# Properties

 Label 12.12.b.a Level 12 Weight 12 Character orbit 12.b Analytic conductor 9.220 Analytic rank 0 Dimension 20 CM No Inner twists 4

# Related objects

## Newspace parameters

 Level: $$N$$ = $$12 = 2^{2} \cdot 3$$ Weight: $$k$$ = $$12$$ Character orbit: $$[\chi]$$ = 12.b (of order $$2$$ and degree $$1$$)

## Newform invariants

 Self dual: No Analytic conductor: $$9.22011816672$$ Analytic rank: $$0$$ Dimension: $$20$$ Coefficient field: $$\mathbb{Q}[x]/(x^{20} - \cdots)$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{104}\cdot 3^{42}\cdot 5^{2}$$ Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q$$ $$+ \beta_{1} q^{2}$$ $$-\beta_{3} q^{3}$$ $$+ ( 99 + \beta_{2} ) q^{4}$$ $$+ ( -5 \beta_{1} + \beta_{6} ) q^{5}$$ $$+ ( 656 - \beta_{8} ) q^{6}$$ $$+ ( -\beta_{1} - 3 \beta_{3} - \beta_{4} ) q^{7}$$ $$+ ( 98 \beta_{1} - 3 \beta_{3} + \beta_{5} ) q^{8}$$ $$+ ( 5182 - 63 \beta_{1} + 3 \beta_{2} - \beta_{3} - \beta_{6} + \beta_{9} ) q^{9}$$ $$+O(q^{10})$$ $$q$$ $$+ \beta_{1} q^{2}$$ $$-\beta_{3} q^{3}$$ $$+ ( 99 + \beta_{2} ) q^{4}$$ $$+ ( -5 \beta_{1} + \beta_{6} ) q^{5}$$ $$+ ( 656 - \beta_{8} ) q^{6}$$ $$+ ( -\beta_{1} - 3 \beta_{3} - \beta_{4} ) q^{7}$$ $$+ ( 98 \beta_{1} - 3 \beta_{3} + \beta_{5} ) q^{8}$$ $$+ ( 5182 - 63 \beta_{1} + 3 \beta_{2} - \beta_{3} - \beta_{6} + \beta_{9} ) q^{9}$$ $$+ ( 10099 + 7 \beta_{1} - 5 \beta_{2} + 22 \beta_{3} + \beta_{4} - \beta_{10} ) q^{10}$$ $$+ ( 1 - 166 \beta_{1} + \beta_{2} - 66 \beta_{3} - \beta_{5} + 2 \beta_{8} + \beta_{12} - \beta_{16} ) q^{11}$$ $$+ ( 53238 + 625 \beta_{1} - 98 \beta_{3} + 3 \beta_{4} - 5 \beta_{6} + \beta_{8} + \beta_{11} + \beta_{12} + \beta_{13} - \beta_{16} ) q^{12}$$ $$+ ( 77189 + 4 \beta_{1} - 45 \beta_{2} + 14 \beta_{3} - \beta_{6} + 12 \beta_{8} + \beta_{9} - 2 \beta_{10} + \beta_{11} + \beta_{13} + \beta_{15} ) q^{13}$$ $$+ ( 2 - 24 \beta_{1} - 3 \beta_{2} - 78 \beta_{3} + 2 \beta_{5} - 34 \beta_{6} - 2 \beta_{8} + 6 \beta_{9} - \beta_{11} + \beta_{12} - \beta_{13} + \beta_{15} - 2 \beta_{16} + \beta_{17} ) q^{14}$$ $$+ ( -12 + 1803 \beta_{1} - 69 \beta_{2} + 27 \beta_{3} + 3 \beta_{4} + 14 \beta_{5} - 5 \beta_{6} + \beta_{8} + 3 \beta_{9} - 6 \beta_{10} + 2 \beta_{12} + \beta_{14} - 2 \beta_{15} - \beta_{16} - \beta_{17} + \beta_{18} - \beta_{19} ) q^{15}$$ $$+ ( 174164 + 54 \beta_{1} + 84 \beta_{2} + 188 \beta_{3} - 34 \beta_{4} + 2 \beta_{5} - \beta_{7} - \beta_{8} + 12 \beta_{9} - 4 \beta_{10} - 2 \beta_{11} + 3 \beta_{12} + \beta_{13} - \beta_{14} - 3 \beta_{15} - \beta_{17} - \beta_{18} + 2 \beta_{19} ) q^{16}$$ $$+ ( -42 - 5420 \beta_{1} - 27 \beta_{2} - 10 \beta_{3} + 33 \beta_{5} - 26 \beta_{6} + 2 \beta_{7} - 94 \beta_{8} + 2 \beta_{9} + \beta_{11} - 2 \beta_{12} - 3 \beta_{13} - 3 \beta_{14} + 3 \beta_{15} - \beta_{17} - 2 \beta_{18} + \beta_{19} ) q^{17}$$ $$+ ( 127861 + 4978 \beta_{1} - 66 \beta_{2} - 571 \beta_{3} - 87 \beta_{4} + 2 \beta_{5} + 232 \beta_{6} + 2 \beta_{7} - 4 \beta_{8} + 2 \beta_{9} + 3 \beta_{10} + 13 \beta_{11} - 14 \beta_{12} + 3 \beta_{13} + \beta_{14} + \beta_{15} + 2 \beta_{16} + 5 \beta_{17} - 4 \beta_{19} ) q^{18}$$ $$+ ( 42 + 69 \beta_{1} + 641 \beta_{2} + 204 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} + 14 \beta_{6} - 6 \beta_{7} - 212 \beta_{8} - 2 \beta_{9} + 12 \beta_{10} - 11 \beta_{11} + 11 \beta_{12} + 10 \beta_{14} + 4 \beta_{15} - 2 \beta_{17} + 4 \beta_{18} - 2 \beta_{19} ) q^{19}$$ $$+ ( -14 + 9558 \beta_{1} + 38 \beta_{2} - 1572 \beta_{3} - 10 \beta_{5} + 710 \beta_{6} - \beta_{7} + 7 \beta_{8} - 40 \beta_{9} - 21 \beta_{11} + 17 \beta_{12} + \beta_{13} - 3 \beta_{14} - \beta_{15} + 2 \beta_{16} + \beta_{17} + \beta_{18} + 10 \beta_{19} ) q^{20}$$ $$+ ( -610265 - 7051 \beta_{1} + 1547 \beta_{2} - 500 \beta_{3} - 104 \beta_{5} - 50 \beta_{6} + 10 \beta_{7} + 38 \beta_{8} - 9 \beta_{9} + 6 \beta_{10} + 24 \beta_{11} - 10 \beta_{12} + \beta_{13} - 16 \beta_{14} - 7 \beta_{15} - 8 \beta_{17} + 6 \beta_{18} - 8 \beta_{19} ) q^{21}$$ $$+ ( -296076 - 996 \beta_{1} - 86 \beta_{2} - 3183 \beta_{3} + 280 \beta_{4} + 10 \beta_{6} - 16 \beta_{7} - 72 \beta_{8} - 40 \beta_{9} + 16 \beta_{10} - 3 \beta_{11} - 45 \beta_{12} - 4 \beta_{13} - 3 \beta_{14} - 4 \beta_{15} + 12 \beta_{17} - 12 \beta_{18} ) q^{22}$$ $$+ ( 360 - 57326 \beta_{1} + 216 \beta_{2} + 3448 \beta_{3} - 132 \beta_{5} - 4 \beta_{6} + 14 \beta_{7} + 782 \beta_{8} - 4 \beta_{9} - 25 \beta_{11} - 16 \beta_{12} - 16 \beta_{13} + 36 \beta_{14} + 16 \beta_{15} + 24 \beta_{16} - 12 \beta_{17} - 14 \beta_{18} + 4 \beta_{19} ) q^{23}$$ $$+ ( 606760 + 53316 \beta_{1} + 676 \beta_{2} - 271 \beta_{3} + 426 \beta_{4} - \beta_{5} - 2308 \beta_{6} + 31 \beta_{7} - 45 \beta_{8} - 12 \beta_{9} + 12 \beta_{10} + 62 \beta_{11} + 55 \beta_{12} + 9 \beta_{13} - 5 \beta_{14} - 23 \beta_{15} + 20 \beta_{16} + 11 \beta_{17} - \beta_{18} + 2 \beta_{19} ) q^{24}$$ $$+ ( -7057393 + 768 \beta_{1} - 7635 \beta_{2} + 2260 \beta_{3} - 13 \beta_{5} - 112 \beta_{6} - 64 \beta_{7} + 1496 \beta_{8} - 26 \beta_{9} + 36 \beta_{10} - 4 \beta_{11} + 64 \beta_{12} - 31 \beta_{13} - 49 \beta_{14} - 5 \beta_{15} - 19 \beta_{17} + 32 \beta_{18} - 13 \beta_{19} ) q^{25}$$ $$+ ( -20 + 84958 \beta_{1} - 238 \beta_{2} + 23338 \beta_{3} - 36 \beta_{5} - 3848 \beta_{6} - 12 \beta_{7} + 20 \beta_{8} - 60 \beta_{9} - 116 \beta_{11} - 292 \beta_{12} + 6 \beta_{13} - 6 \beta_{14} - 6 \beta_{15} + 44 \beta_{16} + 18 \beta_{17} + 12 \beta_{18} + 24 \beta_{19} ) q^{26}$$ $$+ ( -2297 - 104386 \beta_{1} - 11709 \beta_{2} - 2077 \beta_{3} + 6 \beta_{4} + 37 \beta_{5} - 12 \beta_{6} + 106 \beta_{7} - 8 \beta_{8} - 12 \beta_{9} + 96 \beta_{10} - 115 \beta_{11} - 85 \beta_{12} + 92 \beta_{14} + 32 \beta_{15} + 25 \beta_{16} - 20 \beta_{17} + 6 \beta_{18} - 20 \beta_{19} ) q^{27}$$ $$+ ( -180606 + 11472 \beta_{1} - 570 \beta_{2} + 35672 \beta_{3} - 630 \beta_{4} - 2 \beta_{5} + 28 \beta_{6} - 115 \beta_{7} - 149 \beta_{8} - 160 \beta_{9} + 80 \beta_{10} - 9 \beta_{11} + 545 \beta_{12} + 13 \beta_{13} - 5 \beta_{14} + 17 \beta_{15} + 55 \beta_{17} - 53 \beta_{18} - 2 \beta_{19} ) q^{28}$$ $$+ ( -1212 - 236355 \beta_{1} - 686 \beta_{2} - 756 \beta_{3} - 146 \beta_{5} - 589 \beta_{6} + 24 \beta_{7} - 2568 \beta_{8} - 60 \beta_{9} + 352 \beta_{11} - 24 \beta_{12} - 2 \beta_{13} - 90 \beta_{14} + 2 \beta_{15} - 46 \beta_{17} - 24 \beta_{18} - 18 \beta_{19} ) q^{29}$$ $$+ ( 3673378 - 4012 \beta_{1} + 1937 \beta_{2} - 9898 \beta_{3} - 360 \beta_{4} - 86 \beta_{5} + 10678 \beta_{6} + 208 \beta_{7} - 40 \beta_{8} - 42 \beta_{9} - 48 \beta_{10} + 333 \beta_{11} - 1039 \beta_{12} + 7 \beta_{13} - 40 \beta_{14} - 79 \beta_{15} - 42 \beta_{16} + \beta_{17} - 12 \beta_{18} + 64 \beta_{19} ) q^{30}$$ $$+ ( 5478 + 7069 \beta_{1} + 33662 \beta_{2} + 21943 \beta_{3} + 285 \beta_{4} - 14 \beta_{5} + 194 \beta_{6} - 298 \beta_{7} - 2972 \beta_{8} + 82 \beta_{9} - 204 \beta_{10} - 325 \beta_{11} + 388 \beta_{12} - 48 \beta_{13} + 118 \beta_{14} - 20 \beta_{15} - 62 \beta_{17} + 76 \beta_{18} - 14 \beta_{19} ) q^{31}$$ $$+ ( 280 + 132672 \beta_{1} + 1296 \beta_{2} - 126196 \beