Properties

Label 12.21.d.a
Level 12
Weight 21
Character orbit 12.d
Analytic conductor 30.422
Analytic rank 0
Dimension 20
CM No
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 12 = 2^{2} \cdot 3 \)
Weight: \( k \) = \( 21 \)
Character orbit: \([\chi]\) = 12.d (of order \(2\) and degree \(1\))

Newform invariants

Self dual: No
Analytic conductor: \(30.4216518123\)
Analytic rank: \(0\)
Dimension: \(20\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{174}\cdot 3^{82}\cdot 5^{4} \)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(f(q)\) \(=\) \( q + ( 63 - \beta_{1} ) q^{2} + ( 1 - 2 \beta_{1} + \beta_{2} ) q^{3} + ( -24258 - 64 \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{4} + ( 73894 - 147 \beta_{1} - \beta_{3} + \beta_{4} ) q^{5} + ( -1800932 - 126 \beta_{1} + 62 \beta_{2} + 2 \beta_{3} - \beta_{5} ) q^{6} + ( -16517 + 56211 \beta_{1} + 892 \beta_{2} + 53 \beta_{3} + \beta_{4} + 2 \beta_{5} - \beta_{6} ) q^{7} + ( 21699079 + 26349 \beta_{1} - 2490 \beta_{2} + 61 \beta_{3} - 17 \beta_{4} + \beta_{5} - \beta_{6} + \beta_{12} ) q^{8} -1162261467 q^{9} +O(q^{10})\) \( q + ( 63 - \beta_{1} ) q^{2} + ( 1 - 2 \beta_{1} + \beta_{2} ) q^{3} + ( -24258 - 64 \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{4} + ( 73894 - 147 \beta_{1} - \beta_{3} + \beta_{4} ) q^{5} + ( -1800932 - 126 \beta_{1} + 62 \beta_{2} + 2 \beta_{3} - \beta_{5} ) q^{6} + ( -16517 + 56211 \beta_{1} + 892 \beta_{2} + 53 \beta_{3} + \beta_{4} + 2 \beta_{5} - \beta_{6} ) q^{7} + ( 21699079 + 26349 \beta_{1} - 2490 \beta_{2} + 61 \beta_{3} - 17 \beta_{4} + \beta_{5} - \beta_{6} + \beta_{12} ) q^{8} -1162261467 q^{9} + ( 158339249 - 53649 \beta_{1} - 39824 \beta_{2} + 60 \beta_{3} + 293 \beta_{4} + 10 \beta_{5} - 9 \beta_{6} - 12 \beta_{7} - \beta_{9} - \beta_{12} ) q^{10} + ( -761438 + 2606839 \beta_{1} + 55289 \beta_{2} + 7610 \beta_{3} + 111 \beta_{4} + 97 \beta_{5} + 26 \beta_{6} + 32 \beta_{7} - \beta_{8} - \beta_{10} - \beta_{12} + \beta_{13} ) q^{11} + ( 1788895060 + 1921045 \beta_{1} - 21422 \beta_{2} + 150 \beta_{3} + 15 \beta_{4} - 69 \beta_{5} + 18 \beta_{6} + 27 \beta_{7} + 3 \beta_{12} + 3 \beta_{19} ) q^{12} + ( -9872357803 + 6219590 \beta_{1} + 24351 \beta_{2} - 41049 \beta_{3} - 501 \beta_{4} + 790 \beta_{5} - 21 \beta_{6} - 39 \beta_{7} - 2 \beta_{8} + 5 \beta_{9} + 4 \beta_{10} + 5 \beta_{11} - 35 \beta_{12} - 3 \beta_{13} + 6 \beta_{14} + 4 \beta_{15} + 9 \beta_{16} + 6 \beta_{17} + \beta_{18} + 3 \beta_{19} ) q^{13} + ( 58983854470 + 3955089 \beta_{1} - 1472413 \beta_{2} - 57339 \beta_{3} - 6126 \beta_{4} - 1457 \beta_{5} + 3 \beta_{6} - 452 \beta_{7} - 11 \beta_{8} - 4 \beta_{9} - 6 \beta_{10} + 2 \beta_{11} + 62 \beta_{12} + 8 \beta_{13} - 17 \beta_{14} - 5 \beta_{15} - 4 \beta_{16} - 9 \beta_{17} + 7 \beta_{18} - 7 \beta_{19} ) q^{14} + ( -13134763 + 43973988 \beta_{1} + 142975 \beta_{2} - 1256 \beta_{3} - 60 \beta_{4} - 71 \beta_{5} - 45 \beta_{6} + 315 \beta_{7} - 9 \beta_{8} + 9 \beta_{10} + 9 \beta_{11} + 6 \beta_{12} - 9 \beta_{13} + 18 \beta_{14} - 9 \beta_{15} - 9 \beta_{17} + 6 \beta_{19} ) q^{15} + ( 79641596729 - 19032897 \beta_{1} + 7342426 \beta_{2} - 12595 \beta_{3} + 21889 \beta_{4} + 3780 \beta_{5} - 68 \beta_{6} + 349 \beta_{7} - 11 \beta_{8} - 4 \beta_{9} - 7 \beta_{10} - 35 \beta_{11} + 124 \beta_{12} + 14 \beta_{13} - 44 \beta_{14} - 3 \beta_{15} + 6 \beta_{16} - 19 \beta_{17} + 63 \beta_{18} - 19 \beta_{19} ) q^{16} + ( 247615973017 - 62469445 \beta_{1} - 170531 \beta_{2} + 208563 \beta_{3} - 7008 \beta_{4} - 5375 \beta_{5} + 256 \beta_{6} + 518 \beta_{7} + 37 \beta_{8} - 77 \beta_{9} - 16 \beta_{10} + 65 \beta_{11} + 551 \beta_{12} - 62 \beta_{13} + 132 \beta_{14} + 61 \beta_{15} + 179 \beta_{16} + 67 \beta_{17} + 26 \beta_{18} - 54 \beta_{19} ) q^{17} + ( -73222472421 + 1162261467 \beta_{1} ) q^{18} + ( -450659015 + 1472396171 \beta_{1} - 22961013 \beta_{2} - 1250901 \beta_{3} - 6812 \beta_{4} + 7491 \beta_{5} + 5074 \beta_{6} + 13145 \beta_{7} - 35 \beta_{8} - 127 \beta_{9} + 173 \beta_{10} + 25 \beta_{11} + 1409 \beta_{12} + 127 \beta_{13} + 207 \beta_{14} + 223 \beta_{15} + 150 \beta_{16} - 2 \beta_{17} - 157 \beta_{18} + 174 \beta_{19} ) q^{19} + ( -1034356658600 - 219798020 \beta_{1} - 4399463 \beta_{2} + 33257 \beta_{3} - 230600 \beta_{4} + 27124 \beta_{5} + 6232 \beta_{6} - 11413 \beta_{7} - 222 \beta_{8} - 668 \beta_{9} + 228 \beta_{10} - 82 \beta_{11} + 356 \beta_{12} - 272 \beta_{13} + 310 \beta_{14} + 538 \beta_{15} + 480 \beta_{16} + 26 \beta_{17} + 36 \beta_{18} + 92 \beta_{19} ) q^{20} + ( -953433365667 - 1719887037 \beta_{1} - 3017211 \beta_{2} + 988026 \beta_{3} - 46398 \beta_{4} + 60276 \beta_{5} + 2475 \beta_{6} + 15525 \beta_{7} - 216 \beta_{8} - 405 \beta_{9} + 108 \beta_{10} - 405 \beta_{11} - 969 \beta_{12} - 351 \beta_{13} + 54 \beta_{14} + 162 \beta_{15} + 243 \beta_{16} - 108 \beta_{17} + 297 \beta_{18} + 111 \beta_{19} ) q^{21} + ( 2740274885690 + 187415422 \beta_{1} - 81744022 \beta_{2} - 2729630 \beta_{3} + 466100 \beta_{4} - 66202 \beta_{5} + 10078 \beta_{6} - 58420 \beta_{7} + 508 \beta_{8} + 296 \beta_{9} + 446 \beta_{10} + 1028 \beta_{11} + 10428 \beta_{12} + 270 \beta_{13} + 1642 \beta_{14} + 418 \beta_{15} + 1032 \beta_{16} + 316 \beta_{17} - 300 \beta_{18} + 244 \beta_{19} ) q^{22} + ( -4386273012 + 14907004014 \beta_{1} + 216676332 \beta_{2} + 4305136 \beta_{3} + 27476 \beta_{4} - 8528 \beta_{5} - 16450 \beta_{6} + 104694 \beta_{7} - 340 \beta_{8} + 386 \beta_{9} - 44 \beta_{10} + 566 \beta_{11} - 1920 \beta_{12} - 380 \beta_{13} - 1082 \beta_{14} + 442 \beta_{15} - 212 \beta_{16} - 24 \beta_{17} + 1718 \beta_{18} - 1404 \beta_{19} ) q^{23} + ( 2910341125090 - 1626788544 \beta_{1} + 19910888 \beta_{2} - 1840810 \beta_{3} - 135012 \beta_{4} + 5867 \beta_{5} - 16911 \beta_{6} - 69165 \beta_{7} + 333 \beta_{8} - 972 \beta_{9} - 1251 \beta_{10} - 873 \beta_{11} - 2229 \beta_{12} - 936 \beta_{13} + 252 \beta_{14} - 585 \beta_{15} + 1458 \beta_{16} + 441 \beta_{17} + 783 \beta_{18} - 987 \beta_{19} ) q^{24} + ( 25660239940384 - 25998728149 \beta_{1} - 31154583 \beta_{2} - 25947564 \beta_{3} + 255145 \beta_{4} - 306232 \beta_{5} - 27814 \beta_{6} + 189655 \beta_{7} - 1025 \beta_{8} + 209 \beta_{9} - 3196 \beta_{10} - 2988 \beta_{11} + 8152 \beta_{12} + 4242 \beta_{13} - 5002 \beta_{14} - 3962 \beta_{15} - 3366 \beta_{16} - 3197 \beta_{17} + 1603 \beta_{18} + 188 \beta_{19} ) q^{25} + ( -7118377060605 + 9231048764 \beta_{1} + 906393033 \beta_{2} - 4885327 \beta_{3} - 2143542 \beta_{4} - 53672 \beta_{5} - 116880 \beta_{6} - 361095 \beta_{7} - 3459 \beta_{8} + 2314 \beta_{9} + 2037 \beta_{10} + 5118 \beta_{11} - 55018 \beta_{12} - 3405 \beta_{13} + 3314 \beta_{14} - 1710 \beta_{15} - 2184 \beta_{16} - 513 \beta_{17} + 1141 \beta_{18} + 359 \beta_{19} ) q^{26} + ( -1162261467 + 2324522934 \beta_{1} - 1162261467 \beta_{2} ) q^{27} + ( -53197549544400 - 62081719240 \beta_{1} - 1153384919 \beta_{2} - 7780425 \beta_{3} + 4931660 \beta_{4} + 1532192 \beta_{5} + 80600 \beta_{6} - 703333 \beta_{7} - 58 \beta_{8} + 11956 \beta_{9} + 920 \beta_{10} - 4242 \beta_{11} - 44288 \beta_{12} + 5108 \beta_{13} - 7714 \beta_{14} + 9194 \beta_{15} - 6288 \beta_{16} + 4470 \beta_{17} + 2920 \beta_{18} + 6500 \beta_{19} ) q^{28} + ( -55986565989264 - 82607848985 \beta_{1} - 141230838 \beta_{2} + 36910473 \beta_{3} - 1147725 \beta_{4} - 127122 \beta_{5} - 7806 \beta_{6} + 644804 \beta_{7} - 288 \beta_{8} - 642 \beta_{9} + 10904 \beta_{10} + 17556 \beta_{11} - 70408 \beta_{12} - 12910 \beta_{13} + 16120 \beta_{14} + 16182 \beta_{15} - 6716 \beta_{16} + 7156 \beta_{17} - 5624 \beta_{18} + 17786 \beta_{19} ) q^{29} + ( 45977369655965 + 2717165573 \beta_{1} + 178117475 \beta_{2} - 42678309 \beta_{3} + 965814 \beta_{4} - 178791 \beta_{5} + 38385 \beta_{6} + 44010 \beta_{7} - 8424 \beta_{8} - 9396 \beta_{9} - 13473 \beta_{10} - 18522 \beta_{11} + 9282 \beta_{12} + 11043 \beta_{13} - 11313 \beta_{14} - 19629 \beta_{15} - 22356 \beta_{16} - 19332 \beta_{17} - 3024 \beta_{18} + 804 \beta_{19} ) q^{30} + ( -23525456535 + 81608097941 \beta_{1} + 2348772302 \beta_{2} - 89942961 \beta_{3} - 977123 \beta_{4} + 305512 \beta_{5} - 1089593 \beta_{6} + 226738 \beta_{7} + 11746 \beta_{8} - 17550 \beta_{9} - 16710 \beta_{10} - 25378 \beta_{11} - 288922 \beta_{12} - 6690 \beta_{13} + 17526 \beta_{14} - 17262 \beta_{15} - 11700 \beta_{16} + 216 \beta_{17} - 7794 \beta_{18} - 24500 \beta_{19} ) q^{31} + ( 46371959535222 - 78189808202 \beta_{1} + 3734591310 \beta_{2} + 25848784 \beta_{3} - 13214254 \beta_{4} - 7626860 \beta_{5} - 1173588 \beta_{6} - 1745380 \beta_{7} + 39038 \beta_{8} - 26640 \beta_{9} - 37182 \beta_{10} - 16726 \beta_{11} - 110412 \beta_{12} + 30672 \beta_{13} - 49340 \beta_{14} + 4354 \beta_{15} - 42844 \beta_{16} - 3002 \beta_{17} + 12070 \beta_{18} - 41198 \beta_{19} ) q^{32} + ( -62496621214710 - 95696867040 \beta_{1} - 194158860 \beta_{2} + 120655293 \beta_{3} + 19398597 \beta_{4} + 3811683 \beta_{5} + 199386 \beta_{6} + 826749 \beta_{7} + 29466 \beta_{8} + 43254 \beta_{9} + 20052 \beta_{10} - 3897 \beta_{11} + 59709 \beta_{12} + 360 \beta_{13} + 28602 \beta_{14} + 18477 \beta_{15} + 13851 \beta_{16} + 17208 \beta_{17} + 15741 \beta_{18} + 32466 \beta_{19} ) q^{33} + ( 80865576969034 - 242114487836 \beta_{1} - 5700917338 \beta_{2} + 53084606 \beta_{3} + 22501510 \beta_{4} + 1222436 \beta_{5} - 1373378 \beta_{6} + 919902 \beta_{7} - 20378 \beta_{8} - 21350 \beta_{9} + 77958 \beta_{10} + 16692 \beta_{11} + 71618 \beta_{12} - 128214 \beta_{13} + 84396 \beta_{14} + 45868 \beta_{15} - 3504 \beta_{16} - 43006 \beta_{17} + 86598 \beta_{18} + 25602 \beta_{19} ) q^{34} + ( 178843970180 - 586447881339 \beta_{1} + 7032924035 \beta_{2} - 493511100 \beta_{3} - 2952933 \beta_{4} + 2428711 \beta_{5} + 710886 \beta_{6} - 3974914 \beta_{7} - 71047 \beta_{8} + 56054 \beta_{9} + 11865 \beta_{10} + 49478 \beta_{11} + 232641 \beta_{12} - 31761 \beta_{13} + 23690 \beta_{14} + 86794 \beta_{15} - 9948 \beta_{16} - 31868 \beta_{17} + 113618 \beta_{18} - 149948 \beta_{19} ) q^{35} + ( 28194138666486 + 74384733888 \beta_{1} + 2324522934 \beta_{2} - 1162261467 \beta_{3} ) q^{36} + ( 672312345346221 + 1099459715046 \beta_{1} + 2161602455 \beta_{2} - 1223243489 \beta_{3} - 139683311 \beta_{4} - 18501038 \beta_{5} - 1878483 \beta_{6} - 9232879 \beta_{7} - 147184 \beta_{8} - 99055 \beta_{9} - 45532 \beta_{10} - 62081 \beta_{11} - 1165009 \beta_{12} - 12941 \beta_{13} - 17858 \beta_{14} + 31384 \beta_{15} + 23175 \beta_{16} - 88688 \beta_{17} + 140889 \beta_{18} + 212249 \beta_{19} ) q^{37} + ( 1531261275850124 + 95030873486 \beta_{1} - 10427761518 \beta_{2} - 1446974610 \beta_{3} - 94504276 \beta_{4} + 19735542 \beta_{5} - 730078 \beta_{6} + 10102768 \beta_{7} - 110490 \beta_{8} - 121336 \beta_{9} + 56516 \beta_{10} + 16444 \beta_{11} - 1934860 \beta_{12} + 63880 \beta_{13} + 287658 \beta_{14} + 164210 \beta_{15} + 281000 \beta_{16} + 54914 \beta_{17} + 157650 \beta_{18} + 118670 \beta_{19} ) q^{38} + ( 290614041854 - 984039966367 \beta_{1} - 11457256015 \beta_{2} + 71100639 \beta_{3} + 2573829 \beta_{4} + 5668245 \beta_{5} - 1741122 \beta_{6} - 8097543 \beta_{7} - 2403 \beta_{8} - 245754 \beta_{9} + 74439 \beta_{10} + 20763 \beta_{11} - 1375056 \beta_{12} + 40257 \beta_{13} + 11988 \beta_{14} + 99765 \beta_{15} + 57348 \beta_{16} - 2025 \beta_{17} + 128250 \beta_{18} - 103302 \beta_{19} ) q^{39} + ( -2114183825607126 + 907503203150 \beta_{1} + 14185224220 \beta_{2} + 194238886 \beta_{3} + 93077978 \beta_{4} + 15523038 \beta_{5} + 2777154 \beta_{6} + 20212032 \beta_{7} - 396664 \beta_{8} + 9472 \beta_{9} - 32424 \beta_{10} + 362760 \beta_{11} + 240670 \beta_{12} - 161232 \beta_{13} + 532976 \beta_{14} + 519720 \beta_{15} + 423312 \beta_{16} + 136776 \beta_{17} + 336872 \beta_{18} + 53752 \beta_{19} ) q^{40} + ( 1338195021843461 + 2507029891291 \beta_{1} + 4862178957 \beta_{2} - 2905192227 \beta_{3} + 489072970 \beta_{4} + 51724303 \beta_{5} + 980508 \beta_{6} - 18945936 \beta_{7} - 165007 \beta_{8} - 774849 \beta_{9} - 267864 \beta_{10} - 148577 \beta_{11} + 4339361 \beta_{12} - 176630 \beta_{13} + 110472 \beta_{14} + 61103 \beta_{15} - 39767 \beta_{16} - 164229 \beta_{17} + 604684 \beta_{18} + 370550 \beta_{19} ) q^{41} + ( 1737310884816198 + 1044946744725 \beta_{1} + 60116426157 \beta_{2} + 1880398761 \beta_{3} + 59054037 \beta_{4} + 6828798 \beta_{5} + 4297041 \beta_{6} + 10272537 \beta_{7} - 122031 \beta_{8} - 262197 \beta_{9} - 495351 \beta_{10} + 162774 \beta_{11} + 1856775 \beta_{12} + 34623 \beta_{13} - 20646 \beta_{14} + 35514 \beta_{15} - 75816 \beta_{16} - 30501 \beta_{17} + 211977 \beta_{18} - 445245 \beta_{19} ) q^{42} + ( 1214841766701 - 4080407251951 \beta_{1} - 23769519867 \beta_{2} - 1115682633 \beta_{3} - 36128238 \beta_{4} - 48837979 \beta_{5} - 32719026 \beta_{6} - 46796343 \beta_{7} - 303525 \beta_{8} - 406055 \beta_{9} + 654795 \beta_{10} + 189441 \beta_{11} + 2115359 \beta_{12} - 107391 \beta_{13} + 378743 \beta_{14} + 771127 \beta_{15} + 273702 \beta_{16} + 73166 \beta_{17} + 480059 \beta_{18} - 352178 \beta_{19} ) q^{43} + ( -8023245919293104 - 3197184648388 \beta_{1} - 126670762426 \beta_{2} - 613983782 \beta_{3} - 277780820 \beta_{4} + 94108244 \beta_{5} - 36216824 \beta_{6} + 13765886 \beta_{7} + 130484 \beta_{8} - 835048 \beta_{9} + 537296 \beta_{10} - 934684 \beta_{11} - 7881868 \beta_{12} - 120040 \beta_{13} + 414852 \beta_{14} - 346004 \beta_{15} - 134112 \beta_{16} - 774316 \beta_{17} + 549424 \beta_{18} + 566572 \beta_{19} ) q^{44} + ( -85884148842498 + 170852435649 \beta_{1} + 1162261467 \beta_{3} - 1162261467 \beta_{4} ) q^{45} + ( 15635426424393268 + 922348718514 \beta_{1} + 66245504566 \beta_{2} - 14531396374 \beta_{3} + 507981124 \beta_{4} - 208879546 \beta_{5} - 61626186 \beta_{6} - 60375960 \beta_{7} + 505138 \beta_{8} + 1373816 \beta_{9} + 87372 \beta_{10} + 184036 \beta_{11} + 147484 \beta_{12} + 490616 \beta_{13} - 1277762 \beta_{14} + 561814 \beta_{15} - 1255944 \beta_{16} - 7066 \beta_{17} - 720986 \beta_{18} + 284602 \beta_{19} ) q^{46} + ( -1273741840260 + 4517161010114 \beta_{1} + 199725013956 \beta_{2} - 7703873684 \beta_{3} - 114636748 \beta_{4} - 152358936 \beta_{5} + 57414146 \beta_{6} + 61878998 \beta_{7} - 129948 \beta_{8} + 2039646 \beta_{9} - 1633196 \beta_{10} - 683274 \beta_{11} - 1419468 \beta_{12} - 932396 \beta_{13} + 187730 \beta_{14} - 1295334 \beta_{15} - 1282796 \beta_{16} - 273508 \beta_{17} - 66438 \beta_{18} - 1554316 \beta_{19} ) q^{47} + ( -8538616710814631 - 3453995499677 \beta_{1} + 73900897456 \beta_{2} + 1530936003 \beta_{3} + 375842037 \beta_{4} - 7646520 \beta_{5} + 47363544 \beta_{6} - 6996537 \beta_{7} + 228933 \beta_{8} - 563436 \beta_{9} - 146259 \beta_{10} + 604017 \beta_{11} + 4135344 \beta_{12} - 1508814 \beta_{13} + 2808 \beta_{14} - 227367 \beta_{15} - 831546 \beta_{16} - 489915 \beta_{17} + 853875 \beta_{18} - 1170615 \beta_{19} ) q^{48} + ( -9899568687339974 - 15169914146701 \beta_{1} - 27732617007 \beta_{2} + 11280314366 \beta_{3} + 1332267835 \beta_{4} + 264542630 \beta_{5} + 2759542 \beta_{6} + 121077673 \beta_{7} + 1420347 \beta_{8} + 6897213 \beta_{9} + 4461132 \beta_{10} + 1956002 \beta_{11} - 34917822 \beta_{12} - 997494 \beta_{13} + 2973754 \beta_{14} + 3001672 \beta_{15} + 636768 \beta_{16} + 2768747 \beta_{17} - 1609355 \beta_{18} + 3806904 \beta_{19} ) q^{49} + ( 28788078297094815 - 23930952960239 \beta_{1} - 343723971182 \beta_{2} + 26271329186 \beta_{3} - 929154100 \beta_{4} - 122679136 \beta_{5} + 93183336 \beta_{6} - 171629454 \beta_{7} + 3303626 \beta_{8} - 1194372 \beta_{9} - 2707622 \beta_{10} - 2810436 \beta_{11} - 21495596 \beta_{12} + 3046518 \beta_{13} - 3598172 \beta_{14} - 1215516 \beta_{15} - 1521936 \beta_{16} - 54866 \beta_{17} + 60890 \beta_{18} - 1591650 \beta_{19} ) q^{50} + ( -1641966666901 + 5810948478015 \beta_{1} + 255708786769 \beta_{2} + 4808244943 \beta_{3} + 22771920 \beta_{4} - 35462549 \beta_{5} + 3572946 \beta_{6} + 55500381 \beta_{7} + 1994373 \beta_{8} - 4518099 \beta_{9} - 1334331 \beta_{10} - 2015811 \beta_{11} - 46995807 \beta_{12} + 421623 \beta_{13} - 357165 \beta_{14} - 2212389 \beta_{15} - 456354 \beta_{16} + 318510 \beta_{17} - 5049 \beta_{18} - 1098426 \beta_{19} ) q^{51} + ( -44828641606628268 + 4409798042648 \beta_{1} + 157749221822 \beta_{2} - 10313456440 \beta_{3} + 278739696 \beta_{4} - 868294392 \beta_{5} - 16038544 \beta_{6} - 218844146 \beta_{7} - 2886124 \beta_{8} + 4309096 \beta_{9} - 1768216 \beta_{10} - 757300 \beta_{11} - 2063256 \beta_{12} + 3014176 \beta_{13} - 2889380 \beta_{14} - 4877692 \beta_{15} - 5806272 \beta_{16} - 4427196 \beta_{17} - 5936664 \beta_{18} + 2939800 \beta_{19} ) q^{52} + ( 12094647358579572 - 20623708335349 \beta_{1} - 26941397602 \beta_{2} - 14091206131 \beta_{3} - 849463685 \beta_{4} + 1326581746 \beta_{5} + 48188674 \beta_{6} + 201102252 \beta_{7} - 2948708 \beta_{8} - 9068318 \beta_{9} - 3267032 \beta_{10} - 1023064 \beta_{11} + 64070692 \beta_{12} - 710742 \beta_{13} - 5486984 \beta_{14} - 4808678 \beta_{15} - 4698384 \beta_{16} - 1775512 \beta_{17} - 540120 \beta_{18} - 8152606 \beta_{19} ) q^{53} + ( 2093153868287244 + 146444944842 \beta_{1} - 72060210954 \beta_{2} - 2324522934 \beta_{3} + 1162261467 \beta_{5} ) q^{54} + ( -18110601571902 + 58111977587738 \beta_{1} - 1677006468478 \beta_{2} + 33770583086 \beta_{3} - 894729616 \beta_{4} - 2884282106 \beta_{5} - 154505544 \beta_{6} + 267074550 \beta_{7} - 2452142 \beta_{8} + 8788990 \beta_{9} + 