + \beta_{15} ) q^{7} \) \( + ( 22 \beta_{1} - 14 \beta_{2} - 4 \beta_{3} - 8 \beta_{4} + \beta_{5} + 4 \beta_{7} + 4 \beta_{8} + 4 \beta_{10} + 5 \beta_{12} - \beta_{13} + 12 \beta_{14} - \beta_{15} ) q^{8} \) \( + ( -3 - 21 \beta_{1} - 18 \beta_{2} + 11 \beta_{3} + 3 \beta_{4} + 12 \beta_{5} - 6 \beta_{7} - 4 \beta_{9} - 3 \beta_{10} - \beta_{11} - 6 \beta_{12} - 2 \beta_{13} + 3 \beta_{14} ) q^{9} \) \(+O(q^{10})\) \( q\) \( -\beta_{12} q^{2} \) \( + ( -1 - \beta_{8} + \beta_{9} + \beta_{12} - \beta_{14} ) q^{3} \) \( + ( 8 \beta_{3} + \beta_{4} - \beta_{5} - \beta_{11} + \beta_{12} ) q^{4} \) \( + ( 4 - 8 \beta_{1} - 5 \beta_{2} - 2 \beta_{3} + \beta_{5} + \beta_{8} - \beta_{9} - \beta_{10} - 2 \beta_{12} - \beta_{13} - \beta_{14} + \beta_{15} ) q^{5} \) \( + ( -9 + 14 \beta_{1} + 9 \beta_{2} - 13 \beta_{3} - \beta_{4} + \beta_{6} - \beta_{9} + \beta_{10} + 2 \beta_{11} + \beta_{13} + \beta_{14} - 2 \beta_{15} ) q^{6} \) \( + ( 8 - 7 \beta_{1} + 7 \beta_{2} - 6 \beta_{3} - 2 \beta_{6} - \beta_{7} + \beta_{8} + \beta_{11} + 2 \beta_{13} + 2 \beta_{14} + \beta_{15} ) q^{7} \) \( + ( 22 \beta_{1} - 14 \beta_{2} - 4 \beta_{3} - 8 \beta_{4} + \beta_{5} + 4 \beta_{7} + 4 \beta_{8} + 4 \beta_{10} + 5 \beta_{12} - \beta_{13} + 12 \beta_{14} - \beta_{15} ) q^{8} \) \( + ( -3 - 21 \beta_{1} - 18 \beta_{2} + 11 \beta_{3} + 3 \beta_{4} + 12 \beta_{5} - 6 \beta_{7} - 4 \beta_{9} - 3 \beta_{10} - \beta_{11} - 6 \beta_{12} - 2 \beta_{13} + 3 \beta_{14} ) q^{9} \) \( + ( -23 - 20 \beta_{1} + 21 \beta_{2} + 9 \beta_{4} - 20 \beta_{5} + 2 \beta_{6} + 5 \beta_{7} - 3 \beta_{8} - 3 \beta_{9} + 14 \beta_{12} + \beta_{13} - 18 \beta_{14} + 2 \beta_{15} ) q^{10} \) \( + ( -26 + 29 \beta_{1} + 33 \beta_{2} - 8 \beta_{3} - 4 \beta_{4} - 6 \beta_{5} + \beta_{6} - 4 \beta_{7} + 3 \beta_{8} + 4 \beta_{9} - 7 \beta_{10} - 2 \beta_{12} + \beta_{13} - 4 \beta_{14} - \beta_{15} ) q^{11} \) \( + ( -2 - 34 \beta_{2} + 30 \beta_{3} + 19 \beta_{4} + 19 \beta_{5} - \beta_{6} + 2 \beta_{7} - 4 \beta_{8} + 8 \beta_{9} + 2 \beta_{10} - 3 \beta_{11} - 9 \beta_{12} - 3 \beta_{13} - 17 \beta_{14} + 2 \beta_{15} ) q^{12} \) \( + ( 37 - 43 \beta_{1} + 17 \beta_{2} + 31 \beta_{3} - 28 \beta_{4} + 3 \beta_{5} + 3 \beta_{7} - 11 \beta_{8} + 3 \beta_{9} + \beta_{10} - 3 \beta_{11} + 9 \beta_{12} + 12 \beta_{14} - 2 \beta_{15} ) q^{13} \) \( + ( 60 - 28 \beta_{2} - 29 \beta_{3} + 18 \beta_{4} - 18 \beta_{5} - 4 \beta_{6} - 7 \beta_{7} - \beta_{8} + 7 \beta_{10} - \beta_{11} - 7 \beta_{12} + \beta_{13} + 7 \beta_{14} ) q^{14} \) \( + ( -2 + 83 \beta_{1} - 76 \beta_{2} + 68 \beta_{3} - 21 \beta_{4} - 7 \beta_{5} + \beta_{6} + 7 \beta_{7} - \beta_{8} + 7 \beta_{9} + 6 \beta_{10} + 2 \beta_{11} - 25 \beta_{12} + \beta_{13} + 35 \beta_{14} + \beta_{15} ) q^{15} \) \( + ( 16 - 28 \beta_{1} + 74 \beta_{2} - 148 \beta_{3} + 3 \beta_{4} + 25 \beta_{5} - 4 \beta_{7} - 4 \beta_{8} + 4 \beta_{9} - 8 \beta_{10} + 2 \beta_{11} - 26 \beta_{12} + \beta_{13} - 33 \beta_{14} ) q^{16} \) \( + ( 69 - 33 \beta_{1} - 3 \beta_{2} + 81 \beta_{3} + 25 \beta_{4} - 2 \beta_{5} + 6 \beta_{6} + 6 \beta_{8} - 3 \beta_{9} - 3 \beta_{10} + 9 \beta_{11} + 16 \beta_{12} - 17 \beta_{14} - 3 \beta_{15} ) q^{17} \) \( + ( 134 + 32 \beta_{1} + 152 \beta_{2} - 109 \beta_{3} - 16 \beta_{4} + 3 \beta_{5} - 2 \beta_{6} + 3 \beta_{7} + 11 \beta_{8} - 14 \beta_{9} - 11 \beta_{10} + 2 \beta_{11} - 14 \beta_{12} + 7 \beta_{13} - 27 \beta_{14} - 5 \beta_{15} ) q^{18} \) \( + ( -40 - 83 \beta_{1} + 41 \beta_{2} + 94 \beta_{3} + 2 \beta_{4} - 13 \beta_{5} - 8 \beta_{6} + \beta_{7} + \beta_{8} - 10 \beta_{9} + 10 \beta_{10} - 12 \beta_{11} - 8 \beta_{13} + 5 \beta_{14} + 12 \beta_{15} ) q^{19} \) \( + ( -294 + 80 \beta_{1} - 300 \beta_{2} + 88 \beta_{3} + 2 \beta_{4} - 36 \beta_{5} + 13 \beta_{6} - 6 \beta_{7} + 6 \beta_{8} - 2 \beta_{9} - 2 \beta_{10} - 4 \beta_{11} + 72 \beta_{12} - 13 \beta_{13} + 61 \beta_{14} - 4 \beta_{15} ) q^{20} \) \( + ( -96 + 33 \beta_{1} + 56 \beta_{2} - 89 \beta_{3} + 12 \beta_{4} + 37 \beta_{5} - 2 \beta_{6} - 7 \beta_{7} - 7 \beta_{9} - 2 \beta_{11} + 23 \beta_{12} + 5 \beta_{13} - 12 \beta_{14} + 7 \beta_{15} ) q^{21} \) \( + ( 7 - 168 \beta_{1} - 93 \beta_{2} + 43 \beta_{3} - \beta_{4} + 37 \beta_{5} + 14 \beta_{7} + 7 \beta_{9} + 7 \beta_{10} + 6 \beta_{11} + 8 \beta_{12} + 12 \beta_{13} + 52 \beta_{14} - \beta_{15} ) q^{22} \) \( + ( -45 - 56 \beta_{1} - 52 \beta_{2} - 24 \beta_{4} - 34 \beta_{5} - 14 \beta_{6} + 2 \beta_{7} + 11 \beta_{8} + 11 \beta_{9} + 47 \beta_{12} - 4 \beta_{13} - 21 \beta_{14} - 14 \beta_{15} ) q^{23} \) \( + ( -328 + 378 \beta_{1} + 376 \beta_{2} - 60 \beta_{3} - 55 \beta_{4} - 32 \beta_{5} - 3 \beta_{6} + 2 \beta_{7} - 14 \beta_{8} - 2 \beta_{9} + 16 \beta_{10} + 31 \beta_{12} - 3 \beta_{13} + 2 \beta_{14} + 3 \beta_{15} ) q^{24} \) \( + ( -96 + 182 \beta_{1} + 82 \beta_{2} - 74 \beta_{3} + 75 \beta_{4} + 75 \beta_{5} + 6 \beta_{6} - 13 \beta_{7} + 8 \beta_{8} - 16 \beta_{9} - 13 \beta_{10} + 21 \beta_{11} - 87 \beta_{12} + 21 \beta_{13} - 71 \beta_{14} - 12 \beta_{15} ) q^{25} \) \( + ( 362 - 598 \beta_{1} + 114 \beta_{2} - 113 \beta_{3} - 68 \beta_{4} + 7 \beta_{5} - 2 \beta_{6} - 5 \beta_{7} + 27 \beta_{8} - 8 \beta_{9} + 9 \beta_{10} + 16 \beta_{11} - 67 \beta_{12} - 3 \beta_{13} + 65 \beta_{14} + 17 \beta_{15} ) q^{26} \) \( + ( 213 - 305 \beta_{2} - 310 \beta_{3} + 73 \beta_{4} - 73 \beta_{5} + 27 \beta_{6} + 22 \beta_{7} - 5 \beta_{8} - 22 \beta_{10} + 5 \beta_{11} + 6 \beta_{12} - 5 \beta_{13} - 6 \beta_{14} ) q^{27} \) \( + ( 116 + 478 \beta_{1} - 498 \beta_{2} + 636 \beta_{3} + 16 \beta_{4} + 20 \beta_{5} - 4 \beta_{6} - 20 \beta_{7} - 2 \beta_{8} - 20 \beta_{9} - 22 \beta_{10} - 14 \beta_{11} - 54 \beta_{12} - 10 \beta_{13} - 12 \beta_{14} - 10 \beta_{15} ) q^{28} \) \( + ( 146 - 99 \beta_{1} + 247 \beta_{2} - 494 \beta_{3} - 31 \beta_{4} - 23 \beta_{5} - 2 \beta_{6} + 11 \beta_{7} + 25 \beta_{8} - 25 \beta_{9} + 22 \beta_{10} - 16 \beta_{11} - 11 \beta_{12} - 8 \beta_{13} + 59 \beta_{14} + 2 \beta_{15} ) q^{29} \) \( + ( 764 - 392 \beta_{1} + 14 \beta_{2} + 762 \beta_{3} - 38 \beta_{4} + 45 \beta_{5} - 40 \beta_{6} - 28 \beta_{8} + 14 \beta_{9} + 8 \beta_{10} - 30 \beta_{11} - 2 \beta_{12} - 35 \beta_{14} + 20 \beta_{15} ) q^{30} \) \( + ( 181 + 102 \beta_{1} + 355 \beta_{2} - 308 \beta_{3} - 34 \beta_{4} + 4 \beta_{5} + 15 \beta_{6} - 17 \beta_{7} - 55 \beta_{8} + 72 \beta_{9} + 55 \beta_{10} - 15 \beta_{11} + 51 \beta_{12} - 19 \beta_{13} + 21 \beta_{14} + 4 \beta_{15} ) q^{31} \) \( + ( -306 - 102 \beta_{1} + 306 \beta_{2} + 42 \beta_{3} + 120 \beta_{4} - 11 \beta_{5} + 21 \beta_{6} + 60 \beta_{9} - 60 \beta_{10} + 15 \beta_{11} + 21 \beta_{13} - 90 \beta_{14} - 15 \beta_{15} ) q^{32} \) \( + ( -506 + 334 \beta_{1} - 457 \beta_{2} + 269 \beta_{3} - 16 \beta_{4} - 13 \beta_{5} - 25 \beta_{6} + 49 \beta_{7} - 49 \beta_{8} + 16 \beta_{9} + 16 \beta_{10} - 4 \beta_{11} + 26 \beta_{12} + 25 \beta_{13} - 63 \beta_{14} - 4 \beta_{15} ) q^{33} \) \( + ( -456 + 468 \beta_{1} - 78 \beta_{2} - 438 \beta_{3} - 3 \beta_{4} - 16 \beta_{5} + 17 \beta_{6} - 18 \beta_{7} - 48 \beta_{8} + 30 \beta_{9} - 48 \beta_{10} + 17 \beta_{11} - 4 \beta_{12} + \beta_{13} - 45 \beta_{14} - 16 \beta_{15} ) q^{34} \) \( + ( 1 - 651 \beta_{1} - 465 \beta_{2} + 222 \beta_{3} - 5 \beta_{4} - 108 \beta_{5} + 2 \beta_{7} + 21 \beta_{9} + \beta_{10} - 8 \beta_{11} + 6 \beta_{12} - 16 \beta_{13} - 101 \beta_{14} + 6 \beta_{15} ) q^{35} \) \( + ( -672 - 658 \beta_{1} + 262 \beta_{2} + 81 \beta_{4} + 113 \beta_{5} + 28 \beta_{6} - 92 \beta_{7} - 14 \beta_{8} - 14 \beta_{9} - 180 \beta_{12} - 3 \beta_{13} + 7 \beta_{14} + 28 \beta_{15} ) q^{36} \) \( + ( -472 + 860 \beta_{1} + 801 \beta_{2} - 245 \beta_{3} - \beta_{4} + 31 \beta_{5} - 16 \beta_{6} + 59 \beta_{7} + 25 \beta_{8} - 59 \beta_{9} + 34 \beta_{10} + 2 \beta_{11} - 11 \beta_{12} - 14 \beta_{13} + 59 \beta_{14} + 14 \beta_{15} ) q^{37} \) \( + ( -20 + 192 \beta_{1} - 416 \beta_{2} + 447 \beta_{3} - 185 \beta_{4} - 185 \beta_{5} - 5 \beta_{6} + 45 \beta_{7} + 31 \beta_{8} - 62 \beta_{9} + 45 \beta_{10} - 42 \beta_{11} + 188 \beta_{12} - 42 \beta_{13} + 250 \beta_{14} + 10 \beta_{15} ) q^{38} \) \( + ( 805 - 790 \beta_{1} + 94 \beta_{2} + 284 \beta_{3} + 209 \beta_{4} + 91 \beta_{5} + 17 \beta_{6} - 28 \beta_{7} + 55 \beta_{8} - 9 \beta_{9} - 74 \beta_{10} - 7 \beta_{11} - 32 \beta_{12} + 19 \beta_{13} - 100 \beta_{14} - 52 \beta_{15} ) q^{39} \) \( + ( 1146 - 512 \beta_{2} - 454 \beta_{3} - 183 \beta_{4} + 183 \beta_{5} - 53 \beta_{6} + 10 \beta_{7} + 58 \beta_{8} - 10 \beta_{10} + 24 \beta_{12} - 24 \beta_{14} ) q^{40} \) \( + ( -7 + 484 \beta_{1} - 514 \beta_{2} + 499 \beta_{3} + 30 \beta_{4} + 30 \beta_{5} - 8 \beta_{6} - 30 \beta_{7} + 38 \beta_{8} - 30 \beta_{9} + 8 \beta_{10} + 30 \beta_{11} + 110 \beta_{12} + 38 \beta_{13} - 30 \beta_{14} + 38 \beta_{15} ) q^{41} \) \( + ( -161 + 198 \beta_{1} + 763 \beta_{2} - 1526 \beta_{3} + 33 \beta_{4} - 45 \beta_{5} + 12 \beta_{6} + 21 \beta_{7} - 5 \beta_{8} + 5 \beta_{9} + 42 \beta_{10} + 46 \beta_{11} + 125 \beta_{12} + 23 \beta_{13} + 61 \beta_{14} - 12 \beta_{15} ) q^{42} \) \( + ( -44 + 50 \beta_{1} - 21 \beta_{2} + 924 \beta_{3} + 34 \beta_{4} - 181 \beta_{5} + 70 \beta_{6} + 42 \beta_{8} - 21 \beta_{9} + 14 \beta_{10} + 35 \beta_{11} + 6 \beta_{12} + 189 \beta_{14} - 35 \beta_{15} ) q^{43} \) \( + ( 776 - 130 \beta_{1} + 596 \beta_{2} - 712 \beta_{3} + 62 \beta_{4} - 72 \beta_{5} - 36 \beta_{6} + 36 \beta_{7} + 14 \beta_{8} - 50 \beta_{9} - 14 \beta_{10} + 36 \beta_{11} + 58 \beta_{12} + 28 \beta_{13} + 48 \beta_{14} + 8 \beta_{15} ) q^{44} \) \( + ( -606 + 365 \beta_{1} + 583 \beta_{2} - 319 \beta_{3} - 375 \beta_{4} + 134 \beta_{5} - 5 \beta_{6} - 23 \beta_{7} - 23 \beta_{8} - 69 \beta_{9} + 69 \beta_{10} + 38 \beta_{11} - 5 \beta_{13} + 199 \beta_{14} - 38 \beta_{15} ) q^{45} \) \( + ( -819 - 310 \beta_{1} - 909 \beta_{2} - 161 \beta_{3} + 59 \beta_{4} + 212 \beta_{5} + \beta_{6} - 90 \beta_{7} + 90 \beta_{8} - 59 \beta_{9} - 59 \beta_{10} + 26 \beta_{11} - 424 \beta_{12} - \beta_{13} - 199 \beta_{14} + 26 \beta_{15} ) q^{46} \) \( + ( 85 + 56 \beta_{1} - 149 \beta_{2} + 94 \beta_{3} + 130 \beta_{4} - 54 \beta_{5} - 49 \beta_{6} - 9 \beta_{7} + \beta_{8} - 10 \beta_{9} + \beta_{10} - 49 \beta_{11} - 73 \beta_{12} - 43 \beta_{13} - 129 \beta_{14} + 6 \beta_{15} ) q^{47} \) \( + ( 48 - 342 \beta_{1} - 904 \beta_{2} + 454 \beta_{3} - 36 \beta_{4} - 75 \beta_{5} + 96 \beta_{7} - 4 \beta_{9} + 48 \beta_{10} - 17 \beta_{11} + 84 \beta_{12} - 34 \beta_{13} + 57 \beta_{14} - 3 \beta_{15} ) q^{48} \) \( + ( -406 - 448 \beta_{1} - 62 \beta_{2} - 178 \beta_{4} + 129 \beta_{5} + 16 \beta_{6} + 133 \beta_{7} + 42 \beta_{8} + 42 \beta_{9} + 7 \beta_{12} + 19 \beta_{13} + 304 \beta_{14} + 16 \beta_{15} ) q^{49} \) \( + ( -1553 + 1274 \beta_{1} + 1377 \beta_{2} + 34 \beta_{3} + 346 \beta_{4} + 131 \beta_{5} + 79 \beta_{6} - 103 \beta_{7} - 39 \beta_{8} + 103 \beta_{9} - 64 \beta_{10} - 15 \beta_{11} - 78 \beta_{12} + 64 \beta_{13} - 103 \beta_{14} - 64 \beta_{15} ) q^{50} \) \( + ( -103 - 168 \beta_{1} - 463 \beta_{2} + 390 \beta_{3} - 156 \beta_{4} - 156 \beta_{5} - 36 \beta_{6} - 114 \beta_{7} - 73 \beta_{8} + 146 \beta_{9} - 114 \beta_{10} - 24 \beta_{11} + 31 \beta_{12} - 24 \beta_{13} - 115 \beta_{14} + 72 \beta_{15} ) q^{51} \) \( + ( 576 - 906 \beta_{1} + 714 \beta_{2} + 342 \beta_{3} + 252 \beta_{4} - 452 \beta_{5} - 50 \beta_{6} + 54 \beta_{7} - 120 \beta_{8} + 18 \beta_{9} + 146 \beta_{10} - 91 \beta_{11} + 383 \beta_{12} - 23 \beta_{13} - 233 \beta_{14} + 50 \beta_{15} ) q^{52} \) \( + ( 388 - 61 \beta_{2} - 215 \beta_{3} - 144 \beta_{4} + 144 \beta_{5} - 13 \beta_{6} - 35 \beta_{7} - 154 \beta_{8} + 35 \beta_{10} - 36 \beta_{11} + 36 \beta_{12} + 36 \beta_{13} - 36 \beta_{14} ) q^{53} \) \( + ( 55 + 1166 \beta_{1} - 1061 \beta_{2} + 1112 \beta_{3} + 197 \beta_{4} - 105 \beta_{5} + 55 \beta_{6} + 105 \beta_{7} - 101 \beta_{8} + 105 \beta_{9} + 4 \beta_{10} - \beta_{11} + 365 \beta_{12} - 56 \beta_{13} - 499 \beta_{14} - 56 \beta_{15} ) q^{54} \) \( + ( 854 - 1035 \beta_{1} - 46 \beta_{2} + 92 \beta_{3} - 8 \beta_{4} - 147 \beta_{5} - 12 \beta_{6} - 63 \beta_{7} - 55 \beta_{8} + 55 \beta_{9} - 126 \beta_{10} - 48 \beta_{11} + 68 \beta_{12} - 24 \beta_{13} + 29 \beta_{14} + 12 \beta_{15} ) q^{55} \) \( + ( 610 - 352 \beta_{1} + 34 \beta_{2} - 174 \beta_{3} - 328 \beta_{4} + 292 \beta_{5} + 68 \beta_{6} - 68 \beta_{8} + 34 \beta_{9} - 26 \beta_{10} + 34 \beta_{11} - 286 \beta_{12} - 32 \beta_{14} - 34 \beta_{15} ) q^{56} \) \( + ( 936 - 69 \beta_{1} + 770 \beta_{2} - 695 \beta_{3} + 330 \beta_{4} + 293 \beta_{5} + 8 \beta_{6} - 47 \beta_{7} + 144 \beta_{8} - 97 \beta_{9} - 144 \beta_{10} - 8 \beta_{11} - 437 \beta_{12} - 19 \beta_{13} + 186 \beta_{14} + 11 \beta_{15} ) q^{57} \) \( + ( 642 - 766 \beta_{1} - 600 \beta_{2} + 812 \beta_{3} + 340 \beta_{4} - 83 \beta_{5} - 70 \beta_{6} + 42 \beta_{7} + 42 \beta_{8} - 4 \beta_{9} + 4 \beta_{10} - 92 \beta_{11} - 70 \beta_{13} - 126 \beta_{14} + 92 \beta_{15} ) q^{58} \) \( + ( -386 + 555 \beta_{1} - 331 \beta_{2} + 366 \beta_{3} - 134 \beta_{4} + 126 \beta_{5} + 26 \beta_{6} + 55 \beta_{7} - 55 \beta_{8} + 134 \beta_{9} + 134 \beta_{10} + 29 \beta_{11} - 252 \beta_{12} - 26 \beta_{13} - 94 \beta_{14} + 29 \beta_{15} ) q^{59} \) \( + ( -388 + 968 \beta_{1} - 228 \beta_{2} - 518 \beta_{3} - 441 \beta_{4} - 415 \beta_{5} + 25 \beta_{6} + 130 \beta_{7} + 222 \beta_{8} - 