Properties

Label 5.21.c.a
Level 5
Weight 21
Character orbit 5.c
Analytic conductor 12.676
Analytic rank 0
Dimension 18
CM No
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 5 \)
Weight: \( k \) = \( 21 \)
Character orbit: \([\chi]\) = 5.c (of order \(4\) and degree \(2\))

Newform invariants

Self dual: No
Analytic conductor: \(12.6756882551\)
Analytic rank: \(0\)
Dimension: \(18\)
Relative dimension: \(9\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{18} + \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 2^{48}\cdot 3^{16}\cdot 5^{32} \)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

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

\(f(q)\) \(=\) \( q\) \( -\beta_{1} q^{2} \) \( + ( 1636 - \beta_{2} - 1636 \beta_{4} - \beta_{5} ) q^{3} \) \( + ( 31 \beta_{1} - 31 \beta_{2} + 579419 \beta_{4} + 2 \beta_{5} - 2 \beta_{6} - \beta_{7} ) q^{4} \) \( + ( -405698 - 416 \beta_{1} + 576 \beta_{2} + 2 \beta_{3} - 427990 \beta_{4} - 29 \beta_{5} - 13 \beta_{6} + \beta_{7} + \beta_{8} ) q^{5} \) \( + ( 1100905 - 5790 \beta_{1} - 5789 \beta_{2} + 6 \beta_{3} + \beta_{4} + 160 \beta_{5} + 160 \beta_{6} - \beta_{7} + 2 \beta_{9} + \beta_{10} ) q^{6} \) \( + ( -32531412 + 17628 \beta_{1} + 9 \beta_{3} - 32531411 \beta_{4} + 1321 \beta_{6} + 9 \beta_{7} + 2 \beta_{9} - \beta_{13} ) q^{7} \) \( + ( 51607150 - 9 \beta_{1} + 588689 \beta_{2} + 152 \beta_{3} - 51607135 \beta_{4} - 3377 \beta_{5} + 5 \beta_{6} - 155 \beta_{7} + 2 \beta_{8} + 22 \beta_{9} - 3 \beta_{10} + \beta_{11} - \beta_{12} + \beta_{14} + \beta_{15} + \beta_{17} ) q^{8} \) \( + ( 35 - 381757 \beta_{1} + 381720 \beta_{2} - 24 \beta_{3} - 539063154 \beta_{4} - 8309 \beta_{5} + 8311 \beta_{6} - 207 \beta_{7} - 59 \beta_{8} + 20 \beta_{9} - 2 \beta_{10} - 11 \beta_{11} - \beta_{12} - 5 \beta_{13} - 5 \beta_{14} + \beta_{15} - \beta_{16} + 2 \beta_{17} ) q^{9} \) \(+O(q^{10})\) \( q\) \( -\beta_{1} q^{2} \) \( + ( 1636 - \beta_{2} - 1636 \beta_{4} - \beta_{5} ) q^{3} \) \( + ( 31 \beta_{1} - 31 \beta_{2} + 579419 \beta_{4} + 2 \beta_{5} - 2 \beta_{6} - \beta_{7} ) q^{4} \) \( + ( -405698 - 416 \beta_{1} + 576 \beta_{2} + 2 \beta_{3} - 427990 \beta_{4} - 29 \beta_{5} - 13 \beta_{6} + \beta_{7} + \beta_{8} ) q^{5} \) \( + ( 1100905 - 5790 \beta_{1} - 5789 \beta_{2} + 6 \beta_{3} + \beta_{4} + 160 \beta_{5} + 160 \beta_{6} - \beta_{7} + 2 \beta_{9} + \beta_{10} ) q^{6} \) \( + ( -32531412 + 17628 \beta_{1} + 9 \beta_{3} - 32531411 \beta_{4} + 1321 \beta_{6} + 9 \beta_{7} + 2 \beta_{9} - \beta_{13} ) q^{7} \) \( + ( 51607150 - 9 \beta_{1} + 588689 \beta_{2} + 152 \beta_{3} - 51607135 \beta_{4} - 3377 \beta_{5} + 5 \beta_{6} - 155 \beta_{7} + 2 \beta_{8} + 22 \beta_{9} - 3 \beta_{10} + \beta_{11} - \beta_{12} + \beta_{14} + \beta_{15} + \beta_{17} ) q^{8} \) \( + ( 35 - 381757 \beta_{1} + 381720 \beta_{2} - 24 \beta_{3} - 539063154 \beta_{4} - 8309 \beta_{5} + 8311 \beta_{6} - 207 \beta_{7} - 59 \beta_{8} + 20 \beta_{9} - 2 \beta_{10} - 11 \beta_{11} - \beta_{12} - 5 \beta_{13} - 5 \beta_{14} + \beta_{15} - \beta_{16} + 2 \beta_{17} ) q^{9} \) \( + ( -952196829 + 2005720 \beta_{1} - 1570312 \beta_{2} + 931 \beta_{3} + 670601214 \beta_{4} + 43937 \beta_{5} + 12189 \beta_{6} + 1441 \beta_{7} + 65 \beta_{8} + 59 \beta_{9} + 64 \beta_{10} + 23 \beta_{11} - 8 \beta_{12} - 15 \beta_{13} + 20 \beta_{14} - 3 \beta_{15} + 4 \beta_{16} - 6 \beta_{17} ) q^{10} \) \( + ( -360246334 - 5502156 \beta_{1} - 5501824 \beta_{2} + 2563 \beta_{3} + 216 \beta_{4} - 19751 \beta_{5} - 19625 \beta_{6} - 163 \beta_{7} + 440 \beta_{8} + 324 \beta_{9} + 104 \beta_{10} - 16 \beta_{11} - 16 \beta_{12} + 40 \beta_{13} - 40 \beta_{14} - 7 \beta_{15} - 7 \beta_{16} + 2 \beta_{17} ) q^{11} \) \( + ( 7794220662 + 7390057 \beta_{1} + 23 \beta_{2} - 4230 \beta_{3} + 7794219497 \beta_{4} - 745 \beta_{5} - 531841 \beta_{6} - 3141 \beta_{7} + 464 \beta_{8} - 3294 \beta_{9} - 367 \beta_{10} - 471 \beta_{11} - 107 \beta_{12} + 143 \beta_{13} + 7 \beta_{16} + 3 \beta_{17} ) q^{12} \) \( + ( -20518829278 - 1126 \beta_{1} - 8024677 \beta_{2} + 17331 \beta_{3} + 20518832650 \beta_{4} - 127435 \beta_{5} - 245 \beta_{6} - 17668 \beta_{7} - 2311 \beta_{8} + 2538 \beta_{9} - 1297 \beta_{10} + 1034 \beta_{11} - 264 \beta_{12} + 106 \beta_{14} - 26 \beta_{15} - \beta_{17} ) q^{13} \) \( + ( 11188 + 47089399 \beta_{1} - 47095828 \beta_{2} - 7640 \beta_{3} - 28001927132 \beta_{4} + 688944 \beta_{5} - 694538 \beta_{6} + 42363 \beta_{7} - 21508 \beta_{8} - 10670 \beta_{9} - 16 \beta_{10} - 3241 \beta_{11} - 718 \beta_{12} - 165 \beta_{13} - 165 \beta_{14} - 27 \beta_{15} + 27 \beta_{16} + 16 \beta_{17} ) q^{14} \) \( + ( 55488006559 - 109615283 \beta_{1} + 29760975 \beta_{2} - 16634 \beta_{3} - 116083032496 \beta_{4} - 424731 \beta_{5} - 519602 \beta_{6} + 157264 \beta_{7} - 1994 \beta_{8} + 8280 \beta_{9} + 3508 \beta_{10} + 956 \beta_{11} - 1526 \beta_{12} + 170 \beta_{13} - 185 \beta_{14} + 109 \beta_{15} - 137 \beta_{16} + 368 \beta_{17} ) q^{15} \) \( + ( -352572268857 + 106297412 \beta_{1} + 106342702 \beta_{2} + 848073 \beta_{3} + 34189 \beta_{4} + 1502358 \beta_{5} + 1525124 \beta_{6} - 14161 \beta_{7} + 60030 \beta_{8} + 48330 \beta_{9} + 2322 \beta_{10} - 3476 \beta_{11} - 3476 \beta_{12} - 935 \beta_{13} + 935 \beta_{14} + 253 \beta_{15} + 253 \beta_{16} + 302 \beta_{17} ) q^{16} \) \( + ( 361565846185 + 56928115 \beta_{1} + 13815 \beta_{2} - 664399 \beta_{3} + 361565826169 \beta_{4} - 34280 \beta_{5} - 9597554 \beta_{6} - 643489 \beta_{7} + 46285 \beta_{8} - 106007 \beta_{9} - 445 \beta_{10} - 4545 \beta_{11} - 4765 \beta_{12} - 3184 \beta_{13} - 260 \beta_{16} + 665 \beta_{17} ) q^{17} \) \( + ( -625842579188 - 80375 \beta_{1} - 425171344 \beta_{2} + 1508277 \beta_{3} + 625842757156 \beta_{4} - 8501022 \beta_{5} - 1130 \beta_{6} - 1608117 \beta_{7} - 128766 \beta_{8} + 171560 \beta_{9} + 10065 \beta_{10} - 18335 \beta_{11} - 7350 \beta_{12} - 3422 \beta_{14} + 220 \beta_{15} + 920 \beta_{17} ) q^{18} \) \( + ( 78584 - 478310655 \beta_{1} + 478291807 \beta_{2} - 25897 \beta_{3} - 762421469934 \beta_{4} - 11278552 \beta_{5} + 11220906 \beta_{6} + 2143979 \beta_{7} - 143336 \beta_{8} - 124640 \beta_{9} - 1064 \beta_{10} + 25764 \beta_{11} - 7942 \beta_{12} + 6840 \beta_{13} + 6840 \beta_{14} + 247 \beta_{15} - 247 \beta_{16} + 1064 \beta_{17} ) q^{19} \) \( + ( 2130862532300 + 1478149176 \beta_{1} - 1042057524 \beta_{2} - 3756946 \beta_{3} - 2833286595638 \beta_{4} + 14960633 \beta_{5} - 86063089 \beta_{6} + 6028842 \beta_{7} - 49420 \beta_{8} - 204806 \beta_{9} - 70225 \beta_{10} - 49575 \beta_{11} - 4975 \beta_{12} + 2250 \beta_{13} - 3625 \beta_{14} - 1825 \beta_{15} + 2100 \beta_{16} - 3075 \beta_{17} ) q^{20} \) \( + ( -5410889438764 - 1623539941 \beta_{1} - 1623784050 \beta_{2} + 6051357 \beta_{3} - 206008 \beta_{4} + 126070315 \beta_{5} + 125976549 \beta_{6} + 117480 \beta_{7} - 228955 \beta_{8} - 330868 \beta_{9} - 69466 \beta_{10} + 15151 \beta_{11} + 15151 \beta_{12} + 8810 \beta_{13} - 8810 \beta_{14} - 4178 \beta_{15} - 4178 \beta_{16} - 1402 \beta_{17} ) q^{21} \) \( + ( 8947116496200 + 3027756861 \beta_{1} + 8497 \beta_{2} - 12795259 \beta_{3} + 8947116797804 \beta_{4} + 207830 \beta_{5} - 156014410 \beta_{6} - 13091543 \beta_{7} - 93834 \beta_{8} + 802922 \beta_{9} + 79957 \beta_{10} + 101861 \beta_{11} + 28162 \beta_{12} + 30918 \beta_{13} + 4438 \beta_{16} - 6258 \beta_{17} ) q^{22} \) \( + ( -4536397288991 + 606540 \beta_{1} - 671781996 \beta_{2} + 6502215 \beta_{3} + 4536395900026 \beta_{4} - 241463769 \beta_{5} + 20910 \beta_{6} - 5894005 \beta_{7} + 900770 \beta_{8} - 1318140 \beta_{9} + 101100 \beta_{10} - 19240 \beta_{11} + 74310 \beta_{12} + 44755 \beta_{14} + 550 \beta_{15} - 7550 \beta_{17} ) q^{23} \) \( + ( -1755514 - 8357382268 \beta_{1} + 8358137724 \beta_{2} + 941658 \beta_{3} - 13464212276202 \beta_{4} - 618522704 \beta_{5} + 619535812 \beta_{6} + 9087210 \beta_{7} + 3324022 \beta_{8} + 2168330 \beta_{9} + 8842 \beta_{10} + 105910 \beta_{11} + 133346 \beta_{12} - 93770 \beta_{13} - 93770 \beta_{14} + 304 \beta_{15} - 304 \beta_{16} - 8842 \beta_{17} ) q^{24} \) \( + ( 1886476187450 - 23083487045 \beta_{1} + 1555018540 \beta_{2} - 10346790 \beta_{3} - 5390034004575 \beta_{4} + 590920215 \beta_{5} - 504894245 \beta_{6} + 2906895 \beta_{7} + 397235 \beta_{8} + 1653280 \beta_{9} + 381070 \beta_{10} + 400365 \beta_{11} + 196685 \beta_{12} - 63825 \beta_{13} + 80725 \beta_{14} + 18535 \beta_{15} - 18505 \beta_{16} + 12920 \beta_{17} ) q^{25} \) \( + ( 12997429522567 + 39124368395 \beta_{1} + 39122711611 \beta_{2} - 17368429 \beta_{3} - 982881 \beta_{4} + 1499930874 \beta_{5} + 1498805896 \beta_{6} + 72677 \beta_{7} - 3293080 \beta_{8} - 933808 \beta_{9} + 546430 \beta_{10} + 159978 \beta_{11} + 159978 \beta_{12} - 39195 \beta_{13} + 39195 \beta_{14} + 41211 \beta_{15} + 41211 \beta_{16} - 10336 \beta_{17} ) q^{26} \) \( + ( -28443122260050 + 19854495552 \beta_{1} - 1740648 \beta_{2} + 17797080 \beta_{3} - 28443123093000 \beta_{4} + 1791030 \beta_{5} - 575987766 \beta_{6} + 18447486 \beta_{7} - 4500144 \beta_{8} + 4128324 \beta_{9} - 700788 \beta_{10} - 427524 \beta_{11} + 263742 \beta_{12} - 140298 \beta_{13} - 45642 \beta_{16} + 9522 \beta_{17} ) q^{27} \) \( + ( 42905097580679 + 2681519 \beta_{1} - 42694051833 \beta_{2} - 72278747 \beta_{3} - 42905103640894 \beta_{4} - 3824802467 \beta_{5} + 31675 \beta_{6} + 76720510 \beta_{7} + 5750898 \beta_{8} - 5518212 \beta_{9} - 1594117 \beta_{10} + 1728569 \beta_{11} + 118041 \beta_{12} - 332101 \beta_{14} - 28511 \beta_{15} - 16411 \beta_{17} ) q^{28} \) \( + ( 4421152 - 92058965419 \beta_{1} + 92056818721 \beta_{2} - 3112559 \beta_{3} + 96816710013982 \beta_{4} - 431699514 \beta_{5} + 429690520 \beta_{6} - 109569627 \beta_{7} - 9008431 \beta_{8} - 5628785 \beta_{9} + 71429 \beta_{10} - 2140855 \beta_{11} - 204763 \beta_{12} + 684510 \beta_{13} + 684510 \beta_{14} - 28842 \beta_{15} + 28842 \beta_{16} - 71429 \beta_{17} ) q^{29} \) \( + ( -48689492319926 - 75892321052 \beta_{1} - 285987513273 \beta_{2} + 167550459 \beta_{3} + 178186857951800 \beta_{4} + 2949290972 \beta_{5} - 6005441966 \beta_{6} - 207605658 \beta_{7} + 3258162 \beta_{8} - 4880540 \beta_{9} + 414975 \beta_{10} + 171200 \beta_{11} - 651150 \beta_{12} + 648375 \beta_{13} - 665125 \beta_{14} - 126175 \beta_{15} + 97275 \beta_{16} - 93800 \beta_{17} ) q^{30} \) \( + ( 96783702089807 + 41844984842 \beta_{1} + 41855401502 \beta_{2} - 281042222 \beta_{3} + 7328231 \beta_{4} + 3891601789 \beta_{5} + 3897088933 \beta_{6} - 2212824 \beta_{7} + 16290710 \beta_{8} + 9193510 \beta_{9} - 1326672 \beta_{10} - 822804 \beta_{11} - 822804 \beta_{12} + 72385 \beta_{13} - 72385 \beta_{14} - 264408 \beta_{15} - 264408 \beta_{16} + 2688 \beta_{17} ) q^{31} \) \( + ( -226329175859786 + 605650470070 \beta_{1} + 9733550 \beta_{2} + 568184406 \beta_{3} - 226329179384092 \beta_{4} - 16091690 \beta_{5} - 5239798578 \beta_{6} + 573184976 \beta_{7} + 28460520 \beta_{8} - 46265852 \beta_{9} + 1357570 \beta_{10} - 935270 \beta_{11} - 2336790 \beta_{12} + 67286 \beta_{13} + 309790 \beta_{16} + 43950 \beta_{17} ) q^{32} \) \( + ( 87772528296736 - 34448141 \beta_{1} - 383298731089 \beta_{2} - 81128365 \beta_{3} - 87772448615248 \beta_{4} - 286559224 \beta_{5} - 1013950 \beta_{6} + 35638433 \beta_{7} - 64736153 \beta_{8} + 72851343 \beta_{9} + 6741763 \beta_{10} - 10027841 \beta_{11} - 2965449 \beta_{12} + 1543892 \beta_{14} + 292904 \beta_{15} + 320629 \beta_{17} ) q^{33} \) \( + ( 29258744 - 1105555124057 \beta_{1} + 1105540458679 \beta_{2} - 10565432 \beta_{3} - 97775000345168 \beta_{4} - 4712306696 \beta_{5} + 4695592380 \beta_{6} - 186701334 \beta_{7} - 48977526 \beta_{8} - 22986510 \beta_{9} - 1211134 \beta_{10} + 11459964 \beta_{11} - 2917142 \beta_{12} - 2837210 \beta_{13} - 2837210 \beta_{14} + 321992 \beta_{15} - 321992 \beta_{16} + 1211134 \beta_{17} ) q^{34} \) \( + ( -267448062603904 - 777106166670 \beta_{1} - 1081984600147 \beta_{2} + 290961791 \beta_{3} - 19294852806096 \beta_{4} - 251486758 \beta_{5} + 4923267799 \beta_{6} - 310572059 \beta_{7} - 43762080 \beta_{8} - 906396 \beta_{9} - 11334776 \beta_{10} - 15470732 \beta_{11} - 2960628 \beta_{12} - 3824740 \beta_{13} + 2615070 \beta_{14} + 597627 \beta_{15} - 246461 \beta_{16} + 522654 \beta_{17} ) q^{35} \) \( + ( 122623030966170 + 1814382519867 \beta_{1} + 1814413575973 \beta_{2} + 639704304 \beta_{3} + 16489421 \beta_{4} - 18047502844 \beta_{5} - 18021660014 \beta_{6} - 2563761 \beta_{7} + 63132710 \beta_{8} + 16380722 \beta_{9} - 3391414 \beta_{10} - 3305660 \beta_{11} - 3305660 \beta_{12} - 331975 \beta_{13} + 331975 \beta_{14} + 1104325 \beta_{15} + 1104325 \beta_{16} + 1027190 \beta_{17} ) q^{36} \) \( + ( 354443369451716 + 1677782832028 \beta_{1} + 6235219 \beta_{2} - 986625765 \beta_{3} + 354443402860026 \beta_{4} + 16851695 \beta_{5} + 20215428977 \beta_{6} - 1023306538 \beta_{7} + 3492177 \beta_{8} + 73317558 \beta_{9} + 13593859 \beta_{10} + 16168352 \beta_{11} + 2604484 \beta_{12} + 2407564 \beta_{13} - 1409684 \beta_{16} - 29991 \beta_{17} ) q^{37} \) \( + ( -784890013209674 + 58480936 \beta_{1} - 3058682301152 \beta_{2} + 689014556 \beta_{3} + 784889864940400 \beta_{4} + 40086603522 \beta_{5} + 5393150 \beta_{6} - 621039434 \beta_{7} + 124540440 \beta_{8} - 124134828 \beta_{9} + 3349852 \beta_{10} + 4101036 \beta_{11} + 6983754 \beta_{12} - 4685058 \beta_{14} - 1679334 \beta_{15} - 467134 \beta_{17} ) q^{38} \) \( + ( -145407067 - 3742164245280 \beta_{1} + 3742261519544 \beta_{2} + 59124456 \beta_{3} - 498114146649965 \beta_{4} + 45040767081 \beta_{5} - 44998541113 \beta_{6} + 1946481678 \beta_{7} + 241700218 \beta_{8} + 45794450 \beta_{9} + 7161172 \beta_{10} - 28412492 \beta_{11} + 10148876 \beta_{12} + 5427805 \beta_{13} + 5427805 \beta_{14} - 2000996 \beta_{15} + 2000996 \beta_{16} - 7161172 \beta_{17} ) q^{39} \) \( + ( 2406655577382985 - 4624288007585 \beta_{1} - 6426847151085 \beta_{2} - 3584906075 \beta_{3} - 1453024612864250 \beta_{4} - 62765515135 \beta_{5} + 86909376005 \beta_{6} + 3862372660 \beta_{7} + 153621920 \beta_{8} - 8406280 \beta_{9} + 27983955 \beta_{10} + 74023185 \beta_{11} + 18928815 \beta_{12} + 14437325 \beta_{13} - 1495600 \beta_{14} - 1956810 \beta_{15} - 395295 \beta_{16} + 530955 \beta_{17} ) q^{40} \) \( + ( -1700771397303325 + 2828921477422 \beta_{1} + 2828630162891 \beta_{2} + 1806637977 \beta_{3} - 169933121 \beta_{4} - 64618049319 \beta_{5} - 64832356849 \beta_{6} + 45477386 \beta_{7} - 538994410 \beta_{8} - 193620447 \beta_{9} + 12561889 \beta_{10} + 26509660 \beta_{11} + 26509660 \beta_{12} + 5834725 \beta_{13} - 5834725 \beta_{14} - 2624875 \beta_{15} - 2624875 \beta_{16} - 7562415 \beta_{17} ) q^{41} \) \( + ( 2710277611496204 + 12947069243577 \beta_{1} - 218691675 \beta_{2} - 7110150971 \beta_{3} + 2710277409475290 \beta_{4} + 159116480 \beta_{5} + 55168807640 \beta_{6} - 6942911761 \beta_{7} - 532657640 \beta_{8} + 260364002 \beta_{9} - 107664015 \beta_{10} - 83582285 \beta_{11} + 22401130 \beta_{12} - 10095196 \beta_{13} + 3989170 \beta_{16} + 1680600 \beta_{17} ) q^{42} \) \( + ( -2577614195891038 + 345790996 \beta_{1} - 1108557224911 \beta_{2} + 1431201538 \beta_{3} + 2577613518336092 \beta_{4} + 104073266451 \beta_{5} - 34841320 \beta_{6} - 967291506 \beta_{7} + 430445384 \beta_{8} - 744148668 \beta_{9} - 126837068 \beta_{10} + 152960356 \beta_{11} + 19219044 \beta_{12} + 10399470 \beta_{14} + 5730656 \beta_{15} - 6904244 \beta_{17} ) q^{43} \) \( + ( 12772703 - 17276864664044 \beta_{1} + 17276646475214 \beta_{2} + 40986293 \beta_{3} - 4636384559328087 \beta_{4} + 29560585488 \beta_{5} - 29249138646 \beta_{6} - 118365301 \beta_{7} + 98705810 \beta_{8} + 715790090 \beta_{9} - 26742652 \beta_{10} + 15457226 \beta_{11} + 20799514 \beta_{12} + 5003245 \beta_{13} + 5003245 \beta_{14} + 7702591 \beta_{15} - 7702591 \beta_{16} + 26742652 \beta_{17} ) q^{44} \) \( + ( 3776945218796830 - 4962029968376 \beta_{1} - 13572269540721 \beta_{2} + 1379184711 \beta_{3} - 3174663832295232 \beta_{4} - 96183275998 \beta_{5} + 160551824084 \beta_{6} + 4422461853 \beta_{7} - 18556240 \beta_{8} + 421692166 \beta_{9} + 120149675 \beta_{10} - 120081400 \beta_{11} - 6668450 \beta_{12} - 35090500 \beta_{13} - 34647250 \beta_{14} + 4086350 \beta_{15} + 5939200 \beta_{16} - 20897525 \beta_{17} ) q^{45} \) \( + ( 968173701419220 + 9966666643992 \beta_{1} + 9966886283671 \beta_{2} + 1664567047 \beta_{3} + 12849724 \beta_{4} - 51282144914 \beta_{5} - 50951086864 \beta_{6} - 11908904 \beta_{7} + 654464510 \beta_{8} - 45387932 \beta_{9} + 118403499 \beta_{10} - 16832680 \beta_{11} - 16832680 \beta_{12} - 44884425 \beta_{13} + 44884425 \beta_{14} + 511535 \beta_{15} + 511535 \beta_{16} + 28885630 \beta_{17} ) q^{46} \) \( + ( -7618117289138438 + 8475569079274 \beta_{1} + 797555416 \beta_{2} + 11569970179 \beta_{3} - 7618116069204517 \beta_{4} - 316273810 \beta_{5} + 88469419839 \beta_{6} + 10730948377 \beta_{7} + 1824037248 \beta_{8} + 156742134 \beta_{9} + 357740196 \beta_{10} + 279411608 \beta_{11} - 48800914 \beta_{12} - 3111705 \beta_{13} - 4195086 \beta_{16} - 29527674 \beta_{17} ) q^{47} \) \( + ( -5758640660160912 - 1340570380 \beta_{1} - 21542479766868 \beta_{2} + 2331699528 \beta_{3} + 5758643101204444 \beta_{4} - 6266593108 \beta_{5} + 279131300 \beta_{6} - 3927920388 \beta_{7} - 1140493224 \beta_{8} + 2960557840 \beta_{9} + 411590340 \beta_{10} - 534781780 \beta_{11} - 73972020 \beta_{12} - 26656348 \beta_{14} - 8567380 \beta_{15} + 49219420 \beta_{17} ) q^{48} \) \( + ( 537602247 - 953822191133 \beta_{1} + 954368511866 \beta_{2} - 412404972 \beta_{3} - 488255028150568 \beta_{4} + 28196729393 \beta_{5} - 29704205363 \beta_{6} - 4510234981 \beta_{7} - 1410441039 \beta_{8} - 3276417270 \beta_{9} + 82897080 \beta_{10} + 53448213 \beta_{11} - 130248825 \beta_{12} - 44612925 \beta_{13} - 44612925 \beta_{14} - 15948555 \beta_{15} + 15948555 \beta_{16} - 82897080 \beta_{17} ) q^{49} \) \( + ( -2221053175400650 - 12142043254070 \beta_{1} - 5122164375935 \beta_{2} + 13491024035 \beta_{3} + 37312039238882000 \beta_{4} + 351951389440 \beta_{5} - 635460887420 \beta_{6} - 42282769655 \beta_{7} - 931948040 \beta_{8} - 2060617870 \beta_{9} - 849044755 \beta_{10} + 24767715 \beta_{11} - 120961790 \beta_{12} + 50910050 \beta_{13} + 165960350 \beta_{14} - 3304690 \beta_{15} - 22318580 \beta_{16} + 112899220 \beta_{17} ) q^{50} \) \( + ( 38111313206361634 + 11124998045631 \beta_{1} + 11127261618755 \beta_{2} - 20614824282 \beta_{3} + 2124306102 \beta_{4} + 21617922327 \beta_{5} + 22133552699 \beta_{6} - 563120206 \beta_{7} + 2391542420 \beta_{8} + 2898675208 \beta_{9} - 837765012 \beta_{10} - 216062672 \beta_{11} - 216062672 \beta_{12} + 179123930 \beta_{13} - 179123930 \beta_{14} + 19569886 \beta_{15} + 19569886 \beta_{16} - 92700736 \beta_{17} ) q^{51} \) \( + ( -41389157751110159 - 11869398253816 \beta_{1} - 1550206534 \beta_{2} + 46227080349 \beta_{3} - 41389163048889193 \beta_{4} - 1102330370 \beta_{5} - 850066476990 \beta_{6} + 49563380527 \beta_{7} - 2742049572 \beta_{8} - 7294175636 \beta_{9} - 683763274 \beta_{10} - 636145022 \beta_{11} - 129746074 \beta_{12} + 156960870 \beta_{13} - 16743526 \beta_{16} + 177364326 \beta_{17} ) q^{52} \) \( + ( -10228547504149992 + 90801001 \beta_{1} + 5261657637078 \beta_{2} - 55284584498 \beta_{3} + 10228549851156592 \beta_{4} - 841509827015 \beta_{5} - 1501665275 \beta_{6} + 54132476725 \beta_{7} - 5691643628 \beta_{8} - 848457323 \beta_{9} - 174779618 \beta_{10} + 258756501 \beta_{11} - 116575861 \beta_{12} + 94337306 \beta_{14} - 18779794 \beta_{15} - 200552744 \beta_{17} ) q^{53} \) \( + ( 2172885348 + 45794046833328 \beta_{1} - 45796818558828 \beta_{2} - 692100132 \beta_{3} - 32626750664721828 \beta_{4} - 895212653088 \beta_{5} + 896515122960 \beta_{6} - 28404461088 \beta_{7} - 