Properties

Label 8.20.b.a
Level 8
Weight 20
Character orbit 8.b
Analytic conductor 18.305
Analytic rank 0
Dimension 18
CM No
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 8 = 2^{3} \)
Weight: \( k \) = \( 20 \)
Character orbit: \([\chi]\) = 8.b (of order \(2\) and degree \(1\))

Newform invariants

Self dual: No
Analytic conductor: \(18.3053357245\)
Analytic rank: \(0\)
Dimension: \(18\)
Coefficient field: \(\mathbb{Q}[x]/(x^{18} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: multiple of \( 2^{153}\cdot 3^{16}\cdot 5^{4}\cdot 7 \)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(f(q)\) \(=\) \( q\) \( + ( -25 + \beta_{1} ) q^{2} \) \( + ( 2 + 5 \beta_{1} - \beta_{2} ) q^{3} \) \( + ( -22907 - 27 \beta_{1} + \beta_{3} ) q^{4} \) \( + ( -34 - 80 \beta_{1} + 5 \beta_{2} - \beta_{3} - \beta_{6} ) q^{5} \) \( + ( -2624536 - 129 \beta_{1} + 89 \beta_{2} + 5 \beta_{3} + \beta_{5} + \beta_{6} ) q^{6} \) \( + ( -4489606 - 13201 \beta_{1} - 31 \beta_{2} - 21 \beta_{3} - \beta_{4} ) q^{7} \) \( + ( -9639250 - 23397 \beta_{1} - 1410 \beta_{2} - 29 \beta_{3} + \beta_{5} - 4 \beta_{6} - \beta_{7} - \beta_{9} ) q^{8} \) \( + ( -344295579 + 175766 \beta_{1} + 217 \beta_{2} - 232 \beta_{3} - 2 \beta_{4} + 5 \beta_{5} + 4 \beta_{6} + 3 \beta_{7} + \beta_{8} ) q^{9} \) \(+O(q^{10})\) \( q\) \( + ( -25 + \beta_{1} ) q^{2} \) \( + ( 2 + 5 \beta_{1} - \beta_{2} ) q^{3} \) \( + ( -22907 - 27 \beta_{1} + \beta_{3} ) q^{4} \) \( + ( -34 - 80 \beta_{1} + 5 \beta_{2} - \beta_{3} - \beta_{6} ) q^{5} \) \( + ( -2624536 - 129 \beta_{1} + 89 \beta_{2} + 5 \beta_{3} + \beta_{5} + \beta_{6} ) q^{6} \) \( + ( -4489606 - 13201 \beta_{1} - 31 \beta_{2} - 21 \beta_{3} - \beta_{4} ) q^{7} \) \( + ( -9639250 - 23397 \beta_{1} - 1410 \beta_{2} - 29 \beta_{3} + \beta_{5} - 4 \beta_{6} - \beta_{7} - \beta_{9} ) q^{8} \) \( + ( -344295579 + 175766 \beta_{1} + 217 \beta_{2} - 232 \beta_{3} - 2 \beta_{4} + 5 \beta_{5} + 4 \beta_{6} + 3 \beta_{7} + \beta_{8} ) q^{9} \) \( + ( 42159278 + 2214 \beta_{1} + 12169 \beta_{2} - 30 \beta_{3} + \beta_{4} - 6 \beta_{5} - 64 \beta_{6} + 9 \beta_{7} + \beta_{9} + \beta_{12} ) q^{10} \) \( + ( 521436 + 1175468 \beta_{1} - 5387 \beta_{2} - 1355 \beta_{3} - 30 \beta_{5} - 123 \beta_{6} - 18 \beta_{7} - \beta_{9} + \beta_{14} ) q^{11} \) \( + ( -520720591 - 2653173 \beta_{1} - 26134 \beta_{2} - 257 \beta_{3} + 3 \beta_{4} - 88 \beta_{5} - 85 \beta_{6} - 62 \beta_{7} - 2 \beta_{8} - 4 \beta_{9} - \beta_{13} + \beta_{14} ) q^{12} \) \( + ( -1256856 - 2843219 \beta_{1} + 32134 \beta_{2} + 5521 \beta_{3} - 6 \beta_{4} - 154 \beta_{5} - 591 \beta_{6} + 38 \beta_{7} + \beta_{8} + 3 \beta_{9} + 2 \beta_{12} - \beta_{13} - \beta_{16} ) q^{13} \) \( + ( -6805080756 - 4812309 \beta_{1} + 170799 \beta_{2} - 13360 \beta_{3} - 5 \beta_{4} - 37 \beta_{5} - 613 \beta_{6} - 189 \beta_{7} + 10 \beta_{8} + 22 \beta_{9} + 2 \beta_{12} + 5 \beta_{14} + 2 \beta_{16} - \beta_{17} ) q^{14} \) \( + ( 8687769868 + 35206087 \beta_{1} + 60824 \beta_{2} - 2384 \beta_{3} + 72 \beta_{4} + 22 \beta_{5} + 64 \beta_{6} + 540 \beta_{7} + 9 \beta_{8} - 63 \beta_{9} + 22 \beta_{12} + 6 \beta_{13} - 2 \beta_{14} - \beta_{15} - 2 \beta_{16} + 2 \beta_{17} ) q^{15} \) \( + ( -38683597078 - 11510054 \beta_{1} - 148659 \beta_{2} - 29478 \beta_{3} - 463 \beta_{4} + 987 \beta_{5} + 8508 \beta_{6} + 263 \beta_{7} + 80 \beta_{8} + 32 \beta_{9} - \beta_{10} - \beta_{11} + 4 \beta_{12} - 2 \beta_{13} - 16 \beta_{14} - 2 \beta_{15} + 4 \beta_{16} + 2 \beta_{17} ) q^{16} \) \( + ( 761950163 - 50740950 \beta_{1} - 74125 \beta_{2} + 38875 \beta_{3} - 250 \beta_{4} + 2486 \beta_{5} - 115 \beta_{6} - 640 \beta_{7} - 3 \beta_{8} - 326 \beta_{9} - 2 \beta_{10} - \beta_{11} - 84 \beta_{12} + 12 \beta_{13} - 12 \beta_{14} - 2 \beta_{15} - 4 \beta_{16} - 4 \beta_{17} ) q^{17} \) \( + ( 100599334942 - 339948555 \beta_{1} + 2714475 \beta_{2} + 189404 \beta_{3} - 1316 \beta_{4} - 1154 \beta_{5} - 3124 \beta_{6} - 1659 \beta_{7} - 388 \beta_{8} + 272 \beta_{9} + 2 \beta_{10} - 20 \beta_{12} - 8 \beta_{13} - 54 \beta_{14} - 4 \beta_{15} + 4 \beta_{16} - 10 \beta_{17} ) q^{18} \) \( + ( 22817372 + 51492172 \beta_{1} - 101913 \beta_{2} + 206279 \beta_{3} - 116 \beta_{4} + 6288 \beta_{5} + 59707 \beta_{6} - 474 \beta_{7} - 16 \beta_{8} + 1139 \beta_{9} + 4 \beta_{10} + 6 \beta_{11} - 208 \beta_{12} - 8 \beta_{13} - 11 \beta_{14} - 16 \beta_{15} - 8 \beta_{16} + 16 \beta_{17} ) q^{19} \) \( + ( 362086807608 + 59776074 \beta_{1} - 848524 \beta_{2} + 36290 \beta_{3} + 8880 \beta_{4} - 7710 \beta_{5} - 22418 \beta_{6} - 4240 \beta_{7} - 1036 \beta_{8} + 28 \beta_{9} - 22 \beta_{10} - 26 \beta_{11} + 88 \beta_{12} + 30 \beta_{13} + 78 \beta_{14} - 28 \beta_{15} + 8 \beta_{16} + 20 \beta_{17} ) q^{20} \) \( + ( -229811086 - 526447145 \beta_{1} + 18552341 \beta_{2} - 9116 \beta_{3} - 1706 \beta_{4} - 40634 \beta_{5} - 65476 \beta_{6} + 5330 \beta_{7} + 19 \beta_{8} + 6729 \beta_{9} - 40 \beta_{10} - 12 \beta_{11} + 518 \beta_{12} + 29 \beta_{13} - 32 \beta_{15} + 29 \beta_{16} - 32 \beta_{17} ) q^{21} \) \( + ( -615268084350 - 30145597 \beta_{1} + 18263791 \beta_{2} + 1112201 \beta_{3} + 17036 \beta_{4} + 2597 \beta_{5} + 184553 \beta_{6} + 5790 \beta_{7} + 2320 \beta_{8} + 512 \beta_{9} + 44 \beta_{10} - 8 \beta_{11} + 56 \beta_{12} - 16 \beta_{13} - 64 \beta_{14} - 56 \beta_{15} - 64 \beta_{16} - 16 \beta_{17} ) q^{22} \) \( + ( 120249994008 - 1576487219 \beta_{1} - 3013342 \beta_{2} - 719706 \beta_{3} - 642 \beta_{4} - 112610 \beta_{5} - 4616 \beta_{6} - 27324 \beta_{7} - 439 \beta_{8} - 11679 \beta_{9} + 80 \beta_{10} + 136 \beta_{11} + 598 \beta_{12} - 186 \beta_{13} + 126 \beta_{14} - 97 \beta_{15} + 62 \beta_{16} + 2 \beta_{17} ) q^{23} \) \( + ( -2026552502748 - 617387634 \beta_{1} + 1073558 \beta_{2} - 2956194 \beta_{3} - 75570 \beta_{4} - 17096 \beta_{5} + 332352 \beta_{6} + 6712 \beta_{7} + 4208 \beta_{8} + 994 \beta_{9} - 214 \beta_{10} - 318 \beta_{11} + 344 \beta_{12} + 60 \beta_{13} + 392 \beta_{14} - 140 \beta_{15} - 136 \beta_{16} + 28 \beta_{17} ) q^{24} \) \( + ( -2469162919030 - 3803721900 \beta_{1} - 8189246 \beta_{2} - 3749747 \beta_{3} - 10912 \beta_{4} + 292011 \beta_{5} + 123991 \beta_{6} - 52633 \beta_{7} + 96 \beta_{8} - 20858 \beta_{9} - 350 \beta_{10} + 17 \beta_{11} + 1236 \beta_{12} - 396 \beta_{13} + 268 \beta_{14} - 190 \beta_{15} + 132 \beta_{16} + 4 \beta_{17} ) q^{25} \) \( + ( 1487940760982 + 74221122 \beta_{1} + 74528207 \beta_{2} - 2256538 \beta_{3} - 128445 \beta_{4} - 64530 \beta_{5} - 1140792 \beta_{6} - 22489 \beta_{7} - 2720 \beta_{8} - 3237 \beta_{9} + 424 \beta_{10} - 208 \beta_{11} + 875 \beta_{12} + 288 \beta_{13} + 1280 \beta_{14} - 272 \beta_{15} - 128 \beta_{16} + 160 \beta_{17} ) q^{26} \) \( + ( -1443723986 - 3410259991 \beta_{1} + 336894814 \beta_{2} + 19641175 \beta_{3} + 2204 \beta_{4} + 735528 \beta_{5} + 1269627 \beta_{6} + 44310 \beta_{7} - 1872 \beta_{8} + 46283 \beta_{9} + 692 \beta_{10} + 1422 \beta_{11} + 4720 \beta_{12} + 280 \beta_{13} - 3 \beta_{14} - 208 \beta_{15} + 280 \beta_{16} - 304 \beta_{17} ) q^{27} \) \( + ( -5392161296068 - 7071759928 \beta_{1} - 10133448 \beta_{2} - 3638424 \beta_{3} + 219732 \beta_{4} + 828 \beta_{5} - 5205768 \beta_{6} + 24112 \beta_{7} + 9536 \beta_{8} + 13912 \beta_{9} - 1188 \beta_{10} - 2444 \beta_{11} - 880 \beta_{12} - 392 \beta_{13} - 2448 \beta_{14} - 168 \beta_{15} - 272 \beta_{16} - 360 \beta_{17} ) q^{28} \) \( + ( 855679410 + 2165100600 \beta_{1} - 482639729 \beta_{2} + 4047421 \beta_{3} - 42040 \beta_{4} - 1664244 \beta_{5} - 1130907 \beta_{6} - 21040 \beta_{7} + 3448 \beta_{8} + 11032 \beta_{9} - 1720 \beta_{10} + 1276 \beta_{11} - 9328 \beta_{12} - 360 \beta_{13} - 512 \beta_{14} - 352 \beta_{15} - 360 \beta_{16} + 672 \beta_{17} ) q^{29} \) \( + ( 18217969309604 + 9553593987 \beta_{1} - 10542049 \beta_{2} + 32970864 \beta_{3} + 547555 \beta_{4} - 72205 \beta_{5} + 12462147 \beta_{6} - 114941 \beta_{7} - 36966 \beta_{8} - 10970 \beta_{9} + 2288 \beta_{10} - 2496 \beta_{11} + 1106 \beta_{12} + 576 \beta_{13} - 787 \beta_{14} - 224 \beta_{15} + 914 \beta_{16} + 631 \beta_{17} ) q^{30} \) \( + ( 23810824158910 + 11118380364 \beta_{1} + 8776717 \beta_{2} - 29137233 \beta_{3} + 15987 \beta_{4} - 2919326 \beta_{5} - 64968 \beta_{6} + 108332 \beta_{7} - 6849 \beta_{8} + 26791 \beta_{9} + 3280 \beta_{10} + 9160 \beta_{11} - 16038 \beta_{12} + 2538 \beta_{13} - 2318 \beta_{14} + 473 \beta_{15} - 846 \beta_{16} - 626 \beta_{17} ) q^{31} \) \( + ( 6785639689668 - 38391277436 \beta_{1} - 180616066 \beta_{2} - 16902268 \beta_{3} - 332434 \beta_{4} - 345678 \beta_{5} + 16350248 \beta_{6} - 336526 \beta_{7} - 66112 \beta_{8} + 37160 \beta_{9} - 3966 \beta_{10} - 13326 \beta_{11} - 4616 \beta_{12} - 780 \beta_{13} - 3376 \beta_{14} + 1092 \beta_{15} + 2104 \beta_{16} - 1316 \beta_{17} ) q^{32} \) \( + ( -10276473607056 + 50425710550 \beta_{1} + 39449105 \beta_{2} - 121696602 \beta_{3} - 65082 \beta_{4} + 5822371 \beta_{5} + 3039230 \beta_{6} + 895985 \beta_{7} + 19981 \beta_{8} + 208772 \beta_{9} - 4916 \beta_{10} + 12198 \beta_{11} + 5432 \beta_{12} + 5880 \beta_{13} - 2680 \beta_{14} + 1324 \beta_{15} - 1960 \beta_{16} + 1240 \beta_{17} ) q^{33} \) \( + ( -26580378020847 - 474416916 \beta_{1} + 1253851709 \beta_{2} - 37777468 \beta_{3} - 56476 \beta_{4} + 682482 \beta_{5} - 39207564 \beta_{6} + 190739 \beta_{7} + 85732 \beta_{8} - 26704 \beta_{9} + 7086 \beta_{10} - 18304 \beta_{11} - 7436 \beta_{12} - 4792 \beta_{13} - 12282 \beta_{14} + 2724 \beta_{15} + 1820 \beta_{16} - 582 \beta_{17} ) q^{34} \) \( + ( 15779735964 + 35963972226 \beta_{1} - 671822414 \beta_{2} + 411881524 \beta_{3} + 87436 \beta_{4} + 8856354 \beta_{5} + 15358968 \beta_{6} - 125980 \beta_{7} - 