92 \beta_{9} + 222 \beta_{10} + 25 \beta_{11} - 377 \beta_{12} + 53 \beta_{13} + 663 \beta_{14} + 28 \beta_{15} ) q^{60} \) \( + ( -160 - 225 \beta_{1} + 830 \beta_{2} - 360 \beta_{3} - 56 \beta_{4} - 127 \beta_{5} - 320 \beta_{7} - 110 \beta_{9} - 160 \beta_{10} + 53 \beta_{11} - 104 \beta_{12} + 106 \beta_{13} - 391 \beta_{14} - 55 \beta_{15} ) q^{61} \) \( + ( 876 + 964 \beta_{1} + 632 \beta_{2} + 59 \beta_{4} + 96 \beta_{5} - 110 \beta_{6} + 98 \beta_{7} - 88 \beta_{8} - 88 \beta_{9} - 67 \beta_{12} + 38 \beta_{13} + 106 \beta_{14} - 110 \beta_{15} ) q^{62} \) \( + ( 1074 - 674 \beta_{1} - 652 \beta_{2} - 388 \beta_{3} + 210 \beta_{4} + 10 \beta_{5} - 64 \beta_{6} - 22 \beta_{7} + 56 \beta_{8} + 22 \beta_{9} - 78 \beta_{10} + 32 \beta_{11} - 72 \beta_{12} - 32 \beta_{13} - 22 \beta_{14} + 32 \beta_{15} ) q^{63} \) \( + ( 108 + 40 \beta_{1} + 1286 \beta_{2} - 1306 \beta_{3} - 134 \beta_{4} - 134 \beta_{5} + 74 \beta_{6} + 148 \beta_{7} - 20 \beta_{8} + 40 \beta_{9} + 148 \beta_{10} + 169 \beta_{11} + 399 \beta_{12} + 169 \beta_{13} + 359 \beta_{14} - 148 \beta_{15} ) q^{64} \) \( + ( -621 + 27 \beta_{1} - 1645 \beta_{2} - 55 \beta_{3} - 38 \beta_{4} + 478 \beta_{5} + 35 \beta_{6} + 38 \beta_{7} - 187 \beta_{8} + 19 \beta_{9} - 44 \beta_{10} + 114 \beta_{11} + 49 \beta_{12} - 84 \beta_{13} - 237 \beta_{14} + 67 \beta_{15} ) q^{65} \) \( + ( -1532 + 146 \beta_{2} + 295 \beta_{3} - 272 \beta_{4} + 272 \beta_{5} + 129 \beta_{6} - 149 \beta_{7} + 149 \beta_{8} + 149 \beta_{10} + 46 \beta_{11} - 115 \beta_{12} - 46 \beta_{13} + 115 \beta_{14} ) q^{66} \) \( + ( -494 - 489 \beta_{1} + 545 \beta_{2} - 1164 \beta_{3} - 407 \beta_{4} - 56 \beta_{5} - 34 \beta_{6} + 56 \beta_{7} + 69 \beta_{8} + 56 \beta_{9} + 125 \beta_{10} - 62 \beta_{11} + 325 \beta_{12} - 28 \beta_{13} + 758 \beta_{14} - 28 \beta_{15} ) q^{67} \) \( + ( -980 + 858 \beta_{1} + 144 \beta_{2} - 288 \beta_{3} + 354 \beta_{4} + 369 \beta_{5} - 51 \beta_{6} - 82 \beta_{7} + 42 \beta_{8} - 42 \beta_{9} - 164 \beta_{10} - 8 \beta_{11} - 221 \beta_{12} - 4 \beta_{13} - 409 \beta_{14} + 51 \beta_{15} ) q^{68} \) \( + ( 327 - 102 \beta_{1} - 114 \beta_{2} + 693 \beta_{3} + 453 \beta_{4} - 108 \beta_{5} - 264 \beta_{6} + 228 \beta_{8} - 114 \beta_{9} - 105 \beta_{10} - 123 \beta_{11} + 120 \beta_{12} - 117 \beta_{14} + 132 \beta_{15} ) q^{69} \) \( + ( -2406 + 228 \beta_{1} - 2240 \beta_{2} + 2454 \beta_{3} - 190 \beta_{4} - 792 \beta_{5} + 81 \beta_{6} + 76 \beta_{7} - 14 \beta_{8} - 62 \beta_{9} + 14 \beta_{10} - 81 \beta_{11} + 806 \beta_{12} - 67 \beta_{13} - 176 \beta_{14} - 14 \beta_{15} ) q^{70} \) \( + ( -504 + 437 \beta_{1} + 595 \beta_{2} - 220 \beta_{3} - 506 \beta_{4} - 163 \beta_{5} + 98 \beta_{6} + 91 \beta_{7} + 91 \beta_{8} - 126 \beta_{9} + 126 \beta_{10} + 2 \beta_{11} + 98 \beta_{13} + 407 \beta_{14} - 2 \beta_{15} ) q^{71} \) \( + ( 1004 - 222 \beta_{1} + 976 \beta_{2} - 12 \beta_{3} + 182 \beta_{4} - 455 \beta_{5} + 9 \beta_{6} - 28 \beta_{7} + 28 \beta_{8} - 182 \beta_{9} - 182 \beta_{10} - 141 \beta_{11} + 910 \beta_{12} - 9 \beta_{13} + 350 \beta_{14} - 141 \beta_{15} ) q^{72} \) \( + ( -356 - 1826 \beta_{1} + 2129 \beta_{2} - 334 \beta_{3} + 751 \beta_{4} + 330 \beta_{5} + 139 \beta_{6} - 22 \beta_{7} - 31 \beta_{8} + 9 \beta_{9} - 31 \beta_{10} + 139 \beta_{11} + 317 \beta_{12} + 94 \beta_{13} - 782 \beta_{14} - 45 \beta_{15} ) q^{73} \) \( + ( -42 - 198 \beta_{1} + 576 \beta_{2} - 270 \beta_{3} + 439 \beta_{4} + 714 \beta_{5} - 84 \beta_{7} - 36 \beta_{9} - 42 \beta_{10} - 48 \beta_{11} - 481 \beta_{12} - 96 \beta_{13} + 191 \beta_{14} + 144 \beta_{15} ) q^{74} \) \( + ( 113 + 159 \beta_{1} - 1444 \beta_{2} - 135 \beta_{4} - 373 \beta_{5} + 53 \beta_{6} - 65 \beta_{7} - 46 \beta_{8} - 46 \beta_{9} + 554 \beta_{12} - 99 \beta_{13} - 484 \beta_{14} + 53 \beta_{15} ) q^{75} \) \( + ( 1990 - 3040 \beta_{1} - 2992 \beta_{2} + 944 \beta_{3} - 785 \beta_{4} + 542 \beta_{5} - 155 \beta_{6} - 48 \beta_{7} - 10 \beta_{8} + 48 \beta_{9} - 38 \beta_{10} + 24 \beta_{11} - 285 \beta_{12} - 131 \beta_{13} - 48 \beta_{14} + 131 \beta_{15} ) q^{76} \) \( + ( -1346 + 2958 \beta_{1} - 434 \beta_{2} + 483 \beta_{3} + 208 \beta_{4} + 208 \beta_{5} + 49 \beta_{6} + 84 \beta_{7} + 49 \beta_{8} - 98 \beta_{9} + 84 \beta_{10} - 114 \beta_{11} - 395 \beta_{12} - 114 \beta_{13} - 297 \beta_{14} - 98 \beta_{15} ) q^{77} \) \( + ( -2239 + 3512 \beta_{1} + 2215 \beta_{2} - 2809 \beta_{3} - 668 \beta_{4} - 254 \beta_{5} + 100 \beta_{6} + 8 \beta_{7} + 196 \beta_{8} + 171 \beta_{9} - 67 \beta_{10} + 170 \beta_{11} - 353 \beta_{12} + 202 \beta_{13} + 813 \beta_{14} - 173 \beta_{15} ) q^{78} \) \( + ( -774 - 86 \beta_{2} - 90 \beta_{3} + 744 \beta_{4} - 744 \beta_{5} - 68 \beta_{6} + 164 \beta_{7} - 4 \beta_{8} - 164 \beta_{10} + 34 \beta_{11} - 232 \beta_{12} - 34 \beta_{13} + 232 \beta_{14} ) q^{79} \) \( + ( 1932 - 2994 \beta_{1} + 2818 \beta_{2} - 810 \beta_{3} + 263 \beta_{4} + 176 \beta_{5} - 158 \beta_{6} - 176 \beta_{7} + 100 \beta_{8} - 176 \beta_{9} - 76 \beta_{10} + 29 \beta_{11} - 1420 \beta_{12} + 187 \beta_{13} - 350 \beta_{14} + 187 \beta_{15} ) q^{80} \) \( + ( 381 - 141 \beta_{1} - 2410 \beta_{2} + 4820 \beta_{3} - 550 \beta_{4} - 326 \beta_{5} + 78 \beta_{6} + 248 \beta_{7} - 256 \beta_{8} + 256 \beta_{9} + 496 \beta_{10} + 44 \beta_{11} + 528 \beta_{12} + 22 \beta_{13} + 318 \beta_{14} - 78 \beta_{15} ) q^{81} \) \( + ( -508 + 184 \beta_{1} + 128 \beta_{2} - 4428 \beta_{3} - 271 \beta_{4} - 44 \beta_{5} - 16 \beta_{6} - 256 \beta_{8} + 128 \beta_{9} + 116 \beta_{10} + 64 \beta_{11} + 101 \beta_{12} + 59 \beta_{14} + 8 \beta_{15} ) q^{82} \) \( + ( 573 + 546 \beta_{1} + 1335 \beta_{2} - 912 \beta_{3} - 269 \beta_{4} + 779 \beta_{5} - 60 \beta_{6} - 93 \beta_{7} - 123 \beta_{8} + 216 \beta_{9} + 123 \beta_{10} + 60 \beta_{11} - 656 \beta_{12} + 216 \beta_{13} - 146 \beta_{14} - 156 \beta_{15} ) q^{83} \) \( + ( 1328 - 404 \beta_{1} - 1666 \beta_{2} - 90 \beta_{3} + 888 \beta_{4} + 248 \beta_{5} + 30 \beta_{6} - 338 \beta_{7} - 338 \beta_{8} + 156 \beta_{9} - 156 \beta_{10} + 54 \beta_{11} + 30 \beta_{13} - 860 \beta_{14} - 54 \beta_{15} ) q^{84} \) \( + ( 2188 + 432 \beta_{1} + 2111 \beta_{2} + 456 \beta_{3} - 53 \beta_{4} + 224 \beta_{5} + 73 \beta_{6} - 77 \beta_{7} + 77 \beta_{8} + 53 \beta_{9} + 53 \beta_{10} - 44 \beta_{11} - 448 \beta_{12} - 73 \beta_{13} + 553 \beta_{14} - 44 \beta_{15} ) q^{85} \) \( + ( 3752 + 74 \beta_{1} - 4134 \beta_{2} + 3745 \beta_{3} - 908 \beta_{4} + 420 \beta_{5} - 224 \beta_{6} + 7 \beta_{7} - 315 \beta_{8} + 322 \beta_{9} - 315 \beta_{10} - 224 \beta_{11} + 749 \beta_{12} - 195 \beta_{13} + 593 \beta_{14} + 29 \beta_{15} ) q^{86} \) \( + ( 356 + 3972 \beta_{1} + 2259 \beta_{2} - 