# Properties

 Label 7.18.c.a Level 7 Weight 18 Character orbit 7.c Analytic conductor 12.826 Analytic rank 0 Dimension 20 CM No Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ = $$7$$ Weight: $$k$$ = $$18$$ Character orbit: $$[\chi]$$ = 7.c (of order $$3$$ and degree $$2$$)

## Newform invariants

 Self dual: No Analytic conductor: $$12.8255461141$$ Analytic rank: $$0$$ Dimension: $$20$$ Relative dimension: $$10$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{20} - \cdots)$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{50}\cdot 3^{18}\cdot 7^{19}$$ Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q + ( \beta_{1} + 27 \beta_{2} ) q^{2} + ( 656 - 656 \beta_{2} - \beta_{3} + \beta_{4} ) q^{3} + ( -55166 + 55166 \beta_{2} - 75 \beta_{3} - \beta_{4} - \beta_{5} + \beta_{7} ) q^{4} + ( -132 \beta_{1} + 108980 \beta_{2} + 8 \beta_{6} - \beta_{8} + \beta_{10} ) q^{5} + ( -84378 + 1957 \beta_{1} + 1957 \beta_{3} + 69 \beta_{4} - 5 \beta_{5} - 69 \beta_{6} + \beta_{8} - \beta_{11} ) q^{6} + ( 1628746 - 1445 \beta_{1} - 1817956 \beta_{2} + 5996 \beta_{3} - 250 \beta_{4} + 16 \beta_{5} + 312 \beta_{6} - 7 \beta_{7} - 2 \beta_{8} + \beta_{9} + \beta_{10} + \beta_{11} - \beta_{12} + \beta_{16} ) q^{7} + ( -11891162 - 64541 \beta_{1} - 64541 \beta_{3} + 436 \beta_{4} - 96 \beta_{5} - 436 \beta_{6} + 12 \beta_{8} - \beta_{11} - \beta_{15} ) q^{8} + ( 1 - 35222 \beta_{1} - 49888324 \beta_{2} + 3533 \beta_{6} + 16 \beta_{7} + 3 \beta_{8} + 5 \beta_{9} - 3 \beta_{10} + 5 \beta_{11} + \beta_{12} + \beta_{17} ) q^{9} +O(q^{10})$$ $$q + ( \beta_{1} + 27 \beta_{2} ) q^{2} + ( 656 - 656 \beta_{2} - \beta_{3} + \beta_{4} ) q^{3} + ( -55166 + 55166 \beta_{2} - 75 \beta_{3} - \beta_{4} - \beta_{5} + \beta_{7} ) q^{4} + ( -132 \beta_{1} + 108980 \beta_{2} + 8 \beta_{6} - \beta_{8} + \beta_{10} ) q^{5} + ( -84378 + 1957 \beta_{1} + 1957 \beta_{3} + 69 \beta_{4} - 5 \beta_{5} - 69 \beta_{6} + \beta_{8} - \beta_{11} ) q^{6} + ( 1628746 - 1445 \beta_{1} - 1817956 \beta_{2} + 5996 \beta_{3} - 250 \beta_{4} + 16 \beta_{5} + 312 \beta_{6} - 7 \beta_{7} - 2 \beta_{8} + \beta_{9} + \beta_{10} + \beta_{11} - \beta_{12} + \beta_{16} ) q^{7} + ( -11891162 - 64541 \beta_{1} - 64541 \beta_{3} + 436 \beta_{4} - 96 \beta_{5} - 436 \beta_{6} + 12 \beta_{8} - \beta_{11} - \beta_{15} ) q^{8} + ( 1 - 35222 \beta_{1} - 49888324 \beta_{2} + 3533 \beta_{6} + 16 \beta_{7} + 3 \beta_{8} + 5 \beta_{9} - 3 \beta_{10} + 5 \beta_{11} + \beta_{12} + \beta_{17} ) q^{9} + ( 22255369 - 22255370 \beta_{2} - 131354 \beta_{3} + 5199 \beta_{4} - 270 \beta_{5} + \beta_{6} + 271 \beta_{7} + \beta_{8} + 8 \beta_{9} + 35 \beta_{10} + \beta_{11} + \beta_{13} + 3 \beta_{14} + 2 \beta_{16} + \beta_{17} + \beta_{18} ) q^{10} + ( -14825628 - \beta_{1} + 14825623 \beta_{2} + 408912 \beta_{3} - 5299 \beta_{4} + 517 \beta_{5} + 5 \beta_{6} - 512 \beta_{7} - \beta_{8} + 25 \beta_{9} + 39 \beta_{10} + 6 \beta_{11} - 9 \beta_{12} + 5 \beta_{13} - 4 \beta_{14} + 16 \beta_{16} - 2 \beta_{17} - 2 \beta_{18} - \beta_{19} ) q^{11} + ( -24 - 503066 \beta_{1} + 280853296 \beta_{2} - 22 \beta_{3} + 34 \beta_{4} - 34 \beta_{5} - 61067 \beta_{6} - 2277 \beta_{7} + 190 \beta_{8} - 99 \beta_{9} - 174 \beta_{10} - 115 \beta_{11} - 36 \beta_{12} + 16 \beta_{13} - 9 \beta_{14} - 9 \beta_{15} - 20 \beta_{16} - 10 \beta_{17} - 2 \beta_{19} ) q^{12} + ( 178936813 + 2386440 \beta_{1} - 6 \beta_{2} + 2386391 \beta_{3} + 16411 \beta_{4} + 602 \beta_{5} - 16447 \beta_{6} - 36 \beta_{7} - 460 \beta_{8} - 72 \beta_{9} + 30 \beta_{10} - 147 \beta_{11} + 102 \beta_{12} + 36 \beta_{13} + 36 \beta_{15} - 49 \beta_{16} - 5 \beta_{18} - 6 \beta_{19} ) q^{13} + ( 1434054426 + 3721229 \beta_{1} - 299710481 \beta_{2} + 3172370 \beta_{3} + 104654 \beta_{4} + 2359 \beta_{5} - 28172 \beta_{6} + 5860 \beta_{7} - 539 \beta_{8} + 395 \beta_{9} + 1475 \beta_{10} + 542 \beta_{11} + 46 \beta_{12} + 77 \beta_{13} + 42 \beta_{14} + 77 \beta_{15} - 52 \beta_{16} + 14 \beta_{17} + 21 \beta_{18} - 7 \beta_{19} ) q^{14} + ( -1345039571 - 4888353 \beta_{1} - 26 \beta_{2} - 4888553 \beta_{3} + 86626 \beta_{4} - 2216 \beta_{5} - 86777 \beta_{6} - 151 \beta_{7} + 223 \beta_{8} - 302 \beta_{9} + 125 \beta_{10} + 571 \beta_{11} + 427 \beta_{12} + 151 \beta_{13} - 48 \beta_{15} - 200 \beta_{16} - 24 \beta_{18} - 26 \beta_{19} ) q^{15} + ( -256 - 18091218 \beta_{1} - 5020978092 \beta_{2} - 368 \beta_{3} + 576 \beta_{4} - 576 \beta_{5} - 484244 \beta_{6} - 49044 \beta_{7} + 12048 \beta_{8} - 206 \beta_{9} - 11776 \beta_{10} - 478 \beta_{11} - 464 \beta_{12} + 272 \beta_{13} - 198 \beta_{14} - 198 \beta_{15} - 336 \beta_{16} - 16 \beta_{17} - 32 \beta_{19} ) q^{16} + ( 1525918583 - 68 \beta_{1} - 1525919019 \beta_{2} - 9018455 \beta_{3} + 10797 \beta_{4} + 3620 \beta_{5} + 436 \beta_{6} - 3184 \beta_{7} + 73 \beta_{8} - 707 \beta_{9} - 3507 \beta_{10} + 504 \beta_{11} - 567 \beta_{12} + 436 \beta_{13} + 288 \beta_{14} + 1235 \beta_{16} + 5 \beta_{17} + 5 \beta_{18} - 68 \beta_{19} ) q^{17} + ( 8176417288 - 72 \beta_{1} - 8176417921 \beta_{2} + 54230713 \beta_{3} + 1150505 \beta_{4} + 52448 \beta_{5} + 633 \beta_{6} - 51815 \beta_{7} + 97 \beta_{8} + 11030 \beta_{9} + 32865 \beta_{10} + 705 \beta_{11} - 752 \beta_{12} + 633 \beta_{13} - 93 \beta_{14} + 1802 \beta_{16} + 25 \beta_{17} + 25 \beta_{18} - 72 \beta_{19} ) q^{18} + ( -419 + 9072129 \beta_{1} - 7165695418 \beta_{2} - 1133 \beta_{3} + 1656 \beta_{4} - 1656 \beta_{5} - 972852 \beta_{6} + 24500 \beta_{7} - 6743 \beta_{8} - 9329 \beta_{9} + 7510 \beta_{10} - 10096 \beta_{11} - 942 \beta_{12} + 767 \beta_{13} + 132 \beta_{14} + 132 \beta_{15} - 1011 \beta_{16} + 226 \beta_{17} - 122 \beta_{19} ) q^{19} + ( -9032064698 - 32722013 \beta_{1} - 68 \beta_{2} - 32723693 \beta_{3} + 1253887 \beta_{4} - 481093 \beta_{5} - 1254755 \beta_{6} - 868 \beta_{7} - 58288 \beta_{8} - 1736 \beta_{9} + 800 \beta_{10} - 14102 \beta_{11} + 2536 \beta_{12} + 868 \beta_{13} - 374 \beta_{15} - 1680 \beta_{16} + 148 \beta_{18} - 68 \beta_{19} ) q^{20} + ( -54938195991 - 102380106 \beta_{1} + 42961668722 \beta_{2} - 64102016 \beta_{3} - 1158078 \beta_{4} - 167204 \beta_{5} + 1730721 \beta_{6} + 460748 \beta_{7} + 48071 \beta_{8} + 9319 \beta_{9} + 75842 \beta_{10} + 25699 \beta_{11} - 863 \beta_{12} + 658 \beta_{13} - 756 \beta_{14} - 1092 \beta_{15} + 1004 \beta_{16} - 399 \beta_{17} - 574 \beta_{18} - 224 \beta_{19} ) q^{21} + ( 74984654544 + 64815319 \beta_{1} + 205 \beta_{2} + 64814271 \beta_{3} + 3969463 \beta_{4} + 605524 \beta_{5} - 3969362 \beta_{6} + 101 \beta_{7} - 89457 \beta_{8} + 202 \beta_{9} + 104 \beta_{10} + 24012 \beta_{11} - 98 \beta_{12} - 101 \beta_{13} + 1089 \beta_{15} - 1048 \beta_{16} + 635 \beta_{18} + 205 \beta_{19} ) q^{22} + ( 1651 + 310259744 \beta_{1} - 31630719227 \beta_{2} + 1771 \beta_{3} - 3367 \beta_{4} + 3367 \beta_{5} + 5132802 \beta_{6} - 377492 \beta_{7} + 331897 \beta_{8} - 8838 \beta_{9} - 333563 \beta_{10} - 7172 \beta_{11} + 3247 \beta_{12} - 1666 \beta_{13} + 3760 \beta_{14} + 3760 \beta_{15} + 1736 \beta_{16} + 20 \beta_{17} + 35 \beta_{19} ) q^{23} + ( 69449059150 + 784 \beta_{1} - 69449054990 \beta_{2} + 238423835 \beta_{3} - 18301976 \beta_{4} + 1168292 \beta_{5} - 4160 \beta_{6} - 1172452 \beta_{7} - 1360 \beta_{8} - 33659 \beta_{9} - 347788 \beta_{10} - 4944 \beta_{11} + 5152 \beta_{12} - 4160 \beta_{13} - 6579 \beta_{14} - 11120 \beta_{16} - 576 \beta_{17} - 576 \beta_{18} + 784 \beta_{19} ) q^{24} + ( 59132499900 + 780 \beta_{1} - 59132491800 \beta_{2} - 51283802 \beta_{3} + 3069402 \beta_{4} + 2985860 \beta_{5} - 8100 \beta_{6} - 2993960 \beta_{7} - 550 \beta_{8} + 4770 \beta_{9} + 448154 \beta_{10} - 8880 \beta_{11} + 9890 \beta_{12} - 8100 \beta_{13} + 3240 \beta_{14} - 23750 \beta_{16} + 230 \beta_{17} + 230 \beta_{18} + 780 \beta_{19} ) q^{25} + ( 8048 + 416426142 \beta_{1} + 448850634052 \beta_{2} + 18551 \beta_{3} - 26357 \beta_{4} + 26357 \beta_{5} - 10118296 \beta_{6} + 5088470 \beta_{7} - 366569 \beta_{8} - 15672 \beta_{9} + 354465 \beta_{10} - 3568 \beta_{11} + 15854 \beta_{12} - 12104 \beta_{13} - 709 \beta_{14} - 709 \beta_{15} + 16402 \beta_{16} - 1907 \beta_{17} + 2149 \beta_{19} ) q^{26} + ( -573654144129 - 1072215832 \beta_{1} + 1311 \beta_{2} - 1072182407 \beta_{3} - 69736729 \beta_{4} - 8738833 \beta_{5} + 69754393 \beta_{6} + 17664 \beta_{7} - 941390 \beta_{8} + 35328 \beta_{9} - 16353 \beta_{10} + 104830 \beta_{11} - 51681 \beta_{12} - 17664 \beta_{13} - 1932 \beta_{15} + 33425 \beta_{16} - 2030 \beta_{18} + 1311 \beta_{19} ) q^{27} + ( -111750947140 + 1047934644 \beta_{1} + 509328999018 \beta_{2} - 673457338 \beta_{3} + 55397507 \beta_{4} - 5781763 \beta_{5} + 26866448 \beta_{6} + 11637979 \beta_{7} + 412630 \beta_{8} - 64171 \beta_{9} + 406354 \beta_{10} - 186160 \beta_{11} + 9088 \beta_{12} - 21798 \beta_{13} + 5187 \beta_{14} + 1890 \beta_{15} - 10380 \beta_{16} + 5110 \beta_{17} + 6930 \beta_{18} + 5348 \beta_{19} ) q^{28} + ( 14243836383 + 559456292 \beta_{1} - 110 \beta_{2} + 559509035 \beta_{3} + 94077915 \beta_{4} + 5631818 \beta_{5} - 94055323 \beta_{6} + 22592 \beta_{7} - 34630 \beta_{8} + 45184 \beta_{9} - 22702 \beta_{10} - 147751 \beta_{11} - 67886 \beta_{12} - 22592 \beta_{13} - 8212 \beta_{15} + 52743 \beta_{16} - 7229 \beta_{18} - 110 \beta_{19} ) q^{29} + ( 13800 - 1748136533 \beta_{1} - 935783007470 \beta_{2} + 35545 \beta_{3} - 42115 \beta_{4} + 42115 \beta_{5} + 149979097 \beta_{6} - 11176965 \beta_{7} + 470396 \beta_{8} + 43335 \beta_{9} - 488556 \beta_{10} + 61495 \beta_{11} + 20370 \beta_{12} - 18160 \beta_{13} - 26025 \beta_{14} - 26025 \beta_{15} + 29750 \beta_{16} + 1435 \beta_{17} + 5795 \beta_{19} ) q^{30} + ( -160782303947 - 2231 \beta_{1} + 160782311787 \beta_{2} - 3265110741 \beta_{3} - 144855355 \beta_{4} - 1218279 \beta_{5} - 7840 \beta_{6} + 1210439 \beta_{7} + 11187 \beta_{8} + 374907 \beta_{9} + 1087402 \beta_{10} - 5609 \beta_{11} + 12334 \beta_{12} - 7840 \beta_{13} + 54336 \beta_{14} - 34707 \beta_{16} + 8956 \beta_{17} + 8956 \beta_{18} - 2231 \beta_{19} ) q^{31} + ( 1889346513136 + 2080 \beta_{1} - 1889346531968 \beta_{2} + 3959343840 \beta_{3} - 150134856 \beta_{4} + 36176648 \beta_{5} + 18832 \beta_{6} - 36157816 \beta_{7} - 9680 \beta_{8} - 954520 \beta_{9} - 2309984 \beta_{10} + 16752 \beta_{11} - 22272 \beta_{12} + 18832 \beta_{13} - 35384 \beta_{14} + 66176 \beta_{16} - 7600 \beta_{17} - 7600 \beta_{18} + 2080 \beta_{19} ) q^{32} + ( -27798 - 4538556008 \beta_{1} + 958977107265 \beta_{2} - 88164 \beta_{3} + 106008 \beta_{4} - 106008 \beta_{5} - 253316610 \beta_{6} + 27116800 \beta_{7} + 615052 \beta_{8} + 922586 \beta_{9} - 569080 \beta_{10} + 876614 \beta_{11} - 45642 \beta_{12} + 45972 \beta_{13} + 4056 \beta_{14} + 4056 \beta_{15} - 74100 \beta_{16} + 4110 \beta_{17} - 14064 \beta_{19} ) q^{33} + ( -1631415757871 + 1663860285 \beta_{1} + 1775 \beta_{2} + 1663649613 \beta_{3} - 120541890 \beta_{4} - 40390589 \beta_{5} + 120447801 \beta_{6} - 94089 \beta_{7} + 4478614 \beta_{8} - 188178 \beta_{9} + 95864 \beta_{10} + 197181 \beta_{11} + 284042 \beta_{12} + 94089 \beta_{13} + 51309 \beta_{15} - 210672 \beta_{16} + 17169 \beta_{18} + 1775 \beta_{19} ) q^{34} + ( -1003648642164 + 2562543658 \beta_{1} + 1996948545669 \beta_{2} - 3441073413 \beta_{3} + 146813756 \beta_{4} - 7454081 \beta_{5} + 353885302 \beta_{6} + 25423432 \beta_{7} - 2676921 \beta_{8} - 426113 \beta_{9} - 4584647 \beta_{10} - 500446 \beta_{11} - 60163 \beta_{12} + 143339 \beta_{13} - 15316 \beta_{14} + 60872 \beta_{15} + 64446 \beta_{16} - 38262 \beta_{17} - 47446 \beta_{18} - 45451 \beta_{19} ) q^{35} + ( 3836232430428 + 13181189110 \beta_{1} + 7440 \beta_{2} + 13180748438 \beta_{3} + 1754075050 \beta_{4} + 4541242 \beta_{5} - 1754262298 \beta_{6} - 187248 \beta_{7} + 4256624 \beta_{8} - 374496 \beta_{9} + 194688 \beta_{10} - 922360 \beta_{11} + 569184 \beta_{12} + 187248 \beta_{13} + 26328 \beta_{15} - 440672 \beta_{16} + 43856 \beta_{18} + 7440 \beta_{19} ) q^{36} + ( -167579 + 2498473978 \beta_{1} + 552861706478 \beta_{2} - 411786 \beta_{3} + 499732 \beta_{4} - 499732 \beta_{5} + 924098521 \beta_{6} - 20750992 \beta_{7} - 15156506 \beta_{8} + 484193 \beta_{9} + 15373988 \beta_{10} + 266711 \beta_{11} - 255525 \beta_{12} + 217482 \beta_{13} + 42084 \beta_{14} + 42084 \beta_{15} - 347018 \beta_{16} - 14865 \beta_{17} - 64768 \beta_{19} ) q^{37} + ( -1569029815028 + 648 \beta_{1} + 1569029566643 \beta_{2} + 2172074786 \beta_{3} - 1822031282 \beta_{4} + 28798011 \beta_{5} + 248385 \beta_{6} - 28549626 \beta_{7} - 69903 \beta_{8} - 797075 \beta_{9} + 7056366 \beta_{10} + 247737 \beta_{11} - 316344 \beta_{12} + 248385 \beta_{13} - 173507 \beta_{14} + 815058 \beta_{16} - 69255 \beta_{17} - 69255 \beta_{18} + 648 \beta_{19} ) q^{38} + ( 2845155886118 - 70135 \beta_{1} - 2845156060183 \beta_{2} + 12734523245 \beta_{3} - 78135232 \beta_{4} - 51412545 \beta_{5} + 174065 \beta_{6} + 51586610 \beta_{7} + 149619 \beta_{8} + 3906613 \beta_{9} - 8080819 \beta_{10} + 244200 \beta_{11} - 234851 \beta_{12} + 174065 \beta_{13} + 203400 \beta_{14} + 372576 \beta_{16} + 79484 \beta_{17} + 79484 \beta_{18} - 70135 \beta_{19} ) q^{39} + ( -144000 - 54749526801 \beta_{1} - 9102238654226 \beta_{2} + 47872 \beta_{3} + 209376 \beta_{4} - 209376 \beta_{5} - 1580409916 \beta_{6} - 82481192 \beta_{7} + 13085708 \beta_{8} - 2829005 \beta_{9} - 12950508 \beta_{10} - 2964205 \beta_{11} - 401248 \beta_{12} + 135200 \beta_{13} - 60573 \beta_{14} - 60573 \beta_{15} - 13152 \beta_{16} + 52224 \beta_{17} + 61024 \beta_{19} ) q^{40} + ( -3336386393725 + 12473771490 \beta_{1} - 128120 \beta_{2} + 12474307137 \beta_{3} - 1134348471 \beta_{4} + 101720504 \beta_{5} + 1134373751 \beta_{6} + 25280 \beta_{7} + 25247586 \beta_{8} + 50560 \beta_{9} - 153400 \beta_{10} - 3429797 \beta_{11} - 203960 \beta_{12} - 25280 \beta_{13} - 336800 \beta_{15} + 535647 \beta_{16} - 100727 \beta_{18} - 128120 \beta_{19} ) q^{41} + ( 6017459783083 - 86816318159 \beta_{1} - 19474254606464 \beta_{2} - 105931288296 \beta_{3} + 2275591099 \beta_{4} + 22672701 \beta_{5} + 2329442326 \beta_{6} - 142877330 \beta_{7} - 7815313 \beta_{8} + 4468196 \beta_{9} - 1571301 \beta_{10} + 5942046 \beta_{11} + 258106 \beta_{12} - 213752 \beta_{13} + 27069 \beta_{14} - 561183 \beta_{15} - 239858 \beta_{16} + 180915 \beta_{17} + 192360 \beta_{18} + 224035 \beta_{19} ) q^{42} + ( -4651417866356 + 60752800954 \beta_{1} - 173394 \beta_{2} + 60754337876 \beta_{3} + 2979668578 \beta_{4} - 50701398 \beta_{5} - 2979223748 \beta_{6} + 444830 \beta_{7} - 10932714 \beta_{8} + 889660 \beta_{9} - 618224 \beta_{10} + 8143458 \beta_{11} - 1507884 \beta_{12} - 444830 \beta_{13} - 48312 \beta_{15} + 1536922 \beta_{16} - 127080 \beta_{18} - 173394 \beta_{19} ) q^{43} + ( 374024 + 116009217406 \beta_{1} + 16284995569840 \beta_{2} + 1694474 \beta_{3} - 1661918 \beta_{4} + 1661918 \beta_{5} + 4131698149 \beta_{6} + 257995691 \beta_{7} - 5108194 \beta_{8} - 3926699 \beta_{9} + 4449938 \beta_{10} - 3268443 \beta_{11} + 341468 \beta_{12} - 658256 \beta_{13} + 475119 \beta_{14} + 475119 \beta_{15} + 1349068 \beta_{16} + 61174 \beta_{17} + 345406 \beta_{19} ) q^{44} + ( 859230103232 - 113310 \beta_{1} - 859229002070 \beta_{2} - 133629525241 \beta_{3} - 4684203719 \beta_{4} - 98821490 \beta_{5} - 1101162 \beta_{6} + 97720328 \beta_{7} + 412283 \beta_{8} - 3426701 \beta_{9} - 20770017 \beta_{10} - 987852 \beta_{11} + 1173515 \beta_{12} - 1101162 \beta_{13} - 313836 \beta_{14} - 3715769 \beta_{16} + 298973 \beta_{17} + 298973 \beta_{18} - 113310 \beta_{19} ) q^{45} + ( -56308829583688 + 485752 \beta_{1} + 56308830641848 \beta_{2} + 87682682445 \beta_{3} - 2494238873 \beta_{4} - 571159287 \beta_{5} - 1058160 \beta_{6} + 570101127 \beta_{7} - 956056 \beta_{8} + 5597051 \beta_{9} + 28420447 \beta_{10} - 1543912 \beta_{11} + 1559360 \beta_{12} - 1058160 \beta_{13} - 695172 \beta_{14} - 2218424 \beta_{16} - 470304 \beta_{17} - 470304 \beta_{18} + 485752 \beta_{19} ) q^{46} + ( 1283900 - 222164605960 \beta_{1} + 16246019638759 \beta_{2} + 301124 \beta_{3} - 2476943 \beta_{4} + 2476943 \beta_{5} - 2573325127 \beta_{6} - 392896768 \beta_{7} - 33596378 \beta_{8} - 14168775 \beta_{9} + 32170437 \beta_{10} - 12742834 \beta_{11} + 3459719 \beta_{12} - 1425941 \beta_{13} + 616904 \beta_{14} + 616904 \beta_{15} + 676063 \beta_{16} - 516980 \beta_{17} - 374939 \beta_{19} ) q^{47} + ( 81311173830956 + 163815819466 \beta_{1} + 1039264 \beta_{2} + 163815111402 \beta_{3} - 1907995052 \beta_{4} + 990266572 \beta_{5} + 1909418236 \beta_{6} + 1423184 \beta_{7} - 116242240 \beta_{8} + 2846368 \beta_{9} - 383920 \beta_{10} + 7698478 \beta_{11} - 3230288 \beta_{12} - 1423184 \beta_{13} + 888486 \beta_{15} - 708064 \beta_{16} + 436640 \beta_{18} + 1039264 \beta_{19} ) q^{48} + ( 4577651292844 - 11661271846 \beta_{1} - 5436484286982 \beta_{2} - 211219768207 \beta_{3} - 1788425023 \beta_{4} + 674424576 \beta_{5} + 3153577203 \beta_{6} - 819290360 \beta_{7} + 30475802 \beta_{8} - 11518472 \beta_{9} + 32896850 \beta_{10} - 2206953 \beta_{11} - 682528 \beta_{12} - 1403360 \beta_{13} - 301056 \beta_{14} + 2134440 \beta_{15} + 432019 \beta_{16} - 531160 \beta_{17} - 399203 \beta_{18} - 874944 \beta_{19} ) q^{49} + ( -7805277854284 + 449199303372 \beta_{1} + 1342246 \beta_{2} + 449197434252 \beta_{3} + 1428832840 \beta_{4} - 386569774 \beta_{5} - 1427795474 \beta_{6} + 1037366 \beta_{7} - 17862056 \beta_{8} + 2074732 \beta_{9} + 304880 \beta_{10} - 5265586 \beta_{11} - 1769852 \beta_{12} - 1037366 \beta_{13} + 516458 \beta_{15} - 1869120 \beta_{16} - 82886 \beta_{18} + 1342246 \beta_{19} ) q^{50} + ( 2210827 + 138175971463 \beta_{1} - 4157182160180 \beta_{2} - 2668507 \beta_{3} - 774626 \beta_{4} + 774626 \beta_{5} - 2440178366 \beta_{6} + 266870684 \beta_{7} + 255292451 \beta_{8} + 7146535 \beta_{9} - 256290928 \beta_{10} + 8145012 \beta_{11} + 5653960 \beta_{12} - 998477 \beta_{13} - 3300972 \beta_{14} - 3300972 \beta_{15} - 1446179 \beta_{16} - 9978 \beta_{17} - 1222328 \beta_{19} ) q^{51} + ( -66457843931612 + 1420544 \beta_{1} + 66457843253380 \beta_{2} - 875736775030 \beta_{3} + 4918970354 \beta_{4} - 631313206 \beta_{5} + 678232 \beta_{6} + 631991438 \beta_{7} - 1998824 \beta_{8} + 9557052 \beta_{9} - 50351112 \beta_{10} - 742312 \beta_{11} + 1584576 \beta_{12} + 678232 \beta_{13} + 4642692 \beta_{14} + 4033520 \beta_{16} - 578280 \beta_{17} - 578280 \beta_{18} + 1420544 \beta_{19} ) q^{52} + ( 34983795262934 - 1978794 \beta_{1} - 34983795781076 \beta_{2} + 293269829209 \beta_{3} + 1684565503 \beta_{4} - 244628238 \beta_{5} + 518142 \beta_{6} + 245146380 \beta_{7} + 3674889 \beta_{8} - 54777327 \beta_{9} + 102943604 \beta_{10} + 2496936 \beta_{11} - 2779635 \beta_{12} + 518142 \beta_{13} + 1594500 \beta_{14} - 2120463 \beta_{16} + 1696095 \beta_{17} + 1696095 \beta_{18} - 1978794 \beta_{19} ) q^{53} + ( -2947912 - 1866349660825 \beta_{1} - 219762126202400 \beta_{2} + 5407731 \beta_{3} + 2322183 \beta_{4} - 2322183 \beta_{5} + 16339388611 \beta_{6} - 401026063 \beta_{7} - 91066800 \beta_{8} + 88568209 \beta_{9} + 93541656 \beta_{10} + 86093353 \beta_{11} - 10677826 \beta_{12} + 2474856 \beta_{13} - 3588819 \beta_{14} - 3588819 \beta_{15} + 2780202 \beta_{16} + 2154473 \beta_{17} + 2627529 \beta_{19} ) q^{54} + ( -31510481172632 + 366466641424 \beta_{1} - 4484887 \beta_{2} + 366473773899 \beta_{3} + 1907808312 \beta_{4} + 184977743 \beta_{5} - 1911685009 \beta_{6} - 3876697 \beta_{7} - 133936549 \beta_{8} - 7753394 \beta_{9} - 608190 \beta_{10} + 3484317 \beta_{11} + 7145204 \beta_{12} + 3876697 \beta_{13} + 427944 \beta_{15} + 7132475 \beta_{16} - 1431208 \beta_{18} - 4484887 \beta_{19} ) q^{55} + ( -176868851753958 - 816856293609 \beta_{1} + 14618345694876 \beta_{2} - 1585410531439 \beta_{3} + 4416691360 \beta_{4} - 1434610060 \beta_{5} - 21850645232 \beta_{6} + 619944008 \beta_{7} + 146844644 \beta_{8} + 7190358 \beta_{9} + 202439464 \beta_{10} - 73892385 \beta_{11} + 817040 \beta_{12} + 6032656 \beta_{13} + 2582342 \beta_{14} - 2875929 \beta_{15} + 310336 \beta_{16} + 792736 \beta_{17} - 50400 \beta_{18} + 3511648 \beta_{19} ) q^{56} + ( 169373913601201 + 1732393384088 \beta_{1} - 5457700 \beta_{2} + 1732394796198 \beta_{3} - 20428339978 \beta_{4} - 293533492 \beta_{5} + 20421907418 \beta_{6} - 6432560 \beta_{7} + 33929284 \beta_{8} - 12865120 \beta_{9} + 974860 \beta_{10} - 94728262 \beta_{11} + 13839980 \beta_{12} + 6432560 \beta_{13} - 4841112 \beta_{15} + 1412110 \beta_{16} + 2095870 \beta_{18} - 5457700 \beta_{19} ) q^{57} + ( -15424496 + 925754067926 \beta_{1} + 111700968724416 \beta_{2} + 1819125 \beta_{3} + 16546145 \beta_{4} - 16546145 \beta_{5} - 40721857752 \beta_{6} - 1121303006 \beta_{7} - 187784307 \beta_{8} + 824752 \beta_{9} + 198075819 \beta_{10} - 9466760 \beta_{11} - 33789766 \beta_{12} + 10291512 \beta_{13} + 8030385 \beta_{14} + 8030385 \beta_{15} - 2217754 \beta_{16} - 1096105 \beta_{17} + 4036879 \beta_{19} ) q^{58} + ( 289832540340710 - 7667816 \beta_{1} - 289832548018260 \beta_{2} - 402276310735 \beta_{3} + 37117769365 \beta_{4} + 2109828188 \beta_{5} + 7677550 \beta_{6} - 2102150638 \beta_{7} + 6799000 \beta_{8} + 50205396 \beta_{9} + 74387002 \beta_{10} + 15345366 \beta_{11} - 23881998 \beta_{12} + 7677550 \beta_{13} - 15672424 \beta_{14} + 16233650 \beta_{16} - 868816 \beta_{17} - 868816 \beta_{18} - 7667816 \beta_{19} ) q^{59} + ( 185130286572804 + 6703920 \beta_{1} - 185130297236070 \beta_{2} + 2110718860334 \beta_{3} + 22435232751 \beta_{4} + 3778204745 \beta_{5} + 10663266 \beta_{6} - 3767541479 \beta_{7} - 9959574 \beta_{8} + 36842483 \beta_{9} - 162014146 \beta_{10} + 3959346 \beta_{11} - 511080 \beta_{12} + 10663266 \beta_{13} - 3807507 \beta_{14} + 41949372 \beta_{16} - 3255654 \beta_{17} - 3255654 \beta_{18} + 6703920 \beta_{19} ) q^{60} + ( 492767 - 1854993732666 \beta_{1} + 482926361917376 \beta_{2} - 44933146 \beta_{3} + 28111572 \beta_{4} - 28111572 \beta_{5} + 7584435595 \beta_{6} + 4309245504 \beta_{7} + 111236162 \beta_{8} - 93312541 \beta_{9} - 103355848 \beta_{10} - 101192855 \beta_{11} + 17314341 \beta_{12} + 7880314 \beta_{13} + 12675324 \beta_{14} + 12675324 \beta_{15} - 32582202 \beta_{16} - 3977863 \beta_{17} - 12350944 \beta_{19} ) q^{61} + ( -622011847905802 - 726576074641 \beta_{1} + 12602599 \beta_{2} - 726628894345 \beta_{3} + 63162774469 \beta_{4} - 9491765332 \beta_{5} - 63168601310 \beta_{6} - 5826841 \beta_{7} + 656792113 \beta_{8} - 11653682 \beta_{9} + 18429440 \beta_{10} + 31451264 \beta_{11} + 30083122 \beta_{12} + 5826841 \beta_{13} - 7845973 \beta_{15} - 52819704 \beta_{16} + 3358225 \beta_{18} + 12602599 \beta_{19} ) q^{62} + ( -49068646768827 - 1534470007364 \beta_{1} + 223395420397296 \beta_{2} - 3530720176279 \beta_{3} - 67265196110 \beta_{4} - 734241758 \beta_{5} - 49445462263 \beta_{6} + 6078643602 \beta_{7} - 275373625 \beta_{8} - 50131365 \beta_{9} - 1053935496 \beta_{10} + 138352280 \beta_{11} + 953544 \beta_{12} - 1308909 \beta_{13} - 9796920 \beta_{14} - 5375664 \beta_{15} - 3189215 \beta_{16} + 280980 \beta_{17} + 2311400 \beta_{18} - 12868632 \beta_{19} ) q^{63} + ( 112742277731504 + 4513004902168 \beta_{1} + 13252960 \beta_{2} + 4512972209560 \beta_{3} - 146212887456 \beta_{4} + 6110770016 \beta_{5} + 146212909184 \beta_{6} + 21728 \beta_{7} + 365311840 \beta_{8} + 43456 \beta_{9} + 13231232 \beta_{10} + 233000856 \beta_{11} + 13187776 \beta_{12} - 21728 \beta_{13} + 21123672 \beta_{15} - 32692608 \beta_{16} - 7022816 \beta_{18} + 13252960 \beta_{19} ) q^{64} + ( 35803025 + 1540247752894 \beta_{1} + 232360923551262 \beta_{2} - 27661264 \beta_{3} - 17291312 \beta_{4} + 17291312 \beta_{5} - 28664129311 \beta_{6} - 3457126576 \beta_{7} - 1266847375 \beta_{8} + 101834525 \beta_{9} + 1250940335 \beta_{10} + 117741565 \beta_{11} + 80755601 \beta_{12} - 15907040 \beta_{13} + 1038496 \beta_{14} + 1038496 \beta_{15} - 13138496 \beta_{16} + 5373217 \beta_{17} - 14522768 \beta_{19} ) q^{65} + ( 796399376365471 + 24031856 \beta_{1} - 796399360977051 \beta_{2} - 4742514974495 \beta_{3} + 158019021468 \beta_{4} + 2181548448 \beta_{5} - 15388420 \beta_{6} - 2196936868 \beta_{7} - 16058004 \beta_{8} - 229103864 \beta_{9} + 234503732 \beta_{10} - 39420276 \beta_{11} + 71425984 \beta_{12} - 15388420 \beta_{13} + 15108444 \beta_{14} - 30107256 \beta_{16} + 7973852 \beta_{17} + 7973852 \beta_{18} + 24031856 \beta_{19} ) q^{66} + ( 132770523801902 - 22680890 \beta_{1} - 132770497923866 \beta_{2} + 1784712259907 \beta_{3} - 7976793863 \beta_{4} + 8008232494 \beta_{5} - 25878036 \beta_{6} - 8034110530 \beta_{7} + 22173510 \beta_{8} + 176157542 \beta_{9} - 1006109164 \beta_{10} - 3197146 \beta_{11} - 19991124 \beta_{12} - 25878036 \beta_{13} + 12136704 \beta_{14} - 99807618 \beta_{16} - 507380 \beta_{17} - 507380 \beta_{18} - 22680890 \beta_{19} ) q^{67} + ( -7601856 - 6162248448017 \beta_{1} + 56999899362230 \beta_{2} + 133409304 \beta_{3} - 94465608 \beta_{4} + 94465608 \beta_{5} + 46203458183 \beta_{6} + 4890164985 \beta_{7} + 1314535832 \beta_{8} - 220812084 \beta_{9} - 1344533336 \beta_{10} - 190814580 \beta_{11} - 46545552 \beta_{12} - 29997504 \beta_{13} - 26688012 \beta_{14} - 26688012 \beta_{15} + 98938704 \beta_{16} - 3128760 \beta_{17} + 34470600 \beta_{19} ) q^{68} + ( -995468544405400 + 3025844048924 \beta_{1} - 25529620 \beta_{2} + 3026005797868 \beta_{3} - 17232358084 \beta_{4} - 4938821708 \beta_{5} + 17273041328 \beta_{6} + 40683244 \beta_{7} + 1771092049 \beta_{8} + 81366488 \beta_{9} - 66212864 \beta_{10} - 395846760 \beta_{11} - 147579352 \beta_{12} - 40683244 \beta_{13} + 7924776 \beta_{15} + 161748944 \beta_{16} - 3793596 \beta_{18} - 25529620 \beta_{19} ) q^{69} + ( -1124675153589232 - 1684993541192 \beta_{1} + 471553591417825 \beta_{2} - 4861657826437 \beta_{3} - 47169855493 \beta_{4} - 5827980930 \beta_{5} + 12824620535 \beta_{6} + 8468392332 \beta_{7} - 1768302394 \beta_{8} + 371642511 \beta_{9} + 1304356099 \beta_{10} + 218354257 \beta_{11} - 4827650 \beta_{12} - 