8848 \beta_{8} - 659174 \beta_{9} + 8452 \beta_{10} + 41222 \beta_{11} - 31184 \beta_{12} - 4488 \beta_{13} + 462 \beta_{14} + 3568 \beta_{15} - 4488 \beta_{16} + 2064 \beta_{17} ) q^{35} \) \( + ( -58504291409885 + 98128810235 \beta_{1} - 3106749776 \beta_{2} - 323362337 \beta_{3} - 832312 \beta_{4} - 598248 \beta_{5} - 92394000 \beta_{6} + 53088 \beta_{7} + 102912 \beta_{8} - 57232 \beta_{9} - 6952 \beta_{10} - 55224 \beta_{11} + 12448 \beta_{12} + 2800 \beta_{13} + 29856 \beta_{14} + 4592 \beta_{15} + 4192 \beta_{16} + 1904 \beta_{17} ) q^{36} \) \( + ( 3822411560 + 9697995671 \beta_{1} - 1994223350 \beta_{2} + 504642155 \beta_{3} - 293394 \beta_{4} - 6576254 \beta_{5} - 26014357 \beta_{6} + 72914 \beta_{7} + 18195 \beta_{8} - 701959 \beta_{9} - 6368 \beta_{10} + 66544 \beta_{11} + 67750 \beta_{12} + 2349 \beta_{13} + 14848 \beta_{14} + 7808 \beta_{15} + 2349 \beta_{16} - 5504 \beta_{17} ) q^{37} \) \( + ( -27012387376534 - 1367738617 \beta_{1} - 4210089645 \beta_{2} + 7137557 \beta_{3} - 2633220 \beta_{4} - 2447359 \beta_{5} + 100101653 \beta_{6} - 912778 \beta_{7} + 42576 \beta_{8} - 156288 \beta_{9} + 9372 \beta_{10} - 91560 \beta_{11} - 23016 \beta_{12} - 9552 \beta_{13} + 24256 \beta_{14} + 9576 \beta_{15} - 7488 \beta_{16} - 6480 \beta_{17} ) q^{38} \) \( + ( 52476623584118 + 218340010285 \beta_{1} - 20577973 \beta_{2} - 1034089071 \beta_{3} - 80299 \beta_{4} - 3970040 \beta_{5} + 16284264 \beta_{6} + 2999576 \beta_{7} + 52680 \beta_{8} + 857000 \beta_{9} + 6256 \beta_{10} + 140376 \beta_{11} + 147952 \beta_{12} - 19344 \beta_{13} + 20464 \beta_{14} + 5912 \beta_{15} + 6448 \beta_{16} + 7568 \beta_{17} ) q^{39} \) \( + ( -94163062475128 + 357973983684 \beta_{1} + 5081971812 \beta_{2} + 13719204 \beta_{3} + 6751684 \beta_{4} - 1045520 \beta_{5} + 165844096 \beta_{6} + 3296576 \beta_{7} + 413920 \beta_{8} - 371268 \beta_{9} + 1644 \beta_{10} - 181604 \beta_{11} + 35408 \beta_{12} + 5480 \beta_{13} + 3792 \beta_{14} + 3864 \beta_{15} - 19312 \beta_{16} + 15624 \beta_{17} ) q^{40} \) \( + ( 66568733400233 - 252706323652 \beta_{1} - 1093537050 \beta_{2} - 1668025761 \beta_{3} - 1654880 \beta_{4} - 4788615 \beta_{5} + 26792973 \beta_{6} - 4209043 \beta_{7} - 94832 \beta_{8} + 540674 \beta_{9} + 8790 \beta_{10} + 249771 \beta_{11} - 198244 \beta_{12} - 50820 \beta_{13} + 16260 \beta_{14} + 8982 \beta_{15} + 16940 \beta_{16} - 17620 \beta_{17} ) q^{41} \) \( + ( 275810850270188 + 13777722776 \beta_{1} + 17892956952 \beta_{2} - 366101896 \beta_{3} + 12402204 \beta_{4} - 9666600 \beta_{5} - 322021096 \beta_{6} - 3818720 \beta_{7} - 723424 \beta_{8} - 66332 \beta_{9} - 17032 \beta_{10} - 330096 \beta_{11} + 532 \beta_{12} + 48480 \beta_{13} + 50944 \beta_{14} + 2000 \beta_{15} - 14720 \beta_{16} - 5152 \beta_{17} ) q^{42} \) \( + ( -64664913650 - 142720418289 \beta_{1} - 3952827725 \beta_{2} + 3231711102 \beta_{3} - 2185704 \beta_{4} - 21627648 \beta_{5} + 18771046 \beta_{6} + 3336428 \beta_{7} + 16736 \beta_{8} + 1288630 \beta_{9} - 33784 \beta_{10} + 383372 \beta_{11} - 22176 \beta_{12} + 43120 \beta_{13} - 1126 \beta_{14} - 7712 \beta_{15} + 43120 \beta_{16} - 3040 \beta_{17} ) q^{43} \) \( + ( 390383793946941 - 597320270433 \beta_{1} - 23971904510 \beta_{2} + 26397891 \beta_{3} - 7536729 \beta_{4} - 10144936 \beta_{5} - 257949857 \beta_{6} - 1647782 \beta_{7} - 1273594 \beta_{8} - 266196 \beta_{9} + 45584 \beta_{10} - 485264 \beta_{11} - 206912 \beta_{12} - 10461 \beta_{13} - 197443 \beta_{14} - 15712 \beta_{15} - 38080 \beta_{16} + 6176 \beta_{17} ) q^{44} \) \( + ( 14732742244 + 27664626131 \beta_{1} + 15269420536 \beta_{2} + 7384079185 \beta_{3} - 1926370 \beta_{4} + 84988326 \beta_{5} + 295667961 \beta_{6} + 3128250 \beta_{7} - 150705 \beta_{8} + 3116381 \beta_{9} + 63080 \beta_{10} + 694764 \beta_{11} - 206402 \beta_{12} - 6911 \beta_{13} - 196608 \beta_{14} - 31456 \beta_{15} - 6911 \beta_{16} + 17696 \beta_{17} ) q^{45} \) \( + ( -828723870990884 + 80872912801 \beta_{1} - 58939478875 \beta_{2} - 1827729008 \beta_{3} - 11352415 \beta_{4} + 2019697 \beta_{5} + 150670913 \beta_{6} + 427921 \beta_{7} + 1402398 \beta_{8} + 1960930 \beta_{9} - 107536 \beta_{10} - 881088 \beta_{11} + 109574 \beta_{12} + 95808 \beta_{13} - 247249 \beta_{14} - 50912 \beta_{15} + 36678 \beta_{16} + 28829 \beta_{17} ) q^{46} \) \( + ( -63175867098778 - 375344708070 \beta_{1} - 3275092543 \beta_{2} - 6625770189 \beta_{3} - 7160705 \beta_{4} + 145388590 \beta_{5} + 139697632 \beta_{6} - 2440660 \beta_{7} - 683331 \beta_{8} - 5781163 \beta_{9} - 135232 \beta_{10} + 865888 \beta_{11} - 670002 \beta_{12} + 82302 \beta_{13} - 92202 \beta_{14} - 45973 \beta_{15} - 27434 \beta_{16} - 37334 \beta_{17} ) q^{47} \) \( + ( -178539980974268 - 2035668968572 \beta_{1} + 132355134346 \beta_{2} - 363624316 \beta_{3} + 3572354 \beta_{4} - 26050522 \beta_{5} - 18827336 \beta_{6} - 17513618 \beta_{7} - 285088 \beta_{8} + 1472880 \beta_{9} + 134398 \beta_{10} - 1055778 \beta_{11} - 369656 \beta_{12} - 19364 \beta_{13} + 153792 \beta_{14} - 51076 \beta_{15} + 113800 \beta_{16} - 90172 \beta_{17} ) q^{48} \) \( + ( 1754351212205625 + 1027135753328 \beta_{1} - 1783782904 \beta_{2} - 9276527552 \beta_{3} + 4152976 \beta_{4} - 148606168 \beta_{5} + 133001728 \beta_{6} + 9479064 \beta_{7} + 62664 \beta_{8} - 10388832 \beta_{9} + 155328 \beta_{10} + 1468960 \beta_{11} + 1454016 \beta_{12} + 271296 \beta_{13} - 69696 \beta_{14} - 108960 \beta_{15} - 90432 \beta_{16} + 111168 \beta_{17} ) q^{49} \) \( + ( -1930781276540327 - 2561003267983 \beta_{1} + 142546546978 \beta_{2} - 3798745816 \beta_{3} - 22124920 \beta_{4} - 43647020 \beta_{5} + 275317672 \beta_{6} - 38907202 \beta_{7} + 1758760 \beta_{8} + 2914080 \beta_{9} - 226804 \beta_{10} - 1738880 \beta_{11} + 225224 \beta_{12} - 328752 \beta_{13} + 45916 \beta_{14} - 103960 \beta_{15} + 69720 \beta_{16} + 57252 \beta_{17} ) q^{50} \) \( + ( -1626143395270 - 3658303083181 \beta_{1} + 5862033548 \beta_{2} + 15835312175 \beta_{3} - 14757264 \beta_{4} - 256353650 \beta_{5} - 312810385 \beta_{6} + 54630170 \beta_{7} + 670144 \beta_{8} + 13467077 \beta_{9} - 233264 \beta_{10} + 1593912 \beta_{11} + 1220544 \beta_{12} - 271008 \beta_{13} - 8613 \beta_{14} - 67392 \beta_{15} - 271008 \beta_{16} - 33984 \beta_{17} ) q^{51} \) \( + ( -1492605286348792 + 1415193387790 \beta_{1} - 189063795140 \beta_{2} - 755155018 \beta_{3} + 52888688 \beta_{4} - 50876490 \beta_{5} + 936699034 \beta_{6} + 18581776 \beta_{7} + 3908668 \beta_{8} + 3935412 \beta_{9} + 202510 \beta_{10} - 1810718 \beta_{11} + 1926088 \beta_{12} + 5546 \beta_{13} + 723706 \beta_{14} - 57652 \beta_{15} + 219224 \beta_{16} - 125732 \beta_{17} ) q^{52} \) \( + ( 2166926432380 + 4865957044273 \beta_{1} + 30947461904 \beta_{2} + 20836129235 \beta_{3} - 5999926 \beta_{4} + 197305986 \beta_{5} + 569510347 \beta_{6} - 61623618 \beta_{7} - 446459 \beta_{8} + 4709471 \beta_{9} + 195896 \beta_{10} + 2266564 \beta_{11} - 306838 \beta_{12} - 10005 \beta_{13} + 1559040 \beta_{14} - 64672 \beta_{15} - 10005 \beta_{16} + 28512 \beta_{17} ) q^{53} \) \( + ( 1783395851906594 + 82859280230 \beta_{1} - 438184278884 \beta_{2} - 3344017542 \beta_{3} + 112732764 \beta_{4} - 154520990 \beta_{5} - 2970056554 \beta_{6} - 149111706 \beta_{7} - 8030512 \beta_{8} - 10487936 \beta_{9} - 176132 \beta_{10} - 2360808 \beta_{11} + 50648 \beta_{12} - 638416 \beta_{13} + 1354944 \beta_{14} - 4184 \beta_{15} - 89920 \beta_{16} - 25040 \beta_{17} ) q^{54} \) \( + ( -2872048484926254 + 7514700263643 \beta_{1} + 5826789289 \beta_{2} - 18725966821 \beta_{3} - 24827465 \beta_{4} + 183140768 \beta_{5} + 337062536 \beta_{6} + 121134248 \beta_{7} + 8330364 \beta_{8} + 3500860 \beta_{9} - 108752 \beta_{10} + 2176056 \beta_{11} + 906408 \beta_{12} - 118296 \beta_{13} + 122824 \beta_{14} + 16772 \beta_{15} + 39432 \beta_{16} + 43960 \beta_{17} ) q^{55} \) \( + ( 2934386144525616 - 5243253629464 \beta_{1} + 648468665648 \beta_{2} - 4174089048 \beta_{3} - 93581376 \beta_{4} - 34177352 \beta_{5} - 3989165280 \beta_{6} + 155074408 \beta_{7} - 8931968 \beta_{8} - 1689336 \beta_{9} + 44416 \beta_{10} - 2146240 \beta_{11} + 3435008 \beta_{12} + 192 \beta_{13} - 1342784 \beta_{14} + 50176 \beta_{15} - 422144 \beta_{16} + 243328 \beta_{17} ) q^{56} \) \( + ( -435427383568004 - 12108377004898 \beta_{1} - 26679236147 \beta_{2} - 13877299254 \beta_{3} - 58723010 \beta_{4} + 122391683 \beta_{5} + 237533562 \beta_{6} - 181542119 \beta_{7} - 1065663 \beta_{8} + 22354780 \beta_{9} - 59596 \beta_{10} + 1922586 \beta_{11} - 4099576 \beta_{12} - 817080 \beta_{13} + 219320 \beta_{14} + 263668 \beta_{15} + 272360 \beta_{16} - 325400 \beta_{17} ) q^{57} \) \( + ( -1137053081430062 - 45946704582 \beta_{1} + 876294163815 \beta_{2} + 2420184638 \beta_{3} - 150025201 \beta_{4} + 177771430 \beta_{5} + 4924986880 \beta_{6} - 68841433 \beta_{7} + 8914176 \beta_{8} - 17201841 \beta_{9} + 360640 \beta_{10} - 1269632 \beta_{11} - 784177 \beta_{12} + 1539840 \beta_{13} - 1617920 \beta_{14} + 316544 \beta_{15} - 150528 \beta_{16} - 185600 \beta_{17} ) q^{58} \) \( + ( -3275130494102 - 7316811550223 \beta_{1} - 97973748303 \beta_{2} + 5350014558 \beta_{3} + 21440300 \beta_{4} + 793030038 \beta_{5} + 1503395746 \beta_{6} + 134929376 \beta_{7} - 1969808 \beta_{8} - 57251352 \beta_{9} + 566756 \beta_{10} + 1357270 \beta_{11} - 5623888 \beta_{12} + 1120056 \beta_{13} + 36416 \beta_{14} + 283760 \beta_{15} + 1120056 \beta_{16} + 176016 \beta_{17} ) q^{59} \) \( + ( -1427033963734628 + 18145264889480 \beta_{1} - 1606821429768 \beta_{2} + 4066984744 \beta_{3} + 122877684 \beta_{4} + 229212444 \beta_{5} + 7009346232 \beta_{6} + 310992816 \beta_{7} + 4334400 \beta_{8} - 24988392 \beta_{9} - 643524 \beta_{10} - 501228 \beta_{11} - 