1220 \beta_{3} - 188 \beta_{4} - 670 \beta_{5} + 712 \beta_{7} + 181 \beta_{9} + 356 \beta_{10} + 22 \beta_{11} + 544 \beta_{12} + 44 \beta_{13} + 230 \beta_{14} - 54 \beta_{15} ) q^{87} \) \( + ( 1658 + 1528 \beta_{1} + 406 \beta_{2} + 396 \beta_{4} - 334 \beta_{5} + 134 \beta_{6} - 166 \beta_{7} + 130 \beta_{8} + 130 \beta_{9} - 192 \beta_{12} - 160 \beta_{13} - 370 \beta_{14} + 134 \beta_{15} ) q^{88} \) \( + ( 2804 - 2459 \beta_{1} - 2685 \beta_{2} + 116 \beta_{3} - 181 \beta_{4} - 1570 \beta_{5} + 221 \beta_{6} + 226 \beta_{7} + 9 \beta_{8} - 226 \beta_{9} + 217 \beta_{10} - 171 \beta_{11} + 889 \beta_{12} + 50 \beta_{13} + 226 \beta_{14} - 50 \beta_{15} ) q^{89} \) \( + ( 3702 - 8288 \beta_{1} + 730 \beta_{2} - 687 \beta_{3} + 1205 \beta_{4} + 1205 \beta_{5} - 199 \beta_{6} - 485 \beta_{7} + 43 \beta_{8} - 86 \beta_{9} - 485 \beta_{10} - 96 \beta_{11} - 1395 \beta_{12} - 96 \beta_{13} - 1309 \beta_{14} + 398 \beta_{15} ) q^{90} \) \( + ( -442 + 2431 \beta_{1} - 2587 \beta_{2} + 2834 \beta_{3} - 286 \beta_{4} + 676 \beta_{5} - 156 \beta_{6} - 273 \beta_{7} + 377 \beta_{8} - 442 \beta_{9} - 78 \beta_{10} - 195 \beta_{11} - 1118 \beta_{12} + 78 \beta_{13} - 208 \beta_{14} + 91 \beta_{15} ) q^{91} \) \( + ( -3362 + 4450 \beta_{2} + 4298 \beta_{3} + 881 \beta_{4} - 881 \beta_{5} + 73 \beta_{6} + 270 \beta_{7} - 152 \beta_{8} - 270 \beta_{10} - 19 \beta_{11} + 845 \beta_{12} + 19 \beta_{13} - 845 \beta_{14} ) q^{92} \) \( + ( -3276 - 3731 \beta_{1} + 3640 \beta_{2} - 6558 \beta_{3} - 248 \beta_{4} + 91 \beta_{5} + 183 \beta_{6} - 91 \beta_{7} - 267 \beta_{8} - 91 \beta_{9} - 358 \beta_{10} + 24 \beta_{11} - 244 \beta_{12} - 159 \beta_{13} + 587 \beta_{14} - 159 \beta_{15} ) q^{93} \) \( + ( -3472 + 3876 \beta_{1} - 1976 \beta_{2} + 3952 \beta_{3} - 329 \beta_{4} + 164 \beta_{5} + 128 \beta_{6} + 14 \beta_{7} + 376 \beta_{8} - 376 \beta_{9} + 28 \beta_{10} - 148 \beta_{11} - 841 \beta_{12} - 74 \beta_{13} + 226 \beta_{14} - 128 \beta_{15} ) q^{94} \) \( + ( -8673 + 4349 \beta_{1} + 129 \beta_{2} - 2670 \beta_{3} + 685 \beta_{4} - 359 \beta_{5} + 274 \beta_{6} - 258 \beta_{8} + 129 \beta_{9} + 283 \beta_{10} + 67 \beta_{11} + 1226 \beta_{12} - 584 \beta_{14} - 137 \beta_{15} ) q^{95} \) \( + ( -5602 - 1072 \beta_{1} - 6458 \beta_{2} + 5198 \beta_{3} - 593 \beta_{4} + 397 \beta_{5} - 53 \beta_{6} - 28 \beta_{7} - 188 \beta_{8} + 216 \beta_{9} + 188 \beta_{10} + 53 \beta_{11} - 209 \beta_{12} - 83 \beta_{13} - 405 \beta_{14} + 136 \beta_{15} ) q^{96} \) \( + ( 1876 + 2537 \beta_{1} - 1629 \beta_{2} - 2656 \beta_{3} + 354 \beta_{4} - 125 \beta_{5} - 113 \beta_{6} + 247 \beta_{7} + 247 \beta_{8} + 366 \beta_{9} - 366 \beta_{10} + 54 \beta_{11} - 113 \beta_{13} - 113 \beta_{14} - 54 \beta_{15} ) q^{97} \) \( + ( 6143 - 4934 \beta_{1} + 6355 \beta_{2} - 5461 \beta_{3} - 315 \beta_{4} - 851 \beta_{5} - 255 \beta_{6} + 212 \beta_{7} - 212 \beta_{8} + 315 \beta_{9} + 315 \beta_{10} + 311 \beta_{11} + 1702 \beta_{12} + 255 \beta_{13} + 67 \beta_{14} + 311 \beta_{15} ) q^{98} \) \( + ( 608 - 4754 \beta_{1} + 3656 \beta_{2} + 986 \beta_{3} + 142 \beta_{4} + 816 \beta_{5} - 40 \beta_{6} - 378 \beta_{7} - 112 \beta_{8} - 266 \beta_{9} - 112 \beta_{10} - 40 \beta_{11} + 172 \beta_{12} - 26 \beta_{13} - 254 \beta_{14} + 14 \beta_{15} ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(16q \) \(\mathstrut -\mathstrut 4q^{2} \) \(\mathstrut -\mathstrut 2q^{3} \) \(\mathstrut -\mathstrut 6q^{4} \) \(\mathstrut +\mathstrut 8q^{5} \) \(\mathstrut -\mathstrut 38q^{6} \) \(\mathstrut +\mathstrut 56q^{7} \) \(\mathstrut +\mathstrut 90q^{8} \) \(\mathstrut -\mathstrut 164q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(16q \) \(\mathstrut -\mathstrut 4q^{2} \) \(\mathstrut -\mathstrut 2q^{3} \) \(\mathstrut -\mathstrut 6q^{4} \) \(\mathstrut +\mathstrut 8q^{5} \) \(\mathstrut -\mathstrut 38q^{6} \) \(\mathstrut +\mathstrut 56q^{7} \) \(\mathstrut +\mathstrut 90q^{8} \) \(\mathstrut -\mathstrut 164q^{9} \) \(\mathstrut -\mathstrut 486q^{10} \) \(\mathstrut -\mathstrut 100q^{11} \) \(\mathstrut +\mathstrut 294q^{13} \) \(\mathstrut +\mathstrut 808q^{14} \) \(\mathstrut +\mathstrut 346q^{15} \) \(\mathstrut +\mathstrut 230q^{16} \) \(\mathstrut +\mathstrut 984q^{17} \) \(\mathstrut +\mathstrut 2434q^{18} \) \(\mathstrut -\mathstrut 1498q^{19} \) \(\mathstrut -\mathstrut 3962q^{20} \) \(\mathstrut -\mathstrut 1076q^{21} \) \(\mathstrut -\mathstrut 1524q^{22} \) \(\mathstrut -\mathstrut 1014q^{23} \) \(\mathstrut -\mathstrut 2142q^{24} \) \(\mathstrut +\mathstrut 614q^{26} \) \(\mathstrut +\mathstrut 3352q^{27} \) \(\mathstrut +\mathstrut 5764q^{28} \) \(\mathstrut +\mathstrut 814q^{29} \) \(\mathstrut +\mathstrut 9162q^{30} \) \(\mathstrut +\mathstrut 4060q^{31} \) \(\mathstrut -\mathstrut 4996q^{32} \) \(\mathstrut -\mathstrut 5636q^{33} \) \(\mathstrut -\mathstrut 2502q^{34} \) \(\mathstrut -\mathstrut 4892q^{35} \) \(\mathstrut -\mathstrut 15750q^{36} \) \(\mathstrut -\mathstrut 1790q^{37} \) \(\mathstrut +\mathstrut 6982q^{39} \) \(\mathstrut +\mathstrut 18816q^{40} \) \(\mathstrut +\mathstrut 4280q^{41} \) \(\mathstrut -\mathstrut 1204q^{42} \) \(\mathstrut -\mathstrut 1368q^{43} \) \(\mathstrut +\mathstrut 10736q^{44} \) \(\mathstrut -\mathstrut 6806q^{45} \) \(\mathstrut -\mathstrut 15246q^{46} \) \(\mathstrut +\mathstrut 1484q^{47} \) \(\mathstrut -\mathstrut 3002q^{48} \) \(\mathstrut -\mathstrut 11820q^{49} \) \(\mathstrut -\mathstrut 13574q^{50} \) \(\mathstrut +\mathstrut 1432q^{52} \) \(\mathstrut +\mathstrut 7204q^{53} \) \(\mathstrut +\mathstrut 13240q^{54} \) \(\mathstrut +\mathstrut 6936q^{55} \) \(\mathstrut +\mathstrut 8124q^{56} \) \(\mathstrut +\mathstrut 12736q^{57} \) \(\mathstrut +\mathstrut 3030q^{58} \) \(\mathstrut -\mathstrut 2380q^{59} \) \(\mathstrut -\mathstrut 6472q^{60} \) \(\mathstrut -\mathstrut 162q^{61} \) \(\mathstrut +\mathstrut 19614q^{62} \) \(\mathstrut +\mathstrut 12004q^{63} \) \(\mathstrut -\mathstrut 5248q^{65} \) \(\mathstrut -\mathstrut 23872q^{66} \) \(\mathstrut -\mathstrut 14854q^{67} \) \(\mathstrut -\mathstrut 6444q^{68} \) \(\mathstrut +\mathstrut 2412q^{69} \) \(\mathstrut -\mathstrut 34524q^{70} \) \(\mathstrut -\mathstrut 8050q^{71} \) \(\mathstrut +\mathstrut 15420q^{72} \) \(\mathstrut -\mathstrut 15448q^{73} \) \(\mathstrut -\mathstrut 2882q^{74} \) \(\mathstrut +\mathstrut 8280q^{75} \) \(\mathstrut +\mathstrut 10622q^{76} \) \(\mathstrut -\mathstrut 11672q^{78} \) \(\mathstrut -\mathstrut 17064q^{79} \) \(\mathstrut +\mathstrut 2564q^{80} \) \(\mathstrut +\mathstrut 2128q^{81} \) \(\mathstrut -\mathstrut 5346q^{82} \) \(\mathstrut +\mathstrut 12788q^{83} \) \(\mathstrut +\mathstrut 25948q^{84} \) \(\mathstrut +\mathstrut 35382q^{85} \) \(\mathstrut +\mathstrut 67260q^{86} \) \(\mathstrut +\mathstrut 29342q^{87} \) \(\mathstrut +\mathstrut 40836q^{88} \) \(\mathstrut +\mathstrut 20492q^{89} \) \(\mathstrut +\mathstrut 8996q^{91} \) \(\mathstrut -\mathstrut 49884q^{92} \) \(\mathstrut -\mathstrut 78920q^{93} \) \(\mathstrut -\mathstrut 30606q^{94} \) \(\mathstrut -\mathstrut 98574q^{95} \) \(\mathstrut -\mathstrut 94664q^{96} \) \(\mathstrut +\mathstrut 50944q^{97} \) \(\mathstrut +\mathstrut 61484q^{98} \) \(\mathstrut -\mathstrut 21632q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{16}\mathstrut +\mathstrut \) \(152\) \(x^{14}\mathstrut +\mathstrut \) \(9190\) \(x^{12}\mathstrut +\mathstrut \) \(285720\) \(x^{10}\mathstrut +\mathstrut \) \(4862025\) \(x^{8}\mathstrut +\mathstrut \) \(43573680\) \(x^{6}\mathstrut +\mathstrut \) \(169417008\) \(x^{4}\mathstrut +\mathstrut \) \(100636992\) \(x^{2}\mathstrut +\mathstrut \) \(3779136\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\(-\)\(403105\) \(\nu^{15}\mathstrut -\mathstrut \) \(60681659\) \(\nu^{13}\mathstrut -\mathstrut \) \(3635180779\) \(\nu^{11}\mathstrut -\mathstrut \) \(112263516645\) \(\nu^{9}\mathstrut -\mathstrut \) \(1906294858020\) \(\nu^{7}\mathstrut -\mathstrut \) \(17198065117872\) \(\nu^{5}\mathstrut -\mathstrut \) \(68934581954784\) \(\nu^{3}\mathstrut -\mathstrut \) \(50621014857024\) \(\nu\mathstrut +\mathstrut \) \(5545371476736\)\()/\)\(11090742953472\)
\(\beta_{2}\)\(=\)\((\)\(-\)\(450212\) \(\nu^{15}\mathstrut -\mathstrut \) \(12268161\) \(\nu^{14}\mathstrut -\mathstrut \) \(40141192\) \(\nu^{13}\mathstrut -\mathstrut \) \(1670382423\) \(\nu^{12}\mathstrut -\mathstrut \) \(336194096\) \(\nu^{11}\mathstrut -\mathstrut \) \(86530231647\) \(\nu^{10}\mathstrut +\mathstrut \) \(64569965208\) \(\nu^{9}\mathstrut -\mathstrut \) \(2167037463465\) \(\nu^{8}\mathstrut +\mathstrut \) \(2514078351540\) \(\nu^{7}\mathstrut -\mathstrut \) \(26905461929136\) \(\nu^{6}\mathstrut +\mathstrut \) \(35960915423664\) \(\nu^{5}\mathstrut -\mathstrut \) \(144684110994768\) \(\nu^{4}\mathstrut +\mathstrut \) \(189141832996704\) \(\nu^{3}\mathstrut -\mathstrut \) \(182218604617728\) \(\nu^{2}\mathstrut +\mathstrut \) \(90475406179200\) \(\nu\mathstrut -\mathstrut \) \(35916509215296\)\()/\)\(33272228860416\)
\(\beta_{3}\)\(=\)\((\)\(450212\) \(\nu^{15}\mathstrut -\mathstrut \) \(12268161\) \(\nu^{14}\mathstrut +\mathstrut \) \(40141192\) \(\nu^{13}\mathstrut -\mathstrut \) \(1670382423\) \(\nu^{12}\mathstrut +\mathstrut \) \(336194096\) \(\nu^{11}\mathstrut -\mathstrut \) \(86530231647\) \(\nu^{10}\mathstrut -\mathstrut \) \(64569965208\) \(\nu^{9}\mathstrut -\mathstrut \) \(2167037463465\) \(\nu^{8}\mathstrut -\mathstrut \) \(2514078351540\) \(\nu^{7}\mathstrut -\mathstrut \) \(26905461929136\) \(\nu^{6}\mathstrut -\mathstrut \) \(35960915423664\) \(\nu^{5}\mathstrut -\mathstrut \) \(144684110994768\) \(\nu^{4}\mathstrut -\mathstrut \) \(189141832996704\) \(\nu^{3}\mathstrut -\mathstrut \) \(182218604617728\) \(\nu^{2}\mathstrut -\mathstrut \) \(90475406179200\) \(\nu\mathstrut -\mathstrut \) \(35916509215296\)\()/\)\(33272228860416\)
\(\beta_{4}\)\(=\)\((\)\(-\)\(53905\) \(\nu^{15}\mathstrut -\mathstrut \) \(3031335\) \(\nu^{14}\mathstrut -\mathstrut \) \(1946495\) \(\nu^{13}\mathstrut -\mathstrut \) \(399803697\) \(\nu^{12}\mathstrut +\mathstrut \) \(403691033\) \(\nu^{11}\mathstrut -\mathstrut \) \(19722145833\) \(\nu^{10}\mathstrut +\mathstrut \) \(33967214799\) \(\nu^{9}\mathstrut -\mathstrut \) \(456891538383\) \(\nu^{8}\mathstrut +\mathstrut \) \(1036401623208\) \(\nu^{7}\mathstrut -\mathstrut \) \(4928117078304\) \(\nu^{6}\mathstrut +\mathstrut \) \(14113769934816\) \(\nu^{5}\mathstrut -\mathstrut \) \(18429203735856\) \(\nu^{4}\mathstrut +\mathstrut \) \(73741688677440\) \(\nu^{3}\mathstrut +\mathstrut \) \(20576677484160\) \(\nu^{2}\mathstrut +\mathstrut \) \(32386072836672\) \(\nu\mathstrut +\mathstrut \) \(22901572977984\)\()/\)\(3696914317824\)
\(\beta_{5}\)\(=\)\((\)\(-\)\(53905\) \(\nu^{15}\mathstrut +\mathstrut \) \(3031335\) \(\nu^{14}\mathstrut -\mathstrut \) \(1946495\) \(\nu^{13}\mathstrut +\mathstrut \) \(399803697\) \(\nu^{12}\mathstrut +\mathstrut \) \(403691033\) \(\nu^{11}\mathstrut +\mathstrut \) \(19722145833\) \(\nu^{10}\mathstrut +\mathstrut \) \(33967214799\) \(\nu^{9}\mathstrut +\mathstrut \) \(456891538383\) \(\nu^{8}\mathstrut +\mathstrut \) \(1036401623208\) \(\nu^{7}\mathstrut +\mathstrut \) \(4928117078304\) \(\nu^{6}\mathstrut +\mathstrut \) \(14113769934816\) \(\nu^{5}\mathstrut +\mathstrut \) \(18429203735856\) \(\nu^{4}\mathstrut +\mathstrut \) \(73741688677440\) \(\nu^{3}\mathstrut -\mathstrut \) \(20576677484160\) \(\nu^{2}\mathstrut +\mathstrut \) \(32386072836672\) \(\nu\mathstrut -\mathstrut \) \(22901572977984\)\()/\)\(3696914317824\)
\(\beta_{6}\)\(=\)\((\)\(958933\) \(\nu^{14}\mathstrut +\mathstrut \) \(127064675\) \(\nu^{12}\mathstrut +\mathstrut \) \(6269679835\) \(\nu^{10}\mathstrut +\mathstrut \) \(142169221341\) \(\nu^{8}\mathstrut +\mathstrut \) \(1357351279200\) \(\nu^{6}\mathstrut +\mathstrut \) \(1111509749520\) \(\nu^{4}\mathstrut -\mathstrut \) \(41761008777600\) \(\nu^{2}\mathstrut -\mathstrut \) \(17722730943936\)\()/\)\(616152386304\)
\(\beta_{7}\)\(=\)\((\)\(1611011\) \(\nu^{15}\mathstrut +\mathstrut \) \(20386710\) \(\nu^{14}\mathstrut +\mathstrut \) \(235643173\) \(\nu^{13}\mathstrut +\mathstrut \) \(2770864650\) \(\nu^{12}\mathstrut +\mathstrut \) \(13401654797\) \(\nu^{11}\mathstrut +\mathstrut \) \(141965775450\) \(\nu^{10}\mathstrut +\mathstrut \) \(377367280059\) \(\nu^{9}\mathstrut +\mathstrut \) \(3436503751926\) \(\nu^{8}\mathstrut +\mathstrut \) \(5435904315744\) \(\nu^{7}\mathstrut +\mathstrut \) \(38802395601648\) \(\nu^{6}\mathstrut +\mathstrut \) \(35965207159632\) \(\nu^{5}\mathstrut +\mathstrut \) \(147897464997600\) \(\nu^{4}\mathstrut +\mathstrut \) \(69173551698048\) \(\nu^{3}\mathstrut -\mathstrut \) \(236124309230208\) \(\nu^{2}\mathstrut -\mathstrut \) \(38119690075968\) \(\nu\mathstrut -\mathstrut \) \(127810286301312\)\()/\)\(11090742953472\)
\(\beta_{8}\)\(=\)\((\)\(-\)\(64316\) \(\nu^{15}\mathstrut +\mathstrut \) \(10690839\) \(\nu^{14}\mathstrut -\mathstrut \) \(5734456\) \(\nu^{13}\mathstrut +\mathstrut \) \(1454826609\) \(\nu^{12}\mathstrut -\mathstrut \) \(48027728\) \(\nu^{11}\mathstrut +\mathstrut \) \(75309720489\) \(\nu^{10}\mathstrut +\mathstrut \) \(9224280744\) \(\nu^{9}\mathstrut +\mathstrut \) \(1881613092303\) \(\nu^{8}\mathstrut +\mathstrut \) \(359154050220\) \(\nu^{7}\mathstrut +\mathstrut \) \(23121020302128\) \(\nu^{6}\mathstrut +\mathstrut \) \(5137273631952\) \(\nu^{5}\mathstrut +\mathstrut \) \(118652121636336\) \(\nu^{4}\mathstrut +\mathstrut \) \(27020261856672\) \(\nu^{3}\mathstrut +\mathstrut \) \(95067845558784\) \(\nu^{2}\mathstrut +\mathstrut \) \(12925058025600\) \(\nu\mathstrut -\mathstrut \) \(46660774261824\)\()/\)\(4753175551488\)
