Properties

Label 8.22.b.a
Level 8
Weight 22
Character orbit 8.b
Analytic conductor 22.358
Analytic rank 0
Dimension 20
CM No
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 8 = 2^{3} \)
Weight: \( k \) = \( 22 \)
Character orbit: \([\chi]\) = 8.b (of order \(2\) and degree \(1\))

Newform invariants

Self dual: No
Analytic conductor: \(22.358187543\)
Analytic rank: \(0\)
Dimension: \(20\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: multiple of \( 2^{190}\cdot 3^{14}\cdot 5^{4}\cdot 7^{7} \)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(f(q)\) \(=\) \( q\) \( + ( 14 - \beta_{1} ) q^{2} \) \( + ( 1 + 3 \beta_{1} - \beta_{2} ) q^{3} \) \( + ( 20490 - 14 \beta_{1} - \beta_{3} ) q^{4} \) \( + ( -365 - 1221 \beta_{1} - 9 \beta_{2} - \beta_{3} + \beta_{5} ) q^{5} \) \( + ( 5911873 + 69 \beta_{1} - 162 \beta_{2} + 3 \beta_{3} + \beta_{4} ) q^{6} \) \( + ( 28261321 + 46063 \beta_{1} + 31 \beta_{2} + 48 \beta_{3} + \beta_{5} - \beta_{7} ) q^{7} \) \( + ( 182478751 - 19776 \beta_{1} + 2888 \beta_{2} - 13 \beta_{3} + 15 \beta_{5} + \beta_{6} - \beta_{8} ) q^{8} \) \( + ( -3137816611 + 963988 \beta_{1} + 311 \beta_{2} - 467 \beta_{3} - \beta_{5} - 4 \beta_{6} + \beta_{7} + \beta_{9} ) q^{9} \) \(+O(q^{10})\) \( q\) \( + ( 14 - \beta_{1} ) q^{2} \) \( + ( 1 + 3 \beta_{1} - \beta_{2} ) q^{3} \) \( + ( 20490 - 14 \beta_{1} - \beta_{3} ) q^{4} \) \( + ( -365 - 1221 \beta_{1} - 9 \beta_{2} - \beta_{3} + \beta_{5} ) q^{5} \) \( + ( 5911873 + 69 \beta_{1} - 162 \beta_{2} + 3 \beta_{3} + \beta_{4} ) q^{6} \) \( + ( 28261321 + 46063 \beta_{1} + 31 \beta_{2} + 48 \beta_{3} + \beta_{5} - \beta_{7} ) q^{7} \) \( + ( 182478751 - 19776 \beta_{1} + 2888 \beta_{2} - 13 \beta_{3} + 15 \beta_{5} + \beta_{6} - \beta_{8} ) q^{8} \) \( + ( -3137816611 + 963988 \beta_{1} + 311 \beta_{2} - 467 \beta_{3} - \beta_{5} - 4 \beta_{6} + \beta_{7} + \beta_{9} ) q^{9} \) \( + ( -2563473807 - 19819 \beta_{1} + 13754 \beta_{2} - 1238 \beta_{3} + 11 \beta_{4} + 165 \beta_{5} - 6 \beta_{6} - 2 \beta_{7} - \beta_{8} - \beta_{10} ) q^{10} \) \( + ( 2106561 + 7028897 \beta_{1} + 22738 \beta_{2} - 376 \beta_{3} - 6 \beta_{4} + 218 \beta_{5} + 29 \beta_{6} + \beta_{13} ) q^{11} \) \( + ( 3264058365 - 5913035 \beta_{1} + 126695 \beta_{2} + 188 \beta_{3} + 168 \beta_{4} - 155 \beta_{5} + 74 \beta_{6} - 16 \beta_{7} + 4 \beta_{8} - \beta_{12} - \beta_{15} ) q^{12} \) \( + ( -19889612 - 66183789 \beta_{1} + 414929 \beta_{2} - 17821 \beta_{3} - 256 \beta_{4} - 1085 \beta_{5} - 265 \beta_{6} - 8 \beta_{7} + 12 \beta_{8} - \beta_{9} + \beta_{10} + \beta_{12} - \beta_{13} + \beta_{15} - \beta_{18} ) q^{13} \) \( + ( -96224249101 - 29031963 \beta_{1} + 655591 \beta_{2} + 46337 \beta_{3} + 55 \beta_{4} - 1395 \beta_{5} - 387 \beta_{6} + 8 \beta_{7} + 41 \beta_{8} - 10 \beta_{9} - \beta_{11} - 3 \beta_{13} + \beta_{15} - \beta_{18} ) q^{14} \) \( + ( -114311492917 - 91557242 \beta_{1} - 46731 \beta_{2} - 31034 \beta_{3} + 1265 \beta_{4} + 20 \beta_{5} + 329 \beta_{6} - 149 \beta_{7} + 134 \beta_{8} - 20 \beta_{9} + 19 \beta_{10} - \beta_{11} - 4 \beta_{12} + \beta_{13} - 2 \beta_{15} - \beta_{18} + \beta_{19} ) q^{15} \) \( + ( 389620636469 - 180833895 \beta_{1} + 5476436 \beta_{2} - 14229 \beta_{3} - 2424 \beta_{4} + 3518 \beta_{5} - 816 \beta_{6} + 339 \beta_{7} + 3 \beta_{8} - 36 \beta_{9} - 16 \beta_{10} + 12 \beta_{13} + \beta_{15} + \beta_{17} - 4 \beta_{18} - 2 \beta_{19} ) q^{16} \) \( + ( 86899353116 - 455290390 \beta_{1} - 306664 \beta_{2} - 461547 \beta_{3} + 2575 \beta_{4} - 3332 \beta_{5} + 1731 \beta_{6} - 1286 \beta_{7} + 206 \beta_{8} - 39 \beta_{9} - 62 \beta_{10} + 6 \beta_{11} - 12 \beta_{12} + 2 \beta_{13} - 23 \beta_{15} - \beta_{16} - 2 \beta_{17} - 2 \beta_{18} + 2 \beta_{19} ) q^{17} \) \( + ( -2065330319645 + 3124177432 \beta_{1} - 75604 \beta_{2} + 954771 \beta_{3} + 535 \beta_{4} + 5957 \beta_{5} + 3871 \beta_{6} + 2342 \beta_{7} - 269 \beta_{8} + 398 \beta_{9} - 8 \beta_{10} - 6 \beta_{11} - 12 \beta_{12} + 102 \beta_{13} + \beta_{14} - 17 \beta_{15} - 2 \beta_{16} - 2 \beta_{17} + 18 \beta_{18} - 4 \beta_{19} ) q^{18} \) \( + ( -298474717 - 997891669 \beta_{1} + 544984 \beta_{2} - 2369358 \beta_{3} - 13910 \beta_{4} + 184307 \beta_{5} - 2938 \beta_{6} - 1762 \beta_{7} - 2324 \beta_{8} - 119 \beta_{9} - 328 \beta_{10} + 8 \beta_{11} + 64 \beta_{12} - 19 \beta_{13} + 2 \beta_{14} + 75 \beta_{15} - 2 \beta_{16} + 4 \beta_{17} + 8 \beta_{19} ) q^{19} \) \( + ( -3949237404116 + 2534730383 \beta_{1} + 11235812 \beta_{2} - 38861 \beta_{3} - 13546 \beta_{4} + 90319 \beta_{5} - 13293 \beta_{6} - 3198 \beta_{7} - 852 \beta_{8} + 1730 \beta_{9} - 128 \beta_{10} - 8 \beta_{11} - 30 \beta_{12} - 272 \beta_{13} - 2 \beta_{14} + 4 \beta_{15} - 4 \beta_{16} + 14 \beta_{17} + 32 \beta_{18} - 12 \beta_{19} ) q^{20} \) \( + ( 1284800829 + 4268004004 \beta_{1} - 51792288 \beta_{2} + 1972692 \beta_{3} - 8280 \beta_{4} - 137796 \beta_{5} + 17233 \beta_{6} - 1596 \beta_{7} - 6780 \beta_{8} - 297 \beta_{9} + 1435 \beta_{10} + 16 \beta_{11} + 171 \beta_{12} + 21 \beta_{13} - 4 \beta_{14} + 73 \beta_{15} - 20 \beta_{16} - 24 \beta_{17} + 21 \beta_{18} + 16 \beta_{19} ) q^{21} \) \( + ( 14747884583263 + 95639393 \beta_{1} + 21141694 \beta_{2} + 7086563 \beta_{3} - 35905 \beta_{4} - 29592 \beta_{5} + 6920 \beta_{6} - 43756 \beta_{7} - 3110 \beta_{8} - 5728 \beta_{9} - 136 \beta_{10} + 16 \beta_{11} + 56 \beta_{12} - 640 \beta_{13} + 26 \beta_{14} - 74 \beta_{15} - 36 \beta_{16} - 28 \beta_{17} + 32 \beta_{18} - 24 \beta_{19} ) q^{22} \) \( + ( 16786202479365 - 4275867312 \beta_{1} - 4043101 \beta_{2} - 9287710 \beta_{3} + 46593 \beta_{4} - 56246 \beta_{5} + 8757 \beta_{6} - 7631 \beta_{7} + 24886 \beta_{8} - 484 \beta_{9} + 3035 \beta_{10} + 71 \beta_{11} - 132 \beta_{12} - 7 \beta_{13} + 48 \beta_{14} - 90 \beta_{15} - 40 \beta_{16} + 48 \beta_{17} + 7 \beta_{18} - 7 \beta_{19} ) q^{23} \) \( + ( 39335981849256 - 3005017008 \beta_{1} + 40644540 \beta_{2} - 5917134 \beta_{3} - 83860 \beta_{4} + 2192148 \beta_{5} + 133888 \beta_{6} + 93578 \beta_{7} + 448 \beta_{8} - 17372 \beta_{9} + 160 \beta_{10} - 48 \beta_{11} - 248 \beta_{12} + 1048 \beta_{13} - 52 \beta_{14} - 42 \beta_{15} - 72 \beta_{16} + 62 \beta_{17} + 152 \beta_{18} + 36 \beta_{19} ) q^{24} \) \( + ( -83243587229751 + 40491095266 \beta_{1} + 19071795 \beta_{2} + 7673364 \beta_{3} - 720567 \beta_{4} - 36225 \beta_{5} - 158263 \beta_{6} - 16333 \beta_{7} + 73730 \beta_{8} - 1940 \beta_{9} - 5778 \beta_{10} - 182 \beta_{11} + 428 \beta_{12} - 18 \beta_{13} - 96 \beta_{14} + 31 \beta_{15} - 167 \beta_{16} - 78 \beta_{17} + 18 \beta_{18} - 18 \beta_{19} ) q^{25} \) \( + ( -138633840475449 - 1040239689 \beta_{1} + 587029742 \beta_{2} - 65993458 \beta_{3} - 286171 \beta_{4} - 2250473 \beta_{5} - 218250 \beta_{6} + 296258 \beta_{7} - 6847 \beta_{8} + 28752 \beta_{9} + 797 \beta_{10} + 160 \beta_{11} + 624 \beta_{12} - 640 \beta_{13} + 308 \beta_{14} + 348 \beta_{15} - 264 \beta_{16} - 120 \beta_{17} - 576 \beta_{18} + 80 \beta_{19} ) q^{26} \) \( + ( 5255926332 + 18291160278 \beta_{1} + 2424088015 \beta_{2} - 26807670 \beta_{3} - 1830198 \beta_{4} + 7295911 \beta_{5} + 193354 \beta_{6} - 99114 \beta_{7} - 206884 \beta_{8} - 3955 \beta_{9} - 7912 \beta_{10} - 216 \beta_{11} - 2112 \beta_{12} - 19 \beta_{13} + 522 \beta_{14} - 1961 \beta_{15} - 330 \beta_{16} + 148 \beta_{17} - 128 \beta_{18} - 216 \beta_{19} ) q^{27} \) \( + ( -214188672803540 + 94645706634 \beta_{1} + 690547212 \beta_{2} - 29688898 \beta_{3} - 908180 \beta_{4} - 3622238 \beta_{5} - 195482 \beta_{6} - 1052260 \beta_{7} + 33776 \beta_{8} + 43604 \beta_{9} + 2560 \beta_{10} + 176 \beta_{11} + 632 \beta_{12} + 3968 \beta_{13} - 612 \beta_{14} + 436 \beta_{15} - 520 \beta_{16} - 28 \beta_{17} - 1056 \beta_{18} + 344 \beta_{19} ) q^{28} \) \( + ( 22419502821 + 75507785651 \beta_{1} + 1962098625 \beta_{2} + 91131531 \beta_{3} + 4256376 \beta_{4} + 9368015 \beta_{5} + 444292 \beta_{6} - 8940 \beta_{7} - 343776 \beta_{8} - 3474 \beta_{9} - 8280 \beta_{10} - 464 \beta_{11} - 4392 \beta_{12} + 360 \beta_{13} - 1036 \beta_{14} - 6542 \beta_{15} - 700 \beta_{16} + 184 \beta_{17} - 152 \beta_{18} - 464 \beta_{19} ) q^{29} \) \( + ( 190418860847051 + 115119335677 \beta_{1} + 2638796247 \beta_{2} - 94936423 \beta_{3} - 289377 \beta_{4} + 44439221 \beta_{5} + 927573 \beta_{6} - 2803144 \beta_{7} - 58159 \beta_{8} - 87130 \beta_{9} + 5568 \beta_{10} - \beta_{11} - 608 \beta_{12} + 14781 \beta_{13} + 2184 \beta_{14} + 3433 \beta_{15} - 912 \beta_{16} + 112 \beta_{17} - 321 \beta_{18} + 736 \beta_{19} ) q^{30} \) \( + ( -449368676396022 + 314922384113 \beta_{1} + 112806102 \beta_{2} - 128213402 \beta_{3} + 12260607 \beta_{4} + 4847749 \beta_{5} - 1305149 \beta_{6} - 363816 \beta_{7} + 471658 \beta_{8} - 16700 \beta_{9} - 40363 \beta_{10} - 1687 \beta_{11} + 4100 \beta_{12} - 297 \beta_{13} + 3376 \beta_{14} + 1514 \beta_{15} - 1320 \beta_{16} - 464 \beta_{17} + 297 \beta_{18} - 297 \beta_{19} ) q^{31} \) \( + ( -456486238046602 - 392299352886 \beta_{1} - 2839370784 \beta_{2} - 188104442 \beta_{3} - 3999768 \beta_{4} + 50398840 \beta_{5} - 1347172 \beta_{6} + 4391186 \beta_{7} + 90226 \beta_{8} + 8704 \beta_{9} - 608 \beta_{10} + 1568 \beta_{11} + 4848 \beta_{12} - 28888 \beta_{13} - 4264 \beta_{14} + 1654 \beta_{15} - 1680 \beta_{16} - 1082 \beta_{17} - 2424 \beta_{18} - 76 \beta_{19} ) q^{32} \) \( + ( 267918583020058 - 124232576568 \beta_{1} - 177237243 \beta_{2} - 451047359 \beta_{3} - 33777690 \beta_{4} - 15931747 \beta_{5} + 124314 \beta_{6} + 654951 \beta_{7} + 964364 \beta_{8} + 17965 \beta_{9} + 99348 \beta_{10} + 2556 \beta_{11} - 3128 \beta_{12} - 556 \beta_{13} - 6560 \beta_{14} - 18006 \beta_{15} - 1082 \beta_{16} + 1676 \beta_{17} + 556 \beta_{18} - 556 \beta_{19} ) q^{33} \) \( + ( 956215358299787 - 81452120113 \beta_{1} + 5714757172 \beta_{2} - 431574883 \beta_{3} + 1656697 \beta_{4} - 119768949 \beta_{5} + 2072593 \beta_{6} + 6614458 \beta_{7} - 497635 \beta_{8} - 321070 \beta_{9} - 8824 \beta_{10} - 1786 \beta_{11} - 7604 \beta_{12} - 20454 \beta_{13} + 10191 \beta_{14} + 25633 \beta_{15} - 670 \beta_{16} + 2402 \beta_{17} + 8302 \beta_{18} - 316 \beta_{19} ) q^{34} \) \( + ( 128043941398 + 429744313650 \beta_{1} + 6860884776 \beta_{2} + 499169994 \beta_{3} - 55531388 \beta_{4} - 28095187 \beta_{5} + 2274579 \beta_{6} - 855106 \beta_{7} - 664852 \beta_{8} - 20935 \beta_{9} + 216504 \beta_{10} + 2440 \beta_{11} + 24768 \beta_{12} + 4670 \beta_{13} + 14242 \beta_{14} - 4965 \beta_{15} - 1506 \beta_{16} - 3644 \beta_{17} + 4480 \beta_{18} + 2440 \beta_{19} ) q^{35} \) \( + ( 1969529814736942 + 2081274549562 \beta_{1} - 9724670072 \beta_{2} + 3161351821 \beta_{3} - 6523128 \beta_{4} - 405942836 \beta_{5} + 4992740 \beta_{6} - 9944888 \beta_{7} + 200576 \beta_{8} - 704264 \beta_{9} - 15872 \beta_{10} - 992 \beta_{11} - 4848 \beta_{12} + 2432 \beta_{13} - 19160 \beta_{14} + 13272 \beta_{15} - 304 \beta_{16} - 2632 \beta_{17} + 15808 \beta_{18} - 4272 \beta_{19} ) q^{36} \) \( + ( -300196318988 - 1015159142351 \beta_{1} - 51850717149 \beta_{2} + 2054798449 \beta_{3} + 99808864 \beta_{4} - 14291527 \beta_{5} - 5212931 \beta_{6} + 2807288 \beta_{7} + 1990468 \beta_{8} + 98653 \beta_{9} - 179781 \beta_{10} + 5824 \beta_{11} + 48251 \beta_{12} - 