Properties

Label 8.21.d.b
Level 8
Weight 21
Character orbit 8.d
Analytic conductor 20.281
Analytic rank 0
Dimension 18
CM No
Inner twists 2

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) = \( 8 = 2^{3} \)
Weight: \( k \) = \( 21 \)
Character orbit: \([\chi]\) = 8.d (of order \(2\) and degree \(1\))

Newform invariants

Self dual: No
Analytic conductor: \(20.2811012082\)
Analytic rank: \(0\)
Dimension: \(18\)
Coefficient field: \(\mathbb{Q}[x]/(x^{18} + \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: multiple of \( 2^{153}\cdot 3^{15}\cdot 5^{4}\cdot 7^{2} \)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(f(q)\)  \(=\)  \( q\) \( + ( -22 + \beta_{1} ) q^{2} \) \( + ( -6347 - 5 \beta_{1} + \beta_{2} ) q^{3} \) \( + ( 40276 - 21 \beta_{1} - 3 \beta_{2} - \beta_{3} ) q^{4} \) \( + ( -4 - 39 \beta_{1} - 2 \beta_{3} - \beta_{4} ) q^{5} \) \( + ( -5091995 - 6457 \beta_{1} - 98 \beta_{2} + 6 \beta_{3} + \beta_{5} ) q^{6} \) \( + ( 2536 + 22864 \beta_{1} + 28 \beta_{2} - 21 \beta_{3} + 2 \beta_{4} - \beta_{7} ) q^{7} \) \( + ( -172512156 + 36651 \beta_{1} + 83 \beta_{2} + 17 \beta_{3} + 5 \beta_{4} - 4 \beta_{5} + 2 \beta_{6} - \beta_{8} ) q^{8} \) \( + ( 566520475 - 227370 \beta_{1} - 21775 \beta_{2} + 187 \beta_{3} - 4 \beta_{4} - 16 \beta_{5} + 9 \beta_{6} - \beta_{9} ) q^{9} \) \(+O(q^{10})\) \( q\) \( + ( -22 + \beta_{1} ) q^{2} \) \( + ( -6347 - 5 \beta_{1} + \beta_{2} ) q^{3} \) \( + ( 40276 - 21 \beta_{1} - 3 \beta_{2} - \beta_{3} ) q^{4} \) \( + ( -4 - 39 \beta_{1} - 2 \beta_{3} - \beta_{4} ) q^{5} \) \( + ( -5091995 - 6457 \beta_{1} - 98 \beta_{2} + 6 \beta_{3} + \beta_{5} ) q^{6} \) \( + ( 2536 + 22864 \beta_{1} + 28 \beta_{2} - 21 \beta_{3} + 2 \beta_{4} - \beta_{7} ) q^{7} \) \( + ( -172512156 + 36651 \beta_{1} + 83 \beta_{2} + 17 \beta_{3} + 5 \beta_{4} - 4 \beta_{5} + 2 \beta_{6} - \beta_{8} ) q^{8} \) \( + ( 566520475 - 227370 \beta_{1} - 21775 \beta_{2} + 187 \beta_{3} - 4 \beta_{4} - 16 \beta_{5} + 9 \beta_{6} - \beta_{9} ) q^{9} \) \( + ( 41262809 + 836 \beta_{1} + 3175 \beta_{2} + 28 \beta_{3} + 32 \beta_{4} - 5 \beta_{5} + 28 \beta_{6} - 3 \beta_{7} - 2 \beta_{8} - \beta_{11} ) q^{10} \) \( + ( 1547498283 - 3480329 \beta_{1} + 59935 \beta_{2} - 2247 \beta_{3} + 31 \beta_{4} + 109 \beta_{5} + 21 \beta_{6} - 2 \beta_{7} - 4 \beta_{8} - 2 \beta_{9} + \beta_{10} - \beta_{12} ) q^{11} \) \( + ( -11794650405 - 5344199 \beta_{1} + 31476 \beta_{2} + 6112 \beta_{3} + 700 \beta_{4} - 119 \beta_{5} + 16 \beta_{6} + 12 \beta_{7} + 11 \beta_{8} + 9 \beta_{9} - \beta_{10} - \beta_{11} - \beta_{12} + \beta_{14} - \beta_{15} - \beta_{16} ) q^{12} \) \( + ( -902872 - 8177681 \beta_{1} - 11002 \beta_{2} - 1842 \beta_{3} + 842 \beta_{4} + 184 \beta_{5} - 157 \beta_{6} + 91 \beta_{7} + 10 \beta_{8} - \beta_{9} + \beta_{11} - 2 \beta_{13} + \beta_{14} - 2 \beta_{16} + \beta_{17} ) q^{13} \) \( + ( -23958494374 - 502370 \beta_{1} + 490649 \beta_{2} - 30556 \beta_{3} - 2577 \beta_{4} - 62 \beta_{5} + 286 \beta_{6} + 124 \beta_{7} - 20 \beta_{8} + 19 \beta_{9} - 6 \beta_{10} - 7 \beta_{12} + 4 \beta_{13} + \beta_{14} - 3 \beta_{15} + 2 \beta_{16} + 2 \beta_{17} ) q^{14} \) \( + ( -1338421 - 11854237 \beta_{1} - 10650 \beta_{2} + 88749 \beta_{3} + 10487 \beta_{4} + 354 \beta_{5} - 122 \beta_{6} - 107 \beta_{7} + 45 \beta_{8} - 10 \beta_{9} + 14 \beta_{10} - 22 \beta_{11} + \beta_{13} - 2 \beta_{14} + 2 \beta_{15} - 12 \beta_{16} - 2 \beta_{17} ) q^{15} \) \( + ( -53271263488 - 173711755 \beta_{1} + 81467 \beta_{2} - 43062 \beta_{3} - 10512 \beta_{4} - 453 \beta_{5} + 2330 \beta_{6} - 3 \beta_{7} + 14 \beta_{8} - 7 \beta_{9} - 6 \beta_{10} + \beta_{11} - 30 \beta_{12} - 5 \beta_{13} + 2 \beta_{14} - 13 \beta_{15} + 14 \beta_{16} - 4 \beta_{17} ) q^{16} \) \( + ( 241324294053 - 44693065 \beta_{1} - 153140 \beta_{2} - 48703 \beta_{3} + 701 \beta_{4} + 4532 \beta_{5} - 106 \beta_{6} - 15 \beta_{7} - 657 \beta_{8} - 233 \beta_{9} + 47 \beta_{10} - 66 \beta_{11} + 52 \beta_{12} - 62 \beta_{13} - 28 \beta_{14} + 30 \beta_{15} ) q^{17} \) \( + ( -250904250347 + 561515649 \beta_{1} - 12165092 \beta_{2} + 355885 \beta_{3} - 22884 \beta_{4} - 24050 \beta_{5} + 5299 \beta_{6} - 2056 \beta_{7} + 234 \beta_{8} - 92 \beta_{9} - 154 \beta_{10} + 34 \beta_{11} + 42 \beta_{12} + 106 \beta_{13} - 46 \beta_{14} - 36 \beta_{15} + 36 \beta_{16} - 4 \beta_{17} ) q^{18} \) \( + ( -19959248459 + 683581221 \beta_{1} - 6375811 \beta_{2} + 489041 \beta_{3} - 6251 \beta_{4} + 2869 \beta_{5} - 8721 \beta_{6} + 76 \beta_{7} - 1634 \beta_{8} - 494 \beta_{9} + 19 \beta_{10} + 380 \beta_{11} + 95 \beta_{12} + 76 \beta_{13} - 152 \beta_{14} + 76 \beta_{15} ) q^{19} \) \( + ( -1806684681888 + 3068804 \beta_{1} + 19866174 \beta_{2} - 330598 \beta_{3} - 120712 \beta_{4} + 8832 \beta_{5} - 14858 \beta_{6} - 5122 \beta_{7} + 1066 \beta_{8} + 2220 \beta_{9} - 386 \beta_{10} + 96 \beta_{11} + 234 \beta_{12} - 242 \beta_{13} - 82 \beta_{14} - 20 \beta_{15} + 74 \beta_{16} + 8 \beta_{17} ) q^{20} \) \( + ( -223778204 - 2021217540 \beta_{1} - 2538206 \beta_{2} + 920820 \beta_{3} - 74741 \beta_{4} + 82152 \beta_{5} - 47883 \beta_{6} + 5029 \beta_{7} + 9110 \beta_{8} + 41 \beta_{9} - 24 \beta_{10} + 919 \beta_{11} + 128 \beta_{12} - 406 \beta_{13} - 169 \beta_{14} - 112 \beta_{15} - 46 \beta_{16} - 41 \beta_{17} ) q^{21} \) \( + ( -3681856700113 + 1469491795 \beta_{1} + 107788102 \beta_{2} + 2216692 \beta_{3} - 173056 \beta_{4} + 62909 \beta_{5} - 31694 \beta_{6} + 24856 \beta_{7} - 1556 \beta_{8} + 3048 \beta_{9} + 276 \beta_{10} + 68 \beta_{11} - 236 \beta_{12} + 484 \beta_{13} - 604 \beta_{14} - 200 \beta_{15} - 24 \beta_{16} - 72 \beta_{17} ) q^{22} \) \( + ( 127255527 + 1152332159 \beta_{1} + 1547334 \beta_{2} - 196641 \beta_{3} - 428553 \beta_{4} - 112502 \beta_{5} + 37982 \beta_{6} - 4401 \beta_{7} + 14129 \beta_{8} + 1198 \beta_{9} - 1386 \beta_{10} - 1230 \beta_{11} + 453 \beta_{13} - 938 \beta_{14} + 74 \beta_{15} + 260 \beta_{16} + 86 \beta_{17} ) q^{23} \) \( + ( 1237090490812 - 11773416144 \beta_{1} - 237734428 \beta_{2} + 8308178 \beta_{3} + 303050 \beta_{4} - 54106 \beta_{5} + 52868 \beta_{6} + 110786 \beta_{7} + 1698 \beta_{8} - 8278 \beta_{9} - 524 \beta_{10} + 346 \beta_{11} + 780 \beta_{12} - 1690 \beta_{13} - 1060 \beta_{14} + 526 \beta_{15} - 396 \beta_{16} + 152 \beta_{17} ) q^{24} \) \( + ( -21984737481630 + 1743644147 \beta_{1} + 154561513 \beta_{2} + 7520090 \beta_{3} - 147997 \beta_{4} - 759028 \beta_{5} + 157439 \beta_{6} + 23763 \beta_{7} - 44979 \beta_{8} + 5424 \beta_{9} - 2611 \beta_{10} - 1414 \beta_{11} - 868 \beta_{12} - 58 \beta_{13} - 980 \beta_{14} - 2278 \beta_{15} ) q^{25} \) \( + ( 8571752280623 + 179875372 \beta_{1} - 189244623 \beta_{2} + 11730524 \beta_{3} + 3026504 \beta_{4} + 65829 \beta_{5} - 4100 \beta_{6} - 198173 \beta_{7} - 7662 \beta_{8} - 27112 \beta_{9} + 4160 \beta_{10} + 369 \beta_{11} - 184 \beta_{12} - 832 \beta_{13} - 2808 \beta_{14} - 1240 \beta_{15} - 1200 \beta_{16} + 144 \beta_{17} ) q^{26} \) \( + ( -67136783137710 - 32135656656 \beta_{1} - 100894650 \beta_{2} + 32348493 \beta_{3} - 636615 \beta_{4} + 572001 \beta_{5} + 499323 \beta_{6} - 2412 \beta_{7} - 82386 \beta_{8} + 28122 \beta_{9} - 10641 \beta_{10} - 2052 \beta_{11} - 1917 \beta_{12} - 1716 \beta_{13} - 1944 \beta_{14} - 564 \beta_{15} ) q^{27} \) \( + ( -21926098872160 - 24433101120 \beta_{1} + 172175980 \beta_{2} - 3556260 \beta_{3} - 4134080 \beta_{4} + 742376 \beta_{5} + 86068 \beta_{6} - 611316 \beta_{7} - 10532 \beta_{8} + 11776 \beta_{9} + 4900 \beta_{10} + 184 \beta_{11} - 2708 \beta_{12} - 1860 \beta_{13} - 5756 \beta_{14} + 4144 \beta_{15} - 1780 \beta_{16} - 304 \beta_{17} ) q^{28} \) \( + ( 3971251964 + 35914604515 \beta_{1} + 48455320 \beta_{2} - 8437438 \beta_{3} - 3625971 \beta_{4} + 3163232 \beta_{5} - 49848 \beta_{6} + 52768 \beta_{7} + 83360 \beta_{8} - 22792 \beta_{9} - 9576 \beta_{10} - 9912 \beta_{11} - 128 \beta_{12} + 8152 \beta_{13} - 7288 \beta_{14} - 7568 \beta_{15} + 2672 \beta_{16} + 776 \beta_{17} ) q^{29} \) \( + ( 12403002927330 + 260219526 \beta_{1} - 401854427 \beta_{2} + 11108644 \beta_{3} - 20011133 \beta_{4} - 496006 \beta_{5} - 661962 \beta_{6} + 1404428 \beta_{7} + 32284 \beta_{8} - 43209 \beta_{9} + 10578 \beta_{10} + 3680 \beta_{11} + 12133 \beta_{12} - 7436 \beta_{13} - 4403 \beta_{14} - 9191 \beta_{15} - 230 \beta_{16} + 1178 \beta_{17} ) q^{30} \) \( + ( -10385757843 - 94327686899 \beta_{1} - 134130170 \beta_{2} - 114526626 \beta_{3} - 3935093 \beta_{4} - 5352610 \beta_{5} - 881894 \beta_{6} - 59606 \beta_{7} + 249459 \beta_{8} + 63818 \beta_{9} - 33950 \beta_{10} + 33622 \beta_{11} + 4096 \beta_{12} - 22129 \beta_{13} - 2750 \beta_{14} + 12510 \beta_{15} - 1396 \beta_{16} - 1726 \beta_{17} ) q^{31} \) \( + ( -157440768448136 - 56642873254 \beta_{1} + 636932614 \beta_{2} + 178044260 \beta_{3} + 35922904 \beta_{4} - 811810 \beta_{5} + 597436 \beta_{6} + 1302226 \beta_{7} + 73732 \beta_{8} + 201642 \beta_{9} + 1764 \beta_{10} + 3450 \beta_{11} + 932 \beta_{12} + 13102 \beta_{13} - 8700 \beta_{14} + 16078 \beta_{15} + 4668 \beta_{16} - 2664 \beta_{17} ) q^{32} \) \( + ( 247933423878394 + 429276471456 \beta_{1} + 3192450751 \beta_{2} + 141513207 \beta_{3} - 1359546 \beta_{4} - 12963752 \beta_{5} - 3694453 \beta_{6} + 229674 \beta_{7} - 60522 \beta_{8} - 209873 \beta_{9} - 12074 \beta_{10} + 58284 \beta_{11} + 15624 \beta_{12} + 28820 \beta_{13} - 13464 \beta_{14} - 12820 \beta_{15} ) q^{33} \) \( + ( -52167773799215 + 240307177244 \beta_{1} + 4776509108 \beta_{2} + 17418767 \beta_{3} + 89440628 \beta_{4} + 1740058 \beta_{5} - 2635631 \beta_{6} - 1917848 \beta_{7} + 116430 \beta_{8} + 181260 \beta_{9} + 96674 \beta_{10} + 3318 \beta_{11} + 846 \beta_{12} + 10638 \beta_{13} + 15238 \beta_{14} - 43660 \beta_{15} + 18316 \beta_{16} - 2348 \beta_{17} ) q^{34} \) \( + ( 200846288031774 - 