Properties

Label 12.19.d.a
Level 12
Weight 19
Character orbit 12.d
Analytic conductor 24.646
Analytic rank 0
Dimension 18
CM No
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 12 = 2^{2} \cdot 3 \)
Weight: \( k \) = \( 19 \)
Character orbit: \([\chi]\) = 12.d (of order \(2\) and degree \(1\))

Newform invariants

Self dual: No
Analytic conductor: \(24.6463365252\)
Analytic rank: \(0\)
Dimension: \(18\)
Coefficient field: \(\mathbb{Q}[x]/(x^{18} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{139}\cdot 3^{67}\cdot 13^{2} \)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{17}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \( + ( -9 - \beta_{1} ) q^{2} \) \( + ( -\beta_{1} + \beta_{2} ) q^{3} \) \( + ( -24281 + 12 \beta_{1} + 3 \beta_{2} - \beta_{3} ) q^{4} \) \( + ( -95523 - 293 \beta_{1} - \beta_{2} + \beta_{4} ) q^{5} \) \( + ( -373978 + 10 \beta_{1} - 9 \beta_{2} - \beta_{3} - \beta_{6} ) q^{6} \) \( + ( 4639 - 10278 \beta_{1} - 143 \beta_{2} + 23 \beta_{3} - 2 \beta_{6} - \beta_{9} ) q^{7} \) \( + ( 6021531 + 23266 \beta_{1} - 1871 \beta_{2} + 8 \beta_{3} + 23 \beta_{4} - \beta_{5} - 3 \beta_{6} - \beta_{11} ) q^{8} \) \( -129140163 q^{9} \) \(+O(q^{10})\) \( q\) \( + ( -9 - \beta_{1} ) q^{2} \) \( + ( -\beta_{1} + \beta_{2} ) q^{3} \) \( + ( -24281 + 12 \beta_{1} + 3 \beta_{2} - \beta_{3} ) q^{4} \) \( + ( -95523 - 293 \beta_{1} - \beta_{2} + \beta_{4} ) q^{5} \) \( + ( -373978 + 10 \beta_{1} - 9 \beta_{2} - \beta_{3} - \beta_{6} ) q^{6} \) \( + ( 4639 - 10278 \beta_{1} - 143 \beta_{2} + 23 \beta_{3} - 2 \beta_{6} - \beta_{9} ) q^{7} \) \( + ( 6021531 + 23266 \beta_{1} - 1871 \beta_{2} + 8 \beta_{3} + 23 \beta_{4} - \beta_{5} - 3 \beta_{6} - \beta_{11} ) q^{8} \) \( -129140163 q^{9} \) \( + ( 77572599 + 93832 \beta_{1} + 1900 \beta_{2} - 370 \beta_{3} - 122 \beta_{4} + 3 \beta_{6} + 8 \beta_{9} + \beta_{10} - \beta_{11} - \beta_{12} - \beta_{13} + \beta_{17} ) q^{10} \) \( + ( -19274 + 53315 \beta_{1} - 7713 \beta_{2} + 2933 \beta_{3} + 11 \beta_{4} + 4 \beta_{5} + 58 \beta_{6} - \beta_{8} - 10 \beta_{9} + 4 \beta_{11} - 3 \beta_{13} + \beta_{14} - \beta_{15} - 2 \beta_{17} ) q^{11} \) \( + ( -359529483 + 377123 \beta_{1} - 23254 \beta_{2} + 15 \beta_{3} + 2 \beta_{4} + 11 \beta_{5} + 19 \beta_{6} + 2 \beta_{7} - \beta_{8} + 10 \beta_{9} - 2 \beta_{10} - 2 \beta_{11} + \beta_{12} + 2 \beta_{13} + 2 \beta_{14} + \beta_{15} - 2 \beta_{16} - \beta_{17} ) q^{12} \) \( + ( 99693895 + 585022 \beta_{1} + 2213 \beta_{2} - 3371 \beta_{3} + 386 \beta_{4} - 6 \beta_{5} + 62 \beta_{6} + 8 \beta_{7} + 8 \beta_{8} - 2 \beta_{9} - 8 \beta_{10} + 19 \beta_{11} - 2 \beta_{12} - 8 \beta_{14} + 5 \beta_{15} - 4 \beta_{16} ) q^{13} \) \( + ( -2675200410 + 295047 \beta_{1} + 458746 \beta_{2} - 5812 \beta_{3} + 2892 \beta_{4} + 123 \beta_{5} + 124 \beta_{6} + \beta_{7} - 2 \beta_{8} - 135 \beta_{9} - 10 \beta_{10} + 49 \beta_{11} + 8 \beta_{12} - 10 \beta_{13} + 8 \beta_{14} - 17 \beta_{15} + 9 \beta_{16} + 16 \beta_{17} ) q^{14} \) \( + ( 399208 - 800560 \beta_{1} - 98123 \beta_{2} - 356 \beta_{3} + 35 \beta_{4} - 37 \beta_{5} - 286 \beta_{6} + 8 \beta_{7} - 4 \beta_{8} + 121 \beta_{9} + 19 \beta_{10} - 35 \beta_{11} + 4 \beta_{12} + 8 \beta_{13} + 8 \beta_{14} - 23 \beta_{15} - 8 \beta_{16} - 4 \beta_{17} ) q^{15} \) \( + ( -6637949535 - 6080955 \beta_{1} - 650931 \beta_{2} + 23154 \beta_{3} + 198 \beta_{4} + 709 \beta_{5} + 2073 \beta_{6} + 19 \beta_{7} + 27 \beta_{8} - 273 \beta_{9} + 42 \beta_{10} + 14 \beta_{11} - 107 \beta_{12} + 61 \beta_{13} + 24 \beta_{14} - 8 \beta_{15} - 20 \beta_{16} + 12 \beta_{17} ) q^{16} \) \( + ( -15747642577 + 59068605 \beta_{1} + 156182 \beta_{2} + 85716 \beta_{3} + 3037 \beta_{4} - 994 \beta_{5} - 793 \beta_{6} - 16 \beta_{7} - 50 \beta_{8} + 44 \beta_{9} + 8 \beta_{10} - 81 \beta_{11} - 93 \beta_{12} + 32 \beta_{13} + 13 \beta_{14} - 30 \beta_{15} + 50 \beta_{16} + \beta_{17} ) q^{17} \) \( + ( 1162261467 + 129140163 \beta_{1} ) q^{18} \) \( + ( 61259515 - 141671437 \beta_{1} + 3772238 \beta_{2} - 94802 \beta_{3} - 3395 \beta_{4} - 1666 \beta_{5} + 5501 \beta_{6} + 137 \beta_{7} + 250 \beta_{8} - 2914 \beta_{9} - 22 \beta_{10} + 511 \beta_{11} - 151 \beta_{12} + 176 \beta_{13} + 205 \beta_{14} + 18 \beta_{15} - 57 \beta_{16} + 182 \beta_{17} ) q^{19} \) \( + ( 27348391540 - 78795496 \beta_{1} - 322938 \beta_{2} + 78822 \beta_{3} - 12190 \beta_{4} + 6046 \beta_{5} + 598 \beta_{6} + 228 \beta_{7} - 76 \beta_{8} + 3468 \beta_{9} - 72 \beta_{10} - 1048 \beta_{11} + 284 \beta_{12} + 28 \beta_{13} - 128 \beta_{14} - 152 \beta_{15} + 120 \beta_{16} - 304 \beta_{17} ) q^{20} \) \( + ( 14013677676 + 224595093 \beta_{1} + 691618 \beta_{2} - 243015 \beta_{3} + 37895 \beta_{4} - 3406 \beta_{5} - 10976 \beta_{6} - 148 \beta_{7} - 88 \beta_{8} + 70 \beta_{9} + 40 \beta_{10} + 1201 \beta_{11} + 412 \beta_{12} - 580 \beta_{13} + 14 \beta_{14} - 371 \beta_{15} + 148 \beta_{16} + 398 \beta_{17} ) q^{21} \) \( + ( 15150465402 - 8033322 \beta_{1} - 17573932 \beta_{2} - 92 \beta_{3} + 101430 \beta_{4} + 13040 \beta_{5} + 10502 \beta_{6} - 86 \beta_{7} + 560 \beta_{8} + 118 \beta_{9} - 460 \beta_{10} + 4698 \beta_{11} + 420 \beta_{12} - 1264 \beta_{13} - 464 \beta_{14} + 398 \beta_{15} + 130 \beta_{16} + 624 \beta_{17} ) q^{22} \) \( + ( 371815684 - 821940224 \beta_{1} - 13917740 \beta_{2} + 1023728 \beta_{3} - 1900 \beta_{4} - 13856 \beta_{5} - 52974 \beta_{6} + 250 \beta_{7} - 574 \beta_{8} + 22950 \beta_{9} - 784 \beta_{10} - 2254 \beta_{11} + 1450 \beta_{12} - 470 \beta_{13} - 364 \beta_{14} + 486 \beta_{15} + 550 \beta_{16} + 40 \beta_{17} ) q^{23} \) \( + ( 248919011882 + 353381041 \beta_{1} + 7029650 \beta_{2} + 306638 \beta_{3} - 91979 \beta_{4} + 4834 \beta_{5} + 20620 \beta_{6} + 253 \beta_{7} + 805 \beta_{8} - 5539 \beta_{9} + 1070 \beta_{10} - 145 \beta_{11} + 491 \beta_{12} + 307 \beta_{13} + 1144 \beta_{14} - 508 \beta_{15} - 496 \beta_{16} + 76 \beta_{17} ) q^{24} \) \( + ( 1122649815817 + 2123348888 \beta_{1} + 6417651 \beta_{2} - 965123 \beta_{3} - 554347 \beta_{4} - 37057 \beta_{5} - 40330 \beta_{6} - 305 \beta_{7} + 2661 \beta_{8} - 464 \beta_{9} + 2336 \beta_{10} - 6637 \beta_{11} + 217 \beta_{12} - 575 \beta_{13} - 238 \beta_{14} + 1041 \beta_{15} + 1901 \beta_{16} - 14 \beta_{17} ) q^{25} \) \( + ( -153603433191 - 88506192 \beta_{1} + 14634154 \beta_{2} + 521332 \beta_{3} - 560711 \beta_{4} + 15733 \beta_{5} + 7753 \beta_{6} - 2022 \beta_{7} - 706 \beta_{8} - 15736 \beta_{9} - 1814 \beta_{10} - 10128 \beta_{11} + 1128 \beta_{12} + 100 \beta_{13} + 832 \beta_{14} + 2146 \beta_{15} + 6750 \beta_{16} + 1066 \beta_{17} ) q^{26} \) \( + ( 129140163 \beta_{1} - 129140163 \beta_{2} ) q^{27} \) \( + ( 1540946700330 + 2564150852 \beta_{1} - 132319724 \beta_{2} - 594248 \beta_{3} + 1184222 \beta_{4} - 35346 \beta_{5} - 469026 \beta_{6} - 564 \beta_{7} - 216 \beta_{8} + 117548 \beta_{9} + 8144 \beta_{10} + 5544 \beta_{11} + 6568 \beta_{12} + 164 \beta_{13} + 1752 \beta_{14} - 8804 \beta_{15} + 3864 \beta_{16} + 10340 \beta_{17} ) q^{28} \) \( + ( 102382010735 - 3742826117 \beta_{1} - 13605079 \beta_{2} + 14467368 \beta_{3} + 1849411 \beta_{4} + 87314 \beta_{5} - 159354 \beta_{6} - 1214 \beta_{7} + 12290 \beta_{8} + 18260 \beta_{9} + 640 \beta_{10} + 44310 \beta_{11} + 4520 \beta_{12} - 946 \beta_{13} - 4930 \beta_{14} - 6580 \beta_{15} + 9730 \beta_{16} + 9942 \beta_{17} ) q^{29} \) \( + ( -198908945835 + 40624057 \beta_{1} + 78011346 \beta_{2} - 375018 \beta_{3} - 131833 \beta_{4} - 5086 \beta_{5} + 99955 \beta_{6} + 7919 \beta_{7} + 16412 \beta_{8} + 29389 \beta_{9} - 4058 \beta_{10} + 4231 \beta_{11} - 1670 \beta_{12} + 13724 \beta_{13} - 4312 \beta_{14} + 9373 \beta_{15} - 4517 \beta_{16} + 8312 \beta_{17} ) q^{30} \) \( + ( -7436605939 + 16480667644 \beta_{1} + 242900005 \beta_{2} - 12083201 \beta_{3} + 152642 \beta_{4} + 260544 \beta_{5} + 432216 \beta_{6} + 18178 \beta_{7} + 760 \beta_{8} - 38593 \beta_{9} - 22104 \beta_{10} + 47146 \beta_{11} + 24530 \beta_{12} + 5180 \beta_{13} + 23414 \beta_{14} - 7052 \beta_{15} + 12958 \beta_{16} - 4476 \beta_{17} ) q^{31} \) \( + ( 12007935893202 + 6175786638 \beta_{1} - 91003942 \beta_{2} - 11995676 \beta_{3} - 2686512 \beta_{4} - 154462 \beta_{5} + 498298 \beta_{6} + 18146 \beta_{7} + 14994 \beta_{8} - 104702 \beta_{9} - 10772 \beta_{10} + 34752 \beta_{11} + 13806 \beta_{12} - 26498 \beta_{13} + 2864 \beta_{14} - 15768 \beta_{15} + 26720 \beta_{16} + 12632 \beta_{17} ) q^{32} \) \( + ( 949093919889 - 8656437441 \beta_{1} - 22621903 \beta_{2} - 11410491 \beta_{3} - 3445760 \beta_{4} + 172501 \beta_{5} + 159995 \beta_{6} - 5285 \beta_{7} - 3149 \beta_{8} - 4636 \beta_{9} + 8120 \beta_{10} + 23132 \beta_{11} - 4546 \beta_{12} - 31259 \beta_{13} - 5447 \beta_{14} - 18397 \beta_{15} + 26183 \beta_{16} + 22933 \beta_{17} ) q^{33} \) \( + ( -15329172706382 + 16217793134 \beta_{1} - 82825308 \beta_{2} + 67689796 \beta_{3} + 5554762 \beta_{4} - 298826 \beta_{5} - 109412 \beta_{6} + 20940 \beta_{7} + 5892 \beta_{8} + 329920 \beta_{9} + 9094 \beta_{10} + 155718 \beta_{11} - 8362 \beta_{12} - 30370 \beta_{13} + 19328 \beta_{14} - 12612 \beta_{15} + 14020 \beta_{16} - 7162 \beta_{17} ) q^{34} \) \( + ( -24172086856 + 55633774957 \beta_{1} - 1189988661 \beta_{2} + 74929613 \beta_{3} + 1142573 \beta_{4} + 600104 \beta_{5} + 2028656 \beta_{6} + 55670 \beta_{7} + 74047 \beta_{8} + 843978 \beta_{9} + 43260 \beta_{10} - 31954 \beta_{11} - 49994 \beta_{12} + 22729 \beta_{13} + 42795 \beta_{14} + 11983 \beta_{15} - 52118 \beta_{16} + 86922 \beta_{17} ) q^{35} \) \( + ( 3135652297803 - 1549681956 \beta_{1} - 387420489 \beta_{2} + 129140163 \beta_{3} ) q^{36} \) \( + ( 10974256583271 - 12220911580 \beta_{1} - 17651321 \beta_{2} - 104435127 \beta_{3} + 11636990 \beta_{4} + 140684 \beta_{5} - 4732618 \beta_{6} - 26922 \beta_{7} + 4386 \beta_{8} + 139834 \beta_{9} + 46376 \beta_{10} - 181697 \beta_{11} - 72484 \beta_{12} - 137502 \beta_{13} + 8860 \beta_{14} - 53791 \beta_{15} + 122126 \beta_{16} + 47412 \beta_{17} ) q^{37} \) \( + ( -37556848306244 + 685765658 \beta_{1} - 1691861112 \beta_{2} - 144698908 \beta_{3} - 10564776 \beta_{4} - 351398 \beta_{5} - 3558252 \beta_{6} + 11518 \beta_{7} - 19740 \beta_{8} - 1489746 \beta_{9} + 63700 \beta_{10} - 130530 \beta_{11} + 145360 \beta_{12} - 70924 \beta_{13} + 4848 \beta_{14} - 151326 \beta_{15} + 87342 \beta_{16} + 114912 \beta_{17} ) q^{38} \) \( + ( 3188416341 - 7276221152 \beta_{1} + 83315942 \beta_{2} - 25515321 \beta_{3} - 49215 \beta_{4} - 223365 \beta_{5} + 636486 \beta_{6} - 1938 \beta_{7} - 50628 \beta_{8} - 769956 \beta_{9} - 138165 \beta_{10} - 563685 \beta_{11} + 179418 \beta_{12} - 11388 \beta_{13} - 1614 \beta_{14} - 69603 \beta_{15} + 103026 \beta_{16} - 50952 \beta_{17} ) q^{39} \) \( + ( 107852377999466 - 29531427076 \beta_{1} + 3838753726 \beta_{2} - 89221616 \beta_{3} + 21841234 \beta_{4} + 406722 \beta_{5} - 1462298 \beta_{6} + 59360 \beta_{7} + 162016 \beta_{8} - 1323392 \beta_{9} - 51712 \beta_{10} + 304930 \beta_{11} - 138080 \beta_{12} + 274464 \beta_{13} - 48000 \beta_{14} + 351904 \beta_{15} - 50144 \beta_{16} + 64 \beta_{17} ) q^{40} \) \( + ( -11413900100183 + 97561870659 \beta_{1} + 239495896 \beta_{2} + 270076374 \beta_{3} - 2254507 \beta_{4} - 1036328 \beta_{5} - 12209187 \beta_{6} + 32486 \beta_{7} + 232788 \beta_{8} + 566788 \beta_{9} - 130248 \beta_{10} + 579551 \beta_{11} + 191647 \beta_{12} + 265578 \beta_{13} + 16143 \beta_{14} + 72832 \beta_{15} - 189696 \beta_{16} - 101189 \beta_{17} ) q^{41} \) \( + ( -58878113098218 - 13268341752 \beta_{1} - 2764732714 \beta_{2} + 239460006 \beta_{3} + 13158895 \beta_{4} + 453361 \beta_{5} + 903098 \beta_{6} - 44078 \beta_{7} - 42842 \beta_{8} + 3774368 \beta_{9} + 80825 \beta_{10} - 327847 \beta_{11} + 113393 \beta_{12} + 111037 \beta_{13} - 84416 \beta_{14} - 72070 \beta_{15} - 47290 \beta_{16} + 131065 \beta_{17} ) q^{42} \) \( + ( 90791234479 - 206073576943 \beta_{1} + 1456145292 \beta_{2} - 464140744 \beta_{3} - 7309509 \beta_{4} - 2127722 \beta_{5} + 15417393 \beta_{6} - 244183 \beta_{7} - 181328 \beta_{8} + 801858 \beta_{9} - 35558 \beta_{10} + 2512407 \beta_{11} - 265783 \beta_{12} - 228014 \beta_{13} - 15817 \beta_{14} + 107992 \beta_{15} + 60455 \beta_{16} - 409694 \beta_{17} ) q^{43} \) \( + ( 199964524899240 - 24379968020 \beta_{1} - 4855349808 \beta_{2} - 131031116 \beta_{3} - 61353780 \beta_{4} + 3084264 \beta_{5} + 18144488 \beta_{6} - 363456 \beta_{7} - 584940 \beta_{8} + 3031264 \beta_{9} + 73768 \beta_{10} - 878792 \beta_{11} + 334028 \beta_{12} - 316720 \beta_{13} - 178872 \beta_{14} - 86204 \beta_{15} + 230616 \beta_{16} + 11068 \beta_{17} ) q^{44} \) \( + ( 12335855790249 + 37838067759 \beta_{1} + 129140163 \beta_{2} - 129140163 \beta_{4} ) q^{45} \) \( + ( -213744179780636 + 13869645710 \beta_{1} + 13796895276 \beta_{2} - 695680064 \beta_{3} - 16416736 \beta_{4} + 3843742 \beta_{5} + 15937112 \beta_{6} - 11950 \beta_{7} - 259124 \beta_{8} + 1760466 \beta_{9} + 536172 \beta_{10} + 323442 \beta_{11} - 320576 \beta_{12} + 762492 \beta_{13} + 127760 \beta_{14} + 596270 \beta_{15} - 597310 \beta_{16} - 625696 \beta_{17} ) q^{46} \) \( + ( 190808109624 - 438542020748 \beta_{1} + 9591748304 \beta_{2} + 459296568 \beta_{3} - 5513764 \beta_{4} - 9051404 \beta_{5} + 54529398 \beta_{6} - 414874 \beta_{7} + 194602 \beta_{8} + 2135394 \beta_{9} - 70348 \beta_{10} - 3940998 \beta_{11} - 312954 \beta_{12} - 479862 \beta_{13} - 270160 \beta_{14} - 245278 \beta_{15} - 42630 \beta_{16} + 49888 \beta_{17} ) q^{47} \) \( + ( 81081788785245 - 255380157523 \beta_{1} - 7435373319 \beta_{2} + 267565194 \beta_{3} - 74472710 \beta_{4} + 2463241 \beta_{5} - 7824547 \beta_{6} + 176107 \beta_{7} + 357811 \beta_{8} - 9034513 \beta_{9} - 58198 \beta_{10} + 167090 \beta_{11} + 312221 \beta_{12} + 321637 \beta_{13} - 513608 \beta_{14} + 30368 \beta_{15} - 415948 \beta_{16} + 525724 \beta_{17} ) q^{48} \) \( + ( -190588173574407 + 807317746654 \beta_{1} + 2638375669 \beta_{2} - 1721629481 \beta_{3} + 312697589 \beta_{4} - 14706671 \beta_{5} - 158722076 \beta_{6} - 99955 \beta_{7} - 878253 \beta_{8} + 4552552 \beta_{9} - 770800 \beta_{10} - 4288845 \beta_{11} - 788139 \beta_{12} - 278717 \beta_{13} + 1216828 \beta_{14} + 145335 \beta_{15} - 1006781 \beta_{16} - 1381196 \beta_{17} ) q^{49} \) \( + ( -565753030229709 - 1108834500039 \beta_{1} - 10114701084 \beta_{2} + 2355674968 \beta_{3} + 44213850 \beta_{4} + 6015010 \beta_{5} - 4216494 \beta_{6} + 137380 \beta_{7} - 934580 \beta_{8} - 34662000 \beta_{9} - 632324 \beta_{10} - 1721304 \beta_{11} - 384680 \beta_{12} + 637168 \beta_{13} - 1619840 \beta_{14} + 1412852 \beta_{15} - 747892 \beta_{16} - 674692 \beta_{17} ) q^{50} \) \( + ( -15556810769 + 50859925997 \beta_{1} - 15672118376 \beta_{2} + 210226474 \beta_{3} - 417067 \beta_{4} - 2602114 \beta_{5} + 58717841 \beta_{6} - 199339 \beta_{7} + 788534 \beta_{8} - 2705714 \beta_{9} + 394954 \beta_{10} - 4204901 \beta_{11} - 879659 \beta_{12} + 199964 \beta_{13} - 99547 \beta_{14} - 265274 \beta_{15} - 442181 \beta_{16} + 688742 \beta_{17} ) q^{51} \) \( + ( 234831812740818 + 151085608760 \beta_{1} + 13674001374 \beta_{2} + 145674054 \beta_{3} + 597006468 \beta_{4} - 9714756 \beta_{5} - 33201812 \beta_{6} - 101944 \beta_{7} - 2996632 \beta_{8} - 30419112 \beta_{9} + 1982192 \beta_{10} + 4482832 \beta_{11} - 459720 \beta_{12} - 434120 \beta_{13} + 166912 \beta_{14} - 647408 \beta_{15} - 1395664 \beta_{16} - 2145248 \beta_{17} ) q^{52} \) \( + ( -783523497119589 - 1157189879929 \beta_{1} - 2796444175 \beta_{2} - 2531121312 \beta_{3} - 344561849 \beta_{4} + 25282658 \beta_{5} - 235415118 \beta_{6} + 172314 \beta_{7} - 1172862 \beta_{8} + 9071812 \beta_{9} - 1975392 \beta_{10} - 2670966 \beta_{11} - 2411332 \beta_{12} - 1471946 \beta_{13} + 1126050 \beta_{14} - 765028 \beta_{15} - 419502 \beta_{16} - 1027574 \beta_{17} ) q^{53} \) \( + ( 48295579878414 - 1291401630 \beta_{1} + 1162261467 \beta_{2} + 129140163 \beta_{3} + 129140163 \beta_{6} ) q^{54} \) \( + ( -954843365366 + 2016111208434 \beta_{1} + 129494668048 \beta_{2} - 4056504540 \beta_{3} - 19653486 \beta_{4} + 26793184 \beta_{5} + 467309486 \beta_{6} - 583794 \beta_{7} + 1621324 \beta_{8} + 6783448 \beta_{9} + 397576 \beta_{10} + 14471590 \beta_{11} - 4226050 \beta_{12} - 1050256 \beta_{13} + 2723574 \beta_{14} - 1542592 \beta_{15} - 777262 \beta_{16} - 1686044 \beta_{17} ) q^{55} \) \( + ( -365518215701416 - 1510927774020 \beta_{1} + 45838977432 \beta_{2} + 2827723944 \beta_{3} - 372420660 \beta_{4} - 24944040 \beta_{5} + 141714256 \beta_{6} + 2777324 \beta_{7} + 857164 \beta_{8} + 63609100 \beta_{9} + 2837064 \beta_{10} - 6337500 \beta_{11} + 309748 \beta_{12} + 6832724 \beta_{13} - 2486624 \beta_{14} + 2360976 \beta_{15} - 4336672 \beta_{16} + 1746960 \beta_{17} ) q^{56} \) \( + ( -536419861379667 - 829086493671 \beta_{1} - 2997389646 \beta_{2} + 3337060896 \beta_{3} + 357296301 \beta_{4} + 23738706 \beta_{5} - 144612333 \beta_{6} + 3276 \beta_{7} + 1308618 \beta_{8} + 8330220 \beta_{9} - 2248920 \beta_{10} + 13147479 \beta_{11} + 1693323 \beta_{12} + 1044612 \beta_{13} - 22239 \beta_{14} - 1452474 \beta_{15} - 925218 \beta_{16} + 744453 \beta_{17} ) q^{57} \) \( + ( 977008600121295 - 147901684976 \beta_{1} - 32064612312 \beta_{2} - 4699249406 \beta_{3} - 813178388 \beta_{4} - 53535710 \beta_{5} - 3909413 \beta_{6} - 1141852 \beta_{7} - 4364212 \beta_{8} + 18656680 \beta_{9} + 598911 \beta_{10} + 10521509 \beta_{11} + 911573 \beta_{12} + 2521709 \beta_{13} - 4972416 \beta_{14} + 3563508 \beta_{15} + 1689996 \beta_{16} + 3781119 \beta_{17} ) q^{58} \) \( + ( -1382372814842 + 3247345567388 \beta_{1} - 136318638042 \beta_{2} + 390264198 \beta_{3} + 9310516 \beta_{4} + 29933724 \beta_{5} + 812389826 \beta_{6} - 2409274 \beta_{7} - 975206 \beta_{8} - 10112568 \beta_{9} - 7524964 \beta_{10} - 10279182 \beta_{11} + 2147334 \beta_{12} - 8155014 \beta_{13} + 2435456 \beta_{14} - 3731158 \beta_{15} + 4162074 \beta_{16} - 5819936 \beta_{17} ) q^{59} \) \( + ( 10645879576636 + 210508320074 \beta_{1} + 28342272056 \beta_{2} - 445824314 \beta_{3} - 1678641110 \beta_{4} - 19239236 \beta_{5} - 84509124 \beta_{6} - 1253312 \beta_{7} - 3202538 \beta_{8} - 74401168 \beta_{9} + 1488428 \beta_{10} - 14947228 \beta_{11} + 1099130 \beta_{12} + 4085560 \beta_{13} + 174556 \beta_{14} - 368482 \beta_{15} + 3577364 \beta_{16} - 386654 \beta_{17} ) q^{60} \) \( + ( 232245899339027 - 1160672793612 \beta_{1} - 3616630009 \beta_{2} + 3572755765 \beta_{3} - 340306082 \beta_{4} + 46844340 \beta_{5} - 1190036238 \beta_{6} + 357518 \beta_{7} + 11463858 \beta_{8} + 51371906 \beta_{9} - 3775640 \beta_{10} - 9233129 \beta_{11} - 5263840 \beta_{12} + 5390218 \beta_{13} + 3379688 \beta_{14} + 1876949 \beta_{15} + 703710 \beta_{16} - 6564848 \beta_{17} ) q^{61} \) \( + ( 4290527889660690 - 201985915985 \beta_{1} - 96911830022 \beta_{2} + 15617100212 \beta_{3} + 1899857448 \beta_{4} - 47392569 \beta_{5} - 340975064 \beta_{6} + 1356953 \beta_{7} + 1147990 \beta_{8} - 140704503 \beta_{9} + 13085878 \beta_{10} + 7772393 \beta_{11} + 10227696 \beta_{12} + 2220014 \beta_{13} + 285320 \beta_{14} + 2542215 \beta_{15} - 4452495 \beta_{16} + 8546608 \beta_{17} ) q^{62} \) \( + ( -599081216157 + 1327302595314 \beta_{1} + 18467043309 \beta_{2} - 2970223749 \beta_{3} + 258280326 \beta_{6} + 129140163 \beta_{9} ) q^{63} \) \( + ( -2962684862221896 - 11940513117168 \beta_{1} - 70009730600 \beta_{2} + 8272281968 \beta_{3} + 3086245672 \beta_{4} + 57641608 \beta_{5} + 90310600 \beta_{6} - 319696 \beta_{7} - 2005840 \beta_{8} - 134380608 \beta_{9} + 4106112 \beta_{10} + 4453896 \beta_{11} + 10867600 \beta_{12} + 7376080 \beta_{13} - 8969664 \beta_{14} + 7441040 \beta_{15} + 9126864 \beta_{16} + 9314464 \beta_{17} ) q^{64} \) \( + ( 1972301728579065 + 10406199689985 \beta_{1} + 38447089644 \beta_{2} - 42483926610 \beta_{3} - 565716429 \beta_{4} - 139358964 \beta_{5} - 2758784473 \beta_{6} - 8482682 \beta_{7} + 7944808 \beta_{8} + 76212684 \beta_{9} + 3782504 \beta_{10} - 9474631 \beta_{11} + 5072977 \beta_{12} - 46809814 \beta_{13} + 8150461 \beta_{14} - 16923668 \beta_{15} + 18090860 \beta_{16} + 11849633 \beta_{17} ) q^{65} \) \( + ( 2258494245859902 - 1017394282776 \beta_{1} + 13241842768 \beta_{2} - 8442649452 \beta_{3} + 3119901752 \beta_{4} + 15077396 \beta_{5} - 2019410 \beta_{6} + 1817768 \beta_{7} - 4811464 \beta_{8} + 100903312 \beta_{9} - 1887914 \beta_{10} + 3560146 \beta_{11} + 8270002 \beta_{12} + 4375586 \beta_{13} - 5505280 \beta_{14} - 175544 \beta_{15} + 5779384 \beta_{16} - 1810090 \beta_{17} ) q^{66} \) \( + ( 163181559678 - 101908927796 \beta_{1} - 268121448050 \beta_{2} - 5176563802 \beta_{3} - 127243140 \beta_{4} - 99866084 \beta_{5} + 2297707242 \beta_{6} + 4361614 \beta_{7} + 16348282 \beta_{8} + 238944456 \beta_{9} - 22633028 \beta_{10} - 53815078 \beta_{11} + 3102734 \beta_{12} - 2091798 \beta_{13} + 10407064 \beta_{14} - 83718 \beta_{15} + 7566290 \beta_{16} + 10302832 \beta_{17} ) q^{67} \) \( + ( -5084768949821774 + 15451057404184 \beta_{1} - 122015207258 \beta_{2} + 17205082766 \beta_{3} - 1836815252 \beta_{4} + 119730964 \beta_{5} + 177355876 \beta_{6} + 6690456 \beta_{7} + 1399416 \beta_{8} + 8902280 \beta_{9} + 5831888 \beta_{10} + 59455856 \beta_{11} + 13824360 \beta_{12} - 20078744 \beta_{13} - 13882624 \beta_{14} + 20055152 \beta_{15} + 1008080 \beta_{16} + 11459808 \beta_{17} ) q^{68} \) \( + ( 1460523838801872 + 6648658061682 \beta_{1} + 16515437182 \beta_{2} + 16658163264 \beta_{3} + 688181168 \beta_{4} - 106560934 \beta_{5} - 911415548 \beta_{6} - 3277114 \beta_{7} + 4559498 \beta_{8} + 41494276 \beta_{9} + 3107824 \beta_{10} - 40065932 \beta_{11} - 15358094 \beta_{12} + 15010634 \beta_{13} + 6701600 \beta_{14} + 2039248 \beta_{15} + 6393346 \beta_{16} - 9611776 \beta_{17} ) q^{69} \) \( + ( 14720473065172466 - 762920016502 \beta_{1} - 498279599432 \beta_{2} + 43649587424 \beta_{3} - 14310920050 \beta_{4} + 372856932 \beta_{5} + 1249703834 \beta_{6} + 17051390 \beta_{7} + 6958392 \beta_{8} - 114542598 \beta_{9} + 23388428 \beta_{10} - 28182226 \beta_{11} + 26128052 \beta_{12} + 24268984 \beta_{13} - 7666864 \beta_{14} - 15259878 \beta_{15} + 6279894 \beta_{16} + 6832240 \beta_{17} ) q^{70} \) \( + ( 1406611015510 - 3321505764152 \beta_{1} + 189324985714 \beta_{2} + 41597214818 \beta_{3} - 4149840 \beta_{4} - 95278380 \beta_{5} + 2997264856 \beta_{6} + 24519820 \beta_{7} + 21929008 \beta_{8} + 192079838 \beta_{9} - 10834652 \beta_{10} + 40726120 \beta_{11} - 7498116 \beta_{12} - 27280776 \beta_{13} + 43097780 \beta_{14} - 13889244 \beta_{15} - 5155020 \beta_{16} + 3351048 \beta_{17} ) q^{71} \) \( + ( -777621494849553 - 3004575032358 \beta_{1} + 241621244973 \beta_{2} - 1033121304 \beta_{3} - 2970223749 \beta_{4} + 129140163 \beta_{5} + 387420489 \beta_{6} + 129140163 \beta_{11} ) q^{72} \) \( + ( -4665117227251334 + 20303913962818 \beta_{1} + 53787107891 \beta_{2} + 38953182801 \beta_{3} - 3167979357 \beta_{4} - 107119049 \beta_{5} - 4053272036 \beta_{6} - 19825669 \beta_{7} + 7814181 \beta_{8} + 171990072 \beta_{9} - 16135440 \beta_{10} + 227344565 \beta_{11} + 10769875 \beta_{12} - 46704075 \beta_{13} + 1729060 \beta_{14} - 68388495 \beta_{15} + 44216821 \beta_{16} + 54021612 \beta_{17} ) q^{73} \) \( + ( 3114065527016329 - 11644318429160 \beta_{1} - 1330332565746 \beta_{2} - 16429534480 \beta_{3} + 11615707171 \beta_{4} + 290364915 \beta_{5} + 113795561 \beta_{6} + 28242742 \beta_{7} - 25680334 \beta_{8} + 168540200 \beta_{9} + 9231860 \beta_{10} - 53176318 \beta_{11} - 12814902 \beta_{12} - 16594770 \beta_{13} - 17600832 \beta_{14} + 16428334 \beta_{15} + 2138642 \beta_{16} + 1611956 \beta_{17} ) q^{74} \) \( + ( -2195334474243 + 3714002178651 \beta_{1} + 1154112934068 \beta_{2} - 97522868625 \beta_{3} - 681589848 \beta_{4} + 156534162 \beta_{5} + 2291056539 \beta_{6} + 828645 \beta_{7} + 15059229 \beta_{8} - 429989316 \beta_{9} - 16131462 \beta_{10} + 231611871 \beta_{11} - 20800347 \beta_{12} + 31173297 \beta_{13} + 28225518 \beta_{14} - 2373603 \beta_{15} + 8918571 \beta_{16} - 12337644 \beta_{17} ) q^{75} \) \( + ( -7186022116007360 + 38533340114180 \beta_{1} + 1661950792192 \beta_{2} + 20107466828 \beta_{3} + 19439341836 \beta_{4} + 76244112 \beta_{5} + 1588447952 \beta_{6} + 18882800 \beta_{7} + 27842860 \beta_{8} + 72687184 \beta_{9} + 2486104 \beta_{10} - 52085080 \beta_{11} + 25632564 \beta_{12} + 105432640 \beta_{13} - 28247976 \beta_{14} - 3695156 \beta_{15} - 7902776 \beta_{16} + 64099892 \beta_{17} ) q^{76} \) \( + ( -19698746755059726 - 5290506304466 \beta_{1} - 1341132578 \beta_{2} - 82136425698 \beta_{3} + 10631245796 \beta_{4} + 6465350 \beta_{5} - 4847787154 \beta_{6} + 38935474 \beta_{7} + 29586618 \beta_{8} + 158461768 \beta_{9} - 48035856 \beta_{10} - 221422364 \beta_{11} - 64455648 \beta_{12} + 20186894 \beta_{13} + 8950266 \beta_{14} + 61442922 \beta_{15} - 50675838 \beta_{16} - 80088782 \beta_{17} ) q^{77} \) \( + ( -1918550380151273 + 4906580156 \beta_{1} - 156983198940 \beta_{2} + 187518154 \beta_{3} + 19436134119 \beta_{4} - 96339963 \beta_{5} - 408708173 \beta_{6} - 15945708 \beta_{7} + 19726602 \beta_{8} - 412021806 \beta_{9} + 44100228 \beta_{10} + 155023356 \beta_{11} - 65366214 \beta_{12} + 90144258 \beta_{13} + 17599632 \beta_{14} + 12802296 \beta_{15} - 21494760 \beta_{16} + 22499880 \beta_{17} ) q^{78} \) \( + ( -5459944096093 + 13548779773714 \beta_{1} - 1322227447775 \beta_{2} - 79486269549 \beta_{3} - 180634300 \beta_{4} - 67671412 \beta_{5} + 3079821266 \beta_{6} - 25746524 \beta_{7} - 20616108 \beta_{8} - 14737605 \beta_{9} + 26908364 \beta_{10} - 410945592 \beta_{11} + 22152948 \beta_{12} + 30546228 \beta_{13} - 10315816 \beta_{14} - 103771696 \beta_{15} + 3434396 \beta_{16} - 36046816 \beta_{17} ) q^{79} \) \( + ( 2449595180649662 - 106893803127562 \beta_{1} + 2492549677030 \beta_{2} - 25099135972 \beta_{3} - 19131275532 \beta_{4} - 456991626 \beta_{5} - 4194429746 \beta_{6} - 22032806 \beta_{7} - 130167222 \beta_{8} - 1718967902 \beta_{9} + 34670764 \beta_{10} - 310288668 \beta_{11} + 9407062 \beta_{12} - 159987450 \beta_{13} + 41101264 \beta_{14} - 44877040 \beta_{15} + 31164712 \beta_{16} - 44931096 \beta_{17} ) q^{80} \) \( + 16677181699666569 q^{81} \) \( + ( -25464155503032634 + 11017571936886 \beta_{1} - 2895199040364 \beta_{2} + 77344609180 \beta_{3} - 42099880894 \beta_{4} - 766082442 \beta_{5} + 34975352 \beta_{6} - 42042164 \beta_{7} - 12778748 \beta_{8} - 991604640 \beta_{9} - 48914886 \beta_{10} + 82619794 \beta_{11} + 15894178 \beta_{12} + 96247466 \beta_{13} - 352384 \beta_{14} + 48512188 \beta_{15} - 65136956 \beta_{16} + 16884922 \beta_{17} ) q^{82} \) \( + ( -8736935709648 + 20737256601563 \beta_{1} - 746360210699 \beta_{2} + 447670792535 \beta_{3} + 2781288431 \beta_{4} + 371602488 \beta_{5} - 3269294800 \beta_{6} - 75271582 \beta_{7} - 51939291 \beta_{8} + 632028422 \beta_{9} + 37672724 \beta_{10} - 20287286 \beta_{11} - 83702622 \beta_{12} - 282169581 \beta_{13} - 118820967 \beta_{14} + 145878709 \beta_{15} - 47620290 \beta_{16} - 8389906 \beta_{17} ) q^{83} \) \( + ( 17977363075164150 + 58998053778048 \beta_{1} + 1724277707824 \beta_{2} - 20767102056 \beta_{3} - 24425457406 \beta_{4} - 508826050 \beta_{5} + 2721032374 \beta_{6} + 9506468 \beta_{7} + 38935412 \beta_{8} + 1610764300 \beta_{9} + 39450040 \beta_{10} + 64947112 \beta_{11} - 78752420 \beta_{12} + 174223964 \beta_{13} - 8220544 \beta_{14} + 40307752 \beta_{15} - 67378376 \beta_{16} - 43964848 \beta_{17} ) q^{84} \) \( + ( 9267478162397298 - 99259098930842 \beta_{1} - 307790747332 \beta_{2} + 123265586990 \beta_{3} - 32056242428 \beta_{4} + 1604838878 \beta_{5} + 11195203176 \beta_{6} + 18594734 \beta_{7} - 49225406 \beta_{8} - 305624728 \beta_{9} + 2515712 \beta_{10} - 140181498 \beta_{11} + 16247086 \beta_{12} + 212692818 \beta_{13} + 10820848 \beta_{14} + 98550046 \beta_{15} - 122441070 \beta_{16} - 99234816 \beta_{17} ) q^{85} \) \( + ( -54212298140216864 + 357119049118 \beta_{1} - 3939576617264 \beta_{2} - 207181994980 \beta_{3} + 19236727940 \beta_{4} - 2391033998 \beta_{5} - 863517776 \beta_{6} - 102489190 \beta_{7} - 106552332 \beta_{8} + 1314201954 \beta_{9} - 132227092 \beta_{10} - 348453734 \beta_{11} + 105467288 \beta_{12} - 508735868 \beta_{13} - 78368880 \beta_{14} + 64885686 \beta_{15} + 45085242 \beta_{16} - 99182784 \beta_{17} ) q^{86} \) \( + ( -7422950613794 + 16229193864938 \beta_{1} + 209783840665 \beta_{2} - 348586956536 \beta_{3} - 1325770039 \beta_{4} + 208981115 \beta_{5} - 2746793266 \beta_{6} - 60999940 \beta_{7} - 71838994 \beta_{8} + 335278651 \beta_{9} + 29938411 \beta_{10} + 22677841 \beta_{11} + 39569080 \beta_{12} + 155653190 \beta_{13} - 67764574 \beta_{14} - 50535509 \beta_{15} + 34172740 \beta_{16} - 65074360 \beta_{17} ) q^{87} \) \( + ( 64049999783762088 - 200036268219964 \beta_{1} + 5335210356616 \beta_{2} - 1337930632 \beta_{3} + 50082992276 \beta_{4} - 319413560 \beta_{5} + 3884934448 \beta_{6} - 59231948 \beta_{7} - 13803052 \beta_{8} + 2872185460 \beta_{9} + 35520184 \beta_{10} + 103556348 \beta_{11} - 128681620 \beta_{12} + 121552780 \beta_{13} - 7074592 \beta_{14} - 1752624 \beta_{15} - 162938112 \beta_{16} - 232696656 \beta_{17} ) q^{88} \) \( + ( -36662163562525068 - 91630001388846 \beta_{1} - 176905045464 \beta_{2} - 548406154572 \beta_{3} + 21885619350 \beta_{4} + 741538760 \beta_{5} + 12484476198 \beta_{6} - 3715812 \beta_{7} - 245785328 \beta_{8} - 814636952 \beta_{9} + 32336720 \beta_{10} - 256048958 \beta_{11} + 267993578 \beta_{12} - 270265852 \beta_{13} + 48304690 \beta_{14} + 48353408 \beta_{15} - 223073128 \beta_{16} - 8061030 \beta_{17} ) q^{89} \) \( + ( -10017738079193637 - 12117479774616 \beta_{1} - 245366309700 \beta_{2} + 47781860310 \beta_{3} + 15755099886 \beta_{4} - 387420489 \beta_{6} - 1033121304 \beta_{9} - 129140163 \beta_{10} + 129140163 \beta_{11} + 129140163 \beta_{12} + 129140163 \beta_{13} - 129140163 \beta_{17} ) q^{90} \) \( + ( 36730282756733 - 94717209365883 \beta_{1} + 11559226493218 \beta_{2} - 680969756378 \beta_{3} - 4823158769 \beta_{4} - 115435438 \beta_{5} - 15905642101 \beta_{6} - 121006185 \beta_{7} + 51521602 \beta_{8} - 4073167558 \beta_{9} - 102045610 \beta_{10} + 1229273585 \beta_{11} - 104906569 \beta_{12} + 392475972 \beta_{13} - 174507049 \beta_{14} + 328859434 \beta_{15} + 82526745 \beta_{16} + 105022466 \beta_{17} ) q^{91} \) \( + ( 75415722327547244 + 214955624140912 \beta_{1} + 9106630107224 \beta_{2} + 22808418936 \beta_{3} - 36049561492 \beta_{4} - 1624738876 \beta_{5} - 15095418012 \beta_{6} + 63717384 \beta_{7} + 104346584 \beta_{8} - 2494667640 \beta_{9} - 148613456 \beta_{10} - 1083552928 \beta_{11} - 35340344 \beta_{12} - 395331400 \beta_{13} - 4577376 \beta_{14} + 98122800 \beta_{15} - 214375360 \beta_{16} - 178100784 \beta_{17} ) q^{92} \) \( + ( -24833081846687466 - 45397908106257 \beta_{1} - 264151840762 \beta_{2} + 740301780309 \beta_{3} - 3434158115 \beta_{4} + 1132418674 \beta_{5} + 14724118394 \beta_{6} - 107403416 \beta_{7} - 151396448 \beta_{8} - 133522546 \beta_{9} + 51920792 \beta_{10} + 554504543 \beta_{11} - 56118262 \beta_{12} + 381671776 \beta_{13} + 29019052 \beta_{14} - 141821635 \beta_{15} + 91976804 \beta_{16} + 71103748 \beta_{17} ) q^{93} \) \( + ( -115940894542404280 - 1931733568902 \beta_{1} - 14761925229436 \beta_{2} - 495764226920 \beta_{3} - 17448850596 \beta_{4} + 2017920718 \beta_{5} - 7156515420 \beta_{6} - 185876730 \beta_{7} + 135765580 \beta_{8} + 1129033278 \beta_{9} + 6138196 \beta_{10} + 624024070 \beta_{11} - 533757592 \beta_{12} + 230229308 \beta_{13} - 88160400 \beta_{14} - 151450582 \beta_{15} + 202934246 \beta_{16} - 43575488 \beta_{17} ) q^{94} \) \( + ( 86570808724058 - 179432143837536 \beta_{1} - 14389944548702 \beta_{2} + 1320354237466 \beta_{3} + 7512683656 \beta_{4} - 2119968352 \beta_{5} - 53216493758 \beta_{6} + 254934790 \beta_{7} - 136952966 \beta_{8} - 2940428344 \beta_{9} + 479869760 \beta_{10} - 1917923538 \beta_{11} + 221260822 \beta_{12} - 169361462 \beta_{13} - 39029520 \beta_{14} - 210396634 \beta_{15} - 267821606 \beta_{16} + 157011344 \beta_{17} ) q^{95} \) \( + ( 14356694929679606 - 76361992850462 \beta_{1} + 11863479019726 \beta_{2} - 213433759060 \beta_{3} - 29002328056 \beta_{4} + 795926870 \beta_{5} + 7006995774 \beta_{6} - 115410322 \beta_{7} + 30725438 \beta_{8} - 19023170 \beta_{9} - 218535020 \beta_{10} - 39437000 \beta_{11} + 134649346 \beta_{12} + 422257202 \beta_{13} + 91773584 \beta_{14} - 231545336 \beta_{15} + 283327696 \beta_{16} + 150489608 \beta_{17} ) q^{96} \) \( + ( 133891215108647344 + 314572975978786 \beta_{1} + 700664252718 \beta_{2} + 1050935572342 \beta_{3} + 84859188696 \beta_{4} - 6892286002 \beta_{5} + 55501013178 \beta_{6} + 121765074 \beta_{7} - 37946414 \beta_{8} - 1796842008 \beta_{9} + 199977424 \beta_{10} - 588372040 \beta_{11} - 341067748 \beta_{12} + 978174830 \beta_{13} - 219266674 \beta_{14} + 374250362 \beta_{15} - 10645686 \beta_{16} - 187855786 \beta_{17} ) q^{97} \) \( + ( -209339922870383431 + 179782524465291 \beta_{1} - 41646262699148 \beta_{2} + 673955190560 \beta_{3} - 3572706526 \beta_{4} + 5726024066 \beta_{5} + 4773154982 \beta_{6} + 410152676 \beta_{7} + 344212748 \beta_{8} + 4896712816 \beta_{9} + 150432184 \beta_{10} - 1467774036 \beta_{11} - 417537572 \beta_{12} - 419100940 \beta_{13} + 887755904 \beta_{14} - 280830668 \beta_{15} - 561019316 \beta_{16} - 12817864 \beta_{17} ) q^{98} \) \( + ( 2489047501662 - 6885107790345 \beta_{1} + 996058077219 \beta_{2} - 378768098079 \beta_{3} - 1420541793 \beta_{4} - 516560652 \beta_{5} - 7490129454 \beta_{6} + 129140163 \beta_{8} + 1291401630 \beta_{9} - 516560652 \beta_{11} + 387420489 \beta_{13} - 129140163 \beta_{14} + 129140163 \beta_{15} + 258280326 \beta_{17} ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(18q \) \(\mathstrut -\mathstrut 170q^{2} \) \(\mathstrut -\mathstrut 436932q^{4} \) \(\mathstrut -\mathstrut 1721764q^{5} \) \(\mathstrut -\mathstrut 6731586q^{6} \) \(\mathstrut +\mathstrut 108558736q^{8} \) \(\mathstrut -\mathstrut 2324522934q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(18q \) \(\mathstrut -\mathstrut 170q^{2} \) \(\mathstrut -\mathstrut 436932q^{4} \) \(\mathstrut -\mathstrut 1721764q^{5} \) \(\mathstrut -\mathstrut 6731586q^{6} \) \(\mathstrut +\mathstrut 108558736q^{8} \) \(\mathstrut -\mathstrut 2324522934q^{9} \) \(\mathstrut +\mathstrut 1397074644q^{10} \) \(\mathstrut -\mathstrut 6468699852q^{12} \) \(\mathstrut +\mathstrut 1799208612q^{13} \) \(\mathstrut -\mathstrut 48147537912q^{14} \) \(\mathstrut -\mathstrut 119537094672q^{16} \) \(\mathstrut -\mathstrut 282984271180q^{17} \) \(\mathstrut +\mathstrut 21953827710q^{18} \) \(\mathstrut +\mathstrut 491637621128q^{20} \) \(\mathstrut +\mathstrut 254050055640q^{21} \) \(\mathstrut +\mathstrut 272503644408q^{22} \) \(\mathstrut +\mathstrut 4483423343232q^{24} \) \(\mathstrut +\mathstrut 20224739695878q^{25} \) \(\mathstrut -\mathstrut 2765456970196q^{26} \) \(\mathstrut +\mathstrut 27756503338032q^{28} \) \(\mathstrut +\mathstrut 1812741883820q^{29} \) \(\mathstrut -\mathstrut 3579410315772q^{30} \) \(\mathstrut +\mathstrut 216191588070880q^{32} \) \(\mathstrut +\mathstrut 17014318866216q^{33} \) \(\mathstrut -\mathstrut 275796423660708q^{34} \) \(\mathstrut +\mathstrut 56425469699916q^{36} \) \(\mathstrut +\mathstrut 197439381411156q^{37} \) \(\mathstrut -\mathstrut 676030464010008q^{38} \) \(\mathstrut +\mathstrut 1941137842438368q^{40} \) \(\mathstrut -\mathstrut 204669372669676q^{41} \) \(\mathstrut -\mathstrut 1059935698181880q^{42} \) \(\mathstrut +\mathstrut 3599128179401424q^{44} \) \(\mathstrut +\mathstrut 222348883607532q^{45} \) \(\mathstrut -\mathstrut 3847169819651040q^{46} \) \(\mathstrut +\mathstrut 1457367907812912q^{48} \) \(\mathstrut -\mathstrut 3424095824368878q^{49} \) \(\mathstrut -\mathstrut 10192520264009358q^{50} \) \(\mathstrut +\mathstrut 4228291003193208q^{52} \) \(\mathstrut -\mathstrut 14112687314785972q^{53} \) \(\mathstrut +\mathstrut 869318113288518q^{54} \) \(\mathstrut -\mathstrut 6591066595001856q^{56} \) \(\mathstrut -\mathstrut 9662233004693640q^{57} \) \(\mathstrut +\mathstrut 17584741731886020q^{58} \) \(\mathstrut +\mathstrut 193536143011224q^{60} \) \(\mathstrut +\mathstrut 4171094773122132q^{61} \) \(\mathstrut +\mathstrut 77227021488187896q^{62} \) \(\mathstrut -\mathstrut 53424456043095936q^{64} \) \(\mathstrut +\mathstrut 35585253307254712q^{65} \) \(\mathstrut +\mathstrut 40644920520184824q^{66} \) \(\mathstrut -\mathstrut 91403316760947112q^{68} \) \(\mathstrut +\mathstrut 26342656148137824q^{69} \) \(\mathstrut +\mathstrut 264958129332427248q^{70} \) \(\mathstrut -\mathstrut 14019292862113968q^{72} \) \(\mathstrut -\mathstrut 83809473755653692q^{73} \) \(\mathstrut +\mathstrut 55949505142092092q^{74} \) \(\mathstrut -\mathstrut 129026923617832848q^{76} \) \(\mathstrut -\mathstrut 354619239852167328q^{77} \) \(\mathstrut -\mathstrut 34535086926497028q^{78} \) \(\mathstrut +\mathstrut 43257628406200352q^{80} \) \(\mathstrut +\mathstrut 300189270593998242q^{81} \) \(\mathstrut -\mathstrut 458290372754774916q^{82} \) \(\mathstrut +\mathstrut 324078384809786832q^{84} \) \(\mathstrut +\mathstrut 166017223439872920q^{85} \) \(\mathstrut -\mathstrut 975848731010420424q^{86} \) \(\mathstrut +\mathstrut 1151342491040801280q^{88} \) \(\mathstrut -\mathstrut 660650117825905468q^{89} \) \(\mathstrut -\mathstrut 180418447249326972q^{90} \) \(\mathstrut +\mathstrut 1359275349780859584q^{92} \) \(\mathstrut -\mathstrut 447365281400937432q^{93} \) \(\mathstrut -\mathstrut 2087066679318498384q^{94} \) \(\mathstrut +\mathstrut 257905715132447136q^{96} \) \(\mathstrut +\mathstrut 2412557706093917220q^{97} \) \(\mathstrut -\mathstrut 3767017570512990122q^{98} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{18}\mathstrut -\mathstrut \) \(5\) \(x^{17}\mathstrut -\mathstrut \) \(332091\) \(x^{16}\mathstrut -\mathstrut \) \(8722796\) \(x^{15}\mathstrut +\mathstrut \) \(44065046710\) \(x^{14}\mathstrut +\mathstrut \) \(2562387593326\) \(x^{13}\mathstrut -\mathstrut \) \(2939418299568910\) \(x^{12}\mathstrut -\mathstrut \) \(270086571564720600\) \(x^{11}\mathstrut +\mathstrut \) \(99892143837893471971\) \(x^{10}\mathstrut +\mathstrut \) \(13287383366374675095257\) \(x^{9}\mathstrut -\mathstrut \) \(1404426939389796697311653\) \(x^{8}\mathstrut -\mathstrut \) \(302846077765145575240351608\) \(x^{7}\mathstrut -\mathstrut \) \(3048385818268269033396834566\) \(x^{6}\mathstrut +\mathstrut \) \(2387968325606677017735731037126\) \(x^{5}\mathstrut +\mathstrut \) \(189341038803558425525784877687654\) \(x^{4}\mathstrut +\mathstrut \) \(5034939808459428548543033598246780\) \(x^{3}\mathstrut +\mathstrut \) \(5944618618967029844099743191417597\) \(x^{2}\mathstrut -\mathstrut \) \(1074790996286169638673503552568280077\) \(x\mathstrut +\mathstrut \) \(5142216036403070733098077860515333409\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\(12\!\cdots\!05\) \(\nu^{17}\mathstrut -\mathstrut \) \(38\!\cdots\!63\) \(\nu^{16}\mathstrut -\mathstrut \) \(42\!\cdots\!90\) \(\nu^{15}\mathstrut -\mathstrut \) \(15\!\cdots\!60\) \(\nu^{14}\mathstrut +\mathstrut \) \(56\!\cdots\!32\) \(\nu^{13}\mathstrut +\mathstrut \) \(17\!\cdots\!40\) \(\nu^{12}\mathstrut -\mathstrut \) \(38\!\cdots\!00\) \(\nu^{11}\mathstrut -\mathstrut \) \(24\!\cdots\!62\) \(\nu^{10}\mathstrut +\mathstrut \) \(13\!\cdots\!61\) \(\nu^{9}\mathstrut +\mathstrut \) \(13\!\cdots\!43\) \(\nu^{8}\mathstrut -\mathstrut \) \(22\!\cdots\!56\) \(\nu^{7}\mathstrut -\mathstrut \) \(32\!\cdots\!18\) \(\nu^{6}\mathstrut +\mathstrut \) \(70\!\cdots\!12\) \(\nu^{5}\mathstrut +\mathstrut \) \(29\!\cdots\!18\) \(\nu^{4}\mathstrut +\mathstrut \) \(14\!\cdots\!26\) \(\nu^{3}\mathstrut +\mathstrut \) \(13\!\cdots\!90\) \(\nu^{2}\mathstrut -\mathstrut \) \(37\!\cdots\!99\) \(\nu\mathstrut +\mathstrut \) \(13\!\cdots\!31\)\()/\)\(12\!\cdots\!60\)