18811 \beta_{13} - 26416 \beta_{14} - 30045 \beta_{15} + 3216 \beta_{16} + 2016 \beta_{17} + 133 \beta_{18} + 5824 \beta_{19} ) q^{37} \) \( + ( -2093015733301045 - 20174387195 \beta_{1} + 32249646646 \beta_{2} - 1068541521 \beta_{3} + 3502379 \beta_{4} + 799000968 \beta_{5} - 12925528 \beta_{6} - 11394556 \beta_{7} - 1467886 \beta_{8} + 1522848 \beta_{9} - 87400 \beta_{10} - 1840 \beta_{11} + 3288 \beta_{12} - 124032 \beta_{13} + 31954 \beta_{14} + 142654 \beta_{15} + 6444 \beta_{16} + 5012 \beta_{17} - 608 \beta_{18} - 10040 \beta_{19} ) q^{38} \) \( + ( 6086039652457475 - 3864007898079 \beta_{1} - 3363791171 \beta_{2} - 7575987164 \beta_{3} + 213975184 \beta_{4} + 8237647 \beta_{5} + 12739396 \beta_{6} - 3010203 \beta_{7} - 5823024 \beta_{8} + 98448 \beta_{9} + 11832 \beta_{10} + 21048 \beta_{11} - 46336 \beta_{12} + 6280 \beta_{13} + 39696 \beta_{14} - 184200 \beta_{15} + 9032 \beta_{16} - 3056 \beta_{17} - 6280 \beta_{18} + 6280 \beta_{19} ) q^{39} \) \( + ( -7999215896544472 + 3911128283144 \beta_{1} - 89247839112 \beta_{2} + 2293667100 \beta_{3} - 3175944 \beta_{4} + 1436278408 \beta_{5} + 550384 \beta_{6} + 13885564 \beta_{7} + 1889160 \beta_{8} + 2074120 \beta_{9} + 27584 \beta_{10} - 23264 \beta_{11} - 38384 \beta_{12} + 292944 \beta_{13} - 55432 \beta_{14} + 191332 \beta_{15} + 16176 \beta_{16} + 4084 \beta_{17} + 20176 \beta_{18} - 3496 \beta_{19} ) q^{40} \) \( + ( -4232133456987034 - 2714010160738 \beta_{1} - 3294938447 \beta_{2} - 8569715964 \beta_{3} - 200035453 \beta_{4} - 150296787 \beta_{5} + 7614683 \beta_{6} + 10362977 \beta_{7} - 6881610 \beta_{8} + 454644 \beta_{9} - 378982 \beta_{10} - 22258 \beta_{11} - 26268 \beta_{12} + 13658 \beta_{13} - 66368 \beta_{14} - 373547 \beta_{15} + 19891 \beta_{16} - 9882 \beta_{17} - 13658 \beta_{18} + 13658 \beta_{19} ) q^{41} \) \( + ( 8930742999782512 + 60223989996 \beta_{1} + 4490755848 \beta_{2} + 4582204216 \beta_{3} + 24447464 \beta_{4} - 2762533316 \beta_{5} + 7564320 \beta_{6} - 26601296 \beta_{7} - 1406664 \beta_{8} - 1728400 \beta_{9} + 172 \beta_{10} + 9952 \beta_{11} + 11984 \beta_{12} + 347264 \beta_{13} + 62524 \beta_{14} + 488180 \beta_{15} + 25256 \beta_{16} - 13096 \beta_{17} - 69824 \beta_{18} - 6288 \beta_{19} ) q^{42} \) \( + ( -3763320408775 - 12495977469665 \beta_{1} + 90528966397 \beta_{2} + 13678492212 \beta_{3} - 199901932 \beta_{4} - 238616634 \beta_{5} - 47191924 \beta_{6} + 7432348 \beta_{7} + 17822040 \beta_{8} + 1070818 \beta_{9} - 1613968 \beta_{10} - 13680 \beta_{11} - 105216 \beta_{12} - 76838 \beta_{13} + 64996 \beta_{14} - 580538 \beta_{15} + 42460 \beta_{16} + 27208 \beta_{17} - 72320 \beta_{18} - 13680 \beta_{19} ) q^{43} \) \( + ( -1393703560501295 - 14782328376527 \beta_{1} + 137280835555 \beta_{2} + 685670196 \beta_{3} - 20712872 \beta_{4} - 4457438239 \beta_{5} - 44582598 \beta_{6} + 64056736 \beta_{7} + 4741748 \beta_{8} + 73776 \beta_{9} - 88064 \beta_{10} - 10432 \beta_{11} + 30507 \beta_{12} - 449152 \beta_{13} - 87344 \beta_{14} + 1032603 \beta_{15} + 50592 \beta_{16} + 27792 \beta_{17} - 139776 \beta_{18} + 28768 \beta_{19} ) q^{44} \) \( + ( 1230236952994 + 4107663741833 \beta_{1} - 201114365915 \beta_{2} + 55371263085 \beta_{3} + 389594456 \beta_{4} - 1941560609 \beta_{5} + 9134757 \beta_{6} + 35519212 \beta_{7} + 32636724 \beta_{8} + 2364975 \beta_{9} + 2354207 \beta_{10} - 39440 \beta_{11} - 297201 \beta_{12} + 331825 \beta_{13} - 78204 \beta_{14} - 1638735 \beta_{15} + 37012 \beta_{16} - 30952 \beta_{17} + 4145 \beta_{18} - 39440 \beta_{19} ) q^{45} \) \( + ( 9206016661259905 - 16743854200073 \beta_{1} + 89757966069 \beta_{2} - 4799053813 \beta_{3} + 85672989 \beta_{4} + 6542860287 \beta_{5} + 92284159 \beta_{6} + 101107016 \beta_{7} - 2471533 \beta_{8} - 8281550 \beta_{9} + 717760 \beta_{10} + 17997 \beta_{11} - 81248 \beta_{12} + 330407 \beta_{13} + 39624 \beta_{14} + 1499739 \beta_{15} + 31728 \beta_{16} - 65552 \beta_{17} + 43789 \beta_{18} + 78304 \beta_{19} ) q^{46} \) \( + ( 7496677249286640 + 4000165833693 \beta_{1} - 13802788776 \beta_{2} - 66338091906 \beta_{3} + 108045893 \beta_{4} - 468515531 \beta_{5} - 48560499 \beta_{6} + 68879494 \beta_{7} - 12834626 \beta_{8} + 4652956 \beta_{9} + 2478639 \beta_{10} - 157813 \beta_{11} + 304812 \beta_{12} - 56459 \beta_{13} + 6464 \beta_{14} - 1543626 \beta_{15} + 54560 \beta_{16} + 67648 \beta_{17} + 56459 \beta_{18} - 56459 \beta_{19} ) q^{47} \) \( + ( 75479939202625506 - 38697144916094 \beta_{1} - 726230224296 \beta_{2} - 4476271946 \beta_{3} - 124741760 \beta_{4} + 10782596916 \beta_{5} + 25032664 \beta_{6} - 303618882 \beta_{7} - 10316290 \beta_{8} - 16963448 \beta_{9} - 670880 \beta_{10} + 204608 \beta_{11} + 144864 \beta_{12} - 1328072 \beta_{13} + 27248 \beta_{14} + 3021866 \beta_{15} + 29280 \beta_{16} + 25674 \beta_{17} - 75496 \beta_{18} + 42988 \beta_{19} ) q^{48} \) \( + ( 23883601913778681 - 3082622456104 \beta_{1} - 32567318072 \beta_{2} - 133585545584 \beta_{3} + 477586656 \beta_{4} - 841156368 \beta_{5} - 44169144 \beta_{6} + 50807688 \beta_{7} - 9648032 \beta_{8} + 6253480 \beta_{9} - 2756320 \beta_{10} + 136480 \beta_{11} + 569856 \beta_{12} - 140832 \beta_{13} + 113056 \beta_{14} - 3431232 \beta_{15} - 10912 \beta_{16} - 1088 \beta_{17} + 140832 \beta_{18} - 140832 \beta_{19} ) q^{49} \) \( + ( -86082852278708660 + 82940887578869 \beta_{1} - 1477835138456 \beta_{2} + 39929099986 \beta_{3} - 311231926 \beta_{4} - 10844785474 \beta_{5} - 65741990 \beta_{6} - 363598044 \beta_{7} - 12422862 \beta_{8} + 16970900 \beta_{9} + 663760 \beta_{10} - 19268 \beta_{11} + 502584 \beta_{12} - 2264572 \beta_{13} - 201626 \beta_{14} + 5242714 \beta_{15} - 44492 \beta_{16} - 22988 \beta_{17} + 365132 \beta_{18} + 98408 \beta_{19} ) q^{50} \) \( + ( 11409163088654 + 38096641870356 \beta_{1} - 667839309573 \beta_{2} + 216462245184 \beta_{3} + 1616701646 \beta_{4} - 760312286 \beta_{5} + 195953563 \beta_{6} + 91423768 \beta_{7} - 33547152 \beta_{8} + 5952468 \beta_{9} + 2761824 \beta_{10} + 24992 \beta_{11} - 371712 \beta_{12} + 719107 \beta_{13} - 319000 \beta_{14} - 7100228 \beta_{15} - 108904 \beta_{16} - 55600 \beta_{17} + 707840 \beta_{18} + 24992 \beta_{19} ) q^{51} \) \( + ( -81980431370182428 + 137630646131981 \beta_{1} + 2545353921612 \beta_{2} + 1625720985 \beta_{3} - 405125566 \beta_{4} - 10767177907 \beta_{5} + 189793273 \beta_{6} + 261817350 \beta_{7} - 53582844 \beta_{8} + 19068998 \beta_{9} + 2557568 \beta_{10} + 217064 \beta_{11} - 218298 \beta_{12} + 4213456 \beta_{13} + 557562 \beta_{14} + 7246380 \beta_{15} - 191500 \beta_{16} - 91478 \beta_{17} + 788064 \beta_{18} - 99620 \beta_{19} ) q^{52} \) \( + ( -12799937237218 - 41938990328093 \beta_{1} + 1403455728103 \beta_{2} + 195376181007 \beta_{3} - 2554327096 \beta_{4} + 4732764421 \beta_{5} - 11847793 \beta_{6} - 12324188 \beta_{7} - 144477284 \beta_{8} + 3805885 \beta_{9} - 9337923 \beta_{10} + 133712 \beta_{11} + 972045 \beta_{12} - 3524429 \beta_{13} + 765932 \beta_{14} - 8004765 \beta_{15} - 203748 \beta_{16} + 125576 \beta_{17} - 19789 \beta_{18} + 133712 \beta_{19} ) q^{53} \) \( + ( 39268969592467148 - 506155808502 \beta_{1} + 4232883796500 \beta_{2} + 19650786546 \beta_{3} - 1924059156 \beta_{4} + 18618197640 \beta_{5} - 102706520 \beta_{6} + 243382788 \beta_{7} + 2033170 \beta_{8} - 29715808 \beta_{9} - 3252072 \beta_{10} - 67376 \beta_{11} + 1410264 \beta_{12} + 1821568 \beta_{13} - 866606 \beta_{14} + 13131070 \beta_{15} - 258772 \beta_{16} + 319380 \beta_{17} - 463456 \beta_{18} - 364344 \beta_{19} ) q^{54} \) \( + ( -96819445970994947 - 25624221330053 \beta_{1} - 143150925165 \beta_{2} - 550166289972 \beta_{3} - 4461372444 \beta_{4} - 5628618595 \beta_{5} - 129493496 \beta_{6} - 2649901 \beta_{7} + 258245640 \beta_{8} - 8671680 \beta_{9} - 16085676 \beta_{10} + 714436 \beta_{11} - 1293904 \beta_{12} + 272124 \beta_{13} - 1090992 \beta_{14} - 18756688 \beta_{15} - 510104 \beta_{16} - 390576 \beta_{17} - 272124 \beta_{18} + 272124 \beta_{19} ) q^{55} \) \( + ( -187510307425284824 + 213700096958608 \beta_{1} - 4200117071424 \beta_{2} + 89301749320 \beta_{3} - 1007360832 \beta_{4} + 11551146232 \beta_{5} - 259216856 \beta_{6} - 329244592 \beta_{7} - 60101880 \beta_{8} - 20251904 \beta_{9} + 8242432 \beta_{10} - 1157888 \beta_{11} - 164352 \beta_{12} + 504896 \beta_{13} + 1614912 \beta_{14} + 18129648 \beta_{15} - 563072 \beta_{16} - 256656 \beta_{17} - 167104 \beta_{18} - 224480 \beta_{19} ) q^{56} \) \( + ( 20405359466113298 - 209289005288216 \beta_{1} - 289054482859 \beta_{2} - 857206371771 \beta_{3} + 8911682570 \beta_{4} - 3671831563 \beta_{5} + 387592422 \beta_{6} - 31750721 \beta_{7} + 365193204 \beta_{8} - 6158551 \beta_{9} + 30990188 \beta_{10} - 629948 \beta_{11} - 4028168 \beta_{12} + 838508 \beta_{13} + 1876672 \beta_{14} - 21630970 \beta_{15} - 471126 \beta_{16} + 232788 \beta_{17} - 838508 \beta_{18} + 838508 \beta_{19} ) q^{57} \) \( + ( 159119172236426119 + 2597022944035 \beta_{1} - 8458806038858 \beta_{2} + 67880830918 \beta_{3} - 2392513827 \beta_{4} + 19361886739 \beta_{5} + 694399926 \beta_{6} + 369992082 \beta_{7} + 64814025 \beta_{8} - 81619840 \beta_{9} - 5676567 \beta_{10} - 91392 \beta_{11} - 5584768 \beta_{12} + 5884928 \beta_{13} - 1875104 \beta_{14} + 17889568 \beta_{15} - 499136 \beta_{16} + 551360 \beta_{17} - 1103360 \beta_{18} - 652928 \beta_{19} ) q^{58} \) \( + ( -117628027785749 - 391039805580835 \beta_{1} - 348498357383 \beta_{2} + 874096180554 \beta_{3} + 7611006904 \beta_{4} - 11824711311 \beta_{5} - 1362547139 \beta_{6} + 375160382 \beta_{7} - 123775828 \beta_{8} + 17474681 \beta_{9} + 25524792 \beta_{10} + 150408 \beta_{11} + 6647616 \beta_{12} - 4560064 \beta_{13} - 2081182 \beta_{14} - 24648805 \beta_{15} - 684002 \beta_{16} - 334908 \beta_{17} - 4624128 \beta_{18} + 150408 \beta_{19} ) q^{59} \) \( + ( 137526441553634092 - 193273757991542 \beta_{1} + 20875709850636 \beta_{2} + 124367600446 \beta_{3} - 1369071508 \beta_{4} + 54475698850 \beta_{5} - 1354039194 \beta_{6} - 2498006372 \beta_{7} - 93576464 \beta_{8} - 181231020 \beta_{9} - 25757184 \beta_{10} - 1752912 \beta_{11} + 1057912 \beta_{12} - 17362048 \beta_{13} + 2511260 \beta_{14} + 35729076 \beta_{15} - 563720 \beta_{16} - 152860 \beta_{17} - 2764832 \beta_{18} + 35672 \beta_{19} ) q^{60} \) \( + ( 89374936702544 + 300677631878189 \beta_{1} + 4714887636789 \beta_{2} + 896389725577 \beta_{3} - 8006554376 \beta_{4} + 7125730215 \beta_{5} + 1649270019 \beta_{6} + 138879500 \beta_{7} - 236260148 \beta_{8} + 25562069 \beta_{9} - 6003999 \beta_{10} + 6960 \beta_{11} - 98543 \beta_{12} + 25774383 \beta_{13} + 2502580 \beta_{14} - 34268661 \beta_{15} - 395516 \beta_{16} + 90424 \beta_{17} - 19665 \beta_{18} + 6960 \beta_{19} ) q^{61} \) \( + ( -666678466478142268 + 440346779433900 \beta_{1} + 25362964723708 \beta_{2} + 336570468060 \beta_{3} + 331742340 \beta_{4} - 76911977156 \beta_{5} + 1568290508 \beta_{6} - 3349959952 \beta_{7} - 105706644 \beta_{8} + 148640968 \beta_{9} + 6328384 \beta_{10} - 39348 \beta_{11} - 12397728 \beta_{12} - 17088732 \beta_{13} - 2217800 \beta_{14} + 25102988 \beta_{15} - 256752 \beta_{16} - 388848 \beta_{17} + 2520460 \beta_{18} + 856096 \beta_{19} ) q^{62} \) \( + ( -438911350259630235 + 452006356355774 \beta_{1} - 139629392149 \beta_{2} - 1415481595846 \beta_{3} - 