9661210 \beta_{10} + 3098274 \beta_{11} + 170769614 \beta_{12} - 2721394 \beta_{13} - 1497702 \beta_{14} + 3829358 \beta_{15} + 3469908 \beta_{16} + 5281352 \beta_{17} - 4213166 \beta_{18} + 15253604 \beta_{19} ) q^{55} + ( -105010255635785440 + 46930415948040 \beta_{1} + 632500348184 \beta_{2} + 60559791368 \beta_{3} + 751343624 \beta_{4} + 1482442604 \beta_{5} - 363561244 \beta_{6} - 289287044 \beta_{7} + 7453852 \beta_{8} + 3929872 \beta_{9} - 2049076 \beta_{10} + 11983812 \beta_{11} - 36134868 \beta_{12} + 5745360 \beta_{13} + 3708352 \beta_{14} + 2502948 \beta_{15} + 12636248 \beta_{16} + 9095948 \beta_{17} - 5569500 \beta_{18} - 4031220 \beta_{19} ) q^{56} + ( 29776588580324115 - 10525398374493 \beta_{1} - 16307528355 \beta_{2} + 271018647 \beta_{3} - 1626979428 \beta_{4} + 1346527701 \beta_{5} + 73408896 \beta_{6} + 125295282 \beta_{7} + 3873285 \beta_{8} - 2400597 \beta_{9} - 6020352 \beta_{10} - 6509187 \beta_{11} + 120819339 \beta_{12} + 5817690 \beta_{13} - 6460452 \beta_{14} - 8404695 \beta_{15} - 1441233 \beta_{16} - 2010069 \beta_{17} + 3609846 \beta_{18} - 9794214 \beta_{19} ) q^{57} + ( 82800192022218345 + 61268534030013 \beta_{1} - 322230650950 \beta_{2} + 83949121806 \beta_{3} + 9362889491 \beta_{4} + 579878862 \beta_{5} - 519145339 \beta_{6} + 49509254 \beta_{7} - 9196574 \beta_{8} + 7004705 \beta_{9} + 1394034 \beta_{10} + 5321996 \beta_{11} - 6850279 \beta_{12} - 4234978 \beta_{13} + 13509332 \beta_{14} + 1637908 \beta_{15} + 22175280 \beta_{16} + 554294 \beta_{17} - 11317006 \beta_{18} + 10281638 \beta_{19} ) q^{58} + ( -10734398518594 + 39506040719728 \beta_{1} + 2825301895832 \beta_{2} + 67312662454 \beta_{3} - 1229902882 \beta_{4} - 5001635868 \beta_{5} + 759502760 \beta_{6} + 588957102 \beta_{7} - 2091300 \beta_{8} - 6095466 \beta_{9} + 8017852 \beta_{10} + 3489510 \beta_{11} + 27384324 \beta_{12} - 1152500 \beta_{13} - 7502038 \beta_{14} - 652182 \beta_{15} + 3432676 \beta_{16} + 5795876 \beta_{17} + 5447058 \beta_{18} + 10481060 \beta_{19} ) q^{59} + ( 4729364261217848 - 45561684190614 \beta_{1} - 1107800324279 \beta_{2} - 4375684889 \beta_{3} + 4979400882 \beta_{4} - 329012882 \beta_{5} + 929231532 \beta_{6} + 300293037 \beta_{7} + 6714126 \beta_{8} - 12049884 \beta_{9} - 16555464 \beta_{10} - 9614250 \beta_{11} + 52189086 \beta_{12} + 17625636 \beta_{13} - 6590394 \beta_{14} - 8727390 \beta_{15} - 991440 \beta_{16} - 4723074 \beta_{17} - 4400568 \beta_{18} - 8498382 \beta_{19} ) q^{60} + ( 21500361969089469 + 116347705224926 \beta_{1} + 192703599719 \beta_{2} - 34708882053 \beta_{3} + 2098456385 \beta_{4} + 6815257206 \beta_{5} + 144410305 \beta_{6} - 741408699 \beta_{7} - 19381684 \beta_{8} - 1445719 \beta_{9} + 17136164 \beta_{10} - 10004745 \beta_{11} - 150975313 \beta_{12} - 19120697 \beta_{13} - 3520570 \beta_{14} + 3005636 \beta_{15} + 19594935 \beta_{16} + 7647732 \beta_{17} - 2118971 \beta_{18} - 14245339 \beta_{19} ) q^{61} + ( 85912217522191538 + 5109511272745 \beta_{1} + 213328816875 \beta_{2} - 77803927147 \beta_{3} - 8623158670 \beta_{4} - 3139518481 \beta_{5} + 1767330899 \beta_{6} + 1552292388 \beta_{7} - 41431 \beta_{8} - 15152932 \beta_{9} - 6158658 \beta_{10} + 16560258 \beta_{11} + 57119486 \beta_{12} + 10739156 \beta_{13} + 462279 \beta_{14} - 19065453 \beta_{15} - 5861348 \beta_{16} - 25336957 \beta_{17} - 19151869 \beta_{18} - 5934771 \beta_{19} ) q^{62} + ( 19197072650439 - 65331879321537 \beta_{1} - 1036737228564 \beta_{2} - 61599857751 \beta_{3} - 1162261467 \beta_{4} - 2324522934 \beta_{5} + 1162261467 \beta_{6} ) q^{63} + ( -159850864668034904 - 53831703032064 \beta_{1} - 8319785123732 \beta_{2} + 60482722084 \beta_{3} - 13600939664 \beta_{4} - 4410864600 \beta_{5} + 10843800 \beta_{6} + 711904460 \beta_{7} + 37197384 \beta_{8} + 34136528 \beta_{9} + 25900752 \beta_{10} + 10017136 \beta_{11} - 52788536 \beta_{12} + 106776 \beta_{13} - 6801912 \beta_{14} + 42766144 \beta_{15} + 40853040 \beta_{16} + 17170136 \beta_{17} - 25083264 \beta_{18} + 21635456 \beta_{19} ) q^{64} + ( -57412659460453383 + 241068596386389 \beta_{1} + 347898977635 \beta_{2} + 81918940263 \beta_{3} - 25590796214 \beta_{4} + 20545246973 \beta_{5} + 704526364 \beta_{6} - 1330769212 \beta_{7} + 27155175 \beta_{8} + 49227097 \beta_{9} + 39735848 \beta_{10} + 20028717 \beta_{11} + 357898627 \beta_{12} - 18622138 \beta_{13} + 33940128 \beta_{14} + 11659461 \beta_{15} + 31145811 \beta_{16} + 55199701 \beta_{17} - 534552 \beta_{18} - 19384702 \beta_{19} ) q^{65} + ( 96067459675040466 + 67732788693378 \beta_{1} + 2821428873732 \beta_{2} + 104283908412 \beta_{3} - 28766813466 \beta_{4} - 105148644 \beta_{5} - 1993661190 \beta_{6} + 185640876 \beta_{7} + 35698428 \beta_{8} - 28890270 \beta_{9} - 24782868 \beta_{10} - 16827048 \beta_{11} - 114906318 \beta_{12} + 10202868 \beta_{13} - 14048856 \beta_{14} - 37527192 \beta_{15} - 5808672 \beta_{16} - 15971148 \beta_{17} + 537516 \beta_{18} - 32899164 \beta_{19} ) q^{66} + ( 40289106891806 - 118850673994744 \beta_{1} + 11488082775768 \beta_{2} - 179645889682 \beta_{3} - 13559456130 \beta_{4} - 30177525380 \beta_{5} - 61881176 \beta_{6} - 1241347674 \beta_{7} - 17889628 \beta_{8} + 53827534 \beta_{9} + 42071396 \beta_{10} - 15991026 \beta_{11} + 794966492 \beta_{12} - 54013436 \beta_{13} + 42042130 \beta_{14} - 5931390 \beta_{15} - 5971020 \beta_{16} + 30137060 \beta_{17} - 27346662 \beta_{18} + 40733556 \beta_{19} ) q^{67} + ( 224227563518491492 - 68653188665208 \beta_{1} + 3532979651138 \beta_{2} + 251739395652 \beta_{3} + 57324568336 \beta_{4} + 9897557400 \beta_{5} - 3355285936 \beta_{6} + 3068349258 \beta_{7} - 28488324 \beta_{8} - 56352072 \beta_{9} - 143434056 \beta_{10} - 71650972 \beta_{11} - 146769224 \beta_{12} + 75835552 \beta_{13} - 120847660 \beta_{14} - 79308020 \beta_{15} - 104689856 \beta_{16} - 103631476 \beta_{17} - 11944136 \beta_{18} - 42219832 \beta_{19} ) q^{68} + ( -225968024896989600 + 52615412091420 \beta_{1} - 11592241320 \beta_{2} + 256174447296 \beta_{3} + 12756902394 \beta_{4} + 15960044400 \beta_{5} + 747078102 \beta_{6} + 68663340 \beta_{7} - 1201266 \beta_{8} - 49576860 \beta_{9} + 34852608 \beta_{10} + 17132130 \beta_{11} + 267797922 \beta_{12} - 66513078 \beta_{13} + 41973408 \beta_{14} + 38646576 \beta_{15} - 4235490 \beta_{16} + 25389198 \beta_{17} + 2159028 \beta_{18} + 2853186 \beta_{19} ) q^{69} + ( -610675208751122646 - 35521454692214 \beta_{1} - 4429796772386 \beta_{2} + 556682784862 \beta_{3} + 131778948316 \beta_{4} + 1125299054 \beta_{5} - 3962574374 \beta_{6} + 1994325756 \beta_{7} + 32632920 \beta_{8} + 126766168 \beta_{9} - 13956514 \beta_{10} + 46895692 \beta_{11} - 697436524 \beta_{12} + 142609974 \beta_{13} - 43840730 \beta_{14} + 16645838 \beta_{15} - 107559816 \beta_{16} - 54511344 \beta_{17} - 73723032 \beta_{18} + 64642496 \beta_{19} ) q^{70} + ( 168785310110362 - 585091712295178 \beta_{1} - 16628137222264 \beta_{2} + 490340147746 \beta_{3} - 16240246250 \beta_{4} - 50499451028 \beta_{5} - 1652366730 \beta_{6} - 6107949792 \beta_{7} - 3676576 \beta_{8} - 129666324 \beta_{9} + 44787688 \beta_{10} - 54533712 \beta_{11} - 549730852 \beta_{12} - 44579160 \beta_{13} - 3410180 \beta_{14} - 20929296 \beta_{15} + 104264 \beta_{16} + 75092308 \beta_{17} + 63587604 \beta_{18} - 18533264 \beta_{19} ) q^{71} + ( -25220003391088893 - 30624427393983 \beta_{1} + 2894031052830 \beta_{2} - 70897949487 \beta_{3} + 19758444939 \beta_{4} - 1162261467 \beta_{5} + 1162261467 \beta_{6} - 1162261467 \beta_{12} ) q^{72} + ( 628074574502284569 + 206888639904141 \beta_{1} + 618137034591 \beta_{2} - 839043837422 \beta_{3} + 6566979765 \beta_{4} + 53147219034 \beta_{5} + 903982042 \beta_{6} - 515843673 \beta_{7} - 47118827 \beta_{8} + 26286275 \beta_{9} + 55499540 \beta_{10} + 41668750 \beta_{11} + 724154542 \beta_{12} - 16732026 \beta_{13} - 50549306 \beta_{14} - 23303880 \beta_{15} - 195339024 \beta_{16} + 28487717 \beta_{17} + 18995883 \beta_{18} + 86202968 \beta_{19} ) q^{73} + ( -1106582769992570509 - 738683028789682 \beta_{1} - 9934135738969 \beta_{2} - 1152574779481 \beta_{3} - 5852316644 \beta_{4} - 3443999180 \beta_{5} + 2613279634 \beta_{6} - 8458021009 \beta_{7} - 1635661 \beta_{8} + 60058568 \beta_{9} - 253882373 \beta_{10} + 29139938 \beta_{11} - 1155816964 \beta_{12} + 56347357 \beta_{13} - 19005650 \beta_{14} - 42029618 \beta_{15} + 39883272 \beta_{16} - 24819567 \beta_{17} + 27139707 \beta_{18} - 46301207 \beta_{19} ) q^{74} + ( -86126341489566 + 322435170156324 \beta_{1} + 26348472495399 \beta_{2} - 316192005999 \beta_{3} - 13450969377 \beta_{4} - 26854605108 \beta_{5} + 84532788 \beta_{6} + 2334041757 \beta_{7} - 27548316 \beta_{8} + 114043221 \beta_{9} + 13608324 \beta_{10} + 16347285 \beta_{11} + 458322624 \beta_{12} - 