\(\beta_{9}\)\(=\)\((\)\(8483506\) \(\nu^{15}\mathstrut -\mathstrut \) \(22455\) \(\nu^{14}\mathstrut +\mathstrut \) \(1257390830\) \(\nu^{13}\mathstrut -\mathstrut \) \(34398657\) \(\nu^{12}\mathstrut +\mathstrut \) \(73776312574\) \(\nu^{11}\mathstrut -\mathstrut \) \(3085996905\) \(\nu^{10}\mathstrut +\mathstrut \) \(2217207089202\) \(\nu^{9}\mathstrut -\mathstrut \) \(83390031711\) \(\nu^{8}\mathstrut +\mathstrut \) \(36294499625328\) \(\nu^{7}\mathstrut -\mathstrut \) \(575259720336\) \(\nu^{6}\mathstrut +\mathstrut \) \(310555374841728\) \(\nu^{5}\mathstrut +\mathstrut \) \(2193532745904\) \(\nu^{4}\mathstrut +\mathstrut \) \(1153914652058112\) \(\nu^{3}\mathstrut -\mathstrut \) \(10688655293568\) \(\nu^{2}\mathstrut +\mathstrut \) \(921694995453312\) \(\nu\mathstrut -\mathstrut \) \(121083795836352\)\()/\)\(33272228860416\)
\(\beta_{10}\)\(=\)\((\)\(1611011\) \(\nu^{15}\mathstrut -\mathstrut \) \(20386710\) \(\nu^{14}\mathstrut +\mathstrut \) \(235643173\) \(\nu^{13}\mathstrut -\mathstrut \) \(2770864650\) \(\nu^{12}\mathstrut +\mathstrut \) \(13401654797\) \(\nu^{11}\mathstrut -\mathstrut \) \(141965775450\) \(\nu^{10}\mathstrut +\mathstrut \) \(377367280059\) \(\nu^{9}\mathstrut -\mathstrut \) \(3436503751926\) \(\nu^{8}\mathstrut +\mathstrut \) \(5435904315744\) \(\nu^{7}\mathstrut -\mathstrut \) \(38802395601648\) \(\nu^{6}\mathstrut +\mathstrut \) \(35965207159632\) \(\nu^{5}\mathstrut -\mathstrut \) \(147897464997600\) \(\nu^{4}\mathstrut +\mathstrut \) \(69173551698048\) \(\nu^{3}\mathstrut +\mathstrut \) \(236124309230208\) \(\nu^{2}\mathstrut -\mathstrut \) \(38119690075968\) \(\nu\mathstrut +\mathstrut \) \(116719543347840\)\()/\)\(11090742953472\)
\(\beta_{11}\)\(=\)\((\)\(3060949\) \(\nu^{15}\mathstrut +\mathstrut \) \(30779505\) \(\nu^{14}\mathstrut +\mathstrut \) \(538883915\) \(\nu^{13}\mathstrut +\mathstrut \) \(4066826247\) \(\nu^{12}\mathstrut +\mathstrut \) \(37885091971\) \(\nu^{11}\mathstrut +\mathstrut \) \(202355357535\) \(\nu^{10}\mathstrut +\mathstrut \) \(1360934568357\) \(\nu^{9}\mathstrut +\mathstrut \) \(4811620057209\) \(\nu^{8}\mathstrut +\mathstrut \) \(26434213763832\) \(\nu^{7}\mathstrut +\mathstrut \) \(55871559620352\) \(\nu^{6}\mathstrut +\mathstrut \) \(266224281763680\) \(\nu^{5}\mathstrut +\mathstrut \) \(274948280891280\) \(\nu^{4}\mathstrut +\mathstrut \) \(1144577610929088\) \(\nu^{3}\mathstrut +\mathstrut \) \(313787054673792\) \(\nu^{2}\mathstrut +\mathstrut \) \(765185781986496\) \(\nu\mathstrut +\mathstrut \) \(262228572037440\)\()/\)\(11090742953472\)
\(\beta_{12}\)\(=\)\((\)\(1668943\) \(\nu^{15}\mathstrut +\mathstrut \) \(3478479\) \(\nu^{14}\mathstrut +\mathstrut \) \(244791977\) \(\nu^{13}\mathstrut +\mathstrut \) \(476788953\) \(\nu^{12}\mathstrut +\mathstrut \) \(14182744225\) \(\nu^{11}\mathstrut +\mathstrut \) \(24822766977\) \(\nu^{10}\mathstrut +\mathstrut \) \(420766469319\) \(\nu^{9}\mathstrut +\mathstrut \) \(620613013671\) \(\nu^{8}\mathstrut +\mathstrut \) \(6825957638976\) \(\nu^{7}\mathstrut +\mathstrut \) \(7560677757024\) \(\nu^{6}\mathstrut +\mathstrut \) \(58482708936192\) \(\nu^{5}\mathstrut +\mathstrut \) \(37860759720432\) \(\nu^{4}\mathstrut +\mathstrut \) \(218257879045248\) \(\nu^{3}\mathstrut +\mathstrut \) \(28035201382272\) \(\nu^{2}\mathstrut +\mathstrut \) \(118564929748800\) \(\nu\mathstrut -\mathstrut \) \(4839006028608\)\()/\)\(3696914317824\)
\(\beta_{13}\)\(=\)\((\)\(3060949\) \(\nu^{15}\mathstrut -\mathstrut \) \(30779505\) \(\nu^{14}\mathstrut +\mathstrut \) \(538883915\) \(\nu^{13}\mathstrut -\mathstrut \) \(4066826247\) \(\nu^{12}\mathstrut +\mathstrut \) \(37885091971\) \(\nu^{11}\mathstrut -\mathstrut \) \(202355357535\) \(\nu^{10}\mathstrut +\mathstrut \) \(1360934568357\) \(\nu^{9}\mathstrut -\mathstrut \) \(4811620057209\) \(\nu^{8}\mathstrut +\mathstrut \) \(26434213763832\) \(\nu^{7}\mathstrut -\mathstrut \) \(55871559620352\) \(\nu^{6}\mathstrut +\mathstrut \) \(266224281763680\) \(\nu^{5}\mathstrut -\mathstrut \) \(274948280891280\) \(\nu^{4}\mathstrut +\mathstrut \) \(1144577610929088\) \(\nu^{3}\mathstrut -\mathstrut \) \(313787054673792\) \(\nu^{2}\mathstrut +\mathstrut \) \(765185781986496\) \(\nu\mathstrut -\mathstrut \) \(262228572037440\)\()/\)\(11090742953472\)
\(\beta_{14}\)\(=\)\((\)\(1668943\) \(\nu^{15}\mathstrut -\mathstrut \) \(3478479\) \(\nu^{14}\mathstrut +\mathstrut \) \(244791977\) \(\nu^{13}\mathstrut -\mathstrut \) \(476788953\) \(\nu^{12}\mathstrut +\mathstrut \) \(14182744225\) \(\nu^{11}\mathstrut -\mathstrut \) \(24822766977\) \(\nu^{10}\mathstrut +\mathstrut \) \(420766469319\) \(\nu^{9}\mathstrut -\mathstrut \) \(620613013671\) \(\nu^{8}\mathstrut +\mathstrut \) \(6825957638976\) \(\nu^{7}\mathstrut -\mathstrut \) \(7560677757024\) \(\nu^{6}\mathstrut +\mathstrut \) \(58482708936192\) \(\nu^{5}\mathstrut -\mathstrut \) \(37860759720432\) \(\nu^{4}\mathstrut +\mathstrut \) \(218257879045248\) \(\nu^{3}\mathstrut -\mathstrut \) \(28035201382272\) \(\nu^{2}\mathstrut +\mathstrut \) \(118564929748800\) \(\nu\mathstrut +\mathstrut \) \(4839006028608\)\()/\)\(3696914317824\)
\(\beta_{15}\)\(=\)\((\)\(7320853\) \(\nu^{15}\mathstrut +\mathstrut \) \(2876799\) \(\nu^{14}\mathstrut +\mathstrut \) \(1089833795\) \(\nu^{13}\mathstrut +\mathstrut \) \(381194025\) \(\nu^{12}\mathstrut +\mathstrut \) \(64086192235\) \(\nu^{11}\mathstrut +\mathstrut \) \(18809039505\) \(\nu^{10}\mathstrut +\mathstrut \) \(1921681816845\) \(\nu^{9}\mathstrut +\mathstrut \) \(426507664023\) \(\nu^{8}\mathstrut +\mathstrut \) \(31200573373200\) \(\nu^{7}\mathstrut +\mathstrut \) \(4072053837600\) \(\nu^{6}\mathstrut +\mathstrut \) \(262878864690816\) \(\nu^{5}\mathstrut +\mathstrut \) \(3334529248560\) \(\nu^{4}\mathstrut +\mathstrut \) \(938832132172032\) \(\nu^{3}\mathstrut -\mathstrut \) \(125283026332800\) \(\nu^{2}\mathstrut +\mathstrut \) \(452794700306880\) \(\nu\mathstrut -\mathstrut \) \(53168192831808\)\()/\)\(3696914317824\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-\)\(6\) \(\beta_{15}\mathstrut +\mathstrut \) \(8\) \(\beta_{14}\mathstrut +\mathstrut \) \(\beta_{13}\mathstrut +\mathstrut \) \(12\) \(\beta_{12}\mathstrut +\mathstrut \) \(\beta_{11}\mathstrut +\mathstrut \) \(8\) \(\beta_{10}\mathstrut +\mathstrut \) \(4\) \(\beta_{9}\mathstrut -\mathstrut \) \(2\) \(\beta_{8}\mathstrut +\mathstrut \) \(8\) \(\beta_{7}\mathstrut +\mathstrut \) \(3\) \(\beta_{6}\mathstrut -\mathstrut \) \(4\) \(\beta_{5}\mathstrut -\mathstrut \) \(4\) \(\beta_{4}\mathstrut -\mathstrut \) \(18\) \(\beta_{3}\mathstrut +\mathstrut \) \(16\) \(\beta_{2}\mathstrut +\mathstrut \) \(20\) \(\beta_{1}\mathstrut -\mathstrut \) \(4\)\()/26\)