7342106469 \beta_{4} - 12001923040 \beta_{5} - 2411939493 \beta_{6} + 1181995757 \beta_{7} + 95065314 \beta_{8} + 191535876 \beta_{9} + 24165569 \beta_{10} - 1538459 \beta_{11} + 3015252 \beta_{12} - 560613 \beta_{13} - 1956512 \beta_{14} - 38185222 \beta_{15} + 305840 \beta_{16} + 663136 \beta_{17} + 560613 \beta_{18} - 560613 \beta_{19} ) q^{63} \) \( + ( -450939585934353796 + 460581761328884 \beta_{1} - 39524350852000 \beta_{2} - 409538084660 \beta_{3} + 4403952 \beta_{4} - 116138127360 \beta_{5} - 898663352 \beta_{6} + 3687618548 \beta_{7} - 333916812 \beta_{8} + 210080352 \beta_{9} - 65153984 \beta_{10} + 4181440 \beta_{11} - 750048 \beta_{12} + 24219152 \beta_{13} + 1135056 \beta_{14} + 44078748 \beta_{15} + 679456 \beta_{16} + 511228 \beta_{17} + 3556048 \beta_{18} + 406408 \beta_{19} ) q^{64} \) \( + ( 315018936331080076 - 415278889576538 \beta_{1} - 526176311363 \beta_{2} - 1470751733940 \beta_{3} + 2104252159 \beta_{4} - 11822115495 \beta_{5} + 1136575623 \beta_{6} + 409314909 \beta_{7} - 821517778 \beta_{8} + 116819500 \beta_{9} - 100657822 \beta_{10} + 2179046 \beta_{11} + 13763892 \beta_{12} - 2910046 \beta_{13} + 159872 \beta_{14} - 39526663 \beta_{15} + 820719 \beta_{16} - 696354 \beta_{17} + 2910046 \beta_{18} - 2910046 \beta_{19} ) q^{65} \) \( + ( 264509852868197665 - 253464346332141 \beta_{1} - 70552858914868 \beta_{2} - 196033670197 \beta_{3} - 2552259793 \beta_{4} + 218676373965 \beta_{5} + 3223125687 \beta_{6} + 4738097814 \beta_{7} - 504130325 \beta_{8} - 246974146 \beta_{9} + 20974776 \beta_{10} + 648266 \beta_{11} + 30863124 \beta_{12} + 9203222 \beta_{13} + 524649 \beta_{14} + 74523335 \beta_{15} + 1343278 \beta_{16} - 2181330 \beta_{17} + 867874 \beta_{18} + 2205276 \beta_{19} ) q^{66} \) \( + ( -23644283560959 - 79041782316991 \beta_{1} - 6851252735938 \beta_{2} + 1542040294084 \beta_{3} + 2514694882 \beta_{4} + 17768307344 \beta_{5} - 57616241 \beta_{6} + 924752228 \beta_{7} + 1002275240 \beta_{8} + 105080350 \beta_{9} - 149130608 \beta_{10} - 1041040 \beta_{11} - 34220800 \beta_{12} + 20343549 \beta_{13} + 1968412 \beta_{14} - 68405318 \beta_{15} + 2990628 \beta_{16} + 2071992 \beta_{17} + 20707456 \beta_{18} - 1041040 \beta_{19} ) q^{67} \) \( + ( 2044292732592470956 - 955088330504760 \beta_{1} + 69782762276984 \beta_{2} - 54485541974 \beta_{3} + 6493432 \beta_{4} + 307337519156 \beta_{5} - 828841188 \beta_{6} - 4633264840 \beta_{7} + 74286720 \beta_{8} - 18867448 \beta_{9} + 166436352 \beta_{10} + 8115168 \beta_{11} - 1208592 \beta_{12} + 19259008 \beta_{13} - 5630248 \beta_{14} + 36522536 \beta_{15} + 3725616 \beta_{16} + 1786952 \beta_{17} + 4614720 \beta_{18} + 1195184 \beta_{19} ) q^{68} \) \( + ( 191198485635907 + 631905852139480 \beta_{1} - 25582609264080 \beta_{2} + 2322285035616 \beta_{3} + 35789919432 \beta_{4} - 175223399708 \beta_{5} + 961879175 \beta_{6} + 2178672228 \beta_{7} + 2729203964 \beta_{8} + 123360821 \beta_{9} + 160318533 \beta_{10} - 2057648 \beta_{11} - 14539371 \beta_{12} - 136819669 \beta_{13} - 8251156 \beta_{14} - 55583509 \beta_{15} + 3531292 \beta_{16} - 2185592 \beta_{17} + 391723 \beta_{18} - 2057648 \beta_{19} ) q^{69} \) \( + ( 903872789118315900 - 14508583470368 \beta_{1} + 114650781378768 \beta_{2} + 561409513704 \beta_{3} - 2078563396 \beta_{4} - 443060407696 \beta_{5} + 3607164720 \beta_{6} - 3109924008 \beta_{7} + 541818124 \beta_{8} - 220780480 \beta_{9} + 11451280 \beta_{10} + 1308128 \beta_{11} + 63423120 \beta_{12} + 43913472 \beta_{13} + 9144332 \beta_{14} + 85154516 \beta_{15} + 4177864 \beta_{16} - 2623304 \beta_{17} - 6719552 \beta_{18} + 498992 \beta_{19} ) q^{70} \) \( + ( 4822866728283612535 + 1008002670515644 \beta_{1} + 584019192177 \beta_{2} + 477950955798 \beta_{3} + 62490112799 \beta_{4} + 29050867754 \beta_{5} - 3351786013 \beta_{6} + 69139603 \beta_{7} - 2931969750 \beta_{8} - 772233404 \beta_{9} + 140381493 \beta_{10} - 2364983 \beta_{11} + 2701188 \beta_{12} - 1187209 \beta_{13} + 12149040 \beta_{14} + 24803114 \beta_{15} + 6025304 \beta_{16} + 2799920 \beta_{17} + 1187209 \beta_{18} - 1187209 \beta_{19} ) q^{71} \) \( + ( 2491518910127738185 - 1920543114350048 \beta_{1} - 175418285921928 \beta_{2} + 2025848443989 \beta_{3} - 6740676992 \beta_{4} - 581531061063 \beta_{5} - 15237566089 \beta_{6} + 1599974240 \beta_{7} + 1640934441 \beta_{8} - 620719616 \beta_{9} + 365213184 \beta_{10} - 8047104 \beta_{11} + 3972096 \beta_{12} - 113973376 \beta_{13} - 18714752 \beta_{14} - 7850464 \beta_{15} + 6446848 \beta_{16} + 2201888 \beta_{17} - 19306112 \beta_{18} + 1675712 \beta_{19} ) q^{72} \) \( + ( -1074404467137259664 - 4464239688993716 \beta_{1} - 1481646515585 \beta_{2} + 2128494460509 \beta_{3} - 80802309984 \beta_{4} - 32491846513 \beta_{5} + 17333219236 \beta_{6} + 6314690249 \beta_{7} - 1822335904 \beta_{8} - 10287191 \beta_{9} - 24561888 \beta_{10} - 4672992 \beta_{11} - 11987712 \beta_{12} + 3962592 \beta_{13} - 22303200 \beta_{14} + 30066560 \beta_{15} + 5445408 \beta_{16} - 1576128 \beta_{17} - 3962592 \beta_{18} + 3962592 \beta_{19} ) q^{73} \) \( + ( -2147645195831533403 + 24362259903253 \beta_{1} - 215317167089878 \beta_{2} - 1221619164150 \beta_{3} + 35681780383 \beta_{4} + 338657324213 \beta_{5} + 16460478082 \beta_{6} - 8600406106 \beta_{7} + 1950936755 \beta_{8} + 1285560944 \beta_{9} - 15585945 \beta_{10} - 983840 \beta_{11} - 105520176 \beta_{12} - 101421952 \beta_{13} + 23203324 \beta_{14} - 168249036 \beta_{15} + 5139752 \beta_{16} + 498264 \beta_{17} + 7662400 \beta_{18} - 1495440 \beta_{19} ) q^{74} \) \( + ( -2960001247746477 - 9858523208190167 \beta_{1} + 47547424132363 \beta_{2} - 5798015375598 \beta_{3} - 124678406908 \beta_{4} - 80073088215 \beta_{5} - 39489305681 \beta_{6} - 3530353946 \beta_{7} + 1862320828 \beta_{8} - 192552635 \beta_{9} + 147880664 \beta_{10} + 1607400 \beta_{11} + 78466368 \beta_{12} - 61194610 \beta_{13} + 26838202 \beta_{14} + 228479935 \beta_{15} + 3882054 \beta_{16} - 1511564 \beta_{17} - 60976640 \beta_{18} + 1607400 \beta_{19} ) q^{75} \) \( + ( -10347760630552438987 + 1971931745168597 \beta_{1} + 279806296953807 \beta_{2} + 125457695044 \beta_{3} + 1131342904 \beta_{4} + 335249241157 \beta_{5} - 37107934174 \beta_{6} + 25566240416 \beta_{7} + 990641092 \beta_{8} + 1844098928 \beta_{9} - 770750464 \beta_{10} - 20506048 \beta_{11} - 13797193 \beta_{12} + 125075328 \beta_{13} - 31221872 \beta_{14} - 211061401 \beta_{15} + 1728288 \beta_{16} - 2505136 \beta_{17} + 6313472 \beta_{18} - 4598048 \beta_{19} ) q^{76} \) \( + ( 3576217929595765 + 11895805739799480 \beta_{1} - 40349466772016 \beta_{2} - 8574675777136 \beta_{3} + 104303925944 \beta_{4} + 251517032764 \beta_{5} + 44054875889 \beta_{6} - 1301177828 \beta_{7} - 95979164 \beta_{8} - 260589149 \beta_{9} - 365333373 \beta_{10} + 7709360 \beta_{11} + 60801075 \beta_{12} + 535533069 \beta_{13} - 32856364 \beta_{14} + 297522237 \beta_{15} + 1193892 \beta_{16} + 4610552 \beta_{17} - 637939 \beta_{18} + 7709360 \beta_{19} ) q^{77} \) \( + ( 8191582714167898073 - 6110509277531601 \beta_{1} + 437903099508213 \beta_{2} - 3497101832061 \beta_{3} + 42273224229 \beta_{4} - 17967927385 \beta_{5} + 61716186103 \beta_{6} + 28586567832 \beta_{7} - 4976513413 \beta_{8} - 1310396446 \beta_{9} - 96349696 \beta_{10} - 4024643 \beta_{11} - 192986368 \beta_{12} + 56194615 \beta_{13} + 34068416 \beta_{14} - 460871677 \beta_{15} - 775040 \beta_{16} + 10693760 \beta_{17} - 3137347 \beta_{18} - 10278656 \beta_{19} ) q^{78} \) \( + ( -14205757733397839782 + 16636541075659918 \beta_{1} + 9643142310526 \beta_{2} + 9856113549772 \beta_{3} + 141661819260 \beta_{4} + 184776193194 \beta_{5} - 58299014432 \beta_{6} - 11851541714 \beta_{7} + 1049327768 \beta_{8} + 3273763680 \beta_{9} - 616975204 \beta_{10} + 27535404 \beta_{11} - 51565232 \beta_{12} + 10860948 \beta_{13} + 34803280 \beta_{14} + 302407424 \beta_{15} - 10564376 \beta_{16} - 15137456 \beta_{17} - 10860948 \beta_{18} + 10860948 \beta_{19} ) q^{79} \) \( + ( -2492773459668826156 + 8063810612920436 \beta_{1} - 444732570494800 \beta_{2} + 3510939745148 \beta_{3} + 50607152576 \beta_{4} + 565755730472 \beta_{5} - 52987397168 \beta_{6} - 37311185364 \beta_{7} - 789074772 \beta_{8} - 545394160 \beta_{9} - 1503937088 \beta_{10} - 3158656 \beta_{11} - 11600832 \beta_{12} + 144114736 \beta_{13} - 26330592 \beta_{14} - 579947932 \beta_{15} - 15017664 \beta_{16} - 11579356 \beta_{17} + 48244464 \beta_{18} - 11565384 \beta_{19} ) q^{80} \) \( + ( 59463790940881475 - 17063303467935528 \beta_{1} - 2843992789967 \beta_{2} + 19894228635081 \beta_{3} - 150486207750 \beta_{4} + 51180587569 \beta_{5} + 76034060294 \beta_{6} - 34948480029 \beta_{7} + 9458633940 \beta_{8} - 998201027 \beta_{9} + 993135500 \beta_{10} - 1717340 \beta_{11} - 86073288 \beta_{12} + 12956556 \beta_{13} - 21320000 \beta_{14} + 525286550 \beta_{15} - 12739174 \beta_{16} + 11503924 \beta_{17} - 12956556 \beta_{18} + 12956556 \beta_{19} ) q^{81} \) \( + ( 5636264135721333214 + 4347676644119780 \beta_{1} - 424803766639816 \beta_{2} - 2963342986058 \beta_{3} - 51737259234 \beta_{4} - 747156516870 \beta_{5} + 67795557710 \beta_{6} - 37552632340 \beta_{7} - 11168500010 \beta_{8} + 604891196 \beta_{9} - 179663760 \beta_{10} - 4022604 \beta_{11} + 226767016 \beta_{12} + 178282892 \beta_{13} + 22006770 \beta_{14} - 422842290 \beta_{15} - 16731876 \beta_{16} + 21187868 \beta_{17} - 34358940 \beta_{18} - 19524040 \beta_{19} ) q^{82} \) \( + ( -4381000963514869 - 14624190770612475 \beta_{1} + 22208593105097 \beta_{2} - 21248280211514 \beta_{3} - 87443845480 \beta_{4} + 669388359631 \beta_{5} - 55067375653 \beta_{6} - 14205651902 \beta_{7} - 16447893356 \beta_{8} - 1174692665 \beta_{9} + 1071127368 \beta_{10} + 7245432 \beta_{11} + 11313216 \beta_{12} + 96210600 \beta_{13} + 13134046 \beta_{14} + 746230501 \beta_{15} - 31385374 \beta_{16} - 18484164 \beta_{17} + 90864000 \beta_{18} + 7245432 \beta_{19} ) q^{83} \) \( + ( 8931323448498457360 - 8981666318247764 \beta_{1} + 617588073115440 \beta_{2} + 866174535388 \beta_{3} + 43792914552 \beta_{4} - 2549369147348 \beta_{5} - 123381548644 \beta_{6} + 50393625064 \beta_{7} + 4102993648 \beta_{8} - 1701527960 \beta_{9} + 2646113792 \beta_{10} + 5939808 \beta_{11} + 65450696 \beta_{12} - 527463744 \beta_{13} + 16245400 \beta_{14} - 922115024 \beta_{15} - 33852112 \beta_{16} - 12129448 \beta_{17} - 53282688 \beta_{18} + 2643088 \beta_{19} ) q^{84} \) \( + ( 9156912997314481 + 30514131645351508 \beta_{1} + 123060935920154 \beta_{2} - 37273104699144 \beta_{3} - 172352592976 \beta_{4} + 553801064818 \beta_{5} + 118101016783 \beta_{6} - 22345300352 \beta_{7} - 22110908404 \beta_{8} - 1573152325 \beta_{9} - 692569727 \beta_{10} - 3272480 \beta_{11} - 47676319 \beta_{12} - 1515303905 \beta_{13} + 33947464 \beta_{14} + 1024147845 \beta_{15} - 32147992 \beta_{16} + 10945712 \beta_{17} - 3667425 \beta_{18} - 3272480 \beta_{19} ) q^{85} \) \( + ( -26164772751706515767 - 252298588228343 \beta_{1} + 415528909073318 \beta_{2} - 12557283422025 \beta_{3} - 22038115383 \beta_{4} + 2485245511440 \beta_{5} + 112255592272 \beta_{6} + 47320683016 \beta_{7} + 22182294308 \beta_{8} + 5020084544 \beta_{9} + 183759152 \beta_{10} - 3307104 \beta_{11} + 287911344 \beta_{12} - 590776576 \beta_{13} - 37687132 \beta_{14} - 907437188 \beta_{15} - 37495720 \beta_{16} - 1336024 \beta_{17} + 90450752 \beta_{18} + 28954000 \beta_{19} ) q^{86} \) \( + ( 25877150462309450277 + 55069595131402526 \beta_{1} + 37876933020387 \beta_{2} + 59863107809982 \beta_{3} - 458114143017 \beta_{4} + 483651961496 \beta_{5} - 185916204013 \beta_{6} - 18978782203 \beta_{7} + 18438530330 \beta_{8} - 15928867132 \beta_{9} + 76799245 \beta_{10} - 75748159 \beta_{11} + 170258980 \beta_{12} - 24314881 \beta_{13} - 61537392 \beta_{14} + 1447970506 \beta_{15} - 35078008 \beta_{16} + 13531792 \beta_{17} + 24314881 \beta_{18} - 24314881 \beta_{19} ) q^{87} \) \( + ( -6200940655894865400 + 1458318772740432 \beta_{1} - 608119982320660 \beta_{2} - 15634737545830 \beta_{3} - 151770204324 \beta_{4} + 3936855598628 \beta_{5} - 205807030400 \beta_{6} + 13105871186 \beta_{7} + 1896903808 \beta_{8} + 7531203444 \beta_{9} + 4547624224 \beta_{10} + 67518992 \beta_{11} + 20867240 \beta_{12} + 532700152 \beta_{13} + 106453052 \beta_{14} - 1329610098 \beta_{15} - 35461352 \beta_{16} + 4138518 \beta_{17} + 8418936 \beta_{18} + 19578100 \beta_{19} ) q^{88} \) \( + ( 847496568247631168 - 61026488703540844 \beta_{1} - 16608670344677 \beta_{2} + 41550807228005 \beta_{3} + 354181717740 \beta_{4} + 173280810603 \beta_{5} + 231095147904 \beta_{6} + 90479696853 \beta_{7} + 874642936 \beta_{8} - 682813503 \beta_{9} - 1870736952 \beta_{10} + 58895512 \beta_{11} + 344291472 \beta_{12} - 78581752 \beta_{13} + 137583968 \beta_{14} + 1693007572 \beta_{15} - 38828244 \beta_{16} - 5139112 \beta_{17} + 78581752 \beta_{18} - 78581752 \beta_{19} ) q^{89} \) \( + ( 8555102529308682691 + 221137693212923 \beta_{1} - 892892681148138 \beta_{2} + 4158277829846 \beta_{3} + 111673520169 \beta_{4} - 6782535483333 \beta_{5} + 386015432718 \beta_{6} + 49576366442 \beta_{7} + 50333430805 \beta_{8} - 8676882000 \beta_{9} + 733266553 \beta_{10} + 17902944 \beta_{11} - 234822000 \beta_{12} + 475816576 \beta_{13} - 151287924 \beta_{14} - 1972683292 \beta_{15} - 36223608 \beta_{16} - 56167432 \beta_{17} + 51228224 \beta_{18} + 82854576 \beta_{19} ) q^{90} \) \( + ( -25935565371464758 - 86503200103093490 \beta_{1} - 20473607256240 \beta_{2} - 33978114284218 \beta_{3} + 710324335884 \beta_{4} + 807036249635 \beta_{5} - 345150971659 \beta_{6} - 5397322014 \beta_{7} - 7834900460 \beta_{8} - 996628873 \beta_{9} - 3195794616 \beta_{10} - 35196936 \beta_{11} - 524486208 \beta_{12} + 99337674 \beta_{13} - 185808642 \beta_{14} + 990365365 \beta_{15} - 2587134 \beta_{16} + 51088380 \beta_{17} + 105537024 \beta_{18} - 35196936 \beta_{19} ) q^{91} \) \( + ( 41351775480170100676 - 9060169307773570 \beta_{1} + 741599241493604 \beta_{2} - 15637813926054 \beta_{3} - 104367235196 \beta_{4} - 3796745533114 \beta_{5} - 387462617774 \beta_{6} - 147818253036 \beta_{7} - 15707306032 \beta_{8} - 7529528132 \beta_{9} - 6752866816 \beta_{10} + 154331920 \beta_{11} - 35380248 \beta_{12} + 233732736 \beta_{13} + 218530836 \beta_{14} - 1358128676 \beta_{15} + 6714472 \beta_{16} + 37647212 \beta_{17} + 100888992 \beta_{18} + 28500616 \beta_{19} ) q^{92} \) \( + ( 48749255563796680 + 162555166712396596 \beta_{1} + 331106278702960 \beta_{2} - 38536857566820 \beta_{3} - 972875385584 \beta_{4} - 948423725760 \beta_{5} + 626422514992 \beta_{6} - 24356025640 \beta_{7} + 4008920864 \beta_{8} - 1028467108 \beta_{9} + 4040480856 \beta_{10} - 60675424 \beta_{11} - 451727112 \beta_{12} + 2807528584 \beta_{13} + 244910808 \beta_{14} + 978132580 \beta_{15} + 2156984 \beta_{16} - 54073840 \beta_{17} + 13795976 \beta_{18} - 60675424 \beta_{19} ) q^{93} \) \( + ( -8255312081507060678 - 7578297035448314 \beta_{1} + 131260235565482 \beta_{2} + 3326149641582 \beta_{3} - 13817200286 \beta_{4} + 5230533420486 \beta_{5} + 601452074710 \beta_{6} - 150896090208 \beta_{7} - 56627271634 \beta_{8} + 8045573588 \beta_{9} + 128745664 \beta_{10} + 48995594 \beta_{11} + 167133728 \beta_{12} + 1003146206 \beta_{13} - 276450264 \beta_{14} - 1552944770 \beta_{15} + 22151984 \beta_{16} - 66889936 \beta_{17} - 281584950 \beta_{18} - 11757472 \beta_{19} ) q^{94} \) \( + ( -116623470215646541175 + 209401348078407043 \beta_{1} + 98716730685887 \beta_{2} + 33956577668424 \beta_{3} - 1322803783468 \beta_{4} + 437560242485 \beta_{5} - 788286240620 \beta_{6} + 80695725591 \beta_{7} - 18941223688 \beta_{8} + 42419183600 \beta_{9} + 4042097980 \beta_{10} + 8718764 \beta_{11} - 67832400 \beta_{12} - 16416684 \beta_{13} - 303740544 \beta_{14} + 1035901208 \beta_{15} + 76320192 \beta_{16} + 70598528 \beta_{17} + 16416684 \beta_{18} - 16416684 \beta_{19} ) q^{95} \) \( + ( -35795211912951165460 - 75852121475425612 \beta_{1} + 625694926700160 \beta_{2} - 38767844302164 \beta_{3} + 933160254224 \beta_{4} + 7376554636112 \beta_{5} - 999703812520 \beta_{6} + 244790237924 \beta_{7} - 2573414620 \beta_{8} + 570148160 \beta_{9} - 9727244480 \beta_{10} - 187316416 \beta_{11} + 50850400 \beta_{12} - 2228362672 \beta_{13} + 269508336 \beta_{14} + 47900332 \beta_{15} + 112381280 \beta_{16} + 80192716 \beta_{17} - 459413744 \beta_{18} + 50008808 \beta_{19} ) q^{96} \) \( + ( 410667441663984268 - 224474218846963702 \beta_{1} - 90988145577604 \beta_{2} + 26422466122861 \beta_{3} + 1678605867771 \beta_{4} + 221960516424 \beta_{5} + 936692462631 \beta_{6} - 447955047666 \beta_{7} - 54713217978 \beta_{8} - 2394619687 \beta_{9} - 2777709046 \beta_{10} - 216790242 \beta_{11} - 365682044 \beta_{12} + 151759242 \beta_{13} + 262374112 \beta_{14} + 459623997 \beta_{15} + 102190891 \beta_{16} - 88161962 \beta_{17} - 151759242 \beta_{18} + 151759242 \beta_{19} ) q^{97} \) \( + ( 6856374228987556934 - 23992552689787265 \beta_{1} + 854246849092576 \beta_{2} - 1864474953672 \beta_{3} + 110145698264 \beta_{4} - 5557255012472 \beta_{5} + 1023723007768 \beta_{6} + 374286449776 \beta_{7} - 114807444168 \beta_{8} + 12387333808 \beta_{9} - 701669184 \beta_{10} - 4431728 \beta_{11} - 366612192 \beta_{12} - 2173536656 \beta_{13} - 311030808 \beta_{14} - 588174952 \beta_{15} + 127268912 \beta_{16} - 20580304 \beta_{17} + 62537296 \beta_{18} - 93147040 \beta_{19} ) q^{98} \) \( + ( -114749586867117899 - 382565959910796781 \beta_{1} - 290561521472193 \beta_{2} + 20033746003206 \beta_{3} + 943708351640 \beta_{4} + 1448842419783 \beta_{5} - 1510005485677 \beta_{6} + 48993522034 \beta_{7} + 106134233140 \beta_{8} + 2298724207 \beta_{9} - 2286673272 \beta_{10} + 18680248 \beta_{11} + 1038712512 \beta_{12} - 1012523224 \beta_{13} - 293536530 \beta_{14} + 524949917 \beta_{15} + 178075282 \beta_{16} + 46997212 \beta_{17} - 964584704 \beta_{18} + 18680248 \beta_{19} ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(20q \) \(\mathstrut +\mathstrut 286q^{2} \) \(\mathstrut +\mathstrut 409876q^{4} \) \(\mathstrut +\mathstrut 118236748q^{6} \) \(\mathstrut +\mathstrut 564950496q^{7} \) \(\mathstrut +\mathstrut 3649699336q^{8} \) \(\mathstrut -\mathstrut 62762119220q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(20q \) \(\mathstrut +\mathstrut 286q^{2} \) \(\mathstrut +\mathstrut 409876q^{4} \) \(\mathstrut +\mathstrut 118236748q^{6} \) \(\mathstrut +\mathstrut 564950496q^{7} \) \(\mathstrut +\mathstrut 3649699336q^{8} \) \(\mathstrut -\mathstrut 62762119220q^{9} \) \(\mathstrut -\mathstrut 51269339528q^{10} \) \(\mathstrut +\mathstrut 65316900136q^{12} \) \(\mathstrut -\mathstrut 1924309104464q^{14} \) \(\mathstrut -\mathstrut 2285680856096q^{15} \) \(\mathstrut +\mathstrut 7793508571920q^{16} \) \(\mathstrut +\mathstrut 1740714497000q^{17} \) \(\mathstrut -\mathstrut 41325344023822q^{18} \) \(\mathstrut -\mathstrut 78999934156016q^{20} \) \(\mathstrut +\mathstrut 294957216952508q^{22} \) \(\mathstrut +\mathstrut 335749622520224q^{23} \) \(\mathstrut +\mathstrut 786737698607504q^{24} \) \(\mathstrut -\mathstrut 1665114591247004q^{25} \) \(\mathstrut -\mathstrut 2772669922448408q^{26} \) \(\mathstrut -\mathstrut 4284340177681888q^{28} \) \(\mathstrut +\mathstrut 3807691030523312q^{30} \) \(\mathstrut -\mathstrut 8989263821171840q^{31} \) \(\mathstrut -\mathstrut 9127378177656544q^{32} \) \(\mathstrut +\mathstrut 5359113021386256q^{33} \) \(\mathstrut +\mathstrut 19124803780998044q^{34} \) \(\mathstrut +\mathstrut 39378114498954828q^{36} \) \(\mathstrut -\mathstrut 41860137429498580q^{38} \) \(\mathstrut +\mathstrut 121743910075345312q^{39} \) \(\mathstrut -\mathstrut 160007944946664288q^{40} \) \(\mathstrut -\mathstrut 84626460793452344q^{41} \) \(\mathstrut +\mathstrut 178614544452770272q^{42} \) \(\mathstrut -\mathstrut 27785097292304568q^{44} \) \(\mathstrut +\mathstrut 184220935820050448q^{46} \) \(\mathstrut +\mathstrut 149908985865280320q^{47} \) \(\mathstrut +\mathstrut 1509829480382701344q^{48} \) \(\mathstrut +\mathstrut 477689401234988532q^{49} \) \(\mathstrut -\mathstrut 1722157324053237098q^{50} \) \(\mathstrut -\mathstrut 1640429312330235856q^{52} \) \(\mathstrut +\mathstrut 785391046636854392q^{54} \) \(\mathstrut -\mathstrut 1936239868862269088q^{55} \) \(\mathstrut -\mathstrut 3751496032099552448q^{56} \) \(\mathstrut +\mathstrut 409355502574547024q^{57} \) \(\mathstrut +\mathstrut 3182351472522780872q^{58} \) \(\mathstrut +\mathstrut 2751731248604508192q^{60} \) \(\mathstrut -\mathstrut 13336157978726113600q^{62} \) \(\mathstrut -\mathstrut 8780950644120846560q^{63} \) \(\mathstrut -\mathstrut 9021637553716933568q^{64} \) \(\mathstrut +\mathstrut 6302857570713584320q^{65} \) \(\mathstrut +\mathstrut 5291575090585638744q^{66} \) \(\mathstrut +\mathstrut 40891724357029299816q^{68} \) \(\mathstrut +\mathstrut 18077776609913587392q^{70} \) \(\mathstrut +\mathstrut 96451291685825247456q^{71} \) \(\mathstrut +\mathstrut 49841566955976579512q^{72} \) \(\mathstrut -\mathstrut 21461290229157072184q^{73} \) \(\mathstrut -\mathstrut 42953490504627871688q^{74} \) \(\mathstrut -\mathstrut 206966483397015363224q^{76} \) \(\mathstrut +\mathstrut 163869164384378609168q^{78} \) \(\mathstrut -\mathstrut 284214874786393231680q^{79} \) \(\mathstrut -\mathstrut 49904712513208125120q^{80} \) \(\mathstrut +\mathstrut 1291808332257012164q^{81} \) \(\mathstrut +\mathstrut 112698322843931711404q^{82} \) \(\mathstrut +\mathstrut 178681601993395676224q^{84} \) \(\mathstrut -\mathstrut 523293212085532715396q^{86} \) \(\mathstrut +\mathstrut 517213147301817170208q^{87} \) \(\mathstrut -\mathstrut 124028902608265335088q^{88} \) \(\mathstrut +\mathstrut 17316387144455013640q^{89} \) \(\mathstrut +\mathstrut 171098967229297148360q^{90} \) \(\mathstrut +\mathstrut 827091233447044225632q^{92} \) \(\mathstrut -\mathstrut 165060487582514765280q^{94} \) \(\mathstrut -\mathstrut 2333725338330109674784q^{95} \) \(\mathstrut -\mathstrut 715448174391177540544q^{96} \) \(\mathstrut +\mathstrut 9560220979460741928q^{97} \) \(\mathstrut +\mathstrut 137273120368488603630q^{98} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{20}\mathstrut -\mathstrut \) \(10\) \(x^{19}\mathstrut +\mathstrut \) \(8499037025\) \(x^{18}\mathstrut -\mathstrut \) \(76491332940\) \(x^{17}\mathstrut +\mathstrut \) \(29917993634631504354\) \(x^{16}\mathstrut -\mathstrut \) \(23\!\cdots\!84\) \(x^{15}\mathstrut +\mathstrut \) \(56\!\cdots\!38\) \(x^{14}\mathstrut -\mathstrut \) \(39\!\cdots\!76\) \(x^{13}\mathstrut +\mathstrut \) \(63\!\cdots\!69\) \(x^{12}\mathstrut -\mathstrut \) \(37\!