30497383 \beta_{13} + 13629966 \beta_{14} + 18931185 \beta_{15} + 3988484 \beta_{16} - 3383898 \beta_{17} - 3608983 \beta_{18} + 32278085 \beta_{19} ) q^{70} + ( 774033417206522 + 2128632731600 \beta_{1} - 22506280 \beta_{2} + 2128802024348 \beta_{3} - 156203763486 \beta_{4} - 249651164 \beta_{5} + 156257040148 \beta_{6} + 53276662 \beta_{7} - 1714471906 \beta_{8} + 106553324 \beta_{9} - 75782942 \beta_{10} + 179258610 \beta_{11} - 182336266 \beta_{12} - 53276662 \beta_{13} - 41565312 \beta_{15} + 169292748 \beta_{16} + 4779416 \beta_{18} - 22506280 \beta_{19} ) q^{71} + ( -24071808 + 187254704130 \beta_{1} + 1623342494405892 \beta_{2} + 160196544 \beta_{3} - 103405248 \beta_{4} + 103405248 \beta_{5} - 79332573192 \beta_{6} - 2432629488 \beta_{7} - 766749240 \beta_{8} - 745496646 \beta_{9} + 736745400 \beta_{10} - 715492806 \beta_{11} - 80863104 \beta_{12} - 30003840 \beta_{13} - 47767878 \beta_{14} - 47767878 \beta_{15} + 116798976 \beta_{16} - 10678080 \beta_{17} + 43397568 \beta_{19} ) q^{72} + ( -316386298443287 - 48703136 \beta_{1} + 316386327956839 \beta_{2} - 6890391125614 \beta_{3} - 95053913934 \beta_{4} - 12025491152 \beta_{5} - 29513552 \beta_{6} + 11995977600 \beta_{7} + 32929406 \beta_{8} + 238095742 \beta_{9} + 1878595406 \beta_{10} + 19189584 \beta_{11} - 83666450 \beta_{12} - 29513552 \beta_{13} + 44304120 \beta_{14} - 121470062 \beta_{16} - 15773730 \beta_{17} - 15773730 \beta_{18} - 48703136 \beta_{19} ) q^{73} + ( -401863410836449 + 62722040 \beta_{1} + 401863436860679 \beta_{2} + 2816352886889 \beta_{3} + 178929124764 \beta_{4} - 15651296450 \beta_{5} - 26024230 \beta_{6} + 15625272220 \beta_{7} - 41290702 \beta_{8} + 411950158 \beta_{9} + 909597344 \beta_{10} - 88746270 \beta_{11} + 172899648 \beta_{12} - 26024230 \beta_{13} - 22332550 \beta_{14} - 36781988 \beta_{16} + 21431338 \beta_{17} + 21431338 \beta_{18} + 62722040 \beta_{19} ) q^{74} + ( 99092130 - 777571374692 \beta_{1} - 811902178730324 \beta_{2} - 130392322 \beta_{3} + 15547924 \beta_{4} - 15547924 \beta_{5} + 113558982118 \beta_{6} - 8329705608 \beta_{7} - 2366486462 \beta_{8} - 93884190 \beta_{9} + 2349736752 \beta_{10} - 77134480 \beta_{11} + 213936528 \beta_{12} - 16749710 \beta_{13} + 25496928 \beta_{14} + 25496928 \beta_{15} - 81344978 \beta_{16} + 33295076 \beta_{17} - 49047344 \beta_{19} ) q^{75} + ( -732061148232136 + 848187811014 \beta_{1} + 35541510 \beta_{2} + 848026392526 \beta_{3} - 128032832261 \beta_{4} + 24038345961 \beta_{5} + 128000669531 \beta_{6} - 32162730 \beta_{7} - 5630024072 \beta_{8} - 64325460 \beta_{9} + 67704240 \beta_{10} + 1116563451 \beta_{11} + 132029700 \beta_{12} + 32162730 \beta_{13} + 54214275 \beta_{15} - 161418488 \beta_{16} - 9531502 \beta_{18} + 35541510 \beta_{19} ) q^{76} + ( -884722929913907 + 2779584556572 \beta_{1} - 2913112433758030 \beta_{2} + 1843470617605 \beta_{3} + 87260461805 \beta_{4} + 22236332338 \beta_{5} + 92323564651 \beta_{6} - 24274233908 \beta_{7} + 3937289682 \beta_{8} - 752101404 \beta_{9} - 2168936805 \beta_{10} - 217638097 \beta_{11} + 5273042 \beta_{12} + 14906444 \beta_{13} + 19498024 \beta_{14} + 15568700 \beta_{15} + 8251461 \beta_{16} + 4843860 \beta_{17} - 5024635 \beta_{18} - 41256670 \beta_{19} ) q^{77} + ( 2435131140311114 - 3394006986132 \beta_{1} + 42962345 \beta_{2} - 3394328682652 \beta_{3} + 703937153012 \beta_{4} - 17486572501 \beta_{5} - 704016532387 \beta_{6} - 79379375 \beta_{7} + 5247684632 \beta_{8} - 158758750 \beta_{9} + 122341720 \beta_{10} - 791844781 \beta_{11} + 281100470 \beta_{12} + 79379375 \beta_{13} - 6866631 \beta_{15} - 321696520 \beta_{16} + 34050735 \beta_{18} + 42962345 \beta_{19} ) q^{78} + ( -20258463 + 855658179358 \beta_{1} + 4082609089417959 \beta_{2} - 341736499 \beta_{3} + 275691643 \beta_{4} - 275691643 \beta_{5} + 246620943020 \beta_{6} + 17375718692 \beta_{7} + 9282222195 \beta_{8} + 1757663086 \beta_{9} - 9185154509 \beta_{10} + 1660595400 \beta_{11} + 45786393 \beta_{12} + 97067686 \beta_{13} + 73776456 \beta_{14} + 73776456 \beta_{15} - 260180228 \beta_{16} - 4747048 \beta_{17} - 81556271 \beta_{19} ) q^{79} + ( 9081412419472140 + 64653360 \beta_{1} - 9081412519442460 \beta_{2} + 17093963298354 \beta_{3} - 558839957044 \beta_{4} + 24458161140 \beta_{5} + 99970320 \beta_{6} - 24358190820 \beta_{7} - 77650080 \beta_{8} - 266865470 \beta_{9} - 9386432848 \beta_{10} + 35316960 \beta_{11} + 16339680 \beta_{12} + 99970320 \beta_{13} - 127621430 \beta_{14} + 377561040 \beta_{16} - 12996720 \beta_{17} - 12996720 \beta_{18} + 64653360 \beta_{19} ) q^{80} + ( -6915949027935505 - 92343600 \beta_{1} + 6915948879754225 \beta_{2} - 8612806729579 \beta_{3} - 695361564827 \beta_{4} - 16397681648 \beta_{5} + 148181280 \beta_{6} + 16545862928 \beta_{7} + 35571197 \beta_{8} - 2843334023 \beta_{9} + 3911842191 \beta_{10} + 240524880 \beta_{11} - 389640883 \beta_{12} + 148181280 \beta_{13} - 38400360 \beta_{14} + 408972643 \beta_{16} - 56772403 \beta_{17} - 56772403 \beta_{18} - 92343600 \beta_{19} ) q^{81} + ( -252364704 + 8404819406406 \beta_{1} + 2124722716619652 \beta_{2} - 204017087 \beta_{3} + 390405789 \beta_{4} - 390405789 \beta_{5} - 683205223104 \beta_{6} - 41285028510 \beta_{7} - 1235848999 \beta_{8} + 2445376588 \beta_{9} + 1429289055 \beta_{10} + 2251936532 \beta_{11} - 438753406 \beta_{12} + 193440056 \beta_{13} + 28028553 \beta_{14} + 28028553 \beta_{15} - 200491410 \beta_{16} - 62450325 \beta_{17} - 3525677 \beta_{19} ) q^{82} + ( 531153039499490 - 3975681261990 \beta_{1} - 4389418 \beta_{2} - 3975948416892 \beta_{3} + 143102333548 \beta_{4} + 74992222438 \beta_{5} - 143211966036 \beta_{6} - 109632488 \beta_{7} - 736189164 \beta_{8} - 219264976 \beta_{9} + 105243070 \beta_{10} - 1496563844 \beta_{11} + 324508046 \beta_{12} + 109632488 \beta_{13} - 155501720 \beta_{15} - 267154902 \beta_{16} + 61058180 \beta_{18} - 4389418 \beta_{19} ) q^{83} + ( -4213848327035228 + 2431351100031 \beta_{1} - 9455590268345270 \beta_{2} + 30669617765094 \beta_{3} + 774501396298 \beta_{4} - 6317116750 \beta_{5} + 298843453125 \beta_{6} - 90418283179 \beta_{7} + 2894918236 \beta_{8} + 130155718 \beta_{9} + 12124531412 \beta_{10} - 2521577674 \beta_{11} - 3756200 \beta_{12} + 171246768 \beta_{13} - 78759366 \beta_{14} - 96041862 \beta_{15} - 14565992 \beta_{16} - 3559556 \beta_{17} + 11495792 \beta_{18} - 472164 \beta_{19} ) q^{84} + ( 2588474548505897 - 18747183726476 \beta_{1} - 98305338 \beta_{2} - 18747117502191 \beta_{3} + 999326696105 \beta_{4} - 35953709178 \beta_{5} - 999491581893 \beta_{6} - 164885788 \beta_{7} - 13883135477 \beta_{8} - 329771576 \beta_{9} + 66580450 \beta_{10} - 1289393697 \beta_{11} + 396352026 \beta_{12} + 164885788 \beta_{13} + 205523916 \beta_{15} + 66224285 \beta_{16} - 101079847 \beta_{18} - 98305338 \beta_{19} ) q^{85} + ( -18837248 - 4101545522540 \beta_{1} + 11395644217801476 \beta_{2} + 146204476 \beta_{3} + 12790268 \beta_{4} - 12790268 \beta_{5} + 948438873552 \beta_{6} + 73307516056 \beta_{7} - 1642742764 \beta_{8} - 1151217640 \beta_{9} + 1679657820 \beta_{10} - 1188132696 \beta_{11} - 177831992 \beta_{12} + 36915056 \beta_{13} + 92895036 \beta_{14} + 92895036 \beta_{15} + 85164632 \beta_{16} + 79117652 \beta_{17} + 61039844 \beta_{19} ) q^{86} + ( 17155702044955448 - 14685529 \beta_{1} - 17155702115646229 \beta_{2} + 36315322136561 \beta_{3} - 501729690088 \beta_{4} - 86527572907 \beta_{5} + 70690781 \beta_{6} + 86598263688 \beta_{7} + 127923769 \beta_{8} + 2466529563 \beta_{9} + 12623370221 \beta_{10} + 85376310 \beta_{11} + 13176401 \beta_{12} + 70690781 \beta_{13} + 16590912 \beta_{14} + 84148574 \beta_{16} + 113238240 \beta_{17} + 113238240 \beta_{18} - 14685529 \beta_{19} ) q^{87} + ( -11772613067175250 - 60225136 \beta_{1} + 11772613031327730 \beta_{2} - 42984896444757 \beta_{3} + 258897264120 \beta_{4} - 102755721580 \beta_{5} + 35847520 \beta_{6} + 102791569100 \beta_{7} + 112349136 \beta_{8} + 2532863973 \beta_{9} + 3377866964 \beta_{10} + 96072656 \beta_{11} - 104173792 \beta_{12} + 35847520 \beta_{13} + 292609677 \beta_{14} - 4806576 \beta_{16} + 52124000 \beta_{17} + 52124000 \beta_{18} - 60225136 \beta_{19} ) q^{88} + ( -20301798 + 51912552624236 \beta_{1} + 3111518963792951 \beta_{2} + 610767636 \beta_{3} - 441542352 \beta_{4} + 441542352 \beta_{5} - 328224811374 \beta_{6} - 58386055680 \beta_{7} - 5211292496 \beta_{8} - 1973625238 \beta_{9} + 5068520612 \beta_{10} - 1830853354 \beta_{11} - 189527082 \beta_{12} - 142771884 \beta_{13} - 179145784 \beta_{14} - 179145784 \beta_{15} + 454769052 \beta_{16} - 7075098 \beta_{17} + 155998584 \beta_{19} ) q^{89} + ( -25151888409094992 - 24067693621958 \beta_{1} - 130972047 \beta_{2} - 24066969779398 \beta_{3} - 1230587018366 \beta_{4} - 30144378227 \beta_{5} + 1230682007879 \beta_{6} + 94989513 \beta_{7} - 3774380590 \beta_{8} + 189979026 \beta_{9} - 225961560 \beta_{10} + 2054225207 \beta_{11} - 415940586 \beta_{12} - 94989513 \beta_{13} - 71158881 \beta_{15} + 723842560 \beta_{16} - 140947393 \beta_{18} - 130972047 \beta_{19} ) q^{90} + ( -4989456647709331 + 23790276828367 \beta_{1} - 12001043948454869 \beta_{2} + 50079852840206 \beta_{3} + 491212303932 \beta_{4} + 47343609551 \beta_{5} + 365994142206 \beta_{6} + 52891978202 \beta_{7} + 1976528862 \beta_{8} + 138226788 \beta_{9} - 22582341439 \beta_{10} + 3345465144 \beta_{11} + 39514881 \beta_{12} - 137704014 \beta_{13} - 10752168 \beta_{14} - 112497924 \beta_{15} - 65957017 \beta_{16} + 25569572 \beta_{17} + 47992658 \beta_{18} + 49371763 \beta_{19} ) q^{91} + ( 10425084612498284 - 93324772638060 \beta_{1} + 95565426 \beta_{2} - 93324172924980 \beta_{3} + 717417409659 \beta_{4} + 59852250357 \beta_{5} - 716957348025 \beta_{6} + 460061634 \beta_{7} + 26296894840 \beta_{8} + 920123268 \beta_{9} - 364496208 \beta_{10} + 3337297205 \beta_{11} - 1284619476 \beta_{12} - 460061634 \beta_{13} - 360232627 \beta_{15} + 599713080 \beta_{16} + 33713910 \beta_{18} + 95565426 \beta_{19} ) q^{92} + ( -66789399 - 58757571798126 \beta_{1} + 25733231744188194 \beta_{2} + 601927022 \beta_{3} - 592252124 \beta_{4} + 592252124 \beta_{5} + 527304139605 \beta_{6} - 159154268208 \beta_{7} - 3511132592 \beta_{8} - 478519707 \beta_{9} + 3276166722 \beta_{10} - 243553837 \beta_{11} - 76464297 \beta_{12} - 234965870 \beta_{13} - 324600828 \beta_{14} - 324600828 \beta_{15} + 479606638 \beta_{16} - 179434885 \beta_{17} + 122320384 \beta_{19} ) q^{93} + ( 40540008167064286 - 284378752 \beta_{1} - 40540007840781011 \beta_{2} + 42783784646254 \beta_{3} - 2042470459472 \beta_{4} + 181618550141 \beta_{5} - 326283275 \beta_{6} - 181944833416 \beta_{7} + 139890125 \beta_{8} - 4674640161 \beta_{9} - 18495065564 \beta_{10} - 41904523 \beta_{11} - 386962856 \beta_{12} - 326283275 \beta_{13} + 210302097 \beta_{14} - 1118739950 \beta_{16} - 144488627 \beta_{17} - 144488627 \beta_{18} - 284378752 \beta_{19} ) q^{94} + ( -2243106149018097 + 549373220 \beta_{1} + 2243106801591430 \beta_{2} - 11786549317584 \beta_{3} + 91451825879 \beta_{4} + 169363938070 \beta_{5} - 652573333 \beta_{6} - 170016511403 \beta_{7} - 480682888 \beta_{8} + 2817588646 \beta_{9} - 15753779883 \beta_{10} - 1201946553 \beta_{11} + 1820010105 \beta_{12} - 652573333 \beta_{13} - 493303264 \beta_{14} - 1477037111 \beta_{16} + 68690332 \beta_{17} + 68690332 \beta_{18} + 549373220 \beta_{19} ) q^{95} + ( 990589376 + 194350408198632 \beta_{1} + 23262047290820960 \beta_{2} - 491020112 \beta_{3} - 638781616 \beta_{4} + 638781616 \beta_{5} - 162113431912 \beta_{6} + 241854885992 \beta_{7} + 25202711264 \beta_{8} - 5342930096 \beta_{9} - 25684184256 \beta_{10} - 4861457104 \beta_{11} + 2120391104 \beta_{12} - 481472992 \beta_{13} + 547658928 \beta_{14} + 547658928 \beta_{15} - 166855744 \beta_{16} + 184952016 \beta_{17} - 324164368 \beta_{19} ) q^{96} + ( -35140025264258663 + 34159145776958 \beta_{1} + 246852864 \beta_{2} + 34158930731561 \beta_{3} + 13409568169 \beta_{4} - 48932660720 \beta_{5} - 13103860329 \beta_{6} + 305707840 \beta_{7} + 38300838520 \beta_{8} + 611415680 \beta_{9} - 58854976 \beta_{10} - 3291899001 \beta_{11} - 670270656 \beta_{12} - 305707840 \beta_{13} + 922610544 \beta_{15} - 215045397 \beta_{16} + 85902485 \beta_{18} + 246852864 \beta_{19} ) q^{97} + ( -36803619012087004 + 81073421896397 \beta_{1} - 2440087700755369 \beta_{2} + 107691002130341 \beta_{3} - 2588956919507 \beta_{4} - 133282329747 \beta_{5} + 2261438545282 \beta_{6} + 210756809592 \beta_{7} - 49815784151 \beta_{8} + 3390090816 \beta_{9} + 17502190867 \beta_{10} + 6943946646 \beta_{11} - 130927650 \beta_{12} - 784654542 \beta_{13} + 220532781 \beta_{14} + 768968319 \beta_{15} + 174122886 \beta_{16} - 96625305 \beta_{17} - 143067750 \beta_{18} - 8656683 \beta_{19} ) q^{98} + ( 46414023012808905 - 112739776855453 \beta_{1} + 338711415 \beta_{2} - 112740775759412 \beta_{3} - 408287747134 \beta_{4} + 26523716435 \beta_{5} + 408457389292 \beta_{6} + 169642158 \beta_{7} - 40525228676 \beta_{8} + 339284316 \beta_{9} + 169069257 \beta_{10} + 5661938020 \beta_{11} - 170215059 \beta_{12} - 169642158 \beta_{13} + 113235132 \beta_{15} - 998903959 \beta_{16} + 322054030 \beta_{18} + 338711415 \beta_{19} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$20q + 270q^{2} + 6560q^{3} - 551660q^{4} + 1089798q^{5} - 1687556q^{6} + 14395360q^{7} - 237823200q^{8} - 498883228q^{9} + O(q^{10})$$ $$20q + 270q^{2} + 6560q^{3} - 551660q^{4} + 1089798q^{5} - 1687556q^{6} + 14395360q^{7} - 237823200q^{8} - 498883228q^{9} + 222553738q^{10} - 148256184q^{11} + 2808533140q^{12} + 3578734040q^{13} + 25683984606q^{14} - 26900793736q^{15} - 50209759472q^{16} + 15259176570q^{17} + 81764237020q^{18} - 71656970872q^{19} - 180641544216q^{20} - 669146889614q^{21} + 1499692748380q^{22} - 316306503180q^{23} + 694489934496q^{24} + 591325905568q^{25} + 4488505674324q^{26} - 11473086373360q^{27} + 2858273469380q^{28} + 284876854680q^{29} - 9357829097258q^{30} - 1607821082076q^{31} + 18893460602400q^{32} + 9589772011550q^{33} - 32628298247940q^{34} - 103506510744q^{35} + 76724663114416q^{36} + 5528585266950q^{37} - 15690283928010q^{38} + 28451541414128q^{39} - 91022360866896q^{40} - 66727631114424q^{41} - 74393390286880q^{42} - 93028396916240q^{43} + 162849953080692q^{44} + 8592264491540q^{45} - 563088234018354q^{46} + 162460135616460q^{47} + 1626223055746720q^{48} + 37188390461684q^{49} - 156105592028976q^{50} - 41571323179104q^{51} - 664578572037800q^{52} + 349838159677350q^{53} - 2197621449925058q^{54} - 630210277345912q^{55} - 3391192609585536q^{56} + 3387478208982380q^{57} + 1117009291618900q^{58} + 2898325666051656q^{59} + 1851302466771436q^{60} + 4829263771794542q^{61} - 12440234322280740q^{62} + 1252577979478000q^{63} + 2254847316158720q^{64} + 2323606775562324q^{65} + 7963993945620406q^{66} + 1327703510619200q^{67} + 570001815036780q^{68} - 19909363410939516q^{69} - 17777971289904490q^{70} + 15480661755189216q^{71} + 16233423509033760q^{72} - 3163858704730590q^{73} - 4018632851551110q^{74} - 8119026314209112q^{75} - 14641245177256376q^{76} - 46825571816043510q^{77} + 48702643123708840q^{78} + 40826108816616316q^{79} + 90814104963906864q^{80} - 69159481630723930q^{81} + 21247223257167740q^{82} + 10623055038659280q^{83} - 178832833754555092q^{84} + 51769432859611308q^{85} + 113956439761484328q^{86} + 171557044814594000q^{87} - 117726124940340000q^{88} + 31115179936936434q^{89} - 503037784464315176q^{90} - 219799610507722112q^{91} + 208501803198310920q^{92} + 257332309943680890q^{93} + 405400048855155726q^{94} - 22431094320108540q^{95} + 232620528206160416q^{96} - 702800336503765080q^{97} - 760473415304075970q^{98} + 928280307128172080q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{20} - 5 x^{19} + 231900 x^{18} + 2573595 x^{17} + 36222331443 x^{16} + 522056283042 x^{15} + 3095689791534860 x^{14} + 63517964980290950 x^{13} + 191169147185579182773 x^{12} + 4661871029431062841865 x^{11} + 6728981840934861993437384 x^{10} + 251477264592720071695768485 x^{9} + 169785400547236219777139622573 x^{8} + 7884822396560931315753101896470 x^{7} + 2084722393634413874384133666310116 x^{6} + 170299785457406646702831504431152090 x^{5} + 19136307004107023234373216890155693651 x^{4} + 919913048844099315146584754515161210807 x^{3} + 39214983331357496369283116903046958144404 x^{2} + 515992296986415442815810669806567791765095 x + 5495215718943433371703477919782244002731225$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$28\!\cdots\!94$$$$\nu^{19} -$$$$45\!\cdots\!81$$$$\nu^{18} +$$$$65\!\cdots\!26$$$$\nu^{17} -$$$$47\!\cdots\!47$$$$\nu^{16} +$$$$10\!\cdots\!94$$$$\nu^{15} +$$$$33\!\cdots\!30$$$$\nu^{14} +$$$$88\!\cdots\!94$$$$\nu^{13} +$$$$83\!\cdots\!12$$$$\nu^{12} +$$$$54\!\cdots\!74$$$$\nu^{11} +$$$$72\!\cdots\!31$$$$\nu^{10} +$$$$19\!\cdots\!52$$$$\nu^{9} +$$$$50\!\cdots\!83$$$$\nu^{8} +$$$$48\!\cdots\!40$$$$\nu^{7} +$$$$17\!\cdots\!08$$$$\nu^{6} +$$$$59\!\cdots\!94$$$$\nu^{5} +$$$$42\!\cdots\!50$$$$\nu^{4} +$$$$52\!\cdots\!28$$$$\nu^{3} +$$$$21\!\cdots\!41$$$$\nu^{2} +$$$$24\!\cdots\!26$$$$\nu +$$$$29\!\cdots\!75$$$$)/$$$$11\!\cdots\!80$$ $$\beta_{2}$$ $$=$$ $$($$$$-$$$$28\!\cdots\!94$$$$\nu^{19} +$$$$45\!\cdots\!81$$$$\nu^{18} -$$$$65\!\cdots\!26$$$$\nu^{17} +$$$$47\!\cdots\!47$$$$\nu^{16} -$$$$10\!\cdots\!94$$$$\nu^{15} -$$$$33\!\cdots\!30$$$$\nu^{14} -$$$$88\!\cdots\!94$$$$\nu^{13} -$$$$83\!\cdots\!12$$$$\nu^{12} -$$$$54\!\cdots\!74$$$$\nu^{11} -$$$$72\!\cdots\!31$$$$\nu^{10} -$$$$19\!\cdots\!52$$$$\nu^{9} -$$$$50\!\cdots\!83$$$$\nu^{8} -$$$$48\!\cdots\!40$$$$\nu^{7} -$$$$17\!\cdots\!08$$$$\nu^{6} -$$$$59\!\cdots\!94$$$$\nu^{5} -$$$$42\!\cdots\!50$$$$\nu^{4} -$$$$52\!\cdots\!28$$$$\nu^{3} -$$$$21\!\cdots\!41$$$$\nu^{2} -$$$$11\!\cdots\!66$$$$\nu -$$$$29\!\cdots\!75$$$$)/$$$$11\!\cdots\!80$$ $$\beta_{3}$$ $$=$$ $$($$$$-$$$$66\!\cdots\!16$$$$\nu^{19} +$$$$36\!\cdots\!33$$$$\nu^{18} -$$$$15\!\cdots\!80$$$$\nu^{17} +$$$$60\!\cdots\!51$$$$\nu^{16} -$$$$23\!\cdots\!30$$$$\nu^{15} +$$$$87\!\cdots\!78$$$$\nu^{14} -$$$$20\!\cdots\!70$$$$\nu^{13} +$$$$62\!\cdots\!12$$$$\nu^{12} -$$$$12\!\cdots\!32$$$$\nu^{11} +$$$$33\!\cdots\!81$$$$\nu^{10} -$$$$42\!\cdots\!66$$$$\nu^{9} +$$$$61\!\cdots\!73$$$$\nu^{8} -$$$$10\!\cdots\!84$$$$\nu^{7} +$$$$55\!\cdots\!72$$$$\nu^{6} -$$$$11\!\cdots\!14$$$$\nu^{5} -$$$$41\!\cdots\!82$$$$\nu^{4} -$$$$92\!\cdots\!62$$$$\nu^{3} +$$$$57\!\cdots\!39$$$$\nu^{2} -$$$$24\!\cdots\!76$$$$\nu -$$$$32\!\cdots\!55$$$$)/$$$$11\!\cdots\!80$$ $$\beta_{4}$$ $$=$$ $$($$$$48\!\cdots\!33$$$$\nu^{19} -$$$$29\!\cdots\!47$$$$\nu^{18} +$$$$14\!\cdots\!83$$$$\nu^{17} -$$$$68\!\cdots\!45$$$$\nu^{16} +$$$$23\!\cdots\!50$$$$\nu^{15} -$$$$10\!\cdots\!82$$$$\nu^{14} +$$$$23\!\cdots\!86$$$$\nu^{13} -$$$$89\!\cdots\!50$$$$\nu^{12} +$$$$15\!\cdots\!51$$$$\nu^{11} -$$$$54\!\cdots\!05$$$$\nu^{10} +$$$$67\!\cdots\!13$$$$\nu^{9} -$$$$18\!\cdots\!35$$$$\nu^{8} +$$$$17\!\cdots\!82$$$$\nu^{7} -$$$$42\!\cdots\!42$$$$\nu^{6} +$$$$25\!\cdots\!10$$$$\nu^{5} -$$$$38\!\cdots\!10$$$$\nu^{4} -$$$$17\!\cdots\!79$$$$\nu^{3} -$$$$11\!\cdots\!27$$$$\nu^{2} +$$$$47\!\cdots\!59$$$$\nu +$$$$67\!\cdots\!95$$$$)/$$$$13\!\cdots\!60$$ $$\beta_{5}$$ $$=$$ $$($$$$-$$$$75\!\cdots\!30$$$$\nu^{19} +$$$$10\!\cdots\!37$$$$\nu^{18} -$$$$17\!\cdots\!12$$$$\nu^{17} +$$$$20\!\cdots\!89$$$$\nu^{16} -$$$$27\!\cdots\!64$$$$\nu^{15} +$$$$30\!\cdots\!86$$$$\nu^{14} -$$$$23\!\cdots\!68$$$$\nu^{13} +$$$$23\!\cdots\!46$$$$\nu^{12} -$$$$14\!\cdots\!94$$$$\nu^{11} +$$$$13\!\cdots\!43$$$$\nu^{10} -$$$$48\!\cdots\!52$$$$\nu^{9} +$$$$38\!\cdots\!39$$$$\nu^{8} -$$$$11\!\cdots\!16$$$$\nu^{7} +$$$$74\!\cdots\!14$$$$\nu^{6} -$$$$10\!\cdots\!00$$$$\nu^{5} -$$$$80\!\cdots\!30$$$$\nu^{4} -$$$$48\!\cdots\!78$$$$\nu^{3} +$$$$11\!\cdots\!69$$$$\nu^{2} +$$$$16\!\cdots\!20$$$$\nu -$$$$65\!\cdots\!15$$$$)/$$$$34\!\cdots\!60$$ $$\beta_{6}$$ $$=$$ $$($$$$-$$$$50\!\cdots\!89$$$$\nu^{19} +$$$$16\!\cdots\!36$$$$\nu^{18} -$$$$11\!\cdots\!53$$$$\nu^{17} +$$$$16\!\cdots\!86$$$$\nu^{16} -$$$$17\!\cdots\!06$$$$\nu^{15} +$$$$19\!\cdots\!92$$$$\nu^{14} -$$$$14\!\cdots\!18$$$$\nu^{13} +$$$$33\!\cdots\!76$$$$\nu^{12} -$$$$86\!\cdots\!03$$$$\nu^{11} -$$$$29\!\cdots\!12$$$$\nu^{10} -$$$$27\!\cdots\!59$$$$\nu^{9} -$$$$68\!\cdots\!46$$$$\nu^{8} -$$$$63\!\cdots\!66$$$$\nu^{7} -$$$$25\!\cdots\!92$$$$\nu^{6} -$$$$50\!\cdots\!94$$$$\nu^{5} -$$$$73\!\cdots\!08$$$$\nu^{4} -$$$$29\!\cdots\!69$$$$\nu^{3} -$$$$10\!\cdots\!00$$$$\nu^{2} +$$$$77\!\cdots\!83$$$$\nu -$$$$13\!\cdots\!50$$$$)/$$$$19\!\cdots\!20$$ $$\beta_{7}$$ $$=$$ $$($$$$75\!\cdots\!51$$$$\nu^{19} -$$$$11\!\cdots\!56$$$$\nu^{18} +$$$$17\!\cdots\!23$$$$\nu^{17} +$$$$18\!\cdots\!50$$$$\nu^{16} +$$$$27\!\cdots\!78$$$$\nu^{15} +$$$$12\!\cdots\!36$$$$\nu^{14} +$$$$23\!\cdots\!34$$$$\nu^{13} +$$$$24\!\cdots\!24$$$$\nu^{12} +$$$$14\!\cdots\!85$$$$\nu^{11} +$$$$20\!\cdots\!32$$$$\nu^{10} +$$$$52\!\cdots\!65$$$$\nu^{9} +$$$$13\!\cdots\!06$$$$\nu^{8} +$$$$13\!\cdots\!26$$$$\nu^{7} +$$$$47\!\cdots\!32$$$$\nu^{6} +$$$$16\!\cdots\!58$$$$\nu^{5} +$$$$11\!\cdots\!24$$$$\nu^{4} +$$$$14\!\cdots\!63$$$$\nu^{3} +$$$$60\!\cdots\!28$$$$\nu^{2} +$$$$32\!\cdots\!79$$$$\nu +$$$$83\!\cdots\!50$$$$)/$$$$17\!\cdots\!80$$ $$\beta_{8}$$ $$=$$ $$($$$$72\!\cdots\!58$$$$\nu^{19} -$$$$56\!\cdots\!67$$$$\nu^{18} +$$$$16\!\cdots\!16$$$$\nu^{17} -$$$$10\!\cdots\!43$$$$\nu^{16} +$$$$26\!\cdots\!56$$$$\nu^{15} -$$$$15\!\cdots\!78$$$$\nu^{14} +$$$$22\!\cdots\!80$$$$\nu^{13} -$$$$11\!\cdots\!62$$$$\nu^{12} +$$$$13\!\cdots\!90$$$$\nu^{11} -$$$$64\!\cdots\!01$$$$\nu^{10} +$$$$47\!\cdots\!00$$$$\nu^{9} -$$$$15\!\cdots\!73$$$$\nu^{8} +$$$$11\!\cdots\!68$$$$\nu^{7} -$$$$24\!\cdots\!06$$$$\nu^{6} +$$$$12\!\cdots\!36$$$$\nu^{5} +$$$$34\!\cdots\!70$$$$\nu^{4} +$$$$59\!\cdots\!66$$$$\nu^{3} -$$$$52\!\cdots\!99$$$$\nu^{2} -$$$$80\!\cdots\!20$$$$\nu -$$$$32\!\cdots\!55$$$$)/$$$$19\!\cdots\!20$$ $$\beta_{9}$$ $$=$$ $$($$$$10\!\cdots\!67$$$$\nu^{19} +$$$$12\!\cdots\!75$$$$\nu^{18} +$$$$22\!\cdots\!85$$$$\nu^{17} +$$$$32\!\cdots\!33$$$$\nu^{16} +$$$$34\!\cdots\!14$$$$\nu^{15} +$$$$51\!\cdots\!46$$$$\nu^{14} +$$$$28\!\cdots\!38$$$$\nu^{13} +$$$$45\!\cdots\!30$$$$\nu^{12} +$$$$16\!\cdots\!09$$$$\nu^{11} +$$$$28\!\cdots\!45$$$$\nu^{10} +$$$$53\!\cdots\!27$$$$\nu^{9} +$$$$10\!\cdots\!55$$$$\nu^{8} +$$$$13\!\cdots\!78$$$$\nu^{7} +$$$$27\!\cdots\!70$$$$\nu^{6} +$$$$14\!\cdots\!38$$$$\nu^{5} +$$$$37\!\cdots\!98$$$$\nu^{4} +$$$$23\!\cdots\!43$$$$\nu^{3} +$$$$16\!\cdots\!31$$$$\nu^{2} +$$$$27\!\cdots\!85$$$$\nu +$$$$33\!\cdots\!25$$$$)/$$$$22\!\cdots\!60$$ $$\beta_{10}$$ $$=$$ $$($$$$-$$$$15\!\cdots\!55$$$$\nu^{19} -$$$$40\!\cdots\!55$$$$\nu^{18} -$$$$35\!\cdots\!53$$$$\nu^{17} -$$$$15\!\cdots\!89$$$$\nu^{16} -$$$$55\!\cdots\!18$$$$\nu^{15} -$$$$25\!\cdots\!22$$$$\nu^{14} -$$$$46\!\cdots\!66$$$$\nu^{13} -$$$$24\!\cdots\!58$$$$\nu^{12} -$$$$28\!\cdots\!61$$$$\nu^{11} -$$$$16\!\cdots\!89$$$$\nu^{10} -$$$$99\!\cdots\!23$$$$\nu^{9} -$$$$70\!\cdots\!47$$$$\nu^{8} -$$$$25\!\cdots\!74$$$$\nu^{7} -$$$$19\!\cdots\!86$$$$\nu^{6} -$$$$30\!\cdots\!70$$$$\nu^{5} -$$$$34\!\cdots\!22$$$$\nu^{4} -$$$$31\!\cdots\!99$$$$\nu^{3} -$$$$17\!\cdots\!51$$$$\nu^{2} -$$$$56\!\cdots\!49$$$$\nu -$$$$73\!\cdots\!45$$$$)/$$$$29\!\cdots\!80$$ $$\beta_{11}$$ $$=$$ $$($$$$13\!\cdots\!18$$$$\nu^{19} -$$$$12\!\cdots\!91$$$$\nu^{18} +$$$$31\!\cdots\!28$$$$\nu^{17} -$$$$22\!\cdots\!63$$$$\nu^{16} +$$$$48\!\cdots\!36$$$$\nu^{15} -$$$$33\!\cdots\!82$$$$\nu^{14} +$$$$41\!\cdots\!96$$$$\nu^{13} -$$$$25\!\cdots\!46$$$$\nu^{12} +$$$$25\!\cdots\!18$$$$\nu^{11} -$$$$14\!\cdots\!73$$$$\nu^{10} +$$$$88\!\cdots\!84$$$$\nu^{9} -$$$$36\!\cdots\!69$$$$\nu^{8} +$$$$20\!\cdots\!20$$$$\nu^{7} -$$$$63\!\cdots\!38$$$$\nu^{6} +$$$$21\!\cdots\!24$$$$\nu^{5} +$$$$54\!\cdots\!86$$$$\nu^{4} +$$$$85\!\cdots\!54$$$$\nu^{3} -$$$$11\!\cdots\!55$$$$\nu^{2} -$$$$17\!\cdots\!00$$$$\nu -$$$$86\!\cdots\!75$$$$)/$$$$14\!\cdots\!40$$ $$\beta_{12}$$ $$=$$ $$($$$$17\!\cdots\!59$$$$\nu^{19} +$$$$34\!\cdots\!22$$$$\nu^{18} +$$$$36\!\cdots\!71$$$$\nu^{17} +$$$$88\!\cdots\!20$$$$\nu^{16} +$$$$55\!\cdots\!78$$$$\nu^{15} +$$$$13\!\cdots\!44$$$$\nu^{14} +$$$$43\!\cdots\!90$$$$\nu^{13} +$$$$12\!\cdots\!84$$$$\nu^{12} +$$$$24\!\cdots\!81$$$$\nu^{11} +$$$$77\!\cdots\!62$$$$\nu^{10} +$$$$70\!\cdots\!13$$$$\nu^{9} +$$$$29\!\cdots\!16$$$$\nu^{8} +$$$$16\!\cdots\!18$$$$\nu^{7} +$$$$73\!\cdots\!80$$$$\nu^{6} +$$$$11\!\cdots\!98$$$$\nu^{5} +$$$$96\!\cdots\!44$$$$\nu^{4} +$$$$36\!\cdots\!39$$$$\nu^{3} +$$$$47\!\cdots\!86$$$$\nu^{2} +$$$$28\!\cdots\!83$$$$\nu +$$$$33\!\cdots\!60$$$$)/$$$$17\!\cdots\!80$$ $$\beta_{13}$$ $$=$$ $$($$$$-$$$$49\!\cdots\!03$$$$\nu^{19} +$$$$10\!\cdots\!49$$$$\nu^{18} -$$$$11\!\cdots\!61$$$$\nu^{17} +$$$$22\!\cdots\!79$$$$\nu^{16} -$$$$16\!\cdots\!42$$$$\nu^{15} +$$$$34\!\cdots\!66$$$$\nu^{14} -$$$$13\!\cdots\!38$$$$\nu^{13} +$$$$28\!\cdots\!10$$$$\nu^{12} -$$$$77\!\cdots\!29$$$$\nu^{11} +$$$$16\!\cdots\!35$$$$\nu^{10} -$$$$22\!\cdots\!27$$$$\nu^{9} +$$$$52\!\cdots\!45$$$$\nu^{8} -$$$$37\!\cdots\!14$$$$\nu^{7} +$$$$12\!\cdots\!54$$$$\nu^{6} +$$$$25\!\cdots\!86$$$$\nu^{5} +$$$$95\!\cdots\!14$$$$\nu^{4} +$$$$12\!\cdots\!89$$$$\nu^{3} +$$$$13\!\cdots\!01$$$$\nu^{2} +$$$$72\!\cdots\!15$$$$\nu +$$$$11\!\cdots\!15$$$$)/$$$$19\!\cdots\!20$$ $$\beta_{14}$$ $$=$$ $$($$$$18\!\cdots\!75$$$$\nu^{19} +$$$$14\!\cdots\!51$$$$\nu^{18} +$$$$41\!\cdots\!81$$$$\nu^{17} +$$$$40\!\cdots\!05$$$$\nu^{16} +$$$$64\!\cdots\!62$$$$\nu^{15} +$$$$64\!\cdots\!14$$$$\nu^{14} +$$$$54\!\cdots\!22$$$$\nu^{13} +$$$$58\!\cdots\!10$$$$\nu^{12} +$$$$33\!\cdots\!89$$$$\nu^{11} +$$$$37\!\cdots\!05$$$$\nu^{10} +$$$$11\!\cdots\!27$$$$\nu^{9} +$$$$14\!\cdots\!95$$$$\nu^{8} +$$$$28\!\cdots\!06$$$$\nu^{7} +$$$$38\!\cdots\!06$$$$\nu^{6} +$$$$35\!\cdots\!14$$$$\nu^{5} +$$$$57\!\cdots\!90$$$$\nu^{4} +$$$$43\!\cdots\!91$$$$\nu^{3} +$$$$31\!\cdots\!99$$$$\nu^{2} +$$$$63\!\cdots\!09$$$$\nu +$$$$80\!\cdots\!45$$$$)/$$$$44\!\cdots\!20$$ $$\beta_{15}$$ $$=$$ $$($$$$58\!\cdots\!62$$$$\nu^{19} -$$$$45\!\cdots\!09$$$$\nu^{18} +$$$$13\!\cdots\!28$$$$\nu^{17} -$$$$82\!\cdots\!61$$$$\nu^{16} +$$$$21\!\cdots\!48$$$$\nu^{15} -$$$$12\!