Properties

Label 9.12.c.a
Level 9
Weight 12
Character orbit 9.c
Analytic conductor 6.915
Analytic rank 0
Dimension 20
CM No
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 9 = 3^{2} \)
Weight: \( k \) = \( 12 \)
Character orbit: \([\chi]\) = 9.c (of order \(3\) and degree \(2\))

Newform invariants

Self dual: No
Analytic conductor: \(6.91508862504\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(10\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} + \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{24}\cdot 3^{45} \)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( \beta_{1} - 3 \beta_{3} ) q^{2} + ( 24 - 51 \beta_{3} + \beta_{4} + \beta_{6} ) q^{3} + ( -921 + \beta_{2} + 921 \beta_{3} + \beta_{5} - \beta_{6} + \beta_{10} ) q^{4} + ( -723 - \beta_{1} - \beta_{2} + 723 \beta_{3} - 2 \beta_{4} + \beta_{6} + \beta_{12} ) q^{5} + ( 712 + 61 \beta_{1} + 75 \beta_{2} + 676 \beta_{3} + \beta_{4} - \beta_{5} - 3 \beta_{6} + 2 \beta_{10} + \beta_{11} + \beta_{14} ) q^{6} + ( -16 + 120 \beta_{1} - 8 \beta_{2} + 899 \beta_{3} - 2 \beta_{4} + 25 \beta_{6} - \beta_{8} - \beta_{11} - \beta_{14} - \beta_{17} ) q^{7} + ( -1647 - 868 \beta_{1} - 859 \beta_{2} - 38 \beta_{3} - 43 \beta_{4} - 68 \beta_{6} + \beta_{8} - \beta_{9} - \beta_{11} - \beta_{12} - \beta_{13} - \beta_{14} + \beta_{15} ) q^{8} + ( 12264 - 814 \beta_{1} - 253 \beta_{2} - 11295 \beta_{3} + 17 \beta_{4} - 15 \beta_{5} - 18 \beta_{6} - 2 \beta_{7} - \beta_{8} - 8 \beta_{10} - 2 \beta_{11} - 3 \beta_{12} - \beta_{13} + \beta_{15} + \beta_{17} - \beta_{18} + \beta_{19} ) q^{9} +O(q^{10})\) \( q + ( \beta_{1} - 3 \beta_{3} ) q^{2} + ( 24 - 51 \beta_{3} + \beta_{4} + \beta_{6} ) q^{3} + ( -921 + \beta_{2} + 921 \beta_{3} + \beta_{5} - \beta_{6} + \beta_{10} ) q^{4} + ( -723 - \beta_{1} - \beta_{2} + 723 \beta_{3} - 2 \beta_{4} + \beta_{6} + \beta_{12} ) q^{5} + ( 712 + 61 \beta_{1} + 75 \beta_{2} + 676 \beta_{3} + \beta_{4} - \beta_{5} - 3 \beta_{6} + 2 \beta_{10} + \beta_{11} + \beta_{14} ) q^{6} + ( -16 + 120 \beta_{1} - 8 \beta_{2} + 899 \beta_{3} - 2 \beta_{4} + 25 \beta_{6} - \beta_{8} - \beta_{11} - \beta_{14} - \beta_{17} ) q^{7} + ( -1647 - 868 \beta_{1} - 859 \beta_{2} - 38 \beta_{3} - 43 \beta_{4} - 68 \beta_{6} + \beta_{8} - \beta_{9} - \beta_{11} - \beta_{12} - \beta_{13} - \beta_{14} + \beta_{15} ) q^{8} + ( 12264 - 814 \beta_{1} - 253 \beta_{2} - 11295 \beta_{3} + 17 \beta_{4} - 15 \beta_{5} - 18 \beta_{6} - 2 \beta_{7} - \beta_{8} - 8 \beta_{10} - 2 \beta_{11} - 3 \beta_{12} - \beta_{13} + \beta_{15} + \beta_{17} - \beta_{18} + \beta_{19} ) q^{9} + ( -302 - 1615 \beta_{1} - 1648 \beta_{2} + 11 \beta_{3} + 63 \beta_{4} + 18 \beta_{5} - 53 \beta_{6} - 14 \beta_{7} - \beta_{8} - 4 \beta_{9} + 4 \beta_{10} - 8 \beta_{11} - 3 \beta_{12} - 4 \beta_{13} + 4 \beta_{14} + 3 \beta_{15} - 2 \beta_{16} - \beta_{18} ) q^{10} + ( 176 - 32 \beta_{1} + 61 \beta_{2} - 11394 \beta_{3} - 96 \beta_{4} - 12 \beta_{5} - 150 \beta_{6} + 3 \beta_{7} - 5 \beta_{8} + 3 \beta_{9} + \beta_{10} + 12 \beta_{11} + \beta_{13} + 2 \beta_{14} + 3 \beta_{16} - 8 \beta_{17} + 3 \beta_{18} + \beta_{19} ) q^{11} + ( -8178 + 2061 \beta_{1} - 3047 \beta_{2} + 119616 \beta_{3} - 852 \beta_{4} - 81 \beta_{5} + 37 \beta_{6} + 31 \beta_{7} - 5 \beta_{8} - 6 \beta_{9} + 85 \beta_{10} + 27 \beta_{12} - 8 \beta_{14} + 3 \beta_{15} + \beta_{16} - 15 \beta_{17} - 4 \beta_{18} - 3 \beta_{19} ) q^{12} + ( 29791 - 250 \beta_{1} + 7900 \beta_{2} - 30082 \beta_{3} + 180 \beta_{4} - 101 \beta_{5} + 758 \beta_{6} - 5 \beta_{7} - 8 \beta_{8} - 2 \beta_{9} - 97 \beta_{10} + 33 \beta_{11} + 93 \beta_{12} - 6 \beta_{13} + 9 \beta_{14} - 3 \beta_{15} - 8 \beta_{16} + 2 \beta_{17} - \beta_{18} + 3 \beta_{19} ) q^{13} + ( -391559 + 583 \beta_{1} - 2211 \beta_{2} + 391022 \beta_{3} + 1757 \beta_{4} + 579 \beta_{5} - 754 \beta_{6} + 25 \beta_{7} + 9 \beta_{8} + 14 \beta_{9} + 602 \beta_{10} - 13 \beta_{11} - 37 \beta_{12} + 4 \beta_{13} + 72 \beta_{14} + \beta_{15} + 9 \beta_{16} + 17 \beta_{17} + 12 \beta_{18} - \beta_{19} ) q^{14} + ( -491179 + 5856 \beta_{1} + 2324 \beta_{2} + 349467 \beta_{3} - 546 \beta_{4} + 467 \beta_{5} - 112 \beta_{6} - 291 \beta_{7} + 27 \beta_{8} + 6 \beta_{9} + 186 \beta_{10} + 30 \beta_{11} + 216 \beta_{12} + 27 \beta_{13} + \beta_{14} - 6 \beta_{15} + 3 \beta_{16} + 21 \beta_{17} - 15 \beta_{18} + 3 \beta_{19} ) q^{15} + ( -2514 - 4360 \beta_{1} - 938 \beta_{2} - 734092 \beta_{3} + 2819 \beta_{4} + 66 \beta_{5} + 1070 \beta_{6} + 344 \beta_{7} - 9 \beta_{8} - 51 \beta_{9} - 576 \beta_{10} - 113 \beta_{11} - 369 \beta_{12} - 52 \beta_{13} - 132 \beta_{14} - 25 \beta_{16} + 16 \beta_{17} - 5 \beta_{18} - 6 \beta_{19} ) q^{16} + ( 1385694 - 8288 \beta_{1} - 7154 \beta_{2} + 48 \beta_{3} - 1688 \beta_{4} - 1201 \beta_{5} + 2855 \beta_{6} + 77 \beta_{7} + 51 \beta_{8} + 109 \beta_{9} - 122 \beta_{10} - 33 \beta_{11} - 18 \beta_{12} + 47 \beta_{13} - 144 \beta_{14} - 24 \beta_{15} + 30 \beta_{16} + 36 \beta_{18} ) q^{17} + ( 1677369 + 47349 \beta_{1} + 29146 \beta_{2} - 2462746 \beta_{3} - 914 \beta_{4} - 1842 \beta_{5} + 1699 \beta_{6} + 286 \beta_{7} + \beta_{8} - 15 \beta_{9} - 2291 \beta_{10} + 66 \beta_{11} - 747 \beta_{12} + 129 \beta_{13} + 25 \beta_{14} - 21 \beta_{15} + 10 \beta_{16} + 6 \beta_{17} - 16 \beta_{18} - 18 \beta_{19} ) q^{18} + ( 359635 + 19377 \beta_{1} + 18614 \beta_{2} - 1167 \beta_{3} + 70 \beta_{4} + 715 \beta_{5} - 4606 \beta_{6} - 858 \beta_{7} - 35 \beta_{8} - 105 \beta_{9} + 6 \beta_{10} - 77 \beta_{11} + 18 \beta_{12} - 141 \beta_{13} + 69 \beta_{14} - 63 \beta_{15} - 21 \beta_{16} + 12 \beta_{18} ) q^{19} + ( 17910 - 189 \beta_{1} + 6948 \beta_{2} - 3425399 \beta_{3} - 8865 \beta_{4} - 344 \beta_{5} - 17274 \beta_{6} + 110 \beta_{7} + 105 \beta_{8} + 281 \beta_{9} - 4567 \beta_{10} + 183 \beta_{11} - 107 \beta_{12} + 58 \beta_{13} + 108 \beta_{14} + 3 \beta_{16} + 102 \beta_{17} + 21 \beta_{18} - 24 \beta_{19} ) q^{20} + ( 451881 - 73569 \beta_{1} + 30736 \beta_{2} + 4578960 \beta_{3} + 1699 \beta_{4} - 2385 \beta_{5} + 1682 \beta_{6} + 826 \beta_{7} + 94 \beta_{8} + 36 \beta_{9} + 4237 \beta_{10} + 93 \beta_{11} + 477 \beta_{12} + 420 \beta_{13} + 91 \beta_{14} - 63 \beta_{15} - 2 \beta_{16} + 216 \beta_{17} - 7 \beta_{18} + 57 \beta_{19} ) q^{21} + ( 232123 - 13930 \beta_{1} + 11467 \beta_{2} - 232280 \beta_{3} - 19118 \beta_{4} + 822 \beta_{5} + 16517 \beta_{6} + 41 \beta_{7} + 26 \beta_{8} - 628 \beta_{9} + 116 \beta_{10} - 177 \beta_{11} + 996 \beta_{12} - 687 \beta_{13} - 74 \beta_{14} + 57 \beta_{15} + 26 \beta_{16} - 31 \beta_{17} - 8 \beta_{18} - 57 \beta_{19} ) q^{22} + ( -6957919 + 1686 \beta_{1} + 29786 \beta_{2} + 6941070 \beta_{3} + 35896 \beta_{4} + 6445 \beta_{5} + 11801 \beta_{6} - 163 \beta_{7} - 36 \beta_{8} + 924 \beta_{9} + 6465 \beta_{10} + 31 \beta_{11} + 302 \beta_{12} + 180 \beta_{13} - 594 \beta_{14} - 25 \beta_{15} - 36 \beta_{16} - 227 \beta_{17} - 69 \beta_{18} + 25 \beta_{19} ) q^{23} + ( -13188091 + 56281 \beta_{1} - 218996 \beta_{2} + 5214109 \beta_{3} + 4738 \beta_{4} + 13701 \beta_{5} - 6020 \beta_{6} - 660 \beta_{7} - 357 \beta_{8} - 420 \beta_{9} + 5945 \beta_{10} - 431 \beta_{11} + 27 \beta_{12} + 864 \beta_{13} - 117 \beta_{14} + 123 \beta_{15} - 18 \beta_{16} - 336 \beta_{17} + 129 \beta_{18} - 66 \beta_{19} ) q^{24} + ( -46305 + 236648 \beta_{1} - 13951 \beta_{2} - 4440972 \beta_{3} + 67615 \beta_{4} + 933 \beta_{5} + 3403 \beta_{6} + 220 \beta_{7} + 150 \beta_{8} - 1092 \beta_{9} + 4679 \beta_{10} + 1049 \beta_{11} + 9 \beta_{12} - 1526 \beta_{13} + 1119 \beta_{14} + 229 \beta_{16} - 79 \beta_{17} + 53 \beta_{18} + 123 \beta_{19} ) q^{25} + ( 24088701 + 118948 \beta_{1} + 142610 \beta_{2} + 25810 \beta_{3} - 14634 \beta_{4} - 18489 \beta_{5} + 105277 \beta_{6} - 902 \beta_{7} - 837 \beta_{8} + 1713 \beta_{9} - 1023 \beta_{10} + 36 \beta_{11} + 189 \beta_{12} + 24 \beta_{13} + 1575 \beta_{14} + 252 \beta_{15} - 333 \beta_{16} - 387 \beta_{18} ) q^{26} + ( -860265 + 66714 \beta_{1} + 339096 \beta_{2} - 6528948 \beta_{3} + 1569 \beta_{4} - 5625 \beta_{5} + 2085 \beta_{6} - 1161 \beta_{7} + 147 \beta_{8} - 612 \beta_{9} - 12879 \beta_{10} - 528 \beta_{11} + 2448 \beta_{12} + 1842 \beta_{13} - 372 \beta_{14} + 192 \beta_{15} - 129 \beta_{16} - 294 \beta_{17} + 207 \beta_{18} + 129 \beta_{19} ) q^{27} + ( -3162332 - 1149445 \beta_{1} - 1110692 \beta_{2} - 138290 \beta_{3} - 171011 \beta_{4} - 5665 \beta_{5} - 220611 \beta_{6} + 4292 \beta_{7} + 419 \beta_{8} - 3059 \beta_{9} - 1342 \beta_{10} + 1539 \beta_{11} + 21 \beta_{12} - 3653 \beta_{13} - 991 \beta_{14} + 573 \beta_{15} + 374 \beta_{16} - 110 \beta_{18} ) q^{28} + ( 165169 + 85863 \beta_{1} + 60586 \beta_{2} - 2606251 \beta_{3} - 106613 \beta_{4} - 2511 \beta_{5} - 138753 \beta_{6} - 1624 \beta_{7} - 679 \beta_{8} + 3978 \beta_{9} - 14650 \beta_{10} - 2856 \beta_{11} + 1387 \beta_{12} + 635 \beta_{13} - 1136 \beta_{14} - 237 \beta_{16} - 442 \beta_{17} - 489 \beta_{18} + 251 \beta_{19} ) q^{29} + ( -10596684 - 1394940 \beta_{1} - 958546 \beta_{2} + 18106561 \beta_{3} + 11636 \beta_{4} + 5904 \beta_{5} + 6560 \beta_{6} - 5359 \beta_{7} - 700 \beta_{8} - 1851 \beta_{9} + 13577 \beta_{10} - 1068 \beta_{11} - 3870 \beta_{12} + 3147 \beta_{13} - 148 \beta_{14} + 570 \beta_{15} - 43 \beta_{16} - 1167 \beta_{17} + 454 \beta_{18} - 441 \beta_{19} ) q^{30} + ( 11874493 - 82016 \beta_{1} + 1529062 \beta_{2} - 11901572 \beta_{3} - 54002 \beta_{4} + 10359 \beta_{5} + 126167 \beta_{6} + 21 \beta_{7} + 408 \beta_{8} - 3138 \beta_{9} + 5205 \beta_{10} - 729 \beta_{11} - 8262 \beta_{12} - 4590 \beta_{13} + 268 \beta_{14} - 441 \beta_{15} + 408 \beta_{16} + 139 \beta_{17} + 231 \beta_{18} + 441 \beta_{19} ) q^{31} + ( 7853898 + 32765 \beta_{1} + 286529 \beta_{2} - 7993481 \beta_{3} + 319443 \beta_{4} - 18337 \beta_{5} + 141992 \beta_{6} - 294 \beta_{7} - 468 \beta_{8} + 5155 \beta_{9} - 17345 \beta_{10} + 1026 \beta_{11} - 766 \beta_{12} + 635 \beta_{13} + 432 \beta_{14} + 276 \beta_{15} - 468 \beta_{16} + 930 \beta_{17} - 285 \beta_{18} - 276 \beta_{19} ) q^{32} + ( 14975812 + 1130674 \beta_{1} - 987762 \beta_{2} - 26772515 \beta_{3} + 1846 \beta_{4} - 19345 \beta_{5} - 10269 \beta_{6} + 17127 \beta_{7} + 1563 \beta_{8} + 453 \beta_{9} - 2542 \beta_{10} + 1699 \beta_{11} - 7992 \beta_{12} + 6075 \beta_{13} + 652 \beta_{14} - 1074 \beta_{15} - 96 \beta_{16} + 2166 \beta_{17} - 216 \beta_{18} + 636 \beta_{19} ) q^{33} + ( -366066 + 3549538 \beta_{1} - 124825 \beta_{2} - 23126634 \beta_{3} + 334927 \beta_{4} + 7476 \beta_{5} + 204286 \beta_{6} - 17084 \beta_{7} - 552 \beta_{8} - 6843 \beta_{9} - 6477 \beta_{10} - 1492 \beta_{11} + 16686 \beta_{12} - 6905 \beta_{13} - 159 \beta_{14} - 398 \beta_{16} - 154 \beta_{17} - 211 \beta_{18} - 1074 \beta_{19} ) q^{34} + ( -8848356 - 145096 \beta_{1} - 20819 \beta_{2} - 19739 \beta_{3} - 202579 \beta_{4} + 50459 \beta_{5} + 79569 \beta_{6} + 4114 \beta_{7} + 5057 \beta_{8} + 8783 \beta_{9} - 8123 \beta_{10} + 3871 \beta_{11} + 106 \beta_{12} + 3258 \beta_{13} - 5123 \beta_{14} - 1471 \beta_{15} + 1299 \beta_{16} + 1332 \beta_{18} ) q^{35} + ( -58339431 + 4603145 \beta_{1} + 6065418 \beta_{2} + 117761630 \beta_{3} + 23790 \beta_{4} + 33864 \beta_{5} + 59335 \beta_{6} - 760 \beta_{7} - 1161 \beta_{8} - 294 \beta_{9} + 83390 \beta_{10} + 1621 \beta_{11} + 13857 \beta_{12} + 7211 \beta_{13} + 2374 \beta_{14} - 974 \beta_{15} + 703 \beta_{16} + 2752 \beta_{17} - 296 \beta_{18} - 404 \beta_{19} ) q^{36} + ( -7695560 - 3644274 \beta_{1} - 3658372 \beta_{2} - 61725 \beta_{3} - 26384 \beta_{4} + 45724 \beta_{5} - 180649 \beta_{6} + 8400 \beta_{7} - 1025 \beta_{8} - 2838 \beta_{9} + 1407 \beta_{10} - 7247 \beta_{11} - 360 \beta_{12} - 4161 \beta_{13} + 2811 \beta_{14} - 2853 \beta_{15} - 1365 \beta_{16} + 924 \beta_{18} ) q^{37} + ( 409678 - 69086 \beta_{1} + 157843 \beta_{2} + 48674426 \beta_{3} - 189737 \beta_{4} - 11535 \beta_{5} - 400050 \beta_{6} + 7391 \beta_{7} + 1160 \beta_{8} + 5037 \beta_{9} + 85718 \beta_{10} + 9663 \beta_{11} - 6710 \beta_{12} + 1217 \beta_{13} + 1642 \beta_{14} + 681 \beta_{16} + 479 \beta_{17} + 2166 \beta_{18} - 1447 \beta_{19} ) q^{38} + ( -72974265 - 6956397 \beta_{1} - 936767 \beta_{2} - 56253992 \beta_{3} - 66670 \beta_{4} + 1548 \beta_{5} + 51781 \beta_{6} - 3692 \beta_{7} + 2599 \beta_{8} + 3288 \beta_{9} - 110297 \beta_{10} + 4059 \beta_{11} - 1350 \beta_{12} + 6291 \beta_{13} - 2318 \beta_{14} - 2787 \beta_{15} + 13 \beta_{16} + 2325 \beta_{17} - 2626 \beta_{18} + 1590 \beta_{19} ) q^{39} + ( 13445401 - 116031 \beta_{1} + 9096994 \beta_{2} - 13449943 \beta_{3} - 131907 \beta_{4} - 43880 \beta_{5} + 204650 \beta_{6} - 1665 \beta_{7} - 2736 \beta_{8} - 7407 \beta_{9} - 47660 \beta_{10} + 6093 \beta_{11} - 8388 \beta_{12} - 5958 \beta_{13} - 2280 \beta_{14} + 1593 \beta_{15} - 2736 \beta_{16} + 429 \beta_{17} - 1629 \beta_{18} - 1593 \beta_{19} ) q^{40} + ( 32811305 + 120095 \beta_{1} - 986125 \beta_{2} - 32836218 \beta_{3} + 189140 \beta_{4} - 101771 \beta_{5} - 36309 \beta_{6} + 4403 \beta_{7} + 4104 \beta_{8} + 5484 \beta_{9} - 99063 \beta_{10} - 8567 \beta_{11} - 1224 \beta_{12} + 3450 \beta_{13} + 9567 \beta_{14} - 1723 \beta_{15} + 4104 \beta_{16} + 1120 \beta_{17} + 3063 \beta_{18} + 1723 \beta_{19} ) q^{41} + ( 320468963 + 3970089 \beta_{1} - 12984205 \beta_{2} - 226125081 \beta_{3} - 350181 \beta_{4} - 223324 \beta_{5} + 86291 \beta_{6} - 37347 \beta_{7} + 63 \beta_{8} - 7008 \beta_{9} - 73485 \beta_{10} + 627 \beta_{11} + 6507 \beta_{12} - 4725 \beta_{13} - 41 \beta_{14} + 4956 \beta_{15} + 1284 \beta_{16} - 6573 \beta_{17} - 687 \beta_{18} - 3423 \beta_{19} ) q^{42} + ( 280184 + 7237661 \beta_{1} + 94574 \beta_{2} - 109861192 \beta_{3} - 81348 \beta_{4} - 11247 \beta_{5} - 305604 \beta_{6} + 49610 \beta_{7} + 1058 \beta_{8} + 9453 \beta_{9} - 84561 \beta_{10} - 12780 \beta_{11} - 52128 \beta_{12} + 4718 \beta_{13} - 25768 \beta_{14} - 2518 \beta_{16} + 3576 \beta_{17} + 598 \beta_{18} + 4962 \beta_{19} ) q^{43} + ( 13205088 - 1468688 \beta_{1} - 1719065 \beta_{2} + 147011 \beta_{3} + 493519 \beta_{4} + 181686 \beta_{5} - 224935 \beta_{6} - 9876 \beta_{7} - 14842 \beta_{8} - 14285 \beta_{9} + 9735 \beta_{10} - 29528 \beta_{11} - 6002 \beta_{12} - 8096 \beta_{13} - 395 \beta_{14} + 4805 \beta_{15} - 1773 \beta_{16} - 288 \beta_{18} ) q^{44} + ( -45126561 + 11166240 \beta_{1} + 16080662 \beta_{2} + 274003696 \beta_{3} - 416572 \beta_{4} + 129921 \beta_{5} - 287986 \beta_{6} - 15415 \beta_{7} + 1922 \beta_{8} - 2010 \beta_{9} + 135923 \beta_{10} - 3003 \beta_{11} - 41013 \beta_{12} - 17316 \beta_{13} - 8887 \beta_{14} + 2799 \beta_{15} - 2320 \beta_{16} - 12456 \beta_{17} - 4151 \beta_{18} - 3 \beta_{19} ) q^{45} + ( 137118418 - 20482686 \beta_{1} - 20585400 \beta_{2} + 472543 \beta_{3} + 533694 \beta_{4} - 102115 \beta_{5} + 888030 \beta_{6} - 51142 \beta_{7} - 4154 \beta_{8} + 6619 \beta_{9} + 6620 \beta_{10} + 12323 \beta_{11} + 1524 \beta_{12} + 11659 \beta_{13} + 8897 \beta_{14} + 7692 \beta_{15} - 790 \beta_{16} - 5003 \beta_{18} ) q^{46} + ( -1163868 - 1016170 \beta_{1} - 430454 \beta_{2} - 146717477 \beta_{3} + 656426 \beta_{4} + 24846 \beta_{5} + 1048065 \beta_{6} - 6398 \beta_{7} + 4185 \beta_{8} - 22416 \beta_{9} + 319548 \beta_{10} + 13923 \beta_{11} + 10376 \beta_{12} - 2442 \beta_{13} + 21375 \beta_{14} + 3978 \beta_{16} + 207 \beta_{17} - 810 \beta_{18} + 4554 \beta_{19} ) q^{47} + ( -575312586 - 29051469 \beta_{1} - 11961849 \beta_{2} + 205227185 \beta_{3} - 42789 \beta_{4} + 161739 \beta_{5} - 540460 \beta_{6} + 58596 \beta_{7} - 4782 \beta_{8} + 1071 \beta_{9} + 2319 \beta_{10} - 1782 \beta_{11} + 74628 \beta_{12} - 30771 \beta_{13} + 12354 \beta_{14} + 7056 \beta_{15} + 2508 \beta_{16} + 2088 \beta_{17} + 5097 \beta_{18} - 1026 \beta_{19} ) q^{48} + ( -348616359 + 820101 \beta_{1} + 21865264 \beta_{2} + 348686962 \beta_{3} + 836465 \beta_{4} + 79481 \beta_{5} - 1253890 \beta_{6} + 10610 \beta_{7} + 3152 \beta_{8} + 41315 \beta_{9} + 135037 \beta_{10} + 174 \beta_{11} + 152439 \beta_{12} + 45402 \beta_{13} + 20561 \beta_{14} - 1086 \beta_{15} + 3152 \beta_{16} - 7137 \beta_{17} + 5848 \beta_{18} + 1086 \beta_{19} ) q^{49} + ( -662554997 - 729916 \beta_{1} - 1373306 \beta_{2} + 663734709 \beta_{3} - 3241902 \beta_{4} + 36755 \beta_{5} - 457153 \beta_{6} - 5825 \beta_{7} - 11889 \beta_{8} - 64443 \beta_{9} + 27000 \beta_{10} + 29177 \beta_{11} + 10295 \beta_{12} - 21657 \beta_{13} - 28530 \beta_{14} + 6277 \beta_{15} - 11889 \beta_{16} - 22279 \beta_{17} - 6051 \beta_{18} - 6277 \beta_{19} ) q^{50} + ( 435104226 + 12539991 \beta_{1} - 6207660 \beta_{2} - 697000764 \beta_{3} + 1437888 \beta_{4} + 237054 \beta_{5} + 1117782 \beta_{6} - 81249 \beta_{7} - 24456 \beta_{8} + 26565 \beta_{9} + 71289 \beta_{10} - 24171 \beta_{11} + 129006 \beta_{12} - 44793 \beta_{13} - 10665 \beta_{14} - 11067 \beta_{15} - 5472 \beta_{16} + 5916 \beta_{17} - 1623 \beta_{18} + 10479 \beta_{19} ) q^{51} + ( 2915234 + 46910675 \beta_{1} + 947572 \beta_{2} + 375882313 \beta_{3} - 3844451 \beta_{4} - 29268 \beta_{5} - 585038 \beta_{6} + 14406 \beta_{7} - 6925 \beta_{8} + 39627 \beta_{9} + 376621 \beta_{10} + 40265 \beta_{11} - 3681 \beta_{12} + 67614 \beta_{13} + 90176 \beta_{14} + 10725 \beta_{16} - 17650 \beta_{17} - 3453 \beta_{18} - 11196 \beta_{19} ) q^{52} + ( 748354866 + 4907956 \beta_{1} + 4142620 \beta_{2} - 901309 \beta_{3} + 419598 \beta_{4} - 261362 \beta_{5} - 2651883 \beta_{6} + 21172 \beta_{7} + 18965 \beta_{8} - 70356 \beta_{9} + 65675 \beta_{10} + 84571 \beta_{11} + 20980 \beta_{12} - 1339 \beta_{13} + 22735 \beta_{14} - 5773 \beta_{15} + 1671 \beta_{16} - 6768 \beta_{18} ) q^{53} + ( 810546759 + 10913931 \beta_{1} + 32588544 \beta_{2} + 204186813 \beta_{3} + 1482060 \beta_{4} - 297666 \beta_{5} - 861684 \beta_{6} + 193053 \beta_{7} + 13467 \beta_{8} + 30762 \beta_{9} - 336723 \beta_{10} + 19890 \beta_{11} - 190431 \beta_{12} - 50760 \beta_{13} + 25698 \beta_{14} - 3429 \beta_{15} + 6645 \beta_{16} + 28809 \beta_{17} + 20922 \beta_{18} + 3951 \beta_{19} ) q^{54} + ( 235393381 - 29779245 \beta_{1} - 30270081 \beta_{2} + 2715757 \beta_{3} + 2929266 \beta_{4} + 43196 \beta_{5} + 4674594 \beta_{6} - 129658 \beta_{7} + 22708 \beta_{8} + 93262 \beta_{9} - 17707 \beta_{10} - 17326 \beta_{11} - 10626 \beta_{12} + 101137 \beta_{13} - 63787 \beta_{14} - 6159 \beta_{15} + 12791 \beta_{16} + 14788 \beta_{18} ) q^{55} + ( -6559096 - 1645300 \beta_{1} - 2493042 \beta_{2} - 2762923364 \beta_{3} + 3703720 \beta_{4} + 151505 \beta_{5} + 5850864 \beta_{6} - 58941 \beta_{7} - 13118 \beta_{8} - 100016 \beta_{9} - 1473361 \beta_{10} - 149787 \beta_{11} + 32274 \beta_{12} - 34440 \beta_{13} - 112981 \beta_{14} - 26667 \beta_{16} + 13549 \beta_{17} - 17163 \beta_{18} - 4325 \beta_{19} ) q^{56} + ( -744307683 - 23978412 \beta_{1} - 11204793 \beta_{2} + 691359864 \beta_{3} + 463421 \beta_{4} - 637209 \beta_{5} + 981479 \beta_{6} + 44856 \beta_{7} + 3564 \beta_{8} - 9954 \beta_{9} + 255627 \beta_{10} - 22797 \beta_{11} - 245997 \beta_{12} - 95904 \beta_{13} - 3897 \beta_{14} - 2736 \beta_{15} - 15399 \beta_{16} - 10755 \beta_{17} - 279 \beta_{18} - 13293 \beta_{19} ) q^{57} + ( -129660149 + 1916481 \beta_{1} + 24973723 \beta_{2} + 130675463 \beta_{3} + 830847 \beta_{4} - 91685 \beta_{5} - 3431395 \beta_{6} - 38718 \beta_{7} + 17667 \beta_{8} + 91242 \beta_{9} - 70472 \beta_{10} - 83790 \beta_{11} - 246393 \beta_{12} + 108225 \beta_{13} - 95118 \beta_{14} - 12798 \beta_{15} + 17667 \beta_{16} + 30774 \beta_{17} - 12960 \beta_{18} + 12798 \beta_{19} ) q^{58} + ( -2643681459 - 44044 \beta_{1} + 7394069 \beta_{2} + 2646027468 \beta_{3} - 4641491 \beta_{4} + 562089 \beta_{5} - 3284213 \beta_{6} - 31590 \beta_{7} + 10404 \beta_{8} - 77691 \beta_{9} + 510432 \beta_{10} - 52002 \beta_{11} - 17600 \beta_{12} + 11235 \beta_{13} - 21942 \beta_{14} - 10602 \beta_{15} + 10404 \beta_{16} + 72432 \beta_{17} - 10494 \beta_{18} + 10602 \beta_{19} ) q^{59} + ( 1048766942 - 5916697 \beta_{1} - 23161554 \beta_{2} - 3130482337 \beta_{3} - 374947 \beta_{4} + 1161661 \beta_{5} - 4373616 \beta_{6} + 173151 \beta_{7} + 85713 \beta_{8} + 28047 \beta_{9} - 75692 \beta_{10} + 73304 \beta_{11} - 344223 \beta_{12} - 41688 \beta_{13} + 35123 \beta_{14} - 1614 \beta_{15} + 13746 \beta_{16} + 16125 \beta_{17} + 30708 \beta_{18} - 13215 \beta_{19} ) q^{60} + ( 5550627 - 9773141 \beta_{1} + 2002572 \beta_{2} + 93251285 \beta_{3} - 4407603 \beta_{4} - 154413 \beta_{5} - 3767919 \beta_{6} + 35196 \beta_{7} + 33609 \beta_{8} + 99810 \beta_{9} - 368296 \beta_{10} + 57174 \beta_{11} - 38511 \beta_{12} + 94155 \beta_{13} - 40038 \beta_{14} - 3315 \beta_{16} + 36924 \beta_{17} + 18201 \beta_{18} - 423 \beta_{19} ) q^{61} + ( 4554076512 + 9130582 \beta_{1} + 8230198 \beta_{2} - 39235 \beta_{3} + 1359320 \beta_{4} - 1499179 \beta_{5} + 147112 \beta_{6} - 51274 \beta_{7} - 196 \beta_{8} - 54733 \beta_{9} + 65344 \beta_{10} - 63641 \beta_{11} - 7466 \beta_{12} - 27993 \beta_{13} + 47137 \beta_{14} - 18442 \beta_{15} - 19302 \beta_{16} + 3303 \beta_{18} ) q^{62} + ( 2112996372 + 17525905 \beta_{1} + 6878993 \beta_{2} + 934077851 \beta_{3} - 6628 \beta_{4} - 815031 \beta_{5} + 5673001 \beta_{6} - 612861 \beta_{7} - 71710 \beta_{8} - 29166 \beta_{9} - 2453934 \beta_{10} - 104461 \beta_{11} + 758964 \beta_{12} - 93935 \beta_{13} - 75245 \beta_{14} - 4534 \beta_{15} - 18983 \beta_{16} - 19033 \beta_{17} - 32373 \beta_{18} - 10291 \beta_{19} ) q^{63} + ( -363902792 + 39271903 \beta_{1} + 39509701 \beta_{2} - 398311 \beta_{3} - 595806 \beta_{4} - 502159 \beta_{5} + 635162 \beta_{6} + 798916 \beta_{7} - 2311 \beta_{8} + 15368 \beta_{9} + 57811 \beta_{10} + 150673 \beta_{11} + 56787 \beta_{12} + 59369 \beta_{13} + 95242 \beta_{14} - 28212 \beta_{15} - 6905 \beta_{16} - 17740 \beta_{18} ) q^{64} + ( 1237231 + 30691675 \beta_{1} + 308652 \beta_{2} - 3896167455 \beta_{3} - 2525167 \beta_{4} - 63160 \beta_{5} + 795579 \beta_{6} + 228047 \beta_{7} - 31225 \beta_{8} - 45470 \beta_{9} + 1049835 \beta_{10} + 262335 \beta_{11} - 178517 \beta_{12} + 53845 \beta_{13} + 203365 \beta_{14} + 49530 \beta_{16} - 80755 \beta_{17} + 36210 \beta_{18} - 23020 \beta_{19} ) q^{65} + ( -6337972797 + 41172372 \beta_{1} + 37689652 \beta_{2} + 3377891321 \beta_{3} + 594697 \beta_{4} + 748368 \beta_{5} + 522304 \beta_{6} - 582650 \beta_{7} - 314 \beta_{8} + 147909 \beta_{9} + 1241137 \beta_{10} + 14670 \beta_{11} + 450468 \beta_{12} + 112725 \beta_{13} - 134165 \beta_{14} - 38688 \beta_{15} + 41290 \beta_{16} - 25692 \beta_{17} + 6335 \beta_{18} + 48264 \beta_{19} ) q^{66} + ( 634158137 + 84139 \beta_{1} - 58558850 \beta_{2} - 636286069 \beta_{3} + 4268651 \beta_{4} - 323176 \beta_{5} + 2711552 \beta_{6} + 88063 \beta_{7} - 51188 \beta_{8} - 63449 \beta_{9} - 183831 \beta_{10} + 200229 \beta_{11} - 180366 \beta_{12} - 71979 \beta_{13} + 227471 \beta_{14} + 46293 \beta_{15} - 51188 \beta_{16} - 46508 \beta_{17} + 20885 \beta_{18} - 46293 \beta_{19} ) q^{67} + ( -7734393250 + 231057 \beta_{1} + 7958311 \beta_{2} + 7730096915 \beta_{3} + 8499203 \beta_{4} + 2511515 \beta_{5} + 2158528 \beta_{6} + 72488 \beta_{7} + 6822 \beta_{8} + 110453 \beta_{9} + 2576009 \beta_{10} + 70036 \beta_{11} - 18020 \beta_{12} - 36803 \beta_{13} + 175788 \beta_{14} - 12418 \beta_{15} + 6822 \beta_{16} - 52868 \beta_{17} + 42453 \beta_{18} + 12418 \beta_{19} ) q^{68} + ( 3173227115 - 18392220 \beta_{1} + 32318474 \beta_{2} - 5642949699 \beta_{3} - 7805934 \beta_{4} + 566255 \beta_{5} - 2111941 \beta_{6} + 763689 \beta_{7} - 81333 \beta_{8} - 196584 \beta_{9} + 1685178 \beta_{10} - 68214 \beta_{11} - 31239 \beta_{12} - 90099 \beta_{13} - 14288 \beta_{14} + 74382 \beta_{15} - 29019 \beta_{16} - 29568 \beta_{17} - 87249 \beta_{18} - 25473 \beta_{19} ) q^{69} + ( -18104880 - 128594954 \beta_{1} - 6442617 \beta_{2} - 106775654 \beta_{3} + 13655340 \beta_{4} + 320661 \beta_{5} + 12573576 \beta_{6} - 1215795 \beta_{7} - 28245 \beta_{8} - 171144 \beta_{9} - 583327 \beta_{10} - 360612 \beta_{11} + 1174743 \beta_{12} - 187257 \beta_{13} - 262002 \beta_{14} - 41052 \beta_{16} + 12807 \beta_{17} - 49839 \beta_{18} + 68481 \beta_{19} ) q^{70} + ( 