2929972320 \beta_{8} + 3746642280 \beta_{9} - 232266552 \beta_{10} + 317321916 \beta_{11} - 12592056 \beta_{12} + 19422120 \beta_{13} + 19422120 \beta_{14} - 4803504 \beta_{15} + 4803504 \beta_{16} + 232266552 \beta_{17} ) q^{54} \) \( + ( 41821477357531629 + 44732889363743 \beta_{1} + 40027988414877 \beta_{2} + 50917958129 \beta_{3} + 37875487533191945 \beta_{4} + 1435751234467 \beta_{5} - 1338540565851 \beta_{6} + 723905377 \beta_{7} - 154652298 \beta_{8} + 3471070850 \beta_{9} + 1716765000 \beta_{10} - 607937500 \beta_{11} + 43734500 \beta_{12} - 23515625 \beta_{13} - 274278125 \beta_{14} - 11094625 \beta_{15} + 29591375 \beta_{16} - 334477250 \beta_{17} ) q^{55} \) \( + ( 38118120429043214 - 84908387494788 \beta_{1} - 84912702843148 \beta_{2} - 56210407662 \beta_{3} - 5203326878 \beta_{4} + 2688076035172 \beta_{5} + 2688437749120 \beta_{6} + 1562943822 \beta_{7} - 1713649230 \beta_{8} - 7802889790 \beta_{9} + 1951106146 \beta_{10} + 399931482 \beta_{11} + 399931482 \beta_{12} - 344032330 \beta_{13} + 344032330 \beta_{14} - 62661236 \beta_{15} - 62661236 \beta_{16} + 329497546 \beta_{17} ) q^{56} \) \( + ( -43601467735807320 - 97736197025250 \beta_{1} + 4408156788 \beta_{2} - 21604892628 \beta_{3} - 43601452841082942 \beta_{4} + 2817313920 \beta_{5} - 113348558172 \beta_{6} - 30057704664 \beta_{7} + 8070860964 \beta_{8} + 19419316662 \beta_{9} + 1227341328 \beta_{10} + 963342294 \beta_{11} + 308772648 \beta_{12} - 533330190 \beta_{13} + 83133702 \beta_{16} - 572771682 \beta_{17} ) q^{57} \) \( + ( -150092149133917778 + 3521604338 \beta_{1} + 201985966267658 \beta_{2} + 112972278522 \beta_{3} + 150092133291186032 \beta_{4} - 3673398616138 \beta_{5} + 5061076330 \beta_{6} - 102864306256 \beta_{7} + 25831535708 \beta_{8} - 5069461584 \beta_{9} - 1640717374 \beta_{10} + 1525291598 \beta_{11} + 548210382 \beta_{12} - 235494278 \beta_{14} + 132587158 \beta_{15} + 663636158 \beta_{17} ) q^{58} \) \( + ( -10334058406 + 116879722286835 \beta_{1} - 116872196975027 \beta_{2} + 3886892395 \beta_{3} - 186715052633537368 \beta_{4} - 906709124852 \beta_{5} + 909049026750 \beta_{6} - 8414871429 \beta_{7} + 16574573676 \beta_{8} + 804568180 \beta_{9} + 514852352 \beta_{10} - 2606281268 \beta_{11} + 714827226 \beta_{12} + 360885130 \beta_{13} + 360885130 \beta_{14} + 144800899 \beta_{15} - 144800899 \beta_{16} - 514852352 \beta_{17} ) q^{59} \) \( + ( 345453733392832796 + 189023318733985 \beta_{1} + 324255302644923 \beta_{2} - 208154409004 \beta_{3} + 62203928000869599 \beta_{4} - 2332445220393 \beta_{5} - 189431860221 \beta_{6} + 83548471741 \beta_{7} + 12302262460 \beta_{8} + 1490447714 \beta_{9} - 101538271 \beta_{10} + 3833643553 \beta_{11} + 804368337 \beta_{12} - 138365415 \beta_{13} - 356749030 \beta_{14} + 58929842 \beta_{15} + 77078169 \beta_{16} + 749155159 \beta_{17} ) q^{60} \) \( + ( 2810784850416158 - 177522084115920 \beta_{1} - 177527298778707 \beta_{2} + 197716573068 \beta_{3} + 4272484776 \beta_{4} + 1451923316259 \beta_{5} + 1439497799811 \beta_{6} - 3119459475 \beta_{7} - 28717221360 \beta_{8} + 11696748831 \beta_{9} - 289641423 \beta_{10} + 682293288 \beta_{11} + 682293288 \beta_{12} - 138926220 \beta_{13} + 138926220 \beta_{14} + 44217396 \beta_{15} + 44217396 \beta_{16} - 1030415241 \beta_{17} ) q^{61} \) \( + ( -66101406793805464 - 362683240891273 \beta_{1} - 16587835065 \beta_{2} - 310939107461 \beta_{3} - 66101435980388108 \beta_{4} + 5272775610 \beta_{5} + 831670398138 \beta_{6} - 296040981561 \beta_{7} - 37998759150 \beta_{8} - 7615559258 \beta_{9} - 3583066445 \beta_{10} - 1454084565 \beta_{11} + 941530070 \beta_{12} + 415439834 \beta_{13} - 130483070 \beta_{16} + 1187451810 \beta_{17} ) q^{62} \) \( + ( -393463685561953363 + 11016545792 \beta_{1} + 91770142979452 \beta_{2} - 135070240521 \beta_{3} + 393463678065100898 \beta_{4} + 6606926995599 \beta_{5} - 11184076210 \beta_{6} + 135335238675 \beta_{7} - 25915407306 \beta_{8} - 29146681496 \beta_{9} + 1816376424 \beta_{10} + 432528572 \beta_{11} + 535245738 \beta_{12} + 93215207 \beta_{14} - 276292858 \beta_{15} - 1713659258 \beta_{17} ) q^{63} \) \( + ( 2688439454 + 655897434168956 \beta_{1} - 655905101708440 \beta_{2} + 1898947466 \beta_{3} - 618929019531169386 \beta_{4} + 1622183458680 \beta_{5} - 1615279762180 \beta_{6} + 411300816426 \beta_{7} + 816665572 \beta_{8} + 21760634260 \beta_{9} - 900277880 \beta_{10} + 5462015796 \beta_{11} + 78905780 \beta_{12} - 855142950 \beta_{13} - 855142950 \beta_{14} - 435391490 \beta_{15} + 435391490 \beta_{16} + 900277880 \beta_{17} ) q^{64} \) \( + ( 383679663585436964 + 56879596071062 \beta_{1} + 371025945055845 \beta_{2} + 23083644941 \beta_{3} + 124393068889625454 \beta_{4} + 2429159146689 \beta_{5} + 7095046770063 \beta_{6} + 464561382194 \beta_{7} - 30395238934 \beta_{8} - 8868215435 \beta_{9} - 2583051477 \beta_{10} - 5643960564 \beta_{11} - 238181756 \beta_{12} + 836349395 \beta_{13} + 2368823765 \beta_{14} - 175593121 \beta_{15} - 419457497 \beta_{16} - 1725879117 \beta_{17} ) q^{65} \) \( + ( 623803649362904410 - 141079858144532 \beta_{1} - 141076540149810 \beta_{2} + 163928632104 \beta_{3} - 19443236654 \beta_{4} - 15221949152224 \beta_{5} - 15194725521048 \beta_{6} + 14766964422 \beta_{7} + 66541650610 \beta_{8} - 45634665626 \beta_{9} - 7205684216 \beta_{10} - 1577391826 \beta_{11} - 1577391826 \beta_{12} + 2544824440 \beta_{13} - 2544824440 \beta_{14} + 220554938 \beta_{15} + 220554938 \beta_{16} + 2275048762 \beta_{17} ) q^{66} \) \( + ( -310895923971551648 - 605755528336915 \beta_{1} + 31584319248 \beta_{2} - 144818529442 \beta_{3} - 310895877385336006 \beta_{4} - 13620218820 \beta_{5} - 885298063953 \beta_{6} - 169630645078 \beta_{7} + 74077925664 \beta_{8} - 5892211780 \beta_{9} + 6848015208 \beta_{10} + 2678218704 \beta_{11} - 2210104932 \beta_{12} + 2101788086 \beta_{13} - 109175868 \beta_{16} - 1959691572 \beta_{17} ) q^{67} \) \( + ( -1423190720252486135 - 43189771006 \beta_{1} + 64034239752872 \beta_{2} + 790210580353 \beta_{3} + 1423190790495144805 \beta_{4} + 13890818654322 \beta_{5} + 16856862730 \beta_{6} - 828007784775 \beta_{7} - 8495497892 \beta_{8} + 100065519528 \beta_{9} + 6116651018 \beta_{10} - 11464296146 \beta_{11} - 2574410034 \beta_{12} + 1335081674 \beta_{14} - 18627106 \beta_{15} + 2773235094 \beta_{17} ) q^{68} \) \( + ( 15791035372 + 118959919619990 \beta_{1} - 118944183047805 \beta_{2} - 10797585868 \beta_{3} - 968988671769375280 \beta_{4} + 8217384545389 \beta_{5} - 8259246524851 \beta_{6} - 435452630325 \beta_{7} - 40811120180 \beta_{8} - 94009857735 \beta_{9} + 2006980707 \beta_{10} + 2523254834 \beta_{11} - 3625881114 \beta_{12} - 687548670 \beta_{13} - 687548670 \beta_{14} + 406523154 \beta_{15} - 406523154 \beta_{16} - 2006980707 \beta_{17} ) q^{69} \) \( + ( 1761354125107668685 + 672340380989927 \beta_{1} + 293257208813262 \beta_{2} - 114865374987 \beta_{3} + 1267711683927907509 \beta_{4} - 20991325995514 \beta_{5} + 6429262385962 \beta_{6} - 972739484676 \beta_{7} + 11170560690 \beta_{8} - 9787895092 \beta_{9} - 7719372800 \beta_{10} - 6798475475 \beta_{11} - 4360867050 \beta_{12} - 2743747000 \beta_{13} - 3044702750 \beta_{14} + 368537150 \beta_{15} + 644152800 \beta_{16} + 3791677150 \beta_{17} ) q^{70} \) \( + ( 610273665144457295 - 418653946884910 \beta_{1} - 418587065993626 \beta_{2} + 166001375714 \beta_{3} + 88303922903 \beta_{4} - 6269478473563 \beta_{5} - 6279510890067 \beta_{6} - 45654840160 \beta_{7} - 15105525670 \beta_{8} + 153993054018 \beta_{9} + 11417196876 \beta_{10} - 2979175656 \beta_{11} - 2979175656 \beta_{12} - 6080555735 \beta_{13} + 6080555735 \beta_{14} - 556135132 \beta_{15} - 556135132 \beta_{16} - 3627467588 \beta_{17} ) q^{71} \) \( + ( -2307024104185599441 + 709390027760013 \beta_{1} - 8506139955 \beta_{2} + 1790702224563 \beta_{3} - 2307024161670897594 \beta_{4} - 27379268685 \beta_{5} + 24134361253239 \beta_{6} + 1821841403928 \beta_{7} - 6789530910 \beta_{8} - 101729786826 \beta_{9} + 4746229275 \beta_{10} + 4438643205 \beta_{11} - 3554338365 \beta_{12} - 6561670257 \beta_{13} + 747852165 \beta_{16} + 3246752295 \beta_{17} ) q^{72} \) \( + ( -1059632684455642577 - 17150396664 \beta_{1} + 501462479884062 \beta_{2} - 1859694519504 \beta_{3} + 1059632749132684763 \beta_{4} + 1717621712520 \beta_{5} - 17127283500 \beta_{6} + 1827585941976 \beta_{7} - 85770057030 \beta_{8} + 29621934672 \beta_{9} - 4059286098 \beta_{10} + 2169102636 \beta_{11} - 3623795046 \beta_{12} - 4191461898 \beta_{14} + 1368207366 \beta_{15} - 1733611584 \beta_{17} ) q^{73} \) \( + ( 86019216925 - 1363273433351693 \beta_{1} + 1363179080621839 \beta_{2} - 47028178513 \beta_{3} - 2720981422496505501 \beta_{4} + 18478660564544 \beta_{5} - 18447981019930 \beta_{6} - 542454545435 \beta_{7} - 140184178258 \beta_{8} + 74195030270 \beta_{9} - 5658413004 \beta_{10} - 24724758782 \beta_{11} - 130523442 \beta_{12} + 5501316115 \beta_{13} + 5501316115 \beta_{14} + 1113372937 \beta_{15} - 1113372937 \beta_{16} + 5658413004 \beta_{17} ) q^{74} \) \( + ( 2042878709510613000 - 860608727884990 \beta_{1} - 1584363155975345 \beta_{2} + 2502487446120 \beta_{3} + 2380278681824270400 \beta_{4} + 15811213922305 \beta_{5} + 14065009515510 \beta_{6} - 207774170610 \beta_{7} + 61058869520 \beta_{8} + 38472054660 \beta_{9} + 24573495140 \beta_{10} + 22033095480 \beta_{11} + 897574870 \beta_{12} + 3758967350 \beta_{13} - 3840886050 \beta_{14} - 253848930 \beta_{15} + 504487240 \beta_{16} - 5290469910 \beta_{17} ) q^{75} \) \( + ( 4201397345002103358 + 1221797834303264 \beta_{1} + 1221712308661244 \beta_{2} - 3212083421534 \beta_{3} - 109024461526 \beta_{4} + 3042568197244 \beta_{5} + 3051936837400 \beta_{6} + 50779137534 \beta_{7} + 2420342360 \beta_{8} - 183498044960 \beta_{9} - 1659319248 \beta_{10} + 5004673644 \beta_{11} + 5004673644 \beta_{12} + 3996104890 \beta_{13} - 3996104890 \beta_{14} - 178555122 \beta_{15} - 178555122 \beta_{16} + 