106861032 \beta_{13} - 27403029 \beta_{14} + 11539395 \beta_{15} - 46626354 \beta_{16} + 7897662 \beta_{17} + 104524047 \beta_{18} - 119826810 \beta_{19} ) q^{75} + ( 1425213721981687664 - 1450809346866076 \beta_{1} - 2616829086338 \beta_{2} - 53653127838 \beta_{3} - 253958638268 \beta_{4} - 1787734292 \beta_{5} - 1116149032 \beta_{6} - 7993136498 \beta_{7} + 39636772 \beta_{8} - 26105032 \beta_{9} - 82098800 \beta_{10} + 120741652 \beta_{11} - 2296112820 \beta_{12} + 85758648 \beta_{13} + 228329396 \beta_{14} - 7621892 \beta_{15} + 28298400 \beta_{16} - 134605500 \beta_{17} - 22242192 \beta_{18} + 8841380 \beta_{19} ) q^{76} + ( 999638925565921776 - 1222225097151908 \beta_{1} - 1010660593152 \beta_{2} - 2457810026618 \beta_{3} - 106044719378 \beta_{4} + 130548952174 \beta_{5} + 1476508412 \beta_{6} + 12396335602 \beta_{7} - 264507744 \beta_{8} + 5439032 \beta_{9} + 133354208 \beta_{10} - 306262170 \beta_{11} - 143479038 \beta_{12} - 100084432 \beta_{13} - 132281284 \beta_{14} - 10582518 \beta_{15} - 78221914 \beta_{16} - 44468372 \beta_{17} + 372790478 \beta_{18} + 361432748 \beta_{19} ) q^{77} + ( -1031247904188525673 - 59961685875318 \beta_{1} - 7544568401342 \beta_{2} + 957674942620 \beta_{3} - 210621500640 \beta_{4} + 5339354284 \beta_{5} - 3657868200 \beta_{6} - 2215478466 \beta_{7} + 14688621 \beta_{8} - 29305800 \beta_{9} + 73103445 \beta_{10} + 11435040 \beta_{11} - 699856632 \beta_{12} + 36476919 \beta_{13} + 76084218 \beta_{14} + 196518042 \beta_{15} + 20732760 \beta_{16} - 11559645 \beta_{17} - 95544873 \beta_{18} + 91284813 \beta_{19} ) q^{78} + ( -391841525007653 + 1341790688399475 \beta_{1} + 25936378118656 \beta_{2} - 1572059244495 \beta_{3} - 68089549755 \beta_{4} - 140863575346 \beta_{5} + 7882803483 \beta_{6} + 11797859032 \beta_{7} - 63481692 \beta_{8} - 87609196 \beta_{9} + 171001532 \beta_{10} - 39601312 \beta_{11} - 99244056 \beta_{12} - 229423436 \beta_{13} - 57328380 \beta_{14} + 267332608 \beta_{15} - 29210952 \beta_{16} + 142288684 \beta_{17} + 448434812 \beta_{18} - 490138720 \beta_{19} ) q^{79} + ( 2474576717550321918 + 2279441078151602 \beta_{1} - 32138391022948 \beta_{2} - 936543994618 \beta_{3} + 464836630926 \beta_{4} + 930560216 \beta_{5} - 15977660376 \beta_{6} - 18382388474 \beta_{7} - 38907322 \beta_{8} + 124513288 \beta_{9} - 46990946 \beta_{10} + 408763670 \beta_{11} - 579282072 \beta_{12} + 103058212 \beta_{13} + 430412216 \beta_{14} + 78062614 \beta_{15} + 152743060 \beta_{16} - 191058666 \beta_{17} + 3039154 \beta_{18} + 201030070 \beta_{19} ) q^{80} + 1350851717672992089 q^{81} + ( -2535441493592953226 - 1508116674959416 \beta_{1} + 44645212838614 \beta_{2} - 2544496777346 \beta_{3} + 472336116618 \beta_{4} + 19976472428 \beta_{5} - 7762756630 \beta_{6} - 24812639490 \beta_{7} + 352557542 \beta_{8} - 973949850 \beta_{9} - 519608794 \beta_{10} - 396651500 \beta_{11} - 2935047410 \beta_{12} - 63578742 \beta_{13} + 16286796 \beta_{14} + 81128012 \beta_{15} + 395892432 \beta_{16} - 255376830 \beta_{17} + 227827046 \beta_{18} - 595149342 \beta_{19} ) q^{82} + ( -1013914270707112 + 3442242004311175 \beta_{1} + 47413112264301 \beta_{2} + 1182904905468 \beta_{3} - 77493799903 \beta_{4} - 216712195955 \beta_{5} - 4269177830 \beta_{6} + 20175071010 \beta_{7} - 569491885 \beta_{8} + 24874386 \beta_{9} + 516307635 \beta_{10} + 191882274 \beta_{11} + 4739657147 \beta_{12} - 292982683 \beta_{13} + 416545358 \beta_{14} + 247796046 \beta_{15} + 111662476 \beta_{16} + 145929404 \beta_{17} + 312334902 \beta_{18} + 82422716 \beta_{19} ) q^{83} + ( 1227310089232884492 - 1655006425188900 \beta_{1} - 55618140376059 \beta_{2} - 1155278424729 \beta_{3} + 585277441080 \beta_{4} - 44129818284 \beta_{5} - 2786379624 \beta_{6} - 8567092845 \beta_{7} + 241248402 \beta_{8} + 205929540 \beta_{9} + 275001156 \beta_{10} + 162247806 \beta_{11} + 2633251908 \beta_{12} - 106122960 \beta_{13} + 316890630 \beta_{14} + 298268298 \beta_{15} + 213319008 \beta_{16} - 29732022 \beta_{17} - 128664828 \beta_{18} + 98833404 \beta_{19} ) q^{84} + ( -766937403203292210 - 2633965034439176 \beta_{1} - 6061355775102 \beta_{2} + 5609098700882 \beta_{3} - 292250835060 \beta_{4} + 391571670452 \beta_{5} + 18196100048 \beta_{6} + 33043375170 \beta_{7} + 259875130 \beta_{8} - 795202030 \beta_{9} + 576623352 \beta_{10} - 346030872 \beta_{11} + 8943180664 \beta_{12} - 1210661856 \beta_{13} + 1070103812 \beta_{14} + 731850824 \beta_{15} + 1012957740 \beta_{16} + 609025834 \beta_{17} + 818974442 \beta_{18} - 7638116 \beta_{19} ) q^{85} + ( -4269471799735749200 - 269816241862142 \beta_{1} + 46830594137806 \beta_{2} + 4080275981642 \beta_{3} - 370509703740 \beta_{4} + 26893535066 \beta_{5} - 17844715738 \beta_{6} + 1305145192 \beta_{7} - 509368458 \beta_{8} - 5723912 \beta_{9} - 107341616 \beta_{10} + 205499252 \beta_{11} - 3156622788 \beta_{12} + 872808644 \beta_{13} - 416023770 \beta_{14} + 388615294 \beta_{15} - 117700520 \beta_{16} - 474250238 \beta_{17} + 154469250 \beta_{18} + 657677806 \beta_{19} ) q^{86} + ( -153942091738573 + 444299108219736 \beta_{1} - 53873581587467 \beta_{2} - 4521749072474 \beta_{3} - 77667846648 \beta_{4} - 92085147617 \beta_{5} - 6618977307 \beta_{6} - 1200205629 \beta_{7} + 235910925 \beta_{8} - 193314276 \beta_{9} + 474659631 \beta_{10} - 294786099 \beta_{11} + 4409806668 \beta_{12} + 543393 \beta_{13} + 524829366 \beta_{14} + 369645651 \beta_{15} + 237601512 \beta_{16} + 219968991 \beta_{17} - 620602884 \beta_{18} + 466535670 \beta_{19} ) q^{87} + ( 3301412317170765880 + 8263160754691824 \beta_{1} - 39753927320528 \beta_{2} + 3346029814936 \beta_{3} - 1574467899520 \beta_{4} + 59488660860 \beta_{5} + 12129360500 \beta_{6} + 27401239484 \beta_{7} + 341908 \beta_{8} - 1109315312 \beta_{9} - 1741593100 \beta_{10} - 962348004 \beta_{11} - 2392544644 \beta_{12} + 1120464128 \beta_{13} - 1284952944 \beta_{14} - 966396708 \beta_{15} - 1009318776 \beta_{16} - 1099708156 \beta_{17} - 87379588 \beta_{18} - 1042475180 \beta_{19} ) q^{88} + ( 2181776427709548732 + 7287637018094010 \beta_{1} + 16538590596590 \beta_{2} - 14799378108258 \beta_{3} + 366391623812 \beta_{4} + 185792265906 \beta_{5} - 5986983536 \beta_{6} - 55800462968 \beta_{7} - 769922602 \beta_{8} + 42675730 \beta_{9} + 156441824 \beta_{10} + 1804295626 \beta_{11} + 981027078 \beta_{12} - 300264204 \beta_{13} + 462578544 \beta_{14} + 120000434 \beta_{15} + 305296870 \beta_{16} + 974132290 \beta_{17} - 264395144 \beta_{18} + 23711836 \beta_{19} ) q^{89} + ( -184031607826418283 + 62354165443083 \beta_{1} + 46285900661808 \beta_{2} - 69735688020 \beta_{3} - 340542609831 \beta_{4} - 11622614670 \beta_{5} + 10460353203 \beta_{6} + 13947137604 \beta_{7} + 1162261467 \beta_{9} + 1162261467 \beta_{12} ) q^{90} + ( 1734912935041067 - 5800162102195327 \beta_{1} - 14129737246107 \beta_{2} - 2546351494091 \beta_{3} - 120094527944 \beta_{4} - 271798905383 \beta_{5} + 43029048294 \beta_{6} - 25208692465 \beta_{7} + 409036103 \beta_{8} - 1234643065 \beta_{9} + 279824631 \beta_{10} - 362969681 \beta_{11} - 10888499197 \beta_{12} - 555537603 \beta_{13} + 555551785 \beta_{14} - 963047143 \beta_{15} - 137856486 \beta_{16} - 25417166 \beta_{17} - 20345099 \beta_{18} + 37341058 \beta_{19} ) q^{91} + ( -3238861103185371840 - 15814385134235992 \beta_{1} - 270315114884350 \beta_{2} - 1547300259202 \beta_{3} + 638655141056 \beta_{4} - 42250779256 \beta_{5} - 8552699424 \beta_{6} - 3378670338 \beta_{7} - 642665844 \beta_{8} + 624955240 \beta_{9} - 19423440 \beta_{10} + 634036188 \beta_{11} - 16063831544 \beta_{12} + 733565928 \beta_{13} - 1370531140 \beta_{14} + 134620884 \beta_{15} + 805982944 \beta_{16} + 631591404 \beta_{17} + 59917008 \beta_{18} + 844268752 \beta_{19} ) q^{92} + ( -2536643123815424997 + 43383429923685 \beta_{1} - 2748722168037 \beta_{2} + 8002733471436 \beta_{3} - 251046124278 \beta_{4} + 89780194122 \beta_{5} + 4905842283 \beta_{6} + 2628857673 \beta_{7} + 165858786 \beta_{8} + 1476224433 \beta_{9} + 1060241076 \beta_{10} - 907021287 \beta_{11} - 11329813623 \beta_{12} - 155203371 \beta_{13} - 165163050 \beta_{14} + 307023912 \beta_{15} - 55526715 \beta_{16} - 4147038 \beta_{17} - 304219287 \beta_{18} + 433146075 \beta_{19} ) q^{93} + ( 4775161350618811512 + 246960478180366 \beta_{1} + 145922640065146 \beta_{2} - 4282258604386 \beta_{3} + 3129847591596 \beta_{4} - 75755880510 \beta_{5} + 60697879250 \beta_{6} + 90736511184 \beta_{7} + 1387315738 \beta_{8} + 1310310664 \beta_{9} - 476686856 \beta_{10} + 680558028 \beta_{11} + 2681691668 \beta_{12} + 337597412 \beta_{13} + 20525490 \beta_{14} - 2010497542 \beta_{15} - 1430897784 \beta_{16} - 530877010 \beta_{17} - 1483039106 \beta_{18} - 220233550 \beta_{19} ) q^{94} + ( 3079641550167466 - 10022134657586948 \beta_{1} + 195801832916168 \beta_{2} + 27384461736690 \beta_{3} + 139079090642 \beta_{4} - 265712113720 \beta_{5} + 13027293568 \beta_{6} - 78588688510 \beta_{7} - 1379904608 \beta_{8} + 3447571838 \beta_{9} - 1767075624 \beta_{10} + 628180154 \beta_{11} + 18057077228 \beta_{12} + 224476880 \beta_{13} - 3130047494 \beta_{14} - 1080221354 \beta_{15} - 771299372 \beta_{16} + 824699640 \beta_{17} + 801347354 \beta_{18} + 1030379532 \beta_{19} ) q^{95} + ( -4478966745108828302 + 8286609780740634 \beta_{1} + 57936804104438 \beta_{2} + 3352810867820 \beta_{3} + 2057530792494 \beta_{4} + 40256054756 \beta_{5} - 34026687972 \beta_{6} + 53123048280 \beta_{7} + 683696394 \beta_{8} + 216779328 \beta_{9} - 8986914 \beta_{10} - 1131231978 \beta_{11} + 1449388068 \beta_{12} - 530235432 \beta_{13} - 416268972 \beta_{14} + 69414318 \beta_{15} - 969432948 \beta_{16} - 283975182 \beta_{17} - 1024364502 \beta_{18} - 209970210 \beta_{19} ) q^{96} + ( 359116112145889494 + 8110238796549992 \beta_{1} + 7766153037448 \beta_{2} + 13937993793566 \beta_{3} - 1134242875802 \beta_{4} - 22724346310 \beta_{5} + 11860571380 \beta_{6} - 61362273498 \beta_{7} + 1608029396 \beta_{8} + 2293292388 \beta_{9} - 863485832 \beta_{10} - 1604865942 \beta_{11} + 12792963134 \beta_{12} + 3106026152 \beta_{13} - 3947379060 \beta_{14} - 3237587802 \beta_{15} - 4131284526 \beta_{16} - 1369245992 \beta_{17} - 2208717002 \beta_{18} - 3224077900 \beta_{19} ) q^{97} + ( 15229262287889394453 + 10807602618474831 \beta_{1} + 164938159790978 \beta_{2} + 16000345057346 \beta_{3} - 2610118637368 \beta_{4} - 19935872040 \beta_{5} - 172637654980 \beta_{6} + 18330572850 \beta_{7} - 693587702 \beta_{8} + 794065584 \beta_{9} - 509227846 \beta_{10} + 2086624796 \beta_{11} - 8187909752 \beta_{12} + 1132547350 \beta_{13} + 221625796 \beta_{14} - 3293693564 \beta_{15} - 85427856 \beta_{16} + 318799918 \beta_{17} - 2388446022 \beta_{18} + 651750846 \beta_{19} ) q^{98} + ( 884990046909546 - 3029828520372813 \beta_{1} - 64260274248963 \beta_{2} - 8844809763870 \beta_{3} - 129011022837 \beta_{4} - 112739362299 \beta_{5} - 30218798142 \beta_{6} - 37192366944 \beta_{7} + 1162261467 \beta_{8} + 1162261467 \beta_{10} + 1162261467 \beta_{12} - 1162261467 \beta_{13} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20q + 1254q^{2} - 485524q^{4} + 1476984q^{5} - 36019890q^{6} + 434160000q^{8} - 23245229340q^{9} + O(q^{10}) \) \( 20q + 1254q^{2} - 485524q^{4} + 1476984q^{5} - 36019890q^{6} + 434160000q^{8} - 23245229340q^{9} + 3166779028q^{10} + 35789598900q^{12} - 197410187576q^{13} + 1179712421976q^{14} + 1592658749552q^{16} + 4951946877000q^{17} - 1457475879618q^{18} - 20688414141528q^{20} - 19078957703376q^{21} + 54807260127768q^{22} + 58196896476048q^{24} + 513048947332092q^{25} - 142319404376100q^{26} - 1064314324942128q^{28} - 1120225678237608q^{29} + 919562090032260q^{30} + 926940361437984q^{32} - 1250504741372400q^{33} + 1615904447739868q^{34} + 564305836503708q^{36} + 13452822768639208q^{37} + 30625874417040840q^{38} - 42278345025251456q^{40} + 26778887480335560q^{41} + 34752013424992728q^{42} - 160483087306778736q^{44} - 1716641590575528q^{45} + 312713467885194624q^{46} - 170793646915261968q^{48} - 198082137742572844q^{49} + 575620844246091282q^{50} - 896547684604982408q^{52} + 241769380233802968q^{53} + 41864530192578630q^{54} - 2099928347597457216q^{56} + 595468774322860464q^{57} + 1656374273217670084q^{58} + 94322711748139416q^{60} + 430703667006221224q^{61} + 1718273064332222664q^{62} - 3197273374119743872q^{64} - 1146808855391494992q^{65} + 1921733716920631128q^{66} + 4484111570418829944q^{68} - 4519043719923504960q^{69} - 12213680959006457904q^{70} - 504607438512720000q^{72} + 12562724768160620200q^{73} - 22136012572796109780q^{74} + 28495592855672115888q^{76} + 19985445258493440192q^{77} - 20625251537531633940q^{78} + 49505459663687019552q^{80} + 27017034353459841780q^{81} - 50718250504081238180q^{82} + 24536705912722373616q^{84} - 15354475681609135696q^{85} - 85391409389241086952q^{86} + 66078172926918803520q^{88} + 43679060294354361384q^{89} - 3680625238748114076q^{90} - 64869958570030809216q^{92} - 50732545022865845424q^{93} + 95503492747592513424q^{94} - 89530085825201077920q^{96} + 7230988441427384872q^{97} + 304648861287114073350q^{98} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{20} - 8 x^{19} - 1800529 x^{18} + 10471600 x^{17} + 1401323871170 x^{16} - 8255926364216 x^{15} - 618094244121155086 x^{14} + 4776792556997860420 x^{13} + \)\(17\!\cdots\!19\)\( x^{12} - \)\(18\!\cdots\!04\)\( x^{11} - \)\(30\!\cdots\!55\)\( x^{10} + \)\(42\!\cdots\!56\)\( x^{9} + \)\(34\!\cdots\!94\)\( x^{8} - \)\(60\!\cdots\!84\)\( x^{7} - \)\(25\!\cdots\!94\)\( x^{6} + \)\(48\!\cdots\!96\)\( x^{5} + \)\(10\!\cdots\!41\)\( x^{4} - \)\(19\!\cdots\!56\)\( x^{3} - \)\(21\!\cdots\!33\)\( x^{2} + \)\(24\!\cdots\!84\)\( x + \)\(17\!\cdots\!88\)\(\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\(-\)\(28\!\cdots\!53\)\( \nu^{19} - \)\(10\!\cdots\!17\)\( \nu^{18} + \)\(46\!\cdots\!33\)\( \nu^{17} + \)\(17\!\cdots\!22\)\( \nu^{16} - \)\(32\!\cdots\!13\)\( \nu^{15} - \)\(12\!\cdots\!19\)\( \nu^{14} + \)\(12\!\cdots\!44\)\( \nu^{13} + \)\(47\!\cdots\!83\)\( \nu^{12} - \)\(30\!\cdots\!20\)\( \nu^{11} - \)\(11\!\cdots\!51\)\( \nu^{10} + \)\(44\!\cdots\!32\)\( \nu^{9} + \)\(15\!\cdots\!67\)\( \nu^{8} - \)\(39\!\cdots\!23\)\( \nu^{7} - \)\(13\!\cdots\!30\)\( \nu^{6} + \)\(20\!\cdots\!27\)\( \nu^{5} + \)\(64\!\cdots\!21\)\( \nu^{4} - \)\(53\!\cdots\!51\)\( \nu^{3} - \)\(15\!\cdots\!08\)\( \nu^{2} + \)\(53\!\cdots\!80\)\( \nu + \)\(13\!\cdots\!96\)\(\)\()/ \)\(11\!\cdots\!40\)\( \)
\(\beta_{2}\)\(=\)\((\)\(\)\(45\!\cdots\!87\)\( \nu^{19} + \)\(16\!\cdots\!23\)\( \nu^{18} - \)\(75\!\cdots\!07\)\( \nu^{17} - \)\(27\!\cdots\!38\)\( \nu^{16} + \)\(53\!\cdots\!87\)\( \nu^{15} + \)\(18\!\cdots\!81\)\( \nu^{14} - \)\(20\!\cdots\!76\)\( \nu^{13} - \)\(73\!\cdots\!97\)\( \nu^{12} + \)\(48\!\cdots\!00\)\( \nu^{11} + \)\(16\!\cdots\!09\)\( \nu^{10} - \)\(71\!\cdots\!08\)\( \nu^{9} - \)\(24\!\cdots\!33\)\( \nu^{8} + \)\(63\!\cdots\!37\)\( \nu^{7} + \)\(20\!\cdots\!10\)\( \nu^{6} - \)\(33\!\cdots\!73\)\( \nu^{5} - \)\(99\!\cdots\!99\)\( \nu^{4} + \)\(86\!\cdots\!69\)\( \nu^{3} + \)\(23\!\cdots\!72\)\( \nu^{2} - \)\(85\!\cdots\!00\)\( \nu - \)\(20\!\cdots\!84\)\(\)\()/ \)\(55\!\cdots\!20\)\( \)
\(\beta_{3}\)\(=\)\((\)\(-\)\(19\!\cdots\!37\)\( \nu^{19} - \)\(73\!\cdots\!37\)\( \nu^{18} + \)\(32\!\cdots\!29\)\( \nu^{17} + \)\(12\!\cdots\!14\)\( \nu^{16} - \)\(22\!\cdots\!97\)\( \nu^{15} - \)\(85\!\cdots\!55\)\( \nu^{14} + \)\(87\!\cdots\!12\)\( \nu^{13} + \)\(32\!\cdots\!63\)\( \nu^{12} - \)\(20\!\cdots\!48\)\( \nu^{11} - \)\(76\!\cdots\!67\)\( \nu^{10} + \)\(30\!\cdots\!92\)\( \nu^{9} + \)\(10\!\cdots\!67\)\( \nu^{8} - \)\(27\!\cdots\!91\)\( \nu^{7} - \)\(92\!\cdots\!30\)\( \nu^{6} + \)\(13\!\cdots\!63\)\( \nu^{5} + \)\(44\!\cdots\!65\)\( \nu^{4} - \)\(36\!\cdots\!63\)\( \nu^{3} - \)\(10\!\cdots\!16\)\( \nu^{2} + \)\(36\!\cdots\!24\)\( \nu + \)\(91\!\cdots\!12\)\(\)\()/ \)\(27\!\cdots\!60\)\( \)
\(\beta_{4}\)\(=\)\((\)\(-\)\(90\!\cdots\!68\)\( \nu^{19} - \)\(33\!\cdots\!89\)\( \nu^{18} + \)\(15\!\cdots\!99\)\( \nu^{17} + \)\(55\!\cdots\!85\)\( \nu^{16} - \)\(10\!\cdots\!18\)\( \nu^{15} - \)\(39\!\cdots\!21\)\( \nu^{14} + \)\(41\!\cdots\!37\)\( \nu^{13} + \)\(15\!\cdots\!16\)\( \nu^{12} - \)\(96\!\cdots\!89\)\( \nu^{11} - \)\(34\!\cdots\!60\)\( \nu^{10} + \)\(14\!\cdots\!49\)\( \nu^{9} + \)\(49\!\cdots\!04\)\( \nu^{8} - \)\(12\!\cdots\!05\)\( \nu^{7} - \)\(42\!\cdots\!55\)\( \nu^{6} + \)\(65\!\cdots\!02\)\( \nu^{5} + \)\(20\!\cdots\!99\)\( \nu^{4} - \)\(17\!\cdots\!03\)\( \nu^{3} - \)\(47\!\cdots\!75\)\( \nu^{2} + \)\(17\!\cdots\!12\)\( \nu + \)\(41\!\cdots\!80\)\(\)\()/ \)\(11\!\cdots\!80\)\( \)
\(\beta_{5}\)\(=\)\((\)\(\)\(16\!\cdots\!99\)\( \nu^{19} + \)\(70\!\cdots\!13\)\( \nu^{18} - \)\(27\!\cdots\!89\)\( \nu^{17} - \)\(11\!\cdots\!04\)\( \nu^{16} + \)\(19\!\cdots\!67\)\( \nu^{15} + \)\(81\!\cdots\!67\)\( \nu^{14} - \)\(75\!\cdots\!66\)\( \nu^{13} - \)\(31\!\cdots\!65\)\( \nu^{12} + \)\(17\!\cdots\!62\)\( \nu^{11} + \)\(72\!\cdots\!69\)\( \nu^{10} - \)\(26\!\cdots\!42\)\( \nu^{9} - \)\(10\!\cdots\!81\)\( \nu^{8} + \)\(23\!\cdots\!23\)\( \nu^{7} + \)\(88\!\cdots\!40\)\( \nu^{6} - \)\(12\!\cdots\!21\)\( \nu^{5} - \)\(42\!\cdots\!61\)\( \nu^{4} + \)\(32\!\cdots\!51\)\( \nu^{3} + \)\(98\!\cdots\!70\)\( \nu^{2} - \)\(32\!\cdots\!32\)\( \nu - \)\(86\!\cdots\!76\)\(\)\()/ \)\(11\!\cdots\!40\)\( \)