\(\beta_{2}\)\(=\)\((\)\(-\)\(11\!\cdots\!41\) \(\nu^{17}\mathstrut +\mathstrut \) \(73\!\cdots\!23\) \(\nu^{16}\mathstrut +\mathstrut \) \(37\!\cdots\!30\) \(\nu^{15}\mathstrut -\mathstrut \) \(12\!\cdots\!84\) \(\nu^{14}\mathstrut -\mathstrut \) \(49\!\cdots\!20\) \(\nu^{13}\mathstrut +\mathstrut \) \(22\!\cdots\!40\) \(\nu^{12}\mathstrut +\mathstrut \) \(33\!\cdots\!00\) \(\nu^{11}\mathstrut +\mathstrut \) \(10\!\cdots\!86\) \(\nu^{10}\mathstrut -\mathstrut \) \(12\!\cdots\!89\) \(\nu^{9}\mathstrut -\mathstrut \) \(80\!\cdots\!47\) \(\nu^{8}\mathstrut +\mathstrut \) \(20\!\cdots\!28\) \(\nu^{7}\mathstrut +\mathstrut \) \(22\!\cdots\!22\) \(\nu^{6}\mathstrut -\mathstrut \) \(95\!\cdots\!32\) \(\nu^{5}\mathstrut -\mathstrut \) \(21\!\cdots\!66\) \(\nu^{4}\mathstrut -\mathstrut \) \(89\!\cdots\!54\) \(\nu^{3}\mathstrut -\mathstrut \) \(53\!\cdots\!66\) \(\nu^{2}\mathstrut +\mathstrut \) \(22\!\cdots\!99\) \(\nu\mathstrut -\mathstrut \) \(94\!\cdots\!19\)\()/\)\(86\!\cdots\!20\)
\(\beta_{3}\)\(=\)\((\)\(22\!\cdots\!15\) \(\nu^{17}\mathstrut -\mathstrut \) \(15\!\cdots\!93\) \(\nu^{16}\mathstrut -\mathstrut \) \(74\!\cdots\!78\) \(\nu^{15}\mathstrut +\mathstrut \) \(26\!\cdots\!64\) \(\nu^{14}\mathstrut +\mathstrut \) \(98\!\cdots\!16\) \(\nu^{13}\mathstrut -\mathstrut \) \(21\!\cdots\!12\) \(\nu^{12}\mathstrut -\mathstrut \) \(66\!\cdots\!80\) \(\nu^{11}\mathstrut -\mathstrut \) \(20\!\cdots\!26\) \(\nu^{10}\mathstrut +\mathstrut \) \(24\!\cdots\!35\) \(\nu^{9}\mathstrut +\mathstrut \) \(15\!\cdots\!85\) \(\nu^{8}\mathstrut -\mathstrut \) \(41\!\cdots\!04\) \(\nu^{7}\mathstrut -\mathstrut \) \(43\!\cdots\!42\) \(\nu^{6}\mathstrut +\mathstrut \) \(19\!\cdots\!88\) \(\nu^{5}\mathstrut +\mathstrut \) \(42\!\cdots\!66\) \(\nu^{4}\mathstrut +\mathstrut \) \(17\!\cdots\!66\) \(\nu^{3}\mathstrut +\mathstrut \) \(10\!\cdots\!78\) \(\nu^{2}\mathstrut -\mathstrut \) \(42\!\cdots\!29\) \(\nu\mathstrut +\mathstrut \) \(18\!\cdots\!81\)\()/\)\(12\!\cdots\!60\)
\(\beta_{4}\)\(=\)\((\)\(32\!\cdots\!24\) \(\nu^{17}\mathstrut -\mathstrut \) \(20\!\cdots\!27\) \(\nu^{16}\mathstrut -\mathstrut \) \(10\!\cdots\!52\) \(\nu^{15}\mathstrut +\mathstrut \) \(32\!\cdots\!30\) \(\nu^{14}\mathstrut +\mathstrut \) \(14\!\cdots\!54\) \(\nu^{13}\mathstrut +\mathstrut \) \(31\!\cdots\!38\) \(\nu^{12}\mathstrut -\mathstrut \) \(96\!\cdots\!54\) \(\nu^{11}\mathstrut -\mathstrut \) \(33\!\cdots\!58\) \(\nu^{10}\mathstrut +\mathstrut \) \(34\!\cdots\!44\) \(\nu^{9}\mathstrut +\mathstrut \) \(23\!\cdots\!77\) \(\nu^{8}\mathstrut -\mathstrut \) \(59\!\cdots\!46\) \(\nu^{7}\mathstrut -\mathstrut \) \(65\!\cdots\!30\) \(\nu^{6}\mathstrut +\mathstrut \) \(27\!\cdots\!12\) \(\nu^{5}\mathstrut +\mathstrut \) \(62\!\cdots\!88\) \(\nu^{4}\mathstrut +\mathstrut \) \(26\!\cdots\!70\) \(\nu^{3}\mathstrut +\mathstrut \) \(14\!\cdots\!76\) \(\nu^{2}\mathstrut -\mathstrut \) \(65\!\cdots\!46\) \(\nu\mathstrut +\mathstrut \) \(29\!\cdots\!35\)\()/\)\(65\!\cdots\!40\)
\(\beta_{5}\)\(=\)\((\)\(32\!\cdots\!46\) \(\nu^{17}\mathstrut +\mathstrut \) \(13\!\cdots\!47\) \(\nu^{16}\mathstrut -\mathstrut \) \(11\!\cdots\!40\) \(\nu^{15}\mathstrut -\mathstrut \) \(76\!\cdots\!34\) \(\nu^{14}\mathstrut +\mathstrut \) \(15\!\cdots\!30\) \(\nu^{13}\mathstrut +\mathstrut \) \(14\!\cdots\!38\) \(\nu^{12}\mathstrut -\mathstrut \) \(10\!\cdots\!10\) \(\nu^{11}\mathstrut -\mathstrut \) \(12\!\cdots\!34\) \(\nu^{10}\mathstrut +\mathstrut \) \(36\!\cdots\!86\) \(\nu^{9}\mathstrut +\mathstrut \) \(58\!\cdots\!47\) \(\nu^{8}\mathstrut -\mathstrut \) \(55\!\cdots\!42\) \(\nu^{7}\mathstrut -\mathstrut \) \(12\!\cdots\!66\) \(\nu^{6}\mathstrut -\mathstrut \) \(36\!\cdots\!28\) \(\nu^{5}\mathstrut +\mathstrut \) \(10\!\cdots\!84\) \(\nu^{4}\mathstrut +\mathstrut \) \(65\!\cdots\!42\) \(\nu^{3}\mathstrut +\mathstrut \) \(77\!\cdots\!04\) \(\nu^{2}\mathstrut -\mathstrut \) \(16\!\cdots\!40\) \(\nu\mathstrut +\mathstrut \) \(55\!\cdots\!65\)\()/\)\(19\!\cdots\!20\)
\(\beta_{6}\)\(=\)\((\)\(-\)\(67\!\cdots\!85\) \(\nu^{17}\mathstrut +\mathstrut \) \(50\!\cdots\!37\) \(\nu^{16}\mathstrut +\mathstrut \) \(22\!\cdots\!98\) \(\nu^{15}\mathstrut -\mathstrut \) \(96\!\cdots\!28\) \(\nu^{14}\mathstrut -\mathstrut \) \(29\!\cdots\!68\) \(\nu^{13}\mathstrut +\mathstrut \) \(31\!\cdots\!68\) \(\nu^{12}\mathstrut +\mathstrut \) \(19\!\cdots\!68\) \(\nu^{11}\mathstrut +\mathstrut \) \(44\!\cdots\!70\) \(\nu^{10}\mathstrut -\mathstrut \) \(70\!\cdots\!65\) \(\nu^{9}\mathstrut -\mathstrut \) \(40\!\cdots\!41\) \(\nu^{8}\mathstrut +\mathstrut \) \(12\!\cdots\!32\) \(\nu^{7}\mathstrut +\mathstrut \) \(11\!\cdots\!54\) \(\nu^{6}\mathstrut -\mathstrut \) \(63\!\cdots\!44\) \(\nu^{5}\mathstrut -\mathstrut \) \(11\!\cdots\!26\) \(\nu^{4}\mathstrut -\mathstrut \) \(44\!\cdots\!94\) \(\nu^{3}\mathstrut -\mathstrut \) \(17\!\cdots\!10\) \(\nu^{2}\mathstrut +\mathstrut \) \(11\!\cdots\!55\) \(\nu\mathstrut -\mathstrut \) \(51\!\cdots\!81\)\()/\)\(28\!\cdots\!40\)
\(\beta_{7}\)\(=\)\((\)\(-\)\(13\!\cdots\!37\) \(\nu^{17}\mathstrut +\mathstrut \) \(54\!\cdots\!22\) \(\nu^{16}\mathstrut +\mathstrut \) \(39\!\cdots\!06\) \(\nu^{15}\mathstrut -\mathstrut \) \(16\!\cdots\!30\) \(\nu^{14}\mathstrut -\mathstrut \) \(46\!\cdots\!18\) \(\nu^{13}\mathstrut +\mathstrut \) \(19\!\cdots\!94\) \(\nu^{12}\mathstrut +\mathstrut \) \(30\!\cdots\!82\) \(\nu^{11}\mathstrut -\mathstrut \) \(11\!\cdots\!48\) \(\nu^{10}\mathstrut -\mathstrut \) \(12\!\cdots\!73\) \(\nu^{9}\mathstrut +\mathstrut \) \(37\!\cdots\!72\) \(\nu^{8}\mathstrut +\mathstrut \) \(32\!\cdots\!06\) \(\nu^{7}\mathstrut -\mathstrut \) \(61\!\cdots\!56\) \(\nu^{6}\mathstrut -\mathstrut \) \(56\!\cdots\!56\) \(\nu^{5}\mathstrut +\mathstrut \) \(36\!\cdots\!42\) \(\nu^{4}\mathstrut +\mathstrut \) \(43\!\cdots\!16\) \(\nu^{3}\mathstrut +\mathstrut \) \(73\!\cdots\!14\) \(\nu^{2}\mathstrut -\mathstrut \) \(13\!\cdots\!19\) \(\nu\mathstrut +\mathstrut \) \(40\!\cdots\!38\)\()/\)\(97\!\cdots\!60\)
\(\beta_{8}\)\(=\)\((\)\(13\!\cdots\!09\) \(\nu^{17}\mathstrut +\mathstrut \) \(11\!\cdots\!80\) \(\nu^{16}\mathstrut -\mathstrut \) \(46\!\cdots\!42\) \(\nu^{15}\mathstrut -\mathstrut \) \(16\!\cdots\!14\) \(\nu^{14}\mathstrut +\mathstrut \) \(62\!\cdots\!30\) \(\nu^{13}\mathstrut +\mathstrut \) \(40\!\cdots\!22\) \(\nu^{12}\mathstrut -\mathstrut \) \(43\!\cdots\!14\) \(\nu^{11}\mathstrut -\mathstrut \) \(40\!\cdots\!88\) \(\nu^{10}\mathstrut +\mathstrut \) \(15\!\cdots\!57\) \(\nu^{9}\mathstrut +\mathstrut \) \(19\!\cdots\!66\) \(\nu^{8}\mathstrut -\mathstrut \) \(24\!\cdots\!70\) \(\nu^{7}\mathstrut -\mathstrut \) \(44\!\cdots\!52\) \(\nu^{6}\mathstrut +\mathstrut \) \(34\!\cdots\!00\) \(\nu^{5}\mathstrut +\mathstrut \) \(37\!\cdots\!58\) \(\nu^{4}\mathstrut +\mathstrut \) \(21\!\cdots\!20\) \(\nu^{3}\mathstrut +\mathstrut \) \(27\!\cdots\!38\) \(\nu^{2}\mathstrut -\mathstrut \) \(41\!\cdots\!05\) \(\nu\mathstrut +\mathstrut \) \(49\!\cdots\!00\)\()/\)\(97\!\cdots\!60\)
\(\beta_{9}\)\(=\)\((\)\(72\!\cdots\!90\) \(\nu^{17}\mathstrut -\mathstrut \) \(40\!\cdots\!47\) \(\nu^{16}\mathstrut -\mathstrut \) \(23\!\cdots\!72\) \(\nu^{15}\mathstrut +\mathstrut \) \(59\!\cdots\!82\) \(\nu^{14}\mathstrut +\mathstrut \) \(31\!\cdots\!22\) \(\nu^{13}\mathstrut +\mathstrut \) \(22\!\cdots\!66\) \(\nu^{12}\mathstrut -\mathstrut \) \(21\!\cdots\!34\) \(\nu^{11}\mathstrut -\mathstrut \) \(84\!\cdots\!30\) \(\nu^{10}\mathstrut +\mathstrut \) \(76\!\cdots\!98\) \(\nu^{9}\mathstrut +\mathstrut \) \(56\!\cdots\!85\) \(\nu^{8}\mathstrut -\mathstrut \) \(13\!\cdots\!58\) \(\nu^{7}\mathstrut -\mathstrut \) \(15\!\cdots\!86\) \(\nu^{6}\mathstrut +\mathstrut \) \(56\!\cdots\!96\) \(\nu^{5}\mathstrut +\mathstrut \) \(14\!\cdots\!60\) \(\nu^{4}\mathstrut +\mathstrut \) \(61\!\cdots\!26\) \(\nu^{3}\mathstrut +\mathstrut \) \(34\!\cdots\!96\) \(\nu^{2}\mathstrut -\mathstrut \) \(17\!\cdots\!92\) \(\nu\mathstrut +\mathstrut \) \(80\!\cdots\!03\)\()/\)\(19\!\cdots\!20\)