10714864 \beta_{12} + 136248 \beta_{13} - 970768 \beta_{14} + 371224 \beta_{15} - 768400 \beta_{16} + 633688 \beta_{17} ) q^{60} \) \( + ( 5988426517648 + 13588989422461 \beta_{1} - 243320952850 \beta_{2} - 16216818495 \beta_{3} - 1549678 \beta_{4} - 771836702 \beta_{5} - 2519425399 \beta_{6} - 204600410 \beta_{7} + 2328129 \beta_{8} - 55670669 \beta_{9} - 902200 \beta_{10} - 1844292 \beta_{11} + 4788002 \beta_{12} + 158415 \beta_{13} - 8115712 \beta_{14} + 682144 \beta_{15} + 158415 \beta_{16} - 404320 \beta_{17} ) q^{61} \) \( + ( 5219626830927672 + 24066270187292 \beta_{1} - 1486695359116 \beta_{2} + 10425995552 \beta_{3} + 33457676 \beta_{4} + 388251612 \beta_{5} - 6102025780 \beta_{6} - 122295980 \beta_{7} + 11739016 \beta_{8} + 58652248 \beta_{9} + 1543568 \beta_{10} + 3949504 \beta_{11} - 1756696 \beta_{12} + 2888640 \beta_{13} - 3949148 \beta_{14} + 719840 \beta_{15} - 73560 \beta_{16} - 298644 \beta_{17} ) q^{62} \) \( + ( 23915020213513932 + 37677940141313 \beta_{1} + 84279435412 \beta_{2} + 45534593444 \beta_{3} + 182956684 \beta_{4} - 2156338366 \beta_{5} - 1160258384 \beta_{6} + 537157764 \beta_{7} - 52541693 \beta_{8} + 77671851 \beta_{9} + 1889184 \beta_{10} - 3313008 \beta_{11} + 6201906 \beta_{12} - 677502 \beta_{13} + 835242 \beta_{14} + 645141 \beta_{15} + 225834 \beta_{16} + 383574 \beta_{17} ) q^{63} \) \( + ( -14092499439084440 + 6161196620456 \beta_{1} + 1732246708444 \beta_{2} - 33499372888 \beta_{3} + 30587676 \beta_{4} + 51893828 \beta_{5} - 6979112752 \beta_{6} + 576173988 \beta_{7} + 33940224 \beta_{8} - 23368528 \beta_{9} - 1884924 \beta_{10} + 5779876 \beta_{11} - 20395024 \beta_{12} + 310440 \beta_{13} + 4954464 \beta_{14} + 695688 \beta_{15} + 766320 \beta_{16} + 55608 \beta_{17} ) q^{64} \) \( + ( -10243517093242509 - 46917952946244 \beta_{1} - 54086238090 \beta_{2} + 73902363011 \beta_{3} + 296312720 \beta_{4} + 2213459837 \beta_{5} - 728687527 \beta_{6} - 748317999 \beta_{7} + 5546232 \beta_{8} + 85914874 \beta_{9} - 2180546 \beta_{10} - 11672545 \beta_{11} - 2907348 \beta_{12} + 432012 \beta_{13} - 332172 \beta_{14} + 668350 \beta_{15} - 144004 \beta_{16} - 44164 \beta_{17} ) q^{65} \) \( + ( 26638403263557345 - 9005998052822 \beta_{1} + 2891045694555 \beta_{2} + 56425654172 \beta_{3} + 508661468 \beta_{4} - 575776354 \beta_{5} + 6267659212 \beta_{6} - 1296331499 \beta_{7} - 56849476 \beta_{8} + 87086736 \beta_{9} + 2765602 \beta_{10} + 15525120 \beta_{11} - 994772 \beta_{12} - 4830344 \beta_{13} + 8341290 \beta_{14} + 650684 \beta_{15} - 284732 \beta_{16} - 147562 \beta_{17} ) q^{66} \) \( + ( -39067722461876 - 88242846262612 \beta_{1} + 631350551565 \beta_{2} - 114296482243 \beta_{3} + 114154152 \beta_{4} + 2276930590 \beta_{5} + 2396740549 \beta_{6} + 1246664070 \beta_{7} - 2656352 \beta_{8} - 63791237 \beta_{9} + 2900024 \beta_{10} - 15103276 \beta_{11} + 4853152 \beta_{12} - 2700016 \beta_{13} + 87989 \beta_{14} + 443680 \beta_{15} - 2700016 \beta_{16} - 21280 \beta_{17} ) q^{67} \) \( + ( 11176016374244370 - 26102485685158 \beta_{1} - 2980619779376 \beta_{2} - 4082064686 \beta_{3} - 739301000 \beta_{4} - 857616984 \beta_{5} - 5622639216 \beta_{6} + 1802480544 \beta_{7} - 56117760 \beta_{8} + 77923088 \beta_{9} - 2773528 \beta_{10} + 19533304 \beta_{11} + 40215648 \beta_{12} - 601968 \beta_{13} - 3390368 \beta_{14} + 161168 \beta_{15} + 1098400 \beta_{16} - 1090544 \beta_{17} ) q^{68} \) \( + ( 73909569684878 + 166696541159513 \beta_{1} - 977076282589 \beta_{2} - 312747973276 \beta_{3} + 51821562 \beta_{4} - 4065216878 \beta_{5} - 9193579300 \beta_{6} - 2581708786 \beta_{7} + 5356477 \beta_{8} + 7250567 \beta_{9} - 2763848 \beta_{10} - 28420476 \beta_{11} - 12715302 \beta_{12} - 456141 \beta_{13} + 28196352 \beta_{14} - 627360 \beta_{15} - 456141 \beta_{16} + 930144 \beta_{17} ) q^{69} \) \( + ( -18929201530171284 - 985733371408 \beta_{1} - 4775875437572 \beta_{2} + 17316766496 \beta_{3} - 1249832280 \beta_{4} + 655027648 \beta_{5} + 19341895672 \beta_{6} - 2697696668 \beta_{7} + 44758240 \beta_{8} - 315220096 \beta_{9} + 2036200 \beta_{10} + 32871312 \beta_{11} + 3028368 \beta_{12} - 8393440 \beta_{13} + 2582656 \beta_{14} - 1251856 \beta_{15} + 1354880 \beta_{16} + 1128736 \beta_{17} ) q^{70} \) \( + ( -88784463557539056 + 169564424882135 \beta_{1} + 385397914522 \beta_{2} + 224235510910 \beta_{3} - 372920762 \beta_{4} - 819176334 \beta_{5} - 4118794632 \beta_{6} + 2801228556 \beta_{7} + 214621303 \beta_{8} - 194440417 \beta_{9} + 1345104 \beta_{10} - 35690616 \beta_{11} - 34187350 \beta_{12} + 4190394 \beta_{13} - 4560894 \beta_{14} - 1235103 \beta_{15} - 1396798 \beta_{16} - 1767298 \beta_{17} ) q^{71} \) \( + ( -6440118874429678 - 58900617620443 \beta_{1} + 6223142410306 \beta_{2} + 104856561885 \beta_{3} + 947609472 \beta_{4} + 1934775103 \beta_{5} + 26287495940 \beta_{6} + 3997155585 \beta_{7} + 20185856 \beta_{8} + 151095361 \beta_{9} - 1362176 \beta_{10} + 40975488 \beta_{11} + 77456384 \beta_{12} - 1198720 \beta_{13} - 4220544 \beta_{14} - 1703936 \beta_{15} + 1035776 \beta_{16} - 2349824 \beta_{17} ) q^{72} \) \( + ( -5280517882275952 - 255391467194362 \beta_{1} - 330562763551 \beta_{2} + 298301598872 \beta_{3} - 1214882034 \beta_{4} - 846291347 \beta_{5} - 5417823676 \beta_{6} - 3753871653 \beta_{7} + 11402313 \beta_{8} - 346056288 \beta_{9} - 566208 \beta_{10} - 47874592 \beta_{11} + 49944000 \beta_{12} + 7381440 \beta_{13} - 1350720 \beta_{14} - 3791520 \beta_{15} - 2460480 \beta_{16} + 3570240 \beta_{17} ) q^{73} \) \( + ( -5197022958440046 - 224398186986 \beta_{1} + 3384236473765 \beta_{2} + 25805457314 \beta_{3} + 813407745 \beta_{4} + 2550278266 \beta_{5} - 29944096296 \beta_{6} - 4429879763 \beta_{7} + 44922400 \beta_{8} - 501746999 \beta_{9} - 2861192 \beta_{10} + 49903760 \beta_{11} + 10073081 \beta_{12} + 8336736 \beta_{13} - 18956544 \beta_{14} - 4613168 \beta_{15} + 2995840 \beta_{16} + 2706400 \beta_{17} ) q^{74} \) \( + ( -187483565700586 - 423125331585069 \beta_{1} + 2245434923501 \beta_{2} - 563144663284 \beta_{3} + 48382844 \beta_{4} - 5648181862 \beta_{5} - 21914017376 \beta_{6} + 5702371300 \beta_{7} + 4396208 \beta_{8} + 662270586 \beta_{9} - 4480172 \beta_{10} - 59959554 \beta_{11} + 38418160 \beta_{12} + 937816 \beta_{13} - 614898 \beta_{14} - 3626320 \beta_{15} + 937816 \beta_{16} - 2065072 \beta_{17} ) q^{75} \) \( + ( -60590522710258311 - 30017944651853 \beta_{1} - 2457680098086 \beta_{2} + 3400533511 \beta_{3} - 1318065253 \beta_{4} + 5393996792 \beta_{5} - 37693536909 \beta_{6} + 5159397970 \beta_{7} + 95760782 \beta_{8} - 48879172 \beta_{9} + 4383312 \beta_{10} + 63099440 \beta_{11} - 112421184 \beta_{12} + 528871 \beta_{13} + 17883513 \beta_{14} - 3842784 \beta_{15} + 3569728 \beta_{16} - 2873184 \beta_{17} ) q^{76} \) \( + ( 199151621355090 + 447026123662391 \beta_{1} + 2006822894765 \beta_{2} - 354082108036 \beta_{3} + 275885798 \beta_{4} + 7942883886 \beta_{5} + 23481657156 \beta_{6} - 6736768558 \beta_{7} - 18066029 \beta_{8} + 651650793 \beta_{9} + 6541448 \beta_{10} - 57267620 \beta_{11} - 18623290 \beta_{12} - 384771 \beta_{13} - 60453376 \beta_{14} - 5367904 \beta_{15} - 384771 \beta_{16} + 1883040 \beta_{17} ) q^{77} \) \( + ( 112800067863136164 + 57810007389601 \beta_{1} - 2524077607379 \beta_{2} + 194741304432 \beta_{3} + 254279505 \beta_{4} - 2166924943 \beta_{5} + 76420506161 \beta_{6} - 8114955927 \beta_{7} - 202305954 \beta_{8} + 1172275458 \beta_{9} - 11334528 \beta_{10} + 53733888 \beta_{11} + 11012422 \beta_{12} + 10993152 \beta_{13} + 22109743 \beta_{14} - 4337920 \beta_{15} - 4112570 \beta_{16} + 28253 \beta_{17} ) q^{78} \) \( + ( 100966074343363236 + 426348762871142 \beta_{1} + 946397835382 \beta_{2} + 513427949994 \beta_{3} + 1385183306 \beta_{4} + 7865755184 \beta_{5} - 6769222968 \beta_{6} + 7040098696 \beta_{7} - 689894796 \beta_{8} - 396040588 \beta_{9} - 13694800 \beta_{10} - 71121928 \beta_{11} + 40188792 \beta_{12} - 7185096 \beta_{13} + 5893976 \beta_{14} - 4078004 \beta_{15} + 2395032 \beta_{16} + 1103912 \beta_{17} ) q^{79} \) \( + ( 169801330844123512 - 86335263308936 \beta_{1} - 1089126240212 \beta_{2} + 334978708856 \beta_{3} - 1264115780 \beta_{4} - 6874609676 \beta_{5} + 118290638736 \beta_{6} + 10734232292 \beta_{7} - 319401664 \beta_{8} - 307710688 \beta_{9} + 14526532 \beta_{10} + 67170692 \beta_{11} - 194150672 \beta_{12} + 415944 \beta_{13} - 37189760 \beta_{14} - 4266104 \beta_{15} - 10669584 \beta_{16} + 5419512 \beta_{17} ) q^{80} \) \( + ( 99645957533391597 - 1015955338497530 \beta_{1} - 1561768945399 \beta_{2} + 556961467114 \beta_{3} + 6056681990 \beta_{4} - 19664691953 \beta_{5} - 10289689750 \beta_{6} - 17754968979 \beta_{7} - 87716899 \beta_{8} - 239805348 \beta_{9} + 17561652 \beta_{10} - 35868006 \beta_{11} - 105205752 \beta_{12} - 31025592 \beta_{13} + 10347192 \beta_{14} + 936180 \beta_{15} + 10341864 \beta_{16} - 10336536 \beta_{17} ) q^{81} \) \( + ( -134350800085025392 + 60298882190822 \beta_{1} - 3197697219514 \beta_{2} - 246236885768 \beta_{3} - 4751068648 \beta_{4} - 515735236 \beta_{5} - 102450857096 \beta_{6} - 12407303334 \beta_{7} + 341387768 \beta_{8} + 2123006304 \beta_{9} - 19099292 \beta_{10} + 24269440 \beta_{11} - 10042152 \beta_{12} + 3786352 \beta_{13} - 1222316 \beta_{14} + 2295608 \beta_{15} - 7582328 \beta_{16} - 5848148 \beta_{17} ) q^{82} \) \( + ( -515440294382238 - 1157474904209219 \beta_{1} - 4856026453999 \beta_{2} - 622554906406 \beta_{3} - 114059468 \beta_{4} - 18860701974 \beta_{5} + 21798793014 \beta_{6} + 16879464528 \beta_{7} + 23628944 \beta_{8} - 291482576 \beta_{9} - 21325508 \beta_{10} - 39830182 \beta_{11} - 111232304 \beta_{12} + 19058184 \beta_{13} - 410520 \beta_{14} + 36112 \beta_{15} + 19058184 \beta_{16} + 4565232 \beta_{17} ) q^{83} \) \( + ( 7986462657680960 + 276657302152232 \beta_{1} + 12311144060240 \beta_{2} + 71368292360 \beta_{3} + 5976097888 \beta_{4} - 15169819032 \beta_{5} - 77278286216 \beta_{6} + 26538615936 \beta_{7} + 275499728 \beta_{8} - 261859024 \beta_{9} + 23383304 \beta_{10} + 22183224 \beta_{11} + 237374432 \beta_{12} + 4077496 \beta_{13} - 23015624 \beta_{14} + 1706320 \beta_{15} - 23308384 \beta_{16} + 17990416 \beta_{17} ) q^{84} \) \( + ( 726711259382122 + 1633670224887605 \beta_{1} + 3124907535225 \beta_{2} + 408534911410 \beta_{3} + 475226138 \beta_{4} + 22740371214 \beta_{5} + 1485133154 \beta_{6} - 23776970250 \beta_{7} - 42477959 \beta_{8} - 911045045 \beta_{9} + 24326096 \beta_{10} + 37535064 \beta_{11} + 171663154 \beta_{12} + 5657639 \beta_{13} + 43281920 \beta_{14} + 11831872 \beta_{15} + 5657639 \beta_{16} - 12915136 \beta_{17} ) q^{85} \) \( + ( 74021867204752684 + 3723340241427 \beta_{1} + 7481724244809 \beta_{2} - 122941380847 \beta_{3} + 6686110776 \beta_{4} + 6368910845 \beta_{5} - 5364772635 \beta_{6} - 27999774324 \beta_{7} + 49629088 \beta_{8} - 3111712000 \beta_{9} - 18184328 \beta_{10} - 60629200 \beta_{11} - 36773200 \beta_{12} + 23702112 \beta_{13} - 66694784 \beta_{14} + 12428368 \beta_{15} + 1448320 \beta_{16} - 8125856 \beta_{17} ) q^{86} \) \( + ( -728071074094731396 + 2458003151187049 \beta_{1} + 3850255790480 \beta_{2} - 1163604691104 \beta_{3} - 3215476080 \beta_{4} + 26494847314 \beta_{5} + 21494435816 \beta_{6} + 39578900348 \beta_{7} + 1734895855 \beta_{8} + 1726781591 \beta_{9} - 15504656 \beta_{10} + 70030488 \beta_{11} + 190133498 \beta_{12} - 16616310 \beta_{13} + 22355090 \beta_{14} + 11769961 \beta_{15} + 5538770 \beta_{16} + 11277550 \beta_{17} ) q^{87} \) \( + ( -171930604469601516 + 382810748025558 \beta_{1} - 18357468162498 \beta_{2} - 622547031450 \beta_{3} - 6357157610 \beta_{4} + 28221207896 \beta_{5} - 28513736384 \beta_{6} + 34805790936 \beta_{7} + 321351856 \beta_{8} - 56550438 \beta_{9} + 18504194 \beta_{10} - 105824198 \beta_{11} + 303504888 \beta_{12} + 10563404 \beta_{13} + 152064104 \beta_{14} + 16796836 \beta_{15} + 27774872 \beta_{16} + 2681516 \beta_{17} ) q^{88} \) \( + ( 247391611883429836 - 2180743776917306 \beta_{1} - 4361714902095 \beta_{2} - 1392015064940 \beta_{3} - 19409879586 \beta_{4} - 5327114519 \beta_{5} + 15963255256 \beta_{6} - 31255422665 \beta_{7} + 39936257 \beta_{8} + 2200665960 \beta_{9} + 14384024 \beta_{10} + 175698700 \beta_{11} - 78341456 \beta_{12} + 38794800 \beta_{13} - 23574960 \beta_{14} + 18681848 \beta_{15} - 12931600 \beta_{16} + 2288240 \beta_{17} ) q^{89} \) \( + ( -16018388773799914 - 1401341298982 \beta_{1} - 54720823536581 \beta_{2} - 116866021554 \beta_{3} - 4348598601 \beta_{4} - 7026248138 \beta_{5} + 23738211896 \beta_{6} - 54553995165 \beta_{7} - 643622752 \beta_{8} - 5852030177 \beta_{9} + 3189976 \beta_{10} - 210347568 \beta_{11} - 44545009 \beta_{12} - 68235040 \beta_{13} + 127918848 \beta_{14} + 24969232 \beta_{15} - 3274624 \beta_{16} - 8048288 \beta_{17} ) q^{90} \) \( + ( -1413075544157100 - 3176971424977434 \beta_{1} - 3921726093810 \beta_{2} + 2486754549812 \beta_{3} - 587245420 \beta_{4} + 8431747166 \beta_{5} - 176424853424 \beta_{6} + 48285220908 \beta_{7} + 68843664 \beta_{8} - 4050214674 \beta_{9} + 13864284 \beta_{10} + 277787146 \beta_{11} - 62094000 \beta_{12} - 71106744 \beta_{13} + 6938730 \beta_{14} + 25465488 \beta_{15} - 71106744 \beta_{16} + 9586032 \beta_{17} ) q^{91} \) \( + ( -165136706070576460 - 837170131723560 \beta_{1} + 57283138397608 \beta_{2} + 149612929464 \beta_{3} + 6399047292 \beta_{4} + 50346221108 \beta_{5} + 228157779432 \beta_{6} + 43269784720 \beta_{7} - 1176806464 \beta_{8} + 239490056 \beta_{9} - 10733932 \beta_{10} - 330789668 \beta_{11} - 273964112 \beta_{12} - 14672280 \beta_{13} - 42935472 \beta_{14} + 25867016 \beta_{15} + 47566544 \beta_{16} - 26554168 \beta_{17} ) q^{92} \) \( + ( 1934458780575280 + 4366072392280504 \beta_{1} - 25653387262880 \beta_{2} + 2965015228152 \beta_{3} - 1452726528 \beta_{4} - 22410968424 \beta_{5} - 29897829336 \beta_{6} - 61473548208 \beta_{7} + 72842808 \beta_{8} - 3467155896 \beta_{9} - 21885456 \beta_{10} + 368171208 \beta_{11} - 202046160 \beta_{12} - 10436952 \beta_{13} + 195134976 \beta_{14} + 18634944 \beta_{15} - 10436952 \beta_{16} + 11603136 \beta_{17} ) q^{93} \) \( + ( -196329551283081312 - 71727485912478 \beta_{1} + 68682968773138 \beta_{2} - 174806078080 \beta_{3} - 1217166638 \beta_{4} - 11639068158 \beta_{5} - 320525641134 \beta_{6} - 59410127366 \beta_{7} + 1215230780 \beta_{8} + 7158001060 \beta_{9} + 45891504 \beta_{10} - 410381504 \beta_{11} - 29631220 \beta_{12} - 155377856 \beta_{13} - 4555458 \beta_{14} + 17720736 \beta_{15} + 25558348 \beta_{16} + 10439322 \beta_{17} ) q^{94} \) \( + ( 1324646163598321346 + 4233592849547839 \beta_{1} + 6437528881981 \beta_{2} - 2555822476737 \beta_{3} + 3467838115 \beta_{4} - 51892038664 \beta_{5} + 32390283200 \beta_{6} + 66047721584 \beta_{7} - 2790970060 \beta_{8} + 1141496468 \beta_{9} + 57792640 \beta_{10} + 550885184 \beta_{11} - 693523784 \beta_{12} + 99559800 \beta_{13} - 95045928 \beta_{14} + 17820268 \beta_{15} - 33186600 \beta_{16} - 28672728 \beta_{17} ) q^{95} \) \( + ( -166220014364944472 - 186011065113944 \beta_{1} - 80688497444148 \beta_{2} - 2003248231256 \beta_{3} + 6439918956 \beta_{4} - 100802207340 \beta_{5} - 732919271408 \beta_{6} + 77196750932 \beta_{7} + 1173209728 \beta_{8} + 650365520 \beta_{9} - 64976780 \beta_{10} - 615605676 \beta_{11} - 84378064 \beta_{12} - 29902392 \beta_{13} - 137849440 \beta_{14} + 12671848 \beta_{15} - 14117200 \beta_{16} - 26467112 \beta_{17} ) q^{96} \) \( + ( 456618998462816655 - 3407985652362470 \beta_{1} - 7385236399317 \beta_{2} - 3499114493945 \beta_{3} + 50067284838 \beta_{4} + 97513683562 \beta_{5} + 97441243633 \beta_{6} - 64578106172 \beta_{7} + 380504061 \beta_{8} + 371100482 \beta_{9} - 84938890 \beta_{10} + 558977019 \beta_{11} + 621278748 \beta_{12} + 106461180 \beta_{13} - 20249980 \beta_{14} - 11307434 \beta_{15} - 35487060 \beta_{16} + 50724140 \beta_{17} ) q^{97} \) \( + ( 491975385040712455 + 1778624569316313 \beta_{1} - 83400399386056 \beta_{2} + 615753304160 \beta_{3} + 25608927328 \beta_{4} - 2189964624 \beta_{5} + 797305978336 \beta_{6} - 73296526520 \beta_{7} - 1117556384 \beta_{8} + 11469701760 \beta_{9} + 88719696 \beta_{10} - 608704512 \beta_{11} + 100631264 \beta_{12} + 195476160 \beta_{13} - 325860208 \beta_{14} - 34529952 \beta_{15} + 66130592 \beta_{16} + 53834096 \beta_{17} ) q^{98} \) \( + ( -3038488710517962 - 6859303937670513 \beta_{1} + 50992886283831 \beta_{2} + 8296352080554 \beta_{3} - 1471514028 \beta_{4} + 124675136634 \beta_{5} + 828126948966 \beta_{6} + 102896224080 \beta_{7} - 315985776 \beta_{8} + 5098282992 \beta_{9} + 100913820 \beta_{10} + 792673002 \beta_{11} + 634232400 \beta_{12} + 97141704 \beta_{13} - 1871640 \beta_{14} - 17016432 \beta_{15} + 97141704 \beta_{16} - 49655184 \beta_{17} ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(18q \) \(\mathstrut -\mathstrut 458q^{2} \) \(\mathstrut -\mathstrut 412108q^{4} \) \(\mathstrut -\mathstrut 47240948q^{6} \) \(\mathstrut -\mathstrut 80707216q^{7} \) \(\mathstrut -\mathstrut 173313752q^{8} \) \(\mathstrut -\mathstrut 6198727826q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(18q \) \(\mathstrut -\mathstrut 458q^{2} \) \(\mathstrut -\mathstrut 412108q^{4} \) \(\mathstrut -\mathstrut 47240948q^{6} \) \(\mathstrut -\mathstrut 80707216q^{7} \) \(\mathstrut -\mathstrut 173313752q^{8} \) \(\mathstrut -\mathstrut 6198727826q^{9} \) \(\mathstrut +\mathstrut 758800104q^{10} \) \(\mathstrut -\mathstrut 9351642296q^{12} \) \(\mathstrut -\mathstrut 122453668784q^{14} \) \(\mathstrut +\mathstrut 156097960432q^{15} \) \(\mathstrut -\mathstrut 696212072432q^{16} \) \(\mathstrut +\mathstrut 14121426692q^{17} \) \(\mathstrut +\mathstrut 1813497117770q^{18} \) \(\mathstrut +\mathstrut 6517087595632q^{20} \) \(\mathstrut -\mathstrut 11074654117412q^{22} \) \(\mathstrut +\mathstrut 2177121583952q^{23} \) \(\mathstrut -\mathstrut 36473014189168q^{24} \) \(\mathstrut -\mathstrut 44414474211734q^{25} \) \(\mathstrut +\mathstrut 26782030269304q^{26} \) \(\mathstrut -\mathstrut 97002327802784q^{28} \) \(\mathstrut +\mathstrut 327847200544208q^{30} \) \(\mathstrut +\mathstrut 428505770260416q^{31} \) \(\mathstrut +\mathstrut 122449430282912q^{32} \) \(\mathstrut -\mathstrut 185380269683736q^{33} \) \(\mathstrut -\mathstrut 478448330325748q^{34} \) \(\mathstrut -\mathstrut 1053851053424436q^{36} \) \(\mathstrut -\mathstrut 486194587539796q^{38} \) \(\mathstrut +\mathstrut 942830575043152q^{39} \) \(\mathstrut -\mathstrut 1697818250610976q^{40} \) \(\mathstrut +\mathstrut 1200260021141524q^{41} \) \(\mathstrut +\mathstrut 4964410757677984q^{42} \) \(\mathstrut +\mathstrut 7031781193616424q^{44} \) \(\mathstrut -\mathstrut 14917443628457488q^{46} \) \(\mathstrut -\mathstrut 1134160959992928q^{47} \) \(\mathstrut -\mathstrut 3197964747991392q^{48} \) \(\mathstrut +\mathstrut 31570092936329058q^{49} \) \(\mathstrut -\mathstrut 34734151321861826q^{50} \) \(\mathstrut -\mathstrut 26877456911772208q^{52} \) \(\mathstrut +\mathstrut 32102189624395064q^{54} \) \(\mathstrut -\mathstrut 51757047863478736q^{55} \) \(\mathstrut +\mathstrut 52858270161183424q^{56} \) \(\mathstrut -\mathstrut 7740744140670520q^{57} \) \(\mathstrut -\mathstrut 20470056808312424q^{58} \) \(\mathstrut -\mathstrut 25825294995349152q^{60} \) \(\mathstrut +\mathstrut 93766686763697984q^{62} \) \(\mathstrut +\mathstrut 430168668665078416q^{63} \) \(\mathstrut -\mathstrut 253721318369185216q^{64} \) \(\mathstrut -\mathstrut 184007586011668640q^{65} \) \(\mathstrut +\mathstrut 479551889901302712q^{66} \) \(\mathstrut +\mathstrut 201388986348307752q^{68} \) \(\mathstrut -\mathstrut 340698473416023872q^{70} \) \(\mathstrut -\mathstrut 1599477989199799056q^{71} \) \(\mathstrut -\mathstrut 115475454880420648q^{72} \) \(\mathstrut -\mathstrut 93004298989809068q^{73} \) \(\mathstrut -\mathstrut 93558253821761240q^{74} \) \(\mathstrut -\mathstrut 1090379619519933176q^{76} \) \(\mathstrut +\mathstrut 2029949680340770544q^{78} \) \(\mathstrut +\mathstrut 1813975801025264480q^{79} \) \(\mathstrut +\mathstrut 3057120301750869696q^{80} \) \(\mathstrut +\mathstrut 1801762045144484330q^{81} \) \(\mathstrut -\mathstrut 2418785089528546244q^{82} \) \(\mathstrut +\mathstrut 141493317659087296q^{84} \) \(\mathstrut +\mathstrut 1332333723895598620q^{86} \) \(\mathstrut -\mathstrut 13124960835754102512q^{87} \) \(\mathstrut -\mathstrut 3097741184498130608q^{88} \) \(\mathstrut +\mathstrut 4470509771045743540q^{89} \) \(\mathstrut -\mathstrut 288100905697646824q^{90} \) \(\mathstrut -\mathstrut 2965990502259463392q^{92} \) \(\mathstrut -\mathstrut 3533635112749329696q^{94} \) \(\mathstrut +\mathstrut 23809730986995679856q^{95} \) \(\mathstrut -\mathstrut 2990158800543707584q^{96} \) \(\mathstrut +\mathstrut 8246429789464117220q^{97} \) \(\mathstrut +\mathstrut 8841667592735478822q^{98} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{18}\mathstrut -\mathstrut \) \(9\) \(x^{17}\mathstrut +\mathstrut \) \(847482358\) \(x^{16}\mathstrut -\mathstrut \) \(6779858660\) \(x^{15}\mathstrut +\mathstrut \) \(288755694526787238\) \(x^{14}\mathstrut -\mathstrut \) \(2021289743039984830\) \(x^{13}\mathstrut +\mathstrut \) \(51\!\cdots\!32\) \(x^{12}\mathstrut -\mathstrut \) \(30\!\cdots\!80\) \(x^{11}\mathstrut +\mathstrut \) \(50\!\cdots\!97\) \(x^{10}\mathstrut -\mathstrut \) \(25\!\cdots\!93\) \(x^{9}\mathstrut +\mathstrut \) \(27\!\cdots\!38\) \(x^{8}\mathstrut -\mathstrut \) \(11\!\cdots\!44\) \(x^{7}\mathstrut +\mathstrut \) \(78\!\cdots\!96\) \(x^{6}\mathstrut -\mathstrut \) \(23\!\cdots\!48\) \(x^{5}\mathstrut +\mathstrut \) \(88\!\cdots\!84\) \(x^{4}\mathstrut -\mathstrut \) \(17\!\cdots\!80\) \(x^{3}\mathstrut +\mathstrut \) \(27\!\cdots\!68\) \(x^{2}\mathstrut -\mathstrut \) \(27\!\cdots\!68\) \(x\mathstrut +\mathstrut \) \(11\!\cdots\!76\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\(-\)\(10\!\cdots\!25\) \(\nu^{17}\mathstrut -\mathstrut \) \(33\!\cdots\!23\) \(\nu^{16}\mathstrut -\mathstrut \) \(92\!\cdots\!16\) \(\nu^{15}\mathstrut -\mathstrut \) \(27\!\cdots\!84\) \(\nu^{14}\mathstrut -\mathstrut \) \(31\!\cdots\!82\) \(\nu^{13}\mathstrut -\mathstrut \) \(89\!\cdots\!18\) \(\nu^{12}\mathstrut -\mathstrut \) \(55\!\cdots\!04\) \(\nu^{11}\mathstrut -\mathstrut \) \(15\!\cdots\!04\) \(\nu^{10}\mathstrut -\mathstrut \) \(54\!\cdots\!41\) \(\nu^{9}\mathstrut -\mathstrut \) \(13\!\cdots\!79\) \(\nu^{8}\mathstrut -\mathstrut \) \(29\!\cdots\!28\) \(\nu^{7}\mathstrut -\mathstrut \) \(70\!\cdots\!64\) \(\nu^{6}\mathstrut -\mathstrut \) \(83\!\cdots\!56\) \(\nu^{5}\mathstrut -\mathstrut \) \(18\!\cdots\!36\) \(\nu^{4}\mathstrut -\mathstrut \) \(93\!\cdots\!40\) \(\nu^{3}\mathstrut -\mathstrut \) \(19\!\cdots\!60\) \(\nu^{2}\mathstrut -\mathstrut \) \(12\!\cdots\!40\) \(\nu\mathstrut -\mathstrut \) \(70\!\cdots\!64\)\()/\)\(91\!\cdots\!00\)
\(\beta_{2}\)\(=\)\((\)\(-\)\(10\!\cdots\!25\) \(\nu^{17}\mathstrut -\mathstrut \) \(33\!\cdots\!23\) \(\nu^{16}\mathstrut -\mathstrut \) \(92\!\cdots\!16\) \(\nu^{15}\mathstrut -\mathstrut \) \(27\!\cdots\!84\) \(\nu^{14}\mathstrut -\mathstrut \) \(31\!\cdots\!82\) \(\nu^{13}\mathstrut -\mathstrut \) \(89\!\cdots\!18\) \(\nu^{12}\mathstrut -\mathstrut \) \(55\!\cdots\!04\) \(\nu^{11}\mathstrut -\mathstrut \) \(15\!\cdots\!04\) \(\nu^{10}\mathstrut -\mathstrut \) \(54\!\cdots\!41\) \(\nu^{9}\mathstrut -\mathstrut \) \(13\!\cdots\!79\) \(\nu^{8}\mathstrut -\mathstrut \) \(29\!\cdots\!28\) \(\nu^{7}\mathstrut -\mathstrut \) \(70\!\cdots\!64\) \(\nu^{6}\mathstrut -\mathstrut \) \(83\!\cdots\!56\) \(\nu^{5}\mathstrut -\mathstrut \) \(18\!\cdots\!36\) \(\nu^{4}\mathstrut -\mathstrut \) \(93\!\cdots\!40\) \(\nu^{3}\mathstrut -\mathstrut \) \(19\!\cdots\!60\) \(\nu^{2}\mathstrut -\mathstrut \) \(54\!\cdots\!40\) \(\nu\mathstrut -\mathstrut \) \(70\!\cdots\!64\)\()/\)\(18\!\cdots\!00\)
\(\beta_{3}\)\(=\)\((\)\(32\!\cdots\!29\) \(\nu^{17}\mathstrut +\mathstrut \) \(97\!\cdots\!43\) \(\nu^{16}\mathstrut +\mathstrut \) \(27\!\cdots\!92\) \(\nu^{15}\mathstrut +\mathstrut \) \(85\!\cdots\!36\) \(\nu^{14}\mathstrut +\mathstrut \) \(92\!\cdots\!54\) \(\nu^{13}\mathstrut +\mathstrut \) \(30\!\cdots\!82\) \(\nu^{12}\mathstrut +\mathstrut \) \(16\!\cdots\!52\) \(\nu^{11}\mathstrut +\mathstrut \) \(57\!\cdots\!08\) \(\nu^{10}\mathstrut +\mathstrut \) \(16\!\cdots\!17\) \(\nu^{9}\mathstrut +\mathstrut \) \(60\!\cdots\!79\) \(\nu^{8}\mathstrut +\mathstrut \) \(89\!\cdots\!52\) \(\nu^{7}\mathstrut +\mathstrut \) \(35\!\cdots\!08\) \(\nu^{6}\mathstrut +\mathstrut \) \(25\!\cdots\!08\) \(\nu^{5}\mathstrut +\mathstrut \) \(10\!\cdots\!20\) \(\nu^{4}\mathstrut +\mathstrut \) \(28\!\cdots\!00\) \(\nu^{3}\mathstrut +\mathstrut \) \(13\!\cdots\!80\) \(\nu^{2}\mathstrut +\mathstrut \) \(55\!\cdots\!72\) \(\nu\mathstrut +\mathstrut \) \(21\!\cdots\!20\)\()/\)\(33\!\cdots\!00\)
\(\beta_{4}\)\(=\)\((\)\(-\)\(57\!\cdots\!61\) \(\nu^{17}\mathstrut -\mathstrut \) \(16\!\cdots\!43\) \(\nu^{16}\mathstrut -\mathstrut \) \(49\!\cdots\!80\) \(\nu^{15}\mathstrut -\mathstrut \) \(13\!\cdots\!92\) \(\nu^{14}\mathstrut -\mathstrut \) \(16\!\cdots\!50\) \(\nu^{13}\mathstrut -\mathstrut \) \(45\!\cdots\!94\) \(\nu^{12}\mathstrut -\mathstrut \) \(30\!\cdots\!16\) \(\nu^{11}\mathstrut -\mathstrut \) \(78\!\cdots\!60\) \(\nu^{10}\mathstrut -\mathstrut \) \(30\!\cdots\!85\) \(\nu^{9}\mathstrut -\mathstrut \) \(74\!\cdots\!39\) \(\nu^{8}\mathstrut -\mathstrut \) \(16\!\cdots\!24\) \(\nu^{7}\mathstrut -\mathstrut \) \(39\!\cdots\!60\) \(\nu^{6}\mathstrut -\mathstrut \) \(49\!\cdots\!84\) \(\nu^{5}\mathstrut -\mathstrut \) \(10\!\cdots\!72\) \(\nu^{4}\mathstrut -\mathstrut \) \(57\!\cdots\!80\) \(\nu^{3}\mathstrut -\mathstrut \) \(11\!\cdots\!80\) \(\nu^{2}\mathstrut -\mathstrut \) \(21\!\cdots\!68\) \(\nu\mathstrut -\mathstrut \) \(55\!\cdots\!08\)\()/\)\(14\!\cdots\!00\)
\(\beta_{5}\)\(=\)\((\)\(42\!\cdots\!71\) \(\nu^{17}\mathstrut -\mathstrut \) \(24\!\cdots\!23\) \(\nu^{16}\mathstrut +\mathstrut \) \(35\!\cdots\!88\) \(\nu^{15}\mathstrut -\mathstrut \) \(20\!\cdots\!76\) \(\nu^{14}\mathstrut +\mathstrut \) \(12\!\cdots\!06\) \(\nu^{13}\mathstrut -\mathstrut \) \(68\!\cdots\!82\) \(\nu^{12}\mathstrut +\mathstrut \) \(21\!\cdots\!08\) \(\nu^{11}\mathstrut -\mathstrut \) \(11\!\cdots\!88\) \(\nu^{10}\mathstrut +\mathstrut \) \(20\!\cdots\!23\) \(\nu^{9}\mathstrut -\mathstrut \) \(10\!\cdots\!99\) \(\nu^{8}\mathstrut +\mathstrut \) \(11\!\cdots\!68\) \(\nu^{7}\mathstrut -\mathstrut \) \(49\!\cdots\!28\) \(\nu^{6}\mathstrut +\mathstrut \) \(31\!\cdots\!92\) \(\nu^{5}\mathstrut -\mathstrut \) \(11\!\cdots\!80\) \(\nu^{4}\mathstrut +\mathstrut \) \(35\!\cdots\!60\) \(\nu^{3}\mathstrut -\mathstrut \) \(99\!\cdots\!60\) \(\nu^{2}\mathstrut +\mathstrut \) \(11\!\cdots\!88\) \(\nu\mathstrut -\mathstrut \) \(13\!\cdots\!60\)\()/\)\(43\!\cdots\!00\)
\(\beta_{6}\)\(=\)\((\)\(-\)\(28\!\cdots\!28\) \(\nu^{17}\mathstrut -\mathstrut \) \(38\!\cdots\!53\) \(\nu^{16}\mathstrut -\mathstrut \) \(24\!\cdots\!88\) \(\nu^{15}\mathstrut -\mathstrut \) \(37\!\cdots\!68\) \(\nu^{14}\mathstrut -\mathstrut \) \(82\!\cdots\!56\) \(\nu^{13}\mathstrut -\mathstrut \) \(14\!\cdots\!66\) \(\nu^{12}\mathstrut -\mathstrut \) \(14\!\cdots\!80\) \(\nu^{11}\mathstrut -\mathstrut \) \(28\!\cdots\!92\) \(\nu^{10}\mathstrut -\mathstrut \) \(14\!\cdots\!28\) \(\nu^{9}\mathstrut -\mathstrut \) \(32\!\cdots\!89\) \(\nu^{8}\mathstrut -\mathstrut \) \(79\!\cdots\!96\) \(\nu^{7}\mathstrut -\mathstrut \) \(20\!\cdots\!52\) \(\nu^{6}\mathstrut -\mathstrut \) \(22\!\cdots\!20\) \(\nu^{5}\mathstrut -\mathstrut \) \(65\!\cdots\!04\) \(\nu^{4}\mathstrut -\mathstrut \) \(25\!\cdots\!60\) \(\nu^{3}\mathstrut -\mathstrut \) \(82\!\cdots\!40\) \(\nu^{2}\mathstrut -\mathstrut \) \(72\!\cdots\!24\) \(\nu\mathstrut -\mathstrut \) \(13\!\cdots\!16\)\()/\)\(22\!\cdots\!00\)
\(\beta_{7}\)\(=\)\((\)\(70\!\cdots\!91\) \(\nu^{17}\mathstrut -\mathstrut \) \(23\!\cdots\!83\) \(\nu^{16}\mathstrut +\mathstrut \) \(59\!\cdots\!08\) \(\nu^{15}\mathstrut -\mathstrut \) \(19\!\cdots\!96\) \(\nu^{14}\mathstrut +\mathstrut \) \(20\!\cdots\!46\) \(\nu^{13}\mathstrut -\mathstrut \) \(61\!\cdots\!02\) \(\nu^{12}\mathstrut +\mathstrut \) \(35\!\cdots\!68\) \(\nu^{11}\mathstrut -\mathstrut \) \(10\!\cdots\!08\) \(\nu^{10}\mathstrut +\mathstrut \) \(35\!\cdots\!83\) \(\nu^{9}\mathstrut -\mathstrut \) \(94\!\cdots\!99\) \(\nu^{8}\mathstrut +\mathstrut \) \(19\!\cdots\!28\) \(\nu^{7}\mathstrut -\mathstrut \) \(47\!\cdots\!08\) \(\nu^{6}\mathstrut +\mathstrut \) \(54\!\cdots\!72\) \(\nu^{5}\mathstrut -\mathstrut \) \(12\!\cdots\!80\) \(\nu^{4}\mathstrut +\mathstrut \) \(60\!\cdots\!00\) \(\nu^{3}\mathstrut -\mathstrut \) \(12\!\cdots\!80\) \(\nu^{2}\mathstrut +\mathstrut \) \(83\!\cdots\!88\) \(\nu\mathstrut -\mathstrut \) \(46\!\cdots\!60\)\()/\)\(91\!\cdots\!00\)