321934729546 \beta_{1} - 2081662334 \beta_{2} + 473054022 \beta_{3} - 7861880 \beta_{4} + 32420622 \beta_{5} - 433952 \beta_{6} - 927870 \beta_{7} + 270522 \beta_{8} - 24200 \beta_{9} - 115584 \beta_{10} - 37668 \beta_{11} + 35714 \beta_{12} - 56148 \beta_{13} + 6568 \beta_{14} + 81196 \beta_{15} ) q^{35} \) \( + ( 86676621015300 - 249018970551 \beta_{1} - 19332518041 \beta_{2} - 417551811 \beta_{3} - 184580672 \beta_{4} - 14799736 \beta_{5} + 7872736 \beta_{6} - 2227984 \beta_{7} + 448584 \beta_{8} + 23144 \beta_{9} + 69896 \beta_{10} + 7288 \beta_{11} + 11784 \beta_{12} + 9136 \beta_{13} + 45368 \beta_{14} + 48408 \beta_{15} + 22344 \beta_{16} + 5312 \beta_{17} ) q^{36} \) \( + ( 33475190424 + 299321024521 \beta_{1} + 317056538 \beta_{2} - 1008498142 \beta_{3} + 45669566 \beta_{4} + 34360520 \beta_{5} - 4465379 \beta_{6} + 209285 \beta_{7} - 1774026 \beta_{8} - 341823 \beta_{9} - 113056 \beta_{10} - 12225 \beta_{11} - 6656 \beta_{12} + 19618 \beta_{13} + 71487 \beta_{14} - 52032 \beta_{15} - 45182 \beta_{16} - 8897 \beta_{17} ) q^{37} \) \( + ( 716964022885423 - 4786745981 \beta_{1} + 3118047782 \beta_{2} - 796065724 \beta_{3} - 282104704 \beta_{4} - 5510323 \beta_{5} + 6898178 \beta_{6} - 1501608 \beta_{7} + 128972 \beta_{8} - 772312 \beta_{9} + 624948 \beta_{10} - 50844 \beta_{11} - 150860 \beta_{12} + 133380 \beta_{13} + 95684 \beta_{14} - 96520 \beta_{15} + 5928 \beta_{16} - 11400 \beta_{17} ) q^{38} \) \( + ( 86887862032 + 774720836472 \beta_{1} + 724846964 \beta_{2} - 3404978527 \beta_{3} + 11794174 \beta_{4} - 62193744 \beta_{5} + 16055728 \beta_{6} - 2205347 \beta_{7} - 4242248 \beta_{8} + 249232 \beta_{9} - 638560 \beta_{10} - 256272 \beta_{11} - 45056 \beta_{12} + 60200 \beta_{13} + 112464 \beta_{14} + 117904 \beta_{15} - 16160 \beta_{16} + 21328 \beta_{17} ) q^{39} \) \( + ( 656122817384664 - 1793707196284 \beta_{1} + 10731364876 \beta_{2} + 104670480 \beta_{3} + 673774592 \beta_{4} + 24213996 \beta_{5} + 5419120 \beta_{6} - 7026764 \beta_{7} + 594632 \beta_{8} + 228292 \beta_{9} + 967240 \beta_{10} - 106076 \beta_{11} - 110376 \beta_{12} - 188356 \beta_{13} + 203448 \beta_{14} + 130796 \beta_{15} - 26328 \beta_{16} + 28272 \beta_{17} ) q^{40} \) \( + ( 943975093216677 + 565819000809 \beta_{1} + 48864499691 \beta_{2} + 3069998254 \beta_{3} - 48706055 \beta_{4} - 22428124 \beta_{5} + 134925 \beta_{6} - 1069879 \beta_{7} + 2731927 \beta_{8} + 164752 \beta_{9} - 1020457 \beta_{10} - 579954 \beta_{11} - 179692 \beta_{12} - 71310 \beta_{13} + 288068 \beta_{14} - 149266 \beta_{15} ) q^{41} \) \( + ( 2118236695195980 + 44634388960 \beta_{1} - 74465122156 \beta_{2} + 3337081976 \beta_{3} + 1143668824 \beta_{4} - 19744420 \beta_{5} + 3792088 \beta_{6} + 23463684 \beta_{7} + 1082376 \beta_{8} - 1581912 \beta_{9} + 1949888 \beta_{10} - 118692 \beta_{11} + 10488 \beta_{12} + 240448 \beta_{13} + 141624 \beta_{14} + 22552 \beta_{15} - 168144 \beta_{16} + 22512 \beta_{17} ) q^{42} \) \( + ( -1527523026183803 + 1743972148719 \beta_{1} + 48195623913 \beta_{2} + 8277659834 \beta_{3} - 130666686 \beta_{4} - 10889630 \beta_{5} - 6828378 \beta_{6} - 2871976 \beta_{7} + 4523372 \beta_{8} + 1319700 \beta_{9} - 2469938 \beta_{10} + 532568 \beta_{11} - 365914 \beta_{12} + 473336 \beta_{13} + 257296 \beta_{14} - 156808 \beta_{15} ) q^{43} \) \( + ( 1842886064616411 - 3644195643207 \beta_{1} - 48999134476 \beta_{2} - 1239459968 \beta_{3} - 1035443204 \beta_{4} + 97598185 \beta_{5} - 31290032 \beta_{6} + 39636652 \beta_{7} + 2303371 \beta_{8} + 951081 \beta_{9} + 3690303 \beta_{10} - 276705 \beta_{11} + 46143 \beta_{12} - 417440 \beta_{13} + 128577 \beta_{14} + 161375 \beta_{15} - 168769 \beta_{16} - 55936 \beta_{17} ) q^{44} \) \( + ( 202396549880 + 1765034477283 \beta_{1} + 802172226 \beta_{2} - 20559296986 \beta_{3} - 601560460 \beta_{4} - 66877848 \beta_{5} - 11567795 \beta_{6} - 2380531 \beta_{7} - 4737018 \beta_{8} + 874305 \beta_{9} - 4453576 \beta_{10} + 813055 \beta_{11} + 6528 \beta_{12} - 121526 \beta_{13} + 61247 \beta_{14} + 55728 \beta_{15} + 414978 \beta_{16} + 67775 \beta_{17} ) q^{45} \) \( + ( -1207821322517906 - 25891927734 \beta_{1} + 115383226995 \beta_{2} - 2925698020 \beta_{3} - 1323009755 \beta_{4} + 45700470 \beta_{5} - 33751622 \beta_{6} - 51120620 \beta_{7} + 3846340 \beta_{8} + 3764753 \beta_{9} + 4560350 \beta_{10} + 223200 \beta_{11} + 898163 \beta_{12} - 220692 \beta_{13} - 315445 \beta_{14} + 570143 \beta_{15} - 33002 \beta_{16} + 70294 \beta_{17} ) q^{46} \) \( + ( 638504759599 + 5691794458671 \beta_{1} + 5629477962 \beta_{2} - 26192379312 \beta_{3} - 595678723 \beta_{4} + 235545866 \beta_{5} - 28096386 \beta_{6} - 12983524 \beta_{7} + 175281 \beta_{8} - 1283794 \beta_{9} - 6100122 \beta_{10} + 334386 \beta_{11} + 94208 \beta_{12} + 291413 \beta_{13} - 417322 \beta_{14} - 480854 \beta_{15} + 319236 \beta_{16} - 179754 \beta_{17} ) q^{47} \) \( + ( 1575920603563280 + 1269189670102 \beta_{1} + 4457962378 \beta_{2} + 12931704620 \beta_{3} + 2486494224 \beta_{4} - 207246662 \beta_{5} - 15953796 \beta_{6} - 60291754 \beta_{7} + 11692852 \beta_{8} - 7052770 \beta_{9} + 7896492 \beta_{10} + 951182 \beta_{11} + 1012732 \beta_{12} + 591514 \beta_{13} - 774020 \beta_{14} - 620342 \beta_{15} + 24868 \beta_{16} - 199480 \beta_{17} ) q^{48} \) \( + ( -14052809011099279 + 7898735842816 \beta_{1} + 86474099464 \beta_{2} + 46115830408 \beta_{3} - 736356976 \beta_{4} + 312768640 \beta_{5} - 54829592 \beta_{6} - 14772880 \beta_{7} - 25545328 \beta_{8} + 3187336 \beta_{9} - 11919920 \beta_{10} + 2090848 \beta_{11} + 1216192 \beta_{12} - 65184 \beta_{13} - 1195456 \beta_{14} + 1025760 \beta_{15} ) q^{49} \) \( + ( 2314859379768388 - 21941686423515 \beta_{1} - 720621807768 \beta_{2} + 6605936798 \beta_{3} + 1117232168 \beta_{4} + 16505588 \beta_{5} + 221242274 \beta_{6} + 47667856 \beta_{7} + 13896988 \beta_{8} - 334952 \beta_{9} + 10524356 \beta_{10} + 814380 \beta_{11} - 368932 \beta_{12} - 1573156 \beta_{13} - 1186228 \beta_{14} + 896616 \beta_{15} + 1022104 \beta_{16} - 135896 \beta_{17} ) q^{50} \) \( + ( -1897400314666084 + 17291524425762 \beta_{1} + 612036429276 \beta_{2} + 51745119583 \beta_{3} - 826689151 \beta_{4} - 637854645 \beta_{5} - 7242517 \beta_{6} + 8956938 \beta_{7} - 66205716 \beta_{8} - 8236062 \beta_{9} - 12824705 \beta_{10} - 2767248 \beta_{11} + 1807497 \beta_{12} - 1872208 \beta_{13} - 1807200 \beta_{14} - 830032 \beta_{15} ) q^{51} \) \( + ( -2475530719381344 + 8528569426364 \beta_{1} - 896109448830 \beta_{2} + 9459850854 \beta_{3} - 326110968 \beta_{4} - 302548800 \beta_{5} - 57024374 \beta_{6} + 63550146 \beta_{7} + 15578326 \beta_{8} - 2389228 \beta_{9} + 15948034 \beta_{10} + 2800480 \beta_{11} - 1196394 \beta_{12} + 3224626 \beta_{13} - 1829934 \beta_{14} - 3114668 \beta_{15} + 781878 \beta_{16} + 388344 \beta_{17} ) q^{52} \) \( + ( -322007712920 - 3164782078987 \beta_{1} - 10450245954 \beta_{2} - 75440455062 \beta_{3} + 138238292 \beta_{4} - 881203560 \beta_{5} - 114594733 \beta_{6} - 51921709 \beta_{7} + 30082490 \beta_{8} + 7092831 \beta_{9} - 21860088 \beta_{10} - 6114207 \beta_{11} + 164480 \beta_{12} - 347338 \beta_{13} - 2406111 \beta_{14} + 2227152 \beta_{15} - 2320834 \beta_{16} - 349279 \beta_{17} ) q^{53} \) \( + ( -32196830079608532 - 67856036537190 \beta_{1} + 798741057060 \beta_{2} + 24358473426 \beta_{3} + 2867012736 \beta_{4} - 220056354 \beta_{5} + 559067898 \beta_{6} + 129435384 \beta_{7} + 54563484 \beta_{8} + 16226952 \beta_{9} + 18973284 \beta_{10} - 538668 \beta_{11} - 2978844 \beta_{12} - 3085068 \beta_{13} - 253260 \beta_{14} - 134376 \beta_{15} - 152952 \beta_{16} - 266856 \beta_{17} ) q^{54} \) \( + ( 1761343668220 + 15556972064244 \beta_{1} + 12347029652 \beta_{2} - 115075748249 \beta_{3} - 1148279562 \beta_{4} + 1967432344 \beta_{5} - 428759704 \beta_{6} + 139580091 \beta_{7} + 45583452 \beta_{8} - 10373496 \beta_{9} - 26449352 \beta_{10} + 5175672 \beta_{11} + 720896 \beta_{12} - 3002980 \beta_{13} - 619032 \beta_{14} - 3176328 \beta_{15} - 2505616 \beta_{16} + 1075688 \beta_{17} ) q^{55} \) \( + ( -28091158388128944 - 22547082873864 \beta_{1} + 2605559079400 \beta_{2} - 6332404256 \beta_{3} - 6487537152 \beta_{4} + 295810344 \beta_{5} - 161513696 \beta_{6} + 394893976 \beta_{7} - 1691280 \beta_{8} + 14571896 \beta_{9} + 32151664 \beta_{10} - 6061128 \beta_{11} - 4507824 \beta_{12} + 3273096 \beta_{13} - 1461872 \beta_{14} - 4493016 \beta_{15} + 676528 \beta_{16} + 955168 \beta_{17} ) q^{56} \) \( + ( -29024876736105962 + 6909296378276 \beta_{1} + 912329414115 \beta_{2} + 134578654919 \beta_{3} - 2157691718 \beta_{4} + 2039312104 \beta_{5} - 190245993 \beta_{6} - 72253314 \beta_{7} - 93143358 \beta_{8} - 2512313 \beta_{9} - 32090430 \beta_{10} + 2267460 \beta_{11} - 5897448 \beta_{12} - 1136580 \beta_{13} - 1831752 \beta_{14} + 2708868 \beta_{15} ) q^{57} \) \( + ( -37638866802207277 - 766828756116 \beta_{1} - 3210128424851 \beta_{2} - 5709830668 \beta_{3} - 7966448608 \beta_{4} - 196796023 \beta_{5} + 273025012 \beta_{6} - 431680737 \beta_{7} + 49612778 \beta_{8} + 72079936 \beta_{9} + 29775616 \beta_{10} - 2449355 \beta_{11} + 2498240 \beta_{12} - 1679360 \beta_{13} + 1798336 \beta_{14} + 751552 \beta_{15} - 4240512 \beta_{16} + 488832 \beta_{17} ) q^{58} \) \( + ( 5172728943705063 + 1047552110929 \beta_{1} + 536700741687 \beta_{2} + 142889860496 \beta_{3} - 2376141530 \beta_{4} - 1721745056 \beta_{5} + 701225930 \beta_{6} - 15041274 \beta_{7} - 6405222 \beta_{8} - 28688044 \beta_{9} - 40072766 \beta_{10} + 4288716 \beta_{11} - 2766904 \beta_{12} + 4612700 \beta_{13} + 1398088 \beta_{14} - 3538596 \beta_{15} ) q^{59} \) \( + ( 90651319204697632 + 14360492345408 \beta_{1} - 6834188151748 \beta_{2} + 86922011372 \beta_{3} + 30255498560 \beta_{4} - 631308088 \beta_{5} - 297046428 \beta_{6} - 823603876 \beta_{7} + 34947948 \beta_{8} - 57437312 \beta_{9} + 46568340 \beta_{10} - 16677032 \beta_{11} + 8673468 \beta_{12} - 5118900 \beta_{13} + 1450100 \beta_{14} + 2386928 \beta_{15} - 1907748 \beta_{16} - 1798768 \beta_{17} ) q^{60} \) \( + ( -3864161478720 - 35719459587485 \beta_{1} - 64028845922 \beta_{2} - 236981123938 \beta_{3} - 7721167842 \beta_{4} - 106825960 \beta_{5} - 1181989781 \beta_{6} - 123225413 \beta_{7} - 139142614 \beta_{8} + 4511927 \beta_{9} - 45144296 \beta_{10} + 19448713 \beta_{11} - 157824 \beta_{12} - 2324522 \beta_{13} + 5397961 \beta_{14} - 1015440 \beta_{15} + 7532014 \beta_{16} + 1135945 \beta_{17} ) q^{61} \) \( + ( 98891464955999640 + 2028814163320 \beta_{1} + 5973789027780 \beta_{2} + 47197378240 \beta_{3} + 35116598684 \beta_{4} - 104109336 \beta_{5} + 1621792584 \beta_{6} + 800158000 \beta_{7} - 152447600 \beta_{8} - 129872564 \beta_{9} + 55813064 \beta_{10} + 3710880 \beta_{11} + 5659108 \beta_{12} + 8795216 \beta_{13} - 2131708 \beta_{14} + 7990388 \beta_{15} + 3383432 \beta_{16} + 437000 \beta_{17} ) q^{62} \) \( + ( 24021365469617 + 216706971644793 \beta_{1} + 268907078850 \beta_{2} - 190901932549 \beta_{3} - 4148006131 \beta_{4} - 4768450938 \beta_{5} + 3893625970 \beta_{6} - 694348189 \beta_{7} - 373151625 \beta_{8} + 15869442 \beta_{9} - 30925270 \beta_{10} - 26776418 \beta_{11} - 3502080 \beta_{12} + 18629011 \beta_{13} + 2098586 \beta_{14} + 6654630 \beta_{15} + 11278236 \beta_{16} - 4555366 \beta_{17} ) q^{63} \) \( + ( 9725239863790832 - 157369102010420 \beta_{1} + 9103822073332 \beta_{2} - 54888862792 \beta_{3} - 40832261008 \beta_{4} + 2241516516 \beta_{5} - 3826286808 \beta_{6} + 693131708 \beta_{7} + 146706264 \beta_{8} + 5510284 \beta_{9} + 37512440 \beta_{10} + 35572204 \beta_{11} + 10153784 \beta_{12} - 19486076 \beta_{13} + 12557560 \beta_{14} + 16333060 \beta_{15} - 4987128 \beta_{16} - 2919600 \beta_{17} ) q^{64} \) \( + ( 87558376081373571 + 351485691679705 \beta_{1} - 506966645325 \beta_{2} + 21919232518 \beta_{3} + 452192921 \beta_{4} - 7920927708 \beta_{5} - 3773805675 \beta_{6} + 112074121 \beta_{7} + 493962455 \beta_{8} + 60254728 \beta_{9} - 25736553 \beta_{10} - 40625010 \beta_{11} + 26005780 \beta_{12} - 1979918 \beta_{13} + 22901572 \beta_{14} - 13763602 \beta_{15} ) q^{65} \) \( + ( 444537631042659803 + 257562817734218 \beta_{1} - 14144833967092 \beta_{2} - 288405569407 \beta_{3} - 57899181812 \beta_{4} + 2011689350 \beta_{5} + 2935107551 \beta_{6} - 585380776 \beta_{7} - 5430574 \beta_{8} - 258741836 \beta_{9} + 40085182 \beta_{10} + 5984106 \beta_{11} - 456046 \beta_{12} + 23949010 \beta_{13} + 1510362 \beta_{14} + 1878476 \beta_{15} + 11733044 \beta_{16} - 604564 \beta_{17} ) q^{66} \) \( + ( -80749046097585817 - 411675955659989 \beta_{1} + 1895703853299 \beta_{2} - 208095937095 \beta_{3} + 2721468867 \beta_{4} + 13850269133 \beta_{5} + 2417533969 \beta_{6} - 295062846 \beta_{7} + 636663864 \beta_{8} + 308592758 \beta_{9} + 19493837 \beta_{10} + 23289864 \beta_{11} - 13348593 \beta_{12} + 12934120 \beta_{13} + 20191536 \beta_{14} + 17613160 \beta_{15} ) q^{67} \) \( + ( 64099746531740480 - 50709795770690 \beta_{1} - 15735507264838 \beta_{2} - 62412199154 \beta_{3} + 49142621504 \beta_{4} + 4544466008 \beta_{5} - 7845108320 \beta_{6} + 695286864 \beta_{7} + 60343960 \beta_{8} + 206302456 \beta_{9} - 64384808 \beta_{10} + 78039720 \beta_{11} - 28018728 \beta_{12} - 18076528 \beta_{13} + 27291624 \beta_{14} + 24533384 \beta_{15} - 377448 \beta_{16} + 5067840 \beta_{17} ) q^{68} \) \( + ( -49549779931572 - 446260881247468 \beta_{1} - 535959220098 \beta_{2} + 630195609916 \beta_{3} + 17895273885 \beta_{4} + 12833810328 \beta_{5} - 7606894125 \beta_{6} + 1512525971 \beta_{7} - 378932166 \beta_{8} - 124713121 \beta_{9} + 154551624 \beta_{10} - 1266335 \beta_{11} - 2562432 \beta_{12} + 12015606 \beta_{13} + 20796449 \beta_{14} - 23076784 \beta_{15} - 7556034 \beta_{16} - 1451359 \beta_{17} ) q^{69} \) \( + ( -341746744483225188 + 193495945699048 \beta_{1} + 33751490342448 \beta_{2} - 30601570220 \beta_{3} + 59902600320 \beta_{4} + 1239657928 \beta_{5} + 3376034220 \beta_{6} - 1407417840 \beta_{7} + 643597192 \beta_{8} + 318091376 \beta_{9} - 146064264 \beta_{10} - 25851624 \beta_{11} - 5210120 \beta_{12} + 19931096 \beta_{13} + 38934872 \beta_{14} - 50085808 \beta_{15} - 23587728 \beta_{16} + 1252176 \beta_{17} ) q^{70} \) \( + ( 39932276021665 + 363262685072329 \beta_{1} + 514276667578 \beta_{2} + 632351490045 \beta_{3} + 32338879625 \beta_{4} - 16846993882 \beta_{5} + 11122577778 \beta_{6} + 925032829 \beta_{7} + 77958215 \beta_{8} + 122561762 \beta_{9} + 177970074 \beta_{10} + 16908606 \beta_{11} - 1802240 \beta_{12} - 44882573 \beta_{13} + 27258362 \beta_{14} + 38138278 \beta_{15} - 27872548 \beta_{16} + 12669434 \beta_{17} ) q^{71} \) \( + ( -294809467902377908 + 80309605469505 \beta_{1} + 25229756437273 \beta_{2} - 42223632541 \beta_{3} - 35679035777 \beta_{4} - 14714244668 \beta_{5} - 2846167194 \beta_{6} - 4992538128 \beta_{7} - 512510163 \beta_{8} - 28985168 \beta_{9} - 362104224 \beta_{10} - 172759248 \beta_{11} + 1055136 \beta_{12} + 24022224 \beta_{13} + 2760480 \beta_{14} + 23693712 \beta_{15} + 15631968 \beta_{16} + 3812160 \beta_{17} ) q^{72} \) \( + ( 549148336364760026 + 922568504543318 \beta_{1} + 605834841305 \beta_{2} - 1398615006045 \beta_{3} + 25545961420 \beta_{4} - 6415199888 \beta_{5} - 16574946991 \beta_{6} + 535705136 \beta_{7} - 388588336 \beta_{8} - 540410905 \beta_{9} + 513471760 \beta_{10} + 91586016 \beta_{11} - 106469952 \beta_{12} + 42560224 \beta_{13} - 27517632 \beta_{14} - 21052064 \beta_{15} ) q^{73} \) \( + ( -313435585442724855 - 6347854529612 \beta_{1} - 33482437946729 \beta_{2} + 62007423684 \beta_{3} + 14415753400 \beta_{4} - 11951056893 \beta_{5} + 23598259492 \beta_{6} + 6195138869 \beta_{7} - 1181647874 \beta_{8} - 28619288 \beta_{9} - 433531968 \beta_{10} - 30403689 \beta_{11} - 65139528 \beta_{12} - 14623936 \beta_{13} + 34133752 \beta_{14} - 93597480 \beta_{15} - 18830160 \beta_{16} - 3459216 \beta_{17} ) q^{74} \) \( + ( 744985714210491875 - 2630471339672783 \beta_{1} - 34585172889357 \beta_{2} - 1839610219730 \beta_{3} + 23139216368 \beta_{4} + 6874950422 \beta_{5} + 36559536024 \beta_{6} + 162846018 \beta_{7} - 825140214 \beta_{8} - 1088774776 \beta_{9} + 513437784 \beta_{10} - 126204804 \beta_{11} + 74071962 \beta_{12} - 123007668 \beta_{13} - 49673880 \beta_{14} + 70136652 \beta_{15} ) q^{75} \) \( + ( -467503248355711637 + 707655438414569 \beta_{1} - 38841945349964 \beta_{2} + 354223180928 \beta_{3} - 173673714884 \beta_{4} - 516772583 \beta_{5} - 18506126768 \beta_{6} + 4726980588 \beta_{7} - 1200645093 \beta_{8} + 405134169 \beta_{9} - 708849777 \beta_{10} - 313106833 \beta_{11} + 8604815 \beta_{12} + 68203616 \beta_{13} - 69348879 \beta_{14} + 1972143 \beta_{15} + 18417935 \beta_{16} - 4083328 \beta_{17} ) q^{76} \) \( + ( -255696006192220 - 2300489780698564 \beta_{1} - 2724461612102 \beta_{2} + 3963431120628 \beta_{3} + 81980829079 \beta_{4} + 10852064712 \beta_{5} - 31503994823 \beta_{6} - 5985537351 \beta_{7} + 2085945006 \beta_{8} - 101769891 \beta_{9} + 709019160 \beta_{10} - 194734365 \beta_{11} + 2398080 \beta_{12} + 47792418 \beta_{13} - 82209501 \beta_{14} - 18879120 \beta_{15} - 48734406 \beta_{16} - 6009693 \beta_{17} ) q^{77} \) \( + ( -811136386693243250 - 17306368220710 \beta_{1} + 65137871664715 \beta_{2} - 1625506424980 \beta_{3} - 326212733619 \beta_{4} + 346629958 \beta_{5} + 45705694426 \beta_{6} - 8650758284 \beta_{7} - 3665477436 \beta_{8} - 276615495 \beta_{9} - 805880338 \beta_{10} + 77555456 \beta_{11} - 4787349 \beta_{12} - 83167220 \beta_{13} - 107280509 \beta_{14} - 4356745 \beta_{15} + 89838342 \beta_{16} - 9619706 \beta_{17} ) q^{78} \) \( + ( 217305606116844 + 1978538525776940 \beta_{1} + 2907927130896 \beta_{2} + 3722930676526 \beta_{3} - 31767493096 \beta_{4} + 21274940680 \beta_{5} + 38028292664 \beta_{6} + 1715211462 \beta_{7} + 4132478900 \beta_{8} + 34643288 \beta_{9} + 839373192 \beta_{10} + 217835944 \beta_{11} + 38268928 \beta_{12} - 84359788 \beta_{13} - 96710024 \beta_{14} - 39450264 \beta_{15} + 11690960 \beta_{16} - 15233416 \beta_{17} ) q^{79} \) \( + ( -1228013017148829152 + 629881026720280 \beta_{1} + 75525972012264 \beta_{2} + 1028697919280 \beta_{3} + 276777699552 \beta_{4} + 35544394696 \beta_{5} - 78946036976 \beta_{6} - 3725443720 \beta_{7} - 333361616 \beta_{8} + 235086360 \beta_{9} - 944582928 \beta_{10} + 597382872 \beta_{11} - 88452624 \beta_{12} + 42609800 \beta_{13} - 119908752 \beta_{14} - 67935992 \beta_{15} - 7835504 \beta_{16} + 11137696 \beta_{17} ) q^{80} \) \( + ( -1794772312376307585 + 4365028422996444 \beta_{1} - 57619774948773 \beta_{2} - 1689882481113 \beta_{3} + 37944196242 \beta_{4} - 18113318328 \beta_{5} - 62908139457 \beta_{6} + 1685042838 \beta_{7} - 752000598 \beta_{8} + 892615191 \beta_{9} + 589729194 \beta_{10} + 170280468 \beta_{11} + 330624504 \beta_{12} - 9738900 \beta_{13} - 146052072 \beta_{14} + 43706196 \beta_{15} ) q^{81} \) \( + ( 573503454827512994 + 956757014219710 \beta_{1} - 24363390760264 \beta_{2} - 169491100102 \beta_{3} + 357146372344 \beta_{4} + 37502714652 \beta_{5} + 48842783366 \beta_{6} + 1586835632 \beta_{7} + 3676831252 \beta_{8} + 977337928 \beta_{9} - 1340406900 \beta_{10} + 110253892 \beta_{11} + 302553940 \beta_{12} - 146115756 \beta_{13} - 199013212 \beta_{14} + 173770680 \beta_{15} + 3353928 \beta_{16} + 20307960 \beta_{17} ) q^{82} \) \( + ( -2579812279486599141 - 5306252294385059 \beta_{1} + 64134226224915 \beta_{2} - 5664714800280 \beta_{3} + 77359859178 \beta_{4} - 46927308232 \beta_{5} + 70304465734 \beta_{6} + 3769585810 \beta_{7} - 1377619330 \beta_{8} + 3487302316 \beta_{9} + 1060259726 \beta_{10} + 110766276 \beta_{11} - 106079328 \beta_{12} + 147231092 \beta_{13} - 67127272 \beta_{14} - 250823180 \beta_{15} ) q^{83} \) \( + ( 