\(\nu^{2}\)\(=\)\((\)\(7\) \(\beta_{14}\mathstrut -\mathstrut \) \(4\) \(\beta_{13}\mathstrut -\mathstrut \) \(7\) \(\beta_{12}\mathstrut +\mathstrut \) \(4\) \(\beta_{11}\mathstrut -\mathstrut \) \(7\) \(\beta_{10}\mathstrut +\mathstrut \) \(2\) \(\beta_{8}\mathstrut +\mathstrut \) \(7\) \(\beta_{7}\mathstrut -\mathstrut \) \(3\) \(\beta_{6}\mathstrut -\mathstrut \) \(13\) \(\beta_{5}\mathstrut +\mathstrut \) \(13\) \(\beta_{4}\mathstrut +\mathstrut \) \(19\) \(\beta_{3}\mathstrut +\mathstrut \) \(17\) \(\beta_{2}\mathstrut -\mathstrut \) \(242\)\()/13\)
\(\nu^{3}\)\(=\)\((\)\(92\) \(\beta_{15}\mathstrut -\mathstrut \) \(114\) \(\beta_{14}\mathstrut +\mathstrut \) \(15\) \(\beta_{13}\mathstrut -\mathstrut \) \(184\) \(\beta_{12}\mathstrut +\mathstrut \) \(15\) \(\beta_{11}\mathstrut -\mathstrut \) \(140\) \(\beta_{10}\mathstrut -\mathstrut \) \(70\) \(\beta_{9}\mathstrut +\mathstrut \) \(35\) \(\beta_{8}\mathstrut -\mathstrut \) \(140\) \(\beta_{7}\mathstrut -\mathstrut \) \(46\) \(\beta_{6}\mathstrut +\mathstrut \) \(83\) \(\beta_{5}\mathstrut +\mathstrut \) \(83\) \(\beta_{4}\mathstrut +\mathstrut \) \(588\) \(\beta_{3}\mathstrut -\mathstrut \) \(553\) \(\beta_{2}\mathstrut +\mathstrut \) \(274\) \(\beta_{1}\mathstrut -\mathstrut \) \(242\)\()/13\)
\(\nu^{4}\)\(=\)\((\)\(-\)\(407\) \(\beta_{14}\mathstrut +\mathstrut \) \(175\) \(\beta_{13}\mathstrut +\mathstrut \) \(407\) \(\beta_{12}\mathstrut -\mathstrut \) \(175\) \(\beta_{11}\mathstrut +\mathstrut \) \(381\) \(\beta_{10}\mathstrut -\mathstrut \) \(380\) \(\beta_{8}\mathstrut -\mathstrut \) \(381\) \(\beta_{7}\mathstrut +\mathstrut \) \(258\) \(\beta_{6}\mathstrut +\mathstrut \) \(507\) \(\beta_{5}\mathstrut -\mathstrut \) \(507\) \(\beta_{4}\mathstrut -\mathstrut \) \(1855\) \(\beta_{3}\mathstrut -\mathstrut \) \(1475\) \(\beta_{2}\mathstrut +\mathstrut \) \(7318\)\()/13\)
\(\nu^{5}\)\(=\)\((\)\(-\)\(3476\) \(\beta_{15}\mathstrut +\mathstrut \) \(3430\) \(\beta_{14}\mathstrut -\mathstrut \) \(1479\) \(\beta_{13}\mathstrut +\mathstrut \) \(7212\) \(\beta_{12}\mathstrut -\mathstrut \) \(1479\) \(\beta_{11}\mathstrut +\mathstrut \) \(5588\) \(\beta_{10}\mathstrut +\mathstrut \) \(3782\) \(\beta_{9}\mathstrut -\mathstrut \) \(1891\) \(\beta_{8}\mathstrut +\mathstrut \) \(5588\) \(\beta_{7}\mathstrut +\mathstrut \) \(1738\) \(\beta_{6}\mathstrut -\mathstrut \) \(3977\) \(\beta_{5}\mathstrut -\mathstrut \) \(3977\) \(\beta_{4}\mathstrut -\mathstrut \) \(32190\) \(\beta_{3}\mathstrut +\mathstrut \) \(30299\) \(\beta_{2}\mathstrut -\mathstrut \) \(27838\) \(\beta_{1}\mathstrut +\mathstrut \) \(17616\)\()/13\)
\(\nu^{6}\)\(=\)\((\)\(17913\) \(\beta_{14}\mathstrut -\mathstrut \) \(8533\) \(\beta_{13}\mathstrut -\mathstrut \) \(17913\) \(\beta_{12}\mathstrut +\mathstrut \) \(8533\) \(\beta_{11}\mathstrut -\mathstrut \) \(18797\) \(\beta_{10}\mathstrut +\mathstrut \) \(24644\) \(\beta_{8}\mathstrut +\mathstrut \) \(18797\) \(\beta_{7}\mathstrut -\mathstrut \) \(14814\) \(\beta_{6}\mathstrut -\mathstrut \) \(22607\) \(\beta_{5}\mathstrut +\mathstrut \) \(22607\) \(\beta_{4}\mathstrut +\mathstrut \) \(118171\) \(\beta_{3}\mathstrut +\mathstrut \) \(93527\) \(\beta_{2}\mathstrut -\mathstrut \) \(271710\)\()/13\)
\(\nu^{7}\)\(=\)\(11468\) \(\beta_{15}\mathstrut -\mathstrut \) \(8888\) \(\beta_{14}\mathstrut +\mathstrut \) \(7017\) \(\beta_{13}\mathstrut -\mathstrut \) \(24998\) \(\beta_{12}\mathstrut +\mathstrut \) \(7017\) \(\beta_{11}\mathstrut -\mathstrut \) \(19004\) \(\beta_{10}\mathstrut -\mathstrut \) \(16110\) \(\beta_{9}\mathstrut +\mathstrut \) \(8055\) \(\beta_{8}\mathstrut -\mathstrut \) \(19004\) \(\beta_{7}\mathstrut -\mathstrut \) \(5734\) \(\beta_{6}\mathstrut +\mathstrut \) \(15173\) \(\beta_{5}\mathstrut +\mathstrut \) \(15173\) \(\beta_{4}\mathstrut +\mathstrut \) \(130698\) \(\beta_{3}\mathstrut -\mathstrut \) \(122643\) \(\beta_{2}\mathstrut +\mathstrut \) \(124726\) \(\beta_{1}\mathstrut -\mathstrut \) \(73312\)
\(\nu^{8}\)\(=\)\((\)\(-\)\(772903\) \(\beta_{14}\mathstrut +\mathstrut \) \(430867\) \(\beta_{13}\mathstrut +\mathstrut \) \(772903\) \(\beta_{12}\mathstrut -\mathstrut \) \(430867\) \(\beta_{11}\mathstrut +\mathstrut \) \(918997\) \(\beta_{10}\mathstrut -\mathstrut \) \(1362716\) \(\beta_{8}\mathstrut -\mathstrut \) \(918997\) \(\beta_{7}\mathstrut +\mathstrut \) \(772518\) \(\beta_{6}\mathstrut +\mathstrut \) \(1088347\) \(\beta_{5}\mathstrut -\mathstrut \) \(1088347\) \(\beta_{4}\mathstrut -\mathstrut \) \(6639151\) \(\beta_{3}\mathstrut -\mathstrut \) \(5276435\) \(\beta_{2}\mathstrut +\mathstrut \) \(11600462\)\()/13\)
\(\nu^{9}\)\(=\)\((\)\(-\)\(6901292\) \(\beta_{15}\mathstrut +\mathstrut \) \(4408926\) \(\beta_{14}\mathstrut -\mathstrut \) \(5008851\) \(\beta_{13}\mathstrut +\mathstrut \) \(15652900\) \(\beta_{12}\mathstrut -\mathstrut \) \(5008851\) \(\beta_{11}\mathstrut +\mathstrut \) \(11583236\) \(\beta_{10}\mathstrut +\mathstrut \) \(11243974\) \(\beta_{9}\mathstrut -\mathstrut \) \(5621987\) \(\beta_{8}\mathstrut +\mathstrut \) \(11583236\) \(\beta_{7}\mathstrut +\mathstrut \) \(3450646\) \(\beta_{6}\mathstrut -\mathstrut \) \(9941777\) \(\beta_{5}\mathstrut -\mathstrut \) \(9941777\) \(\beta_{4}\mathstrut -\mathstrut \) \(88036518\) \(\beta_{3}\mathstrut +\mathstrut \) \(82414531\) \(\beta_{2}\mathstrut -\mathstrut \) \(84818014\) \(\beta_{1}\mathstrut +\mathstrut \) \(48370256\)\()/13\)
\(\nu^{10}\)\(=\)\((\)\(34597169\) \(\beta_{14}\mathstrut -\mathstrut \) \(21896869\) \(\beta_{13}\mathstrut -\mathstrut \) \(34597169\) \(\beta_{12}\mathstrut +\mathstrut \) \(21896869\) \(\beta_{11}\mathstrut -\mathstrut \) \(45236733\) \(\beta_{10}\mathstrut +\mathstrut \) \(71658740\) \(\beta_{8}\mathstrut +\mathstrut \) \(45236733\) \(\beta_{7}\mathstrut -\mathstrut \) \(39176646\) \(\beta_{6}\mathstrut -\mathstrut \) \(54100527\) \(\beta_{5}\mathstrut +\mathstrut \) \(54100527\) \(\beta_{4}\mathstrut +\mathstrut \) \(353441755\) \(\beta_{3}\mathstrut +\mathstrut \) \(281783015\) \(\beta_{2}\mathstrut -\mathstrut \) \(536617966\)\()/13\)
\(\nu^{11}\)\(=\)\((\)\(332946236\) \(\beta_{15}\mathstrut -\mathstrut \) \(187007200\) \(\beta_{14}\mathstrut +\mathstrut \) \(263030493\) \(\beta_{13}\mathstrut -\mathstrut \) \(775259574\) \(\beta_{12}\mathstrut +\mathstrut \) \(263030493\) \(\beta_{11}\mathstrut -\mathstrut \) \(561086828\) \(\beta_{10}\mathstrut -\mathstrut \) \(588252374\) \(\beta_{9}\mathstrut +\mathstrut \) \(294126187\) \(\beta_{8}\mathstrut -\mathstrut \) \(561086828\) \(\beta_{7}\mathstrut -\mathstrut \) \(166473118\) \(\beta_{6}\mathstrut +\mathstrut \) \(503828801\) \(\beta_{5}\mathstrut +\mathstrut \) \(503828801\) \(\beta_{4}\mathstrut +\mathstrut \) \(4512195906\) \(\beta_{3}\mathstrut -\mathstrut \) \(4218069719\) \(\beta_{2}\mathstrut +\mathstrut \) \(4307663710\) \(\beta_{1}\mathstrut -\mathstrut \) \(2420792496\)\()/13\)