4879946472 \beta_{17} ) q^{76} \) \( + ( -2653908132064388172 + 2109944575956139 \beta_{1} - 32893651445 \beta_{2} - 428321131519 \beta_{3} - 2653908112484838738 \beta_{4} + 54812119550 \beta_{5} - 32674079798052 \beta_{6} - 406882872779 \beta_{7} - 89335129935 \beta_{8} + 130039884853 \beta_{9} - 43356726845 \beta_{10} - 40871799085 \beta_{11} + 7246829605 \beta_{12} + 4793872046 \beta_{13} - 677589530 \beta_{16} - 4761901845 \beta_{17} ) q^{77} \) \( + ( -6068623779211022808 + 161223796317 \beta_{1} - 2417810678074581 \beta_{2} + 777819682787 \beta_{3} + 6068623416386230120 \beta_{4} - 36268810247506 \beta_{5} + 8809377010 \beta_{6} - 543724498123 \beta_{7} + 304261888210 \beta_{8} - 335703966086 \beta_{9} - 50803242761 \beta_{10} + 64062011337 \beta_{11} + 12368411238 \beta_{12} + 5539805794 \beta_{14} - 2673467138 \beta_{15} - 890357338 \beta_{17} ) q^{78} \) \( + ( -240487028252 - 1252318247965320 \beta_{1} + 1252491052870180 \beta_{2} + 123271526168 \beta_{3} - 3591229490117551732 \beta_{4} - 43406931754664 \beta_{5} + 43452675168496 \beta_{6} - 1663088569900 \beta_{7} + 420383820840 \beta_{8} + 70971646740 \beta_{9} + 9528471508 \beta_{10} + 18466262136 \beta_{11} + 11814849944 \beta_{12} - 5168144980 \beta_{13} - 5168144980 \beta_{14} - 4197721664 \beta_{15} + 4197721664 \beta_{16} - 9528471508 \beta_{17} ) q^{79} \) \( + ( 7458334213249857817 - 4693778820732406 \beta_{1} - 4479420207530124 \beta_{2} - 4048815050813 \beta_{3} + 5339811708642845045 \beta_{4} + 30493542271156 \beta_{5} - 58695753069518 \beta_{6} - 2409531046869 \beta_{7} - 119374805954 \beta_{8} + 30421132870 \beta_{9} + 26585602700 \beta_{10} - 4568995850 \beta_{11} + 19085468950 \beta_{12} + 5356709875 \beta_{13} + 13664217875 \beta_{14} - 1561276975 \beta_{15} - 3469627825 \beta_{16} + 529574900 \beta_{17} ) q^{80} \) \( + ( 4122487348468955370 + 3649118925375972 \beta_{1} + 3648998601812301 \beta_{2} + 1790669810727 \beta_{3} - 73521686877 \beta_{4} + 35733523967913 \beta_{5} + 35640781001979 \beta_{6} + 13657646700 \beta_{7} - 223313136360 \beta_{8} - 83504086557 \beta_{9} + 10613908731 \beta_{10} + 10348604394 \beta_{11} + 10348604394 \beta_{12} + 11650422765 \beta_{13} - 11650422765 \beta_{14} + 2346257013 \beta_{15} + 2346257013 \beta_{16} - 3244853193 \beta_{17} ) q^{81} \) \( + ( -4639304766979558868 + 3003537985343587 \beta_{1} - 46520962141 \beta_{2} - 513992040475 \beta_{3} - 4639304724747457568 \beta_{4} + 108792687550 \beta_{5} + 28648063866646 \beta_{6} - 620852934543 \beta_{7} - 160208539968 \beta_{8} + 348430397798 \beta_{9} + 44589168659 \beta_{10} + 64603690237 \beta_{11} + 16331023004 \beta_{12} + 9620260354 \beta_{13} - 1840974904 \beta_{16} + 3683498574 \beta_{17} ) q^{82} \) \( + ( -6651782596400758012 + 21607761340 \beta_{1} - 3683182302169821 \beta_{2} - 2589204631530 \beta_{3} + 6651782515711003282 \beta_{4} - 41816460100123 \beta_{5} + 2252621430 \beta_{6} + 2532864114700 \beta_{7} + 9378970520 \beta_{8} - 60823268380 \beta_{9} + 93637117240 \beta_{10} - 80200899600 \beta_{11} + 12033804010 \beta_{12} - 3291726650 \beta_{14} - 35712830 \beta_{15} - 1402413630 \beta_{17} ) q^{83} \) \( + ( -112343461621 - 8314417550118860 \beta_{1} + 8314471554838606 \beta_{2} + 62282439561 \beta_{3} - 15376386472909948243 \beta_{4} - 48281007392512 \beta_{5} + 48339427057122 \beta_{6} + 6827342609367 \beta_{7} + 211689988462 \beta_{8} + 121792149230 \beta_{9} - 632175800 \beta_{10} + 12772670566 \beta_{11} + 7935177710 \beta_{12} - 9750782375 \beta_{13} - 9750782375 \beta_{14} + 4427405935 \beta_{15} - 4427405935 \beta_{16} + 632175800 \beta_{17} ) q^{84} \) \( + ( 9391529729949378146 + 94115272395325 \beta_{1} - 1496955094403117 \beta_{2} + 2446438934736 \beta_{3} + 1099671214370859184 \beta_{4} + 47481160684452 \beta_{5} - 31907428702156 \beta_{6} + 7924168765016 \beta_{7} + 224165427580 \beta_{8} - 121742916456 \beta_{9} - 152417793881 \beta_{10} + 36901133033 \beta_{11} - 6492645893 \beta_{12} - 27138286190 \beta_{13} - 1485532580 \beta_{14} + 6022752612 \beta_{15} + 4287527434 \beta_{16} + 15807014799 \beta_{17} ) q^{85} \) \( + ( 1860030567926927171 + 4397310973533774 \beta_{1} + 4397375854894939 \beta_{2} + 3524364698156 \beta_{3} + 178783225883 \beta_{4} + 195106158209076 \beta_{5} + 195028872080288 \beta_{6} - 40047441987 \beta_{7} - 190818429540 \beta_{8} + 308165779430 \beta_{9} - 67583405811 \beta_{10} - 5510799832 \beta_{11} - 5510799832 \beta_{12} - 34159647670 \beta_{13} + 34159647670 \beta_{14} - 1129781054 \beta_{15} - 1129781054 \beta_{16} - 14076311236 \beta_{17} ) q^{86} \) \( + ( -1609030435567167700 + 8707067419469234 \beta_{1} + 71697879428 \beta_{2} - 8706044868894 \beta_{3} - 1609030475923086506 \beta_{4} - 131877938140 \beta_{5} - 176693981406168 \beta_{6} - 8708843516130 \beta_{7} + 205913479964 \beta_{8} - 320848161336 \beta_{9} + 62978705948 \beta_{10} + 51776623284 \beta_{11} - 18352876472 \beta_{12} - 20894280346 \beta_{13} + 3742990972 \beta_{16} + 7150793808 \beta_{17} ) q^{87} \) \( + ( -18728949717810716760 - 358162146860 \beta_{1} - 2363603110446252 \beta_{2} + 13043698973408 \beta_{3} + 18728950541301185392 \beta_{4} - 91542552861320 \beta_{5} - 14806669480 \beta_{6} - 13515195241768 \beta_{7} - 660519762664 \beta_{8} + 755751283140 \beta_{9} + 62370752920 \beta_{10} - 98527452020 \beta_{11} - 32540751240 \beta_{12} - 1992457968 \beta_{14} + 7577553260 \beta_{15} + 3615947860 \beta_{17} ) q^{88} \) \( + ( 542306009594 - 6976914952897606 \beta_{1} + 6976550542617178 \beta_{2} - 205285940932 \beta_{3} - 11830958999410162842 \beta_{4} - 150515500409004 \beta_{5} + 150395416934388 \beta_{6} - 5131548765880 \beta_{7} - 899370897526 \beta_{8} - 147735310210 \beta_{9} - 26726274634 \beta_{10} + 100648147814 \beta_{11} - 33924955142 \beta_{12} + 11558385790 \beta_{13} + 11558385790 \beta_{14} + 4520740642 \beta_{15} - 4520740642 \beta_{16} + 26726274634 \beta_{17} ) q^{89} \) \( + ( 22044682245065423534 - 1461301363437733 \beta_{1} - 8807854165941275 \beta_{2} - 9983653499484 \beta_{3} + 8162974203584587229 \beta_{4} - 88836149838381 \beta_{5} - 216772539526027 \beta_{6} + 3449716455864 \beta_{7} - 483225190719 \beta_{8} - 24573305345 \beta_{9} + 116421005208 \beta_{10} - 250180542769 \beta_{11} - 44049324426 \beta_{12} + 21373258170 \beta_{13} - 30263406685 \beta_{14} - 7354482916 \beta_{15} + 2245848063 \beta_{16} - 39833226182 \beta_{17} ) q^{90} \) \( + ( 4794939278181309726 - 7083236674274643 \beta_{1} - 7082960289220143 \beta_{2} + 11868876077088 \beta_{3} - 256275657658 \beta_{4} - 179905488490333 \beta_{5} - 179267537784565 \beta_{6} + 129368990592 \beta_{7} + 1539724668860 \beta_{8} - 654447208880 \beta_{9} + 72970322076 \beta_{10} - 29857348008 \beta_{11} - 29857348008 \beta_{12} + 39399251770 \beta_{13} - 39399251770 \beta_{14} - 8934503436 \beta_{15} - 8934503436 \beta_{16} + 55117201356 \beta_{17} ) q^{91} \) \( + ( -11495624738621156162 + 150412471984453 \beta_{1} + 593331553099 \beta_{2} + 6994707955930 \beta_{3} - 11495623994748906447 \beta_{4} - 387262125305 \beta_{5} + 81616825249055 \beta_{6} + 6831591491147 \beta_{7} + 1483237884692 \beta_{8} - 645601874822 \beta_{9} - 42952963011 \beta_{10} - 155207789383 \beta_{11} - 63039399911 \beta_{12} + 18855058439 \beta_{13} + 4798247611 \beta_{16} - 49215426461 \beta_{17} ) q^{92} \) \( + ( -15860677477648207216 - 286583885667 \beta_{1} + 2458627271342557 \beta_{2} - 5087466611427 \beta_{3} + 15860677956245593684 \beta_{4} + 192843258179800 \beta_{5} + 164852249610 \beta_{6} + 5150727111723 \beta_{7} + 238221599925 \beta_{8} + 738027503121 \beta_{9} - 255912933399 \beta_{10} + 184992136353 \beta_{11} - 42553974663 \beta_{12} + 36558875496 \beta_{14} - 8838467592 \beta_{15} + 28366822383 \beta_{17} ) q^{93} \) \( + ( 313572863758 + 18924461824295903 \beta_{1} - 18924267212633188 \beta_{2} - 313249017322 \beta_{3} - 13749618812547625206 \beta_{4} + 529511009349392 \beta_{5} - 530191778473590 \beta_{6} + 12385440674457 \beta_{7} - 845951146080 \beta_{8} - 1608473294130 \beta_{9} + 50823798448 \beta_{10} - 191164371609 \beta_{11} - 51329300626 \beta_{12} + 31216098245 \beta_{13} + 31216098245 \beta_{14} - 17404035749 \beta_{15} + 17404035749 \beta_{16} - 50823798448 \beta_{17} ) q^{94} \) \( + ( 6560857810934980990 + 2023756567214875 \beta_{1} + 4216259161693465 \beta_{2} + 14402026378745 \beta_{3} + 13650778255730006190 \beta_{4} + 150210145306030 \beta_{5} + 436890022336860 \beta_{6} - 3325719665565 \beta_{7} + 235339838320 \beta_{8} - 141900488820 \beta_{9} + 113256423600 \beta_{10} + 249535983200 \beta_{11} - 13075706900 \beta_{12} + 58773882750 \beta_{13} + 7191870500 \beta_{14} - 8268597175 \beta_{15} - 9317429475 \beta_{16} + 45389499950 \beta_{17} ) q^{95} \) \( + ( 20949416101140538084 + 432520743437616 \beta_{1} + 433397719479784 \beta_{2} - 13511133118308 \beta_{3} + 1369698113068 \beta_{4} - 144473702377112 \beta_{5} - 144912943931792 \beta_{6} - 678565479388 \beta_{7} - 709037544840 \beta_{8} + 2407092175496 \beta_{9} - 4378984472 \beta_{10} - 45292346400 \beta_{11} - 45292346400 \beta_{12} - 3330766500 \beta_{13} + 3330766500 \beta_{14} + 12588658620 \beta_{15} + 12588658620 \beta_{16} - 85727289480 \beta_{17} ) q^{96} \) \( + ( -17698218516894029101 - 20234638590691294 \beta_{1} - 1646399697258 \beta_{2} + 1667792813510 \beta_{3} - 17698221968752234171 \beta_{4} + 116482950090 \beta_{5} - 22099934909830 \beta_{6} + 3607603792556 \beta_{7} - 3570545623254 \beta_{8} - 2197019278026 \beta_{9} - 409894231878 \beta_{10} - 230651006574 \beta_{11} + 39444634392 \beta_{12} - 45351008658 \beta_{13} - 19830899742 \beta_{16} + 139798590912 \beta_{17} ) q^{97} \) \( + ( -1539306985160094584 + 831097575483 \beta_{1} + 5424575466483348 \beta_{2} - 20411996456089 \beta_{3} + 1539306222417658604 \beta_{4} + 112113947551858 \beta_{5} - 790751536330 \beta_{6} + 20575322413765 \beta_{7} - 1430112773194 \beta_{8} - 