\cdots\!22\) \(x^{11}\mathstrut +\mathstrut \) \(42\!\cdots\!97\) \(x^{10}\mathstrut -\mathstrut \) \(21\!\cdots\!48\) \(x^{9}\mathstrut +\mathstrut \) \(16\!\cdots\!32\) \(x^{8}\mathstrut -\mathstrut \) \(67\!\cdots\!48\) \(x^{7}\mathstrut +\mathstrut \) \(38\!\cdots\!28\) \(x^{6}\mathstrut -\mathstrut \) \(11\!\cdots\!60\) \(x^{5}\mathstrut +\mathstrut \) \(44\!\cdots\!08\) \(x^{4}\mathstrut -\mathstrut \) \(89\!\cdots\!64\) \(x^{3}\mathstrut +\mathstrut \) \(21\!\cdots\!00\) \(x^{2}\mathstrut -\mathstrut \) \(21\!\cdots\!00\) \(x\mathstrut +\mathstrut \) \(35\!\cdots\!00\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\(29\!\cdots\!75\) \(\nu^{19}\mathstrut +\mathstrut \) \(30\!\cdots\!72\) \(\nu^{18}\mathstrut +\mathstrut \) \(22\!\cdots\!97\) \(\nu^{17}\mathstrut +\mathstrut \) \(26\!\cdots\!36\) \(\nu^{16}\mathstrut +\mathstrut \) \(72\!\cdots\!10\) \(\nu^{15}\mathstrut +\mathstrut \) \(92\!\cdots\!68\) \(\nu^{14}\mathstrut +\mathstrut \) \(11\!\cdots\!86\) \(\nu^{13}\mathstrut +\mathstrut \) \(17\!\cdots\!04\) \(\nu^{12}\mathstrut +\mathstrut \) \(94\!\cdots\!87\) \(\nu^{11}\mathstrut +\mathstrut \) \(19\!\cdots\!40\) \(\nu^{10}\mathstrut +\mathstrut \) \(34\!\cdots\!57\) \(\nu^{9}\mathstrut +\mathstrut \) \(13\!\cdots\!68\) \(\nu^{8}\mathstrut -\mathstrut \) \(96\!\cdots\!00\) \(\nu^{7}\mathstrut +\mathstrut \) \(55\!\cdots\!08\) \(\nu^{6}\mathstrut -\mathstrut \) \(43\!\cdots\!64\) \(\nu^{5}\mathstrut +\mathstrut \) \(12\!\cdots\!80\) \(\nu^{4}\mathstrut -\mathstrut \) \(10\!\cdots\!28\) \(\nu^{3}\mathstrut +\mathstrut \) \(13\!\cdots\!04\) \(\nu^{2}\mathstrut -\mathstrut \) \(70\!\cdots\!00\) \(\nu\mathstrut +\mathstrut \) \(36\!\cdots\!00\)\()/\)\(16\!\cdots\!00\)
\(\beta_{2}\)\(=\)\((\)\(29\!\cdots\!75\) \(\nu^{19}\mathstrut +\mathstrut \) \(30\!\cdots\!72\) \(\nu^{18}\mathstrut +\mathstrut \) \(22\!\cdots\!97\) \(\nu^{17}\mathstrut +\mathstrut \) \(26\!\cdots\!36\) \(\nu^{16}\mathstrut +\mathstrut \) \(72\!\cdots\!10\) \(\nu^{15}\mathstrut +\mathstrut \) \(92\!\cdots\!68\) \(\nu^{14}\mathstrut +\mathstrut \) \(11\!\cdots\!86\) \(\nu^{13}\mathstrut +\mathstrut \) \(17\!\cdots\!04\) \(\nu^{12}\mathstrut +\mathstrut \) \(94\!\cdots\!87\) \(\nu^{11}\mathstrut +\mathstrut \) \(19\!\cdots\!40\) \(\nu^{10}\mathstrut +\mathstrut \) \(34\!\cdots\!57\) \(\nu^{9}\mathstrut +\mathstrut \) \(13\!\cdots\!68\) \(\nu^{8}\mathstrut -\mathstrut \) \(96\!\cdots\!00\) \(\nu^{7}\mathstrut +\mathstrut \) \(55\!\cdots\!08\) \(\nu^{6}\mathstrut -\mathstrut \) \(43\!\cdots\!64\) \(\nu^{5}\mathstrut +\mathstrut \) \(12\!\cdots\!80\) \(\nu^{4}\mathstrut -\mathstrut \) \(10\!\cdots\!28\) \(\nu^{3}\mathstrut +\mathstrut \) \(13\!\cdots\!04\) \(\nu^{2}\mathstrut +\mathstrut \) \(14\!\cdots\!00\) \(\nu\mathstrut +\mathstrut \) \(36\!\cdots\!00\)\()/\)\(53\!\cdots\!00\)
\(\beta_{3}\)\(=\)\((\)\(14\!\cdots\!18\) \(\nu^{19}\mathstrut +\mathstrut \) \(18\!\cdots\!93\) \(\nu^{18}\mathstrut +\mathstrut \) \(12\!\cdots\!61\) \(\nu^{17}\mathstrut +\mathstrut \) \(15\!\cdots\!68\) \(\nu^{16}\mathstrut +\mathstrut \) \(44\!\cdots\!64\) \(\nu^{15}\mathstrut +\mathstrut \) \(50\!\cdots\!74\) \(\nu^{14}\mathstrut +\mathstrut \) \(82\!\cdots\!78\) \(\nu^{13}\mathstrut +\mathstrut \) \(88\!\cdots\!52\) \(\nu^{12}\mathstrut +\mathstrut \) \(90\!\cdots\!34\) \(\nu^{11}\mathstrut +\mathstrut \) \(88\!\cdots\!65\) \(\nu^{10}\mathstrut +\mathstrut \) \(59\!\cdots\!81\) \(\nu^{9}\mathstrut +\mathstrut \) \(50\!\cdots\!20\) \(\nu^{8}\mathstrut +\mathstrut \) \(22\!\cdots\!52\) \(\nu^{7}\mathstrut +\mathstrut \) \(16\!\cdots\!60\) \(\nu^{6}\mathstrut +\mathstrut \) \(47\!\cdots\!84\) \(\nu^{5}\mathstrut +\mathstrut \) \(27\!\cdots\!76\) \(\nu^{4}\mathstrut +\mathstrut \) \(45\!\cdots\!24\) \(\nu^{3}\mathstrut +\mathstrut \) \(23\!\cdots\!96\) \(\nu^{2}\mathstrut +\mathstrut \) \(12\!\cdots\!00\) \(\nu\mathstrut +\mathstrut \) \(80\!\cdots\!00\)\()/\)\(12\!\cdots\!00\)
\(\beta_{4}\)\(=\)\((\)\(62\!\cdots\!41\) \(\nu^{19}\mathstrut -\mathstrut \) \(82\!\cdots\!04\) \(\nu^{18}\mathstrut +\mathstrut \) \(53\!\cdots\!55\) \(\nu^{17}\mathstrut -\mathstrut \) \(67\!\cdots\!92\) \(\nu^{16}\mathstrut +\mathstrut \) \(19\!\cdots\!66\) \(\nu^{15}\mathstrut -\mathstrut \) \(22\!\cdots\!32\) \(\nu^{14}\mathstrut +\mathstrut \) \(37\!\cdots\!94\) \(\nu^{13}\mathstrut -\mathstrut \) \(39\!\cdots\!00\) \(\nu^{12}\mathstrut +\mathstrut \) \(43\!\cdots\!29\) \(\nu^{11}\mathstrut -\mathstrut \) \(38\!\cdots\!32\) \(\nu^{10}\mathstrut +\mathstrut \) \(30\!\cdots\!03\) \(\nu^{9}\mathstrut -\mathstrut \) \(21\!\cdots\!60\) \(\nu^{8}\mathstrut +\mathstrut \) \(13\!\cdots\!24\) \(\nu^{7}\mathstrut -\mathstrut \) \(66\!\cdots\!76\) \(\nu^{6}\mathstrut +\mathstrut \) \(31\!\cdots\!24\) \(\nu^{5}\mathstrut -\mathstrut \) \(10\!\cdots\!16\) \(\nu^{4}\mathstrut +\mathstrut \) \(34\!\cdots\!00\) \(\nu^{3}\mathstrut -\mathstrut \) \(57\!\cdots\!24\) \(\nu^{2}\mathstrut +\mathstrut \) \(99\!\cdots\!00\) \(\nu\mathstrut -\mathstrut \) \(68\!\cdots\!00\)\()/\)\(16\!\cdots\!00\)
\(\beta_{5}\)\(=\)\((\)\(26\!\cdots\!78\) \(\nu^{19}\mathstrut +\mathstrut \) \(44\!\cdots\!65\) \(\nu^{18}\mathstrut +\mathstrut \) \(22\!\cdots\!93\) \(\nu^{17}\mathstrut +\mathstrut \) \(37\!\cdots\!84\) \(\nu^{16}\mathstrut +\mathstrut \) \(78\!\cdots\!44\) \(\nu^{15}\mathstrut +\mathstrut \) \(13\!\cdots\!82\) \(\nu^{14}\mathstrut +\mathstrut \) \(14\!\cdots\!34\) \(\nu^{13}\mathstrut +\mathstrut \) \(24\!\cdots\!16\) \(\nu^{12}\mathstrut +\mathstrut \) \(15\!\cdots\!66\) \(\nu^{11}\mathstrut +\mathstrut \) \(26\!\cdots\!05\) \(\nu^{10}\mathstrut +\mathstrut \) \(10\!\cdots\!73\) \(\nu^{9}\mathstrut +\mathstrut \) \(17\!\cdots\!08\) \(\nu^{8}\mathstrut +\mathstrut \) \(39\!\cdots\!32\) \(\nu^{7}\mathstrut +\mathstrut \) \(66\!\cdots\!48\) \(\nu^{6}\mathstrut +\mathstrut \) \(81\!\cdots\!20\) \(\nu^{5}\mathstrut +\mathstrut \) \(14\!\cdots\!76\) \(\nu^{4}\mathstrut +\mathstrut \) \(78\!\cdots\!16\) \(\nu^{3}\mathstrut +\mathstrut \) \(14\!\cdots\!60\) \(\nu^{2}\mathstrut +\mathstrut \) \(21\!\cdots\!00\) \(\nu\mathstrut +\mathstrut \) \(42\!\cdots\!00\)\()/\)\(12\!\cdots\!00\)
\(\beta_{6}\)\(=\)\((\)\(19\!\cdots\!79\) \(\nu^{19}\mathstrut -\mathstrut \) \(22\!\cdots\!80\) \(\nu^{18}\mathstrut +\mathstrut \) \(16\!\cdots\!29\) \(\nu^{17}\mathstrut -\mathstrut \) \(19\!\cdots\!68\) \(\nu^{16}\mathstrut +\mathstrut \) \(56\!\cdots\!62\) \(\nu^{15}\mathstrut -\mathstrut \) \(68\!\cdots\!24\) \(\nu^{14}\mathstrut +\mathstrut \) \(10\!\cdots\!82\) \(\nu^{13}\mathstrut -\mathstrut \) \(13\!\cdots\!12\) \(\nu^{12}\mathstrut +\mathstrut \) \(11\!\cdots\!23\) \(\nu^{11}\mathstrut -\mathstrut \) \(14\!\cdots\!80\) \(\nu^{10}\mathstrut +\mathstrut \) \(70\!\cdots\!49\) \(\nu^{9}\mathstrut -\mathstrut \) \(10\!\cdots\!96\) \(\nu^{8}\mathstrut +\mathstrut \) \(25\!\cdots\!16\) \(\nu^{7}\mathstrut -\mathstrut \) \(41\!\cdots\!76\) \(\nu^{6}\mathstrut +\mathstrut \) \(49\!\cdots\!20\) \(\nu^{5}\mathstrut -\mathstrut \) \(95\!\cdots\!12\) \(\nu^{4}\mathstrut +\mathstrut \) \(42\!\cdots\!08\) \(\nu^{3}\mathstrut -\mathstrut \) \(10\!\cdots\!20\) \(\nu^{2}\mathstrut +\mathstrut \) \(83\!\cdots\!00\) \(\nu\mathstrut -\mathstrut \) \(28\!\cdots\!00\)\()/\)\(48\!\cdots\!00\)
\(\beta_{7}\)\(=\)\((\)\(99\!\cdots\!17\) \(\nu^{19}\mathstrut -\mathstrut \) \(10\!\cdots\!67\) \(\nu^{18}\mathstrut +\mathstrut \) \(84\!\cdots\!50\) \(\nu^{17}\mathstrut -\mathstrut \) \(94\!\cdots\!96\) \(\nu^{16}\mathstrut +\mathstrut \) \(29\!\cdots\!54\) \(\nu^{15}\mathstrut -\mathstrut \) \(34\!\cdots\!02\) \(\nu^{14}\mathstrut +\mathstrut \) \(54\!\cdots\!24\) \(\nu^{13}\mathstrut -\mathstrut \) \(69\!\cdots\!04\) \(\nu^{12}\mathstrut +\mathstrut \) \(59\!\cdots\!93\) \(\nu^{11}\mathstrut -\mathstrut \) \(81\!\cdots\!59\) \(\nu^{10}\mathstrut +\mathstrut \) \(39\!\cdots\!62\) \(\nu^{9}\mathstrut -\mathstrut \) \(57\!\cdots\!72\) \(\nu^{8}\mathstrut +\mathstrut \) \(14\!\cdots\!00\) \(\nu^{7}\mathstrut -\mathstrut \) \(24\!\cdots\!40\) \(\nu^{6}\mathstrut +\mathstrut \) \(30\!\cdots\!24\) \(\nu^{5}\mathstrut -\mathstrut \) \(54\!\cdots\!48\) \(\nu^{4}\mathstrut +\mathstrut \) \(29\!\cdots\!92\) \(\nu^{3}\mathstrut -\mathstrut \) \(55\!\cdots\!28\) \(\nu^{2}\mathstrut +\mathstrut \) \(78\!\cdots\!00\) \(\nu\mathstrut -\mathstrut \) \(14\!\cdots\!00\)\()/\)\(12\!\cdots\!00\)
\(\beta_{8}\)\(=\)\((\)\(31\!\cdots\!63\) \(\nu^{19}\mathstrut -\mathstrut \) \(88\!\cdots\!04\) \(\nu^{18}\mathstrut +\mathstrut \) \(26\!\cdots\!09\) \(\nu^{17}\mathstrut -\mathstrut \) \(74\!\cdots\!76\) \(\nu^{16}\mathstrut +\mathstrut \) \(93\!\cdots\!18\) \(\nu^{15}\mathstrut -\mathstrut \) \(25\!\cdots\!80\) \(\nu^{14}\mathstrut +\mathstrut \) \(17\!\cdots\!54\) \(\nu^{13}\mathstrut -\mathstrut \) \(47\!\cdots\!64\) \(\nu^{12}\mathstrut +\mathstrut \) \(19\!\cdots\!15\) \(\nu^{11}\mathstrut -\mathstrut \) \(51\!\cdots\!52\) \(\nu^{10}\mathstrut +\mathstrut \) \(12\!\cdots\!05\) \(\nu^{9}\mathstrut -\mathstrut \) \(33\!\cdots\!56\) \(\nu^{8}\mathstrut +\mathstrut \) \(47\!\cdots\!28\) \(\nu^{7}\mathstrut -\mathstrut \) \(12\!\cdots\!20\) \(\nu^{6}\mathstrut +\mathstrut \) \(97\!\cdots\!68\) \(\nu^{5}\mathstrut -\mathstrut \) \(25\!\cdots\!00\) \(\nu^{4}\mathstrut +\mathstrut \) \(93\!\cdots\!12\) \(\nu^{3}\mathstrut -\mathstrut \) \(24\!\cdots\!20\) \(\nu^{2}\mathstrut +\mathstrut \) \(26\!\cdots\!00\) \(\nu\mathstrut -\mathstrut \) \(64\!\cdots\!00\)\()/\)\(48\!\cdots\!00\)
\(\beta_{9}\)\(=\)\((\)\(10\!\cdots\!53\) \(\nu^{19}\mathstrut -\mathstrut \) \(33\!\cdots\!68\) \(\nu^{18}\mathstrut +\mathstrut \) \(90\!\cdots\!75\) \(\nu^{17}\mathstrut -\mathstrut \) \(28\!\cdots\!32\) \(\nu^{16}\mathstrut +\mathstrut \) \(31\!\cdots\!10\) \(\nu^{15}\mathstrut -\mathstrut \) \(10\!\cdots\!04\) \(\nu^{14}\mathstrut +\mathstrut \) \(59\!\cdots\!18\) \(\nu^{13}\mathstrut -\mathstrut \) \(19\!\cdots\!28\) \(\nu^{12}\mathstrut +\mathstrut \) \(64\!\cdots\!05\) \(\nu^{11}\mathstrut -\mathstrut \) \(22\!\cdots\!88\) \(\nu^{10}\mathstrut +\mathstrut \) \(42\!\cdots\!19\) \(\nu^{9}\mathstrut -\mathstrut \) \(15\!\cdots\!96\) \(\nu^{8}\mathstrut +\mathstrut \) \(16\!\cdots\!76\) \(\nu^{7}\mathstrut -\mathstrut \) \(66\!\cdots\!92\) \(\nu^{6}\mathstrut +\mathstrut \) \(33\!\cdots\!36\) \(\nu^{5}\mathstrut -\mathstrut \) \(15\!\cdots\!08\) \(\nu^{4}\mathstrut +\mathstrut \) \(32\!\cdots\!08\) \(\nu^{3}\mathstrut -\mathstrut \) \(17\!\cdots\!84\) \(\nu^{2}\mathstrut +\mathstrut \) \(87\!\cdots\!00\) \(\nu\mathstrut -\mathstrut \) \(26\!\cdots\!00\)\()/\)\(16\!\cdots\!00\)
\(\beta_{10}\)\(=\)\((\)\(40\!\cdots\!99\) \(\nu^{19}\mathstrut +\mathstrut \) \(78\!\cdots\!20\) \(\nu^{18}\mathstrut +\mathstrut \) \(33\!\cdots\!73\) \(\nu^{17}\mathstrut +\mathstrut \) \(66\!\cdots\!44\) \(\nu^{16}\mathstrut +\mathstrut \) \(11\!\cdots\!18\) \(\nu^{15}\mathstrut +\mathstrut \) \(23\!\cdots\!48\) \(\nu^{14}\mathstrut +\mathstrut \) \(21\!\cdots\!62\) \(\nu^{13}\mathstrut +\mathstrut \) \(44\!\cdots\!40\) \(\nu^{12}\mathstrut +\mathstrut \) \(23\!\cdots\!75\) \(\nu^{11}\mathstrut +\mathstrut \) \(49\!\cdots\!88\) \(\nu^{10}\mathstrut +\mathstrut \) \(14\!\cdots\!85\) \(\nu^{9}\mathstrut +\mathstrut \) \(33\!\cdots\!80\) \(\nu^{8}\mathstrut +\mathstrut \) \(55\!\cdots\!04\) \(\nu^{7}\mathstrut +\mathstrut \) \(13\!\cdots\!80\) \(\nu^{6}\mathstrut +\mathstrut \) \(11\!\cdots\!28\) \(\nu^{5}\mathstrut +\mathstrut \) \(28\!\cdots\!96\) \(\nu^{4}\mathstrut +\mathstrut \) \(10\!\cdots\!24\) \(\nu^{3}\mathstrut +\mathstrut \) \(27\!\cdots\!36\) \(\nu^{2}\mathstrut +\mathstrut \) \(30\!\cdots\!00\) \(\nu\mathstrut +\mathstrut \) \(77\!\cdots\!00\)\()/\)\(48\!\cdots\!00\)