950002682697102720 + 2139587021029776 \beta_{1} - 63703170925608 \beta_{2} + 425208388424 \beta_{3} - 686362665248 \beta_{4} - 50269327168 \beta_{5} - 141122661256 \beta_{6} - 1996225576 \beta_{7} + 1035417224 \beta_{8} - 2160640208 \beta_{9} - 1045667048 \beta_{10} + 969921984 \beta_{11} + 228956936 \beta_{12} - 68941160 \beta_{13} - 134992680 \beta_{14} - 118669648 \beta_{15} - 48606328 \beta_{16} - 32012128 \beta_{17} ) q^{84} \) \( + ( -1076299229975044 - 9719187659566374 \beta_{1} - 12348190279090 \beta_{2} + 5210264992616 \beta_{3} - 249851629189 \beta_{4} - 98561709736 \beta_{5} - 115907027585 \beta_{6} + 27929932039 \beta_{7} + 3877585906 \beta_{8} + 1076637163 \beta_{9} + 618560304 \beta_{10} + 519019157 \beta_{11} + 28133120 \beta_{12} - 165898970 \beta_{13} - 80515819 \beta_{14} + 135719904 \beta_{15} + 243425494 \beta_{16} + 37639189 \beta_{17} ) q^{85} \) \( + ( 1864031450498029401 - 1488676723775057 \beta_{1} - 6398320751602 \beta_{2} - 1832009006222 \beta_{3} - 758857538304 \beta_{4} + 18587047785 \beta_{5} + 139782831412 \beta_{6} + 12273185008 \beta_{7} + 7678591160 \beta_{8} - 1392259568 \beta_{9} - 1191667384 \beta_{10} - 39717336 \beta_{11} + 30961736 \beta_{12} - 5071768 \beta_{13} - 91715608 \beta_{14} + 299995440 \beta_{15} - 163754480 \beta_{16} + 22664752 \beta_{17} ) q^{86} \) \( + ( 1323085190993633 + 11982994503783577 \beta_{1} + 16064808022242 \beta_{2} + 4051687980079 \beta_{3} + 199649962565 \beta_{4} + 104488618182 \beta_{5} + 184967004178 \beta_{6} - 3458359449 \beta_{7} - 1483328553 \beta_{8} - 1073137214 \beta_{9} + 1220139866 \beta_{10} - 534783138 \beta_{11} - 42848256 \beta_{12} + 475663651 \beta_{13} - 117343398 \beta_{14} - 264791930 \beta_{15} + 163846428 \beta_{16} - 44042406 \beta_{17} ) q^{87} \) \( + ( 1797100672969837116 + 1880979526850512 \beta_{1} + 35379336936068 \beta_{2} + 3613035975474 \beta_{3} + 821909113002 \beta_{4} - 13054415962 \beta_{5} - 204059781820 \beta_{6} + 22534972482 \beta_{7} - 3301638590 \beta_{8} - 390916822 \beta_{9} - 131244812 \beta_{10} - 1372069158 \beta_{11} + 260840460 \beta_{12} - 288924698 \beta_{13} + 51140828 \beta_{14} - 126652786 \beta_{15} - 122413708 \beta_{16} - 71996776 \beta_{17} ) q^{88} \) \( + ( -1558825693715670042 + 8028407027979514 \beta_{1} - 30572531291483 \beta_{2} - 1058288874501 \beta_{3} + 43209110896 \beta_{4} + 212937599872 \beta_{5} - 168018241371 \beta_{6} - 4380293996 \beta_{7} + 3094581740 \beta_{8} + 1802317639 \beta_{9} + 647899564 \beta_{10} - 1093828584 \beta_{11} - 603314416 \beta_{12} - 435985368 \beta_{13} + 377245776 \beta_{14} + 155387928 \beta_{15} ) q^{89} \) \( + ( -1844734616536162733 - 39080175350628 \beta_{1} + 102280462853709 \beta_{2} - 2654222816612 \beta_{3} + 784766667640 \beta_{4} - 38611132863 \beta_{5} + 305107530236 \beta_{6} - 48430465241 \beta_{7} - 17531803830 \beta_{8} + 831924072 \beta_{9} - 206618432 \beta_{10} - 66982339 \beta_{11} - 361660104 \beta_{12} + 274867520 \beta_{13} + 93728632 \beta_{14} + 385032024 \beta_{15} + 68214192 \beta_{16} - 37877648 \beta_{17} ) q^{90} \) \( + ( 8945115653338233754 - 17065217765203854 \beta_{1} - 132245201983066 \beta_{2} - 3004487011910 \beta_{3} + 596792800 \beta_{4} - 298572594382 \beta_{5} + 320555137656 \beta_{6} + 2334742566 \beta_{7} + 8904008094 \beta_{8} - 9435943496 \beta_{9} + 341323256 \beta_{10} + 730057972 \beta_{11} - 231842786 \beta_{12} + 675027300 \beta_{13} + 275401016 \beta_{14} - 344595612 \beta_{15} ) q^{91} \) \( + ( -52033312036430624 - 1210289618813760 \beta_{1} + 161666361023076 \beta_{2} - 1937723774156 \beta_{3} - 161079248192 \beta_{4} + 211265274104 \beta_{5} - 283969919748 \beta_{6} - 37963345852 \beta_{7} - 5250312524 \beta_{8} - 1077260928 \beta_{9} + 1262223436 \beta_{10} - 1941582360 \beta_{11} - 642699548 \beta_{12} - 187107884 \beta_{13} + 644089964 \beta_{14} - 307367536 \beta_{15} - 8953148 \beta_{16} + 150250480 \beta_{17} ) q^{92} \) \( + ( -2444577002385072 - 22127646687124944 \beta_{1} - 29453479872544 \beta_{2} - 2542897373360 \beta_{3} + 579807616728 \beta_{4} - 141654123648 \beta_{5} - 370358284936 \beta_{6} - 95822983416 \beta_{7} - 10332731216 \beta_{8} + 662425528 \beta_{9} - 402444848 \beta_{10} + 161845192 \beta_{11} - 25577216 \beta_{12} - 395618784 \beta_{13} + 589085000 \beta_{14} + 421627936 \beta_{15} - 398691728 \beta_{16} - 81391032 \beta_{17} ) q^{93} \) \( + ( -5958776983568163524 - 123226863779740 \beta_{1} - 188752910999010 \beta_{2} - 3692230190712 \beta_{3} + 437707922962 \beta_{4} - 147181285252 \beta_{5} + 466698718036 \beta_{6} + 45056597512 \beta_{7} - 20941935864 \beta_{8} + 5592220906 \beta_{9} + 1199228044 \beta_{10} - 328137248 \beta_{11} - 41728962 \beta_{12} + 276831608 \beta_{13} + 719517358 \beta_{14} + 82325110 \beta_{15} - 145216292 \beta_{16} + 3743836 \beta_{17} ) q^{94} \) \( + ( 3530143946419108 + 31875185964244268 \beta_{1} + 40462570795636 \beta_{2} - 19377700014185 \beta_{3} - 389708718306 \beta_{4} + 37850606952 \beta_{5} + 472314962104 \beta_{6} + 15532454571 \beta_{7} - 19620723996 \beta_{8} - 1319084424 \beta_{9} - 1818008616 \beta_{10} - 672293112 \beta_{11} - 192847872 \beta_{12} + 891252 \beta_{13} + 876086808 \beta_{14} + 12784872 \beta_{15} - 545608560 \beta_{16} + 271861272 \beta_{17} ) q^{95} \) \( + ( 3810490315055563536 + 1660607333998588 \beta_{1} - 587183921753276 \beta_{2} + 4753288571160 \beta_{3} - 1040805578032 \beta_{4} + 27503787796 \beta_{5} - 488506572056 \beta_{6} + 51911696908 \beta_{7} + 8767094424 \beta_{8} - 1832261476 \beta_{9} + 1976298392 \beta_{10} + 1912179836 \beta_{11} - 105929832 \beta_{12} + 839498356 \beta_{13} + 814403800 \beta_{14} - 332524108 \beta_{15} + 446698920 \beta_{16} + 161443600 \beta_{17} ) q^{96} \) \( + ( 7076002819320995705 + 46020828998706179 \beta_{1} + 64937683458840 \beta_{2} + 18878466739281 \beta_{3} - 180240389607 \beta_{4} + 106277991652 \beta_{5} - 679815763174 \beta_{6} - 8834993043 \beta_{7} + 9864386163 \beta_{8} - 4260170369 \beta_{9} - 2874969869 \beta_{10} + 907844358 \beta_{11} + 64222308 \beta_{12} + 359013050 \beta_{13} + 270979284 \beta_{14} + 460797542 \beta_{15} ) q^{97} \) \( + ( 8600368705733329618 - 13879153315908559 \beta_{1} + 382292389875360 \beta_{2} - 12129436389064 \beta_{3} - 1226686162272 \beta_{4} + 23341733712 \beta_{5} + 681947677640 \beta_{6} - 13835995328 \beta_{7} + 50661113840 \beta_{8} - 2614264992 \beta_{9} + 5522282384 \beta_{10} - 709712592 \beta_{11} - 1367568912 \beta_{12} + 179057648 \beta_{13} + 1334521264 \beta_{14} - 822216544 \beta_{15} - 223516832 \beta_{16} - 48972896 \beta_{17} ) q^{98} \) \( + ( 3483745025893127653 - 45431496491426157 \beta_{1} + 527631155060405 \beta_{2} + 8570410032184 \beta_{3} - 243350357638 \beta_{4} + 739197847496 \beta_{5} + 523160539254 \beta_{6} - 17912202030 \beta_{7} + 211209822 \beta_{8} + 11293808588 \beta_{9} - 4733866242 \beta_{10} - 1890022140 \beta_{11} + 1239995088 \beta_{12} - 1622272716 \beta_{13} + 133237656 \beta_{14} + 1487760948 \beta_{15} ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)  \(=\)  \(18q \) \(\mathstrut -\mathstrut 398q^{2} \) \(\mathstrut -\mathstrut 114228q^{3} \) \(\mathstrut +\mathstrut 724980q^{4} \) \(\mathstrut -\mathstrut 91643748q^{6} \) \(\mathstrut -\mathstrut 3105291368q^{8} \) \(\mathstrut +\mathstrut 10197650262q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(18q \) \(\mathstrut -\mathstrut 398q^{2} \) \(\mathstrut -\mathstrut 114228q^{3} \) \(\mathstrut +\mathstrut 724980q^{4} \) \(\mathstrut -\mathstrut 91643748q^{6} \) \(\mathstrut -\mathstrut 3105291368q^{8} \) \(\mathstrut +\mathstrut 10197650262q^{9} \) \(\mathstrut +\mathstrut 742754160q^{10} \) \(\mathstrut +\mathstrut 27862395020q^{11} \) \(\mathstrut -\mathstrut 212292734088q^{12} \) \(\mathstrut -\mathstrut 431248136928q^{14} \) \(\mathstrut -\mathstrut 958534870512q^{16} \) \(\mathstrut +\mathstrut 4343925139172q^{17} \) \(\mathstrut -\mathstrut 4517494524426q^{18} \) \(\mathstrut -\mathstrut 360681653556q^{19} \) \(\mathstrut -\mathstrut 32520172742880q^{20} \) \(\mathstrut -\mathstrut 66275482922148q^{22} \) \(\mathstrut +\mathstrut 22289322475152q^{24} \) \(\mathstrut -\mathstrut 395727477008910q^{25} \) \(\mathstrut +\mathstrut 154289718058128q^{26} \) \(\mathstrut -\mathstrut 1208398438889064q^{27} \) \(\mathstrut -\mathstrut 394619539621440q^{28} \) \(\mathstrut +\mathstrut 223250517248160q^{30} \) \(\mathstrut -\mathstrut 2833814588985248q^{32} \) \(\mathstrut +\mathstrut 4461969540487176q^{33} \) \(\mathstrut -\mathstrut 939462769526748q^{34} \) \(\mathstrut +\mathstrut 3615863153468160q^{35} \) \(\mathstrut +\mathstrut 1560521174396700q^{36} \) \(\mathstrut +\mathstrut 12905383833568412q^{38} \) \(\mathstrut +\mathstrut 11813880412973760q^{40} \) \(\mathstrut +\mathstrut 16990829756398820q^{41} \) \(\mathstrut +\mathstrut 38127588900300480q^{42} \) \(\mathstrut -\mathstrut 27498466356753396q^{43} \) \(\mathstrut +\mathstrut 33178844343953144q^{44} \) \(\mathstrut -\mathstrut 21739818881100192q^{46} \) \(\mathstrut +\mathstrut 28364131397385312q^{48} \) \(\mathstrut -\mathstrut 252965388018862638q^{49} \) \(\mathstrut +\mathstrut 41705619584456530q^{50} \) \(\mathstrut -\mathstrut 34182574557901800q^{51} \) \(\mathstrut -\mathstrut 44583718369992480q^{52} \) \(\mathstrut -\mathstrut 579400710808894920q^{54} \) \(\mathstrut -\mathstrut 505574909383001472q^{56} \) \(\mathstrut -\mathstrut 522453489061123704q^{57} \) \(\mathstrut -\mathstrut 677523738697093680q^{58} \) \(\mathstrut +\mathstrut 93112194700366220q^{59} \) \(\mathstrut +\mathstrut 1631640690429240000q^{60} \) \(\mathstrut +\mathstrut 1780090172849178240q^{62} \) \(\mathstrut +\mathstrut 175441813011570240q^{64} \) \(\mathstrut +\mathstrut 1575343920200472960q^{65} \) \(\mathstrut +\mathstrut 8001047667731567880q^{66} \) \(\mathstrut -\mathstrut 1452645642074335668q^{67} \) \(\mathstrut +\mathstrut 1153770320813215592q^{68} \) \(\mathstrut -\mathstrut 6151558949299572480q^{70} \) \(\mathstrut -\mathstrut 5306529204268842936q^{72} \) \(\mathstrut +\mathstrut 9882821313863175972q^{73} \) \(\mathstrut -\mathstrut 5642095430673385488q^{74} \) \(\mathstrut +\mathstrut 13414715768769378060q^{75} \) \(\mathstrut -\mathstrut 8416781249057544840q^{76} \) \(\mathstrut -\mathstrut 14599907290310144160q^{78} \) \(\mathstrut -\mathstrut 22104885212702947200q^{80} \) \(\mathstrut -\mathstrut 32315102667184036518q^{81} \) \(\mathstrut +\mathstrut 10320950271478847652q^{82} \) \(\mathstrut -\mathstrut 46425529968268557748q^{83} \) \(\mathstrut +\mathstrut 17095266896298568320q^{84} \) \(\mathstrut +\mathstrut 33555485282446075868q^{86} \) \(\mathstrut +\mathstrut 32344350945394345872q^{88} \) \(\mathstrut -\mathstrut 28075170672250840156q^{89} \) \(\mathstrut -\mathstrut 33204348719592139440q^{90} \) \(\mathstrut +\mathstrut 161045136122144660736q^{91} \) \(\mathstrut -\mathstrut 932896292396925120q^{92} \) \(\mathstrut -\mathstrut 107259275077774974528q^{94} \) \(\mathstrut +\mathstrut 68580843917215697472q^{96} \) \(\mathstrut +\mathstrut 127276645173366650724q^{97} \) \(\mathstrut +\mathstrut 154837385163137764402q^{98} \) \(\mathstrut +\mathstrut 62802542261652714084q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{18}\mathstrut +\mathstrut \) \(66008406614424\) \(x^{16}\mathstrut -\mathstrut \) \(170362825164232872\) \(x^{15}\mathstrut +\mathstrut \) \(17\!\cdots\!98\) \(x^{14}\mathstrut -\mathstrut \) \(81\!\cdots\!32\) \(x^{13}\mathstrut +\mathstrut \) \(24\!\cdots\!04\) \(x^{12}\mathstrut -\mathstrut \) \(15\!\cdots\!40\) \(x^{11}\mathstrut +\mathstrut \) \(19\!\cdots\!77\) \(x^{10}\mathstrut -\mathstrut \) \(12\!\cdots\!56\) \(x^{9}\mathstrut +\mathstrut \) \(87\!\cdots\!76\) \(x^{8}\mathstrut -\mathstrut \) \(48\!\cdots\!20\) \(x^{7}\mathstrut +\mathstrut \) \(22\!\cdots\!20\) \(x^{6}\mathstrut -\mathstrut \) \(47\!\cdots\!80\) \(x^{5}\mathstrut +\mathstrut \) \(31\!\cdots\!00\) \(x^{4}\mathstrut +\mathstrut \) \(89\!\cdots\!00\) \(x^{3}\mathstrut +\mathstrut \) \(21\!\cdots\!00\) \(x^{2}\mathstrut +\mathstrut \) \(13\!\cdots\!00\) \(x\mathstrut +\mathstrut \) \(53\!\cdots\!00\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\(-\)\(57\!\cdots\!21\) \(\nu^{17}\mathstrut -\mathstrut \) \(34\!\cdots\!34\) \(\nu^{16}\mathstrut -\mathstrut \) \(37\!\cdots\!80\) \(\nu^{15}\mathstrut -\mathstrut \) \(22\!\cdots\!88\) \(\nu^{14}\mathstrut -\mathstrut \) \(94\!\cdots\!90\) \(\nu^{13}\mathstrut -\mathstrut \) \(56\!\cdots\!08\) \(\nu^{12}\mathstrut -\mathstrut \) \(12\!\cdots\!76\) \(\nu^{11}\mathstrut -\mathstrut \) \(74\!\cdots\!84\) \(\nu^{10}\mathstrut -\mathstrut \) \(89\!\cdots\!53\) \(\nu^{9}\mathstrut -\mathstrut \) \(53\!\cdots\!66\) \(\nu^{8}\mathstrut -\mathstrut \) \(35\!\cdots\!60\) \(\nu^{7}\mathstrut -\mathstrut \) \(21\!\cdots\!20\) \(\nu^{6}\mathstrut -\mathstrut \) \(72\!\cdots\!20\) \(\nu^{5}\mathstrut -\mathstrut \) \(44\!\cdots\!00\) \(\nu^{4}\mathstrut -\mathstrut \) \(66\!\cdots\!00\) \(\nu^{3}\mathstrut -\mathstrut \) \(41\!\cdots\!00\) \(\nu^{2}\mathstrut -\mathstrut \) \(19\!\cdots\!00\) \(\nu\mathstrut -\mathstrut \) \(11\!\cdots\!00\)\()/\)\(33\!\cdots\!00\)
\(\beta_{2}\)\(=\)\((\)\(75\!\cdots\!39\) \(\nu^{17}\mathstrut -\mathstrut \) \(22\!\cdots\!14\) \(\nu^{16}\mathstrut +\mathstrut \) \(48\!\cdots\!40\) \(\nu^{15}\mathstrut -\mathstrut \) \(14\!\cdots\!08\) \(\nu^{14}\mathstrut +\mathstrut \) \(12\!\cdots\!50\) \(\nu^{13}\mathstrut -\mathstrut \) \(34\!\cdots\!28\) \(\nu^{12}\mathstrut +\mathstrut \) \(16\!\cdots\!24\) \(\nu^{11}\mathstrut -\mathstrut \) \(43\!\cdots\!64\) \(\nu^{10}\mathstrut +\mathstrut \) \(11\!\cdots\!47\) \(\nu^{9}\mathstrut -\mathstrut \) \(29\!\cdots\!66\) \(\nu^{8}\mathstrut +\mathstrut \) \(46\!\cdots\!20\) \(\nu^{7}\mathstrut -\mathstrut \) \(10\!\cdots\!20\) \(\nu^{6}\mathstrut +\mathstrut \) \(97\!\cdots\!80\) \(\nu^{5}\mathstrut -\mathstrut \) \(19\!\cdots\!00\) \(\nu^{4}\mathstrut +\mathstrut \) \(90\!\cdots\!00\) \(\nu^{3}\mathstrut -\mathstrut \) \(17\!\cdots\!00\) \(\nu^{2}\mathstrut +\mathstrut \) \(26\!\cdots\!00\) \(\nu\mathstrut -\mathstrut \) \(56\!\cdots\!00\)\()/\)\(66\!\cdots\!00\)
\(\beta_{3}\)\(=\)\((\)\(23\!\cdots\!73\) \(\nu^{17}\mathstrut -\mathstrut \) \(25\!\cdots\!58\) \(\nu^{16}\mathstrut +\mathstrut \) \(15\!\cdots\!60\) \(\nu^{15}\mathstrut -\mathstrut \) \(16\!\cdots\!76\) \(\nu^{14}\mathstrut +\mathstrut \) \(40\!\cdots\!70\) \(\nu^{13}\mathstrut -\mathstrut \) \(42\!\cdots\!36\) \(\nu^{12}\mathstrut +\mathstrut \) \(53\!\cdots\!88\) \(\nu^{11}\mathstrut -\mathstrut \) \(57\!\cdots\!48\) \(\nu^{10}\mathstrut +\mathstrut \) \(39\!\cdots\!09\) \(\nu^{9}\mathstrut -\mathstrut \) \(41\!\cdots\!62\) \(\nu^{8}\mathstrut +\mathstrut \) \(16\!\cdots\!40\) \(\nu^{7}\mathstrut -\mathstrut \) \(17\!\cdots\!20\) \(\nu^{6}\mathstrut +\mathstrut \) \(35\!\cdots\!60\) \(\nu^{5}\mathstrut -\mathstrut \) \(35\!\cdots\!00\) \(\nu^{4}\mathstrut +\mathstrut \) \(33\!\cdots\!00\) \(\nu^{3}\mathstrut -\mathstrut \) \(33\!\cdots\!00\) \(\nu^{2}\mathstrut +\mathstrut \) \(94\!\cdots\!00\) \(\nu\mathstrut -\mathstrut \) \(95\!\cdots\!00\)\()/\)\(16\!\cdots\!00\)
\(\beta_{4}\)\(=\)\((\)\(-\)\(95\!\cdots\!73\) \(\nu^{17}\mathstrut +\mathstrut \) \(10\!\cdots\!58\) \(\nu^{16}\mathstrut -\mathstrut \) \(62\!\cdots\!20\) \(\nu^{15}\mathstrut +\mathstrut \) \(67\!\cdots\!36\) \(\nu^{14}\mathstrut -\mathstrut \) \(16\!\cdots\!70\) \(\nu^{13}\mathstrut +\mathstrut \) \(17\!\cdots\!56\) \(\nu^{12}\mathstrut -\mathstrut \) \(21\!\cdots\!88\) \(\nu^{11}\mathstrut +\mathstrut \) \(23\!\cdots\!68\) \(\nu^{10}\mathstrut -\mathstrut \) \(16\!\cdots\!69\) \(\nu^{9}\mathstrut +\mathstrut \) \(17\!\cdots\!22\) \(\nu^{8}\mathstrut -\mathstrut \) \(66\!\cdots\!20\) \(\nu^{7}\mathstrut +\mathstrut \) \(69\!\cdots\!60\) \(\nu^{6}\mathstrut -\mathstrut \) \(14\!\cdots\!60\) \(\nu^{5}\mathstrut +\mathstrut \) \(14\!\cdots\!00\) \(\nu^{4}\mathstrut -\mathstrut \) \(13\!\cdots\!00\) \(\nu^{3}\mathstrut +\mathstrut \) \(13\!\cdots\!00\) \(\nu^{2}\mathstrut -\mathstrut \) \(37\!\cdots\!00\) \(\nu\mathstrut +\mathstrut \) \(38\!\cdots\!00\)\()/\)\(33\!\cdots\!00\)
\(\beta_{5}\)\(=\)\((\)\(-\)\(13\!\cdots\!33\) \(\nu^{17}\mathstrut -\mathstrut \) \(32\!\cdots\!42\) \(\nu^{16}\mathstrut -\mathstrut \) \(83\!\cdots\!00\) \(\nu^{15}\mathstrut -\mathstrut \) \(21\!\cdots\!04\) \(\nu^{14}\mathstrut -\mathstrut \) \(21\!\cdots\!50\) \(\nu^{13}\mathstrut -\mathstrut \) \(54\!\cdots\!44\) \(\nu^{12}\mathstrut -\mathstrut \) \(28\!\cdots\!28\) \(\nu^{11}\mathstrut -\mathstrut \) \(73\!\cdots\!52\) \(\nu^{10}\mathstrut -\mathstrut \) \(20\!\cdots\!29\) \(\nu^{9}\mathstrut -\mathstrut \) \(54\!\cdots\!78\) \(\nu^{8}\mathstrut -\mathstrut \) \(81\!\cdots\!00\) \(\nu^{7}\mathstrut -\mathstrut \) \(22\!\cdots\!80\) \(\nu^{6}\mathstrut -\mathstrut \) \(17\!\cdots\!60\) \(\nu^{5}\mathstrut -\mathstrut \) \(48\!\cdots\!00\) \(\nu^{4}\mathstrut -\mathstrut \) \(16\!\cdots\!00\) \(\nu^{3}\mathstrut -\mathstrut \) \(45\!\cdots\!00\) \(\nu^{2}\mathstrut -\mathstrut \) \(56\!\cdots\!00\) \(\nu\mathstrut -\mathstrut \) \(13\!\cdots\!00\)\()/\)\(33\!\cdots\!00\)
\(\beta_{6}\)\(=\)\((\)\(16\!\cdots\!77\) \(\nu^{17}\mathstrut -\mathstrut \) \(15\!\cdots\!42\) \(\nu^{16}\mathstrut +\mathstrut \) \(10\!\cdots\!80\) \(\nu^{15}\mathstrut -\mathstrut \) \(10\!\cdots\!64\) \(\nu^{14}\mathstrut +\mathstrut \) \(28\!\cdots\!30\) \(\nu^{13}\mathstrut -\mathstrut \) \(26\!\cdots\!44\) \(\nu^{12}\mathstrut +\mathstrut \) \(37\!\cdots\!12\) \(\nu^{11}\mathstrut -\mathstrut \) \(34\!\cdots\!32\) \(\nu^{10}\mathstrut +\mathstrut \) \(27\!\cdots\!81\) \(\nu^{9}\mathstrut -\mathstrut \) \(24\!\cdots\!78\) \(\nu^{8}\mathstrut +\mathstrut \) \(11\!\cdots\!80\) \(\nu^{7}\mathstrut -\mathstrut \) \(97\!\cdots\!40\) \(\nu^{6}\mathstrut +\mathstrut \) \(23\!\cdots\!40\) \(\nu^{5}\mathstrut -\mathstrut \) \(19\!\cdots\!00\) \(\nu^{4}\mathstrut +\mathstrut \) \(22\!\cdots\!00\) \(\nu^{3}\mathstrut -\mathstrut \) \(18\!\cdots\!00\) \(\nu^{2}\mathstrut +\mathstrut \) \(65\!\cdots\!00\) \(\nu\mathstrut -\mathstrut \) \(52\!\cdots\!00\)\()/\)\(16\!\cdots\!00\)
\(\beta_{7}\)\(=\)\((\)\(-\)\(55\!\cdots\!29\) \(\nu^{17}\mathstrut +\mathstrut \) \(24\!\cdots\!69\) \(\nu^{16}\mathstrut -\mathstrut \) \(35\!\cdots\!00\) \(\nu^{15}\mathstrut +\mathstrut \) \(15\!\cdots\!08\) \(\nu^{14}\mathstrut -\mathstrut \) \(87\!\cdots\!30\) \(\nu^{13}\mathstrut +\mathstrut \) \(41\!\cdots\!98\) \(\nu^{12}\mathstrut -\mathstrut \) \(10\!\cdots\!44\) \(\nu^{11}\mathstrut +\mathstrut \) \(54\!\cdots\!84\) \(\nu^{10}\mathstrut -\mathstrut \) \(74\!\cdots\!77\) \(\nu^{9}\mathstrut +\mathstrut \) \(40\!\cdots\!41\) \(\nu^{8}\mathstrut -\mathstrut \) \(27\!\cdots\!40\) \(\nu^{7}\mathstrut +\mathstrut \) \(16\!\cdots\!00\) \(\nu^{6}\mathstrut -\mathstrut \) \(51\!\cdots\!80\) \(\nu^{5}\mathstrut +\mathstrut \) \(34\!\cdots\!00\) \(\nu^{4}\mathstrut -\mathstrut \) \(41\!\cdots\!00\) \(\nu^{3}\mathstrut +\mathstrut \) \(32\!\cdots\!00\) \(\nu^{2}\mathstrut -\mathstrut \) \(80\!\cdots\!00\) \(\nu\mathstrut +\mathstrut \) \(94\!\cdots\!00\)\()/\)\(34\!\cdots\!00\)
\(\beta_{8}\)\(=\)\((\)\(31\!\cdots\!83\) \(\nu^{17}\mathstrut -\mathstrut \) \(15\!\cdots\!46\) \(\nu^{16}\mathstrut +\mathstrut \) \(19\!\cdots\!12\) \(\nu^{15}\mathstrut -\mathstrut \) \(96\!\cdots\!60\) \(\nu^{14}\mathstrut +\mathstrut \) \(48\!\cdots\!26\) \(\nu^{13}\mathstrut -\mathstrut \) \(24\!\cdots\!64\) \(\nu^{12}\mathstrut +\mathstrut \) \(59\!\cdots\!64\) \(\nu^{11}\mathstrut -\mathstrut \) \(31\!\cdots\!84\) \(\nu^{10}\mathstrut +\mathstrut \) \(39\!\cdots\!51\) \(\nu^{9}\mathstrut -\mathstrut \) \(22\!\cdots\!10\) \(\nu^{8}\mathstrut +\mathstrut \) \(14\!\cdots\!04\) \(\nu^{7}\mathstrut -\mathstrut \) \(87\!\cdots\!76\) \(\nu^{6}\mathstrut +\mathstrut \) \(27\!\cdots\!20\) \(\nu^{5}\mathstrut -\mathstrut \) \(17\!\cdots\!60\) \(\nu^{4}\mathstrut +\mathstrut \) \(24\!\cdots\!40\) \(\nu^{3}\mathstrut -\mathstrut \) \(16\!\cdots\!00\) \(\nu^{2}\mathstrut +\mathstrut \) \(72\!\cdots\!00\) \(\nu\mathstrut -\mathstrut \) \(46\!\cdots\!00\)\()/\)\(66\!\cdots\!00\)
