# Properties

 Label 20.11.b.a Level 20 Weight 11 Character orbit 20.b Analytic conductor 12.707 Analytic rank 0 Dimension 20 CM No Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ = $$20 = 2^{2} \cdot 5$$ Weight: $$k$$ = $$11$$ Character orbit: $$[\chi]$$ = 20.b (of order $$2$$ and degree $$1$$)

## Newform invariants

 Self dual: No Analytic conductor: $$12.7071450535$$ Analytic rank: $$0$$ Dimension: $$20$$ Coefficient field: $$\mathbb{Q}[x]/(x^{20} - \cdots)$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{94}\cdot 3^{4}\cdot 5^{29}$$ 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.

 $$f(q)$$ $$=$$ $$q$$ $$+ ( 1 - \beta_{1} ) q^{2}$$ $$+ ( -\beta_{1} + \beta_{2} ) q^{3}$$ $$+ ( -32 - \beta_{1} - \beta_{2} + \beta_{3} ) q^{4}$$ $$+ ( 2 \beta_{1} + \beta_{4} ) q^{5}$$ $$+ ( -739 - \beta_{1} + 3 \beta_{2} + \beta_{3} + \beta_{4} - \beta_{8} ) q^{6}$$ $$+ ( -10 - 109 \beta_{1} - 4 \beta_{2} + 4 \beta_{3} - \beta_{6} ) q^{7}$$ $$+ ( 177 + 27 \beta_{1} + 13 \beta_{2} + 2 \beta_{3} - 3 \beta_{4} - \beta_{6} + 2 \beta_{7} + 2 \beta_{8} + \beta_{15} ) q^{8}$$ $$+ ( -20721 + 148 \beta_{1} - \beta_{2} + 17 \beta_{3} - 3 \beta_{4} - \beta_{7} - \beta_{8} - \beta_{12} ) q^{9}$$ $$+O(q^{10})$$ $$q$$ $$+ ( 1 - \beta_{1} ) q^{2}$$ $$+ ( -\beta_{1} + \beta_{2} ) q^{3}$$ $$+ ( -32 - \beta_{1} - \beta_{2} + \beta_{3} ) q^{4}$$ $$+ ( 2 \beta_{1} + \beta_{4} ) q^{5}$$ $$+ ( -739 - \beta_{1} + 3 \beta_{2} + \beta_{3} + \beta_{4} - \beta_{8} ) q^{6}$$ $$+ ( -10 - 109 \beta_{1} - 4 \beta_{2} + 4 \beta_{3} - \beta_{6} ) q^{7}$$ $$+ ( 177 + 27 \beta_{1} + 13 \beta_{2} + 2 \beta_{3} - 3 \beta_{4} - \beta_{6} + 2 \beta_{7} + 2 \beta_{8} + \beta_{15} ) q^{8}$$ $$+ ( -20721 + 148 \beta_{1} - \beta_{2} + 17 \beta_{3} - 3 \beta_{4} - \beta_{7} - \beta_{8} - \beta_{12} ) q^{9}$$ $$+ ( -1566 + 6 \beta_{1} - 25 \beta_{2} - 3 \beta_{3} + \beta_{4} + \beta_{9} ) q^{10}$$ $$+ ( -40 - 267 \beta_{1} - 54 \beta_{2} - 21 \beta_{3} - \beta_{6} - \beta_{7} + \beta_{8} - \beta_{10} + \beta_{11} + \beta_{19} ) q^{11}$$ $$+ ( 66543 + 872 \beta_{1} - 240 \beta_{2} - 15 \beta_{3} - 20 \beta_{4} + \beta_{6} - 6 \beta_{7} - 6 \beta_{8} + \beta_{9} + \beta_{12} + 2 \beta_{15} - \beta_{16} - \beta_{17} + \beta_{18} ) q^{12}$$ $$+ ( -14072 - 1236 \beta_{1} - 9 \beta_{2} - 8 \beta_{3} + 15 \beta_{4} + \beta_{5} - 3 \beta_{6} + 10 \beta_{7} + 15 \beta_{8} - \beta_{9} - \beta_{10} - 3 \beta_{12} + 2 \beta_{14} - 5 \beta_{15} + \beta_{16} - 2 \beta_{17} + \beta_{18} ) q^{13}$$ $$+ ( -111950 - 257 \beta_{1} + 331 \beta_{2} + 132 \beta_{3} - 56 \beta_{4} - 3 \beta_{5} + 12 \beta_{6} - 20 \beta_{7} + \beta_{8} + \beta_{9} - \beta_{11} + 3 \beta_{12} - 2 \beta_{13} + \beta_{14} + 9 \beta_{15} - 2 \beta_{16} ) q^{14}$$ $$+ ( 213 + 1912 \beta_{1} + 105 \beta_{2} + 44 \beta_{3} + 2 \beta_{4} - 5 \beta_{5} - 5 \beta_{6} + 10 \beta_{7} + 5 \beta_{8} + 2 \beta_{9} ) q^{15}$$ $$+ ( 213087 + 25 \beta_{1} + 292 \beta_{2} - \beta_{3} + 81 \beta_{4} + 3 \beta_{5} + 27 \beta_{6} - 19 \beta_{7} - 21 \beta_{8} + 2 \beta_{9} + \beta_{10} + 4 \beta_{11} + 3 \beta_{12} + 6 \beta_{14} + 11 \beta_{15} + 4 \beta_{16} + 5 \beta_{17} + 3 \beta_{18} - 3 \beta_{19} ) q^{16}$$ $$+ ( -95926 + 1270 \beta_{1} - 26 \beta_{2} + 154 \beta_{3} + 115 \beta_{4} + 8 \beta_{6} - 10 \beta_{7} + 15 \beta_{8} - 6 \beta_{9} + 6 \beta_{10} + 2 \beta_{11} + 3 \beta_{12} - 8 \beta_{13} + 5 \beta_{14} + 2 \beta_{15} - 11 \beta_{16} + 12 \beta_{17} + 6 \beta_{18} - 13 \beta_{19} ) q^{17}$$ $$+ ( -175917 + 20760 \beta_{1} - 721 \beta_{2} - 220 \beta_{3} - 90 \beta_{4} - 9 \beta_{5} - 48 \beta_{6} + 108 \beta_{7} + 48 \beta_{8} + 2 \beta_{9} + 4 \beta_{10} + 10 \beta_{11} + 4 \beta_{12} + 7 \beta_{13} + 9 \beta_{14} + 21 \beta_{15} + 8 \beta_{16} - 4 \beta_{17} + 4 \beta_{18} - 7 \beta_{19} ) q^{18}$$ $$+ ( 928 + 12939 \beta_{1} - 2328 \beta_{2} - 263 \beta_{3} - 20 \beta_{4} + 5 \beta_{6} + 43 \beta_{7} - 43 \beta_{8} + 10 \beta_{9} + \beta_{10} + 9 \beta_{11} + 16 \beta_{13} + 12 \beta_{14} + 28 \beta_{15} + 8 \beta_{16} + 4 \beta_{18} - 11 \beta_{19} ) q^{19}$$ $$+ ( -59170 + 1352 \beta_{1} + 425 \beta_{2} + 25 \beta_{3} - 39 \beta_{4} + 5 \beta_{5} - 15 \beta_{7} + 25 \beta_{8} - 5 \beta_{9} + 5 \beta_{10} - 10 \beta_{13} - 5 \beta_{15} - 5 \beta_{16} - 5 \beta_{19} ) q^{20}$$ $$+ ( 210873 + 27701 \beta_{1} + 235 \beta_{2} + 117 \beta_{3} - 760 \beta_{4} - 2 \beta_{5} - 8 \beta_{6} - 116 \beta_{7} - 38 \beta_{8} - 31 \beta_{9} - 6 \beta_{10} + 5 \beta_{11} + 34 \beta_{12} - 13 \beta_{13} - 2 \beta_{14} + 53 \beta_{15} - 5 \beta_{16} - 20 \beta_{17} - 7 \beta_{19} ) q^{21}$$ $$+ ( -290717 + 207 \beta_{1} - 1243 \beta_{2} + 236 \beta_{3} + 560 \beta_{4} - 47 \beta_{5} + 12 \beta_{6} + 44 \beta_{7} + 74 \beta_{8} + 31 \beta_{9} + 4 \beta_{10} + 13 \beta_{11} - 18 \beta_{12} - 18 \beta_{13} + 8 \beta_{14} - 22 \beta_{15} - 30 \beta_{16} + 28 \beta_{17} + 26 \beta_{18} + \beta_{19} ) q^{22}$$ $$+ ( -3545 - 44791 \beta_{1} + 2599 \beta_{2} + 1518 \beta_{3} + 52 \beta_{4} - 11 \beta_{5} + 183 \beta_{6} - 237 \beta_{7} - 10 \beta_{8} + 51 \beta_{9} - 32 \beta_{10} - 27 \beta_{11} - 8 \beta_{13} + 14 \beta_{14} + 46 \beta_{15} - 4 \beta_{16} + 18 \beta_{18} + 3 \beta_{19} ) q^{23}$$ $$+ ( -962729 - 67094 \beta_{1} - 1461 \beta_{2} - 971 \beta_{3} + 83 \beta_{4} + 70 \beta_{5} + 41 \beta_{6} - 19 \beta_{7} + 223 \beta_{8} - 12 \beta_{9} - 19 \beta_{10} + 16 \beta_{11} - 41 \beta_{12} + 9 \beta_{13} + 37 \beta_{14} - 28 \beta_{15} + 8 \beta_{16} - 35 \beta_{17} - 9 \beta_{18} + 50 \beta_{19} ) q^{24}$$ $$+ 1953125 q^{25}$$ $$+ ( 1256708 + 11125 \beta_{1} + 12513 \beta_{2} + 2196 \beta_{3} - 550 \beta_{4} - 95 \beta_{5} - 128 \beta_{6} + 92 \beta_{7} + 80 \beta_{8} + 82 \beta_{9} - 36 \beta_{10} + 10 \beta_{11} - 84 \beta_{12} + 13 \beta_{13} + 19 \beta_{14} - 17 \beta_{15} + 128 \beta_{16} - 60 \beta_{17} - 12 \beta_{18} + 95 \beta_{19} ) q^{26}$$ $$+ ( 1515 + 40548 \beta_{1} - 21311 \beta_{2} - 452 \beta_{3} + 8 \beta_{4} + 25 \beta_{5} - 296 \beta_{6} + 129 \beta_{7} - 198 \beta_{8} + 99 \beta_{9} + 18 \beta_{10} - 45 \beta_{11} - 2 \beta_{13} - 36 \beta_{14} - 112 \beta_{15} - 160 \beta_{16} + 42 \beta_{18} - 77 \beta_{19} ) q^{27}$$ $$+ ( -4360007 + 107552 \beta_{1} - 442 \beta_{2} - 57 \beta_{3} - 346 \beta_{4} + 84 \beta_{5} - 117 \beta_{6} - 232 \beta_{7} - 332 \beta_{8} - 7 \beta_{9} + 22 \beta_{10} + 92 \beta_{11} - 33 \beta_{12} + 18 \beta_{13} - 30 \beta_{14} + 182 \beta_{15} + 43 \beta_{16} + 33 \beta_{17} - \beta_{18} + 44 \beta_{19} ) q^{28}$$ $$+ ( -3310985 - 105157 \beta_{1} - 817 \beta_{2} + 899 \beta_{3} - 966 \beta_{4} + 16 \beta_{5} - 194 \beta_{6} + 440 \beta_{7} + 46 \beta_{8} - 307 \beta_{9} - 4 \beta_{10} - 73 \beta_{11} - 30 \beta_{12} + 13 \beta_{13} - 60 \beta_{14} + 161 \beta_{15} - 103 \beta_{16} - 78 \beta_{18} - 47 \beta_{19} ) q^{29}$$ $$+ ( 1993365 + 1729 \beta_{1} + 985 \beta_{2} - 1950 \beta_{3} - 948 \beta_{4} - 35 \beta_{5} - 160 \beta_{6} - 80 \beta_{7} - 65 \beta_{8} + 50 \beta_{11} + 125 \beta_{12} - 25 \beta_{13} - 50 \beta_{14} + 100 \beta_{15} - 50 \beta_{18} + 50 \beta_{19} ) q^{30}$$ $$+ ( 8072 + 78994 \beta_{1} + 8840 \beta_{2} - 1162 \beta_{3} - 156 \beta_{4} + 104 \beta_{5} - 684 \beta_{6} + 820 \beta_{7} + 440 \beta_{8} + 74 \beta_{9} - 62 \beta_{10} + 8 \beta_{11} + 4 \beta_{13} - 80 \beta_{14} - 104 \beta_{15} + 192 \beta_{16} - 108 \beta_{18} - 80 \beta_{19} ) q^{31}$$ $$+ ( -1681166 - 211686 \beta_{1} - 860 \beta_{2} + 202 \beta_{3} + 6650 \beta_{4} + 2 \beta_{5} + 26 \beta_{6} + 94 \beta_{7} - 242 \beta_{8} + 120 \beta_{9} + 178 \beta_{10} - 112 \beta_{11} + 18 \beta_{12} - 20 \beta_{13} - 16 \beta_{14} - 38 \beta_{15} + 12 \beta_{16} + 102 \beta_{17} + 90 \beta_{18} - 34 \beta_{19} ) q^{32}$$ $$+ ( 4291600 - 84794 \beta_{1} - 1630 \beta_{2} + 5012 \beta_{3} + 5325 \beta_{4} + 18 \beta_{5} + 394 \beta_{6} - 708 \beta_{7} - 2055 \beta_{8} - 496 \beta_{9} - 192 \beta_{10} - 78 \beta_{11} - 127 \beta_{12} + 124 \beta_{13} - 93 \beta_{14} - 896 \beta_{15} + 133 \beta_{16} + 32 \beta_{17} + 124 \beta_{18} - 83 \beta_{19} ) q^{33}$$ $$+ ( -1351030 + 89389 \beta_{1} + 11153 \beta_{2} - 658 \beta_{3} - 5312 \beta_{4} - 53 \beta_{5} + 496 \beta_{6} + 508 \beta_{7} - 80 \beta_{8} + 196 \beta_{9} + 436 \beta_{10} - 102 \beta_{11} - 12 \beta_{12} + 67 \beta_{13} - 19 \beta_{14} + 217 \beta_{15} - 200 \beta_{16} + 140 \beta_{17} - 252 \beta_{18} + 5 \beta_{19} ) q^{34}$$ $$+ ( -8257 - 105368 \beta_{1} + 15150 \beta_{2} + 1149 \beta_{3} + 112 \beta_{4} + 65 \beta_{5} + 425 \beta_{6} - 770 \beta_{7} - 775 \beta_{8} + 27 \beta_{9} + 15 \beta_{10} - 50 \beta_{11} - 30 \beta_{13} - 340 \beta_{15} - 40 \beta_{16} - 150 \beta_{18} + 10 \beta_{19} ) q^{35}$$ $$+ ( 17790788 + 184001 \beta_{1} + 65455 \beta_{2} - 16019 \beta_{3} + 6542 \beta_{4} - 138 \beta_{5} + 1504 \beta_{6} + 78 \beta_{7} + 494 \beta_{8} + 234 \beta_{9} - 106 \beta_{10} + 96 \beta_{11} + 576 \beta_{12} - 44 \beta_{13} - 64 \beta_{14} + 74 \beta_{15} + 490 \beta_{16} - 160 \beta_{17} - 96 \beta_{18} + 522 \beta_{19} ) q^{36}$$ $$+ ( -7680454 - 20502 \beta_{1} - 494 \beta_{2} + 3182 \beta_{3} - 4356 \beta_{4} + 268 \beta_{5} - 848 \beta_{6} + 924 \beta_{7} + 3506 \beta_{8} - 1122 \beta_{9} - 48 \beta_{10} - 282 \beta_{11} - 150 \beta_{12} + 222 \beta_{13} + 14 \beta_{14} - 762 \beta_{15} - 304 \beta_{16} + 80 \beta_{17} - 148 \beta_{18} - 248 \beta_{19} ) q^{37}$$ $$+ ( 12520469 - 3141 \beta_{1} + 54693 \beta_{2} - 8682 \beta_{3} - 3930 \beta_{4} + 195 \beta_{5} + 92 \beta_{6} + 1308 \beta_{7} + 2424 \beta_{8} + 191 \beta_{9} + 28 \beta_{10} + 65 \beta_{11} + 690 \beta_{12} - 48 \beta_{13} + 82 \beta_{14} - 492 \beta_{15} + 354 \beta_{16} - 60 \beta_{17} + 234 \beta_{18} + 211 \beta_{19} ) q^{38}$$ $$+ ( -108416 - 938132 \beta_{1} - 122312 \beta_{2} - 12194 \beta_{3} + 