Properties

Label 12.12.b.a
Level 12
Weight 12
Character orbit 12.b
Analytic conductor 9.220
Analytic rank 0
Dimension 20
CM No
Inner twists 4

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Newspace parameters

Level: \( N \) = \( 12 = 2^{2} \cdot 3 \)
Weight: \( k \) = \( 12 \)
Character orbit: \([\chi]\) = 12.b (of order \(2\) and degree \(1\))

Newform invariants

Self dual: No
Analytic conductor: \(9.22011816672\)
Analytic rank: \(0\)
Dimension: \(20\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{104}\cdot 3^{42}\cdot 5^{2} \)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \( + \beta_{1} q^{2} \) \( -\beta_{3} q^{3} \) \( + ( 99 + \beta_{2} ) q^{4} \) \( + ( -5 \beta_{1} + \beta_{6} ) q^{5} \) \( + ( 656 - \beta_{8} ) q^{6} \) \( + ( -\beta_{1} - 3 \beta_{3} - \beta_{4} ) q^{7} \) \( + ( 98 \beta_{1} - 3 \beta_{3} + \beta_{5} ) q^{8} \) \( + ( 5182 - 63 \beta_{1} + 3 \beta_{2} - \beta_{3} - \beta_{6} + \beta_{9} ) q^{9} \) \(+O(q^{10})\) \( q\) \( + \beta_{1} q^{2} \) \( -\beta_{3} q^{3} \) \( + ( 99 + \beta_{2} ) q^{4} \) \( + ( -5 \beta_{1} + \beta_{6} ) q^{5} \) \( + ( 656 - \beta_{8} ) q^{6} \) \( + ( -\beta_{1} - 3 \beta_{3} - \beta_{4} ) q^{7} \) \( + ( 98 \beta_{1} - 3 \beta_{3} + \beta_{5} ) q^{8} \) \( + ( 5182 - 63 \beta_{1} + 3 \beta_{2} - \beta_{3} - \beta_{6} + \beta_{9} ) q^{9} \) \( + ( 10099 + 7 \beta_{1} - 5 \beta_{2} + 22 \beta_{3} + \beta_{4} - \beta_{10} ) q^{10} \) \( + ( 1 - 166 \beta_{1} + \beta_{2} - 66 \beta_{3} - \beta_{5} + 2 \beta_{8} + \beta_{12} - \beta_{16} ) q^{11} \) \( + ( 53238 + 625 \beta_{1} - 98 \beta_{3} + 3 \beta_{4} - 5 \beta_{6} + \beta_{8} + \beta_{11} + \beta_{12} + \beta_{13} - \beta_{16} ) q^{12} \) \( + ( 77189 + 4 \beta_{1} - 45 \beta_{2} + 14 \beta_{3} - \beta_{6} + 12 \beta_{8} + \beta_{9} - 2 \beta_{10} + \beta_{11} + \beta_{13} + \beta_{15} ) q^{13} \) \( + ( 2 - 24 \beta_{1} - 3 \beta_{2} - 78 \beta_{3} + 2 \beta_{5} - 34 \beta_{6} - 2 \beta_{8} + 6 \beta_{9} - \beta_{11} + \beta_{12} - \beta_{13} + \beta_{15} - 2 \beta_{16} + \beta_{17} ) q^{14} \) \( + ( -12 + 1803 \beta_{1} - 69 \beta_{2} + 27 \beta_{3} + 3 \beta_{4} + 14 \beta_{5} - 5 \beta_{6} + \beta_{8} + 3 \beta_{9} - 6 \beta_{10} + 2 \beta_{12} + \beta_{14} - 2 \beta_{15} - \beta_{16} - \beta_{17} + \beta_{18} - \beta_{19} ) q^{15} \) \( + ( 174164 + 54 \beta_{1} + 84 \beta_{2} + 188 \beta_{3} - 34 \beta_{4} + 2 \beta_{5} - \beta_{7} - \beta_{8} + 12 \beta_{9} - 4 \beta_{10} - 2 \beta_{11} + 3 \beta_{12} + \beta_{13} - \beta_{14} - 3 \beta_{15} - \beta_{17} - \beta_{18} + 2 \beta_{19} ) q^{16} \) \( + ( -42 - 5420 \beta_{1} - 27 \beta_{2} - 10 \beta_{3} + 33 \beta_{5} - 26 \beta_{6} + 2 \beta_{7} - 94 \beta_{8} + 2 \beta_{9} + \beta_{11} - 2 \beta_{12} - 3 \beta_{13} - 3 \beta_{14} + 3 \beta_{15} - \beta_{17} - 2 \beta_{18} + \beta_{19} ) q^{17} \) \( + ( 127861 + 4978 \beta_{1} - 66 \beta_{2} - 571 \beta_{3} - 87 \beta_{4} + 2 \beta_{5} + 232 \beta_{6} + 2 \beta_{7} - 4 \beta_{8} + 2 \beta_{9} + 3 \beta_{10} + 13 \beta_{11} - 14 \beta_{12} + 3 \beta_{13} + \beta_{14} + \beta_{15} + 2 \beta_{16} + 5 \beta_{17} - 4 \beta_{19} ) q^{18} \) \( + ( 42 + 69 \beta_{1} + 641 \beta_{2} + 204 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} + 14 \beta_{6} - 6 \beta_{7} - 212 \beta_{8} - 2 \beta_{9} + 12 \beta_{10} - 11 \beta_{11} + 11 \beta_{12} + 10 \beta_{14} + 4 \beta_{15} - 2 \beta_{17} + 4 \beta_{18} - 2 \beta_{19} ) q^{19} \) \( + ( -14 + 9558 \beta_{1} + 38 \beta_{2} - 1572 \beta_{3} - 10 \beta_{5} + 710 \beta_{6} - \beta_{7} + 7 \beta_{8} - 40 \beta_{9} - 21 \beta_{11} + 17 \beta_{12} + \beta_{13} - 3 \beta_{14} - \beta_{15} + 2 \beta_{16} + \beta_{17} + \beta_{18} + 10 \beta_{19} ) q^{20} \) \( + ( -610265 - 7051 \beta_{1} + 1547 \beta_{2} - 500 \beta_{3} - 104 \beta_{5} - 50 \beta_{6} + 10 \beta_{7} + 38 \beta_{8} - 9 \beta_{9} + 6 \beta_{10} + 24 \beta_{11} - 10 \beta_{12} + \beta_{13} - 16 \beta_{14} - 7 \beta_{15} - 8 \beta_{17} + 6 \beta_{18} - 8 \beta_{19} ) q^{21} \) \( + ( -296076 - 996 \beta_{1} - 86 \beta_{2} - 3183 \beta_{3} + 280 \beta_{4} + 10 \beta_{6} - 16 \beta_{7} - 72 \beta_{8} - 40 \beta_{9} + 16 \beta_{10} - 3 \beta_{11} - 45 \beta_{12} - 4 \beta_{13} - 3 \beta_{14} - 4 \beta_{15} + 12 \beta_{17} - 12 \beta_{18} ) q^{22} \) \( + ( 360 - 57326 \beta_{1} + 216 \beta_{2} + 3448 \beta_{3} - 132 \beta_{5} - 4 \beta_{6} + 14 \beta_{7} + 782 \beta_{8} - 4 \beta_{9} - 25 \beta_{11} - 16 \beta_{12} - 16 \beta_{13} + 36 \beta_{14} + 16 \beta_{15} + 24 \beta_{16} - 12 \beta_{17} - 14 \beta_{18} + 4 \beta_{19} ) q^{23} \) \( + ( 606760 + 53316 \beta_{1} + 676 \beta_{2} - 271 \beta_{3} + 426 \beta_{4} - \beta_{5} - 2308 \beta_{6} + 31 \beta_{7} - 45 \beta_{8} - 12 \beta_{9} + 12 \beta_{10} + 62 \beta_{11} + 55 \beta_{12} + 9 \beta_{13} - 5 \beta_{14} - 23 \beta_{15} + 20 \beta_{16} + 11 \beta_{17} - \beta_{18} + 2 \beta_{19} ) q^{24} \) \( + ( -7057393 + 768 \beta_{1} - 7635 \beta_{2} + 2260 \beta_{3} - 13 \beta_{5} - 112 \beta_{6} - 64 \beta_{7} + 1496 \beta_{8} - 26 \beta_{9} + 36 \beta_{10} - 4 \beta_{11} + 64 \beta_{12} - 31 \beta_{13} - 49 \beta_{14} - 5 \beta_{15} - 19 \beta_{17} + 32 \beta_{18} - 13 \beta_{19} ) q^{25} \) \( + ( -20 + 84958 \beta_{1} - 238 \beta_{2} + 23338 \beta_{3} - 36 \beta_{5} - 3848 \beta_{6} - 12 \beta_{7} + 20 \beta_{8} - 60 \beta_{9} - 116 \beta_{11} - 292 \beta_{12} + 6 \beta_{13} - 6 \beta_{14} - 6 \beta_{15} + 44 \beta_{16} + 18 \beta_{17} + 12 \beta_{18} + 24 \beta_{19} ) q^{26} \) \( + ( -2297 - 104386 \beta_{1} - 11709 \beta_{2} - 2077 \beta_{3} + 6 \beta_{4} + 37 \beta_{5} - 12 \beta_{6} + 106 \beta_{7} - 8 \beta_{8} - 12 \beta_{9} + 96 \beta_{10} - 115 \beta_{11} - 85 \beta_{12} + 92 \beta_{14} + 32 \beta_{15} + 25 \beta_{16} - 20 \beta_{17} + 6 \beta_{18} - 20 \beta_{19} ) q^{27} \) \( + ( -180606 + 11472 \beta_{1} - 570 \beta_{2} + 35672 \beta_{3} - 630 \beta_{4} - 2 \beta_{5} + 28 \beta_{6} - 115 \beta_{7} - 149 \beta_{8} - 160 \beta_{9} + 80 \beta_{10} - 9 \beta_{11} + 545 \beta_{12} + 13 \beta_{13} - 5 \beta_{14} + 17 \beta_{15} + 55 \beta_{17} - 53 \beta_{18} - 2 \beta_{19} ) q^{28} \) \( + ( -1212 - 236355 \beta_{1} - 686 \beta_{2} - 756 \beta_{3} - 146 \beta_{5} - 589 \beta_{6} + 24 \beta_{7} - 2568 \beta_{8} - 60 \beta_{9} + 352 \beta_{11} - 24 \beta_{12} - 2 \beta_{13} - 90 \beta_{14} + 2 \beta_{15} - 46 \beta_{17} - 24 \beta_{18} - 18 \beta_{19} ) q^{29} \) \( + ( 3673378 - 4012 \beta_{1} + 1937 \beta_{2} - 9898 \beta_{3} - 360 \beta_{4} - 86 \beta_{5} + 10678 \beta_{6} + 208 \beta_{7} - 40 \beta_{8} - 42 \beta_{9} - 48 \beta_{10} + 333 \beta_{11} - 1039 \beta_{12} + 7 \beta_{13} - 40 \beta_{14} - 79 \beta_{15} - 42 \beta_{16} + \beta_{17} - 12 \beta_{18} + 64 \beta_{19} ) q^{30} \) \( + ( 5478 + 7069 \beta_{1} + 33662 \beta_{2} + 21943 \beta_{3} + 285 \beta_{4} - 14 \beta_{5} + 194 \beta_{6} - 298 \beta_{7} - 2972 \beta_{8} + 82 \beta_{9} - 204 \beta_{10} - 325 \beta_{11} + 388 \beta_{12} - 48 \beta_{13} + 118 \beta_{14} - 20 \beta_{15} - 62 \beta_{17} + 76 \beta_{18} - 14 \beta_{19} ) q^{31} \) \( + ( 280 + 132672 \beta_{1} + 1296 \beta_{2} - 126196 \beta_{3} + 108 \beta_{5} + 13752 \beta_{6} - 52 \beta_{7} + 172 \beta_{8} + 464 \beta_{9} - 476 \beta_{11} + 1492 \beta_{12} + 12 \beta_{13} + 12 \beta_{14} - 12 \beta_{15} - 40 \beta_{16} + 92 \beta_{17} + 52 \beta_{18} - 56 \beta_{19} ) q^{32} \) \( + ( 11499543 + 263019 \beta_{1} + 53840 \beta_{2} - 14861 \beta_{3} + 873 \beta_{5} - 1087 \beta_{6} + 470 \beta_{7} + 158 \beta_{8} + 111 \beta_{9} - 108 \beta_{10} - 467 \beta_{11} - 470 \beta_{12} - 45 \beta_{13} - 75 \beta_{14} + 153 \beta_{15} - 9 \beta_{17} - 22 \beta_{18} + 9 \beta_{19} ) q^{33} \) \( + ( 11107956 - 54608 \beta_{1} - 3868 \beta_{2} - 169848 \beta_{3} - 1044 \beta_{4} + 80 \beta_{5} - 8 \beta_{6} - 488 \beta_{7} + 20 \beta_{8} + 488 \beta_{9} - 100 \beta_{10} - 206 \beta_{11} - 1904 \beta_{12} + 156 \beta_{13} - 4 \beta_{14} - 4 \beta_{15} - 4 \beta_{17} - 76 \beta_{18} + 80 \beta_{19} ) q^{34} \) \( + ( 1021 + 850793 \beta_{1} + 648 \beta_{2} - 26182 \beta_{3} + 1355 \beta_{5} - 648 \beta_{6} - 36 \beta_{7} + 2102 \beta_{8} + 120 \beta_{9} + 1454 \beta_{11} + 312 \beta_{12} + 96 \beta_{13} + 72 \beta_{14} - 96 \beta_{15} - 253 \beta_{16} - 24 \beta_{17} + 36 \beta_{18} - 120 \beta_{19} ) q^{35} \) \( + ( 14905089 + 111286 \beta_{1} + 5403 \beta_{2} - 55380 \beta_{3} - 2256 \beta_{4} + 110 \beta_{5} - 22018 \beta_{6} + 779 \beta_{7} - 133 \beta_{8} + 136 \beta_{9} - 240 \beta_{10} + 1867 \beta_{11} + 3061 \beta_{12} - 99 \beta_{13} - 23 \beta_{14} + 163 \beta_{15} - 166 \beta_{16} + 77 \beta_{17} - 51 \beta_{18} + 50 \beta_{19} ) q^{36} \) \( + ( -2249083 + 10996 \beta_{1} - 115099 \beta_{2} + 34526 \beta_{3} + 58 \beta_{5} - 37 \beta_{6} - 832 \beta_{7} + 172 \beta_{8} + 505 \beta_{9} - 274 \beta_{10} - 355 \beta_{11} + 832 \beta_{12} + 195 \beta_{13} + 34 \beta_{14} + 79 \beta_{15} - 26 \beta_{17} - 32 \beta_{18} + 58 \beta_{19} ) q^{37} \) \( + ( 24 + 141856 \beta_{1} - 5288 \beta_{2} + 436947 \beta_{3} + 676 \beta_{5} - 23718 \beta_{6} - 64 \beta_{7} + 194 \beta_{8} - 148 \beta_{9} - 2257 \beta_{11} - 5113 \beta_{12} + 174 \beta_{13} - 33 \beta_{14} - 174 \beta_{15} - 420 \beta_{16} - 46 \beta_{17} + 64 \beta_{18} - 128 \beta_{19} ) q^{38} \) \( + ( -27448 - 1992112 \beta_{1} - 139337 \beta_{2} - 41152 \beta_{3} - 870 \beta_{4} - 1738 \beta_{5} + 543 \beta_{6} + 1154 \beta_{7} - 37 \beta_{8} + 39 \beta_{9} - 582 \beta_{10} - 2927 \beta_{11} - 922 \beta_{12} - 144 \beta_{13} - 227 \beta_{14} - 50 \beta_{15} - 277 \beta_{16} + 11 \beta_{17} - 33 \beta_{18} + 155 \beta_{19} ) q^{39} \) \( + ( -21984696 + 234204 \beta_{1} + 200 \beta_{2} + 729816 \beta_{3} + 6068 \beta_{4} + 12 \beta_{5} + 48 \beta_{6} - 1334 \beta_{7} - 1374 \beta_{8} + 504 \beta_{9} - 712 \beta_{10} - 904 \beta_{11} + 9634 \beta_{12} - 66 \beta_{13} + 66 \beta_{14} - 90 \beta_{15} - 174 \beta_{17} + 162 \beta_{18} + 12 \beta_{19} ) q^{40} \) \( + ( 4512 - 2156910 \beta_{1} + 2026 \beta_{2} - 9500 \beta_{3} - 1292 \beta_{5} + 3358 \beta_{6} - 138 \beta_{7} + 8838 \beta_{8} + 1020 \beta_{9} + 4949 \beta_{11} + 138 \beta_{12} + 40 \beta_{13} + 348 \beta_{14} - 40 \beta_{15} + 236 \beta_{17} + 138 \beta_{18} - 108 \beta_{19} ) q^{41} \) \( + ( 