2160141446624 \beta_{9} + 25937854971 \beta_{10} + 122979918523 \beta_{11} + 30789615962 \beta_{12} - 147958495622 \beta_{14} - 5355950432 \beta_{15} - 118128157532 \beta_{17} ) q^{98} \) \( + ( 420033902864 + 19240967525394656 \beta_{1} - 19242133632377376 \beta_{2} - 339987201711 \beta_{3} - 3111739963965415890 \beta_{4} - 166550364400985 \beta_{5} + 167612910347671 \beta_{6} - 32461726223331 \beta_{7} - 468451084160 \beta_{8} + 2462786408600 \beta_{9} - 79672849856 \beta_{10} - 496740517532 \beta_{11} + 76769913902 \beta_{12} + 684403360 \beta_{13} + 684403360 \beta_{14} + 14825231833 \beta_{15} - 14825231833 \beta_{16} + 79672849856 \beta_{17} ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(18q \) \(\mathstrut -\mathstrut 2q^{2} \) \(\mathstrut +\mathstrut 29448q^{3} \) \(\mathstrut -\mathstrut 7302140q^{5} \) \(\mathstrut +\mathstrut 19792536q^{6} \) \(\mathstrut -\mathstrut 585532752q^{7} \) \(\mathstrut +\mathstrut 930113700q^{8} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(18q \) \(\mathstrut -\mathstrut 2q^{2} \) \(\mathstrut +\mathstrut 29448q^{3} \) \(\mathstrut -\mathstrut 7302140q^{5} \) \(\mathstrut +\mathstrut 19792536q^{6} \) \(\mathstrut -\mathstrut 585532752q^{7} \) \(\mathstrut +\mathstrut 930113700q^{8} \) \(\mathstrut -\mathstrut 17138778090q^{10} \) \(\mathstrut -\mathstrut 6506343064q^{11} \) \(\mathstrut +\mathstrut 140311795848q^{12} \) \(\mathstrut -\mathstrut 369354655602q^{13} \) \(\mathstrut +\mathstrut 998626204320q^{15} \) \(\mathstrut -\mathstrut 6345876020232q^{16} \) \(\mathstrut +\mathstrut 6508314764998q^{17} \) \(\mathstrut -\mathstrut 11265991705902q^{18} \) \(\mathstrut +\mathstrut 38356516779780q^{20} \) \(\mathstrut -\mathstrut 97402974719064q^{21} \) \(\mathstrut +\mathstrut 161054386758096q^{22} \) \(\mathstrut -\mathstrut 81655963656152q^{23} \) \(\mathstrut +\mathstrut 33913283845350q^{25} \) \(\mathstrut +\mathstrut 234104101103636q^{26} \) \(\mathstrut -\mathstrut 511935279143400q^{27} \) \(\mathstrut +\mathstrut 772213614545352q^{28} \) \(\mathstrut -\mathstrut 877127470680480q^{30} \) \(\mathstrut +\mathstrut 1742256896900736q^{31} \) \(\mathstrut -\mathstrut 4072699683722552q^{32} \) \(\mathstrut +\mathstrut 1579138514189496q^{33} \) \(\mathstrut -\mathstrut 4817791288113440q^{35} \) \(\mathstrut +\mathstrut 2214548622647868q^{36} \) \(\mathstrut +\mathstrut 6383289895587498q^{37} \) \(\mathstrut -\mathstrut 14134212502378200q^{38} \) \(\mathstrut +\mathstrut 43297629431150700q^{40} \) \(\mathstrut -\mathstrut 30602304597564664q^{41} \) \(\mathstrut +\mathstrut 48810732785962896q^{42} \) \(\mathstrut -\mathstrut 46399469942287752q^{43} \) \(\mathstrut +\mathstrut 67947826451186070q^{45} \) \(\mathstrut +\mathstrut 17467214047538136q^{46} \) \(\mathstrut -\mathstrut 137109249757437752q^{47} \) \(\mathstrut -\mathstrut 103698596104819152q^{48} \) \(\mathstrut -\mathstrut 40012852869983150q^{50} \) \(\mathstrut +\mathstrut 686047946059669536q^{51} \) \(\mathstrut -\mathstrut 745026624616846452q^{52} \) \(\mathstrut -\mathstrut 184102027021671302q^{53} \) \(\mathstrut +\mathstrut 752956242406989720q^{55} \) \(\mathstrut +\mathstrut 685775432394721200q^{56} \) \(\mathstrut -\mathstrut 785021725278622800q^{57} \) \(\mathstrut -\mathstrut 2701246502184220800q^{58} \) \(\mathstrut +\mathstrut 6219197632829342760q^{60} \) \(\mathstrut +\mathstrut 49879194030044136q^{61} \) \(\mathstrut -\mathstrut 1190554579738283704q^{62} \) \(\mathstrut -\mathstrut 7082176979112100152q^{63} \) \(\mathstrut +\mathstrut 6907070599410641170q^{65} \) \(\mathstrut +\mathstrut 11227963704975799872q^{66} \) \(\mathstrut -\mathstrut 5597336519263153752q^{67} \) \(\mathstrut -\mathstrut 25617328014464685148q^{68} \) \(\mathstrut +\mathstrut 31706333959370270760q^{70} \) \(\mathstrut +\mathstrut 10983277495574224736q^{71} \) \(\mathstrut -\mathstrut 41525052650392592700q^{72} \) \(\mathstrut -\mathstrut 19072400700441139902q^{73} \) \(\mathstrut +\mathstrut 36766882672050433200q^{75} \) \(\mathstrut +\mathstrut 75630007803532292400q^{76} \) \(\mathstrut -\mathstrut 47766064678699704904q^{77} \) \(\mathstrut -\mathstrut 109239984515466629304q^{78} \) \(\mathstrut +\mathstrut 134231700802559812960q^{80} \) \(\mathstrut +\mathstrut 74219234884008322518q^{81} \) \(\mathstrut -\mathstrut 83501540658117625104q^{82} \) \(\mathstrut -\mathstrut 119739383617210903952q^{83} \) \(\mathstrut +\mathstrut 169044713294843560110q^{85} \) \(\mathstrut +\mathstrut 33497378050769887736q^{86} \) \(\mathstrut -\mathstrut 28944830404542403200q^{87} \) \(\mathstrut -\mathstrut 337125565675837197600q^{88} \) \(\mathstrut +\mathstrut 396784292056608506070q^{90} \) \(\mathstrut +\mathstrut 86281377644254817136q^{91} \) \(\mathstrut -\mathstrut 206921051460419061752q^{92} \) \(\mathstrut -\mathstrut 285487695991878257304q^{93} \) \(\mathstrut +\mathstrut 118106834336527303800q^{95} \) \(\mathstrut +\mathstrut 377091713708556287136q^{96} \) \(\mathstrut -\mathstrut 318608382222604685502q^{97} \) \(\mathstrut -\mathstrut 27697031317753120498q^{98} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{18}\mathstrut +\mathstrut \) \(7326009\) \(x^{16}\mathstrut +\mathstrut \) \(21914129354136\) \(x^{14}\mathstrut +\mathstrut \) \(34716886079362208784\) \(x^{12}\mathstrut +\mathstrut \) \(31708037646565185778430976\) \(x^{10}\mathstrut +\mathstrut \) \(17074068956994774485619232997376\) \(x^{8}\mathstrut +\mathstrut \) \(5289719974975064649188570034899779584\) \(x^{6}\mathstrut +\mathstrut \) \(860456443567863287787024826102919920091136\) \(x^{4}\mathstrut +\mathstrut \) \(56032264901085762369236000491606259366923075584\) \(x^{2}\mathstrut +\mathstrut \) \(797517854172776133766249661927470942176176766976\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\(-\)\(40\!\cdots\!31\) \(\nu^{16}\mathstrut -\mathstrut \) \(28\!\cdots\!03\) \(\nu^{14}\mathstrut -\mathstrut \) \(77\!\cdots\!28\) \(\nu^{12}\mathstrut -\mathstrut \) \(10\!\cdots\!16\) \(\nu^{10}\mathstrut -\mathstrut \) \(84\!\cdots\!20\) \(\nu^{8}\mathstrut -\mathstrut \) \(35\!\cdots\!36\) \(\nu^{6}\mathstrut -\mathstrut \) \(73\!\cdots\!48\) \(\nu^{4}\mathstrut -\mathstrut \) \(58\!\cdots\!08\) \(\nu^{2}\mathstrut +\mathstrut \) \(27\!\cdots\!00\) \(\nu\mathstrut -\mathstrut \) \(18\!\cdots\!36\)\()/\)\(27\!\cdots\!00\)
\(\beta_{2}\)\(=\)\((\)\(-\)\(40\!\cdots\!31\) \(\nu^{16}\mathstrut -\mathstrut \) \(28\!\cdots\!03\) \(\nu^{14}\mathstrut -\mathstrut \) \(77\!\cdots\!28\) \(\nu^{12}\mathstrut -\mathstrut \) \(10\!\cdots\!16\) \(\nu^{10}\mathstrut -\mathstrut \) \(84\!\cdots\!20\) \(\nu^{8}\mathstrut -\mathstrut \) \(35\!\cdots\!36\) \(\nu^{6}\mathstrut -\mathstrut \) \(73\!\cdots\!48\) \(\nu^{4}\mathstrut -\mathstrut \) \(58\!\cdots\!08\) \(\nu^{2}\mathstrut -\mathstrut \) \(27\!\cdots\!00\) \(\nu\mathstrut -\mathstrut \) \(18\!\cdots\!36\)\()/\)\(27\!\cdots\!00\)
\(\beta_{3}\)\(=\)\((\)\(-\)\(62\!\cdots\!67\) \(\nu^{16}\mathstrut -\mathstrut \) \(42\!\cdots\!71\) \(\nu^{14}\mathstrut -\mathstrut \) \(11\!\cdots\!96\) \(\nu^{12}\mathstrut -\mathstrut \) \(15\!\cdots\!12\) \(\nu^{10}\mathstrut -\mathstrut \) \(11\!\cdots\!40\) \(\nu^{8}\mathstrut -\mathstrut \) \(48\!\cdots\!52\) \(\nu^{6}\mathstrut -\mathstrut \) \(95\!\cdots\!36\) \(\nu^{4}\mathstrut -\mathstrut \) \(66\!\cdots\!56\) \(\nu^{2}\mathstrut +\mathstrut \) \(25\!\cdots\!48\)\()/\)\(14\!\cdots\!00\)
\(\beta_{4}\)\(=\)\((\)\(-\)\(32\!\cdots\!49\) \(\nu^{17}\mathstrut -\mathstrut \) \(23\!\cdots\!37\) \(\nu^{15}\mathstrut -\mathstrut \) \(66\!\cdots\!12\) \(\nu^{13}\mathstrut -\mathstrut \) \(10\!\cdots\!64\) \(\nu^{11}\mathstrut -\mathstrut \) \(87\!\cdots\!80\) \(\nu^{9}\mathstrut -\mathstrut \) \(43\!\cdots\!44\) \(\nu^{7}\mathstrut -\mathstrut \) \(12\!\cdots\!92\) \(\nu^{5}\mathstrut -\mathstrut \) \(17\!\cdots\!32\) \(\nu^{3}\mathstrut -\mathstrut \) \(10\!\cdots\!44\) \(\nu\)\()/\)\(37\!\cdots\!00\)
\(\beta_{5}\)\(=\)\((\)\(-\)\(30\!\cdots\!29\) \(\nu^{17}\mathstrut +\mathstrut \) \(49\!\cdots\!80\) \(\nu^{16}\mathstrut -\mathstrut \) \(25\!\cdots\!77\) \(\nu^{15}\mathstrut +\mathstrut \) \(33\!\cdots\!40\) \(\nu^{14}\mathstrut -\mathstrut \) \(86\!\cdots\!52\) \(\nu^{13}\mathstrut +\mathstrut \) \(90\!\cdots\!40\) \(\nu^{12}\mathstrut -\mathstrut \) \(16\!\cdots\!44\) \(\nu^{11}\mathstrut +\mathstrut \) \(12\!\cdots\!80\) \(\nu^{10}\mathstrut -\mathstrut \) \(17\!\cdots\!80\) \(\nu^{9}\mathstrut +\mathstrut \) \(92\!\cdots\!00\) \(\nu^{8}\mathstrut -\mathstrut \) \(10\!\cdots\!24\) \(\nu^{7}\mathstrut +\mathstrut \) \(36\!\cdots\!80\) \(\nu^{6}\mathstrut -\mathstrut \) \(39\!\cdots\!32\) \(\nu^{5}\mathstrut +\mathstrut \) \(71\!\cdots\!40\) \(\nu^{4}\mathstrut -\mathstrut \) \(75\!\cdots\!72\) \(\nu^{3}\mathstrut +\mathstrut \) \(50\!\cdots\!40\) \(\nu^{2}\mathstrut -\mathstrut \) \(57\!\cdots\!24\) \(\nu\mathstrut -\mathstrut \) \(21\!\cdots\!20\)\()/\)\(58\!\cdots\!00\)
\(\beta_{6}\)\(=\)\((\)\(30\!\cdots\!29\) \(\nu^{17}\mathstrut +\mathstrut \) \(49\!\cdots\!80\) \(\nu^{16}\mathstrut +\mathstrut \) \(25\!\cdots\!77\) \(\nu^{15}\mathstrut +\mathstrut \) \(33\!\cdots\!40\) \(\nu^{14}\mathstrut +\mathstrut \) \(86\!\cdots\!52\) \(\nu^{13}\mathstrut +\mathstrut \) \(90\!\cdots\!40\) \(\nu^{12}\mathstrut +\mathstrut \) \(16\!\cdots\!44\) \(\nu^{11}\mathstrut +\mathstrut \) \(12\!\cdots\!80\) \(\nu^{10}\mathstrut +\mathstrut \) \(17\!\cdots\!80\) \(\nu^{9}\mathstrut +\mathstrut \) \(92\!\cdots\!00\) \(\nu^{8}\mathstrut +\mathstrut \) \(10\!\cdots\!24\) \(\nu^{7}\mathstrut +\mathstrut \) \(36\!\cdots\!80\) \(\nu^{6}\mathstrut +\mathstrut \) \(39\!\cdots\!32\) \(\nu^{5}\mathstrut +\mathstrut \) \(71\!\cdots\!40\) \(\nu^{4}\mathstrut +\mathstrut \) \(75\!\cdots\!72\) \(\nu^{3}\mathstrut +\mathstrut \) \(50\!\cdots\!40\) \(\nu^{2}\mathstrut +\mathstrut \) \(57\!\cdots\!24\) \(\nu\mathstrut -\mathstrut \) \(21\!\cdots\!20\)\()/\)\(58\!\cdots\!00\)