\(\nu^{12}\)\(=\)\((\)\(-\)\(1614876399\) \(\beta_{14}\mathstrut +\mathstrut \) \(1111784059\) \(\beta_{13}\mathstrut +\mathstrut \) \(1614876399\) \(\beta_{12}\mathstrut -\mathstrut \) \(1111784059\) \(\beta_{11}\mathstrut +\mathstrut \) \(2244399605\) \(\beta_{10}\mathstrut -\mathstrut \) \(3691353404\) \(\beta_{8}\mathstrut -\mathstrut \) \(2244399605\) \(\beta_{7}\mathstrut +\mathstrut \) \(1971703926\) \(\beta_{6}\mathstrut +\mathstrut \) \(2719026011\) \(\beta_{5}\mathstrut -\mathstrut \) \(2719026011\) \(\beta_{4}\mathstrut -\mathstrut \) \(18332662303\) \(\beta_{3}\mathstrut -\mathstrut \) \(14641308899\) \(\beta_{2}\mathstrut +\mathstrut \) \(25885701678\)\()/13\)
\(\nu^{13}\)\(=\)\((\)\(-\)\(16408659212\) \(\beta_{15}\mathstrut +\mathstrut \) \(8537243750\) \(\beta_{14}\mathstrut -\mathstrut \) \(13536977019\) \(\beta_{13}\mathstrut +\mathstrut \) \(38821589996\) \(\beta_{12}\mathstrut -\mathstrut \) \(13536977019\) \(\beta_{11}\mathstrut +\mathstrut \) \(27670764260\) \(\beta_{10}\mathstrut +\mathstrut \) \(30284346246\) \(\beta_{9}\mathstrut -\mathstrut \) \(15142173123\) \(\beta_{8}\mathstrut +\mathstrut \) \(27670764260\) \(\beta_{7}\mathstrut +\mathstrut \) \(8204329606\) \(\beta_{6}\mathstrut -\mathstrut \) \(25547886401\) \(\beta_{5}\mathstrut -\mathstrut \) \(25547886401\) \(\beta_{4}\mathstrut -\mathstrut \) \(229745363670\) \(\beta_{3}\mathstrut +\mathstrut \) \(214603190547\) \(\beta_{2}\mathstrut -\mathstrut \) \(217025181214\) \(\beta_{1}\mathstrut +\mathstrut \) \(121041181744\)\()/13\)
\(\nu^{14}\)\(=\)\(5983739437\) \(\beta_{14}\mathstrut -\mathstrut \) \(4333135633\) \(\beta_{13}\mathstrut -\mathstrut \) \(5983739437\) \(\beta_{12}\mathstrut +\mathstrut \) \(4333135633\) \(\beta_{11}\mathstrut -\mathstrut \) \(8616856273\) \(\beta_{10}\mathstrut +\mathstrut \) \(14483635364\) \(\beta_{8}\mathstrut +\mathstrut \) \(8616856273\) \(\beta_{7}\mathstrut -\mathstrut \) \(7623329934\) \(\beta_{6}\mathstrut -\mathstrut \) \(10544524843\) \(\beta_{5}\mathstrut +\mathstrut \) \(10544524843\) \(\beta_{4}\mathstrut +\mathstrut \) \(72180049879\) \(\beta_{3}\mathstrut +\mathstrut \) \(57696414515\) \(\beta_{2}\mathstrut -\mathstrut \) \(98119308806\)
\(\nu^{15}\)\(=\)\((\)\(817505384348\) \(\beta_{15}\mathstrut -\mathstrut \) \(407544050424\) \(\beta_{14}\mathstrut +\mathstrut \) \(689836505013\) \(\beta_{13}\mathstrut -\mathstrut \) \(1952234100238\) \(\beta_{12}\mathstrut +\mathstrut \) \(689836505013\) \(\beta_{11}\mathstrut -\mathstrut \) \(1378130038796\) \(\beta_{10}\mathstrut -\mathstrut \) \(1544690049814\) \(\beta_{9}\mathstrut +\mathstrut \) \(772345024907\) \(\beta_{8}\mathstrut -\mathstrut \) \(1378130038796\) \(\beta_{7}\mathstrut -\mathstrut \) \(408752692174\) \(\beta_{6}\mathstrut +\mathstrut \) \(1294054470737\) \(\beta_{5}\mathstrut +\mathstrut \) \(1294054470737\) \(\beta_{4}\mathstrut +\mathstrut \) \(11650966452498\) \(\beta_{3}\mathstrut -\mathstrut \) \(10878621427591\) \(\beta_{2}\mathstrut +\mathstrut \) \(10916665870750\) \(\beta_{1}\mathstrut -\mathstrut \) \(6064117949264\)\()/13\)

Character Values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/13\mathbb{Z}\right)^\times\).

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

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2.1
4.89748i
5.33868i
0.816521i
3.98977i
4.34562i
7.09996i
3.68702i
0.200628i
4.89748i
5.33868i
0.816521i
3.98977i
4.34562i
7.09996i
3.68702i
0.200628i
−6.34141 1.69918i 7.28678 + 12.6211i 23.4699 + 13.5504i −20.2793 + 20.2793i −24.7631 92.4170i 21.3318 5.71584i −51.5321 51.5321i −65.6944 + 113.786i 163.058 94.1415i
2.2 −3.41998 0.916381i −5.16500 8.94604i −2.99988 1.73198i 8.65321 8.65321i 9.46622 + 35.3284i −2.94871 + 0.790103i 48.7300 + 48.7300i −12.8545 + 22.2646i −37.5235 + 21.6642i
2.3 2.38644 + 0.639444i 4.25709 + 7.37349i −8.57021 4.94801i 16.9748 16.9748i 5.44434 + 20.3185i −50.2192 + 13.4562i −45.2402 45.2402i 4.25442 7.36887i 51.3637 29.6548i
2.4 5.50893 + 1.47611i −4.28079 7.41455i 14.3130 + 8.26362i −29.3294 + 29.3294i −12.6379 47.1652i 45.8361 12.2817i 2.12627 + 2.12627i 3.84960 6.66770i −204.867 + 118.280i
6.1 −1.88469 7.03375i −0.939567 + 1.62738i −32.0652 + 18.5128i 26.6717 26.6717i 13.2174 + 3.54158i 7.85376 29.3106i 108.263 + 108.263i 38.7344 + 67.0900i −237.870 137.334i
6.2 −0.387376 1.44571i 3.29308 5.70378i 11.9164 6.87993i −10.5663 + 10.5663i −9.52166 2.55132i −14.3010 + 53.3722i −31.4958 31.4958i 18.8113 + 32.5821i 19.3688 + 11.1826i
6.3 0.540374 + 2.01670i −8.21171 + 14.2231i 10.0813 5.82045i 17.2508 17.2508i −33.1212 8.87479i −2.95016 + 11.0101i 40.8071 + 40.8071i −94.3643 163.444i 44.1117 + 25.4679i
6.4 1.59772 + 5.96275i 2.76012 4.78067i −19.1453 + 11.0536i −5.37551 + 5.37551i 32.9158 + 8.81977i 23.3974 87.3205i −26.6579 26.6579i 25.2635 + 43.7576i −40.6414 23.4643i
7.1 −6.34141 + 1.69918i 7.28678 12.6211i 23.4699 13.5504i −20.2793 20.2793i −24.7631 + 92.4170i 21.3318 + 5.71584i −51.5321 + 51.5321i −65.6944 113.786i 163.058 + 94.1415i
7.2 −3.41998 + 0.916381i −5.16500 + 8.94604i −2.99988 + 1.73198i 8.65321 + 8.65321i 9.46622 35.3284i −2.94871 0.790103i 48.7300 48.7300i −12.8545 22.2646i −37.5235 21.6642i
7.3 2.38644 0.639444i 4.25709 7.37349i −8.57021 + 4.94801i 16.9748 + 16.9748i 5.44434 20.3185i −50.2192 13.4562i −45.2402 + 45.2402i 4.25442 + 7.36887i 51.3637 + 29.6548i
7.4 5.50893 1.47611i −4.28079 + 7.41455i 14.3130 8.26362i −29.3294 29.3294i −12.6379 + 47.1652i 45.8361 + 12.2817i 2.12627 2.12627i 3.84960 + 6.66770i −204.867 118.280i
11.1 −1.88469 + 7.03375i −0.939567 1.62738i −32.0652 18.5128i 26.6717 + 26.6717i 13.2174 3.54158i 7.85376 + 29.3106i 108.263 108.263i 38.7344 67.0900i −237.870 + 137.334i
11.2 −0.387376 + 1.44571i 3.29308 + 5.70378i 11.9164 + 6.87993i −10.5663 10.5663i −9.52166 + 2.55132i −14.3010 53.3722i −31.4958 + 31.4958i 18.8113 32.5821i 19.3688 11.1826i
11.3 0.540374 2.01670i −8.21171 14.2231i 10.0813 + 5.82045i 17.2508 + 17.2508i −33.1212 + 8.87479i −2.95016 11.0101i 40.8071 40.8071i −94.3643 + 163.444i 44.1117 25.4679i
11.4 1.59772 5.96275i 2.76012 + 4.78067i −19.1453 11.0536i −5.37551 5.37551i 32.9158 8.81977i 23.3974 + 87.3205i −26.6579 + 26.6579i 25.2635 43.7576i −40.6414 + 23.4643i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 11.4
Significant digits:
Format:

Inner twists

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

Hecke kernels

There are no other newforms in \(S_{5}^{\mathrm{new}}(13, [\chi])\).