14091001 - 611337 \beta_{1} - 5091 \beta_{2} + 438 \beta_{3} + 6891 \beta_{4} + 1576 \beta_{5} + 13544 \beta_{6} + 1728 \beta_{7} + 60 \beta_{8} + 144 \beta_{9} + 285 \beta_{10} + 7194 \beta_{11} - 13496 \beta_{12} - 72 \beta_{13} + 224 \beta_{14} + 608 \beta_{15} + 376 \beta_{16} + 16 \beta_{17} - 52 \beta_{18} - 128 \beta_{19} ) q^{42} \) \( + ( 47466 - 2135 \beta_{1} + 211017 \beta_{2} - 7760 \beta_{3} - 3526 \beta_{4} + 62 \beta_{5} - 1010 \beta_{6} - 1606 \beta_{7} + 15500 \beta_{8} - 514 \beta_{9} + 1356 \beta_{10} + 165 \beta_{11} + 1043 \beta_{12} + 288 \beta_{13} - 598 \beta_{14} + 164 \beta_{15} + 350 \beta_{17} - 412 \beta_{18} + 62 \beta_{19} ) q^{43} \) \( + ( -2694 - 704930 \beta_{1} + 15554 \beta_{2} - 1278420 \beta_{3} - 582 \beta_{5} + 2998 \beta_{6} + 215 \beta_{7} - 3841 \beta_{8} - 2032 \beta_{9} - 9249 \beta_{11} + 15497 \beta_{12} + 85 \beta_{13} - 135 \beta_{14} - 85 \beta_{15} + 330 \beta_{16} - 515 \beta_{17} - 215 \beta_{18} + 58 \beta_{19} ) q^{44} \) \( + ( 51069034 + 6278151 \beta_{1} + 288930 \beta_{2} - 60610 \beta_{3} - 1146 \beta_{5} + 13013 \beta_{6} + 1782 \beta_{7} - 1338 \beta_{8} - 722 \beta_{9} + 792 \beta_{10} - 11643 \beta_{11} - 1782 \beta_{12} + 78 \beta_{13} + 582 \beta_{14} - 870 \beta_{15} + 42 \beta_{17} - 6 \beta_{18} + 294 \beta_{19} ) q^{45} \) \( + ( -119643400 - 492544 \beta_{1} - 34956 \beta_{2} - 1541110 \beta_{3} - 6464 \beta_{4} - 512 \beta_{5} - 108 \beta_{6} - 2304 \beta_{7} + 1096 \beta_{8} - 2496 \beta_{9} + 256 \beta_{10} - 1626 \beta_{11} - 18418 \beta_{12} - 1120 \beta_{13} + 10 \beta_{14} - 96 \beta_{15} - 224 \beta_{17} + 736 \beta_{18} - 512 \beta_{19} ) q^{46} \) \( + ( -14554 + 9582606 \beta_{1} - 9012 \beta_{2} + 45292 \beta_{3} - 3758 \beta_{5} + 3656 \beta_{6} - 124 \beta_{7} - 30864 \beta_{8} - 568 \beta_{9} + 16514 \beta_{11} - 1828 \beta_{12} - 160 \beta_{13} - 1224 \beta_{14} + 160 \beta_{15} + 1498 \beta_{16} + 408 \beta_{17} + 124 \beta_{18} + 568 \beta_{19} ) q^{47} \) \( + ( 34460572 + 537750 \beta_{1} + 53636 \beta_{2} - 200824 \beta_{3} - 5442 \beta_{4} - 722 \beta_{5} + 47912 \beta_{6} + 2083 \beta_{7} + 1875 \beta_{8} - 1188 \beta_{9} + 2076 \beta_{10} + 20818 \beta_{11} + 25119 \beta_{12} + 341 \beta_{13} + 467 \beta_{14} - 343 \beta_{15} + 680 \beta_{16} - 797 \beta_{17} + 267 \beta_{18} - 230 \beta_{19} ) q^{48} \) \( + ( -50980915 + 27184 \beta_{1} - 271961 \beta_{2} + 80212 \beta_{3} - 195 \beta_{5} + 3588 \beta_{6} - 1984 \beta_{7} - 45000 \beta_{8} - 4122 \beta_{9} + 1092 \beta_{10} - 2192 \beta_{11} + 1984 \beta_{12} - 741 \beta_{13} + 1185 \beta_{14} - 351 \beta_{15} + 867 \beta_{17} - 672 \beta_{18} - 195 \beta_{19} ) q^{49} \) \( + ( 468 - 6040571 \beta_{1} - 37266 \beta_{2} + 3184374 \beta_{3} - 6300 \beta_{5} + 74696 \beta_{6} + 812 \beta_{7} - 2164 \beta_{8} + 3260 \beta_{9} - 28668 \beta_{11} - 36924 \beta_{12} - 1382 \beta_{13} + 486 \beta_{14} + 1382 \beta_{15} + 2196 \beta_{16} - 242 \beta_{17} - 812 \beta_{18} + 40 \beta_{19} ) q^{50} \) \( + ( -44939 - 14893905 \beta_{1} - 229088 \beta_{2} + 17294 \beta_{3} + 10548 \beta_{4} + 8535 \beta_{5} - 2486 \beta_{6} + 1576 \beta_{7} + 848 \beta_{8} - 198 \beta_{9} + 1260 \beta_{10} - 28540 \beta_{11} - 4894 \beta_{12} + 864 \beta_{13} - 1506 \beta_{14} - 444 \beta_{15} + 1749 \beta_{16} + 642 \beta_{17} - 122 \beta_{18} - 222 \beta_{19} ) q^{51} \) \( + ( -197953406 + 1056848 \beta_{1} + 41302 \beta_{2} + 3296160 \beta_{3} - 7136 \beta_{4} - 160 \beta_{5} - 1952 \beta_{6} - 1808 \beta_{7} + 23840 \beta_{8} + 1184 \beta_{9} + 3680 \beta_{10} - 536 \beta_{11} + 44464 \beta_{12} - 96 \beta_{13} - 544 \beta_{14} + 224 \beta_{15} - 640 \beta_{17} + 800 \beta_{18} - 160 \beta_{19} ) q^{52} \) \( + ( 15936 - 18260553 \beta_{1} + 14764 \beta_{2} - 80328 \beta_{3} + 8920 \beta_{5} - 11291 \beta_{6} - 172 \beta_{7} + 41780 \beta_{8} - 7672 \beta_{9} + 40374 \beta_{11} + 172 \beta_{12} + 496 \beta_{13} + 1032 \beta_{14} - 496 \beta_{15} - 152 \beta_{17} + 172 \beta_{18} + 1432 \beta_{19} ) q^{53} \) \( + ( -212475928 - 20876 \beta_{1} - 96660 \beta_{2} - 85739 \beta_{3} - 14424 \beta_{4} - 12556 \beta_{5} - 181530 \beta_{6} + 272 \beta_{7} + 1619 \beta_{8} - 444 \beta_{9} - 528 \beta_{10} + 52135 \beta_{11} - 48947 \beta_{12} - 342 \beta_{13} - 299 \beta_{14} - 1634 \beta_{15} - 1780 \beta_{16} - 754 \beta_{17} + 876 \beta_{18} - 1024 \beta_{19} ) q^{54} \) \( + ( 18150 + 69730 \beta_{1} + 28430 \beta_{2} + 216950 \beta_{3} + 22930 \beta_{4} + 370 \beta_{5} - 2110 \beta_{6} - 170 \beta_{7} + 31780 \beta_{8} + 850 \beta_{9} - 3660 \beta_{10} + 795 \beta_{11} + 660 \beta_{12} - 240 \beta_{13} - 1610 \beta_{14} - 980 \beta_{15} + 130 \beta_{17} - 500 \beta_{18} + 370 \beta_{19} ) q^{55} \) \( + ( 16920 - 1722972 \beta_{1} + 68832 \beta_{2} - 4874698 \beta_{3} + 974 \beta_{5} - 232552 \beta_{6} + 636 \beta_{7} + 34140 \beta_{8} + 1968 \beta_{9} - 69692 \beta_{11} + 60772 \beta_{12} - 1204 \beta_{13} + 1452 \beta_{14} + 1204 \beta_{15} - 1320 \beta_{16} - 68 \beta_{17} - 636 \beta_{18} + 360 \beta_{19} ) q^{56} \) \( + ( 71889345 + 37489209 \beta_{1} - 216807 \beta_{2} + 216213 \beta_{3} - 10518 \beta_{5} - 71829 \beta_{6} - 1230 \beta_{7} - 1854 \beta_{8} + 2211 \beta_{9} - 2736 \beta_{10} - 74517 \beta_{11} + 1230 \beta_{12} + 894 \beta_{13} + 1434 \beta_{14} + 1842 \beta_{15} + 1686 \beta_{17} - 18 \beta_{18} - 1590 \beta_{19} ) q^{57} \) \( + ( 483574445 - 1705503 \beta_{1} - 172923 \beta_{2} - 5317782 \beta_{3} + 33919 \beta_{4} + 480 \beta_{5} - 560 \beta_{6} + 2448 \beta_{7} + 3192 \beta_{8} + 4976 \beta_{9} + 177 \beta_{10} + 2732 \beta_{11} - 69792 \beta_{12} + 2984 \beta_{13} + 744 \beta_{14} + 2024 \beta_{15} - 24 \beta_{17} - 456 \beta_{18} + 480 \beta_{19} ) q^{58} \) \( + ( 3596 + 51970723 \beta_{1} + 1911 \beta_{2} - 336084 \beta_{3} - 6428 \beta_{5} + 656 \beta_{6} + 8 \beta_{7} + 7008 \beta_{8} - 112 \beta_{9} + 98692 \beta_{11} + 2711 \beta_{12} - 64 \beta_{13} - 144 \beta_{14} + 64 \beta_{15} - 5132 \beta_{16} + 48 \beta_{17} - 8 \beta_{18} + 112 \beta_{19} ) q^{59} \) \( + ( 284402990 + 3630138 \beta_{1} - 10602 \beta_{2} - 50844 \beta_{3} + 48984 \beta_{4} + 1198 \beta_{5} + 292394 \beta_{6} - 4587 \beta_{7} + 301 \beta_{8} + 5856 \beta_{9} - 9840 \beta_{10} + 112713 \beta_{11} + 78643 \beta_{12} - 513 \beta_{13} - 2509 \beta_{14} - 391 \beta_{15} - 938 \beta_{16} + 655 \beta_{17} + 419 \beta_{18} - 722 \beta_{19} ) q^{60} \) \( + ( 343849569 - 83452 \beta_{1} + 811797 \beta_{2} - 238074 \beta_{3} + 1894 \beta_{5} - 4633 \beta_{6} + 5440 \beta_{7} + 52540 \beta_{8} + 17797 \beta_{9} - 2026 \beta_{10} + 3105 \beta_{11} - 5440 \beta_{12} + 2907 \beta_{13} - 1538 \beta_{14} - 881 \beta_{15} - 2438 \beta_{17} + 544 \beta_{18} + 1894 \beta_{19} ) q^{61} \) \( + ( -2290 + 1998936 \beta_{1} - 69853 \beta_{2} + 6262612 \beta_{3} + 33062 \beta_{5} + 343670 \beta_{6} - 1216 \beta_{7} + 5478 \beta_{8} - 11278 \beta_{9} - 139825 \beta_{11} - 72131 \beta_{12} + 2669 \beta_{13} - 1218 \beta_{14} - 2669 \beta_{15} - 6182 \beta_{16} - 237 \beta_{17} + 1216 \beta_{18} + 640 \beta_{19} ) q^{62} \) \( + ( 205154 - 86547197 \beta_{1} + 1051578 \beta_{2} + 24535 \beta_{3} - 64419 \beta_{4} - 4846 \beta_{5} + 318 \beta_{6} - 8416 \beta_{7} - 3154 \beta_{8} - 114 \beta_{9} + 2532 \beta_{10} - 164240 \beta_{11} + 21040 \beta_{12} + 3034 \beta_{14} + 844 \beta_{15} - 6580 \beta_{16} - 730 \beta_{17} + 570 \beta_{18} - 730 \beta_{19} ) q^{63} \) \( + ( -72689552 + 1877768 \beta_{1} + 65392 \beta_{2} + 5869264 \beta_{3} - 54520 \beta_{4} - 392 \beta_{5} + 9920 \beta_{6} + 11012 \beta_{7} - 128156 \beta_{8} - 8816 \beta_{9} - 11952 \beta_{10} + 3608 \beta_{11} + 69108 \beta_{12} + 540 \beta_{13} + 4324 \beta_{14} + 1324 \beta_{15} + 2596 \beta_{17} - 2204 \beta_{18} - 392 \beta_{19} ) q^{64} \) \( + ( -49260 - 106360250 \beta_{1} - 50780 \beta_{2} - 412320 \beta_{3} - 3350 \beta_{5} + 127190 \beta_{6} + 1310 \beta_{7} - 136690 \beta_{8} + 30800 \beta_{9} + 203325 \beta_{11} - 1310 \beta_{12} - 3890 \beta_{13} - 3030 \beta_{14} + 3890 \beta_{15} + 1270 \beta_{17} - 1310 \beta_{18} - 3830 \beta_{19} ) q^{65} \) \( + ( -547999329 + 11585803 \beta_{1} + 265050 \beta_{2} + 272447 \beta_{3} - 47445 \beta_{4} + 54870 \beta_{5} - 238496 \beta_{6} - 12882 \beta_{7} - 1328 \beta_{8} + 5982 \beta_{9} - 1935 \beta_{10} + 206749 \beta_{11} - 69722 \beta_{12} + 2317 \beta_{13} - 441 \beta_{14} + 327 \beta_{15} + 4070 \beta_{16} + 915 \beta_{17} - 2052 \beta_{18} + 3876 \beta_{19} ) q^{66} \) \( + ( -432636 - 391477 \beta_{1} - 1867793 \beta_{2} - 1201422 \beta_{3} - 97496 \beta_{4} - 1140 \beta_{5} + 11244 \beta_{6} + 14244 \beta_{7} - 171432 \beta_{8} + 2124 \beta_{9} - 2952 \beta_{10} - 1518 \beta_{11} - 17741 \beta_{12} - 1632 \beta_{13} + 7332 \beta_{14} + 648 \beta_{15} - 2772 \beta_{17} + 3912 \beta_{18} - 1140 \beta_{19} ) q^{67} \) \( + ( -78392 + 9391512 \beta_{1} + 8888 \beta_{2} - 5426224 \beta_{3} + 2712 \beta_{5} - 158248 \beta_{6} - 3108 \beta_{7} - 182788 \beta_{8} + 14880 \beta_{9} - 252756 \beta_{11} + 64996 \beta_{12} + 1924 \beta_{13} - 7116 \beta_{14} - 1924 \beta_{15} + 1544 \beta_{16} + 4292 \beta_{17} + 3108 \beta_{18} - 1944 \beta_{19} ) q^{68} \) \( + ( -665933610 + 135346362 \beta_{1} - 2496572 \beta_{2} + 1311494 \beta_{3} + 36324 \beta_{5} + 344950 \beta_{6} - 17570 \beta_{7} + 9694 \beta_{8} - 3390 \beta_{9} + 2160 \beta_{10} - 298111 \beta_{11} + 17570 \beta_{12} - 720 \beta_{13} - 9156 \beta_{14} - 1440 \beta_{15} - 7380 \beta_{17} - 14 \beta_{18} + 4356 \beta_{19} ) q^{69} \) \( + ( 1759616192 - 1600612 \beta_{1} + 886780 \beta_{2} - 4967070 \beta_{3} + 1720 \beta_{4} + 3072 \beta_{5} + 2668 \beta_{6} + 21296 \beta_{7} - 19764 \beta_{8} + 7544 \beta_{9} - 2864 \beta_{10} + 8196 \beta_{11} - 77026 \beta_{12} - 1396 \beta_{13} - 2994 \beta_{14} - 7540 \beta_{15} + 1116 \beta_{17} - 4188 \beta_{18} + 3072 \beta_{19} ) q^{70} \) \( + ( 119030 + 180819464 \beta_{1} + 81868 \beta_{2} - 1312780 \beta_{3} + 53398 \beta_{5} - 33644 \beta_{6} + 1402 \beta_{7} + 255350 \beta_{8} + 5140 \beta_{9} + 344029 \beta_{11} + 11540 \beta_{12} + 1168 \beta_{13} + 11916 \beta_{14} - 1168 \beta_{15} + 8074 \beta_{16} - 3972 \beta_{17} - 1402 \beta_{18} - 5140 \beta_{19} ) q^{71} \) \( + ( 1521890600 + 14689734 \beta_{1} + 56040 \beta_{2} - 652511 \beta_{3} - 33372 \beta_{4} + 1785 \beta_{5} - 128912 \beta_{6} - 23166 \beta_{7} - 29190 \beta_{8} - 7912 \beta_{9} + 25560 \beta_{10} + 346776 \beta_{11} + 55674 \beta_{12} - 1626 \beta_{13} + 10074 \beta_{14} + 3822 \beta_{15} - 3456 \beta_{16} + 6186 \beta_{17} - 3846 \beta_{18} + 3612 \beta_{19} ) q^{72} \) \( + ( -620604720 - 273544 \beta_{1} + 2985056 \beta_{2} - 902484 \beta_{3} - 8062 \beta_{5} - 22006 \beta_{6} + 24768 \beta_{7} + 308216 \beta_{8} - 45350 \beta_{9} - 1236 \beta_{10} + 25582 \beta_{11} - 24768 \beta_{12} - 7444 \beta_{13} - 7606 \beta_{14} + 8680 \beta_{15} + 1886 \beta_{17} + 6176 \beta_{18} - 8062 \beta_{19} ) q^{73} \) \( + ( 1652 - 1842346 \beta_{1} + 574 \beta_{2} + 1098406 \beta_{3} - 110332 \beta_{5} - 345528 \beta_{6} - 3860 \beta_{7} + 5324 \beta_{8} + 220 \beta_{9} - 437708 \beta_{11} - 1724 \beta_{12} + 3626 \beta_{13} - 810 \beta_{14} - 3626 \beta_{15} + 4660 \beta_{16} + 4094 \beta_{17} + 3860 \beta_{18} + 296 \beta_{19} ) q^{74} \) \( + ( 673469 - 272954053 \beta_{1} + 3408412 \beta_{2} + 5000827 \beta_{3} + 239790 \beta_{4} - 59125 \beta_{5} + 22638 \beta_{6} - 25174 \beta_{7} - 11902 \beta_{8} + 2910 \beta_{9} - 18420 \beta_{10} - 512819 \beta_{11} - 2842 \beta_{12} - 9216 \beta_{13} + 11818 \beta_{14} + 3076 \beta_{15} + 12653 \beta_{16} - 5986 \beta_{17} + 84 \beta_{18} + 3230 \beta_{19} ) q^{75} \) \( + ( -3004179250 - 1654952 \beta_{1} + 291402 \beta_{2} - 5159048 \beta_{3} + 105238 \beta_{4} + 3106 \beta_{5} - 30092 \beta_{6} + 28755 \beta_{7} + 442429 \beta_{8} + 5360 \beta_{9} + 23456 \beta_{10} + 30101 \beta_{11} - 82561 \beta_{12} - 149 \beta_{13} - 19379 \beta_{14} - 6361 \beta_{15} + 2305 \beta_{17} - 5411 \beta_{18} + 3106 \beta_{19} ) q^{76} \) \( + ( -136080 - 267347602 \beta_{1} - 30788 \beta_{2} - 1116680 \beta_{3} - 85328 \beta_{5} - 817718 \beta_{6} + 124 \beta_{7} - 222308 \beta_{8} - 72824 \beta_{9} + 563826 \beta_{11} - 124 \beta_{12} + 11512 \beta_{13} - 11424 \beta_{14} - 11512 \beta_{15} - 11760 \beta_{17} - 124 \beta_{18} + 752 \beta_{19} ) q^{77} \) \( + ( -4052318260 + 5772 \beta_{1} - 1902038 \beta_{2} + 350786 \beta_{3} + 148344 \beta_{4} - 148868 \beta_{5} + 1033080 \beta_{6} - 27632 \beta_{7} - 17938 \beta_{8} - 28500 \beta_{9} + 10896 \beta_{10} + 563858 \beta_{11} + 120552 \beta_{12} - 2034 \beta_{13} + 914 \beta_{14} + 8330 \beta_{15} + 772 \beta_{16} + 7450 \beta_{17} - 2748 \beta_{18} + 1600 \beta_{19} ) q^{78} \) \( + ( -757566 + 643457 \beta_{1} - 3412554 \beta_{2} + 1962119 \beta_{3} + 297877 \beta_{4} - 3066 \beta_{5} + 12726 \beta_{6} + 28690 \beta_{7} - 189588 \beta_{8} - 11802 \beta_{9} + 44604 \beta_{10} + 13001 \beta_{11} + 13912 \beta_{12} + 4368 \beta_{13} + 10962 \beta_{14} + 10500 \beta_{15} + 1302 \beta_{17} + 1764 \beta_{18} - 3066 \beta_{19} ) q^{79} \) \( + ( 280016 - 17032672 \beta_{1} + 41248 \beta_{2} + 15588728 \beta_{3} - 14920 \beta_{5} + 1402896 \beta_{6} - 2776 \beta_{7} + 656232 \beta_{8} - 48160 \beta_{9} - 680136 \beta_{11} - 158568 \beta_{12} + 8936 \beta_{13} + 22632 \beta_{14} - 8936 \beta_{15} + 8272 \beta_{16} - 3384 \beta_{17} + 2776 \beta_{18} + 2800 \beta_{19} ) q^{80} \) \( + ( -2123845653 + 342236538 \beta_{1} - 3585447 \beta_{2} + 2437842 \beta_{3} + 16515 \beta_{5} - 1407192 \beta_{6} - 27000 \beta_{7} + 31464 \beta_{8} + 1266 \beta_{9} + 16632 \beta_{10} - 711954 \beta_{11} + 27000 \beta_{12} - 19017 \beta_{13} - 2817 \beta_{14} + 2385 \beta_{15} + 5661 \beta_{17} + 2520 \beta_{18} - 7677 \beta_{19} ) q^{81} \) \( + ( 4408466582 + 5388158 \beta_{1} - 2476506 \beta_{2} + 16784396 \beta_{3} - 173166 \beta_{4} - 6144 \beta_{5} + 3328 \beta_{6} + 24192 \beta_{7} - 16128 \beta_{8} - 51712 \beta_{9} + 6766 \beta_{10} + 13376 \beta_{11} + 208896 \beta_{12} - 11008 \beta_{13} - 384 \beta_{14} + 1280 \beta_{15} + 5376 \beta_{17} + 768 \beta_{18} - 6144 \beta_{19} ) q^{82} \) \( + ( -26079 + 400912732 \beta_{1} - 79025 \beta_{2} + 4749302 \beta_{3} - 72729 \beta_{5} + 29624 \beta_{6} + 2300 \beta_{7} - 52066 \beta_{8} - 5704 \beta_{9} + 762174 \beta_{11} - 74241 \beta_{12} - 5152 \beta_{13} - 1656 \beta_{14} + 5152 \beta_{15} + 8415 \beta_{16} + 552 \beta_{17} - 2300 \beta_{18} + 5704 \beta_{19} ) q^{83} \) \( + ( 6275029520 + 14330850 \beta_{1} - 726512 \beta_{2} + 512820 \beta_{3} - 244416 \beta_{4} - 9886 \beta_{5} - 1841966 \beta_{6} - 17731 \beta_{7} + 68661 \beta_{8} - 40056 \beta_{9} - 22848 \beta_{10} + 839057 \beta_{11} - 324813 \beta_{12} + 10435 \beta_{13} - 35081 \beta_{14} - 6851 \beta_{15} + 17158 \beta_{16} - 10045 \beta_{17} + 867 \beta_{18} + 4382 \beta_{19} ) q^{84} \) \( + ( 857196228 - 257808 \beta_{1} + 2471740 \beta_{2} - 726200 \beta_{3} + 9928 \beta_{5} + 16532 \beta_{6} + 11904 \beta_{7} - 237296 \beta_{8} + 60796 \beta_{9} + 16904 \beta_{10} - 13076 \beta_{11} - 11904 \beta_{12} + 1476 \beta_{13} + 2344 \beta_{14} - 18380 \beta_{15} - 6536 \beta_{17} - 3392 \beta_{18} + 9928 \beta_{19} ) q^{85} \) \( + ( 14784 - 9625536 \beta_{1} + 353852 \beta_{2} - 30783245 \beta_{3} + 238636 \beta_{5} - 2384382 \beta_{6} + 6592 \beta_{7} - 40406 \beta_{8} + 76804 \beta_{9} - 891373 \beta_{11} + 378259 \beta_{12} - 15894 \beta_{13} + 6711 \beta_{14} + 15894 \beta_{15} + 28884 \beta_{16} + 2710 \beta_{17} - 6592 \beta_{18} - 5248 \beta_{19} ) q^{86} \) \( + ( 137466 - 465634689 \beta_{1} + 555729 \beta_{2} - 1488063 \beta_{3} - 577635 \beta_{4} + 101732 \beta_{5} - 40943 \beta_{6} - 6612 \beta_{7} + 127555 \beta_{8} - 2967 \beta_{9} + 21918 \beta_{10} - 886170 \beta_{11} + 88766 \beta_{12} + 12096 \beta_{13} - 15965 \beta_{14} - 4790 \beta_{15} + 3815 \beta_{16} + 7757 \beta_{17} - 2417 \beta_{18} - 4339 \beta_{19} ) q^{87} \) \( + ( -8304716296 - 11873052 \beta_{1} - 5384 \beta_{2} - 37018072 \beta_{3} + 312540 \beta_{4} + 2532 \beta_{5} + 86960 \beta_{6} - 4642 \beta_{7} - 1281930 \beta_{8} + 53128 \beta_{9} - 15736 \beta_{10} - 60352 \beta_{11} - 479002 \beta_{12} + 1258 \beta_{13} + 48534 \beta_{14} - 3806 \beta_{15} - 13914 \beta_{17} + 11382 \beta_{18} + 2532 \beta_{19} ) q^{88} \) \( + ( 310590 - 470615554 \beta_{1} + 122507 \beta_{2} - 1893790 \beta_{3} + 133121 \beta_{5} + 2789892 \beta_{6} - 2704 \beta_{7} + 581792 \beta_{8} + 91286 \beta_{9} + 942642 \beta_{11} + 2704 \beta_{12} - 14047 \beta_{13} + 24597 \beta_{14} + 14047 \beta_{15} + 19455 \beta_{17} + 2704 \beta_{18} + 2305 \beta_{19} ) q^{89} \) \( + ( -12897582461 + 48942487 \beta_{1} + 6610533 \beta_{2} - 5974828 \beta_{3} + 176997 \beta_{4} + 259220 \beta_{5} + 1935712 \beta_{6} + 9284 \beta_{7} + 6032 \beta_{8} + 22820 \beta_{9} - 8637 \beta_{10} + 1041214 \beta_{11} + 638452 \beta_{12} - 10314 \beta_{13} - 1598 \beta_{14} - 19646 \beta_{15} - 30028 \beta_{16} - 9814 \beta_{17} + 12216 \beta_{18} - 21640 \beta_{19} ) q^{90} \) \( + ( 846894 - 3787504 \beta_{1} + 2467360 \beta_{2} - 11691890 \beta_{3} - 683338 \beta_{4} + 9354 \beta_{5} - 54918 \beta_{6} - 19810 \beta_{7} + 827844 \beta_{8} + 19914 \beta_{9} - 87804 \beta_{10} + 14311 \beta_{11} - 146322 \beta_{12} - 5280 \beta_{13} - 41490 \beta_{14} - 23988 \beta_{15} + 4074 \beta_{17} - 13428 \beta_{18} + 9354 \beta_{19} ) q^{91} \) \( + ( -770428 - 100588692 \beta_{1} - 1161836 \beta_{2} + 59571704 \beta_{3} + 17540 \beta_{5} + 1598172 \beta_{6} + 18118 \beta_{7} - 1676458 \beta_{8} - 8672 \beta_{9} - 1023402 \beta_{11} - 705030 \beta_{12} - 24270 \beta_{13} - 63078 \beta_{14} + 24270 \beta_{15} - 36188 \beta_{16} - 11966 \beta_{17} - 18118 \beta_{18} + 11396 \beta_{19} ) q^{92} \) \( + ( 5720508485 + 533293489 \beta_{1} + 5632705 \beta_{2} + 581558 \beta_{3} - 209314 \beta_{5} + 4138778 \beta_{6} + 37712 \beta_{7} + 316 \beta_{8} + 9777 \beta_{9} - 55986 \beta_{10} - 1041075 \beta_{11} - 37712 \beta_{12} + 58007 \beta_{13} + 40534 \beta_{14} - 2021 \beta_{15} + 10946 \beta_{17} - 4272 \beta_{18} + 3422 \beta_{19} ) q^{93} \) \( + ( 19599834272 + 20790600 \beta_{1} + 8242488 \beta_{2} + 64869524 \beta_{3} - 45168 \beta_{4} - 5120 \beta_{5} - 17704 \beta_{6} - 53600 \beta_{7} + 115912 \beta_{8} + 42256 \beta_{9} - 672 \beta_{10} - 6240 \beta_{11} + 877068 \beta_{12} + 36072 \beta_{13} + 18748 \beta_{14} + 46312 \beta_{15} - 8888 \beta_{17} + 14008 \beta_{18} - 5120 \beta_{19} ) q^{94} \) \( + ( -531104 + 639309198 \beta_{1} - 76832 \beta_{2} - 26240072 \beta_{3} - 99460 \beta_{5} + 98372 \beta_{6} - 12526 \beta_{7} - 1149278 \beta_{8} - 12220 \beta_{9} + 1120169 \beta_{11} + 265032 \beta_{12} + 6416 \beta_{13} - 55908 \beta_{14} - 6416 \beta_{15} - 65248 \beta_{16} + 18636 \beta_{17} + 12526 \beta_{18} + 12220 \beta_{19} ) q^{95} \) \( + ( 22343702024 + 34474056 \beta_{1} + 55168 \beta_{2} + 160508 \beta_{3} + 221832 \beta_{4} + 29700 \beta_{5} - 1175816 \beta_{6} + 64528 \beta_{7} - 32880 \beta_{8} + 160224 \beta_{9} - 66096 \beta_{10} + 1064348 \beta_{11} - 904696 \beta_{12} - 9432 \beta_{13} + 87024 \beta_{14} - 11232 \beta_{15} - 18216 \beta_{16} - 16128 \beta_{17} + 21784 \beta_{18} - 25920 \beta_{19} ) q^{96} \) \( + ( 3785463654 + 1017880 \beta_{1} - 10747697 \beta_{2} + 3226360 \beta_{3} + 11259 \beta_{5} + 108258 \beta_{6} - 71296 \beta_{7} - 1445280 \beta_{8} + 16548 \beta_{9} - 43032 \beta_{10} - 58598 \beta_{11} + 71296 \beta_{12} + 32775 \beta_{13} + 49047 \beta_{14} + 10257 \beta_{15} + 20421 \beta_{17} - 31680 \beta_{18} + 11259 \beta_{19} ) q^{97} \) \( + ( -38148 - 78531389 \beta_{1} + 1089546 \beta_{2} - 86287838 \beta_{3} - 284756 \beta_{5} + 41304 \beta_{6} + 10884 \beta_{7} + 18852 \beta_{8} - 123468 \beta_{9} - 1058996 \beta_{11} + 1085772 \beta_{12} + 7566 \beta_{13} - 8718 \beta_{14} - 7566 \beta_{15} - 103620 \beta_{16} - 29334 \beta_{17} - 10884 \beta_{18} + 3192 \beta_{19} ) q^{98} \) \( + ( -2732786 - 483094015 \beta_{1} - 13419189 \beta_{2} - 9275086 \beta_{3} + 924882 \beta_{4} + 186286 \beta_{5} - 54048 \beta_{6} + 90766 \beta_{7} - 355082 \beta_{8} - 11712 \beta_{9} + 64536 \beta_{10} - 892945 \beta_{11} - 363337 \beta_{12} + 36288 \beta_{13} - 57952 \beta_{14} - 14776 \beta_{15} - 86570 \beta_{16} + 26488 \beta_{17} + 6126 \beta_{18} - 9800 \beta_{19} ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(20q \) \(\mathstrut +\mathstrut 1976q^{4} \) \(\mathstrut +\mathstrut 13128q^{6} \) \(\mathstrut +\mathstrut 