\(\beta_{7}\)\(=\)\((\)\(-\)\(64\!\cdots\!99\) \(\nu^{17}\mathstrut -\mathstrut \) \(47\!\cdots\!87\) \(\nu^{15}\mathstrut -\mathstrut \) \(13\!\cdots\!12\) \(\nu^{13}\mathstrut -\mathstrut \) \(21\!\cdots\!64\) \(\nu^{11}\mathstrut -\mathstrut \) \(19\!\cdots\!80\) \(\nu^{9}\mathstrut -\mathstrut \) \(98\!\cdots\!44\) \(\nu^{7}\mathstrut -\mathstrut \) \(29\!\cdots\!92\) \(\nu^{5}\mathstrut -\mathstrut \) \(44\!\cdots\!32\) \(\nu^{3}\mathstrut -\mathstrut \) \(27\!\cdots\!44\) \(\nu\)\()/\)\(58\!\cdots\!00\)
\(\beta_{8}\)\(=\)\((\)\(94\!\cdots\!85\) \(\nu^{17}\mathstrut -\mathstrut \) \(45\!\cdots\!76\) \(\nu^{16}\mathstrut +\mathstrut \) \(69\!\cdots\!05\) \(\nu^{15}\mathstrut -\mathstrut \) \(39\!\cdots\!88\) \(\nu^{14}\mathstrut +\mathstrut \) \(21\!\cdots\!80\) \(\nu^{13}\mathstrut -\mathstrut \) \(13\!\cdots\!88\) \(\nu^{12}\mathstrut +\mathstrut \) \(34\!\cdots\!60\) \(\nu^{11}\mathstrut -\mathstrut \) \(25\!\cdots\!36\) \(\nu^{10}\mathstrut +\mathstrut \) \(32\!\cdots\!00\) \(\nu^{9}\mathstrut -\mathstrut \) \(25\!\cdots\!20\) \(\nu^{8}\mathstrut +\mathstrut \) \(18\!\cdots\!60\) \(\nu^{7}\mathstrut -\mathstrut \) \(14\!\cdots\!56\) \(\nu^{6}\mathstrut +\mathstrut \) \(62\!\cdots\!80\) \(\nu^{5}\mathstrut -\mathstrut \) \(38\!\cdots\!08\) \(\nu^{4}\mathstrut +\mathstrut \) \(10\!\cdots\!80\) \(\nu^{3}\mathstrut -\mathstrut \) \(42\!\cdots\!68\) \(\nu^{2}\mathstrut +\mathstrut \) \(73\!\cdots\!60\) \(\nu\mathstrut -\mathstrut \) \(79\!\cdots\!56\)\()/\)\(58\!\cdots\!00\)
\(\beta_{9}\)\(=\)\((\)\(-\)\(31\!\cdots\!33\) \(\nu^{17}\mathstrut -\mathstrut \) \(55\!\cdots\!00\) \(\nu^{16}\mathstrut -\mathstrut \) \(23\!\cdots\!29\) \(\nu^{15}\mathstrut -\mathstrut \) \(36\!\cdots\!00\) \(\nu^{14}\mathstrut -\mathstrut \) \(71\!\cdots\!04\) \(\nu^{13}\mathstrut -\mathstrut \) \(94\!\cdots\!00\) \(\nu^{12}\mathstrut -\mathstrut \) \(11\!\cdots\!88\) \(\nu^{11}\mathstrut -\mathstrut \) \(12\!\cdots\!00\) \(\nu^{10}\mathstrut -\mathstrut \) \(10\!\cdots\!60\) \(\nu^{9}\mathstrut -\mathstrut \) \(81\!\cdots\!00\) \(\nu^{8}\mathstrut -\mathstrut \) \(59\!\cdots\!48\) \(\nu^{7}\mathstrut -\mathstrut \) \(27\!\cdots\!00\) \(\nu^{6}\mathstrut -\mathstrut \) \(18\!\cdots\!64\) \(\nu^{5}\mathstrut -\mathstrut \) \(41\!\cdots\!00\) \(\nu^{4}\mathstrut -\mathstrut \) \(31\!\cdots\!44\) \(\nu^{3}\mathstrut -\mathstrut \) \(20\!\cdots\!00\) \(\nu^{2}\mathstrut -\mathstrut \) \(21\!\cdots\!48\) \(\nu\mathstrut -\mathstrut \) \(27\!\cdots\!00\)\()/\)\(58\!\cdots\!00\)
\(\beta_{10}\)\(=\)\((\)\(82\!\cdots\!71\) \(\nu^{17}\mathstrut -\mathstrut \) \(10\!\cdots\!04\) \(\nu^{16}\mathstrut +\mathstrut \) \(61\!\cdots\!23\) \(\nu^{15}\mathstrut -\mathstrut \) \(68\!\cdots\!52\) \(\nu^{14}\mathstrut +\mathstrut \) \(18\!\cdots\!48\) \(\nu^{13}\mathstrut -\mathstrut \) \(18\!\cdots\!52\) \(\nu^{12}\mathstrut +\mathstrut \) \(30\!\cdots\!56\) \(\nu^{11}\mathstrut -\mathstrut \) \(26\!\cdots\!44\) \(\nu^{10}\mathstrut +\mathstrut \) \(28\!\cdots\!20\) \(\nu^{9}\mathstrut -\mathstrut \) \(20\!\cdots\!80\) \(\nu^{8}\mathstrut +\mathstrut \) \(16\!\cdots\!76\) \(\nu^{7}\mathstrut -\mathstrut \) \(85\!\cdots\!24\) \(\nu^{6}\mathstrut +\mathstrut \) \(51\!\cdots\!68\) \(\nu^{5}\mathstrut -\mathstrut \) \(18\!\cdots\!32\) \(\nu^{4}\mathstrut +\mathstrut \) \(86\!\cdots\!28\) \(\nu^{3}\mathstrut -\mathstrut \) \(14\!\cdots\!72\) \(\nu^{2}\mathstrut +\mathstrut \) \(57\!\cdots\!76\) \(\nu\mathstrut +\mathstrut \) \(82\!\cdots\!76\)\()/\)\(85\!\cdots\!00\)
\(\beta_{11}\)\(=\)\((\)\(11\!\cdots\!97\) \(\nu^{17}\mathstrut +\mathstrut \) \(11\!\cdots\!00\) \(\nu^{16}\mathstrut +\mathstrut \) \(76\!\cdots\!61\) \(\nu^{15}\mathstrut +\mathstrut \) \(96\!\cdots\!00\) \(\nu^{14}\mathstrut +\mathstrut \) \(20\!\cdots\!36\) \(\nu^{13}\mathstrut +\mathstrut \) \(32\!\cdots\!00\) \(\nu^{12}\mathstrut +\mathstrut \) \(26\!\cdots\!92\) \(\nu^{11}\mathstrut +\mathstrut \) \(57\!\cdots\!00\) \(\nu^{10}\mathstrut +\mathstrut \) \(18\!\cdots\!40\) \(\nu^{9}\mathstrut +\mathstrut \) \(55\!\cdots\!00\) \(\nu^{8}\mathstrut +\mathstrut \) \(66\!\cdots\!32\) \(\nu^{7}\mathstrut +\mathstrut \) \(29\!\cdots\!00\) \(\nu^{6}\mathstrut +\mathstrut \) \(84\!\cdots\!76\) \(\nu^{5}\mathstrut +\mathstrut \) \(80\!\cdots\!00\) \(\nu^{4}\mathstrut -\mathstrut \) \(62\!\cdots\!04\) \(\nu^{3}\mathstrut +\mathstrut \) \(88\!\cdots\!00\) \(\nu^{2}\mathstrut -\mathstrut \) \(15\!\cdots\!68\) \(\nu\mathstrut +\mathstrut \) \(14\!\cdots\!00\)\()/\)\(58\!\cdots\!00\)
\(\beta_{12}\)\(=\)\((\)\(-\)\(18\!\cdots\!27\) \(\nu^{17}\mathstrut +\mathstrut \) \(49\!\cdots\!16\) \(\nu^{16}\mathstrut -\mathstrut \) \(13\!\cdots\!51\) \(\nu^{15}\mathstrut +\mathstrut \) \(33\!\cdots\!08\) \(\nu^{14}\mathstrut -\mathstrut \) \(38\!\cdots\!76\) \(\nu^{13}\mathstrut +\mathstrut \) \(91\!\cdots\!08\) \(\nu^{12}\mathstrut -\mathstrut \) \(60\!\cdots\!72\) \(\nu^{11}\mathstrut +\mathstrut \) \(12\!\cdots\!76\) \(\nu^{10}\mathstrut -\mathstrut \) \(52\!\cdots\!40\) \(\nu^{9}\mathstrut +\mathstrut \) \(92\!\cdots\!20\) \(\nu^{8}\mathstrut -\mathstrut \) \(27\!\cdots\!12\) \(\nu^{7}\mathstrut +\mathstrut \) \(37\!\cdots\!96\) \(\nu^{6}\mathstrut -\mathstrut \) \(77\!\cdots\!16\) \(\nu^{5}\mathstrut +\mathstrut \) \(74\!\cdots\!28\) \(\nu^{4}\mathstrut -\mathstrut \) \(11\!\cdots\!36\) \(\nu^{3}\mathstrut +\mathstrut \) \(62\!\cdots\!88\) \(\nu^{2}\mathstrut -\mathstrut \) \(71\!\cdots\!12\) \(\nu\mathstrut +\mathstrut \) \(10\!\cdots\!96\)\()/\)\(29\!\cdots\!00\)
\(\beta_{13}\)\(=\)\((\)\(23\!\cdots\!53\) \(\nu^{17}\mathstrut -\mathstrut \) \(59\!\cdots\!32\) \(\nu^{16}\mathstrut +\mathstrut \) \(16\!\cdots\!89\) \(\nu^{15}\mathstrut -\mathstrut \) \(38\!\cdots\!16\) \(\nu^{14}\mathstrut +\mathstrut \) \(45\!\cdots\!64\) \(\nu^{13}\mathstrut -\mathstrut \) \(97\!\cdots\!16\) \(\nu^{12}\mathstrut +\mathstrut \) \(66\!\cdots\!08\) \(\nu^{11}\mathstrut -\mathstrut \) \(12\!\cdots\!52\) \(\nu^{10}\mathstrut +\mathstrut \) \(54\!\cdots\!60\) \(\nu^{9}\mathstrut -\mathstrut \) \(80\!\cdots\!40\) \(\nu^{8}\mathstrut +\mathstrut \) \(26\!\cdots\!68\) \(\nu^{7}\mathstrut -\mathstrut \) \(28\!\cdots\!92\) \(\nu^{6}\mathstrut +\mathstrut \) \(74\!\cdots\!24\) \(\nu^{5}\mathstrut -\mathstrut \) \(51\!\cdots\!56\) \(\nu^{4}\mathstrut +\mathstrut \) \(11\!\cdots\!04\) \(\nu^{3}\mathstrut -\mathstrut \) \(45\!\cdots\!76\) \(\nu^{2}\mathstrut +\mathstrut \) \(80\!\cdots\!68\) \(\nu\mathstrut -\mathstrut \) \(43\!\cdots\!92\)\()/\)\(36\!\cdots\!00\)
\(\beta_{14}\)\(=\)\((\)\(92\!\cdots\!29\) \(\nu^{17}\mathstrut +\mathstrut \) \(16\!\cdots\!52\) \(\nu^{16}\mathstrut +\mathstrut \) \(64\!\cdots\!77\) \(\nu^{15}\mathstrut +\mathstrut \) \(10\!\cdots\!76\) \(\nu^{14}\mathstrut +\mathstrut \) \(18\!\cdots\!52\) \(\nu^{13}\mathstrut +\mathstrut \) \(26\!\cdots\!76\) \(\nu^{12}\mathstrut +\mathstrut \) \(27\!\cdots\!44\) \(\nu^{11}\mathstrut +\mathstrut \) \(33\!\cdots\!72\) \(\nu^{10}\mathstrut +\mathstrut \) \(23\!\cdots\!80\) \(\nu^{9}\mathstrut +\mathstrut \) \(21\!\cdots\!40\) \(\nu^{8}\mathstrut +\mathstrut \) \(11\!\cdots\!24\) \(\nu^{7}\mathstrut +\mathstrut \) \(73\!\cdots\!12\) \(\nu^{6}\mathstrut +\mathstrut \) \(33\!\cdots\!32\) \(\nu^{5}\mathstrut +\mathstrut \) \(13\!\cdots\!16\) \(\nu^{4}\mathstrut +\mathstrut \) \(54\!\cdots\!72\) \(\nu^{3}\mathstrut +\mathstrut \) \(12\!\cdots\!36\) \(\nu^{2}\mathstrut +\mathstrut \) \(37\!\cdots\!24\) \(\nu\mathstrut +\mathstrut \) \(97\!\cdots\!12\)\()/\)\(11\!\cdots\!00\)
\(\beta_{15}\)\(=\)\((\)\(-\)\(10\!\cdots\!39\) \(\nu^{17}\mathstrut +\mathstrut \) \(25\!\cdots\!28\) \(\nu^{16}\mathstrut -\mathstrut \) \(68\!\cdots\!07\) \(\nu^{15}\mathstrut +\mathstrut \) \(17\!\cdots\!64\) \(\nu^{14}\mathstrut -\mathstrut \) \(17\!\cdots\!32\) \(\nu^{13}\mathstrut +\mathstrut \) \(48\!\cdots\!64\) \(\nu^{12}\mathstrut -\mathstrut \) \(23\!\cdots\!04\) \(\nu^{11}\mathstrut +\mathstrut \) \(67\!\cdots\!08\) \(\nu^{10}\mathstrut -\mathstrut \) \(15\!\cdots\!80\) \(\nu^{9}\mathstrut +\mathstrut \) \(52\!\cdots\!60\) \(\nu^{8}\mathstrut -\mathstrut \) \(50\!\cdots\!84\) \(\nu^{7}\mathstrut +\mathstrut \) \(22\!\cdots\!68\) \(\nu^{6}\mathstrut -\mathstrut \) \(45\!\cdots\!12\) \(\nu^{5}\mathstrut +\mathstrut \) \(46\!\cdots\!24\) \(\nu^{4}\mathstrut +\mathstrut \) \(94\!\cdots\!48\) \(\nu^{3}\mathstrut +\mathstrut \) \(36\!\cdots\!04\) \(\nu^{2}\mathstrut +\mathstrut \) \(16\!\cdots\!16\) \(\nu\mathstrut +\mathstrut \) \(56\!\cdots\!68\)\()/\)\(85\!\cdots\!00\)
\(\beta_{16}\)\(=\)\((\)\(71\!\cdots\!01\) \(\nu^{17}\mathstrut +\mathstrut \) \(17\!\cdots\!76\) \(\nu^{16}\mathstrut +\mathstrut \) \(47\!\cdots\!13\) \(\nu^{15}\mathstrut +\mathstrut \) \(12\!\cdots\!88\) \(\nu^{14}\mathstrut +\mathstrut \) \(12\!\cdots\!88\) \(\nu^{13}\mathstrut +\mathstrut \) \(33\!\cdots\!88\) \(\nu^{12}\mathstrut +\mathstrut \) \(16\!\cdots\!36\) \(\nu^{11}\mathstrut +\mathstrut \) \(47\!\cdots\!36\) \(\nu^{10}\mathstrut +\mathstrut \) \(11\!\cdots\!20\) \(\nu^{9}\mathstrut +\mathstrut \) \(36\!\cdots\!20\) \(\nu^{8}\mathstrut +\mathstrut \) \(38\!\cdots\!56\) \(\nu^{7}\mathstrut +\mathstrut \) \(15\!\cdots\!56\) \(\nu^{6}\mathstrut +\mathstrut \) \(45\!\cdots\!08\) \(\nu^{5}\mathstrut +\mathstrut \) \(31\!\cdots\!08\) \(\nu^{4}\mathstrut -\mathstrut \) \(36\!\cdots\!32\) \(\nu^{3}\mathstrut +\mathstrut \) \(25\!\cdots\!68\) \(\nu^{2}\mathstrut -\mathstrut \) \(93\!\cdots\!44\) \(\nu\mathstrut +\mathstrut \) \(46\!\cdots\!56\)\()/\)\(58\!\cdots\!00\)