712 \beta_{4} + 96 \beta_{5} + 2804 \beta_{6} - 6162 \beta_{7} - 1282 \beta_{8} + 520 \beta_{9} + 62 \beta_{10} - 518 \beta_{11} - 436 \beta_{13} - 548 \beta_{14} - 2188 \beta_{15} - 1000 \beta_{16} - 320 \beta_{18} - 194 \beta_{19} ) q^{39}$$ $$+ ( -5613606 + 53631 \beta_{1} + 1340 \beta_{2} - 1663 \beta_{3} + 426 \beta_{4} - 120 \beta_{5} + 1410 \beta_{6} + 45 \beta_{7} - 785 \beta_{8} + 76 \beta_{9} + 95 \beta_{10} - 375 \beta_{12} + 85 \beta_{13} - 75 \beta_{14} + 255 \beta_{15} + 180 \beta_{16} - 125 \beta_{17} - 175 \beta_{18} + 80 \beta_{19} ) q^{40}$$ $$+ ( 23885106 + 185078 \beta_{1} + 2971 \beta_{2} - 9835 \beta_{3} + 2719 \beta_{4} - 68 \beta_{5} + 496 \beta_{6} - 3011 \beta_{7} - 5075 \beta_{8} - 1270 \beta_{9} - 436 \beta_{10} - 238 \beta_{11} + 141 \beta_{12} + 70 \beta_{13} - 192 \beta_{14} - 1102 \beta_{15} + 218 \beta_{16} - 120 \beta_{17} + 216 \beta_{18} - 146 \beta_{19} ) q^{41}$$ $$+ ( -28566730 - 235236 \beta_{1} - 24964 \beta_{2} - 33706 \beta_{3} - 17894 \beta_{4} + 50 \beta_{5} + 5056 \beta_{6} - 2328 \beta_{7} - 2976 \beta_{8} + 154 \beta_{9} - 520 \beta_{10} + 484 \beta_{11} - 648 \beta_{12} + 154 \beta_{13} - 218 \beta_{14} + 990 \beta_{15} + 960 \beta_{16} - 312 \beta_{17} + 136 \beta_{18} + 1038 \beta_{19} ) q^{42}$$ $$+ ( 80225 + 973907 \beta_{1} - 41106 \beta_{2} - 24738 \beta_{3} - 1392 \beta_{4} - 137 \beta_{5} - 4778 \beta_{6} + 7849 \beta_{7} + 6440 \beta_{8} + 877 \beta_{9} - 432 \beta_{10} + 267 \beta_{11} + 302 \beta_{13} - 448 \beta_{14} + 1492 \beta_{15} + 520 \beta_{16} + 710 \beta_{18} - 785 \beta_{19} ) q^{43}$$ $$+ ( 16978106 + 334332 \beta_{1} - 59736 \beta_{2} - 986 \beta_{3} + 20388 \beta_{4} - 492 \beta_{5} - 4362 \beta_{6} + 3432 \beta_{7} + 4224 \beta_{8} + 514 \beta_{9} + 1140 \beta_{10} + 312 \beta_{11} + 366 \beta_{12} - 96 \beta_{13} + 248 \beta_{14} + 200 \beta_{15} - 1362 \beta_{16} + 450 \beta_{17} - 978 \beta_{18} + 268 \beta_{19} ) q^{44}$$ $$+ ( -6984955 + 118911 \beta_{1} - 110 \beta_{2} + 13135 \beta_{3} - 20312 \beta_{4} + 325 \beta_{5} - 1765 \beta_{6} + 2660 \beta_{7} + 7915 \beta_{8} - 510 \beta_{9} + 215 \beta_{10} - 75 \beta_{11} + 125 \beta_{12} + 95 \beta_{13} + 50 \beta_{14} + 1210 \beta_{15} - 540 \beta_{16} - 250 \beta_{17} - 525 \beta_{18} - 65 \beta_{19} ) q^{45}$$ $$+ ( -44867686 - 55403 \beta_{1} - 63303 \beta_{2} + 37694 \beta_{3} - 42178 \beta_{4} + 893 \beta_{5} - 7660 \beta_{6} - 2508 \beta_{7} - 1981 \beta_{8} - 129 \beta_{9} + 464 \beta_{10} + 357 \beta_{11} - 1005 \beta_{12} - 392 \beta_{13} + 415 \beta_{14} + 891 \beta_{15} + 130 \beta_{16} + 560 \beta_{17} + 468 \beta_{18} - 152 \beta_{19} ) q^{46}$$ $$+ ( 104202 + 736553 \beta_{1} + 95440 \beta_{2} + 60756 \beta_{3} + 196 \beta_{4} + 1328 \beta_{5} - 797 \beta_{6} + 1678 \beta_{7} - 11490 \beta_{8} + 906 \beta_{9} - 100 \beta_{10} - 322 \beta_{11} + 780 \beta_{13} + 1416 \beta_{14} + 752 \beta_{15} + 128 \beta_{16} - 396 \beta_{18} - 466 \beta_{19} ) q^{47}$$ $$+ ( -23847028 + 868404 \beta_{1} + 282132 \beta_{2} + 94232 \beta_{3} + 69424 \beta_{4} - 1036 \beta_{5} - 1508 \beta_{6} + 2432 \beta_{7} + 4956 \beta_{8} + 1732 \beta_{9} - 716 \beta_{10} + 504 \beta_{11} - 2904 \beta_{12} + 368 \beta_{13} + 952 \beta_{14} - 2096 \beta_{15} + 1268 \beta_{16} - 176 \beta_{17} + 1296 \beta_{18} + 396 \beta_{19} ) q^{48}$$ $$+ ( 16490681 - 1845298 \beta_{1} - 11783 \beta_{2} - 10485 \beta_{3} + 8311 \beta_{4} - 832 \beta_{5} + 124 \beta_{6} + 4163 \beta_{7} - 12039 \beta_{8} - 1450 \beta_{9} + 500 \beta_{10} - 182 \beta_{11} + 393 \beta_{12} - 1246 \beta_{13} - 710 \beta_{14} + 6062 \beta_{15} - 1348 \beta_{16} + 440 \beta_{17} - 400 \beta_{18} - 648 \beta_{19} ) q^{49}$$ $$+ ( 1953125 - 1953125 \beta_{1} ) q^{50}$$ $$+ ( -59970 - 739442 \beta_{1} - 220284 \beta_{2} + 94262 \beta_{3} + 2660 \beta_{4} - 2006 \beta_{5} + 1558 \beta_{6} - 3806 \beta_{7} + 2516 \beta_{8} + 2108 \beta_{9} - 1026 \beta_{10} - 1134 \beta_{11} - 300 \beta_{13} + 756 \beta_{14} + 3612 \beta_{15} - 888 \beta_{16} + 1872 \beta_{18} + 222 \beta_{19} ) q^{51}$$ $$+ ( -5678404 - 1197356 \beta_{1} - 38874 \beta_{2} - 14802 \beta_{3} + 104850 \beta_{4} - 3238 \beta_{5} - 8672 \beta_{6} - 3950 \beta_{7} - 6926 \beta_{8} - 170 \beta_{9} - 1654 \beta_{10} + 1184 \beta_{11} + 1728 \beta_{12} - 900 \beta_{13} - 848 \beta_{14} + 4870 \beta_{15} + 2742 \beta_{16} - 928 \beta_{17} + 736 \beta_{18} + 1190 \beta_{19} ) q^{52}$$ $$+ ( -83266980 + 1454868 \beta_{1} - 15181 \beta_{2} + 164332 \beta_{3} + 16653 \beta_{4} - 535 \beta_{5} + 7661 \beta_{6} - 15526 \beta_{7} - 23059 \beta_{8} - 545 \beta_{9} - 725 \beta_{10} + 1888 \beta_{11} + 1971 \beta_{12} - 1012 \beta_{13} - 504 \beta_{14} - 821 \beta_{15} + 811 \beta_{16} + 54 \beta_{17} + 1989 \beta_{18} - 1498 \beta_{19} ) q^{53}$$ $$+ ( 35350962 - 117208 \beta_{1} + 236156 \beta_{2} - 28384 \beta_{3} - 187424 \beta_{4} + 3440 \beta_{5} + 1568 \beta_{6} + 832 \beta_{7} + 18538 \beta_{8} - 1228 \beta_{9} + 904 \beta_{10} + 4 \beta_{11} - 754 \beta_{12} - 576 \beta_{13} + 2310 \beta_{14} - 2342 \beta_{15} - 3240 \beta_{16} - 1480 \beta_{17} - 2156 \beta_{18} + 322 \beta_{19} ) q^{54}$$ $$+ ( 70460 + 1013365 \beta_{1} - 101980 \beta_{2} - 46480 \beta_{3} - 1270 \beta_{4} + 1320 \beta_{5} - 1520 \beta_{6} + 1245 \beta_{7} - 11805 \beta_{8} - 205 \beta_{9} + 1090 \beta_{10} + 625 \beta_{11} + 920 \beta_{13} + 450 \beta_{14} - 2190 \beta_{15} - 1740 \beta_{16} - 450 \beta_{18} - 765 \beta_{19} ) q^{55}$$ $$+ ( -67785673 + 4488606 \beta_{1} - 287181 \beta_{2} - 125719 \beta_{3} + 185531 \beta_{4} - 354 \beta_{5} + 9801 \beta_{6} + 15745 \beta_{7} + 5747 \beta_{8} + 172 \beta_{9} + 881 \beta_{10} - 608 \beta_{11} + 363 \beta_{12} + 237 \beta_{13} + 2297 \beta_{14} + 1072 \beta_{15} - 1920 \beta_{16} + 1065 \beta_{17} + 2235 \beta_{18} - 2758 \beta_{19} ) q^{56}$$ $$+ ( 199076476 + 4283378 \beta_{1} + 49958 \beta_{2} - 97766 \beta_{3} - 108405 \beta_{4} + 2072 \beta_{5} - 9304 \beta_{6} + 4682 \beta_{7} + 59407 \beta_{8} + 3702 \beta_{9} + 998 \beta_{10} + 206 \beta_{11} + 243 \beta_{12} + 372 \beta_{13} + 3489 \beta_{14} - 1906 \beta_{15} - 363 \beta_{16} - 1748 \beta_{17} - 610 \beta_{18} + 899 \beta_{19} ) q^{57}$$ $$+ ( 104148038 + 3354604 \beta_{1} - 169350 \beta_{2} + 92968 \beta_{3} - 210668 \beta_{4} + 794 \beta_{5} + 5664 \beta_{6} + 184 \beta_{7} - 6432 \beta_{8} + 1324 \beta_{9} - 1320 \beta_{10} + 2732 \beta_{11} + 1848 \beta_{12} + 3242 \beta_{13} + 694 \beta_{14} + 2110 \beta_{15} + 1200 \beta_{16} + 1192 \beta_{17} + 1400 \beta_{18} - 1722 \beta_{19} ) q^{58}$$ $$+ ( -151506 - 915015 \beta_{1} - 221350 \beta_{2} - 118677 \beta_{3} - 1292 \beta_{4} - 5582 \beta_{5} + 12471 \beta_{6} - 4925 \beta_{7} + 20795 \beta_{8} - 1956 \beta_{9} + 127 \beta_{10} + 2433 \beta_{11} + 244 \beta_{13} - 460 \beta_{14} + 2716 \beta_{15} + 4616 \beta_{16} - 720 \beta_{18} + 1485 \beta_{19} ) q^{59}$$ $$+ ( -34101297 - 1929188 \beta_{1} - 242530 \beta_{2} - 16431 \beta_{3} + 67962 \beta_{4} - 840 \beta_{5} + 16005 \beta_{6} - 10600 \beta_{7} - 6980 \beta_{8} - 353 \beta_{9} - 470 \beta_{10} + 500 \beta_{11} + 625 \beta_{12} - 1310 \beta_{13} - 1750 \beta_{14} + 470 \beta_{15} - 2155 \beta_{16} + 1375 \beta_{17} + 625 \beta_{18} - 280 \beta_{19} ) q^{60}$$ $$+ ( -214483976 - 1662016 \beta_{1} + 33354 \beta_{2} - 341468 \beta_{3} + 123202 \beta_{4} - 878 \beta_{5} - 1590 \beta_{6} + 6944 \beta_{7} + 2104 \beta_{8} + 5450 \beta_{9} + 914 \beta_{10} - 1604 \beta_{11} - 4692 \beta_{12} - 928 \beta_{13} + 2954 \beta_{14} - 6142 \beta_{15} + 792 \beta_{16} + 2340 \beta_{17} + 1686 \beta_{18} + 1034 \beta_{19} ) q^{61}$$ $$+ ( 83207260 + 5616 \beta_{1} - 21164 \beta_{2} - 104788 \beta_{3} - 239720 \beta_{4} - 6932 \beta_{5} + 29880 \beta_{6} - 18600 \beta_{7} - 28954 \beta_{8} - 1186 \beta_{9} + 280 \beta_{10} - 3138 \beta_{11} + 3186 \beta_{12} - 5438 \beta_{13} - 3840 \beta_{14} + 4136 \beta_{15} + 1796 \beta_{16} + 1704 \beta_{17} + 2568 \beta_{18} - 4774 \beta_{19} ) q^{62}$$ $$+ ( -462493 - 4642289 \beta_{1} + 1174143 \beta_{2} - 342306 \beta_{3} - 6772 \beta_{4} + 2817 \beta_{5} + 18769 \beta_{6} - 13025 \beta_{7} + 31674 \beta_{8} - 5993 \beta_{9} - 2240 \beta_{10} + 5009 \beta_{11} - 228 \beta_{13} - 2230 \beta_{14} + 4514 \beta_{15} + 4388 \beta_{16} + 1178 \beta_{18} + 3235 \beta_{19} ) q^{63}$$ $$+ ( 17196352 + 1931580 \beta_{1} - 605900 \beta_{2} + 203528 \beta_{3} + 231264 \beta_{4} + 8380 \beta_{5} - 9096 \beta_{6} - 17904 \beta_{7} + 5240 \beta_{8} - 2016 \beta_{9} + 1976 \beta_{10} - 6016 \beta_{11} + 1824 \beta_{12} + 1564 \beta_{13} + 1940 \beta_{14} - 444 \beta_{15} - 3784 \beta_{16} - 2080 \beta_{17} - 5200 \beta_{18} - 2972 \beta_{19} ) q^{64}$$ $$+ ( 14726180 + 1277140 \beta_{1} - 26340 \beta_{2} + 221390 \beta_{3} - 5225 \beta_{4} + 1850 \beta_{5} - 410 \beta_{6} + 5740 \beta_{7} + 37235 \beta_{8} + 2410 \beta_{9} + 2260 \beta_{10} + 1200 \beta_{11} - 1125 \beta_{12} + 930 \beta_{13} + 1025 \beta_{14} - 2910 \beta_{15} - 2335 \beta_{16} + 3000 \beta_{17} - 300 \beta_{18} - 1835 \beta_{19} ) q^{65}$$ $$+ ( 93709204 - 4010151 \beta_{1} - 1446649 \beta_{2} - 32426 \beta_{3} - 452020 \beta_{4} - 5455 \beta_{5} - 45200 \beta_{6} + 28100 \beta_{7} + 2320 \beta_{8} - 64 \beta_{9} - 2980 \beta_{10} - 7690 \beta_{11} + 2108 \beta_{12} + 2001 \beta_{13} - 225 \beta_{14} - 3021 \beta_{15} - 3656 \beta_{16} - 2140 \beta_{17} - 260 \beta_{18} + 1439 \beta_{19} ) q^{66}$$ $$+ ( 1276471 + 10397037 \beta_{1} + 1129586 \beta_{2} + 381818 \beta_{3} + 8872 \beta_{4} - 2011 \beta_{5} - 43122 \beta_{6} + 47019 \beta_{7} - 37764 \beta_{8} - 7953 \beta_{9} + 5008 \beta_{10} - 999 \beta_{11} - 2290 \beta_{13} + 1828 \beta_{14} - 13320 \beta_{15} - 3920 \beta_{16} - 5614 \beta_{18} + 5409 \beta_{19} ) q^{67}$$ $$+ ( 152319664 + 1353878 \beta_{1} + 1381022 \beta_{2} + 