\cdots\!82$$$$\nu^{14} +$$$$18\!\cdots\!04$$$$\nu^{13} -$$$$91\!\cdots\!38$$$$\nu^{12} +$$$$11\!\cdots\!82$$$$\nu^{11} -$$$$51\!\cdots\!19$$$$\nu^{10} +$$$$38\!\cdots\!96$$$$\nu^{9} -$$$$12\!\cdots\!27$$$$\nu^{8} +$$$$92\!\cdots\!80$$$$\nu^{7} -$$$$19\!\cdots\!10$$$$\nu^{6} +$$$$97\!\cdots\!72$$$$\nu^{5} +$$$$28\!\cdots\!46$$$$\nu^{4} +$$$$52\!\cdots\!90$$$$\nu^{3} -$$$$41\!\cdots\!01$$$$\nu^{2} -$$$$64\!\cdots\!80$$$$\nu -$$$$78\!\cdots\!65$$$$)/$$$$86\!\cdots\!40$$ $$\beta_{16}$$ $$=$$ $$($$$$48\!\cdots\!92$$$$\nu^{19} -$$$$77\!\cdots\!63$$$$\nu^{18} +$$$$11\!\cdots\!62$$$$\nu^{17} +$$$$62\!\cdots\!45$$$$\nu^{16} +$$$$17\!\cdots\!20$$$$\nu^{15} +$$$$69\!\cdots\!02$$$$\nu^{14} +$$$$14\!\cdots\!84$$$$\nu^{13} +$$$$15\!\cdots\!30$$$$\nu^{12} +$$$$90\!\cdots\!84$$$$\nu^{11} +$$$$13\!\cdots\!75$$$$\nu^{10} +$$$$31\!\cdots\!22$$$$\nu^{9} +$$$$92\!\cdots\!75$$$$\nu^{8} +$$$$77\!\cdots\!48$$$$\nu^{7} +$$$$31\!\cdots\!82$$$$\nu^{6} +$$$$89\!\cdots\!40$$$$\nu^{5} +$$$$75\!\cdots\!90$$$$\nu^{4} +$$$$75\!\cdots\!64$$$$\nu^{3} +$$$$34\!\cdots\!97$$$$\nu^{2} +$$$$10\!\cdots\!06$$$$\nu +$$$$15\!\cdots\!85$$$$)/$$$$59\!\cdots\!60$$ $$\beta_{17}$$ $$=$$ $$($$$$-$$$$89\!\cdots\!54$$$$\nu^{19} -$$$$48\!\cdots\!47$$$$\nu^{18} -$$$$21\!\cdots\!76$$$$\nu^{17} -$$$$13\!\cdots\!71$$$$\nu^{16} -$$$$35\!\cdots\!64$$$$\nu^{15} -$$$$21\!\cdots\!18$$$$\nu^{14} -$$$$32\!\cdots\!96$$$$\nu^{13} -$$$$17\!\cdots\!14$$$$\nu^{12} -$$$$21\!\cdots\!22$$$$\nu^{11} -$$$$11\!\cdots\!37$$$$\nu^{10} -$$$$84\!\cdots\!36$$$$\nu^{9} -$$$$39\!\cdots\!41$$$$\nu^{8} -$$$$24\!\cdots\!72$$$$\nu^{7} -$$$$11\!\cdots\!10$$$$\nu^{6} -$$$$40\!\cdots\!00$$$$\nu^{5} -$$$$16\!\cdots\!50$$$$\nu^{4} -$$$$43\!\cdots\!66$$$$\nu^{3} -$$$$20\!\cdots\!95$$$$\nu^{2} -$$$$12\!\cdots\!72$$$$\nu -$$$$41\!\cdots\!95$$$$)/$$$$89\!\cdots\!40$$ $$\beta_{18}$$ $$=$$ $$($$$$-$$$$26\!\cdots\!23$$$$\nu^{19} +$$$$56\!\cdots\!95$$$$\nu^{18} -$$$$63\!\cdots\!81$$$$\nu^{17} +$$$$11\!\cdots\!33$$$$\nu^{16} -$$$$98\!\cdots\!34$$$$\nu^{15} +$$$$17\!\cdots\!74$$$$\nu^{14} -$$$$84\!\cdots\!06$$$$\nu^{13} +$$$$14\!\cdots\!74$$$$\nu^{12} -$$$$51\!\cdots\!65$$$$\nu^{11} +$$$$84\!\cdots\!77$$$$\nu^{10} -$$$$17\!\cdots\!55$$$$\nu^{9} +$$$$25\!\cdots\!51$$$$\nu^{8} -$$$$38\!\cdots\!58$$$$\nu^{7} +$$$$53\!\cdots\!38$$$$\nu^{6} -$$$$29\!\cdots\!86$$$$\nu^{5} +$$$$22\!\cdots\!22$$$$\nu^{4} +$$$$24\!\cdots\!41$$$$\nu^{3} +$$$$19\!\cdots\!51$$$$\nu^{2} +$$$$49\!\cdots\!79$$$$\nu -$$$$28\!\cdots\!35$$$$)/$$$$23\!\cdots\!60$$ $$\beta_{19}$$ $$=$$ $$($$$$21\!\cdots\!25$$$$\nu^{19} +$$$$86\!\cdots\!66$$$$\nu^{18} +$$$$50\!\cdots\!61$$$$\nu^{17} +$$$$27\!\cdots\!76$$$$\nu^{16} +$$$$78\!\cdots\!18$$$$\nu^{15} +$$$$46\!\cdots\!08$$$$\nu^{14} +$$$$66\!\cdots\!18$$$$\nu^{13} +$$$$43\!\cdots\!80$$$$\nu^{12} +$$$$41\!\cdots\!15$$$$\nu^{11} +$$$$27\!\cdots\!90$$$$\nu^{10} +$$$$14\!\cdots\!55$$$$\nu^{9} +$$$$11\!\cdots\!20$$$$\nu^{8} +$$$$37\!\cdots\!90$$$$\nu^{7} +$$$$31\!\cdots\!96$$$$\nu^{6} +$$$$47\!\cdots\!26$$$$\nu^{5} +$$$$51\!\cdots\!16$$$$\nu^{4} +$$$$51\!\cdots\!73$$$$\nu^{3} +$$$$29\!\cdots\!98$$$$\nu^{2} +$$$$97\!\cdots\!73$$$$\nu +$$$$58\!\cdots\!20$$$$)/$$$$17\!\cdots\!80$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{2} + \beta_{1}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{7} - \beta_{5} - \beta_{4} - 23 \beta_{3} + 185510 \beta_{2} - 185510$$$$)/4$$ $$\nu^{3}$$ $$=$$ $$($$$$-\beta_{15} - \beta_{11} + 12 \beta_{8} - 514 \beta_{6} - 18 \beta_{5} + 514 \beta_{4} - 322863 \beta_{3} - 322863 \beta_{1} - 4479666$$$$)/8$$ $$\nu^{4}$$ $$=$$ $$($$$$-16 \beta_{19} - 8 \beta_{17} - 168 \beta_{16} - 47 \beta_{15} - 47 \beta_{14} + 136 \beta_{13} - 232 \beta_{12} - 187 \beta_{11} - 5264 \beta_{10} - 51 \beta_{9} + 5400 \beta_{8} - 218166 \beta_{7} - 20814 \beta_{6} - 288 \beta_{5} + 288 \beta_{4} - 184 \beta_{3} - 29927014606 \beta_{2} - 6920537 \beta_{1} - 128$$$$)/8$$ $$\nu^{5}$$ $$=$$ $$($$$$9360 \beta_{19} - 1380 \beta_{18} - 1380 \beta_{17} + 1464 \beta_{16} - 135173 \beta_{14} - 3092 \beta_{13} + 20432 \beta_{12} - 12452 \beta_{11} - 1779080 \beta_{10} - 366777 \beta_{9} - 10740 \beta_{8} - 7368256 \beta_{7} - 3092 \beta_{6} + 7365164 \beta_{5} - 92515436 \beta_{4} + 29927129977 \beta_{3} - 649774159922 \beta_{2} + 9360 \beta_{1} + 649774163014$$$$)/8$$ $$\nu^{6}$$ $$=$$ $$($$$$-4485289 \beta_{19} - 515163 \beta_{18} + 15507280 \beta_{16} + 2038510 \beta_{15} - 768125 \beta_{13} - 6789664 \beta_{12} + 20782375 \beta_{11} - 5253414 \beta_{10} + 1536250 \beta_{9} - 191709725 \beta_{8} + 768125 \beta_{7} + 7837407009 \beta_{6} + 5761220709 \beta_{5} - 7836638884 \beta_{4} + 282418857656 \beta_{3} - 4485289 \beta_{2} + 282403350376 \beta_{1} + 693023939838225$$$$)/2$$ $$\nu^{7}$$ $$=$$ $$($$$$-172966118 \beta_{19} + 63563632 \beta_{17} - 662251562 \beta_{16} + 3850412022 \beta_{15} + 3850412022 \beta_{14} + 316319326 \beta_{13} - 50176666 \beta_{12} + 14957623647 \beta_{11} + 59614645437 \beta_{10} + 15273942973 \beta_{9} - 59298326111 \beta_{8} + 319082825960 \beta_{7} + 3239752835610 \beta_{6} - 805604770 \beta_{5} + 805604770 \beta_{4} - 835217680 \beta_{3} + 26049040373554246 \beta_{2} + 738101371117808 \beta_{1} - 79789576$$$$)/2$$ $$\nu^{8}$$ $$=$$ $$($$$$1287763873980 \beta_{19} + 173173363242 \beta_{18} + 173173363242 \beta_{17} - 2109686320152 \beta_{16} + 588469246224 \beta_{14} - 1074758943630 \beta_{13} + 3823460054832 \beta_{12} - 2362522817610 \beta_{11} + 47710676778348 \beta_{10} + 1822189532952 \beta_{9} - 1114590510738 \beta_{8} + 1216834381503427 \beta_{7} - 1074758943630 \beta_{6} - 1217909140447057 \beta_{5} + 2935305531711569 \beta_{4} - 83815390907250665 \beta_{3} + 136659769673590360874 \beta_{2} + 1287763873980 \beta_{1} - 136659768598831417244$$$$)/4$$ $$\nu^{9}$$ $$=$$ $$($$$$-73584717689832 \beta_{19} + 37076844790440 \beta_{18} + 287758975043568 \beta_{16} - 1691663687899369 \beta_{15} - 52040833382256 \beta_{13} - 229707217836600 \beta_{12} - 8134193743218565 \beta_{11} - 125625551072088 \beta_{10} + 104081666764512 \beta_{9} + 30144546775302528 \beta_{8} + 52040833382256 \beta_{7} - 1653103957715219458 \beta_{6} - 184883175802868970 \beta_{5} + 1653155998548601714 \beta_{4} - 301634934028892453211 \beta_{3} - 73584717689832 \beta_{2} - 301635221787867496779 \beta_{1} - 15369670974306872418006$$$$)/8$$ $$\nu^{10}$$ $$=$$ $$($$$$-53067390382580596 \beta_{19} - 48169440311102756 \beta_{17} - 401093326387186440 \beta_{16} - 158630661024567719 \beta_{15} - 158630661024567719 \beta_{14} + 294958545622025248 \beta_{13} - 478884360407411464 \beta_{12} - 988135148959064587 \beta_{11} - 10802074859900969864 \beta_{10} - 693176603337039339 \beta_{9} + 11097033405522995112 \beta_{8} - 260173801050668226726 \beta_{7} - 836890826744790165918 \beta_{6} - 642984481626631092 \beta_{5} + 642984481626631092 \beta_{4} - 454160716769767036 \beta_{3} - 27910906513662072579932890 \beta_{2} - 23115649543422055422641 \beta_{1} - 290060595550547408$$$$)/8$$ $$\nu^{11}$$ $$=$$ $$($$$$10504246996500712584 \beta_{19} - 5007963213773696820 \beta_{18} - 5007963213773696820 \beta_{17} - 7614192863871943440 \beta_{16} - 184582394990392796693 \beta_{14} - 7708801024715450948 \beta_{13} + 23709331803943179296 \beta_{12} - 18213048021216163532 \beta_{11} - 3649811667432857192372 \beta_{10} - 1066757414018503309893 \beta_{9} - 15512210210274409404 \beta_{8} - 24904804061953596980248 \beta_{7} - 7708801024715450948 \beta_{6} + 24897095260928881529300 \beta_{5} - 202297585453588224249332 \beta_{4} + 31527178188331229102961541 \beta_{3} - 2115587918414612698145800934 \beta_{2} + 10504246996500712584 \beta_{1} + 2115587926123413722861251882$$$$)/8$$ $$\nu^{12}$$ $$=$$ $$-3327411068742246161018 \beta_{19} - 760682038714209234366 \beta_{18} + 12310270129972979568812 \beta_{16} + 2565485993785599203168 \beta_{15} - 783677442516015925696 \beta_{13} - 5678443396290293938106 \beta_{12} + 23651262367331385950771 \beta_{11} - 4111088511258262086714 \beta_{10} + 1567354885032031851392 \beta_{9} - 155878941701981419891729 \beta_{8} + 783677442516015925696 \beta_{7} + 13367959205239817389386072 \beta_{6} + 3517875550535480222244966 \beta_{5} - 13367175527797301373460376 \beta_{4} + 379822998943628113540964845 \beta_{3} - 3327411068742246161018 \beta_{2} + 379810688673498140561396033 \beta_{1} + 364516462931280780870480611490$$ $$\nu^{13}$$ $$=$$ $$($$$$-6297949603678264953638 \beta_{19} + 162932705582464343942854 \beta_{17} - 188126111189623937039360 \beta_{16} + 5041462053885839746053576 \beta_{15} + 5041462053885839746053576 \beta_{14} + 175530211982267407132084 \beta_{13} - 169233869571035675460400 \beta_{12} + 31426335220988125017554388 \beta_{11} + 108469539013219120833573036 \beta_{10} + 31601865432970392424686472 \beta_{9} - 108294008801236853426440952 \beta_{8} + 801780669935823436623460304 \beta_{7} + 6061233154370099665920821616 \beta_{6} - 357358373568213079217806 \beta_{5} + 357358373568213079217806 \beta_{4} - 194424060793302201992998 \beta_{3} + 69505000608134767965965403375943 \beta_{2} + 837292687048143280101666826673 \beta_{1} - 6299556796124798235592$$$$)/2$$ $$\nu^{14}$$ $$=$$ $$($$$$18\!