103620q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(20q \) \(\mathstrut +\mathstrut 1976q^{4} \) \(\mathstrut +\mathstrut 13128q^{6} \) \(\mathstrut +\mathstrut 103620q^{9} \) \(\mathstrut +\mathstrut 202000q^{10} \) \(\mathstrut +\mathstrut 1064760q^{12} \) \(\mathstrut +\mathstrut 1543864q^{13} \) \(\mathstrut +\mathstrut 3482912q^{16} \) \(\mathstrut +\mathstrut 2557392q^{18} \) \(\mathstrut -\mathstrut 12211752q^{21} \) \(\mathstrut -\mathstrut 5920464q^{22} \) \(\mathstrut +\mathstrut 12133152q^{24} \) \(\mathstrut -\mathstrut 141128700q^{25} \) \(\mathstrut -\mathstrut 3604848q^{28} \) \(\mathstrut +\mathstrut 73456080q^{30} \) \(\mathstrut +\mathstrut 229769760q^{33} \) \(\mathstrut +\mathstrut 222167104q^{34} \) \(\mathstrut +\mathstrut 298087896q^{36} \) \(\mathstrut -\mathstrut 44517800q^{37} \) \(\mathstrut -\mathstrut 439643840q^{40} \) \(\mathstrut +\mathstrut 281776944q^{42} \) \(\mathstrut +\mathstrut 1020227520q^{45} \) \(\mathstrut -\mathstrut 2392795680q^{46} \) \(\mathstrut +\mathstrut 689090592q^{48} \) \(\mathstrut -\mathstrut 1018138084q^{49} \) \(\mathstrut -\mathstrut 3959246384q^{52} \) \(\mathstrut -\mathstrut 4249352520q^{54} \) \(\mathstrut +\mathstrut 1438636392q^{57} \) \(\mathstrut +\mathstrut 9671853040q^{58} \) \(\mathstrut +\mathstrut 5688375360q^{60} \) \(\mathstrut +\mathstrut 6873199864q^{61} \) \(\mathstrut -\mathstrut 1452752512q^{64} \) \(\mathstrut -\mathstrut 10961242896q^{66} \) \(\mathstrut -\mathstrut 13308470976q^{69} \) \(\mathstrut +\mathstrut 35188514400q^{70} \) \(\mathstrut +\mathstrut 30438127680q^{72} \) \(\mathstrut -\mathstrut 12426469112q^{73} \) \(\mathstrut -\mathstrut 60088673808q^{76} \) \(\mathstrut -\mathstrut 81037845456q^{78} \) \(\mathstrut -\mathstrut 42462874764q^{81} \) \(\mathstrut +\mathstrut 88180337440q^{82} \) \(\mathstrut +\mathstrut 125502443664q^{84} \) \(\mathstrut +\mathstrut 17135502080q^{85} \) \(\mathstrut -\mathstrut 166086469440q^{88} \) \(\mathstrut -\mathstrut 257976145200q^{90} \) \(\mathstrut +\mathstrut 114387515256q^{93} \) \(\mathstrut +\mathstrut 391966360512q^{94} \) \(\mathstrut +\mathstrut 446868262272q^{96} \) \(\mathstrut +\mathstrut 75764383528q^{97} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{20}\mathstrut -\mathstrut \) \(247\) \(x^{18}\mathstrut -\mathstrut \) \(23916\) \(x^{16}\mathstrut +\mathstrut \) \(14713536\) \(x^{14}\mathstrut -\mathstrut \) \(45723119616\) \(x^{12}\mathstrut +\mathstrut \) \(40864324780032\) \(x^{10}\mathstrut -\mathstrut \) \(11986041468616704\) \(x^{8}\mathstrut +\mathstrut \) \(1011106494856298496\) \(x^{6}\mathstrut -\mathstrut \) \(430832354752771129344\) \(x^{4}\mathstrut -\mathstrut \) \(1166424521268802367782912\) \(x^{2}\mathstrut +\mathstrut \) \(1237940039285380274899124224\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 2 \nu \)
\(\beta_{2}\)\(=\)\( 4 \nu^{2} - 99 \)
\(\beta_{3}\)\(=\)\((\)\(46131569375\) \(\nu^{19}\mathstrut -\mathstrut \) \(239928354304\) \(\nu^{18}\mathstrut -\mathstrut \) \(22887903126825\) \(\nu^{17}\mathstrut +\mathstrut \) \(259755436976640\) \(\nu^{16}\mathstrut +\mathstrut \) \(19556588970264300\) \(\nu^{15}\mathstrut +\mathstrut \) \(53810623904876544\) \(\nu^{14}\mathstrut -\mathstrut \) \(6529423698712632000\) \(\nu^{13}\mathstrut +\mathstrut \) \(38232749261706461184\) \(\nu^{12}\mathstrut +\mathstrut \) \(897349730929067520000\) \(\nu^{11}\mathstrut -\mathstrut \) \(55782354449881500745728\) \(\nu^{10}\mathstrut +\mathstrut \) \(590330585908631489740800\) \(\nu^{9}\mathstrut +\mathstrut \) \(11627256762088427341479936\) \(\nu^{8}\mathstrut -\mathstrut \) \(1177886442391478235404697600\) \(\nu^{7}\mathstrut +\mathstrut \) \(10971145661586545313891483648\) \(\nu^{6}\mathstrut +\mathstrut \) \(400948207094769548080132915200\) \(\nu^{5}\mathstrut -\mathstrut \) \(1110769065760608792454216286208\) \(\nu^{4}\mathstrut -\mathstrut \) \(271316700506973624203911102464000\) \(\nu^{3}\mathstrut +\mathstrut \) \(2393219287980435280047879833518080\) \(\nu^{2}\mathstrut +\mathstrut \) \(72047149669348558893988904225996800\) \(\nu\mathstrut -\mathstrut \) \(1574683656749145806442103199856852992\)\()/\)\(79\!\cdots\!00\)
\(\beta_{4}\)\(=\)\((\)\(-\)\(9226313875\) \(\nu^{19}\mathstrut -\mathstrut \) \(7553005215232\) \(\nu^{18}\mathstrut +\mathstrut \) \(4577580625365\) \(\nu^{17}\mathstrut +\mathstrut \) \(3427472345510400\) \(\nu^{16}\mathstrut -\mathstrut \) \(3911317794052860\) \(\nu^{15}\mathstrut -\mathstrut \) \(1191761025029265408\) \(\nu^{14}\mathstrut +\mathstrut \) \(1305884739742526400\) \(\nu^{13}\mathstrut +\mathstrut \) \(1656950916513972191232\) \(\nu^{12}\mathstrut -\mathstrut \) \(179469946185813504000\) \(\nu^{11}\mathstrut +\mathstrut \) \(23303313749598573428736\) \(\nu^{10}\mathstrut -\mathstrut \) \(118066117181726297948160\) \(\nu^{9}\mathstrut -\mathstrut \) \(26759412697756590504148992\) \(\nu^{8}\mathstrut +\mathstrut \) \(235577288478295647080939520\) \(\nu^{7}\mathstrut +\mathstrut \) \(53114190156844970000309551104\) \(\nu^{6}\mathstrut -\mathstrut \) \(80189641418953909616026583040\) \(\nu^{5}\mathstrut -\mathstrut \) \(102739084009715981330786291810304\) \(\nu^{4}\mathstrut +\mathstrut \) \(54263340101394724840782220492800\) \(\nu^{3}\mathstrut +\mathstrut \) \(44722723994641766129675232513884160\) \(\nu^{2}\mathstrut -\mathstrut \) \(15472806053487858216995241297182720\) \(\nu\mathstrut -\mathstrut \) \(11229575540777706173909771556755603456\)\()/\)\(53\!\cdots\!80\)
\(\beta_{5}\)\(=\)\((\)\(46131569375\) \(\nu^{19}\mathstrut -\mathstrut \) \(239928354304\) \(\nu^{18}\mathstrut -\mathstrut \) \(22887903126825\) \(\nu^{17}\mathstrut +\mathstrut \) \(259755436976640\) \(\nu^{16}\mathstrut +\mathstrut \) \(19556588970264300\) \(\nu^{15}\mathstrut +\mathstrut \) \(53810623904876544\) \(\nu^{14}\mathstrut -\mathstrut \) \(6529423698712632000\) \(\nu^{13}\mathstrut +\mathstrut \) \(38232749261706461184\) \(\nu^{12}\mathstrut +\mathstrut \) \(897349730929067520000\) \(\nu^{11}\mathstrut -\mathstrut \) \(55782354449881500745728\) \(\nu^{10}\mathstrut +\mathstrut \) \(590330585908631489740800\) \(\nu^{9}\mathstrut +\mathstrut \) \(11627256762088427341479936\) \(\nu^{8}\mathstrut -\mathstrut \) \(1177886442391478235404697600\) \(\nu^{7}\mathstrut +\mathstrut \) \(10971145661586545313891483648\) \(\nu^{6}\mathstrut +\mathstrut \) \(400948207094769548080132915200\) \(\nu^{5}\mathstrut -\mathstrut \) \(1110769065760608792454216286208\) \(\nu^{4}\mathstrut +\mathstrut \) \(20996205691855955139745297937203200\) \(\nu^{3}\mathstrut +\mathstrut \) \(2393219287980435280047879833518080\) \(\nu^{2}\mathstrut -\mathstrut \) \(449007148943543195822766717245849600\) \(\nu\mathstrut -\mathstrut \) \(1574683656749145806442103199856852992\)\()/\)\(26\!\cdots\!00\)
\(\beta_{6}\)\(=\)\((\)\(90366381\) \(\nu^{19}\mathstrut +\mathstrut \) \(142181548565\) \(\nu^{17}\mathstrut +\mathstrut \) \(46580767060484\) \(\nu^{15}\mathstrut -\mathstrut \) \(5363773315802176\) \(\nu^{13}\mathstrut +\mathstrut \) \(22436822557367816192\) \(\nu^{11}\mathstrut +\mathstrut \) \(5254313046384963485696\) \(\nu^{9}\mathstrut -\mathstrut \) \(240863080265417198927872\) \(\nu^{7}\mathstrut +\mathstrut \) \(3469516219226795629516685312\) \(\nu^{5}\mathstrut -\mathstrut \) \(662001721306173609090765291520\) \(\nu^{3}\mathstrut +\mathstrut \) \(76055560082787482507355134885888\) \(\nu\)\()/\)\(44\!\cdots\!00\)
\(\beta_{7}\)\(=\)\((\)\(-\)\(36763771475\) \(\nu^{19}\mathstrut -\mathstrut \) \(298065737291264\) \(\nu^{18}\mathstrut +\mathstrut \) \(158494922939925\) \(\nu^{17}\mathstrut +\mathstrut \) \(81842455582947840\) \(\nu^{16}\mathstrut -\mathstrut \) \(267699088226082300\) \(\nu^{15}\mathstrut +\mathstrut \) \(9099512565774895104\) \(\nu^{14}\mathstrut +\mathstrut \) \(93165443595733310400\) \(\nu^{13}\mathstrut -\mathstrut \) \(2673320262654211424256\) \(\nu^{12}\mathstrut -\mathstrut \) \(37405222621281219686400\) \(\nu^{11}\mathstrut +\mathstrut \) \(10891637640586969535741952\) \(\nu^{10}\mathstrut +\mathstrut \) \(15330136321678026237542400\) \(\nu^{9}\mathstrut -\mathstrut \) \(11301552649261375108335796224\) \(\nu^{8}\mathstrut +\mathstrut \) \(8565022094496776194306867200\) \(\nu^{7}\mathstrut +\mathstrut \) \(3904537820300378081238102048768\) \(\nu^{6}\mathstrut -\mathstrut \) \(4643127697977095954501940019200\) \(\nu^{5}\mathstrut -\mathstrut \) \(336971416754697943806994261475328\) \(\nu^{4}\mathstrut +\mathstrut \) \(3284581484215287182324236969574400\) \(\nu^{3}\mathstrut +\mathstrut \) \(147863901090117001159363811113697280\) \(\nu^{2}\mathstrut -\mathstrut \) \(1593247699565736691208787584037683200\) \(\nu\mathstrut +\mathstrut \) \(239853310276670848164756266046275452928\)\()/\)\(26\!\cdots\!00\)
\(\beta_{8}\)\(=\)\((\)\(-\)\(468610067\) \(\nu^{19}\mathstrut -\mathstrut \) \(22448057600\) \(\nu^{18}\mathstrut +\mathstrut \) \(507334837845\) \(\nu^{17}\mathstrut +\mathstrut \) \(40351311686400\) \(\nu^{16}\mathstrut +\mathstrut \) \(105098874814212\) \(\nu^{15}\mathstrut -\mathstrut \) \(14078480870016000\) \(\nu^{14}\mathstrut +\mathstrut \) \(74673338401770432\) \(\nu^{13}\mathstrut +\mathstrut \) \(5872322256906240000\) \(\nu^{12}\mathstrut -\mathstrut \) \(108949911034924806144\) \(\nu^{11}\mathstrut -\mathstrut \) \(2528915718054582681600\) \(\nu^{10}\mathstrut +\mathstrut \) \(22709485863453959651328\) \(\nu^{9}\mathstrut -\mathstrut \) \(1220608474317109120204800\) \(\nu^{8}\mathstrut +\mathstrut \) \(21428018870286221316194304\) \(\nu^{7}\mathstrut +\mathstrut \) \(692000542347252213586329600\) \(\nu^{6}\mathstrut -\mathstrut \) \(2169470831563689047762141184\) \(\nu^{5}\mathstrut -\mathstrut \) \(491097124696682718509924352000\) \(\nu^{4}\mathstrut +\mathstrut \) \(4674256421836787656343515299840\) \(\nu^{3}\mathstrut +\mathstrut \) \(245812780064377887395567527526400\) \(\nu^{2}\mathstrut -\mathstrut \) \(3075554017088175403207232812220416\) \(\nu\mathstrut -\mathstrut \) \(106430100683834190110937046004531200\)\()/\)\(77\!\cdots\!00\)