\(\beta_{10}\)\(=\)\((\)\(37\!\cdots\!09\) \(\nu^{17}\mathstrut -\mathstrut \) \(12\!\cdots\!13\) \(\nu^{16}\mathstrut -\mathstrut \) \(12\!\cdots\!02\) \(\nu^{15}\mathstrut +\mathstrut \) \(13\!\cdots\!36\) \(\nu^{14}\mathstrut +\mathstrut \) \(16\!\cdots\!68\) \(\nu^{13}\mathstrut +\mathstrut \) \(50\!\cdots\!88\) \(\nu^{12}\mathstrut -\mathstrut \) \(11\!\cdots\!60\) \(\nu^{11}\mathstrut -\mathstrut \) \(70\!\cdots\!58\) \(\nu^{10}\mathstrut +\mathstrut \) \(40\!\cdots\!25\) \(\nu^{9}\mathstrut +\mathstrut \) \(38\!\cdots\!69\) \(\nu^{8}\mathstrut -\mathstrut \) \(66\!\cdots\!36\) \(\nu^{7}\mathstrut -\mathstrut \) \(96\!\cdots\!82\) \(\nu^{6}\mathstrut +\mathstrut \) \(20\!\cdots\!08\) \(\nu^{5}\mathstrut +\mathstrut \) \(87\!\cdots\!54\) \(\nu^{4}\mathstrut +\mathstrut \) \(42\!\cdots\!54\) \(\nu^{3}\mathstrut +\mathstrut \) \(31\!\cdots\!78\) \(\nu^{2}\mathstrut -\mathstrut \) \(12\!\cdots\!15\) \(\nu\mathstrut +\mathstrut \) \(58\!\cdots\!37\)\()/\)\(48\!\cdots\!80\)
\(\beta_{11}\)\(=\)\((\)\(-\)\(24\!\cdots\!87\) \(\nu^{17}\mathstrut +\mathstrut \) \(15\!\cdots\!37\) \(\nu^{16}\mathstrut +\mathstrut \) \(82\!\cdots\!14\) \(\nu^{15}\mathstrut -\mathstrut \) \(26\!\cdots\!12\) \(\nu^{14}\mathstrut -\mathstrut \) \(10\!\cdots\!00\) \(\nu^{13}\mathstrut -\mathstrut \) \(44\!\cdots\!76\) \(\nu^{12}\mathstrut +\mathstrut \) \(73\!\cdots\!12\) \(\nu^{11}\mathstrut +\mathstrut \) \(24\!\cdots\!86\) \(\nu^{10}\mathstrut -\mathstrut \) \(26\!\cdots\!47\) \(\nu^{9}\mathstrut -\mathstrut \) \(17\!\cdots\!97\) \(\nu^{8}\mathstrut +\mathstrut \) \(45\!\cdots\!72\) \(\nu^{7}\mathstrut +\mathstrut \) \(49\!\cdots\!38\) \(\nu^{6}\mathstrut -\mathstrut \) \(20\!\cdots\!64\) \(\nu^{5}\mathstrut -\mathstrut \) \(47\!\cdots\!94\) \(\nu^{4}\mathstrut -\mathstrut \) \(19\!\cdots\!26\) \(\nu^{3}\mathstrut -\mathstrut \) \(11\!\cdots\!22\) \(\nu^{2}\mathstrut +\mathstrut \) \(50\!\cdots\!69\) \(\nu\mathstrut -\mathstrut \) \(21\!\cdots\!13\)\()/\)\(32\!\cdots\!20\)
\(\beta_{12}\)\(=\)\((\)\(-\)\(10\!\cdots\!22\) \(\nu^{17}\mathstrut +\mathstrut \) \(71\!\cdots\!19\) \(\nu^{16}\mathstrut +\mathstrut \) \(36\!\cdots\!64\) \(\nu^{15}\mathstrut -\mathstrut \) \(12\!\cdots\!54\) \(\nu^{14}\mathstrut -\mathstrut \) \(47\!\cdots\!18\) \(\nu^{13}\mathstrut +\mathstrut \) \(72\!\cdots\!50\) \(\nu^{12}\mathstrut +\mathstrut \) \(32\!\cdots\!02\) \(\nu^{11}\mathstrut +\mathstrut \) \(10\!\cdots\!58\) \(\nu^{10}\mathstrut -\mathstrut \) \(11\!\cdots\!06\) \(\nu^{9}\mathstrut -\mathstrut \) \(75\!\cdots\!13\) \(\nu^{8}\mathstrut +\mathstrut \) \(20\!\cdots\!34\) \(\nu^{7}\mathstrut +\mathstrut \) \(21\!\cdots\!10\) \(\nu^{6}\mathstrut -\mathstrut \) \(94\!\cdots\!16\) \(\nu^{5}\mathstrut -\mathstrut \) \(20\!\cdots\!36\) \(\nu^{4}\mathstrut -\mathstrut \) \(83\!\cdots\!34\) \(\nu^{3}\mathstrut -\mathstrut \) \(48\!\cdots\!72\) \(\nu^{2}\mathstrut +\mathstrut \) \(20\!\cdots\!88\) \(\nu\mathstrut -\mathstrut \) \(84\!\cdots\!59\)\()/\)\(92\!\cdots\!20\)
\(\beta_{13}\)\(=\)\((\)\(-\)\(32\!\cdots\!16\) \(\nu^{17}\mathstrut +\mathstrut \) \(21\!\cdots\!63\) \(\nu^{16}\mathstrut +\mathstrut \) \(10\!\cdots\!28\) \(\nu^{15}\mathstrut -\mathstrut \) \(35\!\cdots\!82\) \(\nu^{14}\mathstrut -\mathstrut \) \(14\!\cdots\!14\) \(\nu^{13}\mathstrut +\mathstrut \) \(17\!\cdots\!90\) \(\nu^{12}\mathstrut +\mathstrut \) \(96\!\cdots\!10\) \(\nu^{11}\mathstrut +\mathstrut \) \(31\!\cdots\!62\) \(\nu^{10}\mathstrut -\mathstrut \) \(34\!\cdots\!00\) \(\nu^{9}\mathstrut -\mathstrut \) \(23\!\cdots\!73\) \(\nu^{8}\mathstrut +\mathstrut \) \(59\!\cdots\!78\) \(\nu^{7}\mathstrut +\mathstrut \) \(64\!\cdots\!30\) \(\nu^{6}\mathstrut -\mathstrut \) \(27\!\cdots\!52\) \(\nu^{5}\mathstrut -\mathstrut \) \(62\!\cdots\!72\) \(\nu^{4}\mathstrut -\mathstrut \) \(26\!\cdots\!86\) \(\nu^{3}\mathstrut -\mathstrut \) \(16\!\cdots\!32\) \(\nu^{2}\mathstrut +\mathstrut \) \(61\!\cdots\!38\) \(\nu\mathstrut -\mathstrut \) \(24\!\cdots\!67\)\()/\)\(19\!\cdots\!20\)
\(\beta_{14}\)\(=\)\((\)\(24\!\cdots\!27\) \(\nu^{17}\mathstrut -\mathstrut \) \(15\!\cdots\!89\) \(\nu^{16}\mathstrut -\mathstrut \) \(80\!\cdots\!06\) \(\nu^{15}\mathstrut +\mathstrut \) \(24\!\cdots\!00\) \(\nu^{14}\mathstrut +\mathstrut \) \(10\!\cdots\!96\) \(\nu^{13}\mathstrut +\mathstrut \) \(25\!\cdots\!80\) \(\nu^{12}\mathstrut -\mathstrut \) \(72\!\cdots\!04\) \(\nu^{11}\mathstrut -\mathstrut \) \(25\!\cdots\!06\) \(\nu^{10}\mathstrut +\mathstrut \) \(25\!\cdots\!99\) \(\nu^{9}\mathstrut +\mathstrut \) \(17\!\cdots\!65\) \(\nu^{8}\mathstrut -\mathstrut \) \(44\!\cdots\!80\) \(\nu^{7}\mathstrut -\mathstrut \) \(48\!\cdots\!66\) \(\nu^{6}\mathstrut +\mathstrut \) \(20\!\cdots\!96\) \(\nu^{5}\mathstrut +\mathstrut \) \(46\!\cdots\!66\) \(\nu^{4}\mathstrut +\mathstrut \) \(19\!\cdots\!06\) \(\nu^{3}\mathstrut +\mathstrut \) \(13\!\cdots\!86\) \(\nu^{2}\mathstrut -\mathstrut \) \(44\!\cdots\!97\) \(\nu\mathstrut +\mathstrut \) \(16\!\cdots\!77\)\()/\)\(97\!\cdots\!60\)
\(\beta_{15}\)\(=\)\((\)\(45\!\cdots\!85\) \(\nu^{17}\mathstrut -\mathstrut \) \(34\!\cdots\!66\) \(\nu^{16}\mathstrut -\mathstrut \) \(15\!\cdots\!82\) \(\nu^{15}\mathstrut +\mathstrut \) \(64\!\cdots\!98\) \(\nu^{14}\mathstrut +\mathstrut \) \(19\!\cdots\!34\) \(\nu^{13}\mathstrut -\mathstrut \) \(20\!\cdots\!94\) \(\nu^{12}\mathstrut -\mathstrut \) \(13\!\cdots\!46\) \(\nu^{11}\mathstrut -\mathstrut \) \(30\!\cdots\!44\) \(\nu^{10}\mathstrut +\mathstrut \) \(48\!\cdots\!69\) \(\nu^{9}\mathstrut +\mathstrut \) \(27\!\cdots\!96\) \(\nu^{8}\mathstrut -\mathstrut \) \(84\!\cdots\!90\) \(\nu^{7}\mathstrut -\mathstrut \) \(80\!\cdots\!72\) \(\nu^{6}\mathstrut +\mathstrut \) \(43\!\cdots\!36\) \(\nu^{5}\mathstrut +\mathstrut \) \(79\!\cdots\!70\) \(\nu^{4}\mathstrut +\mathstrut \) \(30\!\cdots\!68\) \(\nu^{3}\mathstrut +\mathstrut \) \(13\!\cdots\!86\) \(\nu^{2}\mathstrut -\mathstrut \) \(72\!\cdots\!29\) \(\nu\mathstrut +\mathstrut \) \(31\!\cdots\!66\)\()/\)\(97\!\cdots\!60\)
\(\beta_{16}\)\(=\)\((\)\(-\)\(24\!\cdots\!81\) \(\nu^{17}\mathstrut +\mathstrut \) \(15\!\cdots\!02\) \(\nu^{16}\mathstrut +\mathstrut \) \(80\!\cdots\!02\) \(\nu^{15}\mathstrut -\mathstrut \) \(25\!\cdots\!58\) \(\nu^{14}\mathstrut -\mathstrut \) \(10\!\cdots\!78\) \(\nu^{13}\mathstrut -\mathstrut \) \(12\!\cdots\!30\) \(\nu^{12}\mathstrut +\mathstrut \) \(71\!\cdots\!22\) \(\nu^{11}\mathstrut +\mathstrut \) \(24\!\cdots\!24\) \(\nu^{10}\mathstrut -\mathstrut \) \(25\!\cdots\!33\) \(\nu^{9}\mathstrut -\mathstrut \) \(17\!\cdots\!76\) \(\nu^{8}\mathstrut +\mathstrut \) \(44\!\cdots\!46\) \(\nu^{7}\mathstrut +\mathstrut \) \(48\!\cdots\!28\) \(\nu^{6}\mathstrut -\mathstrut \) \(20\!\cdots\!96\) \(\nu^{5}\mathstrut -\mathstrut \) \(46\!\cdots\!06\) \(\nu^{4}\mathstrut -\mathstrut \) \(19\!\cdots\!04\) \(\nu^{3}\mathstrut -\mathstrut \) \(11\!\cdots\!86\) \(\nu^{2}\mathstrut +\mathstrut \) \(49\!\cdots\!37\) \(\nu\mathstrut -\mathstrut \) \(21\!\cdots\!78\)\()/\)\(48\!\cdots\!80\)
\(\beta_{17}\)\(=\)\((\)\(12\!\cdots\!09\) \(\nu^{17}\mathstrut -\mathstrut \) \(86\!\cdots\!62\) \(\nu^{16}\mathstrut -\mathstrut \) \(41\!\cdots\!22\) \(\nu^{15}\mathstrut +\mathstrut \) \(14\!\cdots\!46\) \(\nu^{14}\mathstrut +\mathstrut \) \(55\!\cdots\!50\) \(\nu^{13}\mathstrut -\mathstrut \) \(17\!\cdots\!66\) \(\nu^{12}\mathstrut -\mathstrut \) \(37\!\cdots\!98\) \(\nu^{11}\mathstrut -\mathstrut \) \(11\!\cdots\!24\) \(\nu^{10}\mathstrut +\mathstrut \) \(13\!\cdots\!45\) \(\nu^{9}\mathstrut +\mathstrut \) \(85\!\cdots\!12\) \(\nu^{8}\mathstrut -\mathstrut \) \(23\!\cdots\!58\) \(\nu^{7}\mathstrut -\mathstrut \) \(24\!\cdots\!44\) \(\nu^{6}\mathstrut +\mathstrut \) \(11\!\cdots\!16\) \(\nu^{5}\mathstrut +\mathstrut \) \(23\!\cdots\!74\) \(\nu^{4}\mathstrut +\mathstrut \) \(95\!\cdots\!64\) \(\nu^{3}\mathstrut +\mathstrut \) \(50\!\cdots\!70\) \(\nu^{2}\mathstrut -\mathstrut \) \(23\!\cdots\!45\) \(\nu\mathstrut +\mathstrut \) \(10\!\cdots\!18\)\()/\)\(13\!\cdots\!80\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-\)\(32\) \(\beta_{17}\mathstrut +\mathstrut \) \(17\) \(\beta_{16}\mathstrut -\mathstrut \) \(49\) \(\beta_{15}\mathstrut +\mathstrut \) \(64\) \(\beta_{14}\mathstrut -\mathstrut \) \(71\) \(\beta_{13}\mathstrut -\mathstrut \) \(49\) \(\beta_{12}\mathstrut -\mathstrut \) \(37\) \(\beta_{11}\mathstrut -\mathstrut \) \(64\) \(\beta_{10}\mathstrut +\mathstrut \) \(644\) \(\beta_{9}\mathstrut +\mathstrut \) \(49\) \(\beta_{8}\mathstrut -\mathstrut \) \(17\) \(\beta_{7}\mathstrut -\mathstrut \) \(12619\) \(\beta_{6}\mathstrut +\mathstrut \) \(4888\) \(\beta_{5}\mathstrut +\mathstrut \) \(4276\) \(\beta_{4}\mathstrut -\mathstrut \) \(102495\) \(\beta_{3}\mathstrut -\mathstrut \) \(860335\) \(\beta_{2}\mathstrut -\mathstrut \) \(269780532\) \(\beta_{1}\mathstrut +\mathstrut \) \(478565313\)\()/\)\(1289945088\)
\(\nu^{2}\)\(=\)\((\)\(-\)\(29600\) \(\beta_{17}\mathstrut -\mathstrut \) \(41137\) \(\beta_{16}\mathstrut +\mathstrut \) \(18449\) \(\beta_{15}\mathstrut +\mathstrut \) \(12544\) \(\beta_{14}\mathstrut -\mathstrut \) \(7193\) \(\beta_{13}\mathstrut -\mathstrut \) \(4015\) \(\beta_{12}\mathstrut -\mathstrut \) \(82555\) \(\beta_{11}\mathstrut -\mathstrut \) \(19456\) \(\beta_{10}\mathstrut -\mathstrut \) \(10180\) \(\beta_{9}\mathstrut -\mathstrut \) \(27089\) \(\beta_{8}\mathstrut +\mathstrut \) \(10033\) \(\beta_{7}\mathstrut -\mathstrut \) \(339679\) \(\beta_{6}\mathstrut +\mathstrut \) \(156958\) \(\beta_{5}\mathstrut +\mathstrut \) \(241666\) \(\beta_{4}\mathstrut -\mathstrut \) \(59827461\) \(\beta_{3}\mathstrut -\mathstrut \) \(27955513\) \(\beta_{2}\mathstrut -\mathstrut \) \(13232112648\) \(\beta_{1}\mathstrut +\mathstrut \) \(47605348665993\)\()/\)\(1289945088\)