28306 \beta_{3} + 431192 \beta_{4} + 14232 \beta_{5} - 6432 \beta_{6} + 17736 \beta_{7} + 11320 \beta_{8} - 12568 \beta_{9} + 1880 \beta_{10} + 2912 \beta_{11} + 10176 \beta_{12} + 5744 \beta_{13} - 3424 \beta_{14} - 4888 \beta_{15} - 3544 \beta_{16} - 1056 \beta_{17} - 3936 \beta_{18} - 8984 \beta_{19} ) q^{68}$$ $$+ ( -266689101 - 4746965 \beta_{1} - 533 \beta_{2} - 229497 \beta_{3} + 137824 \beta_{4} - 6552 \beta_{5} + 17278 \beta_{6} - 17512 \beta_{7} - 130038 \beta_{8} + 16857 \beta_{9} - 4292 \beta_{10} + 1139 \beta_{11} - 2514 \beta_{12} + 1529 \beta_{13} - 6064 \beta_{14} + 5565 \beta_{15} + 12441 \beta_{16} - 5280 \beta_{17} + 74 \beta_{18} + 11169 \beta_{19} ) q^{69}$$ $$+ ( -103419350 - 294926 \beta_{1} + 422820 \beta_{2} + 123675 \beta_{3} - 110273 \beta_{4} - 2225 \beta_{5} - 2620 \beta_{6} - 4220 \beta_{7} - 16755 \beta_{8} - 495 \beta_{9} + 20 \beta_{10} + 675 \beta_{11} - 1500 \beta_{12} - 690 \beta_{13} - 5050 \beta_{14} - 920 \beta_{15} - 770 \beta_{16} - 500 \beta_{17} + 1450 \beta_{18} - 2595 \beta_{19} ) q^{70}$$ $$+ ( -441524 - 4269944 \beta_{1} - 1291680 \beta_{2} + 266520 \beta_{3} + 1108 \beta_{4} + 6680 \beta_{5} - 16706 \beta_{6} - 16158 \beta_{7} - 8250 \beta_{8} - 16434 \beta_{9} + 6788 \beta_{10} - 2662 \beta_{11} - 2832 \beta_{13} + 1044 \beta_{14} - 76 \beta_{15} + 2888 \beta_{16} - 2004 \beta_{18} + 4046 \beta_{19} ) q^{71}$$ $$+ ( 79526491 - 16857305 \beta_{1} - 1301253 \beta_{2} - 253380 \beta_{3} + 930543 \beta_{4} + 1840 \beta_{5} + 5949 \beta_{6} - 70812 \beta_{7} - 78368 \beta_{8} - 24 \beta_{9} - 8318 \beta_{10} - 4992 \beta_{11} - 9426 \beta_{12} - 5034 \beta_{13} - 9962 \beta_{14} - 16959 \beta_{15} + 280 \beta_{16} + 2746 \beta_{17} + 2718 \beta_{18} + 2848 \beta_{19} ) q^{72}$$ $$+ ( 122634166 - 14645330 \beta_{1} - 289744 \beta_{2} + 1123526 \beta_{3} + 167219 \beta_{4} + 1506 \beta_{5} + 19178 \beta_{6} + 58962 \beta_{7} - 18333 \beta_{8} + 23648 \beta_{9} - 1048 \beta_{10} + 7682 \beta_{11} - 9861 \beta_{12} + 10420 \beta_{13} - 7737 \beta_{14} - 6288 \beta_{15} + 10449 \beta_{16} - 3024 \beta_{17} - 3948 \beta_{18} + 9345 \beta_{19} ) q^{73}$$ $$+ ( 10626440 + 6379546 \beta_{1} + 3328164 \beta_{2} + 201910 \beta_{3} - 790854 \beta_{4} - 3366 \beta_{5} - 31136 \beta_{6} + 28360 \beta_{7} + 4640 \beta_{8} - 1406 \beta_{9} - 4136 \beta_{10} - 2692 \beta_{11} + 2136 \beta_{12} + 8538 \beta_{13} + 902 \beta_{14} - 10546 \beta_{15} + 1392 \beta_{16} + 5160 \beta_{17} - 2280 \beta_{18} - 2490 \beta_{19} ) q^{74}$$ $$+ ( -1953125 \beta_{1} + 1953125 \beta_{2} ) q^{75}$$ $$+ ( 114945368 - 12347844 \beta_{1} + 1195376 \beta_{2} + 108488 \beta_{3} + 830532 \beta_{4} + 5076 \beta_{5} + 14824 \beta_{6} - 68668 \beta_{7} - 63572 \beta_{8} - 14388 \beta_{9} - 4596 \beta_{10} - 19160 \beta_{11} - 15888 \beta_{12} - 6536 \beta_{13} - 2832 \beta_{14} - 14308 \beta_{15} - 3884 \beta_{16} - 960 \beta_{17} - 7696 \beta_{18} + 8396 \beta_{19} ) q^{76}$$ $$+ ( 20321504 + 841436 \beta_{1} + 130679 \beta_{2} - 855336 \beta_{3} - 128897 \beta_{4} + 1841 \beta_{5} - 25923 \beta_{6} + 56242 \beta_{7} + 155213 \beta_{8} + 19795 \beta_{9} + 10723 \beta_{10} - 3628 \beta_{11} + 10707 \beta_{12} + 472 \beta_{13} + 5920 \beta_{14} + 11375 \beta_{15} - 8505 \beta_{16} + 8838 \beta_{17} - 6603 \beta_{18} + 3114 \beta_{19} ) q^{77}$$ $$+ ( -995458846 - 1222184 \beta_{1} - 2180068 \beta_{2} + 779724 \beta_{3} - 1289244 \beta_{4} + 13300 \beta_{5} - 32768 \beta_{6} + 39360 \beta_{7} + 111506 \beta_{8} - 16932 \beta_{9} + 4360 \beta_{10} + 5108 \beta_{11} - 26846 \beta_{12} - 668 \beta_{13} - 11914 \beta_{14} - 37102 \beta_{15} - 18040 \beta_{16} - 6344 \beta_{17} - 14276 \beta_{18} - 10054 \beta_{19} ) q^{78}$$ $$+ ( -602402 - 7229868 \beta_{1} + 711574 \beta_{2} + 49314 \beta_{3} - 596 \beta_{4} + 9826 \beta_{5} + 13978 \beta_{6} - 862 \beta_{7} + 116164 \beta_{8} - 14776 \beta_{9} - 7358 \beta_{10} - 3606 \beta_{11} - 9876 \beta_{13} - 7180 \beta_{14} + 25580 \beta_{15} + 8424 \beta_{16} + 12168 \beta_{18} + 8966 \beta_{19} ) q^{79}$$ $$+ ( 180610795 + 6349257 \beta_{1} - 1482260 \beta_{2} - 135585 \beta_{3} + 214561 \beta_{4} + 3315 \beta_{5} - 21265 \beta_{6} + 66365 \beta_{7} + 18815 \beta_{8} - 410 \beta_{9} - 3795 \beta_{10} + 8700 \beta_{11} + 4875 \beta_{12} + 6540 \beta_{13} - 2350 \beta_{14} - 3105 \beta_{15} + 12720 \beta_{16} - 4875 \beta_{17} + 3075 \beta_{18} - 3155 \beta_{19} ) q^{80}$$ $$+ ( 499130339 + 28981196 \beta_{1} + 601433 \beta_{2} - 2245213 \beta_{3} - 493769 \beta_{4} - 13628 \beta_{5} - 25084 \beta_{6} - 118899 \beta_{7} - 83395 \beta_{8} + 17724 \beta_{9} + 4004 \beta_{10} - 8784 \beta_{11} + 26757 \beta_{12} - 9880 \beta_{13} - 4444 \beta_{14} + 73324 \beta_{15} - 792 \beta_{16} - 6520 \beta_{17} - 10980 \beta_{18} + 16204 \beta_{19} ) q^{81}$$ $$+ ( -162579408 - 23513167 \beta_{1} - 3462435 \beta_{2} - 509652 \beta_{3} - 1012174 \beta_{4} - 8395 \beta_{5} - 21840 \beta_{6} - 68348 \beta_{7} - 59376 \beta_{8} - 682 \beta_{9} - 11316 \beta_{10} - 10546 \beta_{11} - 564 \beta_{12} + 5141 \beta_{13} - 2981 \beta_{14} - 12641 \beta_{15} + 120 \beta_{16} - 6476 \beta_{17} + 1580 \beta_{18} + 11355 \beta_{19} ) q^{82}$$ $$+ ( 524803 + 7505689 \beta_{1} + 3818296 \beta_{2} - 1609332 \beta_{3} - 19512 \beta_{4} - 21147 \beta_{5} + 108164 \beta_{6} - 9375 \beta_{7} - 7454 \beta_{8} + 5835 \beta_{9} + 15602 \beta_{10} + 11107 \beta_{11} + 9786 \beta_{13} - 6456 \beta_{14} - 28284 \beta_{15} - 8760 \beta_{16} - 6534 \beta_{18} - 14921 \beta_{19} ) q^{83}$$ $$+ ( 121073772 + 31711078 \beta_{1} - 8084784 \beta_{2} - 215196 \beta_{3} + 1642170 \beta_{4} - 814 \beta_{5} - 74048 \beta_{6} + 122186 \beta_{7} + 71162 \beta_{8} - 17394 \beta_{9} - 9294 \beta_{10} + 8896 \beta_{11} - 22016 \beta_{12} + 8380 \beta_{13} + 12896 \beta_{14} - 45970 \beta_{15} + 11598 \beta_{16} + 2880 \beta_{17} + 24128 \beta_{18} - 6866 \beta_{19} ) q^{84}$$ $$+ ( 241381270 + 10102450 \beta_{1} - 53035 \beta_{2} + 880060 \beta_{3} - 65005 \beta_{4} + 5075 \beta_{5} + 12035 \beta_{6} - 36040 \beta_{7} + 15615 \beta_{8} + 1815 \beta_{9} - 8835 \beta_{10} + 7550 \beta_{11} + 2625 \beta_{12} + 7570 \beta_{13} - 950 \beta_{14} - 26365 \beta_{15} + 13385 \beta_{16} - 13750 \beta_{17} + 2975 \beta_{18} + 4110 \beta_{19} ) q^{85}$$ $$+ ( 982033483 + 1500419 \beta_{1} - 2771697 \beta_{2} - 1234517 \beta_{3} - 1100661 \beta_{4} + 1666 \beta_{5} + 116424 \beta_{6} + 552 \beta_{7} - 39261 \beta_{8} - 7186 \beta_{9} + 6384 \beta_{10} - 11830 \beta_{11} + 22986 \beta_{12} - 22464 \beta_{13} + 21154 \beta_{14} - 6750 \beta_{15} + 4900 \beta_{16} + 7440 \beta_{17} - 5712 \beta_{18} - 92 \beta_{19} ) q^{86}$$ $$+ ( 1262813 + 16343508 \beta_{1} - 5847809 \beta_{2} + 586918 \beta_{3} - 6928 \beta_{4} + 31077 \beta_{5} + 49456 \beta_{6} + 2753 \beta_{7} - 149092 \beta_{8} - 183 \beta_{9} + 6784 \beta_{10} - 8437 \beta_{11} + 12140 \beta_{13} + 21734 \beta_{14} + 36030 \beta_{15} - 5220 \beta_{16} + 9758 \beta_{18} - 10983 \beta_{19} ) q^{87}$$ $$+ ( 137727958 - 18958180 \beta_{1} + 9372142 \beta_{2} + 424986 \beta_{3} + 1178974 \beta_{4} - 3124 \beta_{5} + 96170 \beta_{6} - 33270 \beta_{7} + 115230 \beta_{8} + 34776 \beta_{9} + 11818 \beta_{10} + 23616 \beta_{11} + 33678 \beta_{12} - 286 \beta_{13} - 17126 \beta_{14} + 11472 \beta_{15} - 32672 \beta_{16} + 10362 \beta_{17} - 13714 \beta_{18} - 9564 \beta_{19} ) q^{88}$$ $$+ ( 150117228 - 1823406 \beta_{1} - 23526 \beta_{2} - 404454 \beta_{3} + 723338 \beta_{4} - 400 \beta_{5} + 57084 \beta_{6} - 62548 \beta_{7} - 69862 \beta_{8} - 12242 \beta_{9} - 2444 \beta_{10} + 5762 \beta_{11} + 14730 \beta_{12} - 15302 \beta_{13} + 21394 \beta_{14} - 89818 \beta_{15} + 444 \beta_{16} + 23160 \beta_{17} + 37152 \beta_{18} - 28776 \beta_{19} ) q^{89}$$ $$+ ( -141217928 + 5029953 \beta_{1} + 6206500 \beta_{2} + 282061 \beta_{3} - 164777 \beta_{4} + 7305 \beta_{5} + 60800 \beta_{6} + 55660 \beta_{7} + 16400 \beta_{8} - 17 \beta_{9} - 7620 \beta_{10} + 18650 \beta_{11} - 4500 \beta_{12} + 7765 \beta_{13} + 1675 \beta_{14} + 14695 \beta_{15} + 18720 \beta_{16} + 6500 \beta_{17} + 2900 \beta_{18} - 905 \beta_{19} ) q^{90}$$ $$+ ( -2722754 - 33459812 \beta_{1} - 3042028 \beta_{2} + 2321560 \beta_{3} + 72364 \beta_{4} - 53214 \beta_{5} - 39520 \beta_{6} - 152416 \beta_{7} - 33338 \beta_{8} + 22888 \beta_{9} - 15552 \beta_{10} - 13556 \beta_{11} - 14132 \beta_{13} + 11028 \beta_{14} - 17140 \beta_{15} + 10216 \beta_{16} - 19192 \beta_{18} + 25736 \beta_{19} ) q^{91}$$ $$+ ( -1362890389 + 42964312 \beta_{1} + 3094174 \beta_{2} + 243221 \beta_{3} + 848398 \beta_{4} - 31748 \beta_{5} + 127817 \beta_{6} - 84796 \beta_{7} + 87840 \beta_{8} - 27865 \beta_{9} + 25110 \beta_{10} - 18884 \beta_{11} + 8349 \beta_{12} - 3734 \beta_{13} + 32394 \beta_{14} + 91418 \beta_{15} - 15667 \beta_{16} - 189 \beta_{17} + 4605 \beta_{18} - 17340 \beta_{19} ) q^{92}$$ $$+ ( -591146510 - 4315722 \beta_{1} - 915212 \beta_{2} + 5895514 \beta_{3} - 532028 \beta_{4} + 19438 \beta_{5} + 20794 \beta_{6} + 86284 \beta_{7} + 244716 \beta_{8} - 48572 \beta_{9} + 23726 \beta_{10} + 30846 \beta_{11} + 12704 \beta_{12} - 8954 \beta_{13} - 11038 \beta_{14} + 84612 \beta_{15} - 54550 \beta_{16} + 20556 \beta_{17} - 9154 \beta_{18} - 52320 \beta_{19} ) q^{93}$$ $$+ ( 785472596 - 3442849 \beta_{1} + 13456779 \beta_{2} + 30304 \beta_{3} - 1212720 \beta_{4} + 4241 \beta_{5} - 48892 \beta_{6} - 162300 \beta_{7} - 110707 \beta_{8} + 19651 \beta_{9} + 6960 \beta_{10} - 3359 \beta_{11} + 19527 \beta_{12} - 14784 \beta_{13} + 1243 \beta_{14} + 52263 \beta_{15} + 7930 \beta_{16} - 3760 \beta_{17} + 44580 \beta_{18} - 10400 \beta_{19} ) q^{94}$$ $$+ ( -1069490 - 7977435 \beta_{1} + 2765690 \beta_{2} - 1446390 \beta_{3} - 45230 \beta_{4} + 12990 \beta_{5} + 10160 \beta_{6} + 15165 \beta_{7} + 95665 \beta_{8} - 1275 \beta_{9} - 13920 \beta_{10} + 25825 \beta_{11} + 12440 \beta_{13} - 3050 \beta_{14} + 30870 \beta_{15} + 24620 \beta_{16} + 4650 \beta_{18} - 2105 \beta_{19} ) q^{95}$$ $$+ ( -97667888 + 23838916 \beta_{1} + 2329916 \beta_{2} - 754664 \beta_{3} + 968904 \beta_{4} - 58868 \beta_{5} - 361944 \beta_{6} + 361280 \beta_{7} - 29952 \beta_{8} + 75608 \beta_{9} + 11648 \beta_{10} + 2800 \beta_{11} + 15216 \beta_{12} - 10604 \beta_{13} + 27436 \beta_{14} + 91956 \beta_{15} + 25824 \beta_{16} - 13920 \beta_{17} - 10352 \beta_{18} + 25812 \beta_{19} ) q^{96}$$ $$+ ( -2008083670 - 80882138 \beta_{1} - 381170 \beta_{2} - 1514072 \beta_{3} + 183857 \beta_{4} - 3342 \beta_{5} - 46166 \beta_{6} + 240760 \beta_{7} - 217979 \beta_{8} - 82976 \beta_{9} - 25184 \beta_{10} - 28894 \beta_{11} - 84723 \beta_{12} - 5300 \beta_{13} + 6579 \beta_{14} - 42704 \beta_{15} + 7397 \beta_{16} - 22432 \beta_{17} + 7676 \beta_{18} - 1075 \beta_{19} ) q^{97}$$ $$+ ( 1918866275 - 10864358 \beta_{1} - 15463931 \beta_{2} + 964368 \beta_{3} - 922738 \beta_{4} + 24385 \beta_{5} + 282352 \beta_{6} - 193196 \beta_{7} - 179888 \beta_{8} + 42978 \beta_{9} + 20252 \beta_{10} + 51542 \beta_{11} + 21852 \beta_{12} + 27553 \beta_{13} + 6287 \beta_{14} + 73987 \beta_{15} + 3416 \beta_{16} + 9828 \beta_{17} + 16764 \beta_{18} - 17905 \beta_{19} ) q^{98}$$ $$+ ( 9603344 + 95513809 \beta_{1} - 3904018 \beta_{2} + 1994945 \beta_{3} + 5220 \beta_{4} - 52528 \beta_{5} - 367371 \beta_{6} + 516031 \beta_{7} - 37743 \beta_{8} + 79870 \beta_{9} - 11903 \beta_{10} - 2851 \beta_{11} + 26192 \beta_{13} + 22436 \beta_{14} + 66036 \beta_{15} - 10056 \beta_{16} + 26828 \beta_{18} - 32799 \beta_{19} ) q^{99}$$ $$+O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$20q$$ $$\mathstrut +\mathstrut 22q^{2}$$ $$\mathstrut -\mathstrut 644q^{4}$$ $$\mathstrut -\mathstrut 14784q^{6}$$ $$\mathstrut +\mathstrut 3448q^{8}$$ $$\mathstrut -\mathstrut 414868q^{9}$$ $$\mathstrut +\mathstrut O(q^{10})$$ $$20q$$ $$\mathstrut +\mathstrut 22q^{2}$$ $$\mathstrut -\mathstrut 644q^{4}$$ $$\mathstrut -\mathstrut 14784q^{6}$$ $$\mathstrut +\mathstrut 3448q^{8}$$ $$\mathstrut -\mathstrut 414868q^{9}$$ $$\mathstrut -\mathstrut 31250q^{10}$$ $$\mathstrut +\mathstrut 1329640q^{12}$$ $$\mathstrut -\mathstrut 278864q^{13}$$ $$\mathstrut -\mathstrut 2240504q^{14}$$ $$\mathstrut +\mathstrut 4261360q^{16}$$ $$\mathstrut -\mathstrut 1921656q^{17}$$ $$\mathstrut -\mathstrut 3556082q^{18}$$ $$\mathstrut -\mathstrut 1187500q^{20}$$ $$\mathstrut +\mathstrut 4157512q^{21}$$ $$\mathstrut -\mathstrut 5811280q^{22}$$ $$\mathstrut -\mathstrut 19112144q^{24}$$ $$\mathstrut +\mathstrut 39062500q^{25}$$ $$\mathstrut +\mathstrut 25066884q^{26}$$ $$\mathstrut -\mathstrut 87415400q^{28}$$ $$\mathstrut -\mathstrut 66014888q^{29}$$ $$\mathstrut +\mathstrut 39875000q^{30}$$ $$\mathstrut -\mathstrut 33171328q^{32}$$ $$\mathstrut +\mathstrut 85980560q^{33}$$ $$\mathstrut -\mathstrut 27236084q^{34}$$ $$\mathstrut +\mathstrut 355456476q^{36}$$ $$\mathstrut -\mathstrut 153620656q^{37}$$ $$\mathstrut +\mathstrut 250352720q^{38}$$ $$\mathstrut -\mathstrut 112375000q^{40}$$ $$\mathstrut +\mathstrut 477406160q^{41}$$ $$\mathstrut -\mathstrut 570662040q^{42}$$ $$\mathstrut +\mathstrut 339141040q^{44}$$ $$\mathstrut -\mathstrut 140125000q^{45}$$ $$\mathstrut -\mathstrut 897549304q^{46}$$ $$\mathstrut -\mathstrut 479727360q^{48}$$ $$\mathstrut +\mathstrut 333772012q^{49}$$ $$\mathstrut +\mathstrut 42968750q^{50}$$ $$\mathstrut -\mathstrut 110465096q^{52}$$ $$\mathstrut -\mathstrut 1669491824q^{53}$$ $$\mathstrut +\mathstrut 706139792q^{54}$$ $$\mathstrut -\mathstrut 1362290224q^{56}$$ $$\mathstrut +\mathstrut 3973032960q^{57}$$ $$\mathstrut +\mathstrut 2075027916q^{58}$$ $$\mathstrut -\mathstrut 677375000q^{60}$$ $$\mathstrut -\mathstrut 4283166080q^{61}$$ $$\mathstrut +\mathstrut 1664032240q^{62}$$ $$\mathstrut +\mathstrut 340459456q^{64}$$ $$\mathstrut +\mathstrut 290125000q^{65}$$ $$\mathstrut +\mathstrut 1884031760q^{66}$$ $$\mathstrut +\mathstrut 3042411896q^{68}$$ $$\mathstrut -\mathstrut 5321669928q^{69}$$ $$\mathstrut -\mathstrut 2070000000q^{70}$$ $$\mathstrut +\mathstrut 1632326712q^{72}$$ $$\mathstrut +\mathstrut 2474287656q^{73}$$ $$\mathstrut +\mathstrut 188682276q^{74}$$ $$\mathstrut +\mathstrut 2323171200q^{76}$$ $$\mathstrut +\mathstrut 410885040q^{77}$$ $$\mathstrut -\mathstrut 19914223760q^{78}$$ $$\mathstrut +\mathstrut 3604750000q^{80}$$ $$\mathstrut +\mathstrut 9939722652q^{81}$$ $$\mathstrut -\mathstrut 3197757116q^{82}$$ $$\mathstrut +\mathstrut 2383099552q^{84}$$ $$\mathstrut +\mathstrut 4799500000q^{85}$$ $$\mathstrut +\mathstrut 19648321456q^{86}$$ $$\mathstrut +\mathstrut 2774318240q^{88}$$ $$\mathstrut +\mathstrut 3011851592q^{89}$$ $$\mathstrut -\mathstrut 2849906250q^{90}$$ $$\mathstrut -\mathstrut 27349072440q^{92}$$ $$\mathstrut -\mathstrut 11861394640q^{93}$$ $$\mathstrut +\mathstrut 15684681576q^{94}$$ $$\mathstrut -\mathstrut 1990377984q^{96}$$ $$\mathstrut -\mathstrut 39984502056q^{97}$$ $$\mathstrut +\mathstrut 38416891998q^{98}$$ $$\mathstrut +\mathstrut O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{20}\mathstrut -\mathstrut$$ $$x^{19}\mathstrut +\mathstrut$$ $$84$$ $$x^{18}\mathstrut -\mathstrut$$ $$86$$ $$x^{17}\mathstrut -\mathstrut$$ $$51565$$ $$x^{16}\mathstrut +\mathstrut$$ $$188134$$ $$x^{15}\mathstrut -\mathstrut$$ $$12946328$$ $$x^{14}\mathstrut +\mathstrut$$ $$95990512$$ $$x^{13}\mathstrut +\mathstrut$$ $$5401091173$$ $$x^{12}\mathstrut -\mathstrut$$ $$59241975423$$ $$x^{11}\mathstrut +\mathstrut$$ $$1752287752812$$ $$x^{10}\mathstrut -\mathstrut$$ $$13855072435862$$ $$x^{9}\mathstrut +\mathstrut$$ $$53025624411849$$ $$x^{8}\mathstrut +\mathstrut$$ $$4768661862568058$$ $$x^{7}\mathstrut -\mathstrut$$ $$71663288570646432$$ $$x^{6}\mathstrut +\mathstrut$$ $$1087383280946073208$$ $$x^{5}\mathstrut -\mathstrut$$ $$9530913414031332129$$ $$x^{4}\mathstrut -\mathstrut$$ $$22591082233500222765$$ $$x^{3}\mathstrut +\mathstrut$$ $$1758197001700115583660$$ $$x^{2}\mathstrut -\mathstrut$$ $$51913151006968686718150$$ $$x\mathstrut +\mathstrut$$ $$1126337396992659918451525$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$29\!\cdots\!84$$ $$\nu^{19}\mathstrut -\mathstrut$$ $$19\!\cdots\!18$$ $$\nu^{18}\mathstrut +\mathstrut$$ $$32\!\cdots\!26$$ $$\nu^{17}\mathstrut +\mathstrut$$ $$85\!\cdots\!00$$ $$\nu^{16}\mathstrut -\mathstrut$$ $$32\!\cdots\!36$$ $$\nu^{15}\mathstrut +\mathstrut$$ $$26\!\cdots\!88$$ $$\nu^{14}\mathstrut -\mathstrut$$ $$29\!\cdots\!72$$ $$\nu^{13}\mathstrut -\mathstrut$$ $$23\!\cdots\!40$$ $$\nu^{12}\mathstrut +\mathstrut$$ $$17\!\cdots\!24$$ $$\nu^{11}\mathstrut -\mathstrut$$ $$18\!\cdots\!98$$ $$\nu^{10}\mathstrut +\mathstrut$$ $$42\!\cdots\!18$$ $$\nu^{9}\mathstrut -\mathstrut$$ $$35\!\cdots\!38$$ $$\nu^{8}\mathstrut -\mathstrut$$ $$68\!\cdots\!16$$ $$\nu^{7}\mathstrut +\mathstrut$$ $$18\!\cdots\!36$$ $$\nu^{6}\mathstrut -\mathstrut$$ $$16\!\cdots\!76$$ $$\nu^{5}\mathstrut +\mathstrut$$ $$17\!\cdots\!40$$ $$\nu^{4}\mathstrut -\mathstrut$$ $$51\!\cdots\!72$$ $$\nu^{3}\mathstrut -\mathstrut$$ $$81\!\cdots\!70$$ $$\nu^{2}\mathstrut -\mathstrut$$ $$20\!\cdots\!60$$ $$\nu\mathstrut -\mathstrut$$ $$38\!\cdots\!75$$$$)/$$$$10\!\cdots\!15$$ $$\beta_{2}$$ $$=$$ $$($$$$33\!\cdots\!39$$ $$\nu^{19}\mathstrut +\mathstrut$$ $$16\!\cdots\!54$$ $$\nu^{18}\mathstrut +\mathstrut$$ $$17\!\cdots\!73$$ $$\nu^{17}\mathstrut +\mathstrut$$ $$14\!\cdots\!35$$ $$\nu^{16}\mathstrut -\mathstrut$$ $$10\!\cdots\!81$$ $$\nu^{15}\mathstrut -\mathstrut$$ $$99\!\cdots\!34$$ $$\nu^{14}\mathstrut +\mathstrut$$ $$81\!\cdots\!61$$ $$\nu^{13}\mathstrut -\mathstrut$$ $$36\!\cdots\!39$$ $$\nu^{12}\mathstrut -\mathstrut$$ $$11\!\cdots\!21$$ $$\nu^{11}\mathstrut +\mathstrut$$ $$50\!\cdots\!21$$ $$\nu^{10}\mathstrut -\mathstrut$$ $$57\!\cdots\!10$$ $$\nu^{9}\mathstrut +\mathstrut$$ $$88\!\cdots\!17$$ $$\nu^{8}\mathstrut +\mathstrut$$ $$16\!\cdots\!16$$ $$\nu^{7}\mathstrut -\mathstrut$$ $$10\!\cdots\!09$$ $$\nu^{6}\mathstrut +\mathstrut$$ $$11\!\cdots\!85$$ $$\nu^{5}\mathstrut -\mathstrut$$ $$21\!\cdots\!10$$ $$\nu^{4}\mathstrut +\mathstrut$$ $$60\!\cdots\!28$$ $$\nu^{3}\mathstrut -\mathstrut$$ $$68\!\cdots\!45$$ $$\nu^{2}\mathstrut -\mathstrut$$ $$32\!\cdots\!35$$ $$\nu\mathstrut +\mathstrut$$ $$53\!\cdots\!40$$$$)/$$$$24\!\cdots\!20$$ $$\beta_{3}$$ $$=$$ $$($$$$69\!\cdots\!65$$ $$\nu^{19}\mathstrut +\mathstrut$$ $$21\!\cdots\!38$$ $$\nu^{18}\mathstrut +\mathstrut$$ $$23\!\cdots\!47$$ $$\nu^{17}\mathstrut +\mathstrut$$ $$43\!\cdots\!97$$ $$\nu^{16}\mathstrut -\mathstrut$$ $$90\!\cdots\!87$$ $$\nu^{15}\mathstrut -\mathstrut$$ $$15\!\cdots\!78$$ $$\nu^{14}\mathstrut +\mathstrut$$ $$18\!\cdots\!83$$ $$\nu^{13}\mathstrut -\mathstrut$$ $$71\!\cdots\!93$$ $$\nu^{12}\mathstrut -\mathstrut$$ $$15\!\cdots\!31$$ $$\nu^{11}\mathstrut +\mathstrut$$ $$85\!\cdots\!75$$ $$\nu^{10}\mathstrut -\mathstrut$$ $$89\!\cdots\!54$$ $$\nu^{9}\mathstrut +\mathstrut$$ $$24\!\cdots\!75$$ $$\nu^{8}\mathstrut +\mathstrut$$ $$16\!\cdots\!76$$ $$\nu^{7}\mathstrut -\mathstrut$$ $$86\!\cdots\!31$$ $$\nu^{6}\mathstrut +\mathstrut$$ $$18\!\cdots\!47$$ $$\nu^{5}\mathstrut -\mathstrut$$ $$62\!\cdots\!98$$ $$\nu^{4}\mathstrut +\mathstrut$$ $$18\!\cdots\!04$$ $$\nu^{3}\mathstrut +\mathstrut$$ $$13\!\cdots\!85$$ $$\nu^{2}\mathstrut -\mathstrut$$ $$55\!\cdots\!05$$ $$\nu\mathstrut +\mathstrut$$ $$13\!\cdots\!00$$$$)/$$$$34\!\cdots\!20$$ $$\beta_{4}$$ $$=$$ $$($$$$26\!\cdots\!76$$ $$\nu^{19}\mathstrut -\mathstrut$$ $$17\!\cdots\!02$$ $$\nu^{18}\mathstrut +\mathstrut$$ $$29\!\cdots\!14$$ $$\nu^{17}\mathstrut +\mathstrut$$ $$76\!\cdots\!00$$ $$\nu^{16}\mathstrut -\mathstrut$$ $$28\!\cdots\!04$$ $$\nu^{15}\mathstrut +\mathstrut$$ $$23\!\cdots\!32$$ $$\nu^{14}\mathstrut -\mathstrut$$ $$26\!\cdots\!08$$ $$\nu^{13}\mathstrut -\mathstrut$$ $$20\!