\(\beta_{9}\)\(=\)\((\)\(-\)\(598262172925\) \(\nu^{19}\mathstrut -\mathstrut \) \(78117971797504\) \(\nu^{18}\mathstrut +\mathstrut \) \(245061855234075\) \(\nu^{17}\mathstrut -\mathstrut \) \(61662761079068160\) \(\nu^{16}\mathstrut +\mathstrut \) \(21718767005754300\) \(\nu^{15}\mathstrut +\mathstrut \) \(51272586309057644544\) \(\nu^{14}\mathstrut -\mathstrut \) \(1680353395577080920000\) \(\nu^{13}\mathstrut +\mathstrut \) \(3684971218613189443584\) \(\nu^{12}\mathstrut -\mathstrut \) \(343690054636713113702400\) \(\nu^{11}\mathstrut -\mathstrut \) \(882593333782537492758528\) \(\nu^{10}\mathstrut -\mathstrut \) \(198864629395050739138560000\) \(\nu^{9}\mathstrut -\mathstrut \) \(1915429115930917298308644864\) \(\nu^{8}\mathstrut -\mathstrut \) \(118575635726017791188061388800\) \(\nu^{7}\mathstrut -\mathstrut \) \(4924509392021741116805666045952\) \(\nu^{6}\mathstrut +\mathstrut \) \(128347953685032807704315088076800\) \(\nu^{5}\mathstrut +\mathstrut \) \(3652037944105187095153747156795392\) \(\nu^{4}\mathstrut -\mathstrut \) \(31630217668136319022817769160704000\) \(\nu^{3}\mathstrut -\mathstrut \) \(64861270229500117352856674862366720\) \(\nu^{2}\mathstrut +\mathstrut \) \(2959773742109028651297735524220928000\) \(\nu\mathstrut +\mathstrut \) \(82100011620472339439472437240012996608\)\()/\)\(79\!\cdots\!00\)
\(\beta_{10}\)\(=\)\((\)\(876499818125\) \(\nu^{19}\mathstrut -\mathstrut \) \(704858789234176\) \(\nu^{18}\mathstrut -\mathstrut \) \(434870159409675\) \(\nu^{17}\mathstrut -\mathstrut \) \(116589674246960640\) \(\nu^{16}\mathstrut +\mathstrut \) \(371575190435021700\) \(\nu^{15}\mathstrut +\mathstrut \) \(7162632924105639936\) \(\nu^{14}\mathstrut -\mathstrut \) \(124059050275540008000\) \(\nu^{13}\mathstrut -\mathstrut \) \(68995303632294469730304\) \(\nu^{12}\mathstrut +\mathstrut \) \(17049644887652282880000\) \(\nu^{11}\mathstrut -\mathstrut \) \(6443033077526701887455232\) \(\nu^{10}\mathstrut +\mathstrut \) \(11216281132263998305075200\) \(\nu^{9}\mathstrut -\mathstrut \) \(3147449340918542980733730816\) \(\nu^{8}\mathstrut -\mathstrut \) \(22379842405438086472689254400\) \(\nu^{7}\mathstrut -\mathstrut \) \(11001634944582836677350877298688\) \(\nu^{6}\mathstrut +\mathstrut \) \(7618015934800621413522525388800\) \(\nu^{5}\mathstrut +\mathstrut \) \(655094595948353532115823307522048\) \(\nu^{4}\mathstrut -\mathstrut \) \(5155017309632498859874310946816000\) \(\nu^{3}\mathstrut +\mathstrut \) \(76764242669034258133694058457989120\) \(\nu^{2}\mathstrut +\mathstrut \) \(1464599694483255798423560620972441600\) \(\nu\mathstrut +\mathstrut \) \(280102002702068699742263678893557809152\)\()/\)\(79\!\cdots\!00\)
\(\beta_{11}\)\(=\)\((\)\(-\)\(1642708321825\) \(\nu^{19}\mathstrut -\mathstrut \) \(239928354304\) \(\nu^{18}\mathstrut +\mathstrut \) \(394255549999575\) \(\nu^{17}\mathstrut +\mathstrut \) \(259755436976640\) \(\nu^{16}\mathstrut +\mathstrut \) \(59946883808203500\) \(\nu^{15}\mathstrut +\mathstrut \) \(53810623904876544\) \(\nu^{14}\mathstrut -\mathstrut \) \(31378230236119915200\) \(\nu^{13}\mathstrut +\mathstrut \) \(38232749261706461184\) \(\nu^{12}\mathstrut +\mathstrut \) \(78116378088539093299200\) \(\nu^{11}\mathstrut -\mathstrut \) \(55782354449881500745728\) \(\nu^{10}\mathstrut -\mathstrut \) \(68422971229562075322777600\) \(\nu^{9}\mathstrut +\mathstrut \) \(11627256762088427341479936\) \(\nu^{8}\mathstrut +\mathstrut \) \(19064618527385844362457907200\) \(\nu^{7}\mathstrut +\mathstrut \) \(10971145661586545313891483648\) \(\nu^{6}\mathstrut -\mathstrut \) \(1306648775669954963539230720000\) \(\nu^{5}\mathstrut -\mathstrut \) \(1110769065760608792454216286208\) \(\nu^{4}\mathstrut +\mathstrut \) \(456290166619136172775910984908800\) \(\nu^{3}\mathstrut +\mathstrut \) \(2393219287980435280047879833518080\) \(\nu^{2}\mathstrut +\mathstrut \) \(1850543709730698476779241510168166400\) \(\nu\mathstrut -\mathstrut \) \(1574683656749145806442103199856852992\)\()/\)\(39\!\cdots\!00\)
\(\beta_{12}\)\(=\)\((\)\(-\)\(599710401875\) \(\nu^{19}\mathstrut -\mathstrut \) \(3358996960256\) \(\nu^{18}\mathstrut +\mathstrut \) \(297542740648725\) \(\nu^{17}\mathstrut +\mathstrut \) \(3636576117672960\) \(\nu^{16}\mathstrut -\mathstrut \) \(254235656613435900\) \(\nu^{15}\mathstrut +\mathstrut \) \(753348734668271616\) \(\nu^{14}\mathstrut +\mathstrut \) \(84882508083264216000\) \(\nu^{13}\mathstrut +\mathstrut \) \(535258489663890456576\) \(\nu^{12}\mathstrut -\mathstrut \) \(11665546502077877760000\) \(\nu^{11}\mathstrut -\mathstrut \) \(780952962298341010440192\) \(\nu^{10}\mathstrut -\mathstrut \) \(7674297616812209366630400\) \(\nu^{9}\mathstrut +\mathstrut \) \(162781594669237982780719104\) \(\nu^{8}\mathstrut +\mathstrut \) \(15312523751089217060261068800\) \(\nu^{7}\mathstrut +\mathstrut \) \(153596039262211634394480771072\) \(\nu^{6}\mathstrut -\mathstrut \) \(5212326692232004125041727897600\) \(\nu^{5}\mathstrut -\mathstrut \) \(15550766920648523094359028006912\) \(\nu^{4}\mathstrut +\mathstrut \) \(3527117106590657114650844332032000\) \(\nu^{3}\mathstrut +\mathstrut \) \(28188189433635361729683015409336320\) \(\nu^{2}\mathstrut -\mathstrut \) \(1003073953177665418009197033186918400\) \(\nu\mathstrut -\mathstrut \) \(21913978399685295668462509067063001088\)\()/\)\(13\!\cdots\!00\)
\(\beta_{13}\)\(=\)\((\)\(-\)\(2059342161337\) \(\nu^{19}\mathstrut -\mathstrut \) \(122888043929344\) \(\nu^{18}\mathstrut -\mathstrut \) \(7256757377112705\) \(\nu^{17}\mathstrut -\mathstrut \) \(74548276279023360\) \(\nu^{16}\mathstrut +\mathstrut \) \(2380369484014821132\) \(\nu^{15}\mathstrut -\mathstrut \) \(31596501599966063616\) \(\nu^{14}\mathstrut +\mathstrut \) \(965034429295281029952\) \(\nu^{13}\mathstrut +\mathstrut \) \(49873315037495650074624\) \(\nu^{12}\mathstrut +\mathstrut \) \(389213420540595849916416\) \(\nu^{11}\mathstrut -\mathstrut \) \(22717862552347329149534208\) \(\nu^{10}\mathstrut +\mathstrut \) \(164586163159111090918391808\) \(\nu^{9}\mathstrut -\mathstrut \) \(8028302850833161666852552704\) \(\nu^{8}\mathstrut -\mathstrut \) \(187451746606807105715155501056\) \(\nu^{7}\mathstrut +\mathstrut \) \(521848681215751088383331401728\) \(\nu^{6}\mathstrut +\mathstrut \) \(44110920201420182273173462450176\) \(\nu^{5}\mathstrut -\mathstrut \) \(1267254250130615885517071201599488\) \(\nu^{4}\mathstrut +\mathstrut \) \(2089598226915952701036641585725440\) \(\nu^{3}\mathstrut +\mathstrut \) \(1438468950404765511202498888681390080\) \(\nu^{2}\mathstrut +\mathstrut \) \(10846674409688296363107785263990964224\) \(\nu\mathstrut -\mathstrut \) \(314902340819552224520706177538470182912\)\()/\)\(19\!\cdots\!00\)
\(\beta_{14}\)\(=\)\((\)\(-\)\(16628286142183\) \(\nu^{19}\mathstrut +\mathstrut \) \(607974004777472\) \(\nu^{18}\mathstrut +\mathstrut \) \(14622089403463905\) \(\nu^{17}\mathstrut -\mathstrut \) \(1085482806128709120\) \(\nu^{16}\mathstrut +\mathstrut \) \(3143284153218307188\) \(\nu^{15}\mathstrut +\mathstrut \) \(372511617855396292608\) \(\nu^{14}\mathstrut +\mathstrut \) \(2038629172992427263168\) \(\nu^{13}\mathstrut -\mathstrut \) \(157988715986125111590912\) \(\nu^{12}\mathstrut -\mathstrut \) \(3039467781032383144390656\) \(\nu^{11}\mathstrut +\mathstrut \) \(69728493318830113846984704\) \(\nu^{10}\mathstrut +\mathstrut \) \(500867039793689454021967872\) \(\nu^{9}\mathstrut +\mathstrut \) \(31997507979448910840648957952\) \(\nu^{8}\mathstrut +\mathstrut \) \(642770852140160869081175556096\) \(\nu^{7}\mathstrut -\mathstrut \) \(18895581702901464383019773067264\) \(\nu^{6}\mathstrut -\mathstrut \) \(73573187175405375995390843682816\) \(\nu^{5}\mathstrut +\mathstrut \) \(13122732917752186875683757247954944\) \(\nu^{4}\mathstrut +\mathstrut \) \(134850384290315088153197380223631360\) \(\nu^{3}\mathstrut -\mathstrut \) \(6870736870936966343083115380620656640\) \(\nu^{2}\mathstrut -\mathstrut \) \(84055983102218869175090724014007517184\) \(\nu\mathstrut +\mathstrut \) \(2906737591377564430123691879676988358656\)\()/\)\(79\!\cdots\!00\)
\(\beta_{15}\)\(=\)\((\)\(19147518521161\) \(\nu^{19}\mathstrut +\mathstrut \) \(1348047059809792\) \(\nu^{18}\mathstrut +\mathstrut \) \(21463383910277265\) \(\nu^{17}\mathstrut -\mathstrut \) \(685295882504133120\) \(\nu^{16}\mathstrut -\mathstrut \) \(10402180382210357196\) \(\nu^{15}\mathstrut +\mathstrut \) \(265514517916036773888\) \(\nu^{14}\mathstrut -\mathstrut \) \(3200878256230621083456\) \(\nu^{13}\mathstrut +\mathstrut \) \(11295489965198363885568\) \(\nu^{12}\mathstrut +\mathstrut \) \(30898934433934614269952\) \(\nu^{11}\mathstrut -\mathstrut \) \(80670937341667042872262656\) \(\nu^{10}\mathstrut -\mathstrut \) \(578157123161926569399681024\) \(\nu^{9}\mathstrut -\mathstrut \) \(6432831683083585429048393728\) \(\nu^{8}\mathstrut +\mathstrut \) \(538247396593968986692169760768\) \(\nu^{7}\mathstrut -\mathstrut \) \(20777602669647249706131866517504\) \(\nu^{6}\mathstrut -\mathstrut \) \(259714444488182396885307362377728\) \(\nu^{5}\mathstrut -\mathstrut \) \(5614499082890471198360357742575616\) \(\nu^{4}\mathstrut -\mathstrut \) \(42769306363831179200323345365073920\) \(\nu^{3}\mathstrut -\mathstrut \) \(1390960470315401954509993751598858240\) \(\nu^{2}\mathstrut -\mathstrut \) \(10262883303858041422809346997928067072\) \(\nu\mathstrut +\mathstrut \) \(235306592015734941495005596577762902016\)\()/\)\(79\!\cdots\!00\)
\(\beta_{16}\)\(=\)\((\)\(-\)\(67315768751491\) \(\nu^{19}\mathstrut -\mathstrut \) \(49572547279360\) \(\nu^{18}\mathstrut +\mathstrut \) \(57887903279997285\) \(\nu^{17}\mathstrut +\mathstrut \) \(86535817888396800\) \(\nu^{16}\mathstrut -\mathstrut \) \(23116739464315131324\) \(\nu^{15}\mathstrut -\mathstrut \) \(28025569463219619840\) \(\nu^{14}\mathstrut +\mathstrut \) \(17156097341881911039936\) \(\nu^{13}\mathstrut +\mathstrut \) \(12600007221069576437760\) \(\nu^{12}\mathstrut -\mathstrut \) \(3943619450344111480614912\) \(\nu^{11}\mathstrut -\mathstrut \) \(6015954707324007843102720\) \(\nu^{10}\mathstrut -\mathstrut \) \(2111080019336329260728057856\) \(\nu^{9}\mathstrut -\mathstrut \) \(2325397303970113068057231360\) \(\nu^{8}\mathstrut +\mathstrut \) \(947536734565477230709196193792\) \(\nu^{7}\mathstrut +\mathstrut \) \(1581784295650970713133175275520\) \(\nu^{6}\mathstrut -\mathstrut \) \(939246874081829721381436372549632\) \(\nu^{5}\mathstrut -\mathstrut \) \(1022428447365215339395138317189120\) \(\nu^{4}\mathstrut +\mathstrut \) \(573574010319336994430638717057105920\) \(\nu^{3}\mathstrut +\mathstrut \) \(539322862891552442586840493876838400\) \(\nu^{2}\mathstrut -\mathstrut \) \(188977894050805310016426886574341357568\) \(\nu\mathstrut -\mathstrut \) \(241581125730832472345544137261742817280\)\()/\)\(79\!\cdots\!00\)