\(\nu^{3}\)\(=\)\((\)\(-\)\(3975712\) \(\beta_{17}\mathstrut -\mathstrut \) \(1037534\) \(\beta_{16}\mathstrut -\mathstrut \) \(1597250\) \(\beta_{15}\mathstrut +\mathstrut \) \(6289088\) \(\beta_{14}\mathstrut -\mathstrut \) \(4135342\) \(\beta_{13}\mathstrut +\mathstrut \) \(1253950\) \(\beta_{12}\mathstrut -\mathstrut \) \(9956714\) \(\beta_{11}\mathstrut -\mathstrut \) \(1934528\) \(\beta_{10}\mathstrut +\mathstrut \) \(40563016\) \(\beta_{9}\mathstrut +\mathstrut \) \(3895490\) \(\beta_{8}\mathstrut -\mathstrut \) \(2466850\) \(\beta_{7}\mathstrut -\mathstrut \) \(847935221\) \(\beta_{6}\mathstrut +\mathstrut \) \(322172465\) \(\beta_{5}\mathstrut -\mathstrut \) \(244500439\) \(\beta_{4}\mathstrut -\mathstrut \) \(9756131172\) \(\beta_{3}\mathstrut -\mathstrut \) \(61145932010\) \(\beta_{2}\mathstrut -\mathstrut \) \(17997069369570\) \(\beta_{1}\mathstrut +\mathstrut \) \(2240335345982961\)\()/\)\(1289945088\)
\(\nu^{4}\)\(=\)\((\)\(-\)\(769389408\) \(\beta_{17}\mathstrut -\mathstrut \) \(1381848273\) \(\beta_{16}\mathstrut +\mathstrut \) \(537387633\) \(\beta_{15}\mathstrut +\mathstrut \) \(295402752\) \(\beta_{14}\mathstrut -\mathstrut \) \(323322489\) \(\beta_{13}\mathstrut +\mathstrut \) \(296245809\) \(\beta_{12}\mathstrut -\mathstrut \) \(1494160155\) \(\beta_{11}\mathstrut -\mathstrut \) \(670569984\) \(\beta_{10}\mathstrut -\mathstrut \) \(323028036\) \(\beta_{9}\mathstrut -\mathstrut \) \(620203185\) \(\beta_{8}\mathstrut +\mathstrut \) \(336729681\) \(\beta_{7}\mathstrut -\mathstrut \) \(8389105769\) \(\beta_{6}\mathstrut +\mathstrut \) \(6870906612\) \(\beta_{5}\mathstrut -\mathstrut \) \(36050697000\) \(\beta_{4}\mathstrut -\mathstrut \) \(1884581486057\) \(\beta_{3}\mathstrut -\mathstrut \) \(1338616397153\) \(\beta_{2}\mathstrut -\mathstrut \) \(498951576952196\) \(\beta_{1}\mathstrut +\mathstrut \) \(1063622436768592819\)\()/\)\(429981696\)
\(\nu^{5}\)\(=\)\((\)\(-\)\(437523926656\) \(\beta_{17}\mathstrut -\mathstrut \) \(316178763047\) \(\beta_{16}\mathstrut -\mathstrut \) \(51322553177\) \(\beta_{15}\mathstrut +\mathstrut \) \(615252070016\) \(\beta_{14}\mathstrut -\mathstrut \) \(172595039455\) \(\beta_{13}\mathstrut +\mathstrut \) \(335796749095\) \(\beta_{12}\mathstrut -\mathstrut \) \(929861432621\) \(\beta_{11}\mathstrut -\mathstrut \) \(116715512960\) \(\beta_{10}\mathstrut +\mathstrut \) \(2930113342948\) \(\beta_{9}\mathstrut +\mathstrut \) \(220906036697\) \(\beta_{8}\mathstrut -\mathstrut \) \(249937157593\) \(\beta_{7}\mathstrut -\mathstrut \) \(63119367482996\) \(\beta_{6}\mathstrut +\mathstrut \) \(24877028946599\) \(\beta_{5}\mathstrut -\mathstrut \) \(39076255213597\) \(\beta_{4}\mathstrut -\mathstrut \) \(793830707727633\) \(\beta_{3}\mathstrut -\mathstrut \) \(5111114992968539\) \(\beta_{2}\mathstrut -\mathstrut \) \(1397409593049322362\) \(\beta_{1}\mathstrut +\mathstrut \) \(239166508033554977034\)\()/\)\(1289945088\)
\(\nu^{6}\)\(=\)\((\)\(-\)\(48342125681296\) \(\beta_{17}\mathstrut -\mathstrut \) \(96510437626505\) \(\beta_{16}\mathstrut +\mathstrut \) \(32657532452089\) \(\beta_{15}\mathstrut +\mathstrut \) \(20564735852192\) \(\beta_{14}\mathstrut -\mathstrut \) \(22467773346241\) \(\beta_{13}\mathstrut +\mathstrut \) \(34885303689337\) \(\beta_{12}\mathstrut -\mathstrut \) \(63780475870931\) \(\beta_{11}\mathstrut -\mathstrut \) \(48439739721632\) \(\beta_{10}\mathstrut +\mathstrut \) \(187845357436\) \(\beta_{9}\mathstrut -\mathstrut \) \(32497669922425\) \(\beta_{8}\mathstrut +\mathstrut \) \(21163787199113\) \(\beta_{7}\mathstrut -\mathstrut \) \(1015944259296371\) \(\beta_{6}\mathstrut +\mathstrut \) \(543724681114418\) \(\beta_{5}\mathstrut -\mathstrut \) \(5126664552476458\) \(\beta_{4}\mathstrut -\mathstrut \) \(124011461887802277\) \(\beta_{3}\mathstrut -\mathstrut \) \(110002982545081841\) \(\beta_{2}\mathstrut -\mathstrut \) \(36307957301670802992\) \(\beta_{1}\mathstrut +\mathstrut \) \(61417883337532678950861\)\()/\)\(322486272\)
\(\nu^{7}\)\(=\)\((\)\(-\)\(45131039259290912\) \(\beta_{17}\mathstrut -\mathstrut \) \(44718503513333431\) \(\beta_{16}\mathstrut -\mathstrut \) \(12366362976361\) \(\beta_{15}\mathstrut +\mathstrut \) \(58235104698957760\) \(\beta_{14}\mathstrut -\mathstrut \) \(2308600217579855\) \(\beta_{13}\mathstrut +\mathstrut \) \(41763876858014231\) \(\beta_{12}\mathstrut -\mathstrut \) \(74559016004408509\) \(\beta_{11}\mathstrut -\mathstrut \) \(12836990242674112\) \(\beta_{10}\mathstrut +\mathstrut \) \(234382743856466660\) \(\beta_{9}\mathstrut +\mathstrut \) \(13073379878956777\) \(\beta_{8}\mathstrut -\mathstrut \) \(22355317726403657\) \(\beta_{7}\mathstrut -\mathstrut \) \(5191512785752541233\) \(\beta_{6}\mathstrut +\mathstrut \) \(2046265673420229466\) \(\beta_{5}\mathstrut -\mathstrut \) \(4788713003117658602\) \(\beta_{4}\mathstrut -\mathstrut \) \(65859230187708057651\) \(\beta_{3}\mathstrut -\mathstrut \) \(451483712145786419023\) \(\beta_{2}\mathstrut -\mathstrut \) \(115034670192235172962440\) \(\beta_{1}\mathstrut +\mathstrut \) \(24530583421621348152671895\)\()/\)\(1289945088\)
\(\nu^{8}\)\(=\)\((\)\(-\)\(5699724632542468256\) \(\beta_{17}\mathstrut -\mathstrut \) \(11852126717983356223\) \(\beta_{16}\mathstrut +\mathstrut \) \(3522548624847163295\) \(\beta_{15}\mathstrut +\mathstrut \) \(2817139204319085568\) \(\beta_{14}\mathstrut -\mathstrut \) \(2386430600137167383\) \(\beta_{13}\mathstrut +\mathstrut \) \(5276127471633383135\) \(\beta_{12}\mathstrut -\mathstrut \) \(5142000983127353749\) \(\beta_{11}\mathstrut -\mathstrut \) \(6067044554050088704\) \(\beta_{10}\mathstrut +\mathstrut \) \(2848902723396953348\) \(\beta_{9}\mathstrut -\mathstrut \) \(3193137734373213791\) \(\beta_{8}\mathstrut +\mathstrut \) \(2241938207067419071\) \(\beta_{7}\mathstrut -\mathstrut \) \(179005266364944382011\) \(\beta_{6}\mathstrut +\mathstrut \) \(74556057533467569976\) \(\beta_{5}\mathstrut -\mathstrut \) \(841455991588659370604\) \(\beta_{4}\mathstrut -\mathstrut \) \(14283176528430436128815\) \(\beta_{3}\mathstrut -\mathstrut \) \(16030304416752138061119\) \(\beta_{2}\mathstrut -\mathstrut \) \(4724964979087230036733748\) \(\beta_{1}\mathstrut +\mathstrut \) \(6739786475551322310185119825\)\()/\)\(429981696\)
\(\nu^{9}\)\(=\)\((\)\(-\)\(4472906701942663055840\) \(\beta_{17}\mathstrut -\mathstrut \) \(5237523999386249411362\) \(\beta_{16}\mathstrut +\mathstrut \) \(263434762473203748674\) \(\beta_{15}\mathstrut +\mathstrut \) \(5437143353500353903424\) \(\beta_{14}\mathstrut +\mathstrut \) \(543733010702095774126\) \(\beta_{13}\mathstrut +\mathstrut \) \(4496282700062359951298\) \(\beta_{12}\mathstrut -\mathstrut \) \(5790381940502874036886\) \(\beta_{11}\mathstrut -\mathstrut \) \(1566875574006923571520\) \(\beta_{10}\mathstrut +\mathstrut \) \(19791403973705052289208\) \(\beta_{9}\mathstrut +\mathstrut \) \(800801274353690150206\) \(\beta_{8}\mathstrut -\mathstrut \) \(1912658396154239013854\) \(\beta_{7}\mathstrut -\mathstrut \) \(452293735091051790282613\) \(\beta_{6}\mathstrut +\mathstrut \) \(174510287797824833270629\) \(\beta_{5}\mathstrut -\mathstrut \) \(546614135276941290510995\) \(\beta_{4}\mathstrut -\mathstrut \) \(5846619601146554891349024\) \(\beta_{3}\mathstrut -\mathstrut \) \(41140413237960872281206238\) \(\beta_{2}\mathstrut -\mathstrut \) \(9810718164634029662788991130\) \(\beta_{1}\mathstrut +\mathstrut \) \(2503097086197661498549260910041\)\()/\)\(1289945088\)
\(\nu^{10}\)\(=\)\((\)\(-\)\(1561588409469884435271584\) \(\beta_{17}\mathstrut -\mathstrut \) \(3278420905201236302193331\) \(\beta_{16}\mathstrut +\mathstrut \) \(869966694579267386119187\) \(\beta_{15}\mathstrut +\mathstrut \) \(881505258153481086653824\) \(\beta_{14}\mathstrut -\mathstrut \) \(525207189985510079969579\) \(\beta_{13}\mathstrut +\mathstrut \) \(1633279371114019669139411\) \(\beta_{12}\mathstrut -\mathstrut \) \(1019115731604750487888273\) \(\beta_{11}\mathstrut -\mathstrut \) \(1687891030793085023094400\) \(\beta_{10}\mathstrut +\mathstrut \) \(1440495536199709308849332\) \(\beta_{9}\mathstrut -\mathstrut \) \(735865177688669459931731\) \(\beta_{8}\mathstrut +\mathstrut \) \(523409021118969267497011\) \(\beta_{7}\mathstrut -\mathstrut \) \(61868871968618689391666821\) \(\beta_{6}\mathstrut +\mathstrut \) \(22789533563337852454661458\) \(\beta_{5}\mathstrut -\mathstrut \) \(269675111448219922554498050\) \(\beta_{4}\mathstrut -\mathstrut \) \(3723899384103200161238895039\) \(\beta_{3}\mathstrut -\mathstrut \) \(5271441947809006467277813675\) \(\beta_{2}\mathstrut -\mathstrut \) \(1402368052172759344181776337352\) \(\beta_{1}\mathstrut +\mathstrut \) \(1732440461746772722756430624430915\)\()/\)\(1289945088\)
\(\nu^{11}\)\(=\)\((\)\(-\)\(434421306449994927195762112\) \(\beta_{17}\mathstrut -\mathstrut \) \(566985220038664512177619019\) \(\beta_{16}\mathstrut +\mathstrut \) \(41626231797973248454765195\) \(\beta_{15}\mathstrut +\mathstrut \) \(505118825559951207036940928\) \(\beta_{14}\mathstrut +\mathstrut \) \(90083560094469395396340509\) \(\beta_{13}\mathstrut +\mathstrut \) \(458811693148805531333558155\) \(\beta_{12}\mathstrut -\mathstrut \) \(454156002392313966256017785\) \(\beta_{11}\mathstrut -\mathstrut \) \(183163029943205299923270272\) \(\beta_{10}\mathstrut +\mathstrut \) \(1718168976613138559790572884\) \(\beta_{9}\mathstrut +\mathstrut \) \(49149402920571303249915509\) \(\beta_{8}\mathstrut -\mathstrut \) \(160331301223421631984331189\) \(\beta_{7}\mathstrut -\mathstrut \) \(40558779497261925004251852110\) \(\beta_{6}\mathstrut +\mathstrut \) \(15249570961603677381124167521\) \(\beta_{5}\mathstrut -\mathstrut \) \(59360670834616368241089699283\) \(\beta_{4}\mathstrut -\mathstrut \) \(550027996998012274585696616913\) \(\beta_{3}\mathstrut -\mathstrut \) \(3830653324395856222150575987767\) \(\beta_{2}\mathstrut -\mathstrut \) \(858427930256674688750275704652686\) \(\beta_{1}\mathstrut +\mathstrut \) \(253721693527016821514776182924307308\)\()/\)\(1289945088\)