\(\beta_{9}\)\(=\)\((\)\(-\)\(29\!\cdots\!47\) \(\nu^{17}\mathstrut +\mathstrut \) \(85\!\cdots\!22\) \(\nu^{16}\mathstrut -\mathstrut \) \(19\!\cdots\!40\) \(\nu^{15}\mathstrut +\mathstrut \) \(54\!\cdots\!04\) \(\nu^{14}\mathstrut -\mathstrut \) \(49\!\cdots\!50\) \(\nu^{13}\mathstrut +\mathstrut \) \(13\!\cdots\!84\) \(\nu^{12}\mathstrut -\mathstrut \) \(64\!\cdots\!52\) \(\nu^{11}\mathstrut +\mathstrut \) \(17\!\cdots\!12\) \(\nu^{10}\mathstrut -\mathstrut \) \(46\!\cdots\!51\) \(\nu^{9}\mathstrut +\mathstrut \) \(11\!\cdots\!38\) \(\nu^{8}\mathstrut -\mathstrut \) \(18\!\cdots\!20\) \(\nu^{7}\mathstrut +\mathstrut \) \(43\!\cdots\!40\) \(\nu^{6}\mathstrut -\mathstrut \) \(38\!\cdots\!40\) \(\nu^{5}\mathstrut +\mathstrut \) \(83\!\cdots\!00\) \(\nu^{4}\mathstrut -\mathstrut \) \(35\!\cdots\!00\) \(\nu^{3}\mathstrut +\mathstrut \) \(74\!\cdots\!00\) \(\nu^{2}\mathstrut -\mathstrut \) \(10\!\cdots\!00\) \(\nu\mathstrut +\mathstrut \) \(22\!\cdots\!00\)\()/\)\(33\!\cdots\!00\)
\(\beta_{10}\)\(=\)\((\)\(-\)\(23\!\cdots\!69\) \(\nu^{17}\mathstrut -\mathstrut \) \(21\!\cdots\!86\) \(\nu^{16}\mathstrut -\mathstrut \) \(15\!\cdots\!20\) \(\nu^{15}\mathstrut -\mathstrut \) \(13\!\cdots\!72\) \(\nu^{14}\mathstrut -\mathstrut \) \(39\!\cdots\!90\) \(\nu^{13}\mathstrut -\mathstrut \) \(35\!\cdots\!92\) \(\nu^{12}\mathstrut -\mathstrut \) \(51\!\cdots\!44\) \(\nu^{11}\mathstrut -\mathstrut \) \(45\!\cdots\!16\) \(\nu^{10}\mathstrut -\mathstrut \) \(36\!\cdots\!17\) \(\nu^{9}\mathstrut -\mathstrut \) \(32\!\cdots\!94\) \(\nu^{8}\mathstrut -\mathstrut \) \(14\!\cdots\!80\) \(\nu^{7}\mathstrut -\mathstrut \) \(12\!\cdots\!40\) \(\nu^{6}\mathstrut -\mathstrut \) \(30\!\cdots\!80\) \(\nu^{5}\mathstrut -\mathstrut \) \(25\!\cdots\!00\) \(\nu^{4}\mathstrut -\mathstrut \) \(27\!\cdots\!00\) \(\nu^{3}\mathstrut -\mathstrut \) \(23\!\cdots\!00\) \(\nu^{2}\mathstrut -\mathstrut \) \(83\!\cdots\!00\) \(\nu\mathstrut -\mathstrut \) \(68\!\cdots\!00\)\()/\)\(23\!\cdots\!00\)
\(\beta_{11}\)\(=\)\((\)\(-\)\(21\!\cdots\!19\) \(\nu^{17}\mathstrut +\mathstrut \) \(20\!\cdots\!94\) \(\nu^{16}\mathstrut -\mathstrut \) \(13\!\cdots\!00\) \(\nu^{15}\mathstrut +\mathstrut \) \(13\!\cdots\!08\) \(\nu^{14}\mathstrut -\mathstrut \) \(32\!\cdots\!30\) \(\nu^{13}\mathstrut +\mathstrut \) \(33\!\cdots\!08\) \(\nu^{12}\mathstrut -\mathstrut \) \(40\!\cdots\!64\) \(\nu^{11}\mathstrut +\mathstrut \) \(43\!\cdots\!64\) \(\nu^{10}\mathstrut -\mathstrut \) \(26\!\cdots\!07\) \(\nu^{9}\mathstrut +\mathstrut \) \(30\!\cdots\!26\) \(\nu^{8}\mathstrut -\mathstrut \) \(94\!\cdots\!80\) \(\nu^{7}\mathstrut +\mathstrut \) \(11\!\cdots\!00\) \(\nu^{6}\mathstrut -\mathstrut \) \(17\!\cdots\!00\) \(\nu^{5}\mathstrut +\mathstrut \) \(23\!\cdots\!00\) \(\nu^{4}\mathstrut -\mathstrut \) \(14\!\cdots\!00\) \(\nu^{3}\mathstrut +\mathstrut \) \(21\!\cdots\!00\) \(\nu^{2}\mathstrut -\mathstrut \) \(29\!\cdots\!00\) \(\nu\mathstrut +\mathstrut \) \(60\!\cdots\!00\)\()/\)\(16\!\cdots\!00\)
\(\beta_{12}\)\(=\)\((\)\(14\!\cdots\!47\) \(\nu^{17}\mathstrut -\mathstrut \) \(14\!\cdots\!22\) \(\nu^{16}\mathstrut +\mathstrut \) \(95\!\cdots\!80\) \(\nu^{15}\mathstrut -\mathstrut \) \(10\!\cdots\!44\) \(\nu^{14}\mathstrut +\mathstrut \) \(24\!\cdots\!50\) \(\nu^{13}\mathstrut -\mathstrut \) \(33\!\cdots\!64\) \(\nu^{12}\mathstrut +\mathstrut \) \(32\!\cdots\!52\) \(\nu^{11}\mathstrut -\mathstrut \) \(54\!\cdots\!92\) \(\nu^{10}\mathstrut +\mathstrut \) \(24\!\cdots\!91\) \(\nu^{9}\mathstrut -\mathstrut \) \(48\!\cdots\!78\) \(\nu^{8}\mathstrut +\mathstrut \) \(97\!\cdots\!40\) \(\nu^{7}\mathstrut -\mathstrut \) \(23\!\cdots\!00\) \(\nu^{6}\mathstrut +\mathstrut \) \(20\!\cdots\!40\) \(\nu^{5}\mathstrut -\mathstrut \) \(56\!\cdots\!00\) \(\nu^{4}\mathstrut +\mathstrut \) \(18\!\cdots\!00\) \(\nu^{3}\mathstrut -\mathstrut \) \(58\!\cdots\!00\) \(\nu^{2}\mathstrut +\mathstrut \) \(52\!\cdots\!00\) \(\nu\mathstrut -\mathstrut \) \(20\!\cdots\!00\)\()/\)\(55\!\cdots\!00\)
\(\beta_{13}\)\(=\)\((\)\(-\)\(40\!\cdots\!77\) \(\nu^{17}\mathstrut -\mathstrut \) \(28\!\cdots\!78\) \(\nu^{16}\mathstrut -\mathstrut \) \(25\!\cdots\!80\) \(\nu^{15}\mathstrut -\mathstrut \) \(18\!\cdots\!16\) \(\nu^{14}\mathstrut -\mathstrut \) \(65\!\cdots\!90\) \(\nu^{13}\mathstrut -\mathstrut \) \(49\!\cdots\!16\) \(\nu^{12}\mathstrut -\mathstrut \) \(84\!\cdots\!72\) \(\nu^{11}\mathstrut -\mathstrut \) \(67\!\cdots\!48\) \(\nu^{10}\mathstrut -\mathstrut \) \(59\!\cdots\!81\) \(\nu^{9}\mathstrut -\mathstrut \) \(51\!\cdots\!62\) \(\nu^{8}\mathstrut -\mathstrut \) \(22\!\cdots\!60\) \(\nu^{7}\mathstrut -\mathstrut \) \(21\!\cdots\!80\) \(\nu^{6}\mathstrut -\mathstrut \) \(46\!\cdots\!40\) \(\nu^{5}\mathstrut -\mathstrut \) \(46\!\cdots\!00\) \(\nu^{4}\mathstrut -\mathstrut \) \(43\!\cdots\!00\) \(\nu^{3}\mathstrut -\mathstrut \) \(44\!\cdots\!00\) \(\nu^{2}\mathstrut -\mathstrut \) \(14\!\cdots\!00\) \(\nu\mathstrut -\mathstrut \) \(13\!\cdots\!00\)\()/\)\(47\!\cdots\!00\)
\(\beta_{14}\)\(=\)\((\)\(78\!\cdots\!45\) \(\nu^{17}\mathstrut +\mathstrut \) \(51\!\cdots\!86\) \(\nu^{16}\mathstrut +\mathstrut \) \(52\!\cdots\!08\) \(\nu^{15}\mathstrut +\mathstrut \) \(33\!\cdots\!76\) \(\nu^{14}\mathstrut +\mathstrut \) \(14\!\cdots\!38\) \(\nu^{13}\mathstrut +\mathstrut \) \(89\!\cdots\!12\) \(\nu^{12}\mathstrut +\mathstrut \) \(19\!\cdots\!08\) \(\nu^{11}\mathstrut +\mathstrut \) \(12\!\cdots\!48\) \(\nu^{10}\mathstrut +\mathstrut \) \(15\!\cdots\!93\) \(\nu^{9}\mathstrut +\mathstrut \) \(94\!\cdots\!14\) \(\nu^{8}\mathstrut +\mathstrut \) \(65\!\cdots\!68\) \(\nu^{7}\mathstrut +\mathstrut \) \(40\!\cdots\!84\) \(\nu^{6}\mathstrut +\mathstrut \) \(14\!\cdots\!00\) \(\nu^{5}\mathstrut +\mathstrut \) \(89\!\cdots\!80\) \(\nu^{4}\mathstrut +\mathstrut \) \(13\!\cdots\!40\) \(\nu^{3}\mathstrut +\mathstrut \) \(87\!\cdots\!00\) \(\nu^{2}\mathstrut +\mathstrut \) \(37\!\cdots\!00\) \(\nu\mathstrut +\mathstrut \) \(26\!\cdots\!00\)\()/\)\(16\!\cdots\!00\)
\(\beta_{15}\)\(=\)\((\)\(-\)\(43\!\cdots\!69\) \(\nu^{17}\mathstrut +\mathstrut \) \(31\!\cdots\!74\) \(\nu^{16}\mathstrut -\mathstrut \) \(20\!\cdots\!60\) \(\nu^{15}\mathstrut +\mathstrut \) \(20\!\cdots\!08\) \(\nu^{14}\mathstrut -\mathstrut \) \(26\!\cdots\!10\) \(\nu^{13}\mathstrut +\mathstrut \) \(51\!\cdots\!68\) \(\nu^{12}\mathstrut +\mathstrut \) \(12\!\cdots\!36\) \(\nu^{11}\mathstrut +\mathstrut \) \(67\!\cdots\!04\) \(\nu^{10}\mathstrut +\mathstrut \) \(54\!\cdots\!43\) \(\nu^{9}\mathstrut +\mathstrut \) \(48\!\cdots\!66\) \(\nu^{8}\mathstrut +\mathstrut \) \(45\!\cdots\!40\) \(\nu^{7}\mathstrut +\mathstrut \) \(18\!\cdots\!80\) \(\nu^{6}\mathstrut +\mathstrut \) \(15\!\cdots\!20\) \(\nu^{5}\mathstrut +\mathstrut \) \(39\!\cdots\!00\) \(\nu^{4}\mathstrut +\mathstrut \) \(23\!\cdots\!00\) \(\nu^{3}\mathstrut +\mathstrut \) \(35\!\cdots\!00\) \(\nu^{2}\mathstrut +\mathstrut \) \(12\!\cdots\!00\) \(\nu\mathstrut +\mathstrut \) \(10\!\cdots\!00\)\()/\)\(47\!\cdots\!00\)
\(\beta_{16}\)\(=\)\((\)\(81\!\cdots\!23\) \(\nu^{17}\mathstrut -\mathstrut \) \(24\!\cdots\!18\) \(\nu^{16}\mathstrut +\mathstrut \) \(51\!\cdots\!00\) \(\nu^{15}\mathstrut -\mathstrut \) \(15\!\cdots\!56\) \(\nu^{14}\mathstrut +\mathstrut \) \(13\!\cdots\!90\) \(\nu^{13}\mathstrut -\mathstrut \) \(39\!\cdots\!96\) \(\nu^{12}\mathstrut +\mathstrut \) \(17\!\cdots\!08\) \(\nu^{11}\mathstrut -\mathstrut \) \(52\!\cdots\!68\) \(\nu^{10}\mathstrut +\mathstrut \) \(12\!\cdots\!99\) \(\nu^{9}\mathstrut -\mathstrut \) \(37\!\cdots\!22\) \(\nu^{8}\mathstrut +\mathstrut \) \(47\!\cdots\!20\) \(\nu^{7}\mathstrut -\mathstrut \) \(14\!\cdots\!40\) \(\nu^{6}\mathstrut +\mathstrut \) \(85\!\cdots\!60\) \(\nu^{5}\mathstrut -\mathstrut \) \(30\!\cdots\!00\) \(\nu^{4}\mathstrut +\mathstrut \) \(42\!\cdots\!00\) \(\nu^{3}\mathstrut -\mathstrut \) \(27\!\cdots\!00\) \(\nu^{2}\mathstrut -\mathstrut \) \(15\!\cdots\!00\) \(\nu\mathstrut -\mathstrut \) \(80\!\cdots\!00\)\()/\)\(33\!\cdots\!00\)
\(\beta_{17}\)\(=\)\((\)\(-\)\(34\!\cdots\!18\) \(\nu^{17}\mathstrut -\mathstrut \) \(29\!\cdots\!17\) \(\nu^{16}\mathstrut -\mathstrut \) \(21\!\cdots\!40\) \(\nu^{15}\mathstrut -\mathstrut \) \(18\!\cdots\!84\) \(\nu^{14}\mathstrut -\mathstrut \) \(49\!\cdots\!80\) \(\nu^{13}\mathstrut -\mathstrut \) \(47\!\cdots\!74\) \(\nu^{12}\mathstrut -\mathstrut \) \(57\!\cdots\!68\) \(\nu^{11}\mathstrut -\mathstrut \) \(62\!\cdots\!52\) \(\nu^{10}\mathstrut -\mathstrut \) \(34\!\cdots\!74\) \(\nu^{9}\mathstrut -\mathstrut \) \(45\!\cdots\!93\) \(\nu^{8}\mathstrut -\mathstrut \) \(10\!\cdots\!60\) \(\nu^{7}\mathstrut -\mathstrut \) \(17\!\cdots\!80\) \(\nu^{6}\mathstrut -\mathstrut \) \(12\!\cdots\!60\) \(\nu^{5}\mathstrut -\mathstrut \) \(37\!\cdots\!00\) \(\nu^{4}\mathstrut -\mathstrut \) \(66\!\cdots\!00\) \(\nu^{3}\mathstrut -\mathstrut \) \(34\!\cdots\!00\) \(\nu^{2}\mathstrut -\mathstrut \) \(33\!\cdots\!00\) \(\nu\mathstrut -\mathstrut \) \(99\!\cdots\!00\)\()/\)\(10\!\cdots\!00\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{4}\mathstrut +\mathstrut \) \(2\) \(\beta_{3}\mathstrut +\mathstrut \) \(\beta_{2}\mathstrut +\mathstrut \) \(34\) \(\beta_{1}\mathstrut +\mathstrut \) \(3\)\()/4\)