\cdots\!72$$$$\beta_{19} +$$$$36\!\cdots\!08$$$$\beta_{18} +$$$$36\!\cdots\!08$$$$\beta_{17} -$$$$29\!\cdots\!88$$$$\beta_{16} +$$$$12\!\cdots\!44$$$$\beta_{14} -$$$$14\!\cdots\!84$$$$\beta_{13} +$$$$54\!\cdots\!36$$$$\beta_{12} -$$$$32\!\cdots\!56$$$$\beta_{11} +$$$$72\!\cdots\!76$$$$\beta_{10} +$$$$56\!\cdots\!00$$$$\beta_{9} -$$$$14\!\cdots\!64$$$$\beta_{8} +$$$$15\!\cdots\!21$$$$\beta_{7} -$$$$14\!\cdots\!84$$$$\beta_{6} -$$$$15\!\cdots\!05$$$$\beta_{5} +$$$$64\!\cdots\!99$$$$\beta_{4} -$$$$19\!\cdots\!11$$$$\beta_{3} +$$$$15\!\cdots\!26$$$$\beta_{2} +$$$$18\!\cdots\!72$$$$\beta_{1} -$$$$15\!\cdots\!42$$$$)/4$$ $$\nu^{15}$$ $$=$$ $$($$$$-$$$$45\!\cdots\!88$$$$\beta_{19} +$$$$82\!\cdots\!36$$$$\beta_{18} +$$$$87\!\cdots\!44$$$$\beta_{16} -$$$$22\!\cdots\!93$$$$\beta_{15} +$$$$23\!\cdots\!92$$$$\beta_{13} +$$$$23\!\cdots\!88$$$$\beta_{12} -$$$$14\!\cdots\!09$$$$\beta_{11} -$$$$22\!\cdots\!96$$$$\beta_{10} -$$$$46\!\cdots\!84$$$$\beta_{9} +$$$$50\!\cdots\!72$$$$\beta_{8} -$$$$23\!\cdots\!92$$$$\beta_{7} -$$$$28\!\cdots\!26$$$$\beta_{6} -$$$$40\!\cdots\!98$$$$\beta_{5} +$$$$28\!\cdots\!34$$$$\beta_{4} -$$$$36\!\cdots\!51$$$$\beta_{3} -$$$$45\!\cdots\!88$$$$\beta_{2} -$$$$36\!\cdots\!95$$$$\beta_{1} -$$$$35\!\cdots\!66$$$$)/8$$ $$\nu^{16}$$ $$=$$ $$($$$$-$$$$81\!\cdots\!00$$$$\beta_{19} -$$$$84\!\cdots\!44$$$$\beta_{17} -$$$$55\!\cdots\!32$$$$\beta_{16} -$$$$31\!\cdots\!67$$$$\beta_{15} -$$$$31\!\cdots\!67$$$$\beta_{14} +$$$$39\!\cdots\!32$$$$\beta_{13} -$$$$63\!\cdots\!08$$$$\beta_{12} -$$$$20\!\cdots\!67$$$$\beta_{11} -$$$$16\!\cdots\!48$$$$\beta_{10} -$$$$16\!\cdots\!35$$$$\beta_{9} +$$$$16\!\cdots\!80$$$$\beta_{8} -$$$$33\!\cdots\!62$$$$\beta_{7} -$$$$15\!\cdots\!18$$$$\beta_{6} -$$$$87\!\cdots\!64$$$$\beta_{5} +$$$$87\!\cdots\!64$$$$\beta_{4} -$$$$63\!\cdots\!32$$$$\beta_{3} -$$$$33\!\cdots\!14$$$$\beta_{2} -$$$$48\!\cdots\!21$$$$\beta_{1} -$$$$40\!\cdots\!76$$$$)/8$$ $$\nu^{17}$$ $$=$$ $$($$$$-$$$$36\!\cdots\!96$$$$\beta_{19} -$$$$10\!\cdots\!84$$$$\beta_{18} -$$$$10\!\cdots\!84$$$$\beta_{17} +$$$$30\!\cdots\!64$$$$\beta_{16} -$$$$24\!\cdots\!69$$$$\beta_{14} -$$$$12\!\cdots\!08$$$$\beta_{13} -$$$$16\!\cdots\!68$$$$\beta_{12} +$$$$24\!\cdots\!88$$$$\beta_{11} -$$$$58\!\cdots\!00$$$$\beta_{10} -$$$$16\!\cdots\!45$$$$\beta_{9} -$$$$66\!\cdots\!88$$$$\beta_{8} -$$$$49\!\cdots\!28$$$$\beta_{7} -$$$$12\!\cdots\!08$$$$\beta_{6} +$$$$49\!\cdots\!20$$$$\beta_{5} -$$$$33\!\cdots\!64$$$$\beta_{4} +$$$$39\!\cdots\!25$$$$\beta_{3} -$$$$44\!\cdots\!98$$$$\beta_{2} -$$$$36\!\cdots\!96$$$$\beta_{1} +$$$$44\!\cdots\!06$$$$)/8$$ $$\nu^{18}$$ $$=$$ $$($$$$-$$$$85\!\cdots\!41$$$$\beta_{19} -$$$$24\!\cdots\!55$$$$\beta_{18} +$$$$32\!\cdots\!32$$$$\beta_{16} +$$$$96\!\cdots\!46$$$$\beta_{15} -$$$$22\!\cdots\!77$$$$\beta_{13} -$$$$15\!\cdots\!72$$$$\beta_{12} +$$$$81\!\cdots\!17$$$$\beta_{11} -$$$$10\!\cdots\!18$$$$\beta_{10} +$$$$45\!\cdots\!54$$$$\beta_{9} -$$$$46\!\cdots\!39$$$$\beta_{8} +$$$$22\!\cdots\!77$$$$\beta_{7} +$$$$43\!\cdots\!57$$$$\beta_{6} +$$$$91\!\cdots\!17$$$$\beta_{5} -$$$$43\!\cdots\!80$$$$\beta_{4} +$$$$14\!\cdots\!38$$$$\beta_{3} -$$$$85\!\cdots\!41$$$$\beta_{2} +$$$$14\!\cdots\!06$$$$\beta_{1} +$$$$90\!\cdots\!99$$$$)/2$$ $$\nu^{19}$$ $$=$$ $$($$$$19\!\cdots\!12$$$$\beta_{19} +$$$$32\!\cdots\!26$$$$\beta_{17} +$$$$65\!\cdots\!42$$$$\beta_{16} +$$$$67\!\cdots\!58$$$$\beta_{15} +$$$$67\!\cdots\!58$$$$\beta_{14} -$$$$27\!\cdots\!18$$$$\beta_{13} +$$$$29\!\cdots\!26$$$$\beta_{12} +$$$$47\!\cdots\!99$$$$\beta_{11} +$$$$16\!\cdots\!77$$$$\beta_{10} +$$$$46\!\cdots\!81$$$$\beta_{9} -$$$$16\!\cdots\!95$$$$\beta_{8} +$$$$14\!\cdots\!76$$$$\beta_{7} +$$$$98\!\cdots\!94$$$$\beta_{6} +$$$$73\!\cdots\!48$$$$\beta_{5} -$$$$73\!\cdots\!48$$$$\beta_{4} +$$$$84\!\cdots\!54$$$$\beta_{3} +$$$$13\!\cdots\!96$$$$\beta_{2} +$$$$10\!\cdots\!28$$$$\beta_{1} +$$$$40\!\cdots\!32$$$$)/2$$

## Character Values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/7\mathbb{Z}\right)^\times$$.

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

## Embeddings

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

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

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
2.1
 −159.672 + 276.560i −139.056 + 240.852i −94.3198 + 163.367i −35.7008 + 61.8356i −28.2933 + 49.0055i −7.21985 + 12.5051i 72.8228 − 126.133i 90.6104 − 156.942i 134.761 − 233.413i 168.567 − 291.966i −159.672 − 276.560i −139.056 − 240.852i −94.3198 − 163.367i −35.7008 − 61.8356i −28.2933 − 49.0055i −7.21985 − 12.5051i 72.8228 + 126.133i 90.6104 + 156.942i 134.761 + 233.413i 168.567 + 291.966i
−306.343 + 530.602i −2643.13 4578.04i −122157. 211581.i 112450. 194769.i 3.23883e6 1.21283e7 + 9.24856e6i 6.93813e7 5.05978e7 8.76379e7i 6.88965e7 + 1.19332e8i
2.2 −265.112 + 459.187i 9523.58 + 16495.3i −75032.3 129960.i 224503. 388850.i −1.00992e7 −1.05114e7 1.10518e7i 1.00703e7 −1.16827e8 + 2.02350e8i 1.19037e8 + 2.06177e8i
2.3 −175.640 + 304.217i −3368.41 5834.25i 3837.49 + 6646.72i −348474. + 603575.i 2.36650e6 −1.46022e7 4.40528e6i −4.87389e7 4.18777e7 7.25344e7i −1.22412e8 2.12023e8i
2.4 −58.4016 + 101.154i −10042.7 17394.4i 58714.5 + 101697.i 526365. 911691.i 2.34603e6 1.15602e7 9.94945e6i −2.90257e7 −1.37140e8 + 2.37533e8i 6.14811e7 + 1.06488e8i
2.5 −43.5867 + 75.4943i 5626.01 + 9744.54i 61736.4 + 106931.i −472563. + 818503.i −980877. 1.52373e7 + 674362.i −2.21895e7 1.26603e6 2.19283e6i −4.11949e7 7.13516e7i
2.6 −1.43970 + 2.49364i 3646.82 + 6316.47i 65531.9 + 113505.i 804501. 1.39344e6i −21001.3 −6.26914e6 + 1.39043e7i −754795. 3.79715e7 6.57686e7i 2.31648e6 + 4.01227e6i
2.7 158.646 274.782i −6861.41 11884.3i 15199.2 + 26325.8i −470055. + 814159.i −4.35413e6 −5.83152e6 + 1.40934e7i 5.12331e7 −2.95879e7 + 5.12478e7i 1.49144e8 + 2.58325e8i
2.8 194.221 336.400i 1157.82 + 2005.40i −9907.51 17160.3i 92672.9 160514.i 899489. −2.00510e6 1.51199e7i 4.32169e7 6.18890e7 1.07195e8i −3.59980e7 6.23504e7i
2.9 282.522 489.343i 11021.2 + 19089.3i −94101.8 162989.i −274540. + 475518.i 1.24550e7 −6.98844e6 + 1.35570e7i −3.22819e7 −1.78365e8 + 3.08938e8i 1.55128e8 + 2.68689e8i
2.10 350.134 606.449i −4779.85 8278.95i −179651. 311165.i 350039. 606285.i −6.69435e6 1.44796e7 + 4.79276e6i −1.59822e8 1.88761e7 3.26943e7i −2.45121e8 4.24562e8i
4.1 −306.343 530.602i −2643.13 + 4578.04i −122157. + 211581.i 112450. + 194769.i 3.23883e6 1.21283e7 9.24856e6i 6.93813e7 5.05978e7 + 8.76379e7i 6.88965e7 1.19332e8i
4.2 −265.112 459.187i 9523.58 16495.3i −75032.3 + 129960.i 224503. + 388850.i −1.00992e7 −1.05114e7 + 1.10518e7i 1.00703e7 −1.16827e8 2.02350e8i 1.19037e8 2.06177e8i
4.3 −175.640 304.217i −3368.41 + 5834.25i 3837.49 6646.72i −348474. 603575.i 2.36650e6 −1.46022e7 + 4.40528e6i −4.87389e7 4.18777e7 + 7.25344e7i −1.22412e8 + 2.12023e8i
4.4 −58.4016 101.154i −10042.7 + 17394.4i 58714.5 101697.i 526365. + 911691.i 2.34603e6 1.15602e7 + 9.94945e6i −2.90257e7 −1.37140e8 2.37533e8i 6.14811e7 1.06488e8i
4.5 −43.5867 75.4943i 5626.01 9744.54i 61736.4 106931.i −472563. 818503.i −980877. 1.52373e7 674362.i −2.21895e7 1.26603e6 + 2.19283e6i −4.11949e7 + 7.13516e7i
4.6 −1.43970 2.49364i 3646.82 6316.47i 65531.9 113505.i 804501. + 1.39344e6i −21001.3 −6.26914e6 1.39043e7i −754795. 3.79715e7 + 6.57686e7i 2.31648e6 4.01227e6i
4.7 158.646 + 274.782i −6861.41 + 11884.3i 15199.2 26325.8i −470055. 814159.i −4.35413e6 −5.83152e6 1.40934e7i 5.12331e7 −2.95879e7 5.12478e7i 1.49144e8 2.58325e8i
4.8 194.221 + 336.400i 1157.82 2005.40i −9907.51 + 17160.3i 92672.9 + 160514.i 899489. −2.00510e6 + 1.51199e7i 4.32169e7 6.18890e7 + 1.07195e8i −3.59980e7 + 6.23504e7i
4.9 282.522 + 489.343i 11021.2 19089.3i −94101.8 + 162989.i −274540. 475518.i 1.24550e7 −6.98844e6 1.35570e7i −3.22819e7 −1.78365e8 3.08938e8i 1.55128e8 2.68689e8i
4.10 350.134 + 606.449i −4779.85 + 8278.95i −179651. + 311165.i 350039. + 606285.i −6.69435e6 1.44796e7 4.79276e6i −1.59822e8 1.88761e7 + 3.26943e7i −2.45121e8 + 4.24562e8i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 4.10 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

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

## Hecke kernels

There are no other newforms in $$S_{18}^{\mathrm{new}}(7, [\chi])$$.