\(\nu^{12}\)\(=\)\((\)\(-\)\(671961953055926017915736304\) \(\beta_{17}\mathstrut -\mathstrut \) \(1404054296656376831116706205\) \(\beta_{16}\mathstrut +\mathstrut \) \(337933573663591617420260781\) \(\beta_{15}\mathstrut +\mathstrut \) \(421605480181080216819765600\) \(\beta_{14}\mathstrut -\mathstrut \) \(167724858260893983399796245\) \(\beta_{13}\mathstrut +\mathstrut \) \(750114197273960965492707117\) \(\beta_{12}\mathstrut -\mathstrut \) \(349646596429259351895743823\) \(\beta_{11}\mathstrut -\mathstrut \) \(720315915156604617007919712\) \(\beta_{10}\mathstrut +\mathstrut \) \(838985617828420557840852972\) \(\beta_{9}\mathstrut -\mathstrut \) \(269607646983510425038807341\) \(\beta_{8}\mathstrut +\mathstrut \) \(186406764453941382844531101\) \(\beta_{7}\mathstrut -\mathstrut \) \(30696637415478086581043300717\) \(\beta_{6}\mathstrut +\mathstrut \) \(10665234587512552496703530652\) \(\beta_{5}\mathstrut -\mathstrut \) \(125487304903164797356265525040\) \(\beta_{4}\mathstrut -\mathstrut \) \(1519135501731567226003140336341\) \(\beta_{3}\mathstrut -\mathstrut \) \(2659467913689669251380571700509\) \(\beta_{2}\mathstrut -\mathstrut \) \(646559322481321023799404926637668\) \(\beta_{1}\mathstrut +\mathstrut \) \(705750036906855682492505398962565807\)\()/5971968\)
\(\nu^{13}\)\(=\)\((\)\(-\)\(41\!\cdots\!16\) \(\beta_{17}\mathstrut -\mathstrut \) \(58\!\cdots\!95\) \(\beta_{16}\mathstrut +\mathstrut \) \(51\!\cdots\!35\) \(\beta_{15}\mathstrut +\mathstrut \) \(46\!\cdots\!24\) \(\beta_{14}\mathstrut +\mathstrut \) \(10\!\cdots\!33\) \(\beta_{13}\mathstrut +\mathstrut \) \(45\!\cdots\!91\) \(\beta_{12}\mathstrut -\mathstrut \) \(36\!\cdots\!41\) \(\beta_{11}\mathstrut -\mathstrut \) \(20\!\cdots\!20\) \(\beta_{10}\mathstrut +\mathstrut \) \(15\!\cdots\!92\) \(\beta_{9}\mathstrut +\mathstrut \) \(28\!\cdots\!57\) \(\beta_{8}\mathstrut -\mathstrut \) \(13\!\cdots\!33\) \(\beta_{7}\mathstrut -\mathstrut \) \(36\!\cdots\!27\) \(\beta_{6}\mathstrut +\mathstrut \) \(13\!\cdots\!60\) \(\beta_{5}\mathstrut -\mathstrut \) \(62\!\cdots\!72\) \(\beta_{4}\mathstrut -\mathstrut \) \(53\!\cdots\!95\) \(\beta_{3}\mathstrut -\mathstrut \) \(36\!\cdots\!19\) \(\beta_{2}\mathstrut -\mathstrut \) \(76\!\cdots\!16\) \(\beta_{1}\mathstrut +\mathstrut \) \(25\!\cdots\!29\)\()/\)\(1289945088\)
\(\nu^{14}\)\(=\)\((\)\(-\)\(13\!\cdots\!84\) \(\beta_{17}\mathstrut -\mathstrut \) \(28\!\cdots\!93\) \(\beta_{16}\mathstrut +\mathstrut \) \(62\!\cdots\!21\) \(\beta_{15}\mathstrut +\mathstrut \) \(92\!\cdots\!28\) \(\beta_{14}\mathstrut -\mathstrut \) \(23\!\cdots\!65\) \(\beta_{13}\mathstrut +\mathstrut \) \(15\!\cdots\!21\) \(\beta_{12}\mathstrut -\mathstrut \) \(62\!\cdots\!15\) \(\beta_{11}\mathstrut -\mathstrut \) \(14\!\cdots\!20\) \(\beta_{10}\mathstrut +\mathstrut \) \(20\!\cdots\!60\) \(\beta_{9}\mathstrut -\mathstrut \) \(47\!\cdots\!89\) \(\beta_{8}\mathstrut +\mathstrut \) \(30\!\cdots\!69\) \(\beta_{7}\mathstrut -\mathstrut \) \(68\!\cdots\!91\) \(\beta_{6}\mathstrut +\mathstrut \) \(23\!\cdots\!34\) \(\beta_{5}\mathstrut -\mathstrut \) \(26\!\cdots\!98\) \(\beta_{4}\mathstrut -\mathstrut \) \(29\!\cdots\!05\) \(\beta_{3}\mathstrut -\mathstrut \) \(61\!\cdots\!77\) \(\beta_{2}\mathstrut -\mathstrut \) \(13\!\cdots\!64\) \(\beta_{1}\mathstrut +\mathstrut \) \(13\!\cdots\!85\)\()/\)\(1289945088\)
\(\nu^{15}\)\(=\)\((\)\(-\)\(39\!\cdots\!96\) \(\beta_{17}\mathstrut -\mathstrut \) \(59\!\cdots\!42\) \(\beta_{16}\mathstrut +\mathstrut \) \(57\!\cdots\!62\) \(\beta_{15}\mathstrut +\mathstrut \) \(43\!\cdots\!60\) \(\beta_{14}\mathstrut +\mathstrut \) \(10\!\cdots\!70\) \(\beta_{13}\mathstrut +\mathstrut \) \(44\!\cdots\!46\) \(\beta_{12}\mathstrut -\mathstrut \) \(30\!\cdots\!14\) \(\beta_{11}\mathstrut -\mathstrut \) \(21\!\cdots\!52\) \(\beta_{10}\mathstrut +\mathstrut \) \(13\!\cdots\!68\) \(\beta_{9}\mathstrut +\mathstrut \) \(15\!\cdots\!10\) \(\beta_{8}\mathstrut -\mathstrut \) \(11\!\cdots\!18\) \(\beta_{7}\mathstrut -\mathstrut \) \(33\!\cdots\!57\) \(\beta_{6}\mathstrut +\mathstrut \) \(12\!\cdots\!49\) \(\beta_{5}\mathstrut -\mathstrut \) \(63\!\cdots\!03\) \(\beta_{4}\mathstrut -\mathstrut \) \(53\!\cdots\!64\) \(\beta_{3}\mathstrut -\mathstrut \) \(34\!\cdots\!14\) \(\beta_{2}\mathstrut -\mathstrut \) \(69\!\cdots\!54\) \(\beta_{1}\mathstrut +\mathstrut \) \(25\!\cdots\!13\)\()/\)\(1289945088\)
\(\nu^{16}\)\(=\)\((\)\(-\)\(42\!\cdots\!48\) \(\beta_{17}\mathstrut -\mathstrut \) \(87\!\cdots\!21\) \(\beta_{16}\mathstrut +\mathstrut \) \(17\!\cdots\!33\) \(\beta_{15}\mathstrut +\mathstrut \) \(31\!\cdots\!76\) \(\beta_{14}\mathstrut -\mathstrut \) \(44\!\cdots\!41\) \(\beta_{13}\mathstrut +\mathstrut \) \(50\!\cdots\!65\) \(\beta_{12}\mathstrut -\mathstrut \) \(19\!\cdots\!75\) \(\beta_{11}\mathstrut -\mathstrut \) \(43\!\cdots\!32\) \(\beta_{10}\mathstrut +\mathstrut \) \(71\!\cdots\!40\) \(\beta_{9}\mathstrut -\mathstrut \) \(12\!\cdots\!89\) \(\beta_{8}\mathstrut +\mathstrut \) \(77\!\cdots\!81\) \(\beta_{7}\mathstrut -\mathstrut \) \(22\!\cdots\!53\) \(\beta_{6}\mathstrut +\mathstrut \) \(76\!\cdots\!16\) \(\beta_{5}\mathstrut -\mathstrut \) \(84\!\cdots\!76\) \(\beta_{4}\mathstrut -\mathstrut \) \(88\!\cdots\!73\) \(\beta_{3}\mathstrut -\mathstrut \) \(22\!\cdots\!33\) \(\beta_{2}\mathstrut -\mathstrut \) \(46\!\cdots\!24\) \(\beta_{1}\mathstrut +\mathstrut \) \(41\!\cdots\!27\)\()/\)\(429981696\)
\(\nu^{17}\)\(=\)\((\)\(-\)\(38\!\cdots\!92\) \(\beta_{17}\mathstrut -\mathstrut \) \(59\!\cdots\!51\) \(\beta_{16}\mathstrut +\mathstrut \) \(60\!\cdots\!63\) \(\beta_{15}\mathstrut +\mathstrut \) \(40\!\cdots\!00\) \(\beta_{14}\mathstrut +\mathstrut \) \(10\!\cdots\!97\) \(\beta_{13}\mathstrut +\mathstrut \) \(43\!\cdots\!51\) \(\beta_{12}\mathstrut -\mathstrut \) \(26\!\cdots\!45\) \(\beta_{11}\mathstrut -\mathstrut \) \(22\!\cdots\!72\) \(\beta_{10}\mathstrut +\mathstrut \) \(12\!\cdots\!24\) \(\beta_{9}\mathstrut +\mathstrut \) \(54\!\cdots\!13\) \(\beta_{8}\mathstrut -\mathstrut \) \(91\!\cdots\!41\) \(\beta_{7}\mathstrut -\mathstrut \) \(31\!\cdots\!52\) \(\beta_{6}\mathstrut +\mathstrut \) \(11\!\cdots\!11\) \(\beta_{5}\mathstrut -\mathstrut \) \(63\!\cdots\!25\) \(\beta_{4}\mathstrut -\mathstrut \) \(52\!\cdots\!13\) \(\beta_{3}\mathstrut -\mathstrut \) \(33\!\cdots\!35\) \(\beta_{2}\mathstrut -\mathstrut \) \(63\!\cdots\!58\) \(\beta_{1}\mathstrut +\mathstrut \) \(25\!\cdots\!90\)\()/\)\(1289945088\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
7.1
−280.970 + 0.866025i
−280.970 0.866025i
−119.916 0.866025i
−119.916 + 0.866025i
−65.2240 0.866025i
−65.2240 + 0.866025i
−213.634 + 0.866025i
−213.634 0.866025i
172.291 0.866025i
172.291 + 0.866025i
−52.7154 + 0.866025i
−52.7154 0.866025i
6.91974 + 0.866025i
6.91974 0.866025i
309.123 0.866025i
309.123 + 0.866025i
246.626 + 0.866025i
246.626 0.866025i
−450.888 242.578i 11364.0i 144456. + 218751.i −780888. −2.75665e6 + 5.12388e6i 2.45477e7i −1.20691e7 1.33674e8i −1.29140e8 3.52093e8 + 1.89426e8i
7.2 −450.888 + 242.578i 11364.0i 144456. 218751.i −780888. −2.75665e6 5.12388e6i 2.45477e7i −1.20691e7 + 1.33674e8i −1.29140e8 3.52093e8 1.89426e8i
7.3 −429.627 278.504i 11364.0i 107015. + 239306.i −3.52810e6 3.16492e6 4.88228e6i 3.84156e7i 2.06713e7 1.32616e8i −1.29140e8 1.51577e9 + 9.82591e8i
7.4 −429.627 + 278.504i 11364.0i 107015. 239306.i −3.52810e6 3.16492e6 + 4.88228e6i 3.84156e7i 2.06713e7 + 1.32616e8i −1.29140e8 1.51577e9 9.82591e8i
7.5 −366.100 357.931i 11364.0i 5914.09 + 262077.i 3.16588e6 4.06753e6 4.16035e6i 2.73339e7i 9.16406e7 9.80633e7i −1.29140e8 −1.15903e9 1.13317e9i
7.6 −366.100 + 357.931i 11364.0i 5914.09 262077.i 3.16588e6 4.06753e6 + 4.16035e6i 2.73339e7i 9.16406e7 + 9.80633e7i −1.29140e8 −1.15903e9 + 1.13317e9i
7.7 −179.406 479.539i 11364.0i −197771. + 172064.i 1.99524e6 −5.44947e6 + 2.03877e6i 1.75473e7i 1.17993e8 + 6.39695e7i −1.29140e8 −3.57959e8 9.56797e8i
7.8 −179.406 + 479.539i 11364.0i −197771. 172064.i 1.99524e6 −5.44947e6 2.03877e6i 1.75473e7i 1.17993e8 6.39695e7i −1.29140e8 −3.57959e8 + 9.56797e8i
7.9 20.2086 511.601i 11364.0i −261327. 20677.5i −429789. 5.81383e6 + 229650.i 2.05285e7i −1.58597e7 + 1.33277e8i −1.29140e8 −8.68544e6 + 2.19881e8i
7.10 20.2086 + 511.601i 11364.0i −261327. + 20677.5i −429789. 5.81383e6 229650.i 2.05285e7i −1.58597e7 1.33277e8i −1.29140e8 −8.68544e6 2.19881e8i
7.11 148.471 490.000i 11364.0i −218057. 145502.i −3.24964e6 −5.56836e6 1.68722e6i 7.85072e7i −1.03671e8 + 8.52451e7i −1.29140e8 −4.82477e8 + 1.59232e9i
7.12 148.471 + 490.000i 11364.0i −218057. + 145502.i −3.24964e6 −5.56836e6 + 1.68722e6i 7.85072e7i −1.03671e8 8.52451e7i −1.29140e8 −4.82477e8 1.59232e9i
7.13 244.545 449.824i 11364.0i −142539. 220005.i 964196. −5.11179e6 2.77901e6i 7.06704e7i −1.33821e8 + 1.03163e7i −1.29140e8 2.35790e8 4.33718e8i
7.14 244.545 + 449.824i 11364.0i −142539. + 220005.i 964196. −5.11179e6 + 2.77901e6i 7.06704e7i −1.33821e8 1.03163e7i −1.29140e8 2.35790e8 + 4.33718e8i
7.15 421.101 291.236i 11364.0i 92507.4 245279.i −1.11240e6 3.30960e6 + 4.78538e6i 6.89054e6i −3.24791e7 1.30229e8i −1.29140e8 −4.68431e8 + 3.23970e8i
7.16 421.101 + 291.236i 11364.0i 92507.4 + 245279.i −1.11240e6 3.30960e6 4.78538e6i 6.89054e6i −3.24791e7 + 1.30229e8i −1.29140e8 −4.68431e8 3.23970e8i
7.17 506.695 73.5121i 11364.0i 251336. 74496.4i 2.11462e6 −835390. 5.75808e6i 4.00973e7i 1.21874e8 5.62232e7i −1.29140e8 1.07147e9 1.55450e8i
7.18 506.695 + 73.5121i 11364.0i 251336. + 74496.4i 2.11462e6 −835390. + 5.75808e6i 4.00973e7i 1.21874e8 + 5.62232e7i −1.29140e8 1.07147e9 + 1.55450e8i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 7.18
Significant digits:
Format:

Inner twists

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

Hecke kernels

There are no other newforms in \(S_{19}^{\mathrm{new}}(12, [\chi])\).