\(\beta_{6}\)\(=\)\((\)\(\)\(31\!\cdots\!55\)\( \nu^{19} + \)\(12\!\cdots\!75\)\( \nu^{18} - \)\(53\!\cdots\!87\)\( \nu^{17} - \)\(20\!\cdots\!46\)\( \nu^{16} + \)\(37\!\cdots\!47\)\( \nu^{15} + \)\(14\!\cdots\!77\)\( \nu^{14} - \)\(14\!\cdots\!00\)\( \nu^{13} - \)\(55\!\cdots\!37\)\( \nu^{12} + \)\(35\!\cdots\!36\)\( \nu^{11} + \)\(13\!\cdots\!77\)\( \nu^{10} - \)\(51\!\cdots\!52\)\( \nu^{9} - \)\(18\!\cdots\!65\)\( \nu^{8} + \)\(47\!\cdots\!57\)\( \nu^{7} + \)\(16\!\cdots\!70\)\( \nu^{6} - \)\(24\!\cdots\!05\)\( \nu^{5} - \)\(78\!\cdots\!35\)\( \nu^{4} + \)\(64\!\cdots\!73\)\( \nu^{3} + \)\(18\!\cdots\!36\)\( \nu^{2} - \)\(64\!\cdots\!56\)\( \nu - \)\(16\!\cdots\!44\)\(\)\()/ \)\(12\!\cdots\!40\)\( \)
\(\beta_{7}\)\(=\)\((\)\(-\)\(14\!\cdots\!17\)\( \nu^{19} - \)\(52\!\cdots\!89\)\( \nu^{18} + \)\(23\!\cdots\!93\)\( \nu^{17} + \)\(88\!\cdots\!02\)\( \nu^{16} - \)\(16\!\cdots\!13\)\( \nu^{15} - \)\(61\!\cdots\!63\)\( \nu^{14} + \)\(63\!\cdots\!32\)\( \nu^{13} + \)\(23\!\cdots\!87\)\( \nu^{12} - \)\(14\!\cdots\!48\)\( \nu^{11} - \)\(55\!\cdots\!75\)\( \nu^{10} + \)\(21\!\cdots\!52\)\( \nu^{9} + \)\(78\!\cdots\!35\)\( \nu^{8} - \)\(19\!\cdots\!47\)\( \nu^{7} - \)\(67\!\cdots\!30\)\( \nu^{6} + \)\(10\!\cdots\!03\)\( \nu^{5} + \)\(32\!\cdots\!73\)\( \nu^{4} - \)\(26\!\cdots\!87\)\( \nu^{3} - \)\(75\!\cdots\!16\)\( \nu^{2} + \)\(26\!\cdots\!76\)\( \nu + \)\(66\!\cdots\!04\)\(\)\()/ \)\(45\!\cdots\!60\)\( \)
\(\beta_{8}\)\(=\)\((\)\(-\)\(16\!\cdots\!16\)\( \nu^{19} - \)\(31\!\cdots\!71\)\( \nu^{18} + \)\(33\!\cdots\!37\)\( \nu^{17} + \)\(69\!\cdots\!11\)\( \nu^{16} - \)\(27\!\cdots\!98\)\( \nu^{15} - \)\(62\!\cdots\!87\)\( \nu^{14} + \)\(12\!\cdots\!99\)\( \nu^{13} + \)\(29\!\cdots\!76\)\( \nu^{12} - \)\(35\!\cdots\!07\)\( \nu^{11} - \)\(83\!\cdots\!20\)\( \nu^{10} + \)\(59\!\cdots\!11\)\( \nu^{9} + \)\(14\!\cdots\!60\)\( \nu^{8} - \)\(60\!\cdots\!87\)\( \nu^{7} - \)\(14\!\cdots\!05\)\( \nu^{6} + \)\(34\!\cdots\!34\)\( \nu^{5} + \)\(75\!\cdots\!65\)\( \nu^{4} - \)\(96\!\cdots\!69\)\( \nu^{3} - \)\(19\!\cdots\!09\)\( \nu^{2} + \)\(99\!\cdots\!16\)\( \nu + \)\(17\!\cdots\!64\)\(\)\()/ \)\(43\!\cdots\!40\)\( \)
\(\beta_{9}\)\(=\)\((\)\(\)\(31\!\cdots\!05\)\( \nu^{19} - \)\(85\!\cdots\!88\)\( \nu^{18} - \)\(56\!\cdots\!66\)\( \nu^{17} + \)\(12\!\cdots\!51\)\( \nu^{16} + \)\(43\!\cdots\!43\)\( \nu^{15} - \)\(76\!\cdots\!10\)\( \nu^{14} - \)\(18\!\cdots\!75\)\( \nu^{13} + \)\(24\!\cdots\!69\)\( \nu^{12} + \)\(48\!\cdots\!71\)\( \nu^{11} - \)\(44\!\cdots\!25\)\( \nu^{10} - \)\(76\!\cdots\!47\)\( \nu^{9} + \)\(44\!\cdots\!57\)\( \nu^{8} + \)\(72\!\cdots\!78\)\( \nu^{7} - \)\(20\!\cdots\!05\)\( \nu^{6} - \)\(37\!\cdots\!05\)\( \nu^{5} + \)\(56\!\cdots\!02\)\( \nu^{4} + \)\(92\!\cdots\!92\)\( \nu^{3} + \)\(20\!\cdots\!81\)\( \nu^{2} - \)\(83\!\cdots\!48\)\( \nu - \)\(35\!\cdots\!36\)\(\)\()/ \)\(62\!\cdots\!20\)\( \)
\(\beta_{10}\)\(=\)\((\)\(-\)\(23\!\cdots\!85\)\( \nu^{19} - \)\(28\!\cdots\!78\)\( \nu^{18} + \)\(42\!\cdots\!04\)\( \nu^{17} + \)\(62\!\cdots\!91\)\( \nu^{16} - \)\(33\!\cdots\!87\)\( \nu^{15} - \)\(54\!\cdots\!60\)\( \nu^{14} + \)\(14\!\cdots\!25\)\( \nu^{13} + \)\(26\!\cdots\!59\)\( \nu^{12} - \)\(37\!\cdots\!89\)\( \nu^{11} - \)\(73\!\cdots\!15\)\( \nu^{10} + \)\(60\!\cdots\!33\)\( \nu^{9} + \)\(12\!\cdots\!07\)\( \nu^{8} - \)\(58\!\cdots\!52\)\( \nu^{7} - \)\(12\!\cdots\!65\)\( \nu^{6} + \)\(32\!\cdots\!65\)\( \nu^{5} + \)\(68\!\cdots\!52\)\( \nu^{4} - \)\(87\!\cdots\!98\)\( \nu^{3} - \)\(17\!\cdots\!59\)\( \nu^{2} + \)\(87\!\cdots\!52\)\( \nu + \)\(16\!\cdots\!24\)\(\)\()/ \)\(20\!\cdots\!40\)\( \)
\(\beta_{11}\)\(=\)\((\)\(\)\(67\!\cdots\!08\)\( \nu^{19} + \)\(23\!\cdots\!43\)\( \nu^{18} - \)\(10\!\cdots\!97\)\( \nu^{17} - \)\(37\!\cdots\!59\)\( \nu^{16} + \)\(73\!\cdots\!54\)\( \nu^{15} + \)\(25\!\cdots\!31\)\( \nu^{14} - \)\(27\!\cdots\!63\)\( \nu^{13} - \)\(95\!\cdots\!48\)\( \nu^{12} + \)\(62\!\cdots\!95\)\( \nu^{11} + \)\(21\!\cdots\!64\)\( \nu^{10} - \)\(86\!\cdots\!19\)\( \nu^{9} - \)\(28\!\cdots\!68\)\( \nu^{8} + \)\(73\!\cdots\!75\)\( \nu^{7} + \)\(23\!\cdots\!65\)\( \nu^{6} - \)\(35\!\cdots\!82\)\( \nu^{5} - \)\(10\!\cdots\!85\)\( \nu^{4} + \)\(88\!\cdots\!21\)\( \nu^{3} + \)\(23\!\cdots\!57\)\( \nu^{2} - \)\(83\!\cdots\!64\)\( \nu - \)\(20\!\cdots\!20\)\(\)\()/ \)\(35\!\cdots\!40\)\( \)
\(\beta_{12}\)\(=\)\((\)\(-\)\(51\!\cdots\!70\)\( \nu^{19} - \)\(19\!\cdots\!43\)\( \nu^{18} + \)\(85\!\cdots\!57\)\( \nu^{17} + \)\(32\!\cdots\!69\)\( \nu^{16} - \)\(59\!\cdots\!32\)\( \nu^{15} - \)\(22\!\cdots\!35\)\( \nu^{14} + \)\(23\!\cdots\!33\)\( \nu^{13} + \)\(88\!\cdots\!74\)\( \nu^{12} - \)\(54\!\cdots\!01\)\( \nu^{11} - \)\(20\!\cdots\!62\)\( \nu^{10} + \)\(79\!\cdots\!21\)\( \nu^{9} + \)\(29\!\cdots\!26\)\( \nu^{8} - \)\(71\!\cdots\!19\)\( \nu^{7} - \)\(24\!\cdots\!95\)\( \nu^{6} + \)\(37\!\cdots\!20\)\( \nu^{5} + \)\(11\!\cdots\!97\)\( \nu^{4} - \)\(97\!\cdots\!05\)\( \nu^{3} - \)\(27\!\cdots\!39\)\( \nu^{2} + \)\(97\!\cdots\!00\)\( \nu + \)\(24\!\cdots\!48\)\(\)\()/ \)\(25\!\cdots\!80\)\( \)
\(\beta_{13}\)\(=\)\((\)\(-\)\(80\!\cdots\!38\)\( \nu^{19} - \)\(16\!\cdots\!29\)\( \nu^{18} + \)\(13\!\cdots\!47\)\( \nu^{17} + \)\(27\!\cdots\!51\)\( \nu^{16} - \)\(93\!\cdots\!72\)\( \nu^{15} - \)\(19\!\cdots\!25\)\( \nu^{14} + \)\(36\!\cdots\!91\)\( \nu^{13} + \)\(74\!\cdots\!70\)\( \nu^{12} - \)\(84\!\cdots\!27\)\( \nu^{11} - \)\(17\!\cdots\!10\)\( \nu^{10} + \)\(12\!\cdots\!87\)\( \nu^{9} + \)\(24\!\cdots\!66\)\( \nu^{8} - \)\(10\!\cdots\!85\)\( \nu^{7} - \)\(20\!\cdots\!85\)\( \nu^{6} + \)\(53\!\cdots\!72\)\( \nu^{5} + \)\(10\!\cdots\!79\)\( \nu^{4} - \)\(13\!\cdots\!15\)\( \nu^{3} - \)\(24\!\cdots\!17\)\( \nu^{2} + \)\(12\!\cdots\!08\)\( \nu + \)\(21\!\cdots\!20\)\(\)\()/ \)\(25\!\cdots\!80\)\( \)
\(\beta_{14}\)\(=\)\((\)\(\)\(11\!\cdots\!81\)\( \nu^{19} + \)\(47\!\cdots\!88\)\( \nu^{18} - \)\(18\!\cdots\!78\)\( \nu^{17} - \)\(79\!\cdots\!13\)\( \nu^{16} + \)\(13\!\cdots\!55\)\( \nu^{15} + \)\(55\!\cdots\!26\)\( \nu^{14} - \)\(51\!\cdots\!15\)\( \nu^{13} - \)\(21\!\cdots\!91\)\( \nu^{12} + \)\(12\!\cdots\!39\)\( \nu^{11} + \)\(49\!\cdots\!83\)\( \nu^{10} - \)\(17\!\cdots\!83\)\( \nu^{9} - \)\(70\!\cdots\!31\)\( \nu^{8} + \)\(16\!\cdots\!58\)\( \nu^{7} + \)\(60\!\cdots\!75\)\( \nu^{6} - \)\(85\!\cdots\!49\)\( \nu^{5} - \)\(29\!\cdots\!58\)\( \nu^{4} + \)\(22\!\cdots\!36\)\( \nu^{3} + \)\(67\!\cdots\!17\)\( \nu^{2} - \)\(22\!\cdots\!72\)\( \nu - \)\(59\!\cdots\!76\)\(\)\()/ \)\(12\!\cdots\!40\)\( \)
\(\beta_{15}\)\(=\)\((\)\(\)\(87\!\cdots\!78\)\( \nu^{19} + \)\(29\!\cdots\!91\)\( \nu^{18} - \)\(14\!\cdots\!81\)\( \nu^{17} - \)\(49\!\cdots\!73\)\( \nu^{16} + \)\(10\!\cdots\!00\)\( \nu^{15} + \)\(35\!\cdots\!95\)\( \nu^{14} - \)\(40\!\cdots\!89\)\( \nu^{13} - \)\(13\!\cdots\!26\)\( \nu^{12} + \)\(97\!\cdots\!25\)\( \nu^{11} + \)\(32\!\cdots\!02\)\( \nu^{10} - \)\(14\!\cdots\!37\)\( \nu^{9} - \)\(46\!\cdots\!98\)\( \nu^{8} + \)\(13\!\cdots\!95\)\( \nu^{7} + \)\(40\!\cdots\!35\)\( \nu^{6} - \)\(68\!\cdots\!92\)\( \nu^{5} - \)\(19\!\cdots\!77\)\( \nu^{4} + \)\(17\!\cdots\!49\)\( \nu^{3} + \)\(47\!\cdots\!23\)\( \nu^{2} - \)\(17\!\cdots\!84\)\( \nu - \)\(41\!\cdots\!80\)\(\)\()/ \)\(83\!\cdots\!60\)\( \)
\(\beta_{16}\)\(=\)\((\)\(-\)\(11\!\cdots\!72\)\( \nu^{19} - \)\(43\!\cdots\!01\)\( \nu^{18} + \)\(19\!\cdots\!07\)\( \nu^{17} + \)\(73\!\cdots\!05\)\( \nu^{16} - \)\(13\!\cdots\!34\)\( \nu^{15} - \)\(51\!\cdots\!01\)\( \nu^{14} + \)\(54\!\cdots\!37\)\( \nu^{13} + \)\(19\!\cdots\!56\)\( \nu^{12} - \)\(12\!\cdots\!53\)\( \nu^{11} - \)\(46\!\cdots\!28\)\( \nu^{10} + \)\(18\!\cdots\!77\)\( \nu^{9} + \)\(65\!\cdots\!68\)\( \nu^{8} - \)\(16\!\cdots\!53\)\( \nu^{7} - \)\(56\!\cdots\!55\)\( \nu^{6} + \)\(87\!\cdots\!18\)\( \nu^{5} + \)\(27\!\cdots\!31\)\( \nu^{4} - \)\(22\!\cdots\!51\)\( \nu^{3} - \)\(63\!\cdots\!71\)\( \nu^{2} + \)\(22\!\cdots\!88\)\( \nu + \)\(56\!\cdots\!36\)\(\)\()/ \)\(83\!\cdots\!60\)\( \)
\(\beta_{17}\)\(=\)\((\)\(-\)\(42\!\cdots\!36\)\( \nu^{19} - \)\(16\!\cdots\!65\)\( \nu^{18} + \)\(71\!\cdots\!95\)\( \nu^{17} + \)\(26\!\cdots\!25\)\( \nu^{16} - \)\(49\!\cdots\!54\)\( \nu^{15} - \)\(18\!\cdots\!17\)\( \nu^{14} + \)\(19\!\cdots\!45\)\( \nu^{13} + \)\(72\!\cdots\!08\)\( \nu^{12} - \)\(45\!\cdots\!33\)\( \nu^{11} - \)\(16\!\cdots\!24\)\( \nu^{10} + \)\(66\!\cdots\!01\)\( \nu^{9} + \)\(23\!\cdots\!24\)\( \nu^{8} - \)\(59\!\cdots\!57\)\( \nu^{7} - \)\(20\!\cdots\!55\)\( \nu^{6} + \)\(30\!\cdots\!34\)\( \nu^{5} + \)\(95\!\cdots\!91\)\( \nu^{4} - \)\(79\!\cdots\!87\)\( \nu^{3} - \)\(22\!\cdots\!71\)\( \nu^{2} + \)\(78\!\cdots\!28\)\( \nu + \)\(19\!\cdots\!44\)\(\)\()/ \)\(25\!\cdots\!80\)\( \)