\(\beta_{17}\)\(=\)\((\)\(77752598588255\) \(\nu^{19}\mathstrut -\mathstrut \) \(1513610837722624\) \(\nu^{18}\mathstrut -\mathstrut \) \(33558837933971625\) \(\nu^{17}\mathstrut +\mathstrut \) \(1010192911603223040\) \(\nu^{16}\mathstrut +\mathstrut \) \(15080693918489719020\) \(\nu^{15}\mathstrut -\mathstrut \) \(784958016157527668736\) \(\nu^{14}\mathstrut +\mathstrut \) \(303187723758517273920\) \(\nu^{13}\mathstrut +\mathstrut \) \(202932145804771167535104\) \(\nu^{12}\mathstrut -\mathstrut \) \(3236441107805450132951040\) \(\nu^{11}\mathstrut -\mathstrut \) \(21558258125752034779987968\) \(\nu^{10}\mathstrut -\mathstrut \) \(742890654813919552623083520\) \(\nu^{9}\mathstrut -\mathstrut \) \(21454678355626107336218640384\) \(\nu^{8}\mathstrut -\mathstrut \) \(450042237173729535477531279360\) \(\nu^{7}\mathstrut +\mathstrut \) \(56883127992053949267051427135488\) \(\nu^{6}\mathstrut +\mathstrut \) \(1139710695646396110383833531023360\) \(\nu^{5}\mathstrut -\mathstrut \) \(24406261697714745713004469549006848\) \(\nu^{4}\mathstrut -\mathstrut \) \(165910551883710751944020159417548800\) \(\nu^{3}\mathstrut +\mathstrut \) \(9219747134140788721780149095529185280\) \(\nu^{2}\mathstrut +\mathstrut \) \(20995905308932208702965773590006333440\) \(\nu\mathstrut -\mathstrut \) \(2676074463828503951306550662363088945152\)\()/\)\(79\!\cdots\!00\)
\(\beta_{18}\)\(=\)\((\)\(20861361537311\) \(\nu^{19}\mathstrut +\mathstrut \) \(664254202777088\) \(\nu^{18}\mathstrut -\mathstrut \) \(9712105016906985\) \(\nu^{17}\mathstrut -\mathstrut \) \(378508764894036480\) \(\nu^{16}\mathstrut +\mathstrut \) \(3659925436052685804\) \(\nu^{15}\mathstrut +\mathstrut \) \(333973837738325317632\) \(\nu^{14}\mathstrut +\mathstrut \) \(1116236301105943450944\) \(\nu^{13}\mathstrut -\mathstrut \) \(112963646743620701749248\) \(\nu^{12}\mathstrut -\mathstrut \) \(265637681025076670926848\) \(\nu^{11}\mathstrut +\mathstrut \) \(21419056014545158894780416\) \(\nu^{10}\mathstrut -\mathstrut \) \(100267392562841143370317824\) \(\nu^{9}\mathstrut +\mathstrut \) \(6227434081898698178253815808\) \(\nu^{8}\mathstrut -\mathstrut \) \(75062984890221276515933356032\) \(\nu^{7}\mathstrut -\mathstrut \) \(18501514133614160826699985453056\) \(\nu^{6}\mathstrut +\mathstrut \) \(197189628916290134997291679875072\) \(\nu^{5}\mathstrut +\mathstrut \) \(6833679502713532408439517663461376\) \(\nu^{4}\mathstrut -\mathstrut \) \(30265381768839622117935619265003520\) \(\nu^{3}\mathstrut -\mathstrut \) \(4540413934350005252197418343868661760\) \(\nu^{2}\mathstrut +\mathstrut \) \(11057913549659291865695803256684412928\) \(\nu\mathstrut +\mathstrut \) \(1453673518702250559917566267443562676224\)\()/\)\(19\!\cdots\!00\)
\(\beta_{19}\)\(=\)\((\)\(52728422385041\) \(\nu^{19}\mathstrut -\mathstrut \) \(64528628019712\) \(\nu^{18}\mathstrut +\mathstrut \) \(8735714394333465\) \(\nu^{17}\mathstrut -\mathstrut \) \(288344739388654080\) \(\nu^{16}\mathstrut -\mathstrut \) \(2076715176723154476\) \(\nu^{15}\mathstrut +\mathstrut \) \(160264440885605701632\) \(\nu^{14}\mathstrut +\mathstrut \) \(3411816834702714588864\) \(\nu^{13}\mathstrut -\mathstrut \) \(16644856777799873691648\) \(\nu^{12}\mathstrut -\mathstrut \) \(1144271951083651373887488\) \(\nu^{11}\mathstrut +\mathstrut \) \(8537469759136614177570816\) \(\nu^{10}\mathstrut -\mathstrut \) \(211428997219881600332857344\) \(\nu^{9}\mathstrut +\mathstrut \) \(1180381335703132787083051008\) \(\nu^{8}\mathstrut +\mathstrut \) \(946834725462405301524662059008\) \(\nu^{7}\mathstrut -\mathstrut \) \(12672481859836240755147046649856\) \(\nu^{6}\mathstrut -\mathstrut \) \(164157668945818972363059225427968\) \(\nu^{5}\mathstrut +\mathstrut \) \(9314498941902225996531690243096576\) \(\nu^{4}\mathstrut -\mathstrut \) \(6040904779116899729561630980177920\) \(\nu^{3}\mathstrut -\mathstrut \) \(1197981035491800412489757689758351360\) \(\nu^{2}\mathstrut -\mathstrut \) \(31184387169193157452512097371561132032\) \(\nu\mathstrut +\mathstrut \) \(600134170201916187129337659532534349824\)\()/\)\(39\!\cdots\!00\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)\(/2\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{2}\mathstrut +\mathstrut \) \(99\)\()/4\)
\(\nu^{3}\)\(=\)\((\)\(\beta_{5}\mathstrut -\mathstrut \) \(3\) \(\beta_{3}\mathstrut +\mathstrut \) \(98\) \(\beta_{1}\)\()/8\)
\(\nu^{4}\)\(=\)\((\)\(2\) \(\beta_{19}\mathstrut -\mathstrut \) \(\beta_{18}\mathstrut -\mathstrut \) \(\beta_{17}\mathstrut -\mathstrut \) \(3\) \(\beta_{15}\mathstrut -\mathstrut \) \(\beta_{14}\mathstrut +\mathstrut \) \(\beta_{13}\mathstrut +\mathstrut \) \(3\) \(\beta_{12}\mathstrut -\mathstrut \) \(2\) \(\beta_{11}\mathstrut -\mathstrut \) \(4\) \(\beta_{10}\mathstrut +\mathstrut \) \(12\) \(\beta_{9}\mathstrut -\mathstrut \) \(\beta_{8}\mathstrut -\mathstrut \) \(\beta_{7}\mathstrut +\mathstrut \) \(2\) \(\beta_{5}\mathstrut -\mathstrut \) \(34\) \(\beta_{4}\mathstrut +\mathstrut \) \(188\) \(\beta_{3}\mathstrut +\mathstrut \) \(84\) \(\beta_{2}\mathstrut +\mathstrut \) \(54\) \(\beta_{1}\mathstrut +\mathstrut \) \(174164\)\()/16\)
\(\nu^{5}\)\(=\)\((\)\(-\)\(14\) \(\beta_{19}\mathstrut +\mathstrut \) \(13\) \(\beta_{18}\mathstrut +\mathstrut \) \(23\) \(\beta_{17}\mathstrut -\mathstrut \) \(10\) \(\beta_{16}\mathstrut -\mathstrut \) \(3\) \(\beta_{15}\mathstrut +\mathstrut \) \(3\) \(\beta_{14}\mathstrut +\mathstrut \) \(3\) \(\beta_{13}\mathstrut +\mathstrut \) \(373\) \(\beta_{12}\mathstrut -\mathstrut \) \(119\) \(\beta_{11}\mathstrut +\mathstrut \) \(116\) \(\beta_{9}\mathstrut +\mathstrut \) \(43\) \(\beta_{8}\mathstrut -\mathstrut \) \(13\) \(\beta_{7}\mathstrut +\mathstrut \) \(3438\) \(\beta_{6}\mathstrut +\mathstrut \) \(27\) \(\beta_{5}\mathstrut -\mathstrut \) \(31549\) \(\beta_{3}\mathstrut +\mathstrut \) \(324\) \(\beta_{2}\mathstrut +\mathstrut \) \(33168\) \(\beta_{1}\mathstrut +\mathstrut \) \(70\)\()/8\)
\(\nu^{6}\)\(=\)\((\)\(-\)\(98\) \(\beta_{19}\mathstrut -\mathstrut \) \(551\) \(\beta_{18}\mathstrut +\mathstrut \) \(649\) \(\beta_{17}\mathstrut +\mathstrut \) \(331\) \(\beta_{15}\mathstrut +\mathstrut \) \(1081\) \(\beta_{14}\mathstrut +\mathstrut \) \(135\) \(\beta_{13}\mathstrut +\mathstrut \) \(17277\) \(\beta_{12}\mathstrut +\mathstrut \) \(902\) \(\beta_{11}\mathstrut -\mathstrut \) \(2988\) \(\beta_{10}\mathstrut -\mathstrut \) \(2204\) \(\beta_{9}\mathstrut -\mathstrut \) \(32039\) \(\beta_{8}\mathstrut +\mathstrut \) \(2753\) \(\beta_{7}\mathstrut +\mathstrut \) \(2480\) \(\beta_{6}\mathstrut -\mathstrut \) \(98\) \(\beta_{5}\mathstrut -\mathstrut \) \(13630\) \(\beta_{4}\mathstrut +\mathstrut \) \(1467316\) \(\beta_{3}\mathstrut +\mathstrut \) \(16348\) \(\beta_{2}\mathstrut +\mathstrut \) \(469442\) \(\beta_{1}\mathstrut -\mathstrut \) \(18172388\)\()/16\)
\(\nu^{7}\)\(=\)\((\)\(9982\) \(\beta_{19}\mathstrut +\mathstrut \) \(1291\) \(\beta_{18}\mathstrut +\mathstrut \) \(2209\) \(\beta_{17}\mathstrut -\mathstrut \) \(21766\) \(\beta_{16}\mathstrut -\mathstrut \) \(373\) \(\beta_{15}\mathstrut +\mathstrut \) \(10197\) \(\beta_{14}\mathstrut +\mathstrut \) \(373\) \(\beta_{13}\mathstrut +\mathstrut \) \(193315\) \(\beta_{12}\mathstrut +\mathstrut \) \(294655\) \(\beta_{11}\mathstrut -\mathstrut \) \(34420\) \(\beta_{9}\mathstrut +\mathstrut \) \(394621\) \(\beta_{8}\mathstrut -\mathstrut \) \(1291\) \(\beta_{7}\mathstrut -\mathstrut \) \(250334\) \(\beta_{6}\mathstrut -\mathstrut \) \(4199\) \(\beta_{5}\mathstrut -\mathstrut \) \(16136303\) \(\beta_{3}\mathstrut +\mathstrut \) \(303532\) \(\beta_{2}\mathstrut -\mathstrut \) \(9707064\) \(\beta_{1}\mathstrut +\mathstrut \) \(169258\)\()/8\)
\(\nu^{8}\)\(=\)\((\)\(-\)\(17846\) \(\beta_{19}\mathstrut -\mathstrut \) \(56333\) \(\beta_{18}\mathstrut +\mathstrut \) \(74179\) \(\beta_{17}\mathstrut -\mathstrut \) \(518631\) \(\beta_{15}\mathstrut +\mathstrut \) \(323731\) \(\beta_{14}\mathstrut -\mathstrut \) \(554323\) \(\beta_{13}\mathstrut +\mathstrut \) \(9963231\) \(\beta_{12}\mathstrut -\mathstrut \) \(1762190\) \(\beta_{11}\mathstrut +\mathstrut \) \(566620\) \(\beta_{10}\mathstrut -\mathstrut \) \(848244\) \(\beta_{9}\mathstrut -\mathstrut \) \(15937805\) \(\beta_{8}\mathstrut -\mathstrut \) \(1895989\) \(\beta_{7}\mathstrut +\mathstrut \) \(1240272\) \(\beta_{6}\mathstrut -\mathstrut \) \(17846\) \(\beta_{5}\mathstrut +\mathstrut \) \(7629014\) \(\beta_{4}\mathstrut +\mathstrut \) \(728802300\) \(\beta_{3}\mathstrut -\mathstrut \) \(26383436\) \(\beta_{2}\mathstrut +\mathstrut \) \(234490102\) \(\beta_{1}\mathstrut +\mathstrut \) \(286478617972\)\()/16\)
\(\nu^{9}\)\(=\)\((\)\(-\)\(1926358\) \(\beta_{19}\mathstrut -\mathstrut \) \(1083367\) \(\beta_{18}\mathstrut -\mathstrut \) \(4659189\) \(\beta_{17}\mathstrut -\mathstrut \) \(6790914\) \(\beta_{16}\mathstrut -\mathstrut \) \(2492455\) \(\beta_{15}\mathstrut +\mathstrut \) \(5772999\) \(\beta_{14}\mathstrut +\mathstrut \) \(2492455\) \(\beta_{13}\mathstrut +\mathstrut \) \(64467233\) \(\beta_{12}\mathstrut -\mathstrut \) \(234549547\) \(\beta_{11}\mathstrut -\mathstrut \) \(13749500\) \(\beta_{9}\mathstrut +\mathstrut \) \(159973567\) \(\beta_{8}\mathstrut +\mathstrut \) \(1083367\) \(\beta_{7}\mathstrut +\mathstrut \) \(618883894\) \(\beta_{6}\mathstrut -\mathstrut \) \(12309437\) \(\beta_{5}\mathstrut -\mathstrut \) \(4443663765\) \(\beta_{3}\mathstrut +\mathstrut \) \(109309348\) \(\beta_{2}\mathstrut +\mathstrut \) \(70170590008\) \(\beta_{1}\mathstrut +\mathstrut \) \(67880718\)\()/8\)