\(\nu^{2}\)\(=\)\((\)\(4\) \(\beta_{17}\mathstrut +\mathstrut \) \(24\) \(\beta_{16}\mathstrut -\mathstrut \) \(2282\) \(\beta_{15}\mathstrut -\mathstrut \) \(976\) \(\beta_{14}\mathstrut -\mathstrut \) \(60\) \(\beta_{13}\mathstrut -\mathstrut \) \(868\) \(\beta_{12}\mathstrut -\mathstrut \) \(1370\) \(\beta_{11}\mathstrut -\mathstrut \) \(2639\) \(\beta_{10}\mathstrut +\mathstrut \) \(5443\) \(\beta_{9}\mathstrut -\mathstrut \) \(45069\) \(\beta_{8}\mathstrut +\mathstrut \) \(23977\) \(\beta_{7}\mathstrut +\mathstrut \) \(157692\) \(\beta_{6}\mathstrut -\mathstrut \) \(759752\) \(\beta_{5}\mathstrut -\mathstrut \) \(156283\) \(\beta_{4}\mathstrut +\mathstrut \) \(7368163\) \(\beta_{3}\mathstrut +\mathstrut \) \(154573730\) \(\beta_{2}\mathstrut +\mathstrut \) \(1767556779\) \(\beta_{1}\mathstrut -\mathstrut \) \(117348153374177\)\()/16\)
\(\nu^{3}\)\(=\)\((\)\(1629969\) \(\beta_{17}\mathstrut +\mathstrut \) \(2425749150\) \(\beta_{16}\mathstrut -\mathstrut \) \(10443921244\) \(\beta_{15}\mathstrut -\mathstrut \) \(10078546727\) \(\beta_{14}\mathstrut +\mathstrut \) \(5368342726\) \(\beta_{13}\mathstrut +\mathstrut \) \(1424850841\) \(\beta_{12}\mathstrut -\mathstrut \) \(2456646531\) \(\beta_{11}\mathstrut -\mathstrut \) \(55655981403\) \(\beta_{10}\mathstrut -\mathstrut \) \(32977917581\) \(\beta_{9}\mathstrut +\mathstrut \) \(129481030176\) \(\beta_{8}\mathstrut +\mathstrut \) \(2300331816283\) \(\beta_{7}\mathstrut +\mathstrut \) \(1592043911510\) \(\beta_{6}\mathstrut +\mathstrut \) \(5041570857835\) \(\beta_{5}\mathstrut -\mathstrut \) \(202380975665952\) \(\beta_{4}\mathstrut -\mathstrut \) \(572933927588211\) \(\beta_{3}\mathstrut -\mathstrut \) \(214120063481500\) \(\beta_{2}\mathstrut +\mathstrut \) \(125519286978274054\) \(\beta_{1}\mathstrut +\mathstrut \) \(1831369458169559968\)\()/64\)
\(\nu^{4}\)\(=\)\((\)\(-\)\(787099348314952\) \(\beta_{17}\mathstrut -\mathstrut \) \(4397834852773200\) \(\beta_{16}\mathstrut +\mathstrut \) \(343122679962936614\) \(\beta_{15}\mathstrut +\mathstrut \) \(138509220812564676\) \(\beta_{14}\mathstrut +\mathstrut \) \(707174811894028\) \(\beta_{13}\mathstrut +\mathstrut \) \(11601857956326864\) \(\beta_{12}\mathstrut +\mathstrut \) \(183898964342216426\) \(\beta_{11}\mathstrut +\mathstrut \) \(285031397610029797\) \(\beta_{10}\mathstrut -\mathstrut \) \(944563736073735106\) \(\beta_{9}\mathstrut +\mathstrut \) \(5738650639022817811\) \(\beta_{8}\mathstrut -\mathstrut \) \(4137329599191497365\) \(\beta_{7}\mathstrut -\mathstrut \) \(18969795394786232659\) \(\beta_{6}\mathstrut +\mathstrut \) \(130384700379212928676\) \(\beta_{5}\mathstrut +\mathstrut \) \(14615343714836581729\) \(\beta_{4}\mathstrut -\mathstrut \) \(533780732780652859160\) \(\beta_{3}\mathstrut -\mathstrut \) \(22869947056621118437833\) \(\beta_{2}\mathstrut -\mathstrut \) \(575093031005934617873135\) \(\beta_{1}\mathstrut +\mathstrut \) \(12013310827358516621954128059\)\()/128\)
\(\nu^{5}\)\(=\)\((\)\(-\)\(38178709141709832251110\) \(\beta_{17}\mathstrut -\mathstrut \) \(19886134874659237154500\) \(\beta_{16}\mathstrut +\mathstrut \) \(481780720145097469361272\) \(\beta_{15}\mathstrut +\mathstrut \) \(358450035953825924795166\) \(\beta_{14}\mathstrut -\mathstrut \) \(114853319862651773771943\) \(\beta_{13}\mathstrut -\mathstrut \) \(65453424728646265751338\) \(\beta_{12}\mathstrut -\mathstrut \) \(25182387821954988985320\) \(\beta_{11}\mathstrut +\mathstrut \) \(2636054464207271180946688\) \(\beta_{10}\mathstrut +\mathstrut \) \(1613909414831286600698912\) \(\beta_{9}\mathstrut -\mathstrut \) \(5350376179030541860423673\) \(\beta_{8}\mathstrut -\mathstrut \) \(94549041673237739297700512\) \(\beta_{7}\mathstrut -\mathstrut \) \(47611751921022678831468350\) \(\beta_{6}\mathstrut -\mathstrut \) \(247692761902209008976779118\) \(\beta_{5}\mathstrut +\mathstrut \) \(6112829191425608246633075173\) \(\beta_{4}\mathstrut +\mathstrut \) \(21251161943591362040590755782\) \(\beta_{3}\mathstrut +\mathstrut \) \(10998000200343310008549330028\) \(\beta_{2}\mathstrut -\mathstrut \) \(4503930271277169575663964245977\) \(\beta_{1}\mathstrut -\mathstrut \) \(110786019509274249075372283843473\)\()/128\)
\(\nu^{6}\)\(=\)\((\)\(17\!\cdots\!28\) \(\beta_{17}\mathstrut +\mathstrut \) \(10\!\cdots\!28\) \(\beta_{16}\mathstrut -\mathstrut \) \(59\!\cdots\!58\) \(\beta_{15}\mathstrut -\mathstrut \) \(21\!\cdots\!84\) \(\beta_{14}\mathstrut -\mathstrut \) \(17\!\cdots\!69\) \(\beta_{13}\mathstrut +\mathstrut \) \(89\!\cdots\!20\) \(\beta_{12}\mathstrut -\mathstrut \) \(35\!\cdots\!64\) \(\beta_{11}\mathstrut -\mathstrut \) \(52\!\cdots\!77\) \(\beta_{10}\mathstrut +\mathstrut \) \(18\!\cdots\!84\) \(\beta_{9}\mathstrut -\mathstrut \) \(88\!\cdots\!08\) \(\beta_{8}\mathstrut +\mathstrut \) \(75\!\cdots\!13\) \(\beta_{7}\mathstrut +\mathstrut \) \(20\!\cdots\!77\) \(\beta_{6}\mathstrut -\mathstrut \) \(23\!\cdots\!00\) \(\beta_{5}\mathstrut -\mathstrut \) \(25\!\cdots\!44\) \(\beta_{4}\mathstrut +\mathstrut \) \(92\!\cdots\!50\) \(\beta_{3}\mathstrut +\mathstrut \) \(41\!\cdots\!63\) \(\beta_{2}\mathstrut +\mathstrut \) \(17\!\cdots\!98\) \(\beta_{1}\mathstrut -\mathstrut \) \(18\!\cdots\!98\)\()/128\)
\(\nu^{7}\)\(=\)\((\)\(11\!\cdots\!58\) \(\beta_{17}\mathstrut -\mathstrut \) \(38\!\cdots\!04\) \(\beta_{16}\mathstrut -\mathstrut \) \(91\!\cdots\!66\) \(\beta_{15}\mathstrut -\mathstrut \) \(56\!\cdots\!62\) \(\beta_{14}\mathstrut +\mathstrut \) \(26\!\cdots\!47\) \(\beta_{13}\mathstrut +\mathstrut \) \(13\!\cdots\!74\) \(\beta_{12}\mathstrut +\mathstrut \) \(31\!\cdots\!74\) \(\beta_{11}\mathstrut -\mathstrut \) \(52\!\cdots\!60\) \(\beta_{10}\mathstrut -\mathstrut \) \(31\!\cdots\!86\) \(\beta_{9}\mathstrut +\mathstrut \) \(97\!\cdots\!23\) \(\beta_{8}\mathstrut +\mathstrut \) \(16\!\cdots\!34\) \(\beta_{7}\mathstrut +\mathstrut \) \(87\!\cdots\!78\) \(\beta_{6}\mathstrut +\mathstrut \) \(48\!\cdots\!72\) \(\beta_{5}\mathstrut -\mathstrut \) \(98\!\cdots\!39\) \(\beta_{4}\mathstrut -\mathstrut \) \(39\!\cdots\!02\) \(\beta_{3}\mathstrut -\mathstrut \) \(26\!\cdots\!46\) \(\beta_{2}\mathstrut +\mathstrut \) \(85\!\cdots\!05\) \(\beta_{1}\mathstrut +\mathstrut \) \(27\!\cdots\!67\)\()/128\)
\(\nu^{8}\)\(=\)\((\)\(-\)\(36\!\cdots\!24\) \(\beta_{17}\mathstrut -\mathstrut \) \(22\!\cdots\!20\) \(\beta_{16}\mathstrut +\mathstrut \) \(10\!\cdots\!38\) \(\beta_{15}\mathstrut +\mathstrut \) \(33\!\cdots\!84\) \(\beta_{14}\mathstrut +\mathstrut \) \(67\!\cdots\!34\) \(\beta_{13}\mathstrut -\mathstrut \) \(26\!\cdots\!52\) \(\beta_{12}\mathstrut +\mathstrut \) \(67\!\cdots\!26\) \(\beta_{11}\mathstrut +\mathstrut \) \(10\!\cdots\!33\) \(\beta_{10}\mathstrut -\mathstrut \) \(35\!\cdots\!34\) \(\beta_{9}\mathstrut +\mathstrut \) \(14\!\cdots\!97\) \(\beta_{8}\mathstrut -\mathstrut \) \(13\!\cdots\!29\) \(\beta_{7}\mathstrut -\mathstrut \) \(14\!\cdots\!95\) \(\beta_{6}\mathstrut +\mathstrut \) \(41\!\cdots\!32\) \(\beta_{5}\mathstrut +\mathstrut \) \(53\!\cdots\!11\) \(\beta_{4}\mathstrut -\mathstrut \) \(19\!\cdots\!84\) \(\beta_{3}\mathstrut -\mathstrut \) \(82\!\cdots\!61\) \(\beta_{2}\mathstrut -\mathstrut \) \(41\!\cdots\!57\) \(\beta_{1}\mathstrut +\mathstrut \) \(29\!\cdots\!37\)\()/128\)
\(\nu^{9}\)\(=\)\((\)\(-\)\(24\!\cdots\!02\) \(\beta_{17}\mathstrut +\mathstrut \) \(12\!\cdots\!40\) \(\beta_{16}\mathstrut +\mathstrut \) \(16\!\cdots\!36\) \(\beta_{15}\mathstrut +\mathstrut \) \(87\!\cdots\!54\) \(\beta_{14}\mathstrut +\mathstrut \) \(21\!\cdots\!33\) \(\beta_{13}\mathstrut -\mathstrut \) \(27\!\cdots\!94\) \(\beta_{12}\mathstrut -\mathstrut \) \(95\!\cdots\!96\) \(\beta_{11}\mathstrut +\mathstrut \) \(98\!\cdots\!24\) \(\beta_{10}\mathstrut +\mathstrut \) \(57\!\cdots\!44\) \(\beta_{9}\mathstrut -\mathstrut \) \(16\!\cdots\!29\) \(\beta_{8}\mathstrut -\mathstrut \) \(28\!\cdots\!04\) \(\beta_{7}\mathstrut -\mathstrut \) \(17\!\cdots\!58\) \(\beta_{6}\mathstrut -\mathstrut \) \(90\!\cdots\!82\) \(\beta_{5}\mathstrut +\mathstrut \) \(16\!\cdots\!73\) \(\beta_{4}\mathstrut +\mathstrut \) \(72\!\cdots\!38\) \(\beta_{3}\mathstrut +\mathstrut \) \(59\!\cdots\!40\) \(\beta_{2}\mathstrut -\mathstrut \) \(16\!\cdots\!09\) \(\beta_{1}\mathstrut -\mathstrut \) \(69\!\cdots\!97\)\()/128\)
\(\nu^{10}\)\(=\)\((\)\(75\!\cdots\!80\) \(\beta_{17}\mathstrut +\mathstrut \) \(48\!\cdots\!00\) \(\beta_{16}\mathstrut -\mathstrut \) \(17\!\cdots\!42\) \(\beta_{15}\mathstrut -\mathstrut \) \(53\!\cdots\!28\) \(\beta_{14}\mathstrut -\mathstrut \) \(16\!\cdots\!39\) \(\beta_{13}\mathstrut +\mathstrut \) \(58\!\cdots\!12\) \(\beta_{12}\mathstrut -\mathstrut \) \(12\!\cdots\!32\) \(\beta_{11}\mathstrut -\mathstrut \) \(19\!\cdots\!93\) \(\beta_{10}\mathstrut +\mathstrut \) \(66\!\cdots\!04\) \(\beta_{9}\mathstrut -\mathstrut \) \(23\!\cdots\!94\) \(\beta_{8}\mathstrut +\mathstrut \) \(23\!\cdots\!77\) \(\beta_{7}\mathstrut -\mathstrut \) \(48\!\cdots\!39\) \(\beta_{6}\mathstrut -\mathstrut \) \(71\!\cdots\!28\) \(\beta_{5}\mathstrut -\mathstrut \) \(11\!\cdots\!22\) \(\beta_{4}\mathstrut +\mathstrut \) \(40\!\cdots\!74\) \(\beta_{3}\mathstrut +\mathstrut \) \(16\!\cdots\!71\) \(\beta_{2}\mathstrut +\mathstrut \) \(87\!\cdots\!96\) \(\beta_{1}\mathstrut -\mathstrut \) \(49\!\cdots\!88\)\()/128\)
\(\nu^{11}\)\(=\)\((\)\(49\!\cdots\!90\) \(\beta_{17}\mathstrut -\mathstrut \) \(25\!\cdots\!56\) \(\beta_{16}\mathstrut -\mathstrut \) \(28\!\cdots\!98\) \(\beta_{15}\mathstrut -\mathstrut \) \(13\!\cdots\!54\) \(\beta_{14}\mathstrut -\mathstrut \) \(72\!\cdots\!13\) \(\beta_{13}\mathstrut +\mathstrut \) \(52\!\cdots\!06\) \(\beta_{12}\mathstrut +\mathstrut \) \(22\!\cdots\!74\) \(\beta_{11}\mathstrut -\mathstrut \) \(17\!\cdots\!52\) \(\beta_{10}\mathstrut -\mathstrut \) \(10\!\cdots\!06\) \(\beta_{9}\mathstrut +\mathstrut \) \(29\!\cdots\!35\) \(\beta_{8}\mathstrut +\mathstrut \) \(49\!\cdots\!14\) \(\beta_{7}\mathstrut +\mathstrut \) \(32\!\cdots\!22\) \(\beta_{6}\mathstrut +\mathstrut \) \(16\!\cdots\!20\) \(\beta_{5}\mathstrut -\mathstrut \) \(27\!\cdots\!15\) \(\beta_{4}\mathstrut -\mathstrut \) \(13\!\cdots\!78\) \(\beta_{3}\mathstrut -\mathstrut \) \(13\!\cdots\!10\) \(\beta_{2}\mathstrut +\mathstrut \) \(29\!\cdots\!73\) \(\beta_{1}\mathstrut +\mathstrut \) \(16\!\cdots\!23\)\()/128\)
\(\nu^{12}\)\(=\)\((\)\(-\)\(15\!\cdots\!88\) \(\beta_{17}\mathstrut -\mathstrut \) \(10\!\cdots\!92\) \(\beta_{16}\mathstrut +\mathstrut \) \(30\!\cdots\!82\) \(\beta_{15}\mathstrut +\mathstrut \) \(86\!\cdots\!56\) \(\beta_{14}\mathstrut +\mathstrut \) \(35\!\cdots\!56\) \(\beta_{13}\mathstrut -\mathstrut \) \(11\!\cdots\!64\) \(\beta_{12}\mathstrut +\mathstrut \) \(22\!\cdots\!90\) \(\beta_{11}\mathstrut +\mathstrut \) \(36\!\cdots\!17\) \(\beta_{10}\mathstrut -\mathstrut \) \(12\!\cdots\!90\) \(\beta_{9}\mathstrut +\mathstrut \) \(38\!\cdots\!03\) \(\beta_{8}\mathstrut -\mathstrut \) \(41\!\cdots\!29\) \(\beta_{7}\mathstrut +\mathstrut \) \(49\!\cdots\!53\) \(\beta_{6}\mathstrut +\mathstrut \) \(12\!\cdots\!32\) \(\beta_{5}\mathstrut +\mathstrut \) \(23\!\cdots\!01\) \(\beta_{4}\mathstrut -\mathstrut \) \(79\!\cdots\!52\) \(\beta_{3}\mathstrut -\mathstrut \) \(31\!\cdots\!93\) \(\beta_{2}\mathstrut -\mathstrut \) \(16\!\cdots\!51\) \(\beta_{1}\mathstrut +\mathstrut \) \(84\!\cdots\!51\)\()/128\)
\(\nu^{13}\)\(=\)\((\)\(-\)\(93\!\cdots\!74\) \(\beta_{17}\mathstrut +\mathstrut \) \(44\!\cdots\!60\) \(\beta_{16}\mathstrut +\mathstrut \) \(50\!\cdots\!92\) \(\beta_{15}\mathstrut +\mathstrut \) \(22\!\cdots\!62\) \(\beta_{14}\mathstrut +\mathstrut \) \(17\!\cdots\!61\) \(\beta_{13}\mathstrut -\mathstrut \) \(99\!\cdots\!78\) \(\beta_{12}\mathstrut -\mathstrut \) \(46\!\cdots\!96\) \(\beta_{11}\mathstrut +\mathstrut \) \(32\!\cdots\!24\) \(\beta_{10}\mathstrut +\mathstrut \) \(18\!\cdots\!20\) \(\beta_{9}\mathstrut -\mathstrut \) \(51\!\cdots\!89\) \(\beta_{8}\mathstrut -\mathstrut \) \(85\!\cdots\!48\) \(\beta_{7}\mathstrut -\mathstrut \) \(60\!\cdots\!62\) \(\beta_{6}\mathstrut -\mathstrut \) \(29\!\cdots\!34\) \(\beta_{5}\mathstrut +\mathstrut \) \(47\!\cdots\!01\) \(\beta_{4}\mathstrut +\mathstrut \) \(23\!\cdots\!74\) \(\beta_{3}\mathstrut +\mathstrut \) \(27\!\cdots\!00\) \(\beta_{2}\mathstrut -\mathstrut \) \(51\!\cdots\!69\) \(\beta_{1}\mathstrut -\mathstrut \) \(38\!\cdots\!61\)\()/128\)
\(\nu^{14}\)\(=\)\((\)\(30\!\cdots\!00\) \(\beta_{17}\mathstrut +\mathstrut \) \(20\!\cdots\!60\) \(\beta_{16}\mathstrut -\mathstrut \) \(52\!\cdots\!94\) \(\beta_{15}\mathstrut -\mathstrut \) \(14\!\cdots\!48\) \(\beta_{14}\mathstrut -\mathstrut \) \(67\!\cdots\!81\) \(\beta_{13}\mathstrut +\mathstrut \) \(21\!\cdots\!60\) \(\beta_{12}\mathstrut -\mathstrut \) \(38\!\cdots\!20\) \(\beta_{11}\mathstrut -\mathstrut \) \(68\!\cdots\!97\) \(\beta_{10}\mathstrut +\mathstrut \) \(22\!\cdots\!60\) \(\beta_{9}\mathstrut -\mathstrut \) \(63\!\cdots\!84\) \(\beta_{8}\mathstrut +\mathstrut \) \(74\!\cdots\!53\) \(\beta_{7}\mathstrut -\mathstrut \) \(14\!\cdots\!87\) \(\beta_{6}\mathstrut -\mathstrut \) \(21\!\cdots\!60\) \(\beta_{5}\mathstrut -\mathstrut \) \(48\!\cdots\!24\) \(\beta_{4}\mathstrut +\mathstrut \) \(15\!\cdots\!74\) \(\beta_{3}\mathstrut +\mathstrut \) \(60\!\cdots\!03\) \(\beta_{2}\mathstrut +\mathstrut \) \(32\!\cdots\!74\) \(\beta_{1}\mathstrut -\mathstrut \) \(14\!\cdots\!42\)\()/128\)
\(\nu^{15}\)\(=\)\((\)\(17\!\cdots\!50\) \(\beta_{17}\mathstrut -\mathstrut \) \(69\!\cdots\!00\) \(\beta_{16}\mathstrut -\mathstrut \) \(87\!\cdots\!06\) \(\beta_{15}\mathstrut -\mathstrut \) \(37\!\cdots\!62\) \(\beta_{14}\mathstrut -\mathstrut \) \(38\!\cdots\!93\) \(\beta_{13}\mathstrut +\mathstrut \) \(18\!\cdots\!26\) \(\beta_{12}\mathstrut +\mathstrut \) \(92\!\cdots\!90\) \(\beta_{11}\mathstrut -\mathstrut \) \(57\!\cdots\!08\) \(\beta_{10}\mathstrut -\mathstrut \) \(32\!\cdots\!06\) \(\beta_{9}\mathstrut +\mathstrut \) \(90\!\cdots\!19\) \(\beta_{8}\mathstrut +\mathstrut \) \(14\!\cdots\!94\) \(\beta_{7}\mathstrut +\mathstrut \) \(10\!\cdots\!38\) \(\beta_{6}\mathstrut +\mathstrut \) \(51\!\cdots\!16\) \(\beta_{5}\mathstrut -\mathstrut \) \(81\!\cdots\!27\) \(\beta_{4}\mathstrut -\mathstrut \) \(41\!\cdots\!78\) \(\beta_{3}\mathstrut -\mathstrut \) \(57\!\cdots\!02\) \(\beta_{2}\mathstrut +\mathstrut \) \(87\!\cdots\!13\) \(\beta_{1}\mathstrut +\mathstrut \) \(84\!\cdots\!75\)\()/128\)
\(\nu^{16}\)\(=\)\((\)\(-\)\(61\!\cdots\!76\) \(\beta_{17}\mathstrut -\mathstrut \) \(40\!\cdots\!08\) \(\beta_{16}\mathstrut +\mathstrut \) \(90\!\cdots\!62\) \(\beta_{15}\mathstrut +\mathstrut \) \(23\!\cdots\!96\) \(\beta_{14}\mathstrut +\mathstrut \) \(12\!\cdots\!46\) \(\beta_{13}\mathstrut -\mathstrut \) \(39\!\cdots\!32\) \(\beta_{12}\mathstrut +\mathstrut \) \(65\!\cdots\!50\) \(\beta_{11}\mathstrut +\mathstrut \) \(12\!\cdots\!09\) \(\beta_{10}\mathstrut -\mathstrut \) \(39\!\cdots\!90\) \(\beta_{9}\mathstrut +\mathstrut \) \(10\!\cdots\!57\) \(\beta_{8}\mathstrut -\mathstrut \) \(13\!\cdots\!37\) \(\beta_{7}\mathstrut +\mathstrut \) \(34\!\cdots\!41\) \(\beta_{6}\mathstrut +\mathstrut \) \(36\!\cdots\!96\) \(\beta_{5}\mathstrut +\mathstrut \) \(97\!\cdots\!91\) \(\beta_{4}\mathstrut -\mathstrut \) \(28\!\cdots\!88\) \(\beta_{3}\mathstrut -\mathstrut \) \(11\!\cdots\!09\) \(\beta_{2}\mathstrut -\mathstrut \) \(59\!\cdots\!13\) \(\beta_{1}\mathstrut +\mathstrut \) \(24\!\cdots\!77\)\()/128\)
\(\nu^{17}\)\(=\)\((\)\(-\)\(31\!\cdots\!22\) \(\beta_{17}\mathstrut +\mathstrut \) \(98\!\cdots\!84\) \(\beta_{16}\mathstrut +\mathstrut \) \(15\!\cdots\!88\) \(\beta_{15}\mathstrut +\mathstrut \) \(62\!\cdots\!82\) \(\beta_{14}\mathstrut +\mathstrut \) \(77\!\cdots\!17\) \(\beta_{13}\mathstrut -\mathstrut \) \(34\!\cdots\!70\) \(\beta_{12}\mathstrut -\mathstrut \) \(17\!\cdots\!64\) \(\beta_{11}\mathstrut +\mathstrut \) \(10\!\cdots\!08\) \(\beta_{10}\mathstrut +\mathstrut \) \(57\!\cdots\!12\) \(\beta_{9}\mathstrut -\mathstrut \) \(15\!\cdots\!89\) \(\beta_{8}\mathstrut -\mathstrut \) \(25\!\cdots\!40\) \(\beta_{7}\mathstrut -\mathstrut \) \(19\!\cdots\!14\) \(\beta_{6}\mathstrut -\mathstrut \) \(91\!\cdots\!74\) \(\beta_{5}\mathstrut +\mathstrut \) \(14\!\cdots\!73\) \(\beta_{4}\mathstrut +\mathstrut \) \(74\!\cdots\!38\) \(\beta_{3}\mathstrut +\mathstrut \) \(11\!\cdots\!28\) \(\beta_{2}\mathstrut -\mathstrut \) \(14\!\cdots\!85\) \(\beta_{1}\mathstrut -\mathstrut \) \(18\!\cdots\!57\)\()/128\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3.1
16622.8 2.05933e6i
16622.8 + 2.05933e6i
−20011.7 4.20507e6i
−20011.7 + 4.20507e6i
−2443.39 + 2.11221e6i
−2443.39 2.11221e6i
22670.7 1.21630e6i
22670.7 + 1.21630e6i
−13803.2 714588.i
−13803.2 + 714588.i
14907.4 + 4.14330e6i
14907.4 4.14330e6i
7663.46 3.49192e6i
7663.46 + 3.49192e6i
−24233.2 + 2.60731e6i
−24233.2 2.60731e6i
−1372.90 1.21540e6i
−1372.90 + 1.21540e6i
−998.751 225.992i 60145.3 946431. + 451420.i 8.23733e6i −6.00701e7 1.35924e7i 5.26995e8i −8.43231e8 6.64742e8i 1.30668e8 1.86157e9 8.22704e9i
3.2 −998.751 + 225.992i 60145.3 946431. 451420.i 8.23733e6i −6.00701e7 + 1.35924e7i 5.26995e8i −8.43231e8 + 6.64742e8i 1.30668e8 1.86157e9 + 8.22704e9i
3.3 −980.704 294.612i −86392.9 874983. + 577854.i 1.68203e7i 8.47258e7 + 2.54524e7i 3.64653e8i −6.87856e8 8.24485e8i 3.97695e9 4.95546e9 1.64957e10i
3.4 −980.704 + 294.612i −86392.9 874983. 577854.i 1.68203e7i 8.47258e7 2.54524e7i 3.64653e8i −6.87856e8 + 8.24485e8i 3.97695e9 4.95546e9 + 1.64957e10i
3.5 −749.926 697.272i −16119.5 76200.7 + 1.04580e6i 8.44883e6i 1.20885e7 + 1.12397e7i 6.17349e7i 6.72064e8 8.37407e8i −3.22694e9 −5.89113e9 + 6.33599e9i
3.6 −749.926 + 697.272i −16119.5 76200.7 1.04580e6i 8.44883e6i 1.20885e7 1.12397e7i 6.17349e7i 6.72064e8 + 8.37407e8i −3.22694e9 −5.89113e9 6.33599e9i
3.7 −340.525 965.722i 84336.7 −816662. + 657705.i 4.86521e6i −2.87188e7 8.14458e7i 1.26224e8i 9.13253e8 + 5.64703e8i 3.62590e9 4.69844e9 1.65673e9i
3.8 −340.525 + 965.722i 84336.7 −816662. 657705.i 4.86521e6i −2.87188e7 + 8.14458e7i 1.26224e8i 9.13253e8 5.64703e8i 3.62590e9 4.69844e9 + 1.65673e9i
3.9 −39.1296 1023.25i −61558.6 −1.04551e6 + 80078.8i 2.85835e6i 2.40876e6 + 6.29900e7i 1.54680e8i 1.22851e8 + 1.06669e9i 3.02682e8 2.92482e9 1.11846e8i
3.10 −39.1296 + 1023.25i −61558.6 −1.04551e6 80078.8i 2.85835e6i 2.40876e6 6.29900e7i 1.54680e8i 1.22851e8 1.06669e9i 3.02682e8 2.92482e9 + 1.11846e8i
3.11 492.862 897.587i 53283.8 −562749. 884774.i 1.65732e7i 2.62616e7 4.78269e7i 1.30215e8i −1.07152e9 + 6.90447e7i −6.47621e8 −1.48759e10 8.16831e9i
3.12 492.862 + 897.587i 53283.8 −562749. + 884774.i 1.65732e7i 2.62616e7 + 4.78269e7i 1.30215e8i −1.07152e9 6.90447e7i −6.47621e8 −1.48759e10 + 8.16831e9i
3.13 607.571 824.278i 24307.8 −310291. 1.00161e6i 1.39677e7i 1.47687e7 2.00364e7i 2.84328e8i −1.01413e9 3.52785e8i −2.89591e9 1.15133e10 + 8.48636e9i
3.14 607.571 + 824.278i 24307.8 −310291. + 1.00161e6i 1.39677e7i 1.47687e7 + 2.00364e7i 2.84328e8i −1.01413e9 + 3.52785e8i −2.89591e9 1.15133e10 8.48636e9i
3.15 829.658 600.204i −103279. 328087. 995927.i 1.04292e7i −8.56862e7 + 6.19884e7i 3.68005e8i −3.25559e8 1.02320e9i 7.17976e9 −6.25967e9 8.65271e9i
3.16 829.658 + 600.204i −103279. 328087. + 995927.i 1.04292e7i −8.56862e7 6.19884e7i 3.68005e8i −3.25559e8 + 1.02320e9i 7.17976e9 −6.25967e9 + 8.65271e9i
3.17 979.944 297.130i −11837.6 872004. 582341.i 4.86160e6i −1.16002e7 + 3.51730e6i 3.96260e8i 6.81484e8 8.29760e8i −3.34666e9 1.44453e9 + 4.76409e9i
3.18 979.944 + 297.130i −11837.6 872004. + 582341.i 4.86160e6i −1.16002e7 3.51730e6i 3.96260e8i 6.81484e8 + 8.29760e8i −3.34666e9 1.44453e9 4.76409e9i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 3.18
Significant digits:
Format:

Inner twists

Char. orbit Parity Mult. Type Proved
1.a Even 1 trivial yes
8.d Odd 1 yes

Hecke kernels

This newform can be constructed as the kernel of the linear operator \(T_{3}^{9} + \cdots\) acting on \(S_{21}^{\mathrm{new}}(8, \chi)\).