\cdots\!60$$ $$\nu^{12}\mathstrut +\mathstrut$$ $$15\!\cdots\!36$$ $$\nu^{11}\mathstrut -\mathstrut$$ $$16\!\cdots\!22$$ $$\nu^{10}\mathstrut +\mathstrut$$ $$37\!\cdots\!02$$ $$\nu^{9}\mathstrut -\mathstrut$$ $$31\!\cdots\!82$$ $$\nu^{8}\mathstrut -\mathstrut$$ $$61\!\cdots\!24$$ $$\nu^{7}\mathstrut +\mathstrut$$ $$16\!\cdots\!04$$ $$\nu^{6}\mathstrut -\mathstrut$$ $$14\!\cdots\!64$$ $$\nu^{5}\mathstrut +\mathstrut$$ $$15\!\cdots\!60$$ $$\nu^{4}\mathstrut -\mathstrut$$ $$45\!\cdots\!08$$ $$\nu^{3}\mathstrut -\mathstrut$$ $$72\!\cdots\!30$$ $$\nu^{2}\mathstrut -\mathstrut$$ $$10\!\cdots\!90$$ $$\nu\mathstrut -\mathstrut$$ $$33\!\cdots\!75$$$$)/$$$$14\!\cdots\!45$$ $$\beta_{5}$$ $$=$$ $$($$$$11\!\cdots\!57$$ $$\nu^{19}\mathstrut +\mathstrut$$ $$32\!\cdots\!42$$ $$\nu^{18}\mathstrut -\mathstrut$$ $$29\!\cdots\!17$$ $$\nu^{17}\mathstrut +\mathstrut$$ $$77\!\cdots\!01$$ $$\nu^{16}\mathstrut +\mathstrut$$ $$11\!\cdots\!97$$ $$\nu^{15}\mathstrut +\mathstrut$$ $$50\!\cdots\!14$$ $$\nu^{14}\mathstrut +\mathstrut$$ $$28\!\cdots\!99$$ $$\nu^{13}\mathstrut -\mathstrut$$ $$43\!\cdots\!97$$ $$\nu^{12}\mathstrut -\mathstrut$$ $$59\!\cdots\!23$$ $$\nu^{11}\mathstrut +\mathstrut$$ $$25\!\cdots\!63$$ $$\nu^{10}\mathstrut -\mathstrut$$ $$17\!\cdots\!70$$ $$\nu^{9}\mathstrut +\mathstrut$$ $$24\!\cdots\!71$$ $$\nu^{8}\mathstrut +\mathstrut$$ $$20\!\cdots\!44$$ $$\nu^{7}\mathstrut +\mathstrut$$ $$86\!\cdots\!57$$ $$\nu^{6}\mathstrut +\mathstrut$$ $$74\!\cdots\!75$$ $$\nu^{5}\mathstrut -\mathstrut$$ $$97\!\cdots\!46$$ $$\nu^{4}\mathstrut +\mathstrut$$ $$23\!\cdots\!08$$ $$\nu^{3}\mathstrut +\mathstrut$$ $$44\!\cdots\!45$$ $$\nu^{2}\mathstrut -\mathstrut$$ $$54\!\cdots\!85$$ $$\nu\mathstrut -\mathstrut$$ $$63\!\cdots\!40$$$$)/$$$$24\!\cdots\!20$$ $$\beta_{6}$$ $$=$$ $$($$$$-$$$$64\!\cdots\!57$$ $$\nu^{19}\mathstrut -\mathstrut$$ $$18\!\cdots\!48$$ $$\nu^{18}\mathstrut -\mathstrut$$ $$31\!\cdots\!79$$ $$\nu^{17}\mathstrut +\mathstrut$$ $$64\!\cdots\!57$$ $$\nu^{16}\mathstrut +\mathstrut$$ $$10\!\cdots\!93$$ $$\nu^{15}\mathstrut -\mathstrut$$ $$27\!\cdots\!96$$ $$\nu^{14}\mathstrut +\mathstrut$$ $$23\!\cdots\!09$$ $$\nu^{13}\mathstrut -\mathstrut$$ $$16\!\cdots\!61$$ $$\nu^{12}\mathstrut -\mathstrut$$ $$57\!\cdots\!91$$ $$\nu^{11}\mathstrut +\mathstrut$$ $$64\!\cdots\!63$$ $$\nu^{10}\mathstrut -\mathstrut$$ $$14\!\cdots\!48$$ $$\nu^{9}\mathstrut +\mathstrut$$ $$54\!\cdots\!25$$ $$\nu^{8}\mathstrut +\mathstrut$$ $$98\!\cdots\!50$$ $$\nu^{7}\mathstrut -\mathstrut$$ $$70\!\cdots\!61$$ $$\nu^{6}\mathstrut +\mathstrut$$ $$57\!\cdots\!79$$ $$\nu^{5}\mathstrut -\mathstrut$$ $$17\!\cdots\!44$$ $$\nu^{4}\mathstrut +\mathstrut$$ $$14\!\cdots\!24$$ $$\nu^{3}\mathstrut +\mathstrut$$ $$19\!\cdots\!75$$ $$\nu^{2}\mathstrut -\mathstrut$$ $$42\!\cdots\!05$$ $$\nu\mathstrut +\mathstrut$$ $$28\!\cdots\!50$$$$)/$$$$82\!\cdots\!40$$ $$\beta_{7}$$ $$=$$ $$($$$$-$$$$13\!\cdots\!45$$ $$\nu^{19}\mathstrut +\mathstrut$$ $$39\!\cdots\!70$$ $$\nu^{18}\mathstrut -\mathstrut$$ $$79\!\cdots\!03$$ $$\nu^{17}\mathstrut +\mathstrut$$ $$15\!\cdots\!43$$ $$\nu^{16}\mathstrut +\mathstrut$$ $$73\!\cdots\!43$$ $$\nu^{15}\mathstrut -\mathstrut$$ $$40\!\cdots\!42$$ $$\nu^{14}\mathstrut +\mathstrut$$ $$16\!\cdots\!69$$ $$\nu^{13}\mathstrut -\mathstrut$$ $$14\!\cdots\!47$$ $$\nu^{12}\mathstrut -\mathstrut$$ $$77\!\cdots\!13$$ $$\nu^{11}\mathstrut +\mathstrut$$ $$90\!\cdots\!73$$ $$\nu^{10}\mathstrut -\mathstrut$$ $$24\!\cdots\!98$$ $$\nu^{9}\mathstrut +\mathstrut$$ $$20\!\cdots\!53$$ $$\nu^{8}\mathstrut -\mathstrut$$ $$37\!\cdots\!40$$ $$\nu^{7}\mathstrut -\mathstrut$$ $$68\!\cdots\!85$$ $$\nu^{6}\mathstrut +\mathstrut$$ $$11\!\cdots\!89$$ $$\nu^{5}\mathstrut -\mathstrut$$ $$14\!\cdots\!50$$ $$\nu^{4}\mathstrut +\mathstrut$$ $$13\!\cdots\!28$$ $$\nu^{3}\mathstrut +\mathstrut$$ $$29\!\cdots\!55$$ $$\nu^{2}\mathstrut -\mathstrut$$ $$25\!\cdots\!35$$ $$\nu\mathstrut +\mathstrut$$ $$69\!\cdots\!60$$$$)/$$$$12\!\cdots\!60$$ $$\beta_{8}$$ $$=$$ $$($$$$-$$$$70\!\cdots\!57$$ $$\nu^{19}\mathstrut -\mathstrut$$ $$65\!\cdots\!54$$ $$\nu^{18}\mathstrut -\mathstrut$$ $$39\!\cdots\!35$$ $$\nu^{17}\mathstrut -\mathstrut$$ $$40\!\cdots\!45$$ $$\nu^{16}\mathstrut +\mathstrut$$ $$37\!\cdots\!75$$ $$\nu^{15}\mathstrut -\mathstrut$$ $$71\!\cdots\!98$$ $$\nu^{14}\mathstrut +\mathstrut$$ $$29\!\cdots\!21$$ $$\nu^{13}\mathstrut +\mathstrut$$ $$14\!\cdots\!93$$ $$\nu^{12}\mathstrut -\mathstrut$$ $$28\!\cdots\!17$$ $$\nu^{11}\mathstrut +\mathstrut$$ $$55\!\cdots\!01$$ $$\nu^{10}\mathstrut -\mathstrut$$ $$54\!\cdots\!98$$ $$\nu^{9}\mathstrut -\mathstrut$$ $$84\!\cdots\!87$$ $$\nu^{8}\mathstrut +\mathstrut$$ $$67\!\cdots\!72$$ $$\nu^{7}\mathstrut -\mathstrut$$ $$66\!\cdots\!53$$ $$\nu^{6}\mathstrut +\mathstrut$$ $$31\!\cdots\!29$$ $$\nu^{5}\mathstrut -\mathstrut$$ $$41\!\cdots\!22$$ $$\nu^{4}\mathstrut +\mathstrut$$ $$28\!\cdots\!60$$ $$\nu^{3}\mathstrut +\mathstrut$$ $$14\!\cdots\!55$$ $$\nu^{2}\mathstrut -\mathstrut$$ $$38\!\cdots\!75$$ $$\nu\mathstrut +\mathstrut$$ $$18\!\cdots\!00$$$$)/$$$$61\!\cdots\!80$$ $$\beta_{9}$$ $$=$$ $$($$$$-$$$$10\!\cdots\!87$$ $$\nu^{19}\mathstrut +\mathstrut$$ $$17\!\cdots\!70$$ $$\nu^{18}\mathstrut +\mathstrut$$ $$14\!\cdots\!71$$ $$\nu^{17}\mathstrut -\mathstrut$$ $$11\!\cdots\!91$$ $$\nu^{16}\mathstrut -\mathstrut$$ $$29\!\cdots\!71$$ $$\nu^{15}\mathstrut -\mathstrut$$ $$12\!\cdots\!10$$ $$\nu^{14}\mathstrut -\mathstrut$$ $$33\!\cdots\!13$$ $$\nu^{13}\mathstrut +\mathstrut$$ $$89\!\cdots\!99$$ $$\nu^{12}\mathstrut -\mathstrut$$ $$16\!\cdots\!19$$ $$\nu^{11}\mathstrut -\mathstrut$$ $$39\!\cdots\!01$$ $$\nu^{10}\mathstrut +\mathstrut$$ $$18\!\cdots\!78$$ $$\nu^{9}\mathstrut -\mathstrut$$ $$57\!\cdots\!81$$ $$\nu^{8}\mathstrut +\mathstrut$$ $$48\!\cdots\!80$$ $$\nu^{7}\mathstrut +\mathstrut$$ $$35\!\cdots\!25$$ $$\nu^{6}\mathstrut -\mathstrut$$ $$28\!\cdots\!53$$ $$\nu^{5}\mathstrut +\mathstrut$$ $$16\!\cdots\!74$$ $$\nu^{4}\mathstrut -\mathstrut$$ $$52\!\cdots\!76$$ $$\nu^{3}\mathstrut -\mathstrut$$ $$50\!\cdots\!95$$ $$\nu^{2}\mathstrut +\mathstrut$$ $$65\!\cdots\!95$$ $$\nu\mathstrut -\mathstrut$$ $$22\!\cdots\!20$$$$)/$$$$61\!\cdots\!80$$ $$\beta_{10}$$ $$=$$ $$($$$$21\!\cdots\!24$$ $$\nu^{19}\mathstrut +\mathstrut$$ $$38\!\cdots\!41$$ $$\nu^{18}\mathstrut -\mathstrut$$ $$16\!\cdots\!02$$ $$\nu^{17}\mathstrut +\mathstrut$$ $$28\!\cdots\!03$$ $$\nu^{16}\mathstrut +\mathstrut$$ $$32\!\cdots\!21$$ $$\nu^{15}\mathstrut -\mathstrut$$ $$97\!\cdots\!31$$ $$\nu^{14}\mathstrut +\mathstrut$$ $$24\!\cdots\!50$$ $$\nu^{13}\mathstrut -\mathstrut$$ $$44\!\cdots\!29$$ $$\nu^{12}\mathstrut -\mathstrut$$ $$24\!\cdots\!49$$ $$\nu^{11}\mathstrut -\mathstrut$$ $$28\!\cdots\!43$$ $$\nu^{10}\mathstrut -\mathstrut$$ $$66\!\cdots\!69$$ $$\nu^{9}\mathstrut +\mathstrut$$ $$22\!\cdots\!42$$ $$\nu^{8}\mathstrut -\mathstrut$$ $$21\!\cdots\!49$$ $$\nu^{7}\mathstrut -\mathstrut$$ $$14\!\cdots\!84$$ $$\nu^{6}\mathstrut +\mathstrut$$ $$15\!\cdots\!77$$ $$\nu^{5}\mathstrut -\mathstrut$$ $$65\!\cdots\!69$$ $$\nu^{4}\mathstrut +\mathstrut$$ $$30\!\cdots\!02$$ $$\nu^{3}\mathstrut +\mathstrut$$ $$49\!\cdots\!80$$ $$\nu^{2}\mathstrut -\mathstrut$$ $$41\!\cdots\!15$$ $$\nu\mathstrut -\mathstrut$$ $$14\!\cdots\!45$$$$)/$$$$12\!\cdots\!60$$ $$\beta_{11}$$ $$=$$ $$($$$$-$$$$82\!\cdots\!63$$ $$\nu^{19}\mathstrut -\mathstrut$$ $$25\!\cdots\!20$$ $$\nu^{18}\mathstrut -\mathstrut$$ $$28\!\cdots\!45$$ $$\nu^{17}\mathstrut -\mathstrut$$ $$73\!\cdots\!13$$ $$\nu^{16}\mathstrut -\mathstrut$$ $$39\!\cdots\!65$$ $$\nu^{15}\mathstrut +\mathstrut$$ $$24\!\cdots\!04$$ $$\nu^{14}\mathstrut -\mathstrut$$ $$39\!\cdots\!29$$ $$\nu^{13}\mathstrut +\mathstrut$$ $$34\!\cdots\!61$$ $$\nu^{12}\mathstrut -\mathstrut$$ $$10\!\cdots\!65$$ $$\nu^{11}\mathstrut -\mathstrut$$ $$86\!\cdots\!51$$ $$\nu^{10}\mathstrut +\mathstrut$$ $$58\!\cdots\!80$$ $$\nu^{9}\mathstrut -\mathstrut$$ $$46\!\cdots\!13$$ $$\nu^{8}\mathstrut +\mathstrut$$ $$47\!\cdots\!54$$ $$\nu^{7}\mathstrut -\mathstrut$$ $$15\!\cdots\!07$$ $$\nu^{6}\mathstrut -\mathstrut$$ $$36\!\cdots\!79$$ $$\nu^{5}\mathstrut +\mathstrut$$ $$19\!\cdots\!84$$ $$\nu^{4}\mathstrut -\mathstrut$$ $$77\!\cdots\!68$$ $$\nu^{3}\mathstrut +\mathstrut$$ $$28\!\cdots\!85$$ $$\nu^{2}\mathstrut -\mathstrut$$ $$45\!\cdots\!15$$ $$\nu\mathstrut -\mathstrut$$ $$13\!\cdots\!30$$$$)/$$$$35\!\cdots\!60$$ $$\beta_{12}$$ $$=$$ $$($$$$-$$$$33\!\cdots\!71$$ $$\nu^{19}\mathstrut +\mathstrut$$ $$16\!\cdots\!26$$ $$\nu^{18}\mathstrut -\mathstrut$$ $$20\!\cdots\!21$$ $$\nu^{17}\mathstrut -\mathstrut$$ $$35\!\cdots\!15$$ $$\nu^{16}\mathstrut +\mathstrut$$ $$65\!\cdots\!41$$ $$\nu^{15}\mathstrut -\mathstrut$$ $$43\!\cdots\!34$$ $$\nu^{14}\mathstrut +\mathstrut$$ $$66\!\cdots\!35$$ $$\nu^{13}\mathstrut +\mathstrut$$ $$23\!\cdots\!79$$ $$\nu^{12}\mathstrut -\mathstrut$$ $$53\!\cdots\!15$$ $$\nu^{11}\mathstrut +\mathstrut$$ $$10\!\cdots\!11$$ $$\nu^{10}\mathstrut -\mathstrut$$ $$75\!\cdots\!06$$ $$\nu^{9}\mathstrut -\mathstrut$$ $$76\!\cdots\!53$$ $$\nu^{8}\mathstrut +\mathstrut$$ $$18\!\cdots\!72$$ $$\nu^{7}\mathstrut -\mathstrut$$ $$47\!\cdots\!35$$ $$\nu^{6}\mathstrut +\mathstrut$$ $$11\!\cdots\!23$$ $$\nu^{5}\mathstrut -\mathstrut$$ $$42\!\cdots\!14$$ $$\nu^{4}\mathstrut -\mathstrut$$ $$19\!\cdots\!68$$ $$\nu^{3}\mathstrut +\mathstrut$$ $$38\!\cdots\!05$$ $$\nu^{2}\mathstrut -\mathstrut$$ $$43\!\cdots\!65$$ $$\nu\mathstrut +\mathstrut$$ $$24\!\cdots\!80$$$$)/$$$$11\!\cdots\!60$$ $$\beta_{13}$$ $$=$$ $$($$$$-$$$$21\!\cdots\!04$$ $$\nu^{19}\mathstrut -\mathstrut$$ $$86\!\cdots\!23$$ $$\nu^{18}\mathstrut +\mathstrut$$ $$26\!\cdots\!74$$ $$\nu^{17}\mathstrut +\mathstrut$$ $$28\!\cdots\!79$$ $$\nu^{16}\mathstrut +\mathstrut$$ $$23\!\cdots\!93$$ $$\nu^{15}\mathstrut +\mathstrut$$ $$12\!\cdots\!73$$ $$\nu^{14}\mathstrut -\mathstrut$$ $$54\!