6079127214 - 78216240 \beta_{1} - 77336676 \beta_{2} + 890887 \beta_{3} - 682010 \beta_{4} + 2240428 \beta_{5} + 599339 \beta_{6} - 33194 \beta_{7} - 2969 \beta_{8} + 95158 \beta_{9} - 172561 \beta_{10} - 140131 \beta_{11} - 87442 \beta_{12} + 55251 \beta_{13} - 318067 \beta_{14} + 90481 \beta_{15} + 66327 \beta_{16} + 31644 \beta_{18} ) q^{71} + ( 2967084699 - 89146074 \beta_{1} - 190107799 \beta_{2} + 10182417916 \beta_{3} - 7168534 \beta_{4} - 20874 \beta_{5} - 6546454 \beta_{6} + 767930 \beta_{7} + 103568 \beta_{8} + 47382 \beta_{9} + 4142240 \beta_{10} + 167112 \beta_{11} - 280710 \beta_{12} + 318369 \beta_{13} + 181541 \beta_{14} + 23361 \beta_{15} + 36125 \beta_{16} - 66090 \beta_{17} - 7889 \beta_{18} - 24 \beta_{19} ) q^{72} + ( -589856884 + 119116050 \beta_{1} + 119450130 \beta_{2} - 2367654 \beta_{3} - 2413530 \beta_{4} + 1744851 \beta_{5} - 6499311 \beta_{6} - 815457 \beta_{7} - 182679 \beta_{8} - 207771 \beta_{9} - 20604 \beta_{10} - 572151 \beta_{11} - 142506 \beta_{12} - 379023 \beta_{13} + 98862 \beta_{14} + 97974 \beta_{15} - 75324 \beta_{16} - 15396 \beta_{18} ) q^{73} + ( 21583036 - 93994712 \beta_{1} + 8859112 \beta_{2} - 10993906468 \beta_{3} - 4093118 \beta_{4} - 527673 \beta_{5} - 27117180 \beta_{6} - 255475 \beta_{7} + 141386 \beta_{8} + 470334 \beta_{9} - 1351003 \beta_{10} + 109239 \beta_{11} + 261646 \beta_{12} - 77866 \beta_{13} - 69041 \beta_{14} + 6171 \beta_{16} + 135215 \beta_{17} + 19113 \beta_{18} + 95057 \beta_{19} ) q^{74} + ( -11321316759 + 189076164 \beta_{1} + 26007988 \beta_{2} + 873374614 \beta_{3} - 1988306 \beta_{4} - 3154527 \beta_{5} - 1558053 \beta_{6} + 380773 \beta_{7} - 19406 \beta_{8} - 381204 \beta_{9} - 2694356 \beta_{10} + 201117 \beta_{11} - 932994 \beta_{12} + 11211 \beta_{13} + 373729 \beta_{14} + 111123 \beta_{15} - 46412 \beta_{16} + 131760 \beta_{17} - 84601 \beta_{18} - 36195 \beta_{19} ) q^{75} + ( 1410392132 - 10966391 \beta_{1} - 180556939 \beta_{2} - 1406954487 \beta_{3} - 21095069 \beta_{4} + 632461 \beta_{5} + 9471154 \beta_{6} - 77256 \beta_{7} - 6822 \beta_{8} - 510183 \beta_{9} + 86359 \beta_{10} + 18180 \beta_{11} - 291528 \beta_{12} - 460899 \beta_{13} - 242564 \beta_{14} - 34074 \beta_{15} - 6822 \beta_{16} - 92588 \beta_{17} - 21591 \beta_{18} + 34074 \beta_{19} ) q^{76} + ( -8900362949 + 4717739 \beta_{1} - 116192321 \beta_{2} + 8899876117 \beta_{3} + 7215710 \beta_{4} - 6762044 \beta_{5} + 1190325 \beta_{6} + 147076 \beta_{7} + 16164 \beta_{8} + 241852 \beta_{9} - 6441392 \beta_{10} - 132532 \beta_{11} + 215721 \beta_{12} + 186104 \beta_{13} - 33408 \beta_{14} + 107476 \beta_{15} + 16164 \beta_{16} - 227356 \beta_{17} + 19800 \beta_{18} - 107476 \beta_{19} ) q^{77} + ( 17633586893 - 82319477 \beta_{1} + 278566552 \beta_{2} - 20206830794 \beta_{3} + 14856475 \beta_{4} - 7406367 \beta_{5} + 21761560 \beta_{6} - 1796727 \beta_{7} - 172842 \beta_{8} + 420168 \beta_{9} - 5024236 \beta_{10} - 103652 \beta_{11} + 558360 \beta_{12} + 868725 \beta_{13} - 139131 \beta_{14} - 158403 \beta_{15} + 54882 \beta_{16} - 50079 \beta_{17} + 36984 \beta_{18} + 136599 \beta_{19} ) q^{78} + ( -4145176 - 196391963 \beta_{1} - 1241007 \beta_{2} - 821460835 \beta_{3} + 13615838 \beta_{4} + 166533 \beta_{5} - 5755813 \beta_{6} + 2267347 \beta_{7} - 233350 \beta_{8} - 325398 \beta_{9} + 914010 \beta_{10} + 124831 \beta_{11} - 2240172 \beta_{12} - 514499 \beta_{13} + 57977 \beta_{14} + 27175 \beta_{16} - 260525 \beta_{17} + 52955 \beta_{18} - 143571 \beta_{19} ) q^{79} + ( 19993489578 + 102630910 \beta_{1} + 116813512 \beta_{2} + 251746 \beta_{3} - 21132662 \beta_{4} + 63658 \beta_{5} + 21192752 \beta_{6} + 746296 \beta_{7} - 48716 \beta_{8} + 801346 \beta_{9} - 761446 \beta_{10} + 120032 \beta_{11} + 105212 \beta_{12} - 22944 \beta_{13} + 206534 \beta_{14} - 122522 \beta_{15} - 31422 \beta_{16} - 7056 \beta_{18} ) q^{80} + ( 3429823500 - 297011646 \beta_{1} - 280039383 \beta_{2} + 15196012227 \beta_{3} + 6643917 \beta_{4} + 5332986 \beta_{5} - 6074451 \beta_{6} - 649467 \beta_{7} + 101358 \beta_{8} - 340443 \beta_{9} + 2036196 \beta_{10} + 369315 \beta_{11} - 827523 \beta_{12} + 208134 \beta_{13} - 121842 \beta_{14} - 30717 \beta_{15} - 11142 \beta_{16} + 167085 \beta_{17} + 41697 \beta_{18} + 39375 \beta_{19} ) q^{81} + ( -2829273494 + 261134938 \beta_{1} + 263393395 \beta_{2} - 28726311 \beta_{3} - 26907279 \beta_{4} + 84247 \beta_{5} - 56101702 \beta_{6} - 145248 \beta_{7} + 351720 \beta_{8} - 811569 \beta_{9} - 15729 \beta_{10} + 569436 \beta_{11} + 88236 \beta_{12} - 620580 \beta_{13} - 276999 \beta_{14} - 95193 \beta_{15} + 103359 \beta_{16} + 55158 \beta_{18} ) q^{82} + ( 23794250 + 61053026 \beta_{1} + 8111438 \beta_{2} - 9897382013 \beta_{3} - 26931748 \beta_{4} - 200625 \beta_{5} - 9425565 \beta_{6} - 521195 \beta_{7} + 50803 \beta_{8} + 483846 \beta_{9} + 9640393 \beta_{10} - 443490 \beta_{11} + 419588 \beta_{12} + 450334 \beta_{13} - 73720 \beta_{14} - 101607 \beta_{16} + 152410 \beta_{17} - 94407 \beta_{18} - 100949 \beta_{19} ) q^{83} + ( -40888746609 + 552395826 \beta_{1} + 181242222 \beta_{2} + 10805360049 \beta_{3} - 3444346 \beta_{4} + 11823786 \beta_{5} - 4352113 \beta_{6} + 1291668 \beta_{7} + 144753 \beta_{8} + 146646 \beta_{9} - 789519 \beta_{10} - 431289 \beta_{11} + 1584171 \beta_{12} + 1092735 \beta_{13} + 62736 \beta_{14} - 69084 \beta_{15} - 31233 \beta_{16} - 37116 \beta_{17} + 129990 \beta_{18} - 158868 \beta_{19} ) q^{84} + ( 1155652722 - 9696589 \beta_{1} - 346108149 \beta_{2} - 1173569599 \beta_{3} + 19678252 \beta_{4} + 976145 \beta_{5} + 31201383 \beta_{6} - 197845 \beta_{7} + 135644 \beta_{8} + 298808 \beta_{9} + 73105 \beta_{10} - 451167 \beta_{11} + 3223860 \beta_{12} - 407298 \beta_{13} - 65165 \beta_{14} - 144627 \beta_{15} + 135644 \beta_{16} + 510404 \beta_{17} - 26609 \beta_{18} + 144627 \beta_{19} ) q^{85} + ( -21429252310 - 3164441 \beta_{1} + 250677169 \beta_{2} + 21393590005 \beta_{3} + 66907703 \beta_{4} - 3018216 \beta_{5} + 49890695 \beta_{6} - 528280 \beta_{7} - 52272 \beta_{8} + 1142131 \beta_{9} - 3767804 \beta_{10} + 97708 \beta_{11} - 779662 \beta_{12} - 165937 \beta_{13} - 523350 \beta_{14} - 208450 \beta_{15} - 52272 \beta_{16} + 540046 \beta_{17} - 159915 \beta_{18} + 208450 \beta_{19} ) q^{86} + ( 17400439558 - 71675525 \beta_{1} + 220111743 \beta_{2} - 24775104896 \beta_{3} - 3678812 \beta_{4} + 2807099 \beta_{5} - 12010818 \beta_{6} - 330489 \beta_{7} + 419859 \beta_{8} - 831132 \beta_{9} + 6653423 \beta_{10} + 349912 \beta_{11} + 930366 \beta_{12} - 105192 \beta_{13} + 211600 \beta_{14} - 495 \beta_{15} - 30996 \beta_{16} + 60417 \beta_{17} + 209421 \beta_{18} - 205749 \beta_{19} ) q^{87} + ( -35069982 - 333004288 \beta_{1} - 10925640 \beta_{2} - 5526022252 \beta_{3} + 40602513 \beta_{4} + 899865 \beta_{5} + 12645786 \beta_{6} + 958713 \beta_{7} + 665751 \beta_{8} - 1058121 \beta_{9} - 276155 \beta_{10} + 1198752 \beta_{11} - 852849 \beta_{12} - 621408 \beta_{13} + 1150761 \beta_{14} + 105864 \beta_{16} + 559887 \beta_{17} + 52752 \beta_{18} - 4041 \beta_{19} ) q^{88} + ( 23839920480 + 202611814 \beta_{1} + 191715832 \beta_{2} + 29786569 \beta_{3} + 40638236 \beta_{4} + 10489914 \beta_{5} + 34339819 \beta_{6} - 1789746 \beta_{7} - 112025 \beta_{8} + 413948 \beta_{9} + 689589 \beta_{10} + 347105 \beta_{11} + 213752 \beta_{12} + 755219 \beta_{13} + 862877 \beta_{14} - 134615 \beta_{15} - 174159 \beta_{16} - 126648 \beta_{18} ) q^{89} + ( 14625227475 - 220973700 \beta_{1} - 525123696 \beta_{2} + 32508889920 \beta_{3} + 7028916 \beta_{4} + 812169 \beta_{5} + 10719459 \beta_{6} + 1599276 \beta_{7} - 365331 \beta_{8} + 503037 \beta_{9} + 7588587 \beta_{10} - 1655790 \beta_{11} - 2362509 \beta_{12} + 1032324 \beta_{13} - 882063 \beta_{14} + 7338 \beta_{15} - 92799 \beta_{16} - 130038 \beta_{17} + 40425 \beta_{18} - 35754 \beta_{19} ) q^{90} + ( 7075339952 + 494509986 \beta_{1} + 511319597 \beta_{2} - 6584962 \beta_{3} - 31405979 \beta_{4} - 7252795 \beta_{5} + 15901028 \beta_{6} - 1922084 \beta_{7} + 255960 \beta_{8} + 803879 \beta_{9} - 1130534 \beta_{10} + 