\(\nu^{10}\)\(=\)\((\)\(-\)\(11602802\) \(\beta_{19}\mathstrut +\mathstrut \) \(33234625\) \(\beta_{18}\mathstrut -\mathstrut \) \(21631823\) \(\beta_{17}\mathstrut -\mathstrut \) \(219962109\) \(\beta_{15}\mathstrut +\mathstrut \) \(92413345\) \(\beta_{14}\mathstrut -\mathstrut \) \(243167713\) \(\beta_{13}\mathstrut -\mathstrut \) \(5805802843\) \(\beta_{12}\mathstrut -\mathstrut \) \(185638394\) \(\beta_{11}\mathstrut -\mathstrut \) \(588698060\) \(\beta_{10}\mathstrut -\mathstrut \) \(224683452\) \(\beta_{9}\mathstrut -\mathstrut \) \(5020829759\) \(\beta_{8}\mathstrut +\mathstrut \) \(286708745\) \(\beta_{7}\mathstrut +\mathstrut \) \(383156976\) \(\beta_{6}\mathstrut -\mathstrut \) \(11602802\) \(\beta_{5}\mathstrut +\mathstrut \) \(4497115090\) \(\beta_{4}\mathstrut -\mathstrut \) \(413099186412\) \(\beta_{3}\mathstrut +\mathstrut \) \(82294907004\) \(\beta_{2}\mathstrut -\mathstrut \) \(132261435214\) \(\beta_{1}\mathstrut -\mathstrut \) \(241062032403524\)\()/16\)
\(\nu^{11}\)\(=\)\((\)\(1120245038\) \(\beta_{19}\mathstrut -\mathstrut \) \(245539837\) \(\beta_{18}\mathstrut -\mathstrut \) \(1709030327\) \(\beta_{17}\mathstrut -\mathstrut \) \(1448893430\) \(\beta_{16}\mathstrut -\mathstrut \) \(1217950653\) \(\beta_{15}\mathstrut -\mathstrut \) \(4766457699\) \(\beta_{14}\mathstrut +\mathstrut \) \(1217950653\) \(\beta_{13}\mathstrut +\mathstrut \) \(10371444443\) \(\beta_{12}\mathstrut +\mathstrut \) \(44631739735\) \(\beta_{11}\mathstrut -\mathstrut \) \(13261923092\) \(\beta_{9}\mathstrut -\mathstrut \) \(105497068603\) \(\beta_{8}\mathstrut +\mathstrut \) \(245539837\) \(\beta_{7}\mathstrut +\mathstrut \) \(424211272434\) \(\beta_{6}\mathstrut +\mathstrut \) \(17782539249\) \(\beta_{5}\mathstrut -\mathstrut \) \(1160922416343\) \(\beta_{3}\mathstrut -\mathstrut \) \(13149897012\) \(\beta_{2}\mathstrut -\mathstrut \) \(60666986586264\) \(\beta_{1}\mathstrut -\mathstrut \) \(54490268230\)\()/8\)
\(\nu^{12}\)\(=\)\((\)\(84428086202\) \(\beta_{19}\mathstrut -\mathstrut \) \(26861954821\) \(\beta_{18}\mathstrut -\mathstrut \) \(57566131381\) \(\beta_{17}\mathstrut -\mathstrut \) \(37109285071\) \(\beta_{15}\mathstrut +\mathstrut \) \(26989162139\) \(\beta_{14}\mathstrut +\mathstrut \) \(131746887333\) \(\beta_{13}\mathstrut -\mathstrut \) \(482249771385\) \(\beta_{12}\mathstrut -\mathstrut \) \(93662384798\) \(\beta_{11}\mathstrut -\mathstrut \) \(535597147140\) \(\beta_{10}\mathstrut +\mathstrut \) \(642157626284\) \(\beta_{9}\mathstrut -\mathstrut \) \(767340864517\) \(\beta_{8}\mathstrut -\mathstrut \) \(85989829997\) \(\beta_{7}\mathstrut +\mathstrut \) \(33521477968\) \(\beta_{6}\mathstrut +\mathstrut \) \(84428086202\) \(\beta_{5}\mathstrut +\mathstrut \) \(3034800033382\) \(\beta_{4}\mathstrut -\mathstrut \) \(23967212175460\) \(\beta_{3}\mathstrut -\mathstrut \) \(61505623044460\) \(\beta_{2}\mathstrut -\mathstrut \) \(7844214361082\) \(\beta_{1}\mathstrut +\mathstrut \) \(54439264050892372\)\()/16\)
\(\nu^{13}\)\(=\)\((\)\(242499472538\) \(\beta_{19}\mathstrut +\mathstrut \) \(1062225369137\) \(\beta_{18}\mathstrut +\mathstrut \) \(348130941779\) \(\beta_{17}\mathstrut +\mathstrut \) \(1932600731470\) \(\beta_{16}\mathstrut -\mathstrut \) \(1776319796495\) \(\beta_{15}\mathstrut -\mathstrut \) \(768644733777\) \(\beta_{14}\mathstrut +\mathstrut \) \(1776319796495\) \(\beta_{13}\mathstrut +\mathstrut \) \(5048915033081\) \(\beta_{12}\mathstrut -\mathstrut \) \(11434652830579\) \(\beta_{11}\mathstrut -\mathstrut \) \(5254564454300\) \(\beta_{9}\mathstrut -\mathstrut \) \(8065373874457\) \(\beta_{8}\mathstrut -\mathstrut \) \(1062225369137\) \(\beta_{7}\mathstrut +\mathstrut \) \(128517737030342\) \(\beta_{6}\mathstrut -\mathstrut \) \(17255167411189\) \(\beta_{5}\mathstrut -\mathstrut \) \(437020238464525\) \(\beta_{3}\mathstrut +\mathstrut \) \(5619228234500\) \(\beta_{2}\mathstrut +\mathstrut \) \(13463303850048248\) \(\beta_{1}\mathstrut -\mathstrut \) \(6422437115170\)\()/8\)
\(\nu^{14}\)\(=\)\((\)\(-\)\(28262175402274\) \(\beta_{19}\mathstrut +\mathstrut \) \(60137703195737\) \(\beta_{18}\mathstrut -\mathstrut \) \(31875527793463\) \(\beta_{17}\mathstrut +\mathstrut \) \(42970685388747\) \(\beta_{15}\mathstrut +\mathstrut \) \(42452345557049\) \(\beta_{14}\mathstrut -\mathstrut \) \(13553665415801\) \(\beta_{13}\mathstrut +\mathstrut \) \(1576445248230813\) \(\beta_{12}\mathstrut +\mathstrut \) \(75007685261462\) \(\beta_{11}\mathstrut -\mathstrut \) \(97635594443500\) \(\beta_{10}\mathstrut -\mathstrut \) \(30975783644508\) \(\beta_{9}\mathstrut -\mathstrut \) \(357875525388967\) \(\beta_{8}\mathstrut +\mathstrut \) \(209116394055265\) \(\beta_{7}\mathstrut +\mathstrut \) \(10058477931888\) \(\beta_{6}\mathstrut -\mathstrut \) \(28262175402274\) \(\beta_{5}\mathstrut +\mathstrut \) \(1151539460476546\) \(\beta_{4}\mathstrut +\mathstrut \) \(134637585821412084\) \(\beta_{3}\mathstrut +\mathstrut \) \(12197447742988828\) \(\beta_{2}\mathstrut +\mathstrut \) \(43186113798062690\) \(\beta_{1}\mathstrut -\mathstrut \) \(11063500004389784740\)\()/16\)
\(\nu^{15}\)\(=\)\((\)\(334300856711806\) \(\beta_{19}\mathstrut +\mathstrut \) \(46520424696523\) \(\beta_{18}\mathstrut -\mathstrut \) \(285404129197407\) \(\beta_{17}\mathstrut +\mathstrut \) \(307896550139898\) \(\beta_{16}\mathstrut -\mathstrut \) \(378444978590453\) \(\beta_{15}\mathstrut +\mathstrut \) \(1168715433262293\) \(\beta_{14}\mathstrut +\mathstrut \) \(378444978590453\) \(\beta_{13}\mathstrut -\mathstrut \) \(18276927096838685\) \(\beta_{12}\mathstrut +\mathstrut \) \(26793702635271679\) \(\beta_{11}\mathstrut -\mathstrut \) \(3328750750210804\) \(\beta_{9}\mathstrut +\mathstrut \) \(34855446163866557\) \(\beta_{8}\mathstrut -\mathstrut \) \(46520424696523\) \(\beta_{7}\mathstrut -\mathstrut \) \(512549952323422\) \(\beta_{6}\mathstrut +\mathstrut \) \(1792618081606921\) \(\beta_{5}\mathstrut +\mathstrut \) \(1526181565647949185\) \(\beta_{3}\mathstrut -\mathstrut \) \(7337976572186644\) \(\beta_{2}\mathstrut -\mathstrut \) \(2277678734468607192\) \(\beta_{1}\mathstrut +\mathstrut \) \(14571372061217322\)\()/8\)
\(\nu^{16}\)\(=\)\((\)\(-\)\(148043133669846\) \(\beta_{19}\mathstrut +\mathstrut \) \(26180663941687299\) \(\beta_{18}\mathstrut -\mathstrut \) \(26032620808017453\) \(\beta_{17}\mathstrut -\mathstrut \) \(46724170236825015\) \(\beta_{15}\mathstrut -\mathstrut \) \(67615707262618461\) \(\beta_{14}\mathstrut -\mathstrut \) \(47020256504164707\) \(\beta_{13}\mathstrut +\mathstrut \) \(145184776446505135\) \(\beta_{12}\mathstrut -\mathstrut \) \(5292671875455086\) \(\beta_{11}\mathstrut -\mathstrut \) \(18269425041927780\) \(\beta_{10}\mathstrut +\mathstrut \) \(30485433385208268\) \(\beta_{9}\mathstrut +\mathstrut \) \(1566991390917243651\) \(\beta_{8}\mathstrut -\mathstrut \) \(26187753860809765\) \(\beta_{7}\mathstrut -\mathstrut \) \(114539865096429360\) \(\beta_{6}\mathstrut -\mathstrut \) \(148043133669846\) \(\beta_{5}\mathstrut +\mathstrut \) \(176630166733831926\) \(\beta_{4}\mathstrut +\mathstrut \) \(12340419527251115068\) \(\beta_{3}\mathstrut -\mathstrut \) \(3004769573127831948\) \(\beta_{2}\mathstrut +\mathstrut \) \(3971324987239654294\) \(\beta_{1}\mathstrut +\mathstrut \) \(23159979889812405966900\)\()/16\)
\(\nu^{17}\)\(=\)\((\)\(-\)\(15441452171127798\) \(\beta_{19}\mathstrut -\mathstrut \) \(194195113220287671\) \(\beta_{18}\mathstrut -\mathstrut \) \(64646036688224613\) \(\beta_{17}\mathstrut +\mathstrut \) \(602679577972900254\) \(\beta_{16}\mathstrut +\mathstrut \) \(323744189752350729\) \(\beta_{15}\mathstrut +\mathstrut \) \(223645379953401495\) \(\beta_{14}\mathstrut -\mathstrut \) \(323744189752350729\) \(\beta_{13}\mathstrut -\mathstrut \) \(5315732609610095151\) \(\beta_{12}\mathstrut -\mathstrut \) \(2074539421055336891\) \(\beta_{11}\mathstrut +\mathstrut \) \(839060267876889540\) \(\beta_{9}\mathstrut +\mathstrut \) \(1960838627298648591\) \(\beta_{8}\mathstrut +\mathstrut \) \(194195113220287671\) \(\beta_{7}\mathstrut +\mathstrut \) \(79322729055914048598\) \(\beta_{6}\mathstrut -\mathstrut \) \(733801132566146925\) \(\beta_{5}\mathstrut +\mathstrut \) \(458073699678311505979\) \(\beta_{3}\mathstrut -\mathstrut \) \(4718567291189623452\) \(\beta_{2}\mathstrut +\mathstrut \) \(5931306687337437075000\) \(\beta_{1}\mathstrut +\mathstrut \) \(1273401624244511406\)\()/8\)
\(\nu^{18}\)\(=\)\((\)\(-\)\(4952817144020814162\) \(\beta_{19}\mathstrut +\mathstrut \) \(6528510030265107249\) \(\beta_{18}\mathstrut -\mathstrut \) \(1575692886244293087\) \(\beta_{17}\mathstrut +\mathstrut \) \(8169676679967086163\) \(\beta_{15}\mathstrut -\mathstrut \) \(11118500381650297839\) \(\beta_{14}\mathstrut -\mathstrut \) \(1735957608074542161\) \(\beta_{13}\mathstrut -\mathstrut \) \(240318150030246699819\) \(\beta_{12}\mathstrut +\mathstrut \) \(4434082807554055590\) \(\beta_{11}\mathstrut -\mathstrut \) \(80808594406875614988\) \(\beta_{10}\mathstrut -\mathstrut \) \(16820147525424919548\) \(\beta_{9}\mathstrut +\mathstrut \) \(428488183966032825393\) \(\beta_{8}\mathstrut -\mathstrut \) \(23179678477861549575\) \(\beta_{7}\mathstrut -\mathstrut \) \(31982370329511974928\) \(\beta_{6}\mathstrut -\mathstrut \) \(4952817144020814162\) \(\beta_{5}\mathstrut -\mathstrut \) \(208609269676507668558\) \(\beta_{4}\mathstrut -\mathstrut \) \(18749298330204427635756\) \(\beta_{3}\mathstrut +\mathstrut \) \(6244323012608557513404\) \(\beta_{2}\mathstrut -\mathstrut \) \(6009899696814312882030\) \(\beta_{1}\mathstrut -\mathstrut \) \(1500963881845785618019844\)\()/16\)
\(\nu^{19}\)\(=\)\((\)\(264353649577719578958\) \(\beta_{19}\mathstrut -\mathstrut \) \(27108892988102479533\) \(\beta_{18}\mathstrut +\mathstrut \) \(87558597300783467577\) \(\beta_{17}\mathstrut +\mathstrut \) \(88271193197533497450\) \(\beta_{16}\mathstrut +\mathstrut \) \(141776383276988426643\) \(\beta_{15}\mathstrut -\mathstrut \) \(240157787123835970227\) \(\beta_{14}\mathstrut -\mathstrut \) \(141776383276988426643\) \(\beta_{13}\mathstrut -\mathstrut \) \(2044640604321636379989\) \(\beta_{12}\mathstrut -\mathstrut \) \(3315354838169745749145\) \(\beta_{11}\mathstrut -\mathstrut \) \(369409656577562633172\) \(\beta_{9}\mathstrut -\mathstrut \) \(5237800633619283587595\) \(\beta_{8}\mathstrut +\mathstrut \) \(27108892988102479533\) \(\beta_{7}\mathstrut +\mathstrut \) \(5318798053804332586194\) \(\beta_{6}\mathstrut +\mathstrut \) \(1784800993036943086689\) \(\beta_{5}\mathstrut +\mathstrut \) \(159556236651584802618393\) \(\beta_{3}\mathstrut -\mathstrut \) \(3181218263009358603252\) \(\beta_{2}\mathstrut -\mathstrut \) \(324203861615639326858520\) \(\beta_{1}\mathstrut -\mathstrut \) \(2549892911480535900390\)\()/8\)