\(\beta_{11}\)\(=\)\((\)\(24\!\cdots\!83\) \(\nu^{19}\mathstrut -\mathstrut \) \(46\!\cdots\!27\) \(\nu^{18}\mathstrut +\mathstrut \) \(21\!\cdots\!04\) \(\nu^{17}\mathstrut -\mathstrut \) \(38\!\cdots\!56\) \(\nu^{16}\mathstrut +\mathstrut \) \(76\!\cdots\!70\) \(\nu^{15}\mathstrut -\mathstrut \) \(13\!\cdots\!58\) \(\nu^{14}\mathstrut +\mathstrut \) \(14\!\cdots\!20\) \(\nu^{13}\mathstrut -\mathstrut \) \(23\!\cdots\!36\) \(\nu^{12}\mathstrut +\mathstrut \) \(16\!\cdots\!51\) \(\nu^{11}\mathstrut -\mathstrut \) \(24\!\cdots\!51\) \(\nu^{10}\mathstrut +\mathstrut \) \(11\!\cdots\!24\) \(\nu^{9}\mathstrut -\mathstrut \) \(14\!\cdots\!16\) \(\nu^{8}\mathstrut +\mathstrut \) \(46\!\cdots\!44\) \(\nu^{7}\mathstrut -\mathstrut \) \(50\!\cdots\!80\) \(\nu^{6}\mathstrut +\mathstrut \) \(10\!\cdots\!20\) \(\nu^{5}\mathstrut -\mathstrut \) \(90\!\cdots\!04\) \(\nu^{4}\mathstrut +\mathstrut \) \(99\!\cdots\!96\) \(\nu^{3}\mathstrut -\mathstrut \) \(73\!\cdots\!84\) \(\nu^{2}\mathstrut +\mathstrut \) \(25\!\cdots\!00\) \(\nu\mathstrut -\mathstrut \) \(17\!\cdots\!00\)\()/\)\(13\!\cdots\!00\)
\(\beta_{12}\)\(=\)\((\)\(16\!\cdots\!83\) \(\nu^{19}\mathstrut +\mathstrut \) \(46\!\cdots\!57\) \(\nu^{18}\mathstrut +\mathstrut \) \(14\!\cdots\!80\) \(\nu^{17}\mathstrut +\mathstrut \) \(40\!\cdots\!24\) \(\nu^{16}\mathstrut +\mathstrut \) \(50\!\cdots\!22\) \(\nu^{15}\mathstrut +\mathstrut \) \(14\!\cdots\!18\) \(\nu^{14}\mathstrut +\mathstrut \) \(95\!\cdots\!84\) \(\nu^{13}\mathstrut +\mathstrut \) \(28\!\cdots\!76\) \(\nu^{12}\mathstrut +\mathstrut \) \(10\!\cdots\!39\) \(\nu^{11}\mathstrut +\mathstrut \) \(33\!\cdots\!41\) \(\nu^{10}\mathstrut +\mathstrut \) \(70\!\cdots\!32\) \(\nu^{9}\mathstrut +\mathstrut \) \(22\!\cdots\!20\) \(\nu^{8}\mathstrut +\mathstrut \) \(27\!\cdots\!64\) \(\nu^{7}\mathstrut +\mathstrut \) \(92\!\cdots\!12\) \(\nu^{6}\mathstrut +\mathstrut \) \(57\!\cdots\!56\) \(\nu^{5}\mathstrut +\mathstrut \) \(19\!\cdots\!64\) \(\nu^{4}\mathstrut +\mathstrut \) \(55\!\cdots\!68\) \(\nu^{3}\mathstrut +\mathstrut \) \(18\!\cdots\!60\) \(\nu^{2}\mathstrut +\mathstrut \) \(14\!\cdots\!00\) \(\nu\mathstrut +\mathstrut \) \(45\!\cdots\!00\)\()/\)\(40\!\cdots\!00\)
\(\beta_{13}\)\(=\)\((\)\(17\!\cdots\!97\) \(\nu^{19}\mathstrut -\mathstrut \) \(31\!\cdots\!44\) \(\nu^{18}\mathstrut +\mathstrut \) \(14\!\cdots\!99\) \(\nu^{17}\mathstrut -\mathstrut \) \(24\!\cdots\!32\) \(\nu^{16}\mathstrut +\mathstrut \) \(51\!\cdots\!62\) \(\nu^{15}\mathstrut -\mathstrut \) \(78\!\cdots\!68\) \(\nu^{14}\mathstrut +\mathstrut \) \(99\!\cdots\!50\) \(\nu^{13}\mathstrut -\mathstrut \) \(12\!\cdots\!72\) \(\nu^{12}\mathstrut +\mathstrut \) \(11\!\cdots\!97\) \(\nu^{11}\mathstrut -\mathstrut \) \(11\!\cdots\!04\) \(\nu^{10}\mathstrut +\mathstrut \) \(74\!\cdots\!55\) \(\nu^{9}\mathstrut -\mathstrut \) \(59\!\cdots\!44\) \(\nu^{8}\mathstrut +\mathstrut \) \(29\!\cdots\!68\) \(\nu^{7}\mathstrut -\mathstrut \) \(15\!\cdots\!96\) \(\nu^{6}\mathstrut +\mathstrut \) \(64\!\cdots\!80\) \(\nu^{5}\mathstrut -\mathstrut \) \(13\!\cdots\!32\) \(\nu^{4}\mathstrut +\mathstrut \) \(65\!\cdots\!04\) \(\nu^{3}\mathstrut +\mathstrut \) \(38\!\cdots\!80\) \(\nu^{2}\mathstrut +\mathstrut \) \(18\!\cdots\!00\) \(\nu\mathstrut +\mathstrut \) \(34\!\cdots\!00\)\()/\)\(40\!\cdots\!00\)
\(\beta_{14}\)\(=\)\((\)\(54\!\cdots\!50\) \(\nu^{19}\mathstrut -\mathstrut \) \(93\!\cdots\!61\) \(\nu^{18}\mathstrut +\mathstrut \) \(45\!\cdots\!19\) \(\nu^{17}\mathstrut -\mathstrut \) \(78\!\cdots\!88\) \(\nu^{16}\mathstrut +\mathstrut \) \(15\!\cdots\!60\) \(\nu^{15}\mathstrut -\mathstrut \) \(27\!\cdots\!94\) \(\nu^{14}\mathstrut +\mathstrut \) \(29\!\cdots\!22\) \(\nu^{13}\mathstrut -\mathstrut \) \(50\!\cdots\!52\) \(\nu^{12}\mathstrut +\mathstrut \) \(31\!\cdots\!54\) \(\nu^{11}\mathstrut -\mathstrut \) \(53\!\cdots\!85\) \(\nu^{10}\mathstrut +\mathstrut \) \(19\!\cdots\!79\) \(\nu^{9}\mathstrut -\mathstrut \) \(34\!\cdots\!24\) \(\nu^{8}\mathstrut +\mathstrut \) \(71\!\cdots\!60\) \(\nu^{7}\mathstrut -\mathstrut \) \(12\!\cdots\!24\) \(\nu^{6}\mathstrut +\mathstrut \) \(13\!\cdots\!32\) \(\nu^{5}\mathstrut -\mathstrut \) \(25\!\cdots\!20\) \(\nu^{4}\mathstrut +\mathstrut \) \(11\!\cdots\!24\) \(\nu^{3}\mathstrut -\mathstrut \) \(23\!\cdots\!52\) \(\nu^{2}\mathstrut +\mathstrut \) \(29\!\cdots\!00\) \(\nu\mathstrut -\mathstrut \) \(59\!\cdots\!00\)\()/\)\(12\!\cdots\!00\)
\(\beta_{15}\)\(=\)\((\)\(-\)\(56\!\cdots\!29\) \(\nu^{19}\mathstrut -\mathstrut \) \(11\!\cdots\!48\) \(\nu^{18}\mathstrut -\mathstrut \) \(47\!\cdots\!79\) \(\nu^{17}\mathstrut -\mathstrut \) \(11\!\cdots\!44\) \(\nu^{16}\mathstrut -\mathstrut \) \(16\!\cdots\!74\) \(\nu^{15}\mathstrut -\mathstrut \) \(48\!\cdots\!88\) \(\nu^{14}\mathstrut -\mathstrut \) \(31\!\cdots\!38\) \(\nu^{13}\mathstrut -\mathstrut \) \(11\!\cdots\!88\) \(\nu^{12}\mathstrut -\mathstrut \) \(34\!\cdots\!65\) \(\nu^{11}\mathstrut -\mathstrut \) \(15\!\cdots\!12\) \(\nu^{10}\mathstrut -\mathstrut \) \(22\!\cdots\!31\) \(\nu^{9}\mathstrut -\mathstrut \) \(12\!\cdots\!16\) \(\nu^{8}\mathstrut -\mathstrut \) \(85\!\cdots\!56\) \(\nu^{7}\mathstrut -\mathstrut \) \(58\!\cdots\!52\) \(\nu^{6}\mathstrut -\mathstrut \) \(17\!\cdots\!08\) \(\nu^{5}\mathstrut -\mathstrut \) \(14\!\cdots\!16\) \(\nu^{4}\mathstrut -\mathstrut \) \(17\!\cdots\!84\) \(\nu^{3}\mathstrut -\mathstrut \) \(15\!\cdots\!72\) \(\nu^{2}\mathstrut -\mathstrut \) \(46\!\cdots\!00\) \(\nu\mathstrut -\mathstrut \) \(34\!\cdots\!00\)\()/\)\(12\!\cdots\!00\)
\(\beta_{16}\)\(=\)\((\)\(12\!\cdots\!95\) \(\nu^{19}\mathstrut +\mathstrut \) \(47\!\cdots\!12\) \(\nu^{18}\mathstrut +\mathstrut \) \(10\!\cdots\!45\) \(\nu^{17}\mathstrut +\mathstrut \) \(39\!\cdots\!28\) \(\nu^{16}\mathstrut +\mathstrut \) \(36\!\cdots\!78\) \(\nu^{15}\mathstrut +\mathstrut \) \(13\!\cdots\!28\) \(\nu^{14}\mathstrut +\mathstrut \) \(68\!\cdots\!14\) \(\nu^{13}\mathstrut +\mathstrut \) \(25\!\cdots\!80\) \(\nu^{12}\mathstrut +\mathstrut \) \(74\!\cdots\!43\) \(\nu^{11}\mathstrut +\mathstrut \) \(27\!\cdots\!68\) \(\nu^{10}\mathstrut +\mathstrut \) \(48\!\cdots\!73\) \(\nu^{9}\mathstrut +\mathstrut \) \(17\!\cdots\!28\) \(\nu^{8}\mathstrut +\mathstrut \) \(18\!\cdots\!00\) \(\nu^{7}\mathstrut +\mathstrut \) \(67\!\cdots\!92\) \(\nu^{6}\mathstrut +\mathstrut \) \(37\!\cdots\!72\) \(\nu^{5}\mathstrut +\mathstrut \) \(13\!\cdots\!32\) \(\nu^{4}\mathstrut +\mathstrut \) \(35\!\cdots\!64\) \(\nu^{3}\mathstrut +\mathstrut \) \(13\!\cdots\!48\) \(\nu^{2}\mathstrut +\mathstrut \) \(99\!\cdots\!00\) \(\nu\mathstrut +\mathstrut \) \(35\!\cdots\!00\)\()/\)\(48\!\cdots\!00\)
\(\beta_{17}\)\(=\)\((\)\(-\)\(87\!\cdots\!49\) \(\nu^{19}\mathstrut +\mathstrut \) \(15\!\cdots\!25\) \(\nu^{18}\mathstrut -\mathstrut \) \(73\!\cdots\!88\) \(\nu^{17}\mathstrut +\mathstrut \) \(12\!\cdots\!20\) \(\nu^{16}\mathstrut -\mathstrut \) \(25\!\cdots\!70\) \(\nu^{15}\mathstrut +\mathstrut \) \(45\!\cdots\!50\) \(\nu^{14}\mathstrut -\mathstrut \) \(46\!\cdots\!28\) \(\nu^{13}\mathstrut +\mathstrut \) \(87\!\cdots\!16\) \(\nu^{12}\mathstrut -\mathstrut \) \(49\!\cdots\!09\) \(\nu^{11}\mathstrut +\mathstrut \) \(98\!\cdots\!73\) \(\nu^{10}\mathstrut -\mathstrut \) \(31\!\cdots\!88\) \(\nu^{9}\mathstrut +\mathstrut \) \(66\!\cdots\!24\) \(\nu^{8}\mathstrut -\mathstrut \) \(11\!\cdots\!12\) \(\nu^{7}\mathstrut +\mathstrut \) \(26\!\cdots\!16\) \(\nu^{6}\mathstrut -\mathstrut \) \(23\!\cdots\!88\) \(\nu^{5}\mathstrut +\mathstrut \) \(58\!\cdots\!52\) \(\nu^{4}\mathstrut -\mathstrut \) \(22\!\cdots\!44\) \(\nu^{3}\mathstrut +\mathstrut \) \(58\!\cdots\!00\) \(\nu^{2}\mathstrut -\mathstrut \) \(64\!\cdots\!00\) \(\nu\mathstrut +\mathstrut \) \(16\!\cdots\!00\)\()/\)\(20\!\cdots\!00\)
\(\beta_{18}\)\(=\)\((\)\(-\)\(83\!\cdots\!59\) \(\nu^{19}\mathstrut +\mathstrut \) \(74\!\cdots\!16\) \(\nu^{18}\mathstrut -\mathstrut \) \(70\!\cdots\!13\) \(\nu^{17}\mathstrut +\mathstrut \) \(69\!\cdots\!48\) \(\nu^{16}\mathstrut -\mathstrut \) \(24\!\cdots\!58\) \(\nu^{15}\mathstrut +\mathstrut \) \(27\!\cdots\!12\) \(\nu^{14}\mathstrut -\mathstrut \) \(46\!\cdots\!42\) \(\nu^{13}\mathstrut +\mathstrut \) \(58\!\cdots\!52\) \(\nu^{12}\mathstrut -\mathstrut \) \(51\!\cdots\!19\) \(\nu^{11}\mathstrut +\mathstrut \) \(73\!\cdots\!48\) \(\nu^{10}\mathstrut -\mathstrut \) \(33\!\cdots\!97\) \(\nu^{9}\mathstrut +\mathstrut \) \(55\!\cdots\!32\) \(\nu^{8}\mathstrut -\mathstrut \) \(12\!\cdots\!80\) \(\nu^{7}\mathstrut +\mathstrut \) \(24\!\cdots\!48\) \(\nu^{6}\mathstrut -\mathstrut \) \(27\!\cdots\!32\) \(\nu^{5}\mathstrut +\mathstrut \) \(55\!\cdots\!56\) \(\nu^{4}\mathstrut -\mathstrut \) \(26\!\cdots\!12\) \(\nu^{3}\mathstrut +\mathstrut \) \(54\!\cdots\!00\) \(\nu^{2}\mathstrut -\mathstrut \) \(72\!\cdots\!00\) \(\nu\mathstrut +\mathstrut \) \(14\!\cdots\!00\)\()/\)\(16\!\cdots\!00\)
\(\beta_{19}\)\(=\)\((\)\(-\)\(32\!\cdots\!15\) \(\nu^{19}\mathstrut +\mathstrut \) \(14\!\cdots\!08\) \(\nu^{18}\mathstrut -\mathstrut \) \(27\!\cdots\!93\) \(\nu^{17}\mathstrut +\mathstrut \) \(11\!\cdots\!24\) \(\nu^{16}\mathstrut -\mathstrut \) \(94\!\cdots\!78\) \(\nu^{15}\mathstrut +\mathstrut \) \(39\!\cdots\!80\) \(\nu^{14}\mathstrut -\mathstrut \) \(17\!\cdots\!98\) \(\nu^{13}\mathstrut +\mathstrut \) \(69\!\cdots\!80\) \(\nu^{12}\mathstrut -\mathstrut \) \(19\!\cdots\!03\) \(\nu^{11}\mathstrut +\mathstrut \) \(70\!\cdots\!60\) \(\nu^{10}\mathstrut -\mathstrut \) \(12\!\cdots\!37\) \(\nu^{9}\mathstrut +\mathstrut \) \(41\!\cdots\!56\) \(\nu^{8}\mathstrut -\mathstrut \) \(48\!\cdots\!88\) \(\nu^{7}\mathstrut +\mathstrut \) \(13\!\cdots\!00\) \(\nu^{6}\mathstrut -\mathstrut \) \(10\!\cdots\!48\) \(\nu^{5}\mathstrut +\mathstrut \) \(24\!\cdots\!04\) \(\nu^{4}\mathstrut -\mathstrut \) \(98\!\cdots\!36\) \(\nu^{3}\mathstrut +\mathstrut \) \(20\!\cdots\!84\) \(\nu^{2}\mathstrut -\mathstrut \) \(27\!\cdots\!00\) \(\nu\mathstrut +\mathstrut \) \(53\!\cdots\!00\)\()/\)\(48\!\cdots\!00\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{2}\mathstrut -\mathstrut \) \(3\) \(\beta_{1}\mathstrut +\mathstrut \) \(1\)\()/4\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{9}\mathstrut +\mathstrut \) \(\beta_{7}\mathstrut -\mathstrut \) \(4\) \(\beta_{6}\mathstrut -\mathstrut \) \(\beta_{5}\mathstrut -\mathstrut \) \(467\) \(\beta_{3}\mathstrut +\mathstrut \) \(315\) \(\beta_{2}\mathstrut +\mathstrut \) \(963976\) \(\beta_{1}\mathstrut -\mathstrut \) \(13598169814\)\()/16\)