1182884 \beta_{11} + 283632 \beta_{12} + 60497 \beta_{13} - 306260 \beta_{14} - 108384 \beta_{15} + 193576 \beta_{16} + 9164 \beta_{18} ) q^{91} + ( 11749610 + 283061217 \beta_{1} + 4044794 \beta_{2} - 46098102341 \beta_{3} + 2286173 \beta_{4} + 71691 \beta_{5} - 17994510 \beta_{6} + 2566939 \beta_{7} - 776915 \beta_{8} + 61059 \beta_{9} - 8827472 \beta_{10} - 594456 \beta_{11} - 2555479 \beta_{12} - 692318 \beta_{13} - 125167 \beta_{14} + 11460 \beta_{16} - 788375 \beta_{17} - 44250 \beta_{18} - 225119 \beta_{19} ) q^{92} + ( -32693456787 + 12105234 \beta_{1} + 11146270 \beta_{2} + 8086174909 \beta_{3} + 4084496 \beta_{4} - 11464371 \beta_{5} + 11076859 \beta_{6} + 388273 \beta_{7} - 332969 \beta_{8} - 106404 \beta_{9} - 4854308 \beta_{10} - 364212 \beta_{11} - 225963 \beta_{12} - 730107 \beta_{13} - 2017106 \beta_{14} - 248406 \beta_{15} + 197335 \beta_{16} - 516534 \beta_{17} + 202433 \beta_{18} + 396321 \beta_{19} ) q^{93} + ( 1273511863 - 655691 \beta_{1} - 433724687 \beta_{2} - 1239254364 \beta_{3} - 67466917 \beta_{4} - 648565 \beta_{5} - 39179326 \beta_{6} + 648643 \beta_{7} + 18577 \beta_{8} - 1548308 \beta_{9} - 145182 \beta_{10} - 90831 \beta_{11} - 2040261 \beta_{12} - 266136 \beta_{13} + 616544 \beta_{14} + 332583 \beta_{15} + 18577 \beta_{16} - 627065 \beta_{17} + 158030 \beta_{18} - 332583 \beta_{19} ) q^{94} + ( -41804113616 + 9664142 \beta_{1} + 91750462 \beta_{2} + 41840091402 \beta_{3} - 57589356 \beta_{4} - 7350730 \beta_{5} - 46769494 \beta_{6} - 119270 \beta_{7} - 161532 \beta_{8} - 594744 \beta_{9} - 6356910 \beta_{10} + 559646 \beta_{11} + 940700 \beta_{12} + 841464 \beta_{13} - 99630 \beta_{14} - 16394 \beta_{15} - 161532 \beta_{16} - 97672 \beta_{17} - 51438 \beta_{18} + 16394 \beta_{19} ) q^{95} + ( 35150892804 - 477546969 \beta_{1} - 119762391 \beta_{2} - 58600593111 \beta_{3} - 10017024 \beta_{4} - 3884799 \beta_{5} - 7824810 \beta_{6} + 497640 \beta_{7} - 30105 \beta_{8} + 553098 \beta_{9} - 11562897 \beta_{10} - 404445 \beta_{11} - 1479519 \beta_{12} + 209601 \beta_{13} + 291606 \beta_{14} + 459564 \beta_{15} - 197895 \beta_{16} + 444876 \beta_{17} - 273666 \beta_{18} - 42024 \beta_{19} ) q^{96} + ( 89936582 - 398464081 \beta_{1} + 32526401 \beta_{2} + 12178792799 \beta_{3} - 71224632 \beta_{4} - 1629015 \beta_{5} - 63785010 \beta_{6} - 2641069 \beta_{7} - 110551 \beta_{8} + 2318682 \beta_{9} + 3563634 \beta_{10} - 912876 \beta_{11} + 2647584 \beta_{12} + 991907 \beta_{13} - 119008 \beta_{14} + 6515 \beta_{16} - 117066 \beta_{17} - 229889 \beta_{18} + 368607 \beta_{19} ) q^{97} + ( 65447323998 - 625101573 \beta_{1} - 599753629 \beta_{2} - 73951098 \beta_{3} - 97967648 \beta_{4} - 23402111 \beta_{5} - 94379669 \beta_{6} - 399702 \beta_{7} + 818233 \beta_{8} - 1398317 \beta_{9} - 908203 \beta_{10} + 498026 \beta_{11} - 395365 \beta_{12} - 1968378 \beta_{13} - 858451 \beta_{14} + 652690 \beta_{15} + 169071 \beta_{16} - 44127 \beta_{18} ) q^{98} + ( 22458395685 + 130083129 \beta_{1} + 283671274 \beta_{2} + 17316263516 \beta_{3} + 18309676 \beta_{4} - 1040802 \beta_{5} - 8713739 \beta_{6} - 774101 \beta_{7} - 56888 \beta_{8} - 1031199 \beta_{9} - 15730601 \beta_{10} + 1164009 \beta_{11} + 3238362 \beta_{12} - 2442615 \beta_{13} + 2934655 \beta_{14} - 5691 \beta_{15} + 141976 \beta_{16} + 120696 \beta_{17} + 95429 \beta_{18} - 67977 \beta_{19} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20q - 33q^{2} - 12q^{3} - 9217q^{4} - 7230q^{5} + 20583q^{6} + 8512q^{7} - 29118q^{8} + 135504q^{9} + O(q^{10}) \) \( 20q - 33q^{2} - 12q^{3} - 9217q^{4} - 7230q^{5} + 20583q^{6} + 8512q^{7} - 29118q^{8} + 135504q^{9} + 4092q^{10} - 112776q^{11} + 1027860q^{12} + 279706q^{13} - 3901584q^{14} - 6358608q^{15} - 7342081q^{16} + 27765792q^{17} + 8682876q^{18} + 7029400q^{19} - 34163508q^{20} + 55012206q^{21} + 2274591q^{22} - 69371616q^{23} - 211100355q^{24} - 45286204q^{25} + 481929144q^{26} - 83699352q^{27} - 61345796q^{28} - 25437246q^{29} - 23582592q^{30} + 114575368q^{31} + 80396559q^{32} + 31338342q^{33} - 243855063q^{34} - 178147464q^{35} - 19984653q^{36} - 134218328q^{37} + 489799995q^{38} - 1999064976q^{39} + 107425416q^{40} + 331873026q^{41} + 4171968882q^{42} - 1118847584q^{43} + 278477274q^{44} + 1749349170q^{45} + 2882537592q^{46} - 1469650704q^{47} - 9335236125q^{48} - 3553434720q^{49} - 6643771701q^{50} + 1736777052q^{51} + 3632448874q^{52} + 14914261944q^{53} + 18127857753q^{54} + 4981449984q^{55} - 27669139026q^{56} - 7855424196q^{57} - 1387480560q^{58} - 26505032592q^{59} - 10283356116q^{60} + 990409066q^{61} + 91044996180q^{62} + 51565206888q^{63} - 7516709566q^{64} - 39045315390q^{65} - 93201828246q^{66} + 6557215720q^{67} - 77299152993q^{68} + 6907292550q^{69} - 785437278q^{70} + 122053719744q^{71} + 161899013547q^{72} - 12612893936q^{73} - 109519086216q^{74} - 218383044348q^{75} + 14574055597q^{76} - 88616208018q^{77} + 150319870614q^{78} - 7621233248q^{79} + 399166683072q^{80} + 222307104312q^{81} - 59168477334q^{82} - 99007044180q^{83} - 711968015814q^{84} + 12911595156q^{85} - 214357830519q^{86} + 99715491216q^{87} - 54423523605q^{88} + 476597704824q^{89} + 620021743884q^{90} + 138211652216q^{91} - 461776423998q^{92} - 572981484354q^{93} + 13393667064q^{94} - 418952909328q^{95} + 118587589272q^{96} + 123483551938q^{97} + 1310123604078q^{98} + 621154334268q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{20} + 9863 x^{18} + 40416552 x^{16} + 89424581388 x^{14} + 116167273852206 x^{12} + 90155066123992770 x^{10} + 40615438428348034476 x^{8} + 9750032167348768665084 x^{6} + 1016547159346572680888121 x^{4} + 22772285013675084656406191 x^{2} + 59348899944549565171293124\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\(31285236839951669629877 \nu^{18} + 306165294234660048396729540 \nu^{16} + 1248856157739744317892914467260 \nu^{14} + 2757154538934650640796654672929672 \nu^{12} + 3561953767047321278890139438297032302 \nu^{10} + 2694956424448654794709530402831092777856 \nu^{8} + 1117280057517996170054159781948717061704732 \nu^{6} + 215319004597386877272192167363465215730471448 \nu^{4} + 13418460491308855372445877668935436240028188805 \nu^{2} - 46511484941820046638418322741100684388388044800 \nu + 84766046618479519445647956789987018374677180828\)\()/ \)\(31\!\cdots\!00\)\( \)
\(\beta_{2}\)\(=\)\((\)\(31285236839951669629877 \nu^{18} + 306165294234660048396729540 \nu^{16} + 1248856157739744317892914467260 \nu^{14} + 2757154538934650640796654672929672 \nu^{12} + 3561953767047321278890139438297032302 \nu^{10} + 2694956424448654794709530402831092777856 \nu^{8} + 1117280057517996170054159781948717061704732 \nu^{6} + 215319004597386877272192167363465215730471448 \nu^{4} + 13418460491308855372445877668935436240028188805 \nu^{2} + 46511484941820046638418322741100684388388044800 \nu + 84766046618479519445647956789987018374677180828\)\()/ \)\(31\!\cdots\!00\)\( \)
\(\beta_{3}\)\(=\)\((\)\(-\)\(55\!\cdots\!73\)\( \nu^{19} - \)\(54\!\cdots\!56\)\( \nu^{17} - \)\(22\!\cdots\!36\)\( \nu^{15} - \)\(48\!\cdots\!84\)\( \nu^{13} - \)\(62\!\cdots\!90\)\( \nu^{11} - \)\(48\!\cdots\!92\)\( \nu^{9} - \)\(21\!\cdots\!44\)\( \nu^{7} - \)\(49\!\cdots\!44\)\( \nu^{5} - \)\(47\!\cdots\!01\)\( \nu^{3} - \)\(73\!\cdots\!48\)\( \nu + \)\(59\!\cdots\!00\)\(\)\()/ \)\(11\!\cdots\!00\)\( \)
\(\beta_{4}\)\(=\)\((\)\(\)\(47\!\cdots\!59\)\( \nu^{19} - \)\(58\!\cdots\!62\)\( \nu^{18} + \)\(49\!\cdots\!28\)\( \nu^{17} - \)\(57\!\cdots\!28\)\( \nu^{16} + \)\(21\!\cdots\!68\)\( \nu^{15} - \)\(22\!\cdots\!48\)\( \nu^{14} + \)\(49\!\cdots\!52\)\( \nu^{13} - \)\(48\!\cdots\!00\)\( \nu^{12} + \)\(68\!\cdots\!30\)\( \nu^{11} - \)\(59\!\cdots\!88\)\( \nu^{10} + \)\(57\!\cdots\!76\)\( \nu^{9} - \)\(41\!\cdots\!80\)\( \nu^{8} + \)\(28\!\cdots\!32\)\( \nu^{7} - \)\(16\!\cdots\!20\)\( \nu^{6} + \)\(78\!\cdots\!12\)\( \nu^{5} - \)\(34\!\cdots\!64\)\( \nu^{4} + \)\(10\!\cdots\!63\)\( \nu^{3} - \)\(42\!\cdots\!98\)\( \nu^{2} + \)\(55\!\cdots\!64\)\( \nu - \)\(13\!\cdots\!96\)\(\)\()/ \)\(41\!\cdots\!00\)\( \)
\(\beta_{5}\)\(=\)\((\)\(\)\(62\!\cdots\!23\)\( \nu^{19} + \)\(72\!\cdots\!39\)\( \nu^{18} + \)\(62\!\cdots\!32\)\( \nu^{17} + \)\(70\!\cdots\!24\)\( \nu^{16} + \)\(26\!\cdots\!12\)\( \nu^{15} + \)\(28\!\cdots\!64\)\( \nu^{14} + \)\(58\!\cdots\!20\)\( \nu^{13} + \)\(60\!\cdots\!88\)\( \nu^{12} + \)\(78\!\cdots\!42\)\( \nu^{11} + \)\(74\!\cdots\!02\)\( \nu^{10} + \)\(62\!\cdots\!80\)\( \nu^{9} + \)\(54\!\cdots\!64\)\( \nu^{8} + \)\(28\!\cdots\!00\)\( \nu^{7} + \)\(22\!\cdots\!88\)\( \nu^{6} + \)\(70\!\cdots\!96\)\( \nu^{5} + \)\(45\!\cdots\!24\)\( \nu^{4} + \)\(73\!\cdots\!87\)\( \nu^{3} + \)\(39\!\cdots\!19\)\( \nu^{2} + \)\(96\!\cdots\!04\)\( \nu + \)\(28\!\cdots\!60\)\(\)\()/ \)\(82\!\cdots\!00\)\( \)
\(\beta_{6}\)\(=\)\((\)\(\)\(62\!\cdots\!23\)\( \nu^{19} + \)\(63\!\cdots\!59\)\( \nu^{18} + \)\(62\!\cdots\!32\)\( \nu^{17} + \)\(61\!\cdots\!24\)\( \nu^{16} + \)\(26\!\cdots\!12\)\( \nu^{15} + \)\(24\!\cdots\!64\)\( \nu^{14} + \)\(58\!\cdots\!20\)\( \nu^{13} + \)\(52\!\cdots\!08\)\( \nu^{12} + \)\(78\!\cdots\!42\)\( \nu^{11} + \)\(65\!\cdots\!22\)\( \nu^{10} + \)\(62\!\cdots\!80\)\( \nu^{9} + \)\(47\!\cdots\!24\)\( \nu^{8} + \)\(28\!\cdots\!00\)\( \nu^{7} + \)\(19\!\cdots\!08\)\( \nu^{6} + \)\(70\!\cdots\!96\)\( \nu^{5} + \)\(39\!\cdots\!04\)\( \nu^{4} + \)\(73\!\cdots\!87\)\( \nu^{3} + \)\(33\!\cdots\!19\)\( \nu^{2} + \)\(96\!\cdots\!04\)\( \nu + \)\(38\!\cdots\!40\)\(\)\()/ \)\(82\!\cdots\!00\)\( \)
\(\beta_{7}\)\(=\)\((\)\(\)\(76\!\cdots\!52\)\( \nu^{19} + \)\(16\!\cdots\!41\)\( \nu^{18} + \)\(77\!\cdots\!60\)\( \nu^{17} + \)\(10\!\cdots\!72\)\( \nu^{16} + \)\(32\!\cdots\!80\)\( \nu^{15} + \)\(99\!\cdots\!32\)\( \nu^{14} + \)\(73\!\cdots\!92\)\( \nu^{13} - \)\(70\!\cdots\!52\)\( \nu^{12} + \)\(98\!\cdots\!92\)\( \nu^{11} - \)\(24\!\cdots\!30\)\( \nu^{10} + \)\(79\!\cdots\!16\)\( \nu^{9} - \)\(35\!\cdots\!76\)\( \nu^{8} + \)\(37\!\cdots\!52\)\( \nu^{7} - \)\(24\!\cdots\!32\)\( \nu^{6} + \)\(94\!\cdots\!88\)\( \nu^{5} - \)\(87\!\cdots\!12\)\( \nu^{4} + \)\(10\!\cdots\!00\)\( \nu^{3} - \)\(13\!\cdots\!63\)\( \nu^{2} + \)\(26\!\cdots\!48\)\( \nu - \)\(47\!\cdots\!64\)\(\)\()/ \)\(41\!\cdots\!00\)\( \)
\(\beta_{8}\)\(=\)\((\)\(\)\(13\!\cdots\!76\)\( \nu^{19} + \)\(30\!\cdots\!09\)\( \nu^{18} + \)\(15\!\cdots\!36\)\( \nu^{17} + \)\(30\!\cdots\!00\)\( \nu^{16} + \)\(73\!\cdots\!96\)\( \nu^{15} + \)\(12\!\cdots\!40\)\( \nu^{14} + \)\(19\!\cdots\!12\)\( \nu^{13} + \)\(29\!\cdots\!44\)\( \nu^{12} + \)\(30\!\cdots\!08\)\( \nu^{11} + \)\(39\!\cdots\!74\)\( \nu^{10} + \)\(29\!\cdots\!36\)\( \nu^{9} + \)\(32\!\cdots\!12\)\( \nu^{8} + \)\(16\!\cdots\!12\)\( \nu^{7} + \)\(15\!\cdots\!64\)\( \nu^{6} + \)\(48\!\cdots\!56\)\( \nu^{5} + \)\(36\!\cdots\!56\)\( \nu^{4} + \)\(61\!\cdots\!16\)\( \nu^{3} + \)\(36\!\cdots\!05\)\( \nu^{2} + \)\(16\!\cdots\!60\)\( \nu + \)\(35\!\cdots\!96\)\(\)\()/ \)\(41\!\cdots\!00\)\( \)
\(\beta_{9}\)\(=\)\((\)\(-\)\(70\!\cdots\!90\)\( \nu^{19} + \)\(33\!\cdots\!05\)\( \nu^{18} - \)\(72\!\cdots\!60\)\( \nu^{17} + \)\(32\!\cdots\!04\)\( \nu^{16} - \)\(30\!\cdots\!60\)\( \nu^{15} + \)\(12\!\cdots\!04\)\( \nu^{14} - \)\(71\!\cdots\!00\)\( \nu^{13} + \)\(27\!\cdots\!24\)\( \nu^{12} - \)\(99\!\cdots\!60\)\( \nu^{11} + \)\(33\!\cdots\!38\)\( \nu^{10} - \)\(82\!\cdots\!00\)\( \nu^{9} + \)\(24\!\cdots\!92\)\( \nu^{8} - \)\(39\!\cdots\!00\)\( \nu^{7} + \)\(97\!\cdots\!04\)\( \nu^{6} - \)\(10\!\cdots\!80\)\( \nu^{5} + \)\(20\!\cdots\!28\)\( \nu^{4} - \)\(96\!\cdots\!10\)\( \nu^{3} + \)\(18\!\cdots\!69\)\( \nu^{2} + \)\(24\!\cdots\!80\)\( \nu + \)\(32\!\cdots\!44\)\(\)\()/ \)\(13\!\cdots\!00\)\( \)
\(\beta_{10}\)\(=\)\((\)\(\)\(15\!\cdots\!51\)\( \nu^{19} - \)\(60\!\cdots\!15\)\( \nu^{18} + \)\(15\!\cdots\!80\)\( \nu^{17} - \)\(58\!\cdots\!00\)\( \nu^{16} + \)\(64\!\cdots\!40\)\( \nu^{15} - \)\(24\!\cdots\!00\)\( \nu^{14} + \)\(14\!\cdots\!96\)\( \nu^{13} - \)\(53\!\cdots\!40\)\( \nu^{12} + \)\(18\!\cdots\!46\)\( \nu^{11} - \)\(68\!\cdots\!90\)\( \nu^{10} + \)\(13\!\cdots\!08\)\( \nu^{9} - \)\(51\!\cdots\!20\)\( \nu^{8} + \)\(61\!\cdots\!76\)\( \nu^{7} - \)\(21\!\cdots\!40\)\( \nu^{6} + \)\(14\!\cdots\!44\)\( \nu^{5} - \)\(41\!\cdots\!60\)\( \nu^{4} + \)\(13\!\cdots\!75\)\( \nu^{3} - \)\(43\!\cdots\!75\)\( \nu^{2} + \)\(21\!\cdots\!24\)\( \nu - \)\(17\!\cdots\!60\)\(\)\()/ \)\(11\!\cdots\!00\)\( \)
\(\beta_{11}\)\(=\)\((\)\(\)\(16\!\cdots\!73\)\( \nu^{19} + \)\(50\!\cdots\!67\)\( \nu^{18} + \)\(16\!\cdots\!40\)\( \nu^{17} + \)\(45\!\cdots\!72\)\( \nu^{16} + \)\(66\!\cdots\!20\)\( \nu^{15} + \)\(16\!\cdots\!92\)\( \nu^{14} + \)\(14\!\cdots\!08\)\( \nu^{13} + \)\(31\!\cdots\!64\)\( \nu^{12} + \)\(18\!\cdots\!58\)\( \nu^{11} + \)\(33\!\cdots\!06\)\( \nu^{10} + \)\(13\!\cdots\!84\)\( \nu^{9} + \)\(19\!\cdots\!92\)\( \nu^{8} + \)\(59\!\cdots\!48\)\( \nu^{7} + \)\(55\!\cdots\!64\)\( \nu^{6} + \)\(13\!\cdots\!12\)\( \nu^{5} + \)\(33\!\cdots\!72\)\( \nu^{4} + \)\(12\!\cdots\!25\)\( \nu^{3} - \)\(89\!\cdots\!93\)\( \nu^{2} + \)\(21\!\cdots\!52\)\( \nu + \)\(51\!\cdots\!80\)\(\)\()/ \)\(82\!\cdots\!00\)\( \)
\(\beta_{12}\)\(=\)\((\)\(-\)\(29\!\cdots\!63\)\( \nu^{19} - \)\(11\!\cdots\!87\)\( \nu^{18} - \)\(29\!\cdots\!64\)\( \nu^{17} - \)\(11\!\cdots\!64\)\( \nu^{16} - \)\(11\!\cdots\!44\)\( \nu^{15} - \)\(50\!\cdots\!84\)\( \nu^{14} - \)\(25\!\cdots\!12\)\( \nu^{13} - \)\(11\!\cdots\!96\)\( \nu^{12} - \)\(33\!\cdots\!46\)\( \nu^{11} - \)\(16\!\cdots\!10\)\( \nu^{10} - \)\(25\!\cdots\!16\)\( \nu^{9} - \)\(13\!\cdots\!48\)\( \nu^{8} - \)\(11\!\cdots\!32\)\( \nu^{7} - \)\(64\!\cdots\!36\)\( \nu^{6} - \)\(27\!\cdots\!20\)\( \nu^{5} - \)\(16\!\cdots\!36\)\( \nu^{4} - \)\(27\!\cdots\!39\)\( \nu^{3} - \)\(20\!\cdots\!19\)\( \nu^{2} - \)\(49\!\cdots\!56\)\( \nu - \)\(56\!\cdots\!12\)\(\)\()/ \)\(82\!\cdots\!00\)\( \)
\(\beta_{13}\)\(=\)\((\)\(-\)\(34\!\cdots\!53\)\( \nu^{19} + \)\(93\!\cdots\!43\)\( \nu^{18} - \)\(34\!\cdots\!08\)\( \nu^{17} + \)\(90\!\cdots\!80\)\( \nu^{16} - \)\(14\!\cdots\!88\)\( \nu^{15} + \)\(36\!\cdots\!60\)\( \nu^{14} - \)\(32\!\cdots\!36\)\( \nu^{13} + \)\(77\!\cdots\!68\)\( \nu^{12} - \)\(42\!\cdots\!74\)\( \nu^{11} + \)\(97\!\cdots\!58\)\( \nu^{10} - \)\(33\!\cdots\!08\)\( \nu^{9} + \)\(71\!\cdots\!64\)\( \nu^{8} - \)\(15\!\cdots\!36\)\( \nu^{7} + \)\(29\!\cdots\!08\)\( \nu^{6} - \)\(38\!\cdots\!68\)\( \nu^{5} + \)\(60\!\cdots\!72\)\( \nu^{4} - \)\(41\!\cdots\!73\)\( \nu^{3} + \)\(36\!\cdots\!15\)\( \nu^{2} - \)\(10\!\cdots\!80\)\( \nu - \)\(46\!\cdots\!28\)\(\)\()/ \)\(82\!\cdots\!00\)\( \)
\(\beta_{14}\)\(=\)\((\)\(-\)\(40\!\cdots\!27\)\( \nu^{19} - \)\(19\!\cdots\!31\)\( \nu^{18} - \)\(39\!\cdots\!48\)\( \nu^{17} - \)\(18\!\cdots\!08\)\( \nu^{16} - \)\(15\!\cdots\!68\)\( \nu^{15} - \)\(67\!\cdots\!68\)\( \nu^{14} - \)\(34\!\cdots\!60\)\( \nu^{13} - \)\(13\!\cdots\!84\)\( \nu^{12} - \)\(44\!\cdots\!18\)\( \nu^{11} - \)\(14\!\cdots\!82\)\( \nu^{10} - \)\(33\!\cdots\!60\)\( \nu^{9} - \)\(86\!\cdots\!12\)\( \nu^{8} - \)\(14\!\cdots\!80\)\( \nu^{7} - \)\(26\!\cdots\!24\)\( \nu^{6} - \)\(33\!\cdots\!64\)\( \nu^{5} - \)\(32\!\cdots\!20\)\( \nu^{4} - \)\(31\!\cdots\!43\)\( \nu^{3} - \)\(99\!\cdots\!83\)\( \nu^{2} - \)\(46\!\cdots\!76\)\( \nu - \)\(11\!\cdots\!12\)\(\)\()/ \)\(82\!\cdots\!00\)\( \)
\(\beta_{15}\)\(=\)\((\)\(-\)\(11\!\cdots\!87\)\( \nu^{19} + \)\(17\!\cdots\!90\)\( \nu^{18} - \)\(11\!\cdots\!64\)\( \nu^{17} + \)\(16\!\cdots\!88\)\( \nu^{16} - \)\(47\!\cdots\!84\)\( \nu^{15} + \)\(67\!\cdots\!88\)\( \nu^{14} - \)\(10\!\cdots\!96\)\( \nu^{13} + \)\(14\!\cdots\!08\)\( \nu^{12} - \)\(13\!\cdots\!10\)\( \nu^{11} + \)\(17\!\cdots\!16\)\( \nu^{10} - \)\(10\!\cdots\!48\)\( \nu^{9} + \)\(11\!\cdots\!64\)\( \nu^{8} - \)\(49\!\cdots\!36\)\( \nu^{7} + \)\(38\!\cdots\!68\)\( \nu^{6} - \)\(11\!\cdots\!36\)\( \nu^{5} + \)\(43\!\cdots\!36\)\( \nu^{4} - \)\(11\!\cdots\!19\)\( \nu^{3} - \)\(21\!\cdots\!82\)\( \nu^{2} - \)\(20\!\cdots\!12\)\( \nu - \)\(72\!\cdots\!12\)\(\)\()/ \)\(20\!\cdots\!00\)\( \)
\(\beta_{16}\)\(=\)\((\)\(-\)\(37\!\cdots\!29\)\( \nu^{19} + \)\(28\!\cdots\!65\)\( \nu^{18} - \)\(36\!\cdots\!60\)\( \nu^{17} + \)\(22\!\cdots\!88\)\( \nu^{16} - \)\(14\!\cdots\!00\)\( \nu^{15} + \)\(66\!\cdots\!88\)\( \nu^{14} - \)\(29\!\cdots\!24\)\( \nu^{13} + \)\(75\!\cdots\!08\)\( \nu^{12} - \)\(35\!\cdots\!14\)\( \nu^{11} - \)\(10\!\cdots\!34\)\( \nu^{10} - \)\(24\!\cdots\!52\)\( \nu^{9} - \)\(10\!\cdots\!36\)\( \nu^{8} - \)\(89\!\cdots\!44\)\( \nu^{7} - \)\(96\!\cdots\!32\)\( \nu^{6} - \)\(15\!\cdots\!56\)\( \nu^{5} - \)\(35\!\cdots\!64\)\( \nu^{4} - \)\(75\!\cdots\!65\)\( \nu^{3} - \)\(46\!\cdots\!07\)\( \nu^{2} + \)\(24\!\cdots\!64\)\( \nu - \)\(42\!\cdots\!12\)\(\)\()/ \)\(59\!\cdots\!00\)\( \)
\(\beta_{17}\)\(=\)\((\)\(-\)\(55\!\cdots\!53\)\( \nu^{19} - \)\(40\!\cdots\!21\)\( \nu^{18} - \)\(55\!\cdots\!20\)\( \nu^{17} - \)\(60\!\cdots\!72\)\( \nu^{16} - \)\(22\!\cdots\!00\)\( \nu^{15} - \)\(34\!\cdots\!32\)\( \nu^{14} - \)\(50\!\cdots\!68\)\( \nu^{13} - \)\(10\!\cdots\!28\)\( \nu^{12} - \)\(65\!\cdots\!98\)\( \nu^{11} - \)\(17\!\cdots\!50\)\( \nu^{10} - \)\(51\!\cdots\!64\)\( \nu^{9} - \)\(17\!\cdots\!64\)\( \nu^{8} - \)\(23\!\cdots\!08\)\( \nu^{7} - \)\(10\!\cdots\!48\)\( \nu^{6} - \)\(60\!\cdots\!92\)\( \nu^{5} - \)\(28\!\cdots\!08\)\( \nu^{4} - \)\(72\!\cdots\!05\)\( \nu^{3} - \)\(31\!\cdots\!37\)\( \nu^{2} - \)\(28\!\cdots\!52\)\( \nu - \)\(29\!\cdots\!56\)\(\)\()/ \)\(82\!\cdots\!00\)\( \)
\(\beta_{18}\)\(=\)\((\)\(-\)\(41\!\cdots\!33\)\( \nu^{19} + \)\(29\!\cdots\!21\)\( \nu^{18} - \)\(41\!\cdots\!88\)\( \nu^{17} + \)\(28\!\cdots\!32\)\( \nu^{16} - \)\(16\!\cdots\!68\)\( \nu^{15} + \)\(10\!\cdots\!92\)\( \nu^{14} - \)\(37\!\cdots\!96\)\( \nu^{13} + \)\(22\!\cdots\!88\)\( \nu^{12} - \)\(49\!\cdots\!14\)\( \nu^{11} + \)\(26\!\cdots\!70\)\( \nu^{10} - \)\(39\!\cdots\!88\)\( \nu^{9} + \)\(17\!\cdots\!44\)\( \nu^{8} - \)\(18\!\cdots\!96\)\( \nu^{7} + \)\(66\!\cdots\!08\)\( \nu^{6} - \)\(44\!\cdots\!48\)\( \nu^{5} + \)\(12\!\cdots\!28\)\( \nu^{4} - \)\(45\!\cdots\!53\)\( \nu^{3} + \)\(11\!\cdots\!97\)\( \nu^{2} - \)\(93\!\cdots\!80\)\( \nu + \)\(90\!\cdots\!16\)\(\)\()/ \)\(41\!\cdots\!00\)\( \)
\(\beta_{19}\)\(=\)\((\)\(\)\(15\!\cdots\!35\)\( \nu^{19} - \)\(32\!\cdots\!13\)\( \nu^{18} + \)\(15\!\cdots\!80\)\( \nu^{17} - \)\(29\!\cdots\!20\)\( \nu^{16} + \)\(61\!\cdots\!80\)\( \nu^{15} - \)\(10\!\cdots\!00\)\( \nu^{14} + \)\(13\!\cdots\!40\)\( \nu^{13} - \)\(20\!\cdots\!28\)\( \nu^{12} + \)\(17\!\cdots\!70\)\( \nu^{11} - \)\(23\!\cdots\!58\)\( \nu^{10} + \)\(13\!\cdots\!20\)\( \nu^{9} - \)\(15\!\cdots\!44\)\( \nu^{8} + \)\(63\!\cdots\!40\)\( \nu^{7} - \)\(69\!\cdots\!68\)\( \nu^{6} + \)\(15\!\cdots\!00\)\( \nu^{5} - \)\(19\!\cdots\!32\)\( \nu^{4} + \)\(16\!\cdots\!55\)\( \nu^{3} - \)\(25\!\cdots\!05\)\( \nu^{2} + \)\(33\!\cdots\!20\)\( \nu - \)\(42\!\cdots\!92\)\(\)\()/ \)\(10\!\cdots\!00\)\( \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{2} - \beta_{1}\)\()/3\)
\(\nu^{2}\)\(=\)\((\)\(-\beta_{6} + \beta_{5} - 5 \beta_{2} - 5 \beta_{1} - 2960\)\()/3\)
\(\nu^{3}\)\(=\)\((\)\(-2 \beta_{19} - 2 \beta_{17} + \beta_{15} + \beta_{14} - \beta_{13} - 3 \beta_{12} - \beta_{11} + 20 \beta_{10} - 3 \beta_{9} - \beta_{8} + 2 \beta_{7} + 7 \beta_{6} + 11 \beta_{5} - 89 \beta_{4} + 32176 \beta_{3} - 4989 \beta_{2} + 4966 \beta_{1} - 16054\)\()/9\)
\(\nu^{4}\)\(=\)\((\)\(-20 \beta_{18} + 5 \beta_{16} - 6 \beta_{15} + 56 \beta_{14} + 13 \beta_{13} + 51 \beta_{12} + 149 \beta_{11} - 55 \beta_{10} + 58 \beta_{9} - 23 \beta_{8} - 388 \beta_{7} + 9352 \beta_{6} - 6815 \beta_{5} - 708 \beta_{4} + 775 \beta_{3} + 57719 \beta_{2} + 56903 \beta_{1} + 14814645\)\()/9\)
\(\nu^{5}\)\(=\)\((\)\(15832 \beta_{19} - 552 \beta_{18} + 18904 \beta_{17} - 1428 \beta_{16} - 7916 \beta_{15} - 12212 \beta_{14} + 2576 \beta_{13} + 12352 \beta_{12} + 8120 \beta_{11} - 255496 \beta_{10} + 29796 \beta_{9} + 8024 \beta_{8} - 11400 \beta_{7} - 26310 \beta_{6} - 135790 \beta_{5} + 1245188 \beta_{4} - 361469896 \beta_{3} + 28852195 \beta_{2} - 28605015 \beta_{1} + 180221954\)\()/27\)
\(\nu^{6}\)\(=\)\((\)\(77888 \beta_{18} - 13892 \beta_{16} + 36330 \beta_{15} - 242210 \beta_{14} - 105166 \beta_{13} - 205998 \beta_{12} - 578622 \beta_{11} + 163276 \beta_{10} - 240658 \beta_{9} + 94822 \beta_{8} + 1083040 \beta_{7} - 26535969 \beta_{6} + 15263893 \beta_{5} + 1128290 \beta_{4} - 3923656 \beta_{3} - 185810527 \beta_{2} - 183183017 \beta_{1} - 28707727592\)\()/9\)
\(\nu^{7}\)\(=\)\((\)\(-12108962 \beta_{19} + 662708 \beta_{18} - 16789658 \beta_{17} + 2157862 \beta_{16} + 6054481 \beta_{15} + 11993357 \beta_{14} + 1552549 \beta_{13} + 785851 \beta_{12} - 7239151 \beta_{11} + 281142674 \beta_{10} - 25076727 \beta_{9} - 6236967 \beta_{8} + 4375954 \beta_{7} - 45104623 \beta_{6} + 146448683 \beta_{5} - 1274347625 \beta_{4} + 384415295490 \beta_{3} - 20077240801 \beta_{2} + 19876160026 \beta_{1} - 191653437692\)\()/9\)
\(\nu^{8}\)\(=\)\((\)\(-227714548 \beta_{18} + 29592577 \beta_{16} - 138269400 \beta_{15} + 762227890 \beta_{14} + 393077451 \beta_{13} + 623291073 \beta_{12} + 1683733059 \beta_{11} - 431435883 \beta_{10} + 765328180 \beta_{9} - 305177885 \beta_{8} - 2547232580 \beta_{7} + 71559960762 \beta_{6} - 34761494029 \beta_{5} - 975629150 \beta_{4} + 13410166279 \beta_{3} + 559042169979 \beta_{2} + 551805927309 \beta_{1} + 60229425527857\)\()/9\)
\(\nu^{9}\)\(=\)\((\)\(27427343600 \beta_{19} - 1927847452 \beta_{18} + 42161993096 \beta_{17} - 7076732714 \beta_{16} - 13713671800 \beta_{15} - 33069495516 \beta_{14} - 10173477658 \beta_{13} - 22431910778 \beta_{12} + 18518492002 \beta_{11} - 837138976078 \beta_{10} + 59306784124 \beta_{9} + 14004263834 \beta_{8} - 1107094320 \beta_{7} + 296434865904 \beta_{6} - 430715659830 \beta_{5} + 3443825863008 \beta_{4} - 1150174263397570 \beta_{3} + 44051853455269 \beta_{2} - 43619744524245 \beta_{1} + 573524051995116\)\()/9\)
\(\nu^{10}\)\(=\)\((\)\(604759247088 \beta_{18} - 56014019556 \beta_{16} + 437997118332 \beta_{15} - 2140246820484 \beta_{14} - 1210090128760 \beta_{13} - 1703529632064 \beta_{12} - 4438399359160 \beta_{11} + 1088129311372 \beta_{10} - 2186191401020 \beta_{9} + 894965685344 \beta_{8} + 5863403976432 \beta_{7} - 187996850703911 \beta_{6} + 80514971222275 \beta_{5} - 2773672584684 \beta_{4} - 40554986012420 \beta_{3} - 1603725822737835 \beta_{2} - 1585348698616607 \beta_{1} - 132714896271803432\)\()/9\)
\(\nu^{11}\)\(=\)\((\)\(-188491657070390 \beta_{19} + 16429507804632 \beta_{18} - 310436022233990 \beta_{17} + 61866209965812 \beta_{16} + 94245828535195 \beta_{15} + 269054826648739 \beta_{14} + 107056435219865 \beta_{13} + 268577596829203 \beta_{12} - 134936682059647 \beta_{11} + 7070653296146384 \beta_{10} - 414387125682729 \beta_{9} - 93351801151183 \beta_{8} - 39803271359466 \beta_{7} - 3488178478352559 \beta_{6} + 3607323851321345 \beta_{5} - 26513541449653507 \beta_{4} + 9863258499455443388 \beta_{3} - 299840910609360263 \beta_{2} + 297241927051731066 \beta_{1} - 4919158308803647150\)\()/27\)
\(\nu^{12}\)\(=\)\((\)\(-1541998142391092 \beta_{18} + 95655799280645 \beta_{16} - 1270382007870546 \beta_{15} + 5705064836482532 \beta_{14} + 3450054267296809 \beta_{13} + 4450034091933375 \beta_{12} + 11216883687970449 \beta_{11} - 2660725924551031 \beta_{10} + 5919468593286550 \beta_{9} - 2500490521630867 \beta_{8} - 13661407575190372 \beta_{7} + 486989002550949660 \beta_{6} - 189142697055216355 \beta_{5} + 21432584832251824 \beta_{4} + 116153139348723055 \beta_{3} + 4438493479148464591 \beta_{2} + 4394249788712291963 \beta_{1} + 302167449595104619685\)\()/9\)
\(\nu^{13}\)\(=\)\((\)\(146151572834102216 \beta_{19} - 15556972004131568 \beta_{18} + 251890459124844824 \beta_{17} - 56830820307569920 \beta_{16} - 73075786417051108 \beta_{15} - 241331097368982884 \beta_{14} - 106012902447818452 \beta_{13} - 276560637869807572 \beta_{12} + 104991189849334492 \beta_{11} - 6411524459936247020 \beta_{10} + 321866774514074340 \beta_{9} + 69114409254852492 \beta_{8} + 57770043568461704 \beta_{7} + 3821723776538970898 \beta_{6} - 3250773283879375562 \beta_{5} + 22160046160652402348 \beta_{4} - 9078893752469272423524 \beta_{3} + 232151367985508455081 \beta_{2} - 230512583683449500197 \beta_{1} + 4528699219890118556042\)\()/9\)