\cdots\!90$$ $$\nu^{13}\mathstrut -\mathstrut$$ $$51\!\cdots\!93$$ $$\nu^{12}\mathstrut +\mathstrut$$ $$10\!\cdots\!07$$ $$\nu^{11}\mathstrut -\mathstrut$$ $$82\!\cdots\!31$$ $$\nu^{10}\mathstrut +\mathstrut$$ $$15\!\cdots\!59$$ $$\nu^{9}\mathstrut -\mathstrut$$ $$11\!\cdots\!26$$ $$\nu^{8}\mathstrut -\mathstrut$$ $$78\!\cdots\!81$$ $$\nu^{7}\mathstrut +\mathstrut$$ $$29\!\cdots\!24$$ $$\nu^{6}\mathstrut -\mathstrut$$ $$36\!\cdots\!87$$ $$\nu^{5}\mathstrut +\mathstrut$$ $$15\!\cdots\!07$$ $$\nu^{4}\mathstrut +\mathstrut$$ $$92\!\cdots\!54$$ $$\nu^{3}\mathstrut -\mathstrut$$ $$75\!\cdots\!40$$ $$\nu^{2}\mathstrut +\mathstrut$$ $$47\!\cdots\!45$$ $$\nu\mathstrut -\mathstrut$$ $$31\!\cdots\!05$$$$)/$$$$61\!\cdots\!80$$ $$\beta_{14}$$ $$=$$ $$($$$$-$$$$91\!\cdots\!49$$ $$\nu^{19}\mathstrut +\mathstrut$$ $$42\!\cdots\!94$$ $$\nu^{18}\mathstrut +\mathstrut$$ $$78\!\cdots\!89$$ $$\nu^{17}\mathstrut +\mathstrut$$ $$34\!\cdots\!83$$ $$\nu^{16}\mathstrut +\mathstrut$$ $$91\!\cdots\!99$$ $$\nu^{15}\mathstrut -\mathstrut$$ $$31\!\cdots\!06$$ $$\nu^{14}\mathstrut -\mathstrut$$ $$53\!\cdots\!83$$ $$\nu^{13}\mathstrut +\mathstrut$$ $$52\!\cdots\!93$$ $$\nu^{12}\mathstrut -\mathstrut$$ $$70\!\cdots\!21$$ $$\nu^{11}\mathstrut +\mathstrut$$ $$20\!\cdots\!29$$ $$\nu^{10}\mathstrut +\mathstrut$$ $$76\!\cdots\!62$$ $$\nu^{9}\mathstrut +\mathstrut$$ $$19\!\cdots\!61$$ $$\nu^{8}\mathstrut +\mathstrut$$ $$35\!\cdots\!48$$ $$\nu^{7}\mathstrut -\mathstrut$$ $$59\!\cdots\!17$$ $$\nu^{6}\mathstrut +\mathstrut$$ $$71\!\cdots\!97$$ $$\nu^{5}\mathstrut +\mathstrut$$ $$39\!\cdots\!98$$ $$\nu^{4}\mathstrut -\mathstrut$$ $$16\!\cdots\!72$$ $$\nu^{3}\mathstrut +\mathstrut$$ $$89\!\cdots\!35$$ $$\nu^{2}\mathstrut -\mathstrut$$ $$45\!\cdots\!35$$ $$\nu\mathstrut -\mathstrut$$ $$78\!\cdots\!40$$$$)/$$$$24\!\cdots\!20$$ $$\beta_{15}$$ $$=$$ $$($$$$49\!\cdots\!22$$ $$\nu^{19}\mathstrut -\mathstrut$$ $$29\!\cdots\!73$$ $$\nu^{18}\mathstrut +\mathstrut$$ $$43\!\cdots\!40$$ $$\nu^{17}\mathstrut +\mathstrut$$ $$30\!\cdots\!27$$ $$\nu^{16}\mathstrut -\mathstrut$$ $$22\!\cdots\!03$$ $$\nu^{15}\mathstrut +\mathstrut$$ $$51\!\cdots\!99$$ $$\nu^{14}\mathstrut -\mathstrut$$ $$25\!\cdots\!16$$ $$\nu^{13}\mathstrut -\mathstrut$$ $$68\!\cdots\!77$$ $$\nu^{12}\mathstrut +\mathstrut$$ $$26\!\cdots\!63$$ $$\nu^{11}\mathstrut -\mathstrut$$ $$52\!\cdots\!39$$ $$\nu^{10}\mathstrut +\mathstrut$$ $$77\!\cdots\!53$$ $$\nu^{9}\mathstrut -\mathstrut$$ $$21\!\cdots\!48$$ $$\nu^{8}\mathstrut -\mathstrut$$ $$51\!\cdots\!63$$ $$\nu^{7}\mathstrut +\mathstrut$$ $$38\!\cdots\!30$$ $$\nu^{6}\mathstrut -\mathstrut$$ $$44\!\cdots\!23$$ $$\nu^{5}\mathstrut +\mathstrut$$ $$38\!\cdots\!81$$ $$\nu^{4}\mathstrut +\mathstrut$$ $$88\!\cdots\!62$$ $$\nu^{3}\mathstrut -\mathstrut$$ $$66\!\cdots\!70$$ $$\nu^{2}\mathstrut +\mathstrut$$ $$23\!\cdots\!85$$ $$\nu\mathstrut -\mathstrut$$ $$24\!\cdots\!15$$$$)/$$$$12\!\cdots\!60$$ $$\beta_{16}$$ $$=$$ $$($$$$-$$$$69\!\cdots\!11$$ $$\nu^{19}\mathstrut +\mathstrut$$ $$84\!\cdots\!19$$ $$\nu^{18}\mathstrut -\mathstrut$$ $$14\!\cdots\!99$$ $$\nu^{17}\mathstrut -\mathstrut$$ $$28\!\cdots\!60$$ $$\nu^{16}\mathstrut -\mathstrut$$ $$40\!\cdots\!14$$ $$\nu^{15}\mathstrut -\mathstrut$$ $$11\!\cdots\!93$$ $$\nu^{14}\mathstrut +\mathstrut$$ $$14\!\cdots\!77$$ $$\nu^{13}\mathstrut +\mathstrut$$ $$16\!\cdots\!74$$ $$\nu^{12}\mathstrut -\mathstrut$$ $$10\!\cdots\!16$$ $$\nu^{11}\mathstrut +\mathstrut$$ $$12\!\cdots\!68$$ $$\nu^{10}\mathstrut -\mathstrut$$ $$11\!\cdots\!99$$ $$\nu^{9}\mathstrut +\mathstrut$$ $$17\!\cdots\!93$$ $$\nu^{8}\mathstrut -\mathstrut$$ $$73\!\cdots\!85$$ $$\nu^{7}\mathstrut -\mathstrut$$ $$47\!\cdots\!19$$ $$\nu^{6}\mathstrut +\mathstrut$$ $$76\!\cdots\!56$$ $$\nu^{5}\mathstrut -\mathstrut$$ $$69\!\cdots\!87$$ $$\nu^{4}\mathstrut -\mathstrut$$ $$94\!\cdots\!94$$ $$\nu^{3}\mathstrut -\mathstrut$$ $$78\!\cdots\!35$$ $$\nu^{2}\mathstrut -\mathstrut$$ $$25\!\cdots\!20$$ $$\nu\mathstrut +\mathstrut$$ $$77\!\cdots\!15$$$$)/$$$$12\!\cdots\!60$$ $$\beta_{17}$$ $$=$$ $$($$$$-$$$$13\!\cdots\!55$$ $$\nu^{19}\mathstrut -\mathstrut$$ $$48\!\cdots\!18$$ $$\nu^{18}\mathstrut -\mathstrut$$ $$37\!\cdots\!45$$ $$\nu^{17}\mathstrut +\mathstrut$$ $$38\!\cdots\!41$$ $$\nu^{16}\mathstrut -\mathstrut$$ $$41\!\cdots\!99$$ $$\nu^{15}\mathstrut +\mathstrut$$ $$11\!\cdots\!38$$ $$\nu^{14}\mathstrut +\mathstrut$$ $$79\!\cdots\!91$$ $$\nu^{13}\mathstrut +\mathstrut$$ $$35\!\cdots\!83$$ $$\nu^{12}\mathstrut +\mathstrut$$ $$12\!\cdots\!93$$ $$\nu^{11}\mathstrut +\mathstrut$$ $$17\!\cdots\!51$$ $$\nu^{10}\mathstrut -\mathstrut$$ $$54\!\cdots\!70$$ $$\nu^{9}\mathstrut -\mathstrut$$ $$16\!\cdots\!73$$ $$\nu^{8}\mathstrut -\mathstrut$$ $$13\!\cdots\!48$$ $$\nu^{7}\mathstrut -\mathstrut$$ $$41\!\cdots\!59$$ $$\nu^{6}\mathstrut +\mathstrut$$ $$10\!\cdots\!63$$ $$\nu^{5}\mathstrut +\mathstrut$$ $$10\!\cdots\!62$$ $$\nu^{4}\mathstrut +\mathstrut$$ $$80\!\cdots\!40$$ $$\nu^{3}\mathstrut -\mathstrut$$ $$18\!\cdots\!55$$ $$\nu^{2}\mathstrut +\mathstrut$$ $$34\!\cdots\!75$$ $$\nu\mathstrut +\mathstrut$$ $$21\!\cdots\!60$$$$)/$$$$24\!\cdots\!20$$ $$\beta_{18}$$ $$=$$ $$($$$$18\!\cdots\!65$$ $$\nu^{19}\mathstrut +\mathstrut$$ $$98\!\cdots\!02$$ $$\nu^{18}\mathstrut +\mathstrut$$ $$46\!\cdots\!27$$ $$\nu^{17}\mathstrut +\mathstrut$$ $$97\!\cdots\!73$$ $$\nu^{16}\mathstrut +\mathstrut$$ $$23\!\cdots\!09$$ $$\nu^{15}\mathstrut -\mathstrut$$ $$15\!\cdots\!86$$ $$\nu^{14}\mathstrut -\mathstrut$$ $$53\!\cdots\!41$$ $$\nu^{13}\mathstrut -\mathstrut$$ $$66\!\cdots\!33$$ $$\nu^{12}\mathstrut -\mathstrut$$ $$45\!\cdots\!11$$ $$\nu^{11}\mathstrut -\mathstrut$$ $$73\!\cdots\!05$$ $$\nu^{10}\mathstrut +\mathstrut$$ $$35\!\cdots\!06$$ $$\nu^{9}\mathstrut +\mathstrut$$ $$61\!\cdots\!51$$ $$\nu^{8}\mathstrut +\mathstrut$$ $$11\!\cdots\!16$$ $$\nu^{7}\mathstrut -\mathstrut$$ $$43\!\cdots\!47$$ $$\nu^{6}\mathstrut +\mathstrut$$ $$23\!\cdots\!91$$ $$\nu^{5}\mathstrut +\mathstrut$$ $$70\!\cdots\!70$$ $$\nu^{4}\mathstrut +\mathstrut$$ $$16\!\cdots\!48$$ $$\nu^{3}\mathstrut +\mathstrut$$ $$95\!\cdots\!05$$ $$\nu^{2}\mathstrut -\mathstrut$$ $$67\!\cdots\!85$$ $$\nu\mathstrut -\mathstrut$$ $$14\!\cdots\!80$$$$)/$$$$24\!\cdots\!20$$ $$\beta_{19}$$ $$=$$ $$($$$$17\!\cdots\!45$$ $$\nu^{19}\mathstrut +\mathstrut$$ $$37\!\cdots\!72$$ $$\nu^{18}\mathstrut -\mathstrut$$ $$28\!\cdots\!73$$ $$\nu^{17}\mathstrut +\mathstrut$$ $$15\!\cdots\!87$$ $$\nu^{16}\mathstrut +\mathstrut$$ $$40\!\cdots\!67$$ $$\nu^{15}\mathstrut -\mathstrut$$ $$28\!\cdots\!28$$ $$\nu^{14}\mathstrut +\mathstrut$$ $$13\!\cdots\!19$$ $$\nu^{13}\mathstrut -\mathstrut$$ $$88\!\cdots\!19$$ $$\nu^{12}\mathstrut +\mathstrut$$ $$53\!\cdots\!35$$ $$\nu^{11}\mathstrut +\mathstrut$$ $$18\!\cdots\!09$$ $$\nu^{10}\mathstrut -\mathstrut$$ $$16\!\cdots\!88$$ $$\nu^{9}\mathstrut +\mathstrut$$ $$73\!\cdots\!75$$ $$\nu^{8}\mathstrut -\mathstrut$$ $$86\!\cdots\!18$$ $$\nu^{7}\mathstrut +\mathstrut$$ $$29\!\cdots\!73$$ $$\nu^{6}\mathstrut +\mathstrut$$ $$23\!\cdots\!69$$ $$\nu^{5}\mathstrut -\mathstrut$$ $$32\!\cdots\!44$$ $$\nu^{4}\mathstrut +\mathstrut$$ $$73\!\cdots\!44$$ $$\nu^{3}\mathstrut +\mathstrut$$ $$29\!\cdots\!25$$ $$\nu^{2}\mathstrut -\mathstrut$$ $$75\!\cdots\!55$$ $$\nu\mathstrut -\mathstrut$$ $$24\!\cdots\!50$$$$)/$$$$22\!\cdots\!20$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{4}\mathstrut -\mathstrut$$ $$623$$ $$\beta_{1}$$$$)/1250$$ $$\nu^{2}$$ $$=$$ $$($$$$2$$ $$\beta_{9}\mathstrut +\mathstrut$$ $$619$$ $$\beta_{3}\mathstrut -\mathstrut$$ $$675$$ $$\beta_{2}\mathstrut +\mathstrut$$ $$633$$ $$\beta_{1}\mathstrut -\mathstrut$$ $$20632$$$$)/2500$$ $$\nu^{3}$$ $$=$$ $$($$$$-$$$$15$$ $$\beta_{19}\mathstrut -\mathstrut$$ $$15$$ $$\beta_{16}\mathstrut +\mathstrut$$ $$610$$ $$\beta_{15}\mathstrut -\mathstrut$$ $$30$$ $$\beta_{13}\mathstrut +\mathstrut$$ $$15$$ $$\beta_{10}\mathstrut -\mathstrut$$ $$21$$ $$\beta_{9}\mathstrut +\mathstrut$$ $$1325$$ $$\beta_{8}\mathstrut +\mathstrut$$ $$1205$$ $$\beta_{7}\mathstrut -\mathstrut$$ $$625$$ $$\beta_{6}\mathstrut +\mathstrut$$ $$15$$ $$\beta_{5}\mathstrut -\mathstrut$$ $$1990$$ $$\beta_{4}\mathstrut -\mathstrut$$ $$532$$ $$\beta_{3}\mathstrut +\mathstrut$$ $$11425$$ $$\beta_{2}\mathstrut +\mathstrut$$ $$11536$$ $$\beta_{1}\mathstrut +\mathstrut$$ $$3761$$$$)/5000$$ $$\nu^{4}$$ $$=$$ $$($$$$-$$$$1495$$ $$\beta_{19}\mathstrut +\mathstrut$$ $$1175$$ $$\beta_{18}\mathstrut +\mathstrut$$ $$2625$$ $$\beta_{17}\mathstrut +\mathstrut$$ $$3280$$ $$\beta_{16}\mathstrut +\mathstrut$$ $$5455$$ $$\beta_{15}\mathstrut +\mathstrut$$ $$3450$$ $$\beta_{14}\mathstrut +\mathstrut$$ $$460$$ $$\beta_{13}\mathstrut +\mathstrut$$ $$375$$ $$\beta_{12}\mathstrut +\mathstrut$$ $$2500$$ $$\beta_{11}\mathstrut +\mathstrut$$ $$945$$ $$\beta_{10}\mathstrut +\mathstrut$$ $$1646$$ $$\beta_{9}\mathstrut -\mathstrut$$ $$21565$$ $$\beta_{8}\mathstrut -\mathstrut$$ $$16515$$ $$\beta_{7}\mathstrut +\mathstrut$$ $$25015$$ $$\beta_{6}\mathstrut +\mathstrut$$ $$1335$$ $$\beta_{5}\mathstrut +\mathstrut$$ $$60305$$ $$\beta_{4}\mathstrut +\mathstrut$$ $$9827$$ $$\beta_{3}\mathstrut +\mathstrut$$ $$126960$$ $$\beta_{2}\mathstrut +\mathstrut$$ $$164069$$ $$\beta_{1}\mathstrut +\mathstrut$$ $$110217379$$$$)/10000$$ $$\nu^{5}$$ $$=$$ $$($$$$-$$$$2965$$ $$\beta_{19}\mathstrut +\mathstrut$$ $$6575$$ $$\beta_{18}\mathstrut +\mathstrut$$ $$2625$$ $$\beta_{17}\mathstrut +\mathstrut$$ $$5460$$ $$\beta_{16}\mathstrut -\mathstrut$$ $$4165$$ $$\beta_{15}\mathstrut -\mathstrut$$ $$3900$$ $$\beta_{14}\mathstrut +\mathstrut$$ $$1770$$ $$\beta_{13}\mathstrut +\mathstrut$$ $$3375$$ $$\beta_{12}\mathstrut -\mathstrut$$ $$3900$$ $$\beta_{11}\mathstrut +\mathstrut$$ $$8765$$ $$\beta_{10}\mathstrut +\mathstrut$$ $$6466$$ $$\beta_{9}\mathstrut +\mathstrut$$ $$10115$$ $$\beta_{8}\mathstrut +\mathstrut$$ $$52285$$ $$\beta_{7}\mathstrut -\mathstrut$$ $$24015$$ $$\beta_{6}\mathstrut +\mathstrut$$ $$1125$$ $$\beta_{5}\mathstrut +\mathstrut$$ $$485175$$ $$\beta_{4}\mathstrut -\mathstrut$$ $$53793$$ $$\beta_{3}\mathstrut -\mathstrut$$ $$826360$$ $$\beta_{2}\mathstrut -\mathstrut$$ $$10066311$$ $$\beta_{1}\mathstrut -\mathstrut$$ $$69594271$$$$)/2000$$ $$\nu^{6}$$ $$=$$ $$($$$$-$$$$129360$$ $$\beta_{19}\mathstrut -\mathstrut$$ $$826175$$ $$\beta_{18}\mathstrut -\mathstrut$$ $$463125$$ $$\beta_{17}\mathstrut -\mathstrut$$ $$317710$$ $$\beta_{16}\mathstrut -\mathstrut$$ $$119060$$ $$\beta_{15}\mathstrut +\mathstrut$$ $$366875$$ $$\beta_{14}\mathstrut -\mathstrut$$ $$21445$$ $$\beta_{13}\mathstrut +\mathstrut$$ $$428625$$ $$\beta_{12}\mathstrut -\mathstrut$$ $$1147600$$ $$\beta_{11}\mathstrut -\mathstrut$$ $$87215$$ $$\beta_{10}\mathstrut -\mathstrut$$ $$48386$$ $$\beta_{9}\mathstrut +\mathstrut$$ $$2193155$$ $$\beta_{8}\mathstrut -\mathstrut$$ $$1681195$$ $$\beta_{7}\mathstrut +\mathstrut$$ $$876545$$ $$\beta_{6}\mathstrut +\mathstrut$$ $$1061080$$ $$\beta_{5}\mathstrut +\mathstrut$$ $$27921185$$ $$\beta_{4}\mathstrut +\mathstrut$$ $$21408303$$ $$\beta_{3}\mathstrut -\mathstrut$$ $$118793945$$ $$\beta_{2}\mathstrut +\mathstrut$$ $$214803526$$ $$\beta_{1}\mathstrut +\mathstrut$$ $$24855714571$$$$)/10000$$ $$\nu^{7}$$ $$=$$ $$($$$$-$$$$5860815$$ $$\beta_{19}\mathstrut +\mathstrut$$ $$8491100$$ $$\beta_{18}\mathstrut +\mathstrut$$ $$114000$$ $$\beta_{17}\mathstrut +\mathstrut$$ $$13611860$$ $$\beta_{16}\mathstrut +\mathstrut$$ $$4616085$$ $$\beta_{15}\mathstrut -\mathstrut$$ $$6997725$$ $$\beta_{14}\mathstrut -\mathstrut$$ $$1035355$$ $$\beta_{13}\mathstrut -\mathstrut$$ $$16107000$$ $$\beta_{12}\mathstrut +\mathstrut$$ $$2024000$$ $$\beta_{11}\mathstrut -\mathstrut$$ $$5952160$$ $$\beta_{10}\mathstrut +\mathstrut$$ $$14947362$$ $$\beta_{9}\mathstrut +\mathstrut$$ $$27803760$$ $$\beta_{8}\mathstrut -\mathstrut$$ $$114801850$$ $$\beta_{7}\mathstrut -\mathstrut$$ $$6915610$$ $$\beta_{6}\mathstrut +\mathstrut$$ $$9322855$$ $$\beta_{5}\mathstrut -\mathstrut$$ $$804932350$$ $$\beta_{4}\mathstrut +\mathstrut$$ $$103482994$$ $$\beta_{3}\mathstrut +\mathstrut$$ $$3650517135$$ $$\beta_{2}\mathstrut -\mathstrut$$ $$16105112677$$ $$\beta_{1}\mathstrut -\mathstrut$$ $$251574266952$$$$)/10000$$ $$\nu^{8}$$ $$=$$ $$($$$$-$$$$83867960$$ $$\beta_{19}\mathstrut +\mathstrut$$ $$14307300$$ $$\beta_{18}\mathstrut -\mathstrut$$ $$6304500$$ $$\beta_{17}\mathstrut +\mathstrut$$ $$43201190$$ $$\beta_{16}\mathstrut +\mathstrut$$ $$71702115$$ $$\beta_{15}\mathstrut +\mathstrut$$ $$89344950$$ $$\beta_{14}\mathstrut -\mathstrut$$ $$45705970$$ $$\beta_{13}\mathstrut -\mathstrut$$ $$33399000$$ $$\beta_{12}\mathstrut -\mathstrut$$ $$64756000$$ $$\beta_{11}\mathstrut +\mathstrut$$ $$46918410$$ $$\beta_{10}\mathstrut -\mathstrut$$ $$258441434$$ $$\beta_{9}\mathstrut -\mathstrut$$ $$812858320$$ $$\beta_{8}\mathstrut +\mathstrut$$ $$553814930$$ $$\beta_{7}\mathstrut +\mathstrut$$ $$80845345$$ $$\beta_{6}\mathstrut -\mathstrut$$ $$36434120$$ $$\beta_{5}\mathstrut -\mathstrut$$ $$838790975$$ $$\beta_{4}\mathstrut +\mathstrut$$ $$2467428822$$ $$\beta_{3}\mathstrut -\mathstrut$$ $$21598044495$$ $$\beta_{2}\mathstrut +\mathstrut$$ $$51226141139$$ $$\beta_{1}\mathstrut -\mathstrut$$ $$9705580637361$$$$)/5000$$ $$\nu^{9}$$ $$=$$ $$($$$$3293850425$$ $$\beta_{19}\mathstrut -\mathstrut$$ $$432657750$$ $$\beta_{18}\mathstrut -\mathstrut$$ $$2727291750$$ $$\beta_{17}\mathstrut +\mathstrut$$ $$6280359300$$ $$\beta_{16}\mathstrut +\mathstrut$$ $$7323898925$$ $$\beta_{15}\mathstrut -\mathstrut$$ $$1886394375$$ $$\beta_{14}\mathstrut +\mathstrut$$ $$3553556475$$ $$\beta_{13}\mathstrut +\mathstrut$$ $$2037872250$$ $$\beta_{12}\mathstrut +\mathstrut$$ $$456882500$$ $$\beta_{11}\mathstrut -\mathstrut$$ $$776628050$$ $$\beta_{10}\mathstrut +\mathstrut$$ $$1636912734$$ $$\beta_{9}\mathstrut +\mathstrut$$ $$14613021330$$ $$\beta_{8}\mathstrut -\mathstrut$$ $$22026294190$$ $$\beta_{7}\mathstrut +\mathstrut$$ $$2424062420$$ $$\beta_{6}\mathstrut -\mathstrut$$ $$1244819505$$ $$\beta_{5}\mathstrut -\mathstrut$$ $$112220156640$$ $$\beta_{4}\mathstrut -\mathstrut$$ $$46034537252$$ $$\beta_{3}\mathstrut -\mathstrut$$ $$77996228995$$ $$\beta_{2}\mathstrut +\mathstrut$$ $$10498290842181$$ $$\beta_{1}\mathstrut +\mathstrut$$ $$238415301771326$$$$)/10000$$ $$\nu^{10}$$ $$=$$ $$($$$$-$$$$1040701415$$ $$\beta_{19}\mathstrut +\mathstrut$$ $$5069226800$$ $$\beta_{18}\mathstrut +\mathstrut$$ $$249825500$$ $$\beta_{17}\mathstrut +\mathstrut$$ $$9149902760$$ $$\beta_{16}\mathstrut -\mathstrut$$ $$6043711965$$ $$\beta_{15}\mathstrut +\mathstrut$$ $$1305874675$$ $$\beta_{14}\mathstrut -\mathstrut$$ $$1550060955$$ $$\beta_{13}\mathstrut +\mathstrut$$ $$7101919500$$ $$\beta_{12}\mathstrut -\mathstrut$$ $$3279288800$$ $$\beta_{11}\mathstrut -\mathstrut$$ $$13637336760$$ $$\beta_{10}\mathstrut -\mathstrut$$ $$21677108802$$ $$\beta_{9}\mathstrut +\mathstrut$$ $$20136145020$$ $$\beta_{8}\mathstrut +\mathstrut$$ $$18206843370$$ $$\beta_{7}\mathstrut -\mathstrut$$ $$7803143220$$ $$\beta_{6}\mathstrut +\mathstrut$$ $$11742160895$$ $$\beta_{5}\mathstrut +\mathstrut$$ $$619355836080$$ $$\beta_{4}\mathstrut -\mathstrut$$ $$1198926865614$$ $$\beta_{3}\mathstrut -\mathstrut$$ $$271584373755$$ $$\beta_{2}\mathstrut -\mathstrut$$ $$24788161193923$$ $$\beta_{1}\mathstrut -\mathstrut$$ $$735576109152838$$$$)/2000$$ $$\nu^{11}$$ $$=$$ $$($$$$22585590495$$ $$\beta_{19}\mathstrut +\mathstrut$$ $$568484609750$$ $$\beta_{18}\mathstrut +\mathstrut$$ $$602744039750$$ $$\beta_{17}\mathstrut +\mathstrut$$ $$126199061920$$ $$\beta_{16}\mathstrut -\mathstrut$$ $$2978907070755$$ $$\beta_{15}\mathstrut +\mathstrut$$ $$171843722075$$ $$\beta_{14}\mathstrut -\mathstrut$$ $$224364746535$$ $$\beta_{13}\mathstrut -\mathstrut$$ $$1046871170250$$ $$\beta_{12}\mathstrut -\mathstrut$$ $$961018531100$$ $$\beta_{11}\mathstrut -\mathstrut$$ $$85555210370$$ $$\beta_{10}\mathstrut +\mathstrut$$ $$1938781132606$$ $$\beta_{9}\mathstrut -\mathstrut$$ $$6259659480290$$ $$\beta_{8}\mathstrut -\mathstrut$$ $$12660062432170$$ $$\beta_{7}\mathstrut +\mathstrut$$ $$3753900243690$$ $$\beta_{6}\mathstrut -\mathstrut$$ $$889422262055$$ $$\beta_{5}\mathstrut -\mathstrut$$ $$38144955284170$$ $$\beta_{4}\mathstrut +\mathstrut$$ $$52783753906092$$ $$\beta_{3}\mathstrut -\mathstrut$$ $$214749811374915$$ $$\beta_{2}\mathstrut +\mathstrut$$ $$2029134009362509$$ $$\beta_{1}\mathstrut +\mathstrut$$ $$12707546386070644$$$$)/10000$$ $$\nu^{12}$$ $$=$$ $$($$$$-$$$$4490857413325$$ $$\beta_{19}\mathstrut -\mathstrut$$ $$5369267873375$$ $$\beta_{18}\mathstrut -\mathstrut$$ $$1997166635125$$ $$\beta_{17}\mathstrut -\mathstrut$$ $$8173326219825$$ $$\beta_{16}\mathstrut +\mathstrut$$ $$7109293046675$$ $$\beta_{15}\mathstrut -\mathstrut$$ $$1263853236375$$ $$\beta_{14}\mathstrut -\mathstrut$$ $$2852779229025$$ $$\beta_{13}\mathstrut +\mathstrut$$ $$3305324061375$$ $$\beta_{12}\mathstrut -\mathstrut$$ $$5154128898750$$ $$\beta_{11}\mathstrut +\mathstrut$$ $$5766158288450$$ $$\beta_{10}\mathstrut -\mathstrut$$ $$24352739677461$$ $$\beta_{9}\mathstrut +\mathstrut$$ $$87936947410900$$ $$\beta_{8}\mathstrut -\mathstrut$$ $$2605943014425$$ $$\beta_{7}\mathstrut -\mathstrut$$ $$85528873925275$$ $$\beta_{6}\mathstrut +\mathstrut$$ $$2559047681925$$ $$\beta_{5}\mathstrut -\mathstrut$$ $$224551755667620$$ $$\beta_{4}\mathstrut -\mathstrut$$ $$599558829311217$$ $$\beta_{3}\mathstrut +\mathstrut$$ $$319955363530675$$ $$\beta_{2}\mathstrut -\mathstrut$$ $$5637089134912584$$ $$\beta_{1}\mathstrut -\mathstrut$$ $$510269747049017099$$$$)/5000$$ $$\nu^{13}$$ $$=$$ $$($$$$104775524271145$$ $$\beta_{19}\mathstrut -\mathstrut$$ $$153232175321875$$ $$\beta_{18}\mathstrut -\mathstrut$$ $$19407191189125$$ $$\beta_{17}\mathstrut +\mathstrut$$ $$198692796861270$$ $$\beta_{16}\mathstrut -\mathstrut$$ $$176636713697855$$ $$\beta_{15}\mathstrut +\mathstrut$$ $$6395862912750$$ $$\beta_{14}\mathstrut +\mathstrut$$ $$176818599497040$$ $$\beta_{13}\mathstrut +\mathstrut$$ $$71035102670625$$ $$\beta_{12}\mathstrut +\mathstrut$$ $$50920194194000$$ $$\beta_{11}\mathstrut -\mathstrut$$ $$170775101653895$$ $$\beta_{10}\mathstrut +\mathstrut$$ $$68810212562288$$ $$\beta_{9}\mathstrut -\mathstrut$$ $$1975433932289525$$ $$\beta_{8}\mathstrut -\mathstrut$$ $$3458905775380665$$ $$\beta_{7}\mathstrut +\mathstrut$$ $$2978067898355175$$ $$\beta_{6}\mathstrut -\mathstrut$$ $$109839644034345$$ $$\beta_{5}\mathstrut -\mathstrut$$ $$7306100520842965$$ $$\beta_{4}\mathstrut +\mathstrut$$ $$10877026748115221$$ $$\beta_{3}\mathstrut +\mathstrut$$ $$54576420587851050$$ $$\beta_{2}\mathstrut +\mathstrut$$ $$511332906628420297$$ $$\beta_{1}\mathstrut -\mathstrut$$ $$15867021491155160333$$$$)/10000$$ $$\nu^{14}$$ $$=$$ $$($$$$-$$$$1339060679902795$$ $$\beta_{19}\mathstrut +\mathstrut$$ $$4077537265419800$$ $$\beta_{18}\mathstrut +\mathstrut$$ $$1350521202755500$$ $$\beta_{17}\mathstrut +\mathstrut$$ $$1064486522733230$$ $$\beta_{16}\mathstrut -\mathstrut$$ $$1810325517140095$$ $$\beta_{15}\mathstrut -\mathstrut$$ $$4628516566373675$$ $$\beta_{14}\mathstrut +\mathstrut$$ $$228336778958735$$ $$\beta_{13}\mathstrut +\mathstrut$$ $$726522223108500$$ $$\beta_{12}\mathstrut +\mathstrut$$ $$1820886583880000$$ $$\beta_{11}\mathstrut -\mathstrut$$ $$1223158560457630$$ $$\beta_{10}\mathstrut -\mathstrut$$ $$5706406896785864$$ $$\beta_{9}\mathstrut -\mathstrut$$ $$9987832178943790$$ $$\beta_{8}\mathstrut +\mathstrut$$ $$44830616109001760$$ $$\beta_{7}\mathstrut -\mathstrut$$ $$40074901964449010$$ $$\beta_{6}\mathstrut +\mathstrut$$ $$1014843823561235$$ $$\beta_{5}\mathstrut -\mathstrut$$ $$36718510257417720$$ $$\beta_{4}\mathstrut -\mathstrut$$ $$320570118039952918$$ $$\beta_{3}\mathstrut -\mathstrut$$ $$1705624652116515$$ $$\beta_{2}\mathstrut +\mathstrut$$ $$7755106402042063979$$ $$\beta_{1}\mathstrut -\mathstrut$$ $$233252588179061773736$$$$)/10000$$ $$\nu^{15}$$ $$=$$ $$($$$$273154966312280$$ $$\beta_{19}\mathstrut -\mathstrut$$ $$6301347290116025$$ $$\beta_{18}\mathstrut +\mathstrut$$ $$161257869890125$$ $$\beta_{17}\mathstrut -\mathstrut$$ $$19845663077302070$$ $$\beta_{16}\mathstrut -\mathstrut$$ $$16542415104178320$$ $$\beta_{15}\mathstrut +\mathstrut$$ $$16032500892032425$$ $$\beta_{14}\mathstrut -\mathstrut$$ $$4573171932768215$$ $$\beta_{13}\mathstrut +\mathstrut$$ $$9869571368097375$$ $$\beta_{12}\mathstrut -\mathstrut$$ $$17723356156385200$$ $$\beta_{11}\mathstrut -\mathstrut$$ $$11551288217005$$ $$\beta_{10}\mathstrut -\mathstrut$$ $$9604454546364262$$ $$\beta_{9}\mathstrut -\mathstrut$$ $$77103811814063255$$ $$\beta_{8}\mathstrut -\mathstrut$$ $$92586548626751445$$ $$\beta_{7}\mathstrut +\mathstrut$$ $$116587509747265555$$ $$\beta_{6}\mathstrut -\mathstrut$$ $$8759449691975200$$ $$\beta_{5}\mathstrut -\mathstrut$$ $$153284514950701845$$ $$\beta_{4}\mathstrut -\mathstrut$$ $$997766842627688499$$ $$\beta_{3}\mathstrut -\mathstrut$$ $$2906022595845122555$$ $$\beta_{2}\mathstrut +\mathstrut$$ $$25699034648618401062$$ $$\beta_{1}\mathstrut +\mathstrut$$ $$905155679947580981597$$$$)/2000$$ $$\nu^{16}$$ $$=$$ $$($$$$821779322468729165$$ $$\beta_{19}\mathstrut +\mathstrut$$ $$3542162213489450$$ $$\beta_{18}\mathstrut -\mathstrut$$ $$49624088540542750$$ $$\beta_{17}\mathstrut +\mathstrut$$ $$325366482244570190$$ $$\beta_{16}\mathstrut -\mathstrut$$ $$4087771644600051035$$ $$\beta_{15}\mathstrut -\mathstrut$$ $$1249581684138318625$$ $$\beta_{14}\mathstrut +\mathstrut$$ $$831206469230632105$$ $$\beta_{13}\mathstrut -\mathstrut$$ $$915072820762448250$$ $$\beta_{12}\mathstrut +\mathstrut$$ $$996789626740254400$$ $$\beta_{11}\mathstrut -\mathstrut$$ $$94616116266941240$$ $$\beta_{10}\mathstrut -\mathstrut$$ $$109907376549484896$$ $$\beta_{9}\mathstrut +\mathstrut$$ $$4430378912941814580$$ $$\beta_{8}\mathstrut -\mathstrut$$ $$13564798892107007770$$ $$\beta_{7}\mathstrut -\mathstrut$$ $$6333385351600560380$$ $$\beta_{6}\mathstrut +\mathstrut$$ $$1011344101806315755$$ $$\beta_{5}\mathstrut -\mathstrut$$ $$44404385154167848590$$ $$\beta_{4}\mathstrut -\mathstrut$$ $$51471004864552738092$$ $$\beta_{3}\mathstrut +\mathstrut$$ $$287380621116097829855$$ $$\beta_{2}\mathstrut -\mathstrut$$ $$2390500563458819617689$$ $$\beta_{1}\mathstrut +\mathstrut$$ $$3111087605890877614306$$$$)/10000$$ $$\nu^{17}$$ $$=$$ $$($$$$-$$$$12126412681319286890$$ $$\beta_{19}\mathstrut -\mathstrut$$ $$4601097432636548275$$ $$\beta_{18}\mathstrut -\mathstrut$$ $$921257754041830625$$ $$\beta_{17}\mathstrut -\mathstrut$$ $$30814803816768160840$$ $$\beta_{16}\mathstrut -\mathstrut$$ $$42786105296115395740$$ $$\beta_{15}\mathstrut +\mathstrut$$ $$14559009901705323775$$ $$\beta_{14}\mathstrut -\mathstrut$$ $$6775678422584956005$$ $$\beta_{13}\mathstrut +\mathstrut$$ $$452031205324666125$$ $$\beta_{12}\mathstrut -\mathstrut$$ $$18947374294985127000$$ $$\beta_{11}\mathstrut -\mathstrut$$ $$7464716849475205585$$ $$\beta_{10}\mathstrut -\mathstrut$$ $$34633881184768322328$$ $$\beta_{9}\mathstrut -\mathstrut$$ $$86511426967096742315$$ $$\beta_{8}\mathstrut +\mathstrut$$ $$159218347724899825775$$ $$\beta_{7}\mathstrut +\mathstrut$$ $$28565410855930833715$$ $$\beta_{6}\mathstrut -\mathstrut$$ $$16536635068304401870$$ $$\beta_{5}\mathstrut +\mathstrut$$ $$849603525212838000525$$ $$\beta_{4}\mathstrut +\mathstrut$$ $$1145764696421820661339$$ $$\beta_{3}\mathstrut -\mathstrut$$ $$899726969819763134315$$ $$\beta_{2}\mathstrut -\mathstrut$$ $$1955281696870846847062$$ $$\beta_{1}\mathstrut -\mathstrut$$ $$2381294790659358339853087$$$$)/10000$$ $$\nu^{18}$$ $$=$$ $$($$$$355041762624936898930$$ $$\beta_{19}\mathstrut -\mathstrut$$ $$80124328419555648275$$ $$\beta_{18}\mathstrut -\mathstrut$$ $$49800048080140922625$$ $$\beta_{17}\mathstrut +\mathstrut$$ $$74105026426661360480$$ $$\beta_{16}\mathstrut +\mathstrut$$ $$450963973065595200580$$ $$\beta_{15}\mathstrut -\mathstrut$$ $$256360397535811781725$$ $$\beta_{14}\mathstrut +\mathstrut$$ $$151683649006613454135$$ $$\beta_{13}\mathstrut -\mathstrut$$ $$11175561158056302375$$ $$\beta_{12}\mathstrut +\mathstrut$$ $$302432171669502639500$$ $$\beta_{11}\mathstrut -\mathstrut$$ $$2885123124967183405$$ $$\beta_{10}\mathstrut +\mathstrut$$ $$746001102411148864772$$ $$\beta_{9}\mathstrut +\mathstrut$$ $$209043557469638080185$$ $$\beta_{8}\mathstrut -\mathstrut$$ $$1240333802043229355315$$ $$\beta_{7}\mathstrut -\mathstrut$$ $$1803526891084836416385$$ $$\beta_{6}\mathstrut -\mathstrut$$ $$181501840341490329290$$ $$\beta_{5}\mathstrut -\mathstrut$$ $$32944190128708816460075$$ $$\beta_{4}\mathstrut +\mathstrut$$ $$4237795922590652998399$$ $$\beta_{3}\mathstrut +\mathstrut$$ $$66604497368759191064585$$ $$\beta_{2}\mathstrut +\mathstrut$$ $$1220181278738092586167088$$ $$\beta_{1}\mathstrut +\mathstrut$$ $$30404091533996977147184713$$$$)/10000$$ $$\nu^{19}$$ $$=$$ $$($$$$-$$$$3394491025708634762825$$ $$\beta_{19}\mathstrut +\mathstrut$$ $$1390069587941412306375$$ $$\beta_{18}\mathstrut +\mathstrut$$ $$1642963784647447000625$$ $$\beta_{17}\mathstrut -\mathstrut$$ $$7401990148642859888450$$ $$\beta_{16}\mathstrut +\mathstrut$$ $$4931337466572375854675$$ $$\beta_{15}\mathstrut +\mathstrut$$ $$2872032270013229550250$$ $$\beta_{14}\mathstrut -\mathstrut$$ $$4769210780307171675400$$ $$\beta_{13}\mathstrut +\mathstrut$$ $$5413477038190455390375$$ $$\beta_{12}\mathstrut +\mathstrut$$ $$1637081919320633958500$$ $$\beta_{11}\mathstrut -\mathstrut$$ $$328354267490587478925$$ $$\beta_{10}\mathstrut -\mathstrut$$ $$19159847027598078195556$$ $$\beta_{9}\mathstrut -\mathstrut$$ $$20500213350586263429095$$ $$\beta_{8}\mathstrut +\mathstrut$$ $$23060309359545651643835$$ $$\beta_{7}\mathstrut +\mathstrut$$ $$11215001066929965487345$$ $$\beta_{6}\mathstrut -\mathstrut$$ $$181327434133196370455$$ $$\beta_{5}\mathstrut +\mathstrut$$ $$335857030531346337587385$$ $$\beta_{4}\mathstrut -\mathstrut$$ $$619554265834681317961757$$ $$\beta_{3}\mathstrut +\mathstrut$$ $$919367445673676985672830$$ $$\beta_{2}\mathstrut -\mathstrut$$ $$16711487076384740246534179$$ $$\beta_{1}\mathstrut -\mathstrut$$ $$331345756140820009151117859$$$$)/10000$$

## Character Values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/20\mathbb{Z}\right)^\times$$.

 $$n$$ $$11$$ $$17$$ $$\chi(n)$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
11.1
 −15.1364 − 2.79258i −15.1364 + 2.79258i −16.5762 − 5.67923i −16.5762 + 5.67923i −7.66078 − 13.6916i −7.66078 + 13.6916i −8.12576 − 14.6168i −8.12576 + 14.6168i −2.00318 − 15.9954i −2.00318 + 15.9954i 4.14605 − 15.6062i 4.14605 + 15.6062i 5.61235 − 15.2006i 5.61235 + 15.2006i 10.5132 − 10.4323i 10.5132 + 10.4323i 13.6469 − 4.79406i 13.6469 + 4.79406i 16.0838 − 4.09984i 16.0838 + 4.09984i
−31.5088 5.58516i 275.626i 961.612 + 351.964i 1397.54 −1539.42 + 8684.66i 19327.7i −28333.5 16460.7i −16920.9 −44034.9 7805.50i
11.2 −31.5088 + 5.58516i 275.626i 961.612 351.964i 1397.54 −1539.42 8684.66i 19327.7i −28333.5 + 16460.7i −16920.9 −44034.9 + 7805.50i
11.3 −29.9163 11.3585i 137.165i 765.971 + 679.606i −1397.54 −1557.98 + 4103.46i 24910.8i −15195.7 29031.6i 40234.9 41809.3 + 15873.9i
11.4 −29.9163 + 11.3585i 137.165i 765.971 679.606i −1397.54 −1557.98 4103.46i 24910.8i −15195.7 + 29031.6i 40234.9 41809.3 15873.9i
11.5 −16.5576 27.3833i 17.1314i −475.690 + 906.805i 1397.54 −469.114 + 283.655i 2883.30i 32707.6 1988.60i 58755.5 −23140.0 38269.3i
11.6 −16.5576 + 27.3833i 17.1314i −475.690 906.805i 1397.54 −469.114 283.655i 2883.30i 32707.6 + 1988.60i 58755.5 −23140.0 + 38269.3i
11.7 −13.0154 29.2335i 448.707i −685.197 + 760.974i −1397.54 −13117.3 + 5840.12i 19334.7i 31164.1 + 10126.3i −142289. 18189.6 + 40855.1i
11.8 −13.0154 + 29.2335i 448.707i −685.197 760.974i −1397.54 −13117.3 5840.12i 19334.7i 31164.1 10126.3i −142289. 18189.6 40855.1i
11.9 −0.770283 31.9907i 80.5620i −1022.81 + 49.2838i −1397.54 2577.24 62.0555i 345.112i 2364.48 + 32682.6i 52558.8 1076.50 + 44708.4i
11.10 −0.770283 + 31.9907i 80.5620i −1022.81 49.2838i −1397.54 2577.24 + 62.0555i 345.112i 2364.48 32682.6i 52558.8 1076.50 44708.4i
11.11 7.05603 31.2124i 442.423i −924.425 440.471i 1397.54 13809.1 + 3121.75i 2455.96i −20270.9 + 25745.5i −136689. 9861.10 43620.6i
11.12 7.05603 + 31.2124i 442.423i −924.425 + 440.471i 1397.54 13809.1 3121.75i 2455.96i −20270.9 25745.5i −136689. 9861.10 + 43620.6i
11.13 9.98863 30.4011i 330.781i −824.454 607.331i 1397.54 −10056.1 3304.05i 29449.0i −26698.7 + 18997.9i −50366.8 13959.5 42486.8i
11.14 9.98863 + 30.4011i 330.781i −824.454 + 607.331i 1397.54 −10056.1 + 3304.05i 29449.0i −26698.7 18997.9i −50366.8 13959.5 + 42486.8i
11.15 24.2624 20.8647i 208.777i 153.329 1012.46i −1397.54 4356.06 + 5065.43i 17557.3i −17404.4 27763.8i 15461.2 −33907.8 + 29159.3i
11.16 24.2624 + 20.8647i 208.777i 153.329 + 1012.46i −1397.54 4356.06 5065.43i 17557.3i −17404.4 + 27763.8i 15461.2 −33907.8 29159.3i
11.17 30.5298 9.58811i 321.971i 840.136 585.446i −1397.54 −3087.10 9829.71i 9880.78i 20035.9 25928.9i −44616.5 −42666.7 + 13399.8i
11.18 30.5298 + 9.58811i 321.971i 840.136 + 585.446i −1397.54 −3087.10 + 9829.71i 9880.78i 20035.9 + 25928.9i −44616.5 −42666.7 13399.8i
11.19 30.9316 8.19968i 206.425i 889.530 507.259i 1397.54 1692.62 + 6385.07i 1527.38i 23355.3 22984.2i 16437.6 43228.3 11459.4i
11.20 30.9316 + 8.19968i 206.425i 889.530 + 507.259i 1397.54 1692.62 6385.07i 1527.38i 23355.3 + 22984.2i 16437.6 43228.3 + 11459.4i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 11.20 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

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

## Hecke kernels

There are no other newforms in $$S_{11}^{\mathrm{new}}(20, [\chi])$$.