\(\beta_{18}\)\(=\)\((\)\(\)\(45\!\cdots\!88\)\( \nu^{19} + \)\(16\!\cdots\!69\)\( \nu^{18} - \)\(75\!\cdots\!67\)\( \nu^{17} - \)\(27\!\cdots\!09\)\( \nu^{16} + \)\(52\!\cdots\!06\)\( \nu^{15} + \)\(19\!\cdots\!69\)\( \nu^{14} - \)\(20\!\cdots\!77\)\( \nu^{13} - \)\(74\!\cdots\!64\)\( \nu^{12} + \)\(48\!\cdots\!73\)\( \nu^{11} + \)\(17\!\cdots\!48\)\( \nu^{10} - \)\(70\!\cdots\!77\)\( \nu^{9} - \)\(24\!\cdots\!44\)\( \nu^{8} + \)\(62\!\cdots\!93\)\( \nu^{7} + \)\(20\!\cdots\!55\)\( \nu^{6} - \)\(32\!\cdots\!02\)\( \nu^{5} - \)\(99\!\cdots\!99\)\( \nu^{4} + \)\(84\!\cdots\!15\)\( \nu^{3} + \)\(23\!\cdots\!59\)\( \nu^{2} - \)\(83\!\cdots\!96\)\( \nu - \)\(20\!\cdots\!16\)\(\)\()/ \)\(25\!\cdots\!80\)\( \)
\(\beta_{19}\)\(=\)\((\)\(-\)\(41\!\cdots\!67\)\( \nu^{19} - \)\(15\!\cdots\!50\)\( \nu^{18} + \)\(68\!\cdots\!32\)\( \nu^{17} + \)\(25\!\cdots\!69\)\( \nu^{16} - \)\(48\!\cdots\!49\)\( \nu^{15} - \)\(17\!\cdots\!20\)\( \nu^{14} + \)\(18\!\cdots\!71\)\( \nu^{13} + \)\(69\!\cdots\!37\)\( \nu^{12} - \)\(44\!\cdots\!03\)\( \nu^{11} - \)\(15\!\cdots\!13\)\( \nu^{10} + \)\(65\!\cdots\!99\)\( \nu^{9} + \)\(22\!\cdots\!41\)\( \nu^{8} - \)\(58\!\cdots\!44\)\( \nu^{7} - \)\(19\!\cdots\!95\)\( \nu^{6} + \)\(30\!\cdots\!43\)\( \nu^{5} + \)\(93\!\cdots\!12\)\( \nu^{4} - \)\(79\!\cdots\!82\)\( \nu^{3} - \)\(21\!\cdots\!45\)\( \nu^{2} + \)\(78\!\cdots\!20\)\( \nu + \)\(19\!\cdots\!28\)\(\)\()/ \)\(17\!\cdots\!20\)\( \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{19} + 9 \beta_{18} + 12 \beta_{17} + 3 \beta_{15} + 3 \beta_{14} - 15 \beta_{13} + 118 \beta_{12} - 3 \beta_{11} + 15 \beta_{10} - 6 \beta_{8} - 4812 \beta_{7} + 294 \beta_{6} + 10846 \beta_{5} - 4348 \beta_{4} - 62461 \beta_{3} + 1094801 \beta_{2} + 648227387 \beta_{1} + 837932855\)\()/ 2579890176 \)
\(\nu^{2}\)\(=\)\((\)\(26968 \beta_{19} + 11736 \beta_{18} - 3384 \beta_{17} - 2592 \beta_{16} + 17712 \beta_{15} + 16272 \beta_{14} - 12456 \beta_{13} - 20480 \beta_{12} - 7344 \beta_{11} + 7272 \beta_{10} - 7344 \beta_{9} + 6840 \beta_{8} - 118377 \beta_{7} + 82896 \beta_{6} - 401856 \beta_{5} - 2376880 \beta_{4} + 142036741 \beta_{3} - 27848577 \beta_{2} + 14392253158 \beta_{1} + 929045568739400\)\()/ 5159780352 \)
\(\nu^{3}\)\(=\)\((\)\(880964 \beta_{19} - 233052 \beta_{18} + 688676 \beta_{17} - 2508624 \beta_{16} + 17696 \beta_{15} - 564976 \beta_{14} - 645628 \beta_{13} + 5043728 \beta_{12} - 102368 \beta_{11} + 1821532 \beta_{10} + 1463112 \beta_{9} - 129556 \beta_{8} - 390089599 \beta_{7} + 24881400 \beta_{6} + 903229872 \beta_{5} - 244257432 \beta_{4} - 1136650197 \beta_{3} + 93788641825 \beta_{2} + 52511536775538 \beta_{1} + 491593836890236\)\()/ 859963392 \)
\(\nu^{4}\)\(=\)\((\)\(4618748563 \beta_{19} + 1291489011 \beta_{18} - 299669784 \beta_{17} - 1340154720 \beta_{16} + 3664976565 \beta_{15} + 3465167061 \beta_{14} - 2954040621 \beta_{13} + 358839178 \beta_{12} - 240602421 \beta_{11} + 1596853869 \beta_{10} - 2088748944 \beta_{9} + 1438690530 \beta_{8} - 47371277373 \beta_{7} + 20770640466 \beta_{6} - 37051355694 \beta_{5} - 411057840004 \beta_{4} + 31369497398698 \beta_{3} - 1665863926386 \beta_{2} + 5992835152485379 \beta_{1} + 113337894536114911037\)\()/ 2579890176 \)
\(\nu^{5}\)\(=\)\((\)\(2489752913992 \beta_{19} - 1386065948328 \beta_{18} + 719148588120 \beta_{17} - 7164868006656 \beta_{16} - 244576362240 \beta_{15} - 1646137898496 \beta_{14} - 217648523736 \beta_{13} + 7858079975392 \beta_{12} + 338400873216 \beta_{11} + 3233643529176 \beta_{10} + 3641451078528 \beta_{9} + 447497601192 \beta_{8} - 667100056962033 \beta_{7} + 39461522453808 \beta_{6} + 1393631850397120 \beta_{5} - 1185874153059232 \beta_{4} + 2784698879676845 \beta_{3} + 166714406653747175 \beta_{2} + 89330677746075788918 \beta_{1} + 5296101843735665848040\)\()/ 5159780352 \)
\(\nu^{6}\)\(=\)\((\)\(252080348387418 \beta_{19} + 47811515662410 \beta_{18} - 12631288723820 \beta_{17} - 132652756118448 \beta_{16} + 207138198721450 \beta_{15} + 194446181389498 \beta_{14} - 178773024177926 \beta_{13} + 312526237440228 \beta_{12} + 20094209807126 \beta_{11} + 92131809671078 \beta_{10} - 144596879240904 \beta_{9} + 85173982937584 \beta_{8} - 4296325285833665 \beta_{7} + 1379012743885692 \beta_{6} + 2644895062743860 \beta_{5} - 22599654875930864 \beta_{4} + 1843066961946188887 \beta_{3} + 273836284554322741 \beta_{2} + 553944905529019331384 \beta_{1} + 5253158585242726513664470\)\()/ 429981696 \)
\(\nu^{7}\)\(=\)\((\)\(531171056201097661 \beta_{19} - 293534596328699691 \beta_{18} + 80293303427672100 \beta_{17} - 1383244601399520768 \beta_{16} - 25873557059792049 \beta_{15} - 267139299004641969 \beta_{14} + 21364356398168973 \beta_{13} + 1296469937720190958 \beta_{12} + 144362032026946737 \beta_{11} + 511136312707598451 \beta_{10} + 632099795995494144 \beta_{9} + 134497611578291322 \beta_{8} - 101651930838440644038 \beta_{7} + 5691677031133987470 \beta_{6} + 199622566519455202198 \beta_{5} - 310425886461603047884 \beta_{4} + 850147666320857433329 \beta_{3} + 26420583048847371490187 \beta_{2} + 13567196602055812404953123 \beta_{1} + 1880165166632855305457109131\)\()/ 2579890176 \)
\(\nu^{8}\)\(=\)\((\)\(1010157868334760050752 \beta_{19} + 135646328335385872800 \beta_{18} - 53856057921327458136 \beta_{17} - 763761509709794439264 \beta_{16} + 803294379754201846728 \beta_{15} + 736065255676625021160 \beta_{14} - 724586558385367575936 \beta_{13} + 2381878967526987493552 \beta_{12} + 154468779226936229304 \beta_{11} + 370633725442745797824 \beta_{10} - 638517263697322188048 \beta_{9} + 353120974530528066120 \beta_{8} - 24243184854844608855141 \beta_{7} + 6223743966891518911680 \beta_{6} + 34810380561591805767792 \beta_{5} - 97738136583916395375280 \beta_{4} + 7391311589837315292896617 \beta_{3} + 3037164671453998543642251 \beta_{2} + 3150687087303979782902765446 \beta_{1} + 18994497517844593797439427730752\)\()/ 5159780352 \)
\(\nu^{9}\)\(=\)\((\)\(72022709397695859104752 \beta_{19} - 35513074896345532242384 \beta_{18} + 7735788710403268199804 \beta_{17} - 167687469741196948137840 \beta_{16} + 1965927856970439813164 \beta_{15} - 24858098961083346311044 \beta_{14} + 2367240499408809523184 \beta_{13} + 154604937073728175791880 \beta_{12} + 24701033667711270124948 \beta_{11} + 56786960023564507355440 \beta_{10} + 69183301317476420894616 \beta_{9} + 18802080287724482652620 \beta_{8} - 10798418226151882336356265 \beta_{7} + 585811635187788284941632 \beta_{6} + 20607597211553503609703048 \beta_{5} - 44587553755247939588972024 \beta_{4} + 121303471612898617632847897 \beta_{3} + 2920003623296055742032705675 \beta_{2} + 1437433565468338219091427330026 \beta_{1} + 335342022452471831951814525360816\)\()/ 859963392 \)
\(\nu^{10}\)\(=\)\((\)\(\)\(17\!\cdots\!15\)\( \beta_{19} + \)\(16\!\cdots\!31\)\( \beta_{18} - \)\(10\!\cdots\!44\)\( \beta_{17} - \)\(16\!\cdots\!80\)\( \beta_{16} + \)\(12\!\cdots\!97\)\( \beta_{15} + \)\(11\!\cdots\!57\)\( \beta_{14} - \)\(12\!\cdots\!17\)\( \beta_{13} + \)\(56\!\cdots\!70\)\( \beta_{12} + \)\(33\!\cdots\!07\)\( \beta_{11} + \)\(63\!\cdots\!81\)\( \beta_{10} - \)\(11\!\cdots\!84\)\( \beta_{9} + \)\(61\!\cdots\!90\)\( \beta_{8} - \)\(53\!\cdots\!39\)\( \beta_{7} + \)\(11\!\cdots\!18\)\( \beta_{6} + \)\(10\!\cdots\!18\)\( \beta_{5} - \)\(19\!\cdots\!32\)\( \beta_{4} + \)\(12\!\cdots\!92\)\( \beta_{3} + \)\(89\!\cdots\!20\)\( \beta_{2} + \)\(69\!\cdots\!19\)\( \beta_{1} + \)\(30\!\cdots\!21\)\(\)\()/ 2579890176 \)
\(\nu^{11}\)\(=\)\((\)\(\)\(17\!\cdots\!08\)\( \beta_{19} - \)\(73\!\cdots\!08\)\( \beta_{18} + \)\(15\!\cdots\!20\)\( \beta_{17} - \)\(35\!\cdots\!20\)\( \beta_{16} + \)\(16\!\cdots\!80\)\( \beta_{15} - \)\(37\!\cdots\!24\)\( \beta_{14} - \)\(12\!\cdots\!60\)\( \beta_{13} + \)\(34\!\cdots\!04\)\( \beta_{12} + \)\(64\!\cdots\!68\)\( \beta_{11} + \)\(11\!\cdots\!00\)\( \beta_{10} + \)\(13\!\cdots\!44\)\( \beta_{9} + \)\(43\!\cdots\!72\)\( \beta_{8} - \)\(21\!\cdots\!65\)\( \beta_{7} + \)\(11\!\cdots\!16\)\( \beta_{6} + \)\(40\!\cdots\!68\)\( \beta_{5} - \)\(10\!\cdots\!80\)\( \beta_{4} + \)\(29\!\cdots\!29\)\( \beta_{3} + \)\(59\!\cdots\!83\)\( \beta_{2} + \)\(28\!\cdots\!02\)\( \beta_{1} + \)\(94\!\cdots\!60\)\(\)\()/ 5159780352 \)
\(\nu^{12}\)\(=\)\((\)\(\)\(49\!\cdots\!66\)\( \beta_{19} + \)\(28\!\cdots\!02\)\( \beta_{18} - \)\(32\!\cdots\!28\)\( \beta_{17} - \)\(56\!\cdots\!84\)\( \beta_{16} + \)\(34\!\cdots\!50\)\( \beta_{15} + \)\(29\!\cdots\!54\)\( \beta_{14} - \)\(33\!\cdots\!02\)\( \beta_{13} + \)\(19\!\cdots\!16\)\( \beta_{12} + \)\(10\!\cdots\!50\)\( \beta_{11} + \)\(18\!\cdots\!86\)\( \beta_{10} - \)\(30\!\cdots\!80\)\( \beta_{9} + \)\(17\!\cdots\!60\)\( \beta_{8} - \)\(18\!\cdots\!75\)\( \beta_{7} + \)\(34\!\cdots\!56\)\( \beta_{6} + \)\(40\!\cdots\!16\)\( \beta_{5} - \)\(66\!\cdots\!48\)\( \beta_{4} + \)\(34\!\cdots\!21\)\( \beta_{3} + \)\(37\!\cdots\!23\)\( \beta_{2} + \)\(24\!\cdots\!84\)\( \beta_{1} + \)\(82\!\cdots\!78\)\(\)\()/ 214990848 \)
\(\nu^{13}\)\(=\)\((\)\(\)\(32\!\cdots\!85\)\( \beta_{19} - \)\(12\!\cdots\!59\)\( \beta_{18} + \)\(26\!\cdots\!20\)\( \beta_{17} - \)\(63\!\cdots\!36\)\( \beta_{16} + \)\(49\!\cdots\!47\)\( \beta_{15} - \)\(40\!\cdots\!41\)\( \beta_{14} - \)\(16\!\cdots\!67\)\( \beta_{13} + \)\(65\!\cdots\!18\)\( \beta_{12} + \)\(12\!\cdots\!37\)\( \beta_{11} + \)\(20\!\cdots\!91\)\( \beta_{10} + \)\(21\!\cdots\!04\)\( \beta_{9} + \)\(81\!\cdots\!46\)\( \beta_{8} - \)\(35\!\cdots\!08\)\( \beta_{7} + \)\(19\!\cdots\!22\)\( \beta_{6} + \)\(67\!\cdots\!74\)\( \beta_{5} - \)\(19\!\cdots\!04\)\( \beta_{4} + \)\(59\!\cdots\!03\)\( \beta_{3} + \)\(10\!\cdots\!65\)\( \beta_{2} + \)\(47\!\cdots\!35\)\( \beta_{1} + \)\(20\!\cdots\!47\)\(\)\()/ 2579890176 \)
\(\nu^{14}\)\(=\)\((\)\(\)\(42\!\cdots\!08\)\( \beta_{19} + \)\(10\!\cdots\!20\)\( \beta_{18} - \)\(27\!\cdots\!28\)\( \beta_{17} - \)\(52\!\cdots\!16\)\( \beta_{16} + \)\(27\!\cdots\!80\)\( \beta_{15} + \)\(22\!\cdots\!60\)\( \beta_{14} - \)\(26\!\cdots\!40\)\( \beta_{13} + \)\(18\!\cdots\!64\)\( \beta_{12} + \)\(99\!\cdots\!84\)\( \beta_{11} + \)\(15\!\cdots\!28\)\( \beta_{10} - \)\(24\!\cdots\!68\)\( \beta_{9} + \)\(15\!\cdots\!48\)\( \beta_{8} - \)\(18\!\cdots\!09\)\( \beta_{7} + \)\(30\!\cdots\!72\)\( \beta_{6} + \)\(41\!\cdots\!64\)\( \beta_{5} - \)\(66\!\cdots\!96\)\( \beta_{4} + \)\(27\!\cdots\!05\)\( \beta_{3} + \)\(41\!\cdots\!31\)\( \beta_{2} + \)\(23\!\cdots\!10\)\( \beta_{1} + \)\(66\!\cdots\!68\)\(\)\()/ 5159780352 \)
\(\nu^{15}\)\(=\)\((\)\(\)\(41\!\cdots\!64\)\( \beta_{19} - \)\(14\!\cdots\!72\)\( \beta_{18} + \)\(30\!\cdots\!40\)\( \beta_{17} - \)\(75\!\cdots\!56\)\( \beta_{16} + \)\(80\!\cdots\!40\)\( \beta_{15} - \)\(21\!\cdots\!60\)\( \beta_{14} - \)\(37\!\cdots\!08\)\( \beta_{13} + \)\(83\!\cdots\!44\)\( \beta_{12} + \)\(16\!\cdots\!68\)\( \beta_{11} + \)\(24\!\cdots\!60\)\( \beta_{10} + \)\(22\!\cdots\!88\)\( \beta_{9} + \)\(10\!\cdots\!12\)\( \beta_{8} - \)\(41\!\cdots\!67\)\( \beta_{7} + \)\(22\!\cdots\!76\)\( \beta_{6} + \)\(77\!\cdots\!84\)\( \beta_{5} - \)\(23\!\cdots\!36\)\( \beta_{4} + \)\(79\!\cdots\!39\)\( \beta_{3} + \)\(12\!\cdots\!01\)\( \beta_{2} + \)\(54\!\cdots\!22\)\( \beta_{1} + \)\(28\!\cdots\!36\)\(\)\()/ 859963392 \)
\(\nu^{16}\)\(=\)\((\)\(\)\(75\!\cdots\!55\)\( \beta_{19} - \)\(40\!\cdots\!17\)\( \beta_{18} - \)\(48\!\cdots\!40\)\( \beta_{17} - \)\(10\!\cdots\!12\)\( \beta_{16} + \)\(46\!\cdots\!41\)\( \beta_{15} + \)\(36\!\cdots\!73\)\( \beta_{14} - \)\(44\!\cdots\!09\)\( \beta_{13} + \)\(35\!\cdots\!18\)\( \beta_{12} + \)\(18\!\cdots\!59\)\( \beta_{11} + \)\(29\!\cdots\!13\)\( \beta_{10} - \)\(39\!\cdots\!08\)\( \beta_{9} + \)\(27\!\cdots\!98\)\( \beta_{8} - \)\(36\!\cdots\!49\)\( \beta_{7} + \)\(53\!\cdots\!34\)\( \beta_{6} + \)\(84\!\cdots\!58\)\( \beta_{5} - \)\(13\!\cdots\!20\)\( \beta_{4} + \)\(47\!\cdots\!86\)\( \beta_{3} + \)\(90\!\cdots\!42\)\( \beta_{2} + \)\(46\!\cdots\!59\)\( \beta_{1} + \)\(11\!\cdots\!33\)\(\)\()/ 2579890176 \)
\(\nu^{17}\)\(=\)\((\)\(\)\(92\!\cdots\!16\)\( \beta_{19} - \)\(29\!\cdots\!52\)\( \beta_{18} + \)\(63\!\cdots\!12\)\( \beta_{17} - \)\(16\!\cdots\!12\)\( \beta_{16} + \)\(21\!\cdots\!88\)\( \beta_{15} + \)\(20\!\cdots\!68\)\( \beta_{14} - \)\(11\!\cdots\!88\)\( \beta_{13} + \)\(19\!\cdots\!48\)\( \beta_{12} + \)\(36\!\cdots\!56\)\( \beta_{11} + \)\(51\!\cdots\!44\)\( \beta_{10} + \)\(42\!\cdots\!40\)\( \beta_{9} + \)\(23\!\cdots\!64\)\( \beta_{8} - \)\(85\!\cdots\!81\)\( \beta_{7} + \)\(47\!\cdots\!76\)\( \beta_{6} + \)\(16\!\cdots\!48\)\( \beta_{5} - \)\(51\!\cdots\!48\)\( \beta_{4} + \)\(18\!\cdots\!97\)\( \beta_{3} + \)\(26\!\cdots\!39\)\( \beta_{2} + \)\(11\!\cdots\!70\)\( \beta_{1} + \)\(68\!\cdots\!72\)\(\)\()/ 5159780352 \)
\(\nu^{18}\)\(=\)\((\)\(\)\(45\!\cdots\!58\)\( \beta_{19} - \)\(14\!\cdots\!10\)\( \beta_{18} - \)\(27\!\cdots\!00\)\( \beta_{17} - \)\(64\!\cdots\!40\)\( \beta_{16} + \)\(26\!\cdots\!94\)\( \beta_{15} + \)\(19\!\cdots\!18\)\( \beta_{14} - \)\(25\!\cdots\!34\)\( \beta_{13} + \)\(21\!\cdots\!12\)\( \beta_{12} + \)\(11\!\cdots\!22\)\( \beta_{11} + \)\(17\!\cdots\!14\)\( \beta_{10} - \)\(21\!\cdots\!48\)\( \beta_{9} + \)\(16\!\cdots\!16\)\( \beta_{8} - \)\(23\!\cdots\!63\)\( \beta_{7} + \)\(32\!\cdots\!84\)\( \beta_{6} + \)\(55\!\cdots\!56\)\( \beta_{5} - \)\(93\!\cdots\!96\)\( \beta_{4} + \)\(27\!\cdots\!81\)\( \beta_{3} + \)\(64\!\cdots\!91\)\( \beta_{2} + \)\(30\!\cdots\!16\)\( \beta_{1} + \)\(65\!\cdots\!66\)\(\)\()/ 429981696 \)
\(\nu^{19}\)\(=\)\((\)\(\)\(17\!\cdots\!13\)\( \beta_{19} - \)\(51\!\cdots\!95\)\( \beta_{18} + \)\(10\!\cdots\!04\)\( \beta_{17} - \)\(29\!\cdots\!60\)\( \beta_{16} + \)\(44\!\cdots\!35\)\( \beta_{15} + \)\(76\!\cdots\!91\)\( \beta_{14} - \)\(26\!\cdots\!91\)\( \beta_{13} + \)\(37\!\cdots\!54\)\( \beta_{12} + \)\(67\!\cdots\!65\)\( \beta_{11} + \)\(91\!\cdots\!43\)\( \beta_{10} + \)\(66\!\cdots\!88\)\( \beta_{9} + \)\(43\!\cdots\!98\)\( \beta_{8} - \)\(14\!\cdots\!42\)\( \beta_{7} + \)\(84\!\cdots\!38\)\( \beta_{6} + \)\(28\!\cdots\!62\)\( \beta_{5} - \)\(92\!\cdots\!16\)\( \beta_{4} + \)\(37\!\cdots\!65\)\( \beta_{3} + \)\(48\!\cdots\!39\)\( \beta_{2} + \)\(19\!\cdots\!99\)\( \beta_{1} + \)\(13\!\cdots\!79\)\(\)\()/ 2579890176 \)

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
470.592 + 0.866025i
470.592 0.866025i
604.769 0.866025i
604.769 + 0.866025i
165.483 + 0.866025i
165.483 0.866025i
382.240 0.866025i
382.240 + 0.866025i
374.026 0.866025i
374.026 + 0.866025i
−360.777 + 0.866025i
−360.777 0.866025i
−516.519 + 0.866025i
−516.519 0.866025i
−144.957 0.866025i
−144.957 + 0.866025i
−558.687 + 0.866025i
−558.687 0.866025i
−412.169 0.866025i
−412.169 + 0.866025i
−1009.00 174.621i 34092.0i 987591. + 352385.i 5.90557e6 5.95317e6 3.43988e7i 2.05952e8i −9.34947e8 5.28011e8i −1.16226e9 −5.95872e9 1.03123e9i
7.2 −1009.00 + 174.621i 34092.0i 987591. 352385.i 5.90557e6 5.95317e6 + 3.43988e7i 2.05952e8i −9.34947e8 + 5.28011e8i −1.16226e9 −5.95872e9 + 1.03123e9i
7.3 −831.424 597.755i 34092.0i 333954. + 993975.i −5.52569e6 −2.03786e7 + 2.83449e7i 2.40780e8i 3.16496e8 1.02604e9i −1.16226e9 4.59419e9 + 3.30301e9i
7.4 −831.424 + 597.755i 34092.0i 333954. 993975.i −5.52569e6 −2.03786e7 2.83449e7i 2.40780e8i 3.16496e8 + 1.02604e9i −1.16226e9 4.59419e9 3.30301e9i
7.5 −718.951 729.168i 34092.0i −14796.1 + 1.04847e6i −6.68527e6 2.48588e7 2.45104e7i 4.90135e8i 7.75150e8 7.43010e8i −1.16226e9 4.80638e9 + 4.87468e9i
7.6 −718.951 + 729.168i 34092.0i −14796.1 1.04847e6i −6.68527e6 2.48588e7 + 2.45104e7i 4.90135e8i 7.75150e8 + 7.43010e8i −1.16226e9 4.80638e9 4.87468e9i
7.7 −146.341 1013.49i 34092.0i −1.00574e6 + 296630.i 1.07961e7 −3.45518e7 + 4.98905e6i 1.91267e8i 4.47813e8 + 9.75902e8i −1.16226e9 −1.57992e9 1.09418e10i
7.8 −146.341 + 1013.49i 34092.0i −1.00574e6 296630.i 1.07961e7 −3.45518e7 4.98905e6i 1.91267e8i 4.47813e8 9.75902e8i −1.16226e9 −1.57992e9 + 1.09418e10i
7.9 −128.521 1015.90i 34092.0i −1.01554e6 + 261130.i −1.38143e7 −3.46341e7 + 4.38153e6i 4.88171e8i 3.95801e8 + 9.98130e8i −1.16226e9 1.77543e9 + 1.40340e10i
7.10 −128.521 + 1015.90i 34092.0i −1.01554e6 261130.i −1.38143e7 −3.46341e7 4.38153e6i 4.88171e8i 3.95801e8 9.98130e8i −1.16226e9 1.77543e9 1.40340e10i
7.11 171.720 1009.50i 34092.0i −989600. 346703.i 1.74429e7 3.44158e7 + 5.85428e6i 3.10877e8i −5.19931e8 + 9.39464e8i −1.16226e9 2.99530e9 1.76086e10i
7.12 171.720 + 1009.50i 34092.0i −989600. + 346703.i 1.74429e7 3.44158e7 5.85428e6i 3.10877e8i −5.19931e8 9.39464e8i −1.16226e9 2.99530e9 + 1.76086e10i
7.13 578.030 845.256i 34092.0i −380338. 977166.i −1.01328e7 2.88164e7 + 1.97062e7i 2.56091e7i −1.04580e9 2.43348e8i −1.16226e9 −5.85703e9 + 8.56477e9i
7.14 578.030 + 845.256i 34092.0i −380338. + 977166.i −1.01328e7 2.88164e7 1.97062e7i 2.56091e7i −1.04580e9 + 2.43348e8i −1.16226e9 −5.85703e9 8.56477e9i
7.15 734.722 713.274i 34092.0i 31057.2 1.04812e6i 6.80556e6 −2.43169e7 2.50481e7i 4.46066e7i −7.24775e8 7.92226e8i −1.16226e9 5.00020e9 4.85423e9i
7.16 734.722 + 713.274i 34092.0i 31057.2 + 1.04812e6i 6.80556e6 −2.43169e7 + 2.50481e7i 4.46066e7i −7.24775e8 + 7.92226e8i −1.16226e9 5.00020e9 + 4.85423e9i
7.17 981.162 293.082i 34092.0i 876782. 575122.i 1.08427e7 9.99175e6 + 3.34497e7i 4.06904e8i 6.91707e8 8.21257e8i −1.16226e9 1.06384e10 3.17779e9i
7.18 981.162 + 293.082i 34092.0i 876782. + 575122.i 1.08427e7 9.99175e6 3.34497e7i 4.06904e8i 6.91707e8 + 8.21257e8i −1.16226e9 1.06384e10 + 3.17779e9i
7.19 995.603 239.481i 34092.0i 933874. 476855.i −1.48963e7 −8.16437e6 3.39420e7i 1.28787e8i 8.15570e8 6.98403e8i −1.16226e9 −1.48308e10 + 3.56738e9i
7.20 995.603 + 239.481i 34092.0i 933874. + 476855.i −1.48963e7 −8.16437e6 + 3.39420e7i 1.28787e8i 8.15570e8 + 6.98403e8i −1.16226e9 −1.48308e10 3.56738e9i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 7.20
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 \(S_{21}^{\mathrm{new}}(12, [\chi])\).