\(\nu^{14}\)\(=\)\((\)\(3858798748686670720 \beta_{18} - 139571538153908812 \beta_{16} + 3510251348023525830 \beta_{15} - 14807504517775720366 \beta_{14} - 9463405127674563282 \beta_{13} - 11367420383550776082 \beta_{12} - 27806281449818088546 \beta_{11} + 6382678073927221092 \beta_{10} - 15566940125293966750 \beta_{9} + 6784098639651293402 \beta_{8} + 32409760670019631904 \beta_{7} - 1251014407783504763805 \beta_{6} + 449441920173716486809 \beta_{5} - 90997643853472402258 \beta_{4} - 322315669849098120904 \beta_{3} - 11963827315213345907511 \beta_{2} - 11860861342674096562557 \beta_{1} - 703857818356209081676432\)\()/9\)
\(\nu^{15}\)\(=\)\((\)\(-344784617193027471762 \beta_{19} + 43865378700544561132 \beta_{18} - 612069316381675041882 \beta_{17} + 151915592195708267018 \beta_{16} + 172392308596513735881 \beta_{15} + 644366114292332917229 \beta_{14} + 293126848893022262897 \beta_{13} + 773633765193190257767 \beta_{12} - 238623356883822982299 \beta_{11} + 17051205027223376138150 \beta_{10} - 754351668947411275255 \beta_{9} - 154119065995129253923 \beta_{8} - 180797553499939566302 \beta_{7} - 11494381020177065780011 \beta_{6} + 8604142071136443422995 \beta_{5} - 54981082367211079581985 \beta_{4} + 24438155508231015601194798 \beta_{3} - 548417746760495864794201 \beta_{2} + 545524602825064801936202 \beta_{1} - 12191692953886758643682576\)\()/9\)
\(\nu^{16}\)\(=\)\((\)\(-9574588164909994646548 \beta_{18} + 133409690805863624545 \beta_{16} - 9417628550590273191156 \beta_{15} + 37874573663708529881710 \beta_{14} + 25361380406443554224535 \beta_{13} + 28700214571216126108797 \beta_{12} + 68341982491400974327919 \beta_{11} - 15156524989792364310987 \beta_{10} + 40251086014624191286432 \beta_{9} - 18049030527372989078409 \beta_{8} - 78113304123385124228900 \beta_{7} + 3195863815503695546378254 \beta_{6} - 1077879467964732633446513 \beta_{5} + 317760765569111634796038 \beta_{4} + 875004845826387250919023 \beta_{3} + 31634283309813493164799267 \beta_{2} + 31399463853284642632697697 \beta_{1} + 1666575483405638022130641233\)\()/9\)
\(\nu^{17}\)\(=\)\((\)\(2468749543218389273315056 \beta_{19} - 365786966314435126961628 \beta_{18} + 4466843628470035483733992 \beta_{17} - 1195988822701858477788282 \beta_{16} - 1234374771609194636657528 \beta_{15} - 5117596876108287718276604 \beta_{14} - 2343063592632246634410394 \beta_{13} - 6184317832779969247437962 \beta_{12} + 1600475805364700584847138 \beta_{11} - 133927959793054411413872566 \beta_{10} + 5342734602563156552802060 \beta_{9} + 1037432991533159264078714 \beta_{8} + 1511190152920022939534448 \beta_{7} + 98407699329871146093109164 \beta_{6} - 67328068553642019126377986 \beta_{5} + 407447068602570428307743984 \beta_{4} - 193692279687963681326849791834 \beta_{3} + 3935166397740825714704850151 \beta_{2} - 3921418026639023535057749319 \beta_{1} + 96638460789150303091144186448\)\()/27\)
\(\nu^{18}\)\(=\)\((\)\(7891785953700763815218000 \beta_{18} + 41924337308893523102764 \beta_{16} + 8261587618390276057230936 \beta_{15} - 32020794154110354549090464 \beta_{14} - 22300361696850471215036596 \beta_{13} - 24003235188482910164564172 \beta_{12} - 55813255551708852004315860 \beta_{11} + 11950682623770219587437084 \beta_{10} - 34334892995238477848679208 \beta_{9} + 15784724011741472724260956 \beta_{8} + 63502339206275765476685584 \beta_{7} - 2710220407704892577369549313 \beta_{6} + 868139265456436055526122653 \beta_{5} - 333879497601754045123399216 \beta_{4} - 778751129881844172357802972 \beta_{3} - 27494486103893677073789300929 \beta_{2} - 27317901227645934404316172481 \beta_{1} - 1331203180761043927374031739936\)\()/3\)
\(\nu^{19}\)\(=\)\((\)\(-\)\(66\!\cdots\!70\)\( \beta_{19} + \)\(11\!\cdots\!36\)\( \beta_{18} - \)\(12\!\cdots\!22\)\( \beta_{17} + \)\(34\!\cdots\!60\)\( \beta_{16} + \)\(33\!\cdots\!35\)\( \beta_{15} + \)\(14\!\cdots\!31\)\( \beta_{14} + \)\(67\!\cdots\!13\)\( \beta_{13} + \)\(17\!\cdots\!11\)\( \beta_{12} - \)\(39\!\cdots\!91\)\( \beta_{11} + \)\(38\!\cdots\!00\)\( \beta_{10} - \)\(14\!\cdots\!45\)\( \beta_{9} - \)\(26\!\cdots\!51\)\( \beta_{8} - \)\(44\!\cdots\!22\)\( \beta_{7} - \)\(30\!\cdots\!83\)\( \beta_{6} + \)\(19\!\cdots\!33\)\( \beta_{5} - \)\(11\!\cdots\!79\)\( \beta_{4} + \)\(56\!\cdots\!12\)\( \beta_{3} - \)\(10\!\cdots\!71\)\( \beta_{2} + \)\(10\!\cdots\!62\)\( \beta_{1} - \)\(27\!\cdots\!50\)\(\)\()/3\)

Character Values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/9\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(-1 + \beta_{3}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
4.1
44.9985i
39.8650i
26.4646i
20.7390i
5.09438i
1.73070i
15.2468i
31.4671i
37.0247i
49.9602i
44.9985i
39.8650i
26.4646i
20.7390i
5.09438i
1.73070i
15.2468i
31.4671i
37.0247i
49.9602i
−40.4699 70.0959i −411.436 88.6985i −2251.62 + 3899.92i −156.497 + 271.061i 10433.4 + 32429.6i −40821.5 70704.9i 198726. 161412. + 72987.5i 25333.7
4.2 −36.0241 62.3956i 417.750 + 51.3055i −1571.47 + 2721.87i −5310.88 + 9198.71i −11847.8 27914.0i 26579.3 + 46036.7i 78888.8 171882. + 42865.7i 765278.
4.3 −24.4190 42.2949i 129.541 400.457i −168.574 + 291.979i 5966.72 10334.7i −20100.6 + 4299.85i 19533.0 + 33832.1i −83554.5 −143585. 103751.i −582805.
4.4 −19.4605 33.7066i 5.51306 + 420.852i 266.578 461.726i 1691.47 2929.71i 14078.2 8375.82i −16014.5 27738.0i −100461. −177086. + 4640.36i −131667.
4.5 −5.91186 10.2396i −417.157 + 55.9222i 954.100 1652.55i −2014.61 + 3489.40i 3038.79 + 3940.93i 34855.3 + 60371.1i −46777.0 170892. 46656.6i 47640.3
4.6 −0.00116679 0.00202094i 35.8201 419.361i 1024.00 1773.62i −5478.49 + 9489.02i −0.889299 + 0.416917i −31655.6 54829.1i −9.55835 −174581. 30043.1i 25.5690
4.7 11.7041 + 20.2721i 409.205 + 98.4789i 750.028 1299.09i 1606.12 2781.88i 2793.01 + 9448.06i −185.973 322.114i 83053.6 157751. + 80596.1i 75192.8
4.8 25.7513 + 44.6025i −329.417 261.976i −302.258 + 523.527i 4464.02 7731.91i 3201.87 21439.1i −5876.40 10178.2i 74343.1 39884.4 + 172599.i 459817.
4.9 30.5643 + 52.9389i −141.830 + 396.272i −844.353 + 1462.46i −1684.05 + 2916.87i −25313.1 + 4603.46i −5398.44 9350.38i 21963.1 −136916. 112406.i −205888.
4.10 41.7668 + 72.3422i 296.011 299.207i −2464.93 + 4269.38i −2698.81 + 4674.47i 34008.7 + 8917.18i 23240.9 + 40254.5i −240732. −1902.15 177137.i −450882.
7.1 −40.4699 + 70.0959i −411.436 + 88.6985i −2251.62 3899.92i −156.497 271.061i 10433.4 32429.6i −40821.5 + 70704.9i 198726. 161412. 72987.5i 25333.7
7.2 −36.0241 + 62.3956i 417.750 51.3055i −1571.47 2721.87i −5310.88 9198.71i −11847.8 + 27914.0i 26579.3 46036.7i 78888.8 171882. 42865.7i 765278.
7.3 −24.4190 + 42.2949i 129.541 + 400.457i −168.574 291.979i 5966.72 + 10334.7i −20100.6 4299.85i 19533.0 33832.1i −83554.5 −143585. + 103751.i −582805.
7.4 −19.4605 + 33.7066i 5.51306 420.852i 266.578 + 461.726i 1691.47 + 2929.71i 14078.2 + 8375.82i −16014.5 + 27738.0i −100461. −177086. 4640.36i −131667.
7.5 −5.91186 + 10.2396i −417.157 55.9222i 954.100 + 1652.55i −2014.61 3489.40i 3038.79 3940.93i 34855.3 60371.1i −46777.0 170892. + 46656.6i 47640.3
7.6 −0.00116679 + 0.00202094i 35.8201 + 419.361i 1024.00 + 1773.62i −5478.49 9489.02i −0.889299 0.416917i −31655.6 + 54829.1i −9.55835 −174581. + 30043.1i 25.5690
7.7 11.7041 20.2721i 409.205 98.4789i 750.028 + 1299.09i 1606.12 + 2781.88i 2793.01 9448.06i −185.973 + 322.114i 83053.6 157751. 80596.1i 75192.8
7.8 25.7513 44.6025i −329.417 + 261.976i −302.258 523.527i 4464.02 + 7731.91i 3201.87 + 21439.1i −5876.40 + 10178.2i 74343.1 39884.4 172599.i 459817.
7.9 30.5643 52.9389i −141.830 396.272i −844.353 1462.46i −1684.05 2916.87i −25313.1 4603.46i −5398.44 + 9350.38i 21963.1 −136916. + 112406.i −205888.
7.10 41.7668 72.3422i 296.011 + 299.207i −2464.93 4269.38i −2698.81 4674.47i 34008.7 8917.18i 23240.9 40254.5i −240732. −1902.15 + 177137.i −450882.
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 7.10
Significant digits:
Format:

Inner twists

Char. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
9.c Even 1 yes

Hecke kernels

There are no other newforms in \(S_{12}^{\mathrm{new}}(9, [\chi])\).