Character Values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/12\mathbb{Z}\right)^\times\).

\(n\) \(5\) \(7\)
\(\chi(n)\) \(-1\) \(-1\)

Embeddings

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

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

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
11.1
−22.4409 2.89952i
−22.4409 + 2.89952i
−20.8511 8.78821i
−20.8511 + 8.78821i
−16.0196 15.9804i
−16.0196 + 15.9804i
−12.1023 19.1190i
−12.1023 + 19.1190i
−0.544378 22.6209i
−0.544378 + 22.6209i
0.544378 22.6209i
0.544378 + 22.6209i
12.1023 19.1190i
12.1023 + 19.1190i
16.0196 15.9804i
16.0196 + 15.9804i
20.8511 8.78821i
20.8511 + 8.78821i
22.4409 2.89952i
22.4409 + 2.89952i
−44.8817 5.79904i −353.249 228.828i 1980.74 + 520.542i 12652.7i 14527.4 + 12318.7i 32182.6i −85880.5 34849.2i 72422.0 + 161667.i −73373.6 + 567877.i
11.2 −44.8817 + 5.79904i −353.249 + 228.828i 1980.74 520.542i 12652.7i 14527.4 12318.7i 32182.6i −85880.5 + 34849.2i 72422.0 161667.i −73373.6 567877.i
11.3 −41.7022 17.5764i 364.303 210.784i 1430.14 + 1465.95i 505.366i −18897.1 + 2387.01i 77554.9i −33873.8 86269.9i 88286.9 153579.i −8882.52 + 21074.9i
11.4 −41.7022 + 17.5764i 364.303 + 210.784i 1430.14 1465.95i 505.366i −18897.1 2387.01i 77554.9i −33873.8 + 86269.9i 88286.9 + 153579.i −8882.52 21074.9i
11.5 −32.0392 31.9607i 70.6484 + 414.917i 5.02409 + 2047.99i 1842.04i 10997.5 15551.6i 16302.4i 65294.4 65776.7i −167165. + 58626.4i −58872.8 + 59017.4i
11.6 −32.0392 + 31.9607i 70.6484 414.917i 5.02409 2047.99i 1842.04i 10997.5 + 15551.6i 16302.4i 65294.4 + 65776.7i −167165. 58626.4i −58872.8 59017.4i
11.7 −24.2046 38.2379i −183.786 378.642i −876.277 + 1851.07i 9922.43i −10030.0 + 16192.5i 35930.2i 91990.8 11297.3i −109593. + 139178.i 379413. 240168.i
11.8 −24.2046 + 38.2379i −183.786 + 378.642i −876.277 1851.07i 9922.43i −10030.0 16192.5i 35930.2i 91990.8 + 11297.3i −109593. 139178.i 379413. + 240168.i
11.9 −1.08876 45.2417i −399.437 + 132.653i −2045.63 + 98.5144i 4150.68i 6436.35 + 17926.8i 39165.5i 6684.15 + 92440.6i 141953. 105973.i −187784. + 4519.08i
11.10 −1.08876 + 45.2417i −399.437 132.653i −2045.63 98.5144i 4150.68i 6436.35 17926.8i 39165.5i 6684.15 92440.6i 141953. + 105973.i −187784. 4519.08i
11.11 1.08876 45.2417i 399.437 132.653i −2045.63 98.5144i 4150.68i −5566.58 18215.7i 39165.5i −6684.15 + 92440.6i 141953. 105973.i −187784. 4519.08i
11.12 1.08876 + 45.2417i 399.437 + 132.653i −2045.63 + 98.5144i 4150.68i −5566.58 + 18215.7i 39165.5i −6684.15 92440.6i 141953. + 105973.i −187784. + 4519.08i
11.13 24.2046 38.2379i 183.786 + 378.642i −876.277 1851.07i 9922.43i 18926.9 + 2137.29i 35930.2i −91990.8 11297.3i −109593. + 139178.i 379413. + 240168.i
11.14 24.2046 + 38.2379i 183.786 378.642i −876.277 + 1851.07i 9922.43i 18926.9 2137.29i 35930.2i −91990.8 + 11297.3i −109593. 139178.i 379413. 240168.i
11.15 32.0392 31.9607i −70.6484 414.917i 5.02409 2047.99i 1842.04i −15524.6 11035.6i 16302.4i −65294.4 65776.7i −167165. + 58626.4i −58872.8 59017.4i
11.16 32.0392 + 31.9607i −70.6484 + 414.917i 5.02409 + 2047.99i 1842.04i −15524.6 + 11035.6i 16302.4i −65294.4 + 65776.7i −167165. 58626.4i −58872.8 + 59017.4i
11.17 41.7022 17.5764i −364.303 + 210.784i 1430.14 1465.95i 505.366i −11487.4 + 15193.3i 77554.9i 33873.8 86269.9i 88286.9 153579.i −8882.52 21074.9i
11.18 41.7022 + 17.5764i −364.303 210.784i 1430.14 + 1465.95i 505.366i −11487.4 15193.3i 77554.9i 33873.8 + 86269.9i 88286.9 + 153579.i −8882.52 + 21074.9i
11.19 44.8817 5.79904i 353.249 + 228.828i 1980.74 520.542i 12652.7i 17181.4 + 8221.72i 32182.6i 85880.5 34849.2i 72422.0 + 161667.i −73373.6 567877.i
11.20 44.8817 + 5.79904i 353.249 228.828i 1980.74 + 520.542i 12652.7i 17181.4 8221.72i 32182.6i 85880.5 + 34849.2i 72422.0 161667.i −73373.6 + 567877.i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 11.20
Significant digits:
Format:

Inner twists

Char. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
3.b Odd 1 yes
4.b Odd 1 yes
12.b Even 1 yes

Hecke kernels

There are no other newforms in \(S_{12}^{\mathrm{new}}(12, [\chi])\).