\(\beta_{17}\)\(=\)\((\)\(-\)\(38\!\cdots\!81\) \(\nu^{17}\mathstrut +\mathstrut \) \(92\!\cdots\!44\) \(\nu^{16}\mathstrut -\mathstrut \) \(28\!\cdots\!53\) \(\nu^{15}\mathstrut +\mathstrut \) \(44\!\cdots\!72\) \(\nu^{14}\mathstrut -\mathstrut \) \(87\!\cdots\!28\) \(\nu^{13}\mathstrut +\mathstrut \) \(52\!\cdots\!72\) \(\nu^{12}\mathstrut -\mathstrut \) \(14\!\cdots\!16\) \(\nu^{11}\mathstrut -\mathstrut \) \(63\!\cdots\!16\) \(\nu^{10}\mathstrut -\mathstrut \) \(13\!\cdots\!20\) \(\nu^{9}\mathstrut -\mathstrut \) \(19\!\cdots\!20\) \(\nu^{8}\mathstrut -\mathstrut \) \(74\!\cdots\!36\) \(\nu^{7}\mathstrut -\mathstrut \) \(15\!\cdots\!36\) \(\nu^{6}\mathstrut -\mathstrut \) \(23\!\cdots\!48\) \(\nu^{5}\mathstrut -\mathstrut \) \(54\!\cdots\!48\) \(\nu^{4}\mathstrut -\mathstrut \) \(40\!\cdots\!08\) \(\nu^{3}\mathstrut -\mathstrut \) \(63\!\cdots\!08\) \(\nu^{2}\mathstrut -\mathstrut \) \(27\!\cdots\!36\) \(\nu\mathstrut -\mathstrut \) \(26\!\cdots\!36\)\()/\)\(29\!\cdots\!00\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-\)\(\beta_{2}\mathstrut +\mathstrut \) \(\beta_{1}\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(2\) \(\beta_{6}\mathstrut +\mathstrut \) \(2\) \(\beta_{5}\mathstrut +\mathstrut \) \(\beta_{3}\mathstrut -\mathstrut \) \(31\) \(\beta_{2}\mathstrut -\mathstrut \) \(31\) \(\beta_{1}\mathstrut -\mathstrut \) \(1627995\)\()/2\)
\(\nu^{3}\)\(=\)\((\)\(2\) \(\beta_{17}\mathstrut -\mathstrut \) \(\beta_{16}\mathstrut +\mathstrut \) \(\beta_{15}\mathstrut +\mathstrut \) \(\beta_{14}\mathstrut +\mathstrut \) \(\beta_{13}\mathstrut +\mathstrut \) \(4\) \(\beta_{11}\mathstrut -\mathstrut \) \(2\) \(\beta_{10}\mathstrut +\mathstrut \) \(24\) \(\beta_{9}\mathstrut -\mathstrut \) \(20\) \(\beta_{8}\mathstrut -\mathstrut \) \(307\) \(\beta_{7}\mathstrut +\mathstrut \) \(3382\) \(\beta_{6}\mathstrut -\mathstrut \) \(3372\) \(\beta_{5}\mathstrut -\mathstrut \) \(103214285\) \(\beta_{4}\mathstrut -\mathstrut \) \(3\) \(\beta_{3}\mathstrut +\mathstrut \) \(2685832\) \(\beta_{2}\mathstrut -\mathstrut \) \(2685850\) \(\beta_{1}\mathstrut +\mathstrut \) \(15\)\()/4\)
\(\nu^{4}\)\(=\)\((\)\(-\)\(302\) \(\beta_{17}\mathstrut -\mathstrut \) \(253\) \(\beta_{16}\mathstrut -\mathstrut \) \(253\) \(\beta_{15}\mathstrut -\mathstrut \) \(935\) \(\beta_{14}\mathstrut +\mathstrut \) \(935\) \(\beta_{13}\mathstrut +\mathstrut \) \(3476\) \(\beta_{12}\mathstrut +\mathstrut \) \(3476\) \(\beta_{11}\mathstrut -\mathstrut \) \(2322\) \(\beta_{10}\mathstrut -\mathstrut \) \(48330\) \(\beta_{9}\mathstrut -\mathstrut \) \(60030\) \(\beta_{8}\mathstrut +\mathstrut \) \(14161\) \(\beta_{7}\mathstrut -\mathstrut \) \(7816580\) \(\beta_{6}\mathstrut -\mathstrut \) \(7793814\) \(\beta_{5}\mathstrut -\mathstrut \) \(34189\) \(\beta_{4}\mathstrut -\mathstrut \) \(3993801\) \(\beta_{3}\mathstrut -\mathstrut \) \(8825134\) \(\beta_{2}\mathstrut -\mathstrut \) \(8779844\) \(\beta_{1}\mathstrut +\mathstrut \) \(4374290096441\)\()/4\)
\(\nu^{5}\)\(=\)\((\)\(-\)\(5340724\) \(\beta_{17}\mathstrut +\mathstrut \) \(2252047\) \(\beta_{16}\mathstrut -\mathstrut \) \(2252047\) \(\beta_{15}\mathstrut -\mathstrut \) \(2063509\) \(\beta_{14}\mathstrut -\mathstrut \) \(2063509\) \(\beta_{13}\mathstrut -\mathstrut \) \(1190370\) \(\beta_{12}\mathstrut -\mathstrut \) \(8177458\) \(\beta_{11}\mathstrut +\mathstrut \) \(5340724\) \(\beta_{10}\mathstrut -\mathstrut \) \(87694834\) \(\beta_{9}\mathstrut +\mathstrut \) \(33040374\) \(\beta_{8}\mathstrut +\mathstrut \) \(1214510355\) \(\beta_{7}\mathstrut -\mathstrut \) \(9720513198\) \(\beta_{6}\mathstrut +\mathstrut \) \(9683449988\) \(\beta_{5}\mathstrut -\mathstrut \) \(9873133405619\) \(\beta_{4}\mathstrut +\mathstrut \) \(3791171\) \(\beta_{3}\mathstrut -\mathstrut \) \(4286150877060\) \(\beta_{2}\mathstrut +\mathstrut \) \(4286198359346\) \(\beta_{1}\mathstrut -\mathstrut \) \(29695127\)\()/4\)
\(\nu^{6}\)\(=\)\((\)\(752221990\) \(\beta_{17}\mathstrut +\mathstrut \) \(880920065\) \(\beta_{16}\mathstrut +\mathstrut \) \(880920065\) \(\beta_{15}\mathstrut +\mathstrut \) \(2878617875\) \(\beta_{14}\mathstrut -\mathstrut \) \(2878617875\) \(\beta_{13}\mathstrut -\mathstrut \) \(8661986500\) \(\beta_{12}\mathstrut -\mathstrut \) \(8661986500\) \(\beta_{11}\mathstrut +\mathstrut \) \(8817991578\) \(\beta_{10}\mathstrut +\mathstrut \) \(127102527986\) \(\beta_{9}\mathstrut +\mathstrut \) \(146484726070\) \(\beta_{8}\mathstrut -\mathstrut \) \(38071685573\) \(\beta_{7}\mathstrut +\mathstrut \) \(13082533979204\) \(\beta_{6}\mathstrut +\mathstrut \) \(13026306124414\) \(\beta_{5}\mathstrut +\mathstrut \) \(88280192433\) \(\beta_{4}\mathstrut +\mathstrut \) \(7376625218325\) \(\beta_{3}\mathstrut +\mathstrut \) \(453337857740606\) \(\beta_{2}\mathstrut +\mathstrut \) \(453222966492748\) \(\beta_{1}\mathstrut -\mathstrut \) \(6982865995099735861\)\()/4\)
\(\nu^{7}\)\(=\)\((\)\(11639024494588\) \(\beta_{17}\mathstrut -\mathstrut \) \(4319612645399\) \(\beta_{16}\mathstrut +\mathstrut \) \(4319612645399\) \(\beta_{15}\mathstrut +\mathstrut \) \(3821660837909\) \(\beta_{14}\mathstrut +\mathstrut \) \(3821660837909\) \(\beta_{13}\mathstrut +\mathstrut \) \(3405273609610\) \(\beta_{12}\mathstrut +\mathstrut \) \(16177338720906\) \(\beta_{11}\mathstrut -\mathstrut \) \(11639024494588\) \(\beta_{10}\mathstrut +\mathstrut \) \(210147571322746\) \(\beta_{9}\mathstrut -\mathstrut \) \(58085016065630\) \(\beta_{8}\mathstrut -\mathstrut \) \(3292693708783203\) \(\beta_{7}\mathstrut +\mathstrut \) \(21367080022662830\) \(\beta_{6}\mathstrut -\mathstrut \) \(21278642912109220\) \(\beta_{5}\mathstrut +\mathstrut \) \(730423918911235782531\) \(\beta_{4}\mathstrut -\mathstrut \) \(5627504954387\) \(\beta_{3}\mathstrut +\mathstrut \) \(7343353110445477812\) \(\beta_{2}\mathstrut -\mathstrut \) \(7343457340880166514\) \(\beta_{1}\mathstrut +\mathstrut \) \(57937332066375\)\()/4\)
\(\nu^{8}\)\(=\)\((\)\(-\)\(1558304960746918\) \(\beta_{17}\mathstrut -\mathstrut \) \(2262681425410817\) \(\beta_{16}\mathstrut -\mathstrut \) \(2262681425410817\) \(\beta_{15}\mathstrut -\mathstrut \) \(6555282095399315\) \(\beta_{14}\mathstrut +\mathstrut \) \(6555282095399315\) \(\beta_{13}\mathstrut +\mathstrut \) \(17886814326073924\) \(\beta_{12}\mathstrut +\mathstrut \) \(17886814326073924\) \(\beta_{11}\mathstrut -\mathstrut \) \(21664062717155610\) \(\beta_{10}\mathstrut -\mathstrut \) \(269604323163637874\) \(\beta_{9}\mathstrut -\mathstrut \) \(297796707321122230\) \(\beta_{8}\mathstrut +\mathstrut \) \(81598808831558981\) \(\beta_{7}\mathstrut -\mathstrut \) \(22621515157362967812\) \(\beta_{6}\mathstrut -\mathstrut \) \(22506253194571370558\) \(\beta_{5}\mathstrut -\mathstrut \) \(185420770654256433\) \(\beta_{4}\mathstrut -\mathstrut \) \(13531838991904104437\) \(\beta_{3}\mathstrut -\mathstrut \) \(1735509722012319292062\) \(\beta_{2}\mathstrut -\mathstrut \) \(1735270813308907806380\) \(\beta_{1}\mathstrut +\mathstrut \) \(11965940071090991953338069\)\()/4\)
\(\nu^{9}\)\(=\)\((\)\(-\)\(23863122329098745436\) \(\beta_{17}\mathstrut +\mathstrut \) \(8054059900275062343\) \(\beta_{16}\mathstrut -\mathstrut \) \(8054059900275062343\) \(\beta_{15}\mathstrut -\mathstrut \) \(7072864687458005061\) \(\beta_{14}\mathstrut -\mathstrut \) \(7072864687458005061\) \(\beta_{13}\mathstrut -\mathstrut \) \(7445123267136723850\) \(\beta_{12}\mathstrut -\mathstrut \) \(31708437520199099722\) \(\beta_{11}\mathstrut +\mathstrut \) \(23863122329098745436\) \(\beta_{10}\mathstrut -\mathstrut \) \(446216843291749745146\) \(\beta_{9}\mathstrut +\mathstrut \) \(112356858531478921246\) \(\beta_{8}\mathstrut +\mathstrut \) \(7810743434745587545075\) \(\beta_{7}\mathstrut -\mathstrut \) \(42245681443191944526030\) \(\beta_{6}\mathstrut +\mathstrut \) \(42059049842880808387300\) \(\beta_{5}\mathstrut -\mathstrut \) \(2812153335699271989537428755\) \(\beta_{4}\mathstrut +\mathstrut \) \(11019252293100476579\) \(\beta_{3}\mathstrut -\mathstrut \) \(13010195446341265777990964\) \(\beta_{2}\mathstrut +\mathstrut \) \(13010411961761354215437138\) \(\beta_{1}\mathstrut -\mathstrut \) \(115991174719192272663\)\()/4\)
\(\nu^{10}\)\(=\)\((\)\(3133366599252183549510\) \(\beta_{17}\mathstrut +\mathstrut \) \(5203497326004704215665\) \(\beta_{16}\mathstrut +\mathstrut \) \(5203497326004704215665\) \(\beta_{15}\mathstrut +\mathstrut \) \(13648318563412577128675\) \(\beta_{14}\mathstrut -\mathstrut \) \(13648318563412577128675\) \(\beta_{13}\mathstrut -\mathstrut \) \(35373000751335076875780\) \(\beta_{12}\mathstrut -\mathstrut \) \(35373000751335076875780\) \(\beta_{11}\mathstrut +\mathstrut \) \(45399473155097313952890\) \(\beta_{10}\mathstrut +\mathstrut \) \(537109900719405595453010\) \(\beta_{9}\mathstrut +\mathstrut \) \(582550262625868455697750\) \(\beta_{8}\mathstrut -\mathstrut \) \(162887779238919815943285\) \(\beta_{7}\mathstrut +\mathstrut \) \(39594058477629350954114500\) \(\beta_{6}\mathstrut +\mathstrut \) \(39367160534979332432895070\) \(\beta_{5}\mathstrut +\mathstrut \) \(367903771988077232363425\) \(\beta_{4}\mathstrut +\mathstrut \) \(24982594726713306915237285\) \(\beta_{3}\mathstrut +\mathstrut \) \(4833175763507878243413330750\) \(\beta_{2}\mathstrut +\mathstrut \) \(4832703862480656817651053420\) \(\beta_{1}\mathstrut -\mathstrut \) \(21202160566213628333296242831749\)\()/4\)
\(\nu^{11}\)\(=\)\((\)\(47799894363283767916337180\) \(\beta_{17}\mathstrut -\mathstrut \) \(15005660059603066558965015\) \(\beta_{16}\mathstrut +\mathstrut \) \(15005660059603066558965015\) \(\beta_{15}\mathstrut +\mathstrut \) \(13356421749439843881551925\) \(\beta_{14}\mathstrut +\mathstrut \) \(13356421749439843881551925\) \(\beta_{13}\mathstrut +\mathstrut \) \(15003183624940456784655850\) \(\beta_{12}\mathstrut +\mathstrut \) \(60542899966311623419850410\) \(\beta_{11}\mathstrut -\mathstrut \) \(47799894363283767916337180\) \(\beta_{10}\mathstrut +\mathstrut \) \(905891018199167142897401690\) \(\beta_{9}\mathstrut -\mathstrut \) \(229747388409273466156420670\) \(\beta_{8}\mathstrut -\mathstrut \) \(17366804487333556073039068035\) \(\beta_{7}\mathstrut +\mathstrut \) \(79460192934899719529511625070\) \(\beta_{6}\mathstrut -\mathstrut \) \(79083379419839699400854285220\) \(\beta_{5}\mathstrut +\mathstrut \) \(7845051905607360211366867874478755\) \(\beta_{4}\mathstrut -\mathstrut \) \(25631915677577640978598195\) \(\beta_{3}\mathstrut +\mathstrut \) \(23509832787469854876656729327092\) \(\beta_{2}\mathstrut -\mathstrut \) \(23510273607821873079922847376562\) \(\beta_{1}\mathstrut +\mathstrut \) \(235766125422414741875928935\)\()/4\)