\(\nu^{3}\)\(=\)\((\)\(216\) \(\beta_{19}\mathstrut +\mathstrut \) \(128\) \(\beta_{18}\mathstrut -\mathstrut \) \(148\) \(\beta_{17}\mathstrut +\mathstrut \) \(330\) \(\beta_{16}\mathstrut +\mathstrut \) \(1961\) \(\beta_{15}\mathstrut -\mathstrut \) \(522\) \(\beta_{14}\mathstrut +\mathstrut \) \(19\) \(\beta_{13}\mathstrut +\mathstrut \) \(2112\) \(\beta_{12}\mathstrut +\mathstrut \) \(216\) \(\beta_{11}\mathstrut +\mathstrut \) \(7912\) \(\beta_{10}\mathstrut +\mathstrut \) \(3961\) \(\beta_{9}\mathstrut +\mathstrut \) \(206884\) \(\beta_{8}\mathstrut +\mathstrut \) \(99120\) \(\beta_{7}\mathstrut -\mathstrut \) \(193378\) \(\beta_{6}\mathstrut -\mathstrut \) \(7295917\) \(\beta_{5}\mathstrut +\mathstrut \) \(1830198\) \(\beta_{4}\mathstrut +\mathstrut \) \(26804868\) \(\beta_{3}\mathstrut -\mathstrut \) \(23344792543\) \(\beta_{2}\mathstrut +\mathstrut \) \(44476742832\) \(\beta_{1}\mathstrut -\mathstrut \) \(65924238814\)\()/64\)
\(\nu^{4}\)\(=\)\((\)\(6479142\) \(\beta_{19}\mathstrut -\mathstrut \) \(6477766\) \(\beta_{18}\mathstrut +\mathstrut \) \(5751370\) \(\beta_{17}\mathstrut -\mathstrut \) \(6368267\) \(\beta_{16}\mathstrut +\mathstrut \) \(262651119\) \(\beta_{15}\mathstrut -\mathstrut \) \(10662088\) \(\beta_{14}\mathstrut +\mathstrut \) \(6478354\) \(\beta_{13}\mathstrut -\mathstrut \) \(43028196\) \(\beta_{12}\mathstrut -\mathstrut \) \(857806\) \(\beta_{11}\mathstrut +\mathstrut \) \(496599398\) \(\beta_{10}\mathstrut -\mathstrut \) \(16189614486\) \(\beta_{9}\mathstrut +\mathstrut \) \(4730144506\) \(\beta_{8}\mathstrut -\mathstrut \) \(33164373351\) \(\beta_{7}\mathstrut +\mathstrut \) \(100778375901\) \(\beta_{6}\mathstrut +\mathstrut \) \(41251639933\) \(\beta_{5}\mathstrut -\mathstrut \) \(75235783083\) \(\beta_{4}\mathstrut +\mathstrut \) \(17274698961318\) \(\beta_{3}\mathstrut -\mathstrut \) \(6395130338119\) \(\beta_{2}\mathstrut -\mathstrut \) \(23656956283744542\) \(\beta_{1}\mathstrut +\mathstrut \) \(158682725982414565016\)\()/128\)
\(\nu^{5}\)\(=\)\((\)\(-\)\(4492587005148\) \(\beta_{19}\mathstrut +\mathstrut \) \(1507086129732\) \(\beta_{18}\mathstrut +\mathstrut \) \(3284020257464\) \(\beta_{17}\mathstrut -\mathstrut \) \(5808600630872\) \(\beta_{16}\mathstrut -\mathstrut \) \(29199067410659\) \(\beta_{15}\mathstrut +\mathstrut \) \(11987208480794\) \(\beta_{14}\mathstrut +\mathstrut \) \(27433450503444\) \(\beta_{13}\mathstrut -\mathstrut \) \(48509372845992\) \(\beta_{12}\mathstrut -\mathstrut \) \(4492660374628\) \(\beta_{11}\mathstrut -\mathstrut \) \(182302511629292\) \(\beta_{10}\mathstrut -\mathstrut \) \(79927459779087\) \(\beta_{9}\mathstrut -\mathstrut \) \(3900403348064352\) \(\beta_{8}\mathstrut -\mathstrut \) \(2118636161514656\) \(\beta_{7}\mathstrut +\mathstrut \) \(7256623103811795\) \(\beta_{6}\mathstrut +\mathstrut \) \(234626737990460935\) \(\beta_{5}\mathstrut -\mathstrut \) \(41726167208278614\) \(\beta_{4}\mathstrut -\mathstrut \) \(1050352791124920002\) \(\beta_{3}\mathstrut +\mathstrut \) \(321124548825789247016\) \(\beta_{2}\mathstrut +\mathstrut \) \(341309367902293326178\) \(\beta_{1}\mathstrut +\mathstrut \) \(1657606209561209759430\)\()/512\)
\(\nu^{6}\)\(=\)\((\)\(-\)\(95365066130568996\) \(\beta_{19}\mathstrut +\mathstrut \) \(95356109627928988\) \(\beta_{18}\mathstrut -\mathstrut \) \(62928788203295776\) \(\beta_{17}\mathstrut +\mathstrut \) \(74077246141512212\) \(\beta_{16}\mathstrut -\mathstrut \) \(2610418283964817269\) \(\beta_{15}\mathstrut +\mathstrut \) \(210881115452226982\) \(\beta_{14}\mathstrut -\mathstrut \) \(95269288018035340\) \(\beta_{13}\mathstrut +\mathstrut \) \(606577962203714040\) \(\beta_{12}\mathstrut -\mathstrut \) \(4101745927596284\) \(\beta_{11}\mathstrut -\mathstrut \) \(6012742351877177156\) \(\beta_{10}\mathstrut +\mathstrut \) \(125693220653236904883\) \(\beta_{9}\mathstrut -\mathstrut \) \(59536240922624242912\) \(\beta_{8}\mathstrut +\mathstrut \) \(351709962607126824012\) \(\beta_{7}\mathstrut -\mathstrut \) \(947599307238065851953\) \(\beta_{6}\mathstrut -\mathstrut \) \(227381960675654798497\) \(\beta_{5}\mathstrut +\mathstrut \) \(1214866059771963727194\) \(\beta_{4}\mathstrut -\mathstrut \) \(163357266091112674604868\) \(\beta_{3}\mathstrut +\mathstrut \) \(59698742176933679054768\) \(\beta_{2}\mathstrut +\mathstrut \) \(221149293912473436122153412\) \(\beta_{1}\mathstrut -\mathstrut \) \(1090455952173526959235286924690\)\()/512\)
\(\nu^{7}\)\(=\)\((\)\(2435553630144813656523\) \(\beta_{19}\mathstrut -\mathstrut \) \(1957309372521586870545\) \(\beta_{18}\mathstrut -\mathstrut \) \(1813433528662869506972\) \(\beta_{17}\mathstrut +\mathstrut \) \(2949400828896842881202\) \(\beta_{16}\mathstrut +\mathstrut \) \(11273220439378086734501\) \(\beta_{15}\mathstrut -\mathstrut \) \(6944198305747334739356\) \(\beta_{14}\mathstrut -\mathstrut \) \(24515649206030256690273\) \(\beta_{13}\mathstrut +\mathstrut \) \(27505414167980766704706\) \(\beta_{12}\mathstrut +\mathstrut \) \(2435633485550055455941\) \(\beta_{11}\mathstrut +\mathstrut \) \(102150013817160779151935\) \(\beta_{10}\mathstrut +\mathstrut \) \(45459035407073459934187\) \(\beta_{9}\mathstrut +\mathstrut \) \(2054021489955985114100652\) \(\beta_{8}\mathstrut +\mathstrut \) \(1205718074019989973300678\) \(\beta_{7}\mathstrut -\mathstrut \) \(5644425215250787629557083\) \(\beta_{6}\mathstrut -\mathstrut \) \(151859486690198866299628535\) \(\beta_{5}\mathstrut +\mathstrut \) \(24252002727015900485511729\) \(\beta_{4}\mathstrut +\mathstrut \) \(761500323492340543949591457\) \(\beta_{3}\mathstrut -\mathstrut \) \(148483743019678514992185000252\) \(\beta_{2}\mathstrut -\mathstrut \) \(682935346161481623593012015466\) \(\beta_{1}\mathstrut -\mathstrut \) \(1144548908064795109068575933498\)\()/128\)
\(\nu^{8}\)\(=\)\((\)\(12\!\cdots\!16\) \(\beta_{19}\mathstrut -\mathstrut \) \(12\!\cdots\!52\) \(\beta_{18}\mathstrut +\mathstrut \) \(72\!\cdots\!84\) \(\beta_{17}\mathstrut -\mathstrut \) \(86\!\cdots\!02\) \(\beta_{16}\mathstrut +\mathstrut \) \(27\!\cdots\!63\) \(\beta_{15}\mathstrut -\mathstrut \) \(31\!\cdots\!90\) \(\beta_{14}\mathstrut +\mathstrut \) \(12\!\cdots\!40\) \(\beta_{13}\mathstrut -\mathstrut \) \(77\!\cdots\!88\) \(\beta_{12}\mathstrut +\mathstrut \) \(16\!\cdots\!52\) \(\beta_{11}\mathstrut +\mathstrut \) \(72\!\cdots\!72\) \(\beta_{10}\mathstrut -\mathstrut \) \(12\!\cdots\!47\) \(\beta_{9}\mathstrut +\mathstrut \) \(72\!\cdots\!12\) \(\beta_{8}\mathstrut -\mathstrut \) \(38\!\cdots\!02\) \(\beta_{7}\mathstrut +\mathstrut \) \(10\!\cdots\!55\) \(\beta_{6}\mathstrut +\mathstrut \) \(70\!\cdots\!63\) \(\beta_{5}\mathstrut -\mathstrut \) \(16\!\cdots\!96\) \(\beta_{4}\mathstrut +\mathstrut \) \(17\!\cdots\!96\) \(\beta_{3}\mathstrut -\mathstrut \) \(66\!\cdots\!26\) \(\beta_{2}\mathstrut -\mathstrut \) \(23\!\cdots\!08\) \(\beta_{1}\mathstrut +\mathstrut \) \(10\!\cdots\!62\)\()/256\)
\(\nu^{9}\)\(=\)\((\)\(-\)\(20\!\cdots\!12\) \(\beta_{19}\mathstrut +\mathstrut \) \(22\!\cdots\!24\) \(\beta_{18}\mathstrut +\mathstrut \) \(14\!\cdots\!68\) \(\beta_{17}\mathstrut -\mathstrut \) \(24\!\cdots\!68\) \(\beta_{16}\mathstrut -\mathstrut \) \(71\!\cdots\!45\) \(\beta_{15}\mathstrut +\mathstrut \) \(60\!\cdots\!26\) \(\beta_{14}\mathstrut +\mathstrut \) \(26\!\cdots\!36\) \(\beta_{13}\mathstrut -\mathstrut \) \(23\!\cdots\!52\) \(\beta_{12}\mathstrut -\mathstrut \) \(20\!\cdots\!36\) \(\beta_{11}\mathstrut -\mathstrut \) \(84\!\cdots\!20\) \(\beta_{10}\mathstrut -\mathstrut \) \(39\!\cdots\!09\) \(\beta_{9}\mathstrut -\mathstrut \) \(16\!\cdots\!28\) \(\beta_{8}\mathstrut -\mathstrut \) \(10\!\cdots\!00\) \(\beta_{7}\mathstrut +\mathstrut \) \(57\!\cdots\!57\) \(\beta_{6}\mathstrut +\mathstrut \) \(13\!\cdots\!65\) \(\beta_{5}\mathstrut -\mathstrut \) \(21\!\cdots\!86\) \(\beta_{4}\mathstrut -\mathstrut \) \(72\!\cdots\!34\) \(\beta_{3}\mathstrut +\mathstrut \) \(11\!\cdots\!52\) \(\beta_{2}\mathstrut +\mathstrut \) \(87\!\cdots\!30\) \(\beta_{1}\mathstrut +\mathstrut \) \(11\!\cdots\!94\)\()/512\)
\(\nu^{10}\)\(=\)\((\)\(-\)\(55\!\cdots\!64\) \(\beta_{19}\mathstrut +\mathstrut \) \(55\!\cdots\!88\) \(\beta_{18}\mathstrut -\mathstrut \) \(31\!\cdots\!48\) \(\beta_{17}\mathstrut +\mathstrut \) \(37\!\cdots\!68\) \(\beta_{16}\mathstrut -\mathstrut \) \(10\!\cdots\!55\) \(\beta_{15}\mathstrut +\mathstrut \) \(15\!\cdots\!46\) \(\beta_{14}\mathstrut -\mathstrut \) \(55\!\cdots\!32\) \(\beta_{13}\mathstrut +\mathstrut \) \(35\!\cdots\!28\) \(\beta_{12}\mathstrut -\mathstrut \) \(97\!\cdots\!44\) \(\beta_{11}\mathstrut -\mathstrut \) \(32\!\cdots\!76\) \(\beta_{10}\mathstrut +\mathstrut \) \(48\!\cdots\!45\) \(\beta_{9}\mathstrut -\mathstrut \) \(32\!\cdots\!16\) \(\beta_{8}\mathstrut +\mathstrut \) \(15\!\cdots\!64\) \(\beta_{7}\mathstrut -\mathstrut \) \(42\!\cdots\!83\) \(\beta_{6}\mathstrut +\mathstrut \) \(28\!\cdots\!89\) \(\beta_{5}\mathstrut +\mathstrut \) \(82\!\cdots\!26\) \(\beta_{4}\mathstrut -\mathstrut \) \(70\!\cdots\!92\) \(\beta_{3}\mathstrut +\mathstrut \) \(28\!\cdots\!16\) \(\beta_{2}\mathstrut +\mathstrut \) \(10\!\cdots\!56\) \(\beta_{1}\mathstrut -\mathstrut \) \(38\!\cdots\!78\)\()/512\)
\(\nu^{11}\)\(=\)\((\)\(20\!\cdots\!58\) \(\beta_{19}\mathstrut -\mathstrut \) \(28\!\cdots\!74\) \(\beta_{18}\mathstrut -\mathstrut \) \(14\!\cdots\!32\) \(\beta_{17}\mathstrut +\mathstrut \) \(24\!\cdots\!52\) \(\beta_{16}\mathstrut +\mathstrut \) \(62\!\cdots\!77\) \(\beta_{15}\mathstrut -\mathstrut \) \(63\!\cdots\!62\) \(\beta_{14}\mathstrut -\mathstrut \) \(30\!\cdots\!30\) \(\beta_{13}\mathstrut +\mathstrut \) \(24\!\cdots\!04\) \(\beta_{12}\mathstrut +\mathstrut \) \(20\!\cdots\!34\) \(\beta_{11}\mathstrut +\mathstrut \) \(84\!\cdots\!14\) \(\beta_{10}\mathstrut +\mathstrut \) \(41\!\cdots\!33\) \(\beta_{9}\mathstrut +\mathstrut \) \(17\!\cdots\!36\) \(\beta_{8}\mathstrut +\mathstrut \) \(10\!\cdots\!40\) \(\beta_{7}\mathstrut -\mathstrut \) \(68\!\cdots\!93\) \(\beta_{6}\mathstrut -\mathstrut \) \(14\!\cdots\!97\) \(\beta_{5}\mathstrut +\mathstrut \) \(22\!\cdots\!72\) \(\beta_{4}\mathstrut +\mathstrut \) \(79\!\cdots\!80\) \(\beta_{3}\mathstrut -\mathstrut \) \(11\!\cdots\!36\) \(\beta_{2}\mathstrut -\mathstrut \) \(11\!\cdots\!30\) \(\beta_{1}\mathstrut -\mathstrut \) \(13\!\cdots\!78\)\()/256\)
\(\nu^{12}\)\(=\)\((\)\(29\!\cdots\!82\) \(\beta_{19}\mathstrut -\mathstrut \) \(29\!\cdots\!70\) \(\beta_{18}\mathstrut +\mathstrut \) \(17\!\cdots\!66\) \(\beta_{17}\mathstrut -\mathstrut \) \(19\!\cdots\!51\) \(\beta_{16}\mathstrut +\mathstrut \) \(54\!\cdots\!55\) \(\beta_{15}\mathstrut -\mathstrut \) \(92\!\cdots\!44\) \(\beta_{14}\mathstrut +\mathstrut \) \(29\!\cdots\!18\) \(\beta_{13}\mathstrut -\mathstrut \) \(18\!\cdots\!84\) \(\beta_{12}\mathstrut +\mathstrut \) \(57\!\cdots\!06\) \(\beta_{11}\mathstrut +\mathstrut \) \(17\!\cdots\!74\) \(\beta_{10}\mathstrut -\mathstrut \) \(24\!\cdots\!06\) \(\beta_{9}\mathstrut +\mathstrut \) \(17\!\cdots\!46\) \(\beta_{8}\mathstrut -\mathstrut \) \(78\!\cdots\!79\) \(\beta_{7}\mathstrut +\mathstrut \) \(22\!\cdots\!49\) \(\beta_{6}\mathstrut -\mathstrut \) \(36\!\cdots\!27\) \(\beta_{5}\mathstrut -\mathstrut \) \(47\!\cdots\!11\) \(\beta_{4}\mathstrut +\mathstrut \) \(35\!\cdots\!34\) \(\beta_{3}\mathstrut -\mathstrut \) \(14\!\cdots\!23\) \(\beta_{2}\mathstrut -\mathstrut \) \(52\!\cdots\!06\) \(\beta_{1}\mathstrut +\mathstrut \) \(18\!\cdots\!64\)\()/128\)