\(\beta_{8}\)\(=\)\((\)\(17\!\cdots\!25\) \(\nu^{17}\mathstrut +\mathstrut \) \(67\!\cdots\!39\) \(\nu^{16}\mathstrut +\mathstrut \) \(14\!\cdots\!48\) \(\nu^{15}\mathstrut +\mathstrut \) \(76\!\cdots\!92\) \(\nu^{14}\mathstrut +\mathstrut \) \(51\!\cdots\!26\) \(\nu^{13}\mathstrut +\mathstrut \) \(32\!\cdots\!74\) \(\nu^{12}\mathstrut +\mathstrut \) \(90\!\cdots\!32\) \(\nu^{11}\mathstrut +\mathstrut \) \(71\!\cdots\!12\) \(\nu^{10}\mathstrut +\mathstrut \) \(89\!\cdots\!33\) \(\nu^{9}\mathstrut +\mathstrut \) \(84\!\cdots\!47\) \(\nu^{8}\mathstrut +\mathstrut \) \(49\!\cdots\!24\) \(\nu^{7}\mathstrut +\mathstrut \) \(54\!\cdots\!32\) \(\nu^{6}\mathstrut +\mathstrut \) \(13\!\cdots\!48\) \(\nu^{5}\mathstrut +\mathstrut \) \(17\!\cdots\!48\) \(\nu^{4}\mathstrut +\mathstrut \) \(15\!\cdots\!20\) \(\nu^{3}\mathstrut +\mathstrut \) \(26\!\cdots\!80\) \(\nu^{2}\mathstrut +\mathstrut \) \(28\!\cdots\!40\) \(\nu\mathstrut +\mathstrut \) \(17\!\cdots\!72\)\()/\)\(91\!\cdots\!00\)
\(\beta_{9}\)\(=\)\((\)\(33\!\cdots\!22\) \(\nu^{17}\mathstrut +\mathstrut \) \(26\!\cdots\!33\) \(\nu^{16}\mathstrut +\mathstrut \) \(28\!\cdots\!44\) \(\nu^{15}\mathstrut +\mathstrut \) \(22\!\cdots\!80\) \(\nu^{14}\mathstrut +\mathstrut \) \(98\!\cdots\!48\) \(\nu^{13}\mathstrut +\mathstrut \) \(77\!\cdots\!90\) \(\nu^{12}\mathstrut +\mathstrut \) \(17\!\cdots\!08\) \(\nu^{11}\mathstrut +\mathstrut \) \(13\!\cdots\!16\) \(\nu^{10}\mathstrut +\mathstrut \) \(17\!\cdots\!74\) \(\nu^{9}\mathstrut +\mathstrut \) \(13\!\cdots\!09\) \(\nu^{8}\mathstrut +\mathstrut \) \(94\!\cdots\!60\) \(\nu^{7}\mathstrut +\mathstrut \) \(70\!\cdots\!16\) \(\nu^{6}\mathstrut +\mathstrut \) \(26\!\cdots\!12\) \(\nu^{5}\mathstrut +\mathstrut \) \(19\!\cdots\!48\) \(\nu^{4}\mathstrut +\mathstrut \) \(30\!\cdots\!40\) \(\nu^{3}\mathstrut +\mathstrut \) \(20\!\cdots\!00\) \(\nu^{2}\mathstrut +\mathstrut \) \(11\!\cdots\!36\) \(\nu\mathstrut +\mathstrut \) \(32\!\cdots\!52\)\()/\)\(75\!\cdots\!00\)
\(\beta_{10}\)\(=\)\((\)\(22\!\cdots\!07\) \(\nu^{17}\mathstrut +\mathstrut \) \(80\!\cdots\!49\) \(\nu^{16}\mathstrut +\mathstrut \) \(18\!\cdots\!96\) \(\nu^{15}\mathstrut +\mathstrut \) \(66\!\cdots\!28\) \(\nu^{14}\mathstrut +\mathstrut \) \(61\!\cdots\!82\) \(\nu^{13}\mathstrut +\mathstrut \) \(21\!\cdots\!66\) \(\nu^{12}\mathstrut +\mathstrut \) \(10\!\cdots\!96\) \(\nu^{11}\mathstrut +\mathstrut \) \(36\!\cdots\!04\) \(\nu^{10}\mathstrut +\mathstrut \) \(98\!\cdots\!71\) \(\nu^{9}\mathstrut +\mathstrut \) \(33\!\cdots\!17\) \(\nu^{8}\mathstrut +\mathstrut \) \(51\!\cdots\!16\) \(\nu^{7}\mathstrut +\mathstrut \) \(16\!\cdots\!64\) \(\nu^{6}\mathstrut +\mathstrut \) \(13\!\cdots\!84\) \(\nu^{5}\mathstrut +\mathstrut \) \(38\!\cdots\!20\) \(\nu^{4}\mathstrut +\mathstrut \) \(14\!\cdots\!20\) \(\nu^{3}\mathstrut +\mathstrut \) \(34\!\cdots\!60\) \(\nu^{2}\mathstrut +\mathstrut \) \(12\!\cdots\!16\) \(\nu\mathstrut +\mathstrut \) \(45\!\cdots\!80\)\()/\)\(91\!\cdots\!00\)
\(\beta_{11}\)\(=\)\((\)\(22\!\cdots\!11\) \(\nu^{17}\mathstrut -\mathstrut \) \(84\!\cdots\!67\) \(\nu^{16}\mathstrut +\mathstrut \) \(18\!\cdots\!00\) \(\nu^{15}\mathstrut -\mathstrut \) \(73\!\cdots\!88\) \(\nu^{14}\mathstrut +\mathstrut \) \(64\!\cdots\!10\) \(\nu^{13}\mathstrut -\mathstrut \) \(25\!\cdots\!66\) \(\nu^{12}\mathstrut +\mathstrut \) \(11\!\cdots\!96\) \(\nu^{11}\mathstrut -\mathstrut \) \(47\!\cdots\!60\) \(\nu^{10}\mathstrut +\mathstrut \) \(11\!\cdots\!15\) \(\nu^{9}\mathstrut -\mathstrut \) \(49\!\cdots\!91\) \(\nu^{8}\mathstrut +\mathstrut \) \(62\!\cdots\!44\) \(\nu^{7}\mathstrut -\mathstrut \) \(29\!\cdots\!60\) \(\nu^{6}\mathstrut +\mathstrut \) \(17\!\cdots\!64\) \(\nu^{5}\mathstrut -\mathstrut \) \(88\!\cdots\!48\) \(\nu^{4}\mathstrut +\mathstrut \) \(19\!\cdots\!20\) \(\nu^{3}\mathstrut -\mathstrut \) \(10\!\cdots\!20\) \(\nu^{2}\mathstrut +\mathstrut \) \(41\!\cdots\!88\) \(\nu\mathstrut -\mathstrut \) \(16\!\cdots\!52\)\()/\)\(27\!\cdots\!00\)
\(\beta_{12}\)\(=\)\((\)\(-\)\(96\!\cdots\!13\) \(\nu^{17}\mathstrut +\mathstrut \) \(10\!\cdots\!37\) \(\nu^{16}\mathstrut -\mathstrut \) \(82\!\cdots\!48\) \(\nu^{15}\mathstrut +\mathstrut \) \(87\!\cdots\!12\) \(\nu^{14}\mathstrut -\mathstrut \) \(27\!\cdots\!06\) \(\nu^{13}\mathstrut +\mathstrut \) \(29\!\cdots\!54\) \(\nu^{12}\mathstrut -\mathstrut \) \(49\!\cdots\!80\) \(\nu^{11}\mathstrut +\mathstrut \) \(50\!\cdots\!68\) \(\nu^{10}\mathstrut -\mathstrut \) \(48\!\cdots\!53\) \(\nu^{9}\mathstrut +\mathstrut \) \(48\!\cdots\!61\) \(\nu^{8}\mathstrut -\mathstrut \) \(26\!\cdots\!56\) \(\nu^{7}\mathstrut +\mathstrut \) \(25\!\cdots\!88\) \(\nu^{6}\mathstrut -\mathstrut \) \(75\!\cdots\!60\) \(\nu^{5}\mathstrut +\mathstrut \) \(71\!\cdots\!56\) \(\nu^{4}\mathstrut -\mathstrut \) \(85\!\cdots\!40\) \(\nu^{3}\mathstrut +\mathstrut \) \(80\!\cdots\!80\) \(\nu^{2}\mathstrut -\mathstrut \) \(23\!\cdots\!24\) \(\nu\mathstrut +\mathstrut \) \(14\!\cdots\!64\)\()/\)\(91\!\cdots\!00\)
\(\beta_{13}\)\(=\)\((\)\(17\!\cdots\!37\) \(\nu^{17}\mathstrut +\mathstrut \) \(25\!\cdots\!63\) \(\nu^{16}\mathstrut +\mathstrut \) \(14\!\cdots\!64\) \(\nu^{15}\mathstrut +\mathstrut \) \(20\!\cdots\!20\) \(\nu^{14}\mathstrut +\mathstrut \) \(50\!\cdots\!58\) \(\nu^{13}\mathstrut +\mathstrut \) \(62\!\cdots\!30\) \(\nu^{12}\mathstrut +\mathstrut \) \(88\!\cdots\!08\) \(\nu^{11}\mathstrut +\mathstrut \) \(98\!\cdots\!96\) \(\nu^{10}\mathstrut +\mathstrut \) \(88\!\cdots\!89\) \(\nu^{9}\mathstrut +\mathstrut \) \(82\!\cdots\!79\) \(\nu^{8}\mathstrut +\mathstrut \) \(48\!\cdots\!60\) \(\nu^{7}\mathstrut +\mathstrut \) \(35\!\cdots\!76\) \(\nu^{6}\mathstrut +\mathstrut \) \(13\!\cdots\!72\) \(\nu^{5}\mathstrut +\mathstrut \) \(67\!\cdots\!48\) \(\nu^{4}\mathstrut +\mathstrut \) \(15\!\cdots\!60\) \(\nu^{3}\mathstrut +\mathstrut \) \(38\!\cdots\!80\) \(\nu^{2}\mathstrut +\mathstrut \) \(12\!\cdots\!56\) \(\nu\mathstrut -\mathstrut \) \(12\!\cdots\!88\)\()/\)\(91\!\cdots\!00\)
\(\beta_{14}\)\(=\)\((\)\(88\!\cdots\!31\) \(\nu^{17}\mathstrut +\mathstrut \) \(14\!\cdots\!87\) \(\nu^{16}\mathstrut +\mathstrut \) \(75\!\cdots\!28\) \(\nu^{15}\mathstrut +\mathstrut \) \(13\!\cdots\!44\) \(\nu^{14}\mathstrut +\mathstrut \) \(25\!\cdots\!86\) \(\nu^{13}\mathstrut +\mathstrut \) \(52\!\cdots\!98\) \(\nu^{12}\mathstrut +\mathstrut \) \(45\!\cdots\!48\) \(\nu^{11}\mathstrut +\mathstrut \) \(10\!\cdots\!72\) \(\nu^{10}\mathstrut +\mathstrut \) \(44\!\cdots\!23\) \(\nu^{9}\mathstrut +\mathstrut \) \(11\!\cdots\!71\) \(\nu^{8}\mathstrut +\mathstrut \) \(24\!\cdots\!88\) \(\nu^{7}\mathstrut +\mathstrut \) \(70\!\cdots\!32\) \(\nu^{6}\mathstrut +\mathstrut \) \(68\!\cdots\!92\) \(\nu^{5}\mathstrut +\mathstrut \) \(22\!\cdots\!00\) \(\nu^{4}\mathstrut +\mathstrut \) \(76\!\cdots\!60\) \(\nu^{3}\mathstrut +\mathstrut \) \(27\!\cdots\!80\) \(\nu^{2}\mathstrut +\mathstrut \) \(81\!\cdots\!68\) \(\nu\mathstrut +\mathstrut \) \(48\!\cdots\!40\)\()/\)\(41\!\cdots\!00\)
\(\beta_{15}\)\(=\)\((\)\(-\)\(34\!\cdots\!51\) \(\nu^{17}\mathstrut +\mathstrut \) \(26\!\cdots\!65\) \(\nu^{16}\mathstrut -\mathstrut \) \(29\!\cdots\!04\) \(\nu^{15}\mathstrut +\mathstrut \) \(22\!\cdots\!92\) \(\nu^{14}\mathstrut -\mathstrut \) \(10\!\cdots\!58\) \(\nu^{13}\mathstrut +\mathstrut \) \(74\!\cdots\!54\) \(\nu^{12}\mathstrut -\mathstrut \) \(17\!\cdots\!52\) \(\nu^{11}\mathstrut +\mathstrut \) \(13\!\cdots\!84\) \(\nu^{10}\mathstrut -\mathstrut \) \(17\!\cdots\!99\) \(\nu^{9}\mathstrut +\mathstrut \) \(12\!\cdots\!05\) \(\nu^{8}\mathstrut -\mathstrut \) \(96\!\cdots\!36\) \(\nu^{7}\mathstrut +\mathstrut \) \(68\!\cdots\!64\) \(\nu^{6}\mathstrut -\mathstrut \) \(27\!\cdots\!28\) \(\nu^{5}\mathstrut +\mathstrut \) \(18\!\cdots\!84\) \(\nu^{4}\mathstrut -\mathstrut \) \(30\!\cdots\!40\) \(\nu^{3}\mathstrut +\mathstrut \) \(20\!\cdots\!80\) \(\nu^{2}\mathstrut -\mathstrut \) \(10\!\cdots\!08\) \(\nu\mathstrut +\mathstrut \) \(31\!\cdots\!96\)\()/\)\(45\!\cdots\!00\)
\(\beta_{16}\)\(=\)\((\)\(66\!\cdots\!77\) \(\nu^{17}\mathstrut +\mathstrut \) \(22\!\cdots\!68\) \(\nu^{16}\mathstrut +\mathstrut \) \(56\!\cdots\!44\) \(\nu^{15}\mathstrut +\mathstrut \) \(20\!\cdots\!60\) \(\nu^{14}\mathstrut +\mathstrut \) \(19\!\cdots\!78\) \(\nu^{13}\mathstrut +\mathstrut \) \(78\!\cdots\!00\) \(\nu^{12}\mathstrut +\mathstrut \) \(34\!\cdots\!08\) \(\nu^{11}\mathstrut +\mathstrut \) \(15\!\cdots\!16\) \(\nu^{10}\mathstrut +\mathstrut \) \(33\!\cdots\!29\) \(\nu^{9}\mathstrut +\mathstrut \) \(16\!\cdots\!64\) \(\nu^{8}\mathstrut +\mathstrut \) \(18\!\cdots\!00\) \(\nu^{7}\mathstrut +\mathstrut \) \(10\!\cdots\!16\) \(\nu^{6}\mathstrut +\mathstrut \) \(52\!\cdots\!52\) \(\nu^{5}\mathstrut +\mathstrut \) \(31\!\cdots\!88\) \(\nu^{4}\mathstrut +\mathstrut \) \(59\!\cdots\!00\) \(\nu^{3}\mathstrut +\mathstrut \) \(40\!\cdots\!40\) \(\nu^{2}\mathstrut +\mathstrut \) \(13\!\cdots\!36\) \(\nu\mathstrut +\mathstrut \) \(81\!\cdots\!72\)\()/\)\(75\!\cdots\!00\)
\(\beta_{17}\)\(=\)\((\)\(12\!\cdots\!27\) \(\nu^{17}\mathstrut +\mathstrut \) \(37\!\cdots\!29\) \(\nu^{16}\mathstrut +\mathstrut \) \(10\!\cdots\!76\) \(\nu^{15}\mathstrut +\mathstrut \) \(31\!\cdots\!28\) \(\nu^{14}\mathstrut +\mathstrut \) \(34\!\cdots\!02\) \(\nu^{13}\mathstrut +\mathstrut \) \(10\!\cdots\!26\) \(\nu^{12}\mathstrut +\mathstrut \) \(61\!\cdots\!36\) \(\nu^{11}\mathstrut +\mathstrut \) \(19\!\cdots\!24\) \(\nu^{10}\mathstrut +\mathstrut \) \(60\!\cdots\!11\) \(\nu^{9}\mathstrut +\mathstrut \) \(18\!\cdots\!17\) \(\nu^{8}\mathstrut +\mathstrut \) \(33\!\cdots\!36\) \(\nu^{7}\mathstrut +\mathstrut \) \(10\!\cdots\!84\) \(\nu^{6}\mathstrut +\mathstrut \) \(94\!\cdots\!84\) \(\nu^{5}\mathstrut +\mathstrut \) \(27\!\cdots\!40\) \(\nu^{4}\mathstrut +\mathstrut \) \(10\!\cdots\!40\) \(\nu^{3}\mathstrut +\mathstrut \) \(30\!\cdots\!60\) \(\nu^{2}\mathstrut +\mathstrut \) \(29\!\cdots\!16\) \(\nu\mathstrut +\mathstrut \) \(51\!\cdots\!20\)\()/\)\(82\!\cdots\!00\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{2}\mathstrut -\mathstrut \) \(5\) \(\beta_{1}\)\()/4\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{8}\mathstrut +\mathstrut \) \(3\) \(\beta_{7}\mathstrut +\mathstrut \) \(4\) \(\beta_{6}\mathstrut +\mathstrut \) \(5\) \(\beta_{5}\mathstrut -\mathstrut \) \(2\) \(\beta_{4}\mathstrut -\mathstrut \) \(232\) \(\beta_{3}\mathstrut +\mathstrut \) \(221\) \(\beta_{2}\mathstrut +\mathstrut \) \(175746\) \(\beta_{1}\mathstrut -\mathstrut \) \(1506557050\)\()/16\)