\(\nu^{12}\)\(=\)\((\)\(-\)\(64\!\cdots\!10\) \(\beta_{17}\mathstrut -\mathstrut \) \(11\!\cdots\!65\) \(\beta_{16}\mathstrut -\mathstrut \) \(11\!\cdots\!65\) \(\beta_{15}\mathstrut -\mathstrut \) \(27\!\cdots\!75\) \(\beta_{14}\mathstrut +\mathstrut \) \(27\!\cdots\!75\) \(\beta_{13}\mathstrut +\mathstrut \) \(69\!\cdots\!80\) \(\beta_{12}\mathstrut +\mathstrut \) \(69\!\cdots\!80\) \(\beta_{11}\mathstrut -\mathstrut \) \(88\!\cdots\!70\) \(\beta_{10}\mathstrut -\mathstrut \) \(10\!\cdots\!70\) \(\beta_{9}\mathstrut -\mathstrut \) \(11\!\cdots\!50\) \(\beta_{8}\mathstrut +\mathstrut \) \(31\!\cdots\!65\) \(\beta_{7}\mathstrut -\mathstrut \) \(68\!\cdots\!24\) \(\beta_{6}\mathstrut -\mathstrut \) \(68\!\cdots\!94\) \(\beta_{5}\mathstrut -\mathstrut \) \(71\!\cdots\!05\) \(\beta_{4}\mathstrut -\mathstrut \) \(46\!\cdots\!77\) \(\beta_{3}\mathstrut -\mathstrut \) \(11\!\cdots\!58\) \(\beta_{2}\mathstrut -\mathstrut \) \(11\!\cdots\!48\) \(\beta_{1}\mathstrut +\mathstrut \) \(38\!\cdots\!25\)\()/4\)
\(\nu^{13}\)\(=\)\((\)\(-\)\(94\!\cdots\!52\) \(\beta_{17}\mathstrut +\mathstrut \) \(28\!\cdots\!51\) \(\beta_{16}\mathstrut -\mathstrut \) \(28\!\cdots\!51\) \(\beta_{15}\mathstrut -\mathstrut \) \(25\!\cdots\!61\) \(\beta_{14}\mathstrut -\mathstrut \) \(25\!\cdots\!61\) \(\beta_{13}\mathstrut -\mathstrut \) \(29\!\cdots\!50\) \(\beta_{12}\mathstrut -\mathstrut \) \(11\!\cdots\!54\) \(\beta_{11}\mathstrut +\mathstrut \) \(94\!\cdots\!52\) \(\beta_{10}\mathstrut -\mathstrut \) \(18\!\cdots\!54\) \(\beta_{9}\mathstrut +\mathstrut \) \(48\!\cdots\!90\) \(\beta_{8}\mathstrut +\mathstrut \) \(37\!\cdots\!87\) \(\beta_{7}\mathstrut -\mathstrut \) \(14\!\cdots\!42\) \(\beta_{6}\mathstrut +\mathstrut \) \(14\!\cdots\!32\) \(\beta_{5}\mathstrut -\mathstrut \) \(19\!\cdots\!55\) \(\beta_{4}\mathstrut +\mathstrut \) \(62\!\cdots\!03\) \(\beta_{3}\mathstrut -\mathstrut \) \(43\!\cdots\!72\) \(\beta_{2}\mathstrut +\mathstrut \) \(43\!\cdots\!90\) \(\beta_{1}\mathstrut -\mathstrut \) \(48\!\cdots\!75\)\()/4\)
\(\nu^{14}\)\(=\)\((\)\(13\!\cdots\!82\) \(\beta_{17}\mathstrut +\mathstrut \) \(23\!\cdots\!73\) \(\beta_{16}\mathstrut +\mathstrut \) \(23\!\cdots\!73\) \(\beta_{15}\mathstrut +\mathstrut \) \(54\!\cdots\!35\) \(\beta_{14}\mathstrut -\mathstrut \) \(54\!\cdots\!35\) \(\beta_{13}\mathstrut -\mathstrut \) \(13\!\cdots\!16\) \(\beta_{12}\mathstrut -\mathstrut \) \(13\!\cdots\!16\) \(\beta_{11}\mathstrut +\mathstrut \) \(16\!\cdots\!82\) \(\beta_{10}\mathstrut +\mathstrut \) \(20\!\cdots\!90\) \(\beta_{9}\mathstrut +\mathstrut \) \(22\!\cdots\!30\) \(\beta_{8}\mathstrut -\mathstrut \) \(60\!\cdots\!81\) \(\beta_{7}\mathstrut +\mathstrut \) \(11\!\cdots\!00\) \(\beta_{6}\mathstrut +\mathstrut \) \(11\!\cdots\!94\) \(\beta_{5}\mathstrut +\mathstrut \) \(13\!\cdots\!29\) \(\beta_{4}\mathstrut +\mathstrut \) \(87\!\cdots\!81\) \(\beta_{3}\mathstrut +\mathstrut \) \(26\!\cdots\!14\) \(\beta_{2}\mathstrut +\mathstrut \) \(26\!\cdots\!44\) \(\beta_{1}\mathstrut -\mathstrut \) \(70\!\cdots\!61\)\()/4\)
\(\nu^{15}\)\(=\)\((\)\(18\!\cdots\!04\) \(\beta_{17}\mathstrut -\mathstrut \) \(53\!\cdots\!87\) \(\beta_{16}\mathstrut +\mathstrut \) \(53\!\cdots\!87\) \(\beta_{15}\mathstrut +\mathstrut \) \(50\!\cdots\!29\) \(\beta_{14}\mathstrut +\mathstrut \) \(50\!\cdots\!29\) \(\beta_{13}\mathstrut +\mathstrut \) \(56\!\cdots\!70\) \(\beta_{12}\mathstrut +\mathstrut \) \(20\!\cdots\!18\) \(\beta_{11}\mathstrut -\mathstrut \) \(18\!\cdots\!04\) \(\beta_{10}\mathstrut +\mathstrut \) \(35\!\cdots\!34\) \(\beta_{9}\mathstrut -\mathstrut \) \(10\!\cdots\!34\) \(\beta_{8}\mathstrut -\mathstrut \) \(77\!\cdots\!75\) \(\beta_{7}\mathstrut +\mathstrut \) \(26\!\cdots\!58\) \(\beta_{6}\mathstrut -\mathstrut \) \(26\!\cdots\!48\) \(\beta_{5}\mathstrut +\mathstrut \) \(43\!\cdots\!39\) \(\beta_{4}\mathstrut -\mathstrut \) \(15\!\cdots\!91\) \(\beta_{3}\mathstrut +\mathstrut \) \(79\!\cdots\!80\) \(\beta_{2}\mathstrut -\mathstrut \) \(79\!\cdots\!86\) \(\beta_{1}\mathstrut +\mathstrut \) \(99\!\cdots\!07\)\()/4\)
\(\nu^{16}\)\(=\)\((\)\(-\)\(28\!\cdots\!10\) \(\beta_{17}\mathstrut -\mathstrut \) \(49\!\cdots\!45\) \(\beta_{16}\mathstrut -\mathstrut \) \(49\!\cdots\!45\) \(\beta_{15}\mathstrut -\mathstrut \) \(10\!\cdots\!75\) \(\beta_{14}\mathstrut +\mathstrut \) \(10\!\cdots\!75\) \(\beta_{13}\mathstrut +\mathstrut \) \(26\!\cdots\!60\) \(\beta_{12}\mathstrut +\mathstrut \) \(26\!\cdots\!60\) \(\beta_{11}\mathstrut -\mathstrut \) \(29\!\cdots\!38\) \(\beta_{10}\mathstrut -\mathstrut \) \(38\!\cdots\!66\) \(\beta_{9}\mathstrut -\mathstrut \) \(43\!\cdots\!70\) \(\beta_{8}\mathstrut +\mathstrut \) \(11\!\cdots\!73\) \(\beta_{7}\mathstrut -\mathstrut \) \(19\!\cdots\!04\) \(\beta_{6}\mathstrut -\mathstrut \) \(19\!\cdots\!54\) \(\beta_{5}\mathstrut -\mathstrut \) \(26\!\cdots\!13\) \(\beta_{4}\mathstrut -\mathstrut \) \(16\!\cdots\!25\) \(\beta_{3}\mathstrut -\mathstrut \) \(58\!\cdots\!86\) \(\beta_{2}\mathstrut -\mathstrut \) \(58\!\cdots\!88\) \(\beta_{1}\mathstrut +\mathstrut \) \(13\!\cdots\!61\)\()/4\)
\(\nu^{17}\)\(=\)\((\)\(-\)\(36\!\cdots\!08\) \(\beta_{17}\mathstrut +\mathstrut \) \(10\!\cdots\!59\) \(\beta_{16}\mathstrut -\mathstrut \) \(10\!\cdots\!59\) \(\beta_{15}\mathstrut -\mathstrut \) \(10\!\cdots\!89\) \(\beta_{14}\mathstrut -\mathstrut \) \(10\!\cdots\!89\) \(\beta_{13}\mathstrut -\mathstrut \) \(10\!\cdots\!10\) \(\beta_{12}\mathstrut -\mathstrut \) \(36\!\cdots\!46\) \(\beta_{11}\mathstrut +\mathstrut \) \(36\!\cdots\!08\) \(\beta_{10}\mathstrut -\mathstrut \) \(69\!\cdots\!46\) \(\beta_{9}\mathstrut +\mathstrut \) \(21\!\cdots\!70\) \(\beta_{8}\mathstrut +\mathstrut \) \(16\!\cdots\!83\) \(\beta_{7}\mathstrut -\mathstrut \) \(47\!\cdots\!50\) \(\beta_{6}\mathstrut +\mathstrut \) \(46\!\cdots\!40\) \(\beta_{5}\mathstrut -\mathstrut \) \(94\!\cdots\!51\) \(\beta_{4}\mathstrut +\mathstrut \) \(36\!\cdots\!67\) \(\beta_{3}\mathstrut -\mathstrut \) \(14\!\cdots\!92\) \(\beta_{2}\mathstrut +\mathstrut \) \(14\!\cdots\!74\) \(\beta_{1}\mathstrut -\mathstrut \) \(20\!\cdots\!95\)\()/4\)

Character Values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/5\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(\beta_{4}\)

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
1286.49i
1196.41i
670.386i
439.799i
3.77310i
569.040i
782.816i
836.109i
1400.36i
1286.49i
1196.41i
670.386i
439.799i
3.77310i
569.040i
782.816i
836.109i
1400.36i
−1286.49 1286.49i 59231.4 59231.4i 2.26156e6i 2.84523e6 9.34195e6i −1.52402e8 −2.38026e8 2.38026e8i 1.56049e9 1.56049e9i 3.52992e9i −1.56787e10 + 8.35799e9i
2.2 −1196.41 1196.41i −49580.3 + 49580.3i 1.81424e6i −9.20993e6 + 3.24723e6i 1.18637e8 2.14321e8 + 2.14321e8i 9.16051e8 9.16051e8i 1.42964e9i 1.49039e10 + 7.13386e9i
2.3 −670.386 670.386i 17254.5 17254.5i 149742.i 4.96780e6 + 8.40764e6i −2.31343e7 9.51724e7 + 9.51724e7i −8.03335e8 + 8.03335e8i 2.89135e9i 2.30602e9 8.96670e9i
2.4 −439.799 439.799i −48235.4 + 48235.4i 661730.i 5.04058e6 8.36421e6i 4.24277e7 −2.17438e8 2.17438e8i −7.52190e8 + 7.52190e8i 1.16652e9i −5.89541e9 + 1.46173e9i
2.5 3.77310 + 3.77310i 38858.9 38858.9i 1.04855e6i −8.90462e6 4.00939e6i 293237. 4.46902e7 + 4.46902e7i 7.91266e6 7.91266e6i 4.66753e8i −1.84702e7 4.87259e7i
2.6 569.040 + 569.040i −56179.4 + 56179.4i 400963.i −4.35767e6 + 8.73946e6i −6.39367e7 −1.35277e8 1.35277e8i 8.24846e8 8.24846e8i 2.82548e9i −7.45279e9 + 2.49341e9i
2.7 782.816 + 782.816i −12801.5 + 12801.5i 177026.i 7.95503e6 5.66435e6i −2.00424e7 3.15497e8 + 3.15497e8i 6.82263e8 6.82263e8i 3.15903e9i 1.06615e10 + 1.79318e9i
2.8 836.109 + 836.109i 68486.6 68486.6i 349580.i 7.10218e6 + 6.70272e6i 1.14524e8 −2.53016e8 2.53016e8i 5.84437e8 5.84437e8i 5.89403e9i 3.33994e8 + 1.15424e10i
2.9 1400.36 + 1400.36i −2310.64 + 2310.64i 2.87341e6i −9.08967e6 3.57006e6i −6.47145e6 −1.18690e8 1.18690e8i −2.55542e9 + 2.55542e9i 3.47611e9i −7.72941e9 1.77281e10i
3.1 −1286.49 + 1286.49i 59231.4 + 59231.4i 2.26156e6i 2.84523e6 + 9.34195e6i −1.52402e8 −2.38026e8 + 2.38026e8i 1.56049e9 + 1.56049e9i 3.52992e9i −1.56787e10 8.35799e9i
3.2 −1196.41 + 1196.41i −49580.3 49580.3i 1.81424e6i −9.20993e6 3.24723e6i 1.18637e8 2.14321e8 2.14321e8i 9.16051e8 + 9.16051e8i 1.42964e9i 1.49039e10 7.13386e9i
3.3 −670.386 + 670.386i 17254.5 + 17254.5i 149742.i 4.96780e6 8.40764e6i −2.31343e7 9.51724e7 9.51724e7i −8.03335e8 8.03335e8i 2.89135e9i 2.30602e9 + 8.96670e9i
3.4 −439.799 + 439.799i −48235.4 48235.4i 661730.i 5.04058e6 + 8.36421e6i 4.24277e7 −2.17438e8 + 2.17438e8i −7.52190e8 7.52190e8i 1.16652e9i −5.89541e9 1.46173e9i
3.5 3.77310 3.77310i 38858.9 + 38858.9i 1.04855e6i −8.90462e6 + 4.00939e6i 293237. 4.46902e7 4.46902e7i 7.91266e6 + 7.91266e6i 4.66753e8i −1.84702e7 + 4.87259e7i
3.6 569.040 569.040i −56179.4 56179.4i 400963.i −4.35767e6 8.73946e6i −6.39367e7 −1.35277e8 + 1.35277e8i 8.24846e8 + 8.24846e8i 2.82548e9i −7.45279e9 2.49341e9i
3.7 782.816 782.816i −12801.5 12801.5i 177026.i 7.95503e6 + 5.66435e6i −2.00424e7 3.15497e8 3.15497e8i 6.82263e8 + 6.82263e8i 3.15903e9i 1.06615e10 1.79318e9i
3.8 836.109 836.109i 68486.6 + 68486.6i 349580.i 7.10218e6 6.70272e6i 1.14524e8 −2.53016e8 + 2.53016e8i 5.84437e8 + 5.84437e8i 5.89403e9i 3.33994e8 1.15424e10i
3.9 1400.36 1400.36i −2310.64 2310.64i 2.87341e6i −9.08967e6 + 3.57006e6i −6.47145e6 −1.18690e8 + 1.18690e8i −2.55542e9 2.55542e9i 3.47611e9i −7.72941e9 + 1.77281e10i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 3.9
Significant digits:
Format:

Inner twists

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

Hecke kernels

There are no other newforms in \(S_{21}^{\mathrm{new}}(5, [\chi])\).