\(\nu^{13}\)\(=\)\((\)\(-\)\(83\!\cdots\!96\) \(\beta_{19}\mathstrut +\mathstrut \) \(13\!\cdots\!36\) \(\beta_{18}\mathstrut +\mathstrut \) \(59\!\cdots\!44\) \(\beta_{17}\mathstrut -\mathstrut \) \(10\!\cdots\!80\) \(\beta_{16}\mathstrut -\mathstrut \) \(24\!\cdots\!71\) \(\beta_{15}\mathstrut +\mathstrut \) \(26\!\cdots\!22\) \(\beta_{14}\mathstrut +\mathstrut \) \(13\!\cdots\!16\) \(\beta_{13}\mathstrut -\mathstrut \) \(10\!\cdots\!64\) \(\beta_{12}\mathstrut -\mathstrut \) \(83\!\cdots\!68\) \(\beta_{11}\mathstrut -\mathstrut \) \(33\!\cdots\!32\) \(\beta_{10}\mathstrut -\mathstrut \) \(17\!\cdots\!43\) \(\beta_{9}\mathstrut -\mathstrut \) \(71\!\cdots\!36\) \(\beta_{8}\mathstrut -\mathstrut \) \(44\!\cdots\!20\) \(\beta_{7}\mathstrut +\mathstrut \) \(30\!\cdots\!11\) \(\beta_{6}\mathstrut +\mathstrut \) \(61\!\cdots\!39\) \(\beta_{5}\mathstrut -\mathstrut \) \(91\!\cdots\!70\) \(\beta_{4}\mathstrut -\mathstrut \) \(33\!\cdots\!50\) \(\beta_{3}\mathstrut +\mathstrut \) \(44\!\cdots\!36\) \(\beta_{2}\mathstrut +\mathstrut \) \(52\!\cdots\!50\) \(\beta_{1}\mathstrut +\mathstrut \) \(64\!\cdots\!30\)\()/512\)
\(\nu^{14}\)\(=\)\((\)\(-\)\(23\!\cdots\!64\) \(\beta_{19}\mathstrut +\mathstrut \) \(23\!\cdots\!44\) \(\beta_{18}\mathstrut -\mathstrut \) \(14\!\cdots\!84\) \(\beta_{17}\mathstrut +\mathstrut \) \(16\!\cdots\!36\) \(\beta_{16}\mathstrut -\mathstrut \) \(43\!\cdots\!85\) \(\beta_{15}\mathstrut +\mathstrut \) \(82\!\cdots\!30\) \(\beta_{14}\mathstrut -\mathstrut \) \(23\!\cdots\!96\) \(\beta_{13}\mathstrut +\mathstrut \) \(15\!\cdots\!20\) \(\beta_{12}\mathstrut -\mathstrut \) \(48\!\cdots\!24\) \(\beta_{11}\mathstrut -\mathstrut \) \(14\!\cdots\!84\) \(\beta_{10}\mathstrut +\mathstrut \) \(19\!\cdots\!27\) \(\beta_{9}\mathstrut -\mathstrut \) \(15\!\cdots\!88\) \(\beta_{8}\mathstrut +\mathstrut \) \(62\!\cdots\!48\) \(\beta_{7}\mathstrut -\mathstrut \) \(18\!\cdots\!73\) \(\beta_{6}\mathstrut +\mathstrut \) \(42\!\cdots\!07\) \(\beta_{5}\mathstrut +\mathstrut \) \(41\!\cdots\!66\) \(\beta_{4}\mathstrut -\mathstrut \) \(29\!\cdots\!56\) \(\beta_{3}\mathstrut +\mathstrut \) \(12\!\cdots\!32\) \(\beta_{2}\mathstrut +\mathstrut \) \(42\!\cdots\!00\) \(\beta_{1}\mathstrut -\mathstrut \) \(15\!\cdots\!62\)\()/512\)
\(\nu^{15}\)\(=\)\((\)\(42\!\cdots\!79\) \(\beta_{19}\mathstrut -\mathstrut \) \(73\!\cdots\!01\) \(\beta_{18}\mathstrut -\mathstrut \) \(29\!\cdots\!36\) \(\beta_{17}\mathstrut +\mathstrut \) \(51\!\cdots\!54\) \(\beta_{16}\mathstrut +\mathstrut \) \(13\!\cdots\!20\) \(\beta_{15}\mathstrut -\mathstrut \) \(13\!\cdots\!22\) \(\beta_{14}\mathstrut -\mathstrut \) \(75\!\cdots\!09\) \(\beta_{13}\mathstrut +\mathstrut \) \(51\!\cdots\!50\) \(\beta_{12}\mathstrut +\mathstrut \) \(42\!\cdots\!49\) \(\beta_{11}\mathstrut +\mathstrut \) \(16\!\cdots\!67\) \(\beta_{10}\mathstrut +\mathstrut \) \(87\!\cdots\!02\) \(\beta_{9}\mathstrut +\mathstrut \) \(36\!\cdots\!36\) \(\beta_{8}\mathstrut +\mathstrut \) \(22\!\cdots\!74\) \(\beta_{7}\mathstrut -\mathstrut \) \(16\!\cdots\!06\) \(\beta_{6}\mathstrut -\mathstrut \) \(31\!\cdots\!94\) \(\beta_{5}\mathstrut +\mathstrut \) \(46\!\cdots\!51\) \(\beta_{4}\mathstrut +\mathstrut \) \(16\!\cdots\!07\) \(\beta_{3}\mathstrut -\mathstrut \) \(22\!\cdots\!08\) \(\beta_{2}\mathstrut -\mathstrut \) \(29\!\cdots\!68\) \(\beta_{1}\mathstrut -\mathstrut \) \(36\!\cdots\!92\)\()/128\)
\(\nu^{16}\)\(=\)\((\)\(23\!\cdots\!64\) \(\beta_{19}\mathstrut -\mathstrut \) \(23\!\cdots\!44\) \(\beta_{18}\mathstrut +\mathstrut \) \(15\!\cdots\!40\) \(\beta_{17}\mathstrut -\mathstrut \) \(17\!\cdots\!82\) \(\beta_{16}\mathstrut +\mathstrut \) \(44\!\cdots\!93\) \(\beta_{15}\mathstrut -\mathstrut \) \(89\!\cdots\!90\) \(\beta_{14}\mathstrut +\mathstrut \) \(23\!\cdots\!92\) \(\beta_{13}\mathstrut -\mathstrut \) \(15\!\cdots\!84\) \(\beta_{12}\mathstrut +\mathstrut \) \(48\!\cdots\!64\) \(\beta_{11}\mathstrut +\mathstrut \) \(14\!\cdots\!64\) \(\beta_{10}\mathstrut -\mathstrut \) \(19\!\cdots\!05\) \(\beta_{9}\mathstrut +\mathstrut \) \(15\!\cdots\!20\) \(\beta_{8}\mathstrut -\mathstrut \) \(61\!\cdots\!30\) \(\beta_{7}\mathstrut +\mathstrut \) \(18\!\cdots\!93\) \(\beta_{6}\mathstrut -\mathstrut \) \(52\!\cdots\!59\) \(\beta_{5}\mathstrut -\mathstrut \) \(45\!\cdots\!72\) \(\beta_{4}\mathstrut +\mathstrut \) \(29\!\cdots\!20\) \(\beta_{3}\mathstrut -\mathstrut \) \(12\!\cdots\!42\) \(\beta_{2}\mathstrut -\mathstrut \) \(43\!\cdots\!28\) \(\beta_{1}\mathstrut +\mathstrut \) \(15\!\cdots\!18\)\()/256\)
\(\nu^{17}\)\(=\)\((\)\(-\)\(35\!\cdots\!92\) \(\beta_{19}\mathstrut +\mathstrut \) \(64\!\cdots\!72\) \(\beta_{18}\mathstrut +\mathstrut \) \(23\!\cdots\!48\) \(\beta_{17}\mathstrut -\mathstrut \) \(42\!\cdots\!32\) \(\beta_{16}\mathstrut -\mathstrut \) \(11\!\cdots\!97\) \(\beta_{15}\mathstrut +\mathstrut \) \(10\!\cdots\!06\) \(\beta_{14}\mathstrut +\mathstrut \) \(64\!\cdots\!40\) \(\beta_{13}\mathstrut -\mathstrut \) \(42\!\cdots\!16\) \(\beta_{12}\mathstrut -\mathstrut \) \(35\!\cdots\!76\) \(\beta_{11}\mathstrut -\mathstrut \) \(13\!\cdots\!00\) \(\beta_{10}\mathstrut -\mathstrut \) \(70\!\cdots\!33\) \(\beta_{9}\mathstrut -\mathstrut \) \(29\!\cdots\!12\) \(\beta_{8}\mathstrut -\mathstrut \) \(18\!\cdots\!32\) \(\beta_{7}\mathstrut +\mathstrut \) \(14\!\cdots\!21\) \(\beta_{6}\mathstrut +\mathstrut \) \(25\!\cdots\!57\) \(\beta_{5}\mathstrut -\mathstrut \) \(37\!\cdots\!34\) \(\beta_{4}\mathstrut -\mathstrut \) \(13\!\cdots\!94\) \(\beta_{3}\mathstrut +\mathstrut \) \(18\!\cdots\!44\) \(\beta_{2}\mathstrut +\mathstrut \) \(26\!\cdots\!78\) \(\beta_{1}\mathstrut +\mathstrut \) \(33\!\cdots\!50\)\()/512\)
\(\nu^{18}\)\(=\)\((\)\(-\)\(94\!\cdots\!56\) \(\beta_{19}\mathstrut +\mathstrut \) \(94\!\cdots\!16\) \(\beta_{18}\mathstrut -\mathstrut \) \(64\!\cdots\!72\) \(\beta_{17}\mathstrut +\mathstrut \) \(71\!\cdots\!56\) \(\beta_{16}\mathstrut -\mathstrut \) \(18\!\cdots\!51\) \(\beta_{15}\mathstrut +\mathstrut \) \(38\!\cdots\!34\) \(\beta_{14}\mathstrut -\mathstrut \) \(93\!\cdots\!80\) \(\beta_{13}\mathstrut +\mathstrut \) \(61\!\cdots\!32\) \(\beta_{12}\mathstrut -\mathstrut \) \(18\!\cdots\!32\) \(\beta_{11}\mathstrut -\mathstrut \) \(61\!\cdots\!44\) \(\beta_{10}\mathstrut +\mathstrut \) \(80\!\cdots\!33\) \(\beta_{9}\mathstrut -\mathstrut \) \(65\!\cdots\!44\) \(\beta_{8}\mathstrut +\mathstrut \) \(24\!\cdots\!44\) \(\beta_{7}\mathstrut -\mathstrut \) \(73\!\cdots\!71\) \(\beta_{6}\mathstrut +\mathstrut \) \(24\!\cdots\!41\) \(\beta_{5}\mathstrut +\mathstrut \) \(19\!\cdots\!82\) \(\beta_{4}\mathstrut -\mathstrut \) \(12\!\cdots\!56\) \(\beta_{3}\mathstrut +\mathstrut \) \(49\!\cdots\!08\) \(\beta_{2}\mathstrut +\mathstrut \) \(17\!\cdots\!00\) \(\beta_{1}\mathstrut -\mathstrut \) \(60\!\cdots\!58\)\()/512\)
\(\nu^{19}\)\(=\)\((\)\(36\!\cdots\!66\) \(\beta_{19}\mathstrut -\mathstrut \) \(70\!\cdots\!06\) \(\beta_{18}\mathstrut -\mathstrut \) \(23\!\cdots\!16\) \(\beta_{17}\mathstrut +\mathstrut \) \(43\!\cdots\!28\) \(\beta_{16}\mathstrut +\mathstrut \) \(13\!\cdots\!79\) \(\beta_{15}\mathstrut -\mathstrut \) \(10\!\cdots\!42\) \(\beta_{14}\mathstrut -\mathstrut \) \(68\!\cdots\!38\) \(\beta_{13}\mathstrut +\mathstrut \) \(44\!\cdots\!28\) \(\beta_{12}\mathstrut +\mathstrut \) \(36\!\cdots\!86\) \(\beta_{11}\mathstrut +\mathstrut \) \(13\!\cdots\!86\) \(\beta_{10}\mathstrut +\mathstrut \) \(70\!\cdots\!43\) \(\beta_{9}\mathstrut +\mathstrut \) \(30\!\cdots\!40\) \(\beta_{8}\mathstrut +\mathstrut \) \(18\!\cdots\!08\) \(\beta_{7}\mathstrut -\mathstrut \) \(15\!\cdots\!67\) \(\beta_{6}\mathstrut -\mathstrut \) \(25\!\cdots\!71\) \(\beta_{5}\mathstrut +\mathstrut \) \(37\!\cdots\!08\) \(\beta_{4}\mathstrut +\mathstrut \) \(13\!\cdots\!04\) \(\beta_{3}\mathstrut -\mathstrut \) \(18\!\cdots\!80\) \(\beta_{2}\mathstrut -\mathstrut \) \(28\!\cdots\!86\) \(\beta_{1}\mathstrut -\mathstrut \) \(37\!\cdots\!90\)\()/256\)

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 + 19382.6i
0.500000 19382.6i
0.500000 32835.5i
0.500000 + 32835.5i
0.500000 + 21396.5i
0.500000 21396.5i
0.500000 6359.74i
0.500000 + 6359.74i
0.500000 37074.8i
0.500000 + 37074.8i
0.500000 + 45387.1i
0.500000 45387.1i
0.500000 6610.70i
0.500000 + 6610.70i
0.500000 44665.4i
0.500000 + 44665.4i
0.500000 + 15078.6i
0.500000 15078.6i
0.500000 + 29091.7i
0.500000 29091.7i
−1394.30 391.258i 77530.5i 1.79099e6 + 1.09106e6i 2.27953e7i −3.03344e7 + 1.08101e8i 1.10984e9 −2.07029e9 2.22200e9i 4.44938e9 −8.91883e9 + 3.17834e10i
5.2 −1394.30 + 391.258i 77530.5i 1.79099e6 1.09106e6i 2.27953e7i −3.03344e7 1.08101e8i 1.10984e9 −2.07029e9 + 2.22200e9i 4.44938e9 −8.91883e9 3.17834e10i
5.3 −1358.45 501.755i 131342.i 1.59363e6 + 1.36322e6i 4.61121e6i 6.59016e7 1.78422e8i −9.67293e8 −1.48087e9 2.65149e9i −6.79041e9 −2.31370e9 + 6.26411e9i
5.4 −1358.45 + 501.755i 131342.i 1.59363e6 1.36322e6i 4.61121e6i 6.59016e7 + 1.78422e8i −9.67293e8 −1.48087e9 + 2.65149e9i −6.79041e9 −2.31370e9 6.26411e9i
5.5 −962.766 1081.77i 85585.8i −243314. + 2.08299e6i 2.31314e7i −9.25844e7 + 8.23991e7i −3.19239e8 2.48758e9 1.74222e9i 3.13542e9 2.50229e10 2.22701e10i
5.6 −962.766 + 1081.77i 85585.8i −243314. 2.08299e6i 2.31314e7i −9.25844e7 8.23991e7i −3.19239e8 2.48758e9 + 1.74222e9i 3.13542e9 2.50229e10 + 2.22701e10i
5.7 −476.143 1367.64i 25439.0i −1.64373e6 + 1.30238e6i 3.81460e7i 3.47914e7 1.21126e7i 7.30446e7 2.56384e9 + 1.62791e9i 9.81321e9 −5.21701e10 + 1.81630e10i
5.8 −476.143 + 1367.64i 25439.0i −1.64373e6 1.30238e6i 3.81460e7i 3.47914e7 + 1.21126e7i 7.30446e7 2.56384e9 1.62791e9i 9.81321e9 −5.21701e10 1.81630e10i
5.9 −377.837 1398.00i 148299.i −1.81163e6 + 1.05643e6i 2.51716e7i 2.07321e8 5.60329e7i 8.96078e8 2.16139e9 + 2.13349e9i −1.15323e10 3.51898e10 9.51079e9i
5.10 −377.837 + 1398.00i 148299.i −1.81163e6 1.05643e6i 2.51716e7i 2.07321e8 + 5.60329e7i 8.96078e8 2.16139e9 2.13349e9i −1.15323e10 3.51898e10 + 9.51079e9i
5.11 228.196 1430.06i 181549.i −1.99301e6 652669.i 6.89272e6i −2.59626e8 4.14287e7i 1.45891e8 −1.38815e9 + 2.70119e9i −2.24995e10 −9.85702e9 1.57289e9i
5.12 228.196 + 1430.06i 181549.i −1.99301e6 + 652669.i 6.89272e6i −2.59626e8 + 4.14287e7i 1.45891e8 −1.38815e9 2.70119e9i −2.24995e10 −9.85702e9 + 1.57289e9i
5.13 582.905 1325.66i 26442.8i −1.41760e6 1.54547e6i 7.17621e6i 3.50542e7 + 1.54136e7i −5.46394e8 −2.87509e9 + 9.78391e8i 9.76113e9 9.51321e9 + 4.18305e9i
5.14 582.905 + 1325.66i 26442.8i −1.41760e6 + 1.54547e6i 7.17621e6i 3.50542e7 1.54136e7i −5.46394e8 −2.87509e9 9.78391e8i 9.76113e9 9.51321e9 4.18305e9i
5.15 1174.53 847.130i 178662.i 661894. 1.98996e6i 3.12530e7i 1.51350e8 + 2.09844e8i 2.98846e8 −9.08340e8 2.89798e9i −2.14596e10 −2.64753e10 3.67076e10i
5.16 1174.53 + 847.130i 178662.i 661894. + 1.98996e6i 3.12530e7i 1.51350e8 2.09844e8i 2.98846e8 −9.08340e8 + 2.89798e9i −2.14596e10 −2.64753e10 + 3.67076e10i
5.17 1282.51 672.555i 60314.6i 1.19249e6 1.72511e6i 1.19399e7i −4.05649e7 7.73538e7i 8.57498e8 3.69144e8 3.01448e9i 6.82251e9 8.03026e9 + 1.53130e10i
5.18 1282.51 + 672.555i 60314.6i 1.19249e6 + 1.72511e6i 1.19399e7i −4.05649e7 + 7.73538e7i 8.57498e8 3.69144e8 + 3.01448e9i 6.82251e9 8.03026e9 1.53130e10i
5.19 1444.36 104.758i 116367.i 2.07520e6 302616.i 3.48989e7i −1.21903e7 1.68076e8i −1.26580e9 2.96564e9 6.54481e8i −3.08088e9 −3.65593e9 5.04066e10i
5.20 1444.36 + 104.758i 116367.i 2.07520e6 + 302616.i 3.48989e7i −1.21903e7 + 1.68076e8i −1.26580e9 2.96564e9 + 6.54481e8i −3.08088e9 −3.65593e9 + 5.04066e10i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 5.20
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 \(S_{22}^{\mathrm{new}}(8, [\chi])\).