\(\nu^{3}\)\(=\)\((\)\(304\) \(\beta_{17}\mathstrut -\mathstrut \) \(280\) \(\beta_{16}\mathstrut +\mathstrut \) \(208\) \(\beta_{15}\mathstrut +\mathstrut \) \(3\) \(\beta_{14}\mathstrut -\mathstrut \) \(280\) \(\beta_{13}\mathstrut -\mathstrut \) \(4720\) \(\beta_{12}\mathstrut -\mathstrut \) \(1422\) \(\beta_{11}\mathstrut -\mathstrut \) \(692\) \(\beta_{10}\mathstrut -\mathstrut \) \(46283\) \(\beta_{9}\mathstrut +\mathstrut \) \(1878\) \(\beta_{8}\mathstrut -\mathstrut \) \(44292\) \(\beta_{7}\mathstrut -\mathstrut \) \(1269603\) \(\beta_{6}\mathstrut -\mathstrut \) \(735498\) \(\beta_{5}\mathstrut -\mathstrut \) \(2216\) \(\beta_{4}\mathstrut -\mathstrut \) \(19642567\) \(\beta_{3}\mathstrut -\mathstrut \) \(2661416434\) \(\beta_{2}\mathstrut +\mathstrut \) \(15033929197\) \(\beta_{1}\mathstrut -\mathstrut \) \(2946572438\)\()/64\)
\(\nu^{4}\)\(=\)\((\)\(-\)\(5167052\) \(\beta_{17}\mathstrut +\mathstrut \) \(5169812\) \(\beta_{16}\mathstrut +\mathstrut \) \(468922\) \(\beta_{15}\mathstrut +\mathstrut \) \(5173608\) \(\beta_{14}\mathstrut -\mathstrut \) \(15513916\) \(\beta_{13}\mathstrut -\mathstrut \) \(52621756\) \(\beta_{12}\mathstrut -\mathstrut \) \(17939691\) \(\beta_{11}\mathstrut +\mathstrut \) \(8778058\) \(\beta_{10}\mathstrut -\mathstrut \) \(120087806\) \(\beta_{9}\mathstrut -\mathstrut \) \(1787243150\) \(\beta_{8}\mathstrut -\mathstrut \) \(14107838295\) \(\beta_{7}\mathstrut -\mathstrut \) \(12123492137\) \(\beta_{6}\mathstrut -\mathstrut \) \(18552249031\) \(\beta_{5}\mathstrut +\mathstrut \) \(6515116556\) \(\beta_{4}\mathstrut +\mathstrut \) \(682869156589\) \(\beta_{3}\mathstrut -\mathstrut \) \(1169846248580\) \(\beta_{2}\mathstrut -\mathstrut \) \(814346609154092\) \(\beta_{1}\mathstrut +\mathstrut \) \(2000916929827314317\)\()/128\)
\(\nu^{5}\)\(=\)\((\)\(-\)\(339623258964\) \(\beta_{17}\mathstrut +\mathstrut \) \(361957444368\) \(\beta_{16}\mathstrut -\mathstrut \) \(273742380294\) \(\beta_{15}\mathstrut -\mathstrut \) \(1008021542328\) \(\beta_{14}\mathstrut +\mathstrut \) \(361854025728\) \(\beta_{13}\mathstrut +\mathstrut \) \(5256921929676\) \(\beta_{12}\mathstrut +\mathstrut \) \(2328563090376\) \(\beta_{11}\mathstrut +\mathstrut \) \(883371924828\) \(\beta_{10}\mathstrut +\mathstrut \) \(57871079044494\) \(\beta_{9}\mathstrut -\mathstrut \) \(2411268605078\) \(\beta_{8}\mathstrut +\mathstrut \) \(150976603557573\) \(\beta_{7}\mathstrut +\mathstrut \) \(3221711956455328\) \(\beta_{6}\mathstrut +\mathstrut \) \(986123362042964\) \(\beta_{5}\mathstrut +\mathstrut \) \(1276036993546\) \(\beta_{4}\mathstrut +\mathstrut \) \(32158067002639892\) \(\beta_{3}\mathstrut +\mathstrut \) \(2105891937520680963\) \(\beta_{2}\mathstrut -\mathstrut \) \(18833884042875609632\) \(\beta_{1}\mathstrut +\mathstrut \) \(2098699620702960865\)\()/256\)
\(\nu^{6}\)\(=\)\((\)\(4558370694653092\) \(\beta_{17}\mathstrut -\mathstrut \) \(4943331546200512\) \(\beta_{16}\mathstrut -\mathstrut \) \(304156137159002\) \(\beta_{15}\mathstrut -\mathstrut \) \(5332469337269412\) \(\beta_{14}\mathstrut +\mathstrut \) \(14834337817700432\) \(\beta_{13}\mathstrut +\mathstrut \) \(43906169105653076\) \(\beta_{12}\mathstrut +\mathstrut \) \(13422999138120738\) \(\beta_{11}\mathstrut -\mathstrut \) \(7681204133328104\) \(\beta_{10}\mathstrut +\mathstrut \) \(118907391856768414\) \(\beta_{9}\mathstrut +\mathstrut \) \(822573183300699766\) \(\beta_{8}\mathstrut +\mathstrut \) \(8556184064124686121\) \(\beta_{7}\mathstrut +\mathstrut \) \(7343946605578202746\) \(\beta_{6}\mathstrut +\mathstrut \) \(12579280320816316646\) \(\beta_{5}\mathstrut -\mathstrut \) \(3326285630381572174\) \(\beta_{4}\mathstrut -\mathstrut \) \(385365471385965828578\) \(\beta_{3}\mathstrut +\mathstrut \) \(723233956739937857325\) \(\beta_{2}\mathstrut +\mathstrut \) \(495668474555355802770012\) \(\beta_{1}\mathstrut -\mathstrut \) \(793482891177599212133804523\)\()/256\)
\(\nu^{7}\)\(=\)\((\)\(42307637094527250510\) \(\beta_{17}\mathstrut -\mathstrut \) \(48000176167595499900\) \(\beta_{16}\mathstrut +\mathstrut \) \(37295629692394139289\) \(\beta_{15}\mathstrut +\mathstrut \) \(224959493751746936868\) \(\beta_{14}\mathstrut -\mathstrut \) \(47965565246027690628\) \(\beta_{13}\mathstrut -\mathstrut \) \(655326996592454715978\) \(\beta_{12}\mathstrut -\mathstrut \) \(389942018798264302680\) \(\beta_{11}\mathstrut -\mathstrut \) \(120870486725972158380\) \(\beta_{10}\mathstrut -\mathstrut \) \(7700201032631251901649\) \(\beta_{9}\mathstrut +\mathstrut \) \(328441279626721338377\) \(\beta_{8}\mathstrut -\mathstrut \) \(28696587362866504676349\) \(\beta_{7}\mathstrut -\mathstrut \) \(585727079188356691004218\) \(\beta_{6}\mathstrut -\mathstrut \) \(139141803163035320595218\) \(\beta_{5}\mathstrut +\mathstrut \) \(61208927834874724271\) \(\beta_{4}\mathstrut -\mathstrut \) \(5272928865982883298975989\) \(\beta_{3}\mathstrut -\mathstrut \) \(230705435585280083381707154\) \(\beta_{2}\mathstrut +\mathstrut \) \(2801142103469196963846858564\) \(\beta_{1}\mathstrut -\mathstrut \) \(194458475996688925650484815\)\()/128\)
\(\nu^{8}\)\(=\)\((\)\(-\)\(736976645003181291638416\) \(\beta_{17}\mathstrut +\mathstrut \) \(849070250688613857506236\) \(\beta_{16}\mathstrut +\mathstrut \) \(43148265663049324911260\) \(\beta_{15}\mathstrut +\mathstrut \) \(962278465219762258337892\) \(\beta_{14}\mathstrut -\mathstrut \) \(2547978616450970913854564\) \(\beta_{13}\mathstrut -\mathstrut \) \(6787379974795975941727856\) \(\beta_{12}\mathstrut -\mathstrut \) \(1953100285029115007149647\) \(\beta_{11}\mathstrut +\mathstrut \) \(1223153213879391792310190\) \(\beta_{10}\mathstrut -\mathstrut \) \(19909509876007177946586388\) \(\beta_{9}\mathstrut -\mathstrut \) \(98662112041950887438711484\) \(\beta_{8}\mathstrut -\mathstrut \) \(1176412829708703804887591298\) \(\beta_{7}\mathstrut -\mathstrut \) \(1014898991811778641582217833\) \(\beta_{6}\mathstrut -\mathstrut \) \(1849140193227078518857179663\) \(\beta_{5}\mathstrut +\mathstrut \) \(406926954053008185991884998\) \(\beta_{4}\mathstrut +\mathstrut \) \(50858187670562145037459417345\) \(\beta_{3}\mathstrut -\mathstrut \) \(100702557921776003462922325629\) \(\beta_{2}\mathstrut -\mathstrut \) \(68390183643609983562077099983064\) \(\beta_{1}\mathstrut +\mathstrut \) \(87128303417588077281080276788408934\)\()/128\)
\(\nu^{9}\)\(=\)\((\)\(-\)\(20\!\cdots\!04\) \(\beta_{17}\mathstrut +\mathstrut \) \(24\!\cdots\!96\) \(\beta_{16}\mathstrut -\mathstrut \) \(19\!\cdots\!06\) \(\beta_{15}\mathstrut -\mathstrut \) \(14\!\cdots\!28\) \(\beta_{14}\mathstrut +\mathstrut \) \(24\!\cdots\!44\) \(\beta_{13}\mathstrut +\mathstrut \) \(32\!\cdots\!64\) \(\beta_{12}\mathstrut +\mathstrut \) \(23\!\cdots\!64\) \(\beta_{11}\mathstrut +\mathstrut \) \(63\!\cdots\!84\) \(\beta_{10}\mathstrut +\mathstrut \) \(40\!\cdots\!06\) \(\beta_{9}\mathstrut -\mathstrut \) \(17\!\cdots\!98\) \(\beta_{8}\mathstrut +\mathstrut \) \(17\!\cdots\!03\) \(\beta_{7}\mathstrut +\mathstrut \) \(35\!\cdots\!84\) \(\beta_{6}\mathstrut +\mathstrut \) \(75\!\cdots\!28\) \(\beta_{5}\mathstrut -\mathstrut \) \(13\!\cdots\!42\) \(\beta_{4}\mathstrut +\mathstrut \) \(31\!\cdots\!60\) \(\beta_{3}\mathstrut +\mathstrut \) \(10\!\cdots\!29\) \(\beta_{2}\mathstrut -\mathstrut \) \(15\!\cdots\!72\) \(\beta_{1}\mathstrut +\mathstrut \) \(11\!\cdots\!43\)\()/256\)
\(\nu^{10}\)\(=\)\((\)\(42\!\cdots\!40\) \(\beta_{17}\mathstrut -\mathstrut \) \(51\!\cdots\!68\) \(\beta_{16}\mathstrut -\mathstrut \) \(25\!\cdots\!06\) \(\beta_{15}\mathstrut -\mathstrut \) \(59\!\cdots\!36\) \(\beta_{14}\mathstrut +\mathstrut \) \(15\!\cdots\!76\) \(\beta_{13}\mathstrut +\mathstrut \) \(38\!\cdots\!08\) \(\beta_{12}\mathstrut +\mathstrut \) \(10\!\cdots\!50\) \(\beta_{11}\mathstrut -\mathstrut \) \(69\!\cdots\!64\) \(\beta_{10}\mathstrut +\mathstrut \) \(11\!\cdots\!58\) \(\beta_{9}\mathstrut +\mathstrut \) \(48\!\cdots\!66\) \(\beta_{8}\mathstrut +\mathstrut \) \(62\!\cdots\!31\) \(\beta_{7}\mathstrut +\mathstrut \) \(54\!\cdots\!82\) \(\beta_{6}\mathstrut +\mathstrut \) \(10\!\cdots\!94\) \(\beta_{5}\mathstrut -\mathstrut \) \(20\!\cdots\!98\) \(\beta_{4}\mathstrut -\mathstrut \) \(26\!\cdots\!34\) \(\beta_{3}\mathstrut +\mathstrut \) \(54\!\cdots\!71\) \(\beta_{2}\mathstrut +\mathstrut \) \(36\!\cdots\!72\) \(\beta_{1}\mathstrut -\mathstrut \) \(40\!\cdots\!33\)\()/256\)
\(\nu^{11}\)\(=\)\((\)\(26\!\cdots\!22\) \(\beta_{17}\mathstrut -\mathstrut \) \(30\!\cdots\!24\) \(\beta_{16}\mathstrut +\mathstrut \) \(25\!\cdots\!81\) \(\beta_{15}\mathstrut +\mathstrut \) \(21\!\cdots\!56\) \(\beta_{14}\mathstrut -\mathstrut \) \(30\!\cdots\!36\) \(\beta_{13}\mathstrut -\mathstrut \) \(41\!\cdots\!06\) \(\beta_{12}\mathstrut -\mathstrut \) \(33\!\cdots\!76\) \(\beta_{11}\mathstrut -\mathstrut \) \(83\!\cdots\!96\) \(\beta_{10}\mathstrut -\mathstrut \) \(51\!\cdots\!05\) \(\beta_{9}\mathstrut +\mathstrut \) \(22\!\cdots\!05\) \(\beta_{8}\mathstrut -\mathstrut \) \(24\!\cdots\!16\) \(\beta_{7}\mathstrut -\mathstrut \) \(51\!\cdots\!70\) \(\beta_{6}\mathstrut -\mathstrut \) \(99\!\cdots\!74\) \(\beta_{5}\mathstrut +\mathstrut \) \(27\!\cdots\!31\) \(\beta_{4}\mathstrut -\mathstrut \) \(44\!\cdots\!43\) \(\beta_{3}\mathstrut -\mathstrut \) \(13\!\cdots\!37\) \(\beta_{2}\mathstrut +\mathstrut \) \(21\!\cdots\!28\) \(\beta_{1}\mathstrut -\mathstrut \) \(21\!\cdots\!58\)\()/128\)
\(\nu^{12}\)\(=\)\((\)\(-\)\(59\!\cdots\!08\) \(\beta_{17}\mathstrut +\mathstrut \) \(72\!\cdots\!88\) \(\beta_{16}\mathstrut +\mathstrut \) \(36\!\cdots\!46\) \(\beta_{15}\mathstrut +\mathstrut \) \(86\!\cdots\!48\) \(\beta_{14}\mathstrut -\mathstrut \) \(21\!\cdots\!08\) \(\beta_{13}\mathstrut -\mathstrut \) \(51\!\cdots\!40\) \(\beta_{12}\mathstrut -\mathstrut \) \(13\!\cdots\!19\) \(\beta_{11}\mathstrut +\mathstrut \) \(95\!\cdots\!54\) \(\beta_{10}\mathstrut -\mathstrut \) \(15\!\cdots\!58\) \(\beta_{9}\mathstrut -\mathstrut \) \(61\!\cdots\!78\) \(\beta_{8}\mathstrut -\mathstrut \) \(83\!\cdots\!87\) \(\beta_{7}\mathstrut -\mathstrut \) \(70\!\cdots\!05\) \(\beta_{6}\mathstrut -\mathstrut \) \(13\!\cdots\!87\) \(\beta_{5}\mathstrut +\mathstrut \) \(25\!\cdots\!96\) \(\beta_{4}\mathstrut +\mathstrut \) \(33\!\cdots\!85\) \(\beta_{3}\mathstrut -\mathstrut \) \(72\!\cdots\!24\) \(\beta_{2}\mathstrut -\mathstrut \) \(48\!\cdots\!20\) \(\beta_{1}\mathstrut +\mathstrut \) \(49\!\cdots\!49\)\()/128\)
\(\nu^{13}\)\(=\)\((\)\(-\)\(13\!\cdots\!88\) \(\beta_{17}\mathstrut +\mathstrut \) \(15\!\cdots\!28\) \(\beta_{16}\mathstrut -\mathstrut \) \(13\!\cdots\!66\) \(\beta_{15}\mathstrut -\mathstrut \) \(11\!\cdots\!80\) \(\beta_{14}\mathstrut +\mathstrut \) \(15\!\cdots\!68\) \(\beta_{13}\mathstrut +\mathstrut \) \(20\!\cdots\!64\) \(\beta_{12}\mathstrut +\mathstrut \) \(18\!\cdots\!44\) \(\beta_{11}\mathstrut +\mathstrut \) \(43\!\cdots\!72\) \(\beta_{10}\mathstrut +\mathstrut \) \(26\!\cdots\!34\) \(\beta_{9}\mathstrut -\mathstrut \) \(11\!\cdots\!06\) \(\beta_{8}\mathstrut +\mathstrut \) \(13\!\cdots\!53\) \(\beta_{7}\mathstrut +\mathstrut \) \(28\!\cdots\!32\) \(\beta_{6}\mathstrut +\mathstrut \) \(52\!\cdots\!64\) \(\beta_{5}\mathstrut -\mathstrut \) \(17\!\cdots\!90\) \(\beta_{4}\mathstrut +\mathstrut \) \(24\!\cdots\!72\) \(\beta_{3}\mathstrut +\mathstrut \) \(65\!\cdots\!71\) \(\beta_{2}\mathstrut -\mathstrut \) \(11\!\cdots\!04\) \(\beta_{1}\mathstrut +\mathstrut \) \(16\!\cdots\!81\)\()/256\)
\(\nu^{14}\)\(=\)\((\)\(31\!\cdots\!48\) \(\beta_{17}\mathstrut -\mathstrut \) \(39\!\cdots\!00\) \(\beta_{16}\mathstrut -\mathstrut \) \(20\!\cdots\!22\) \(\beta_{15}\mathstrut -\mathstrut \) \(47\!\cdots\!32\) \(\beta_{14}\mathstrut +\mathstrut \) \(11\!\cdots\!32\) \(\beta_{13}\mathstrut +\mathstrut \) \(27\!\cdots\!16\) \(\beta_{12}\mathstrut +\mathstrut \) \(71\!\cdots\!54\) \(\beta_{11}\mathstrut -\mathstrut \) \(50\!\cdots\!96\) \(\beta_{10}\mathstrut +\mathstrut \) \(83\!\cdots\!38\) \(\beta_{9}\mathstrut +\mathstrut \) \(31\!\cdots\!62\) \(\beta_{8}\mathstrut +\mathstrut \) \(43\!\cdots\!33\) \(\beta_{7}\mathstrut +\mathstrut \) \(37\!\cdots\!98\) \(\beta_{6}\mathstrut +\mathstrut \) \(72\!\cdots\!06\) \(\beta_{5}\mathstrut -\mathstrut \) \(12\!\cdots\!94\) \(\beta_{4}\mathstrut -\mathstrut \) \(17\!\cdots\!74\) \(\beta_{3}\mathstrut +\mathstrut \) \(38\!\cdots\!37\) \(\beta_{2}\mathstrut +\mathstrut \) \(25\!\cdots\!00\) \(\beta_{1}\mathstrut -\mathstrut \) \(24\!\cdots\!91\)\()/256\)
\(\nu^{15}\)\(=\)\((\)\(17\!\cdots\!30\) \(\beta_{17}\mathstrut -\mathstrut \) \(19\!\cdots\!44\) \(\beta_{16}\mathstrut +\mathstrut \) \(17\!\cdots\!13\) \(\beta_{15}\mathstrut +\mathstrut \) \(15\!\cdots\!00\) \(\beta_{14}\mathstrut -\mathstrut \) \(19\!\cdots\!36\) \(\beta_{13}\mathstrut -\mathstrut \) \(26\!\cdots\!18\) \(\beta_{12}\mathstrut -\mathstrut \) \(24\!\cdots\!88\) \(\beta_{11}\mathstrut -\mathstrut \) \(57\!\cdots\!16\) \(\beta_{10}\mathstrut -\mathstrut \) \(35\!\cdots\!65\) \(\beta_{9}\mathstrut +\mathstrut \) \(15\!\cdots\!41\) \(\beta_{8}\mathstrut -\mathstrut \) \(17\!\cdots\!23\) \(\beta_{7}\mathstrut -\mathstrut \) \(37\!\cdots\!54\) \(\beta_{6}\mathstrut -\mathstrut \) \(68\!\cdots\!62\) \(\beta_{5}\mathstrut +\mathstrut \) \(25\!\cdots\!95\) \(\beta_{4}\mathstrut -\mathstrut \) \(32\!\cdots\!77\) \(\beta_{3}\mathstrut -\mathstrut \) \(83\!\cdots\!44\) \(\beta_{2}\mathstrut +\mathstrut \) \(14\!\cdots\!68\) \(\beta_{1}\mathstrut -\mathstrut \) \(30\!\cdots\!09\)\()/128\)
\(\nu^{16}\)\(=\)\((\)\(-\)\(42\!\cdots\!52\) \(\beta_{17}\mathstrut +\mathstrut \) \(53\!\cdots\!80\) \(\beta_{16}\mathstrut +\mathstrut \) \(28\!\cdots\!44\) \(\beta_{15}\mathstrut +\mathstrut \) \(64\!\cdots\!00\) \(\beta_{14}\mathstrut -\mathstrut \) \(16\!\cdots\!68\) \(\beta_{13}\mathstrut -\mathstrut \) \(36\!\cdots\!36\) \(\beta_{12}\mathstrut -\mathstrut \) \(92\!\cdots\!95\) \(\beta_{11}\mathstrut +\mathstrut \) \(66\!\cdots\!50\) \(\beta_{10}\mathstrut -\mathstrut \) \(10\!\cdots\!00\) \(\beta_{9}\mathstrut -\mathstrut \) \(40\!\cdots\!52\) \(\beta_{8}\mathstrut -\mathstrut \) \(57\!\cdots\!08\) \(\beta_{7}\mathstrut -\mathstrut \) \(48\!\cdots\!01\) \(\beta_{6}\mathstrut -\mathstrut \) \(95\!\cdots\!07\) \(\beta_{5}\mathstrut +\mathstrut \) \(16\!\cdots\!14\) \(\beta_{4}\mathstrut +\mathstrut \) \(22\!\cdots\!17\) \(\beta_{3}\mathstrut -\mathstrut \) \(50\!\cdots\!87\) \(\beta_{2}\mathstrut -\mathstrut \) \(33\!\cdots\!52\) \(\beta_{1}\mathstrut +\mathstrut \) \(31\!\cdots\!36\)\()/128\)
\(\nu^{17}\)\(=\)\((\)\(-\)\(88\!\cdots\!52\) \(\beta_{17}\mathstrut +\mathstrut \) \(10\!\cdots\!16\) \(\beta_{16}\mathstrut -\mathstrut \) \(92\!\cdots\!34\) \(\beta_{15}\mathstrut -\mathstrut \) \(83\!\cdots\!92\) \(\beta_{14}\mathstrut +\mathstrut \) \(10\!\cdots\!08\) \(\beta_{13}\mathstrut +\mathstrut \) \(13\!\cdots\!48\) \(\beta_{12}\mathstrut +\mathstrut \) \(13\!\cdots\!12\) \(\beta_{11}\mathstrut +\mathstrut \) \(29\!\cdots\!28\) \(\beta_{10}\mathstrut +\mathstrut \) \(18\!\cdots\!62\) \(\beta_{9}\mathstrut -\mathstrut \) \(79\!\cdots\!10\) \(\beta_{8}\mathstrut +\mathstrut \) \(93\!\cdots\!79\) \(\beta_{7}\mathstrut +\mathstrut \) \(20\!\cdots\!84\) \(\beta_{6}\mathstrut +\mathstrut \) \(35\!\cdots\!40\) \(\beta_{5}\mathstrut -\mathstrut \) \(14\!\cdots\!46\) \(\beta_{4}\mathstrut +\mathstrut \) \(17\!\cdots\!00\) \(\beta_{3}\mathstrut +\mathstrut \) \(43\!\cdots\!29\) \(\beta_{2}\mathstrut -\mathstrut \) \(76\!\cdots\!88\) \(\beta_{1}\mathstrut +\mathstrut \) \(20\!\cdots\!67\)\()/256\)

Character Values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/8\mathbb{Z}\right)^\times\).

\(n\) \(5\) \(7\)
\(\chi(n)\) \(-1\) \(1\)

Embeddings

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

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

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
5.1
0.500000 + 222.384i
0.500000 222.384i
0.500000 + 16156.1i
0.500000 16156.1i
0.500000 12463.1i
0.500000 + 12463.1i
0.500000 520.665i
0.500000 + 520.665i
0.500000 + 7755.46i
0.500000 7755.46i
0.500000 11170.2i
0.500000 + 11170.2i
0.500000 + 5761.07i
0.500000 5761.07i
0.500000 + 12610.3i
0.500000 12610.3i
0.500000 7326.87i
0.500000 + 7326.87i
−696.194 199.004i 889.536i 445083. + 277090.i 5.56401e6i −177021. + 619289.i −4.51327e6 −2.54722e8 2.81482e8i 1.16147e9 1.10726e9 3.87363e9i
5.2 −696.194 + 199.004i 889.536i 445083. 277090.i 5.56401e6i −177021. 619289.i −4.51327e6 −2.54722e8 + 2.81482e8i 1.16147e9 1.10726e9 + 3.87363e9i
5.3 −600.469 404.629i 64624.3i 196838. + 485935.i 4.52012e6i −2.61489e7 + 3.88049e7i −6.45051e7 7.84279e7 3.71435e8i −3.01404e9 −1.82897e9 + 2.71419e9i
5.4 −600.469 + 404.629i 64624.3i 196838. 485935.i 4.52012e6i −2.61489e7 3.88049e7i −6.45051e7 7.84279e7 + 3.71435e8i −3.01404e9 −1.82897e9 2.71419e9i
5.5 −540.302 482.039i 49852.5i 59564.6 + 520893.i 3.24834e6i 2.40309e7 2.69354e7i 1.08353e8 2.18908e8 3.10152e8i −1.32301e9 −1.56583e9 + 1.75508e9i
5.6 −540.302 + 482.039i 49852.5i 59564.6 520893.i 3.24834e6i 2.40309e7 + 2.69354e7i 1.08353e8 2.18908e8 + 3.10152e8i −1.32301e9 −1.56583e9 1.75508e9i
5.7 −238.377 683.714i 2082.66i −410641. + 325963.i 1.39037e6i 1.42394e6 496458.i −1.59434e8 3.20753e8 + 2.03059e8i 1.15792e9 −9.50615e8 + 3.31432e8i
5.8 −238.377 + 683.714i 2082.66i −410641. 325963.i 1.39037e6i 1.42394e6 + 496458.i −1.59434e8 3.20753e8 2.03059e8i 1.15792e9 −9.50615e8 3.31432e8i
5.9 −97.3835 717.499i 31021.8i −505321. + 139745.i 4.13771e6i −2.22581e7 + 3.02101e6i 1.99482e8 1.49477e8 + 3.48958e8i 1.99908e8 2.96880e9 4.02945e8i
5.10 −97.3835 + 717.499i 31021.8i −505321. 139745.i 4.13771e6i −2.22581e7 3.02101e6i 1.99482e8 1.49477e8 3.48958e8i 1.99908e8 2.96880e9 + 4.02945e8i
5.11 255.450 677.520i 44680.9i −393778. 346145.i 2.26130e6i 3.02722e7 + 1.14138e7i −2.45061e7 −3.35111e8 + 1.78370e8i −8.34122e8 1.53207e9 + 5.77649e8i
5.12 255.450 + 677.520i 44680.9i −393778. + 346145.i 2.26130e6i 3.02722e7 1.14138e7i −2.45061e7 −3.35111e8 1.78370e8i −8.34122e8 1.53207e9 5.77649e8i
5.13 424.854 586.333i 23044.3i −163286. 498212.i 7.37490e6i −1.35116e7 9.79045e6i 5.30965e7 −3.61491e8 1.15928e8i 6.31223e8 −4.32415e9 3.13326e9i
5.14 424.854 + 586.333i 23044.3i −163286. + 498212.i 7.37490e6i −1.35116e7 + 9.79045e6i 5.30965e7 −3.61491e8 + 1.15928e8i 6.31223e8 −4.32415e9 + 3.13326e9i
5.15 564.947 452.905i 50441.2i 114042. 511735.i 7.26212e6i −2.28451e7 2.84966e7i −1.80984e8 −1.67340e8 3.40753e8i −1.38205e9 3.28905e9 + 4.10271e9i
5.16 564.947 + 452.905i 50441.2i 114042. + 511735.i 7.26212e6i −2.28451e7 + 2.84966e7i −1.80984e8 −1.67340e8 + 3.40753e8i −1.38205e9 3.28905e9 4.10271e9i
5.17 698.474 190.846i 29307.5i 451443. 266602.i 795283.i 5.59322e6 + 2.04705e7i 3.26570e7 2.64441e8 2.72371e8i 3.03333e8 1.51777e8 + 5.55484e8i
5.18 698.474 + 190.846i 29307.5i 451443. + 266602.i 795283.i 5.59322e6 2.04705e7i 3.26570e7 2.64441e8 + 2.72371e8i 3.03333e8 1.51777e8 5.55484e8i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 5.18
Significant digits:
Format:

Inner twists

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

Hecke kernels

There are no other newforms in \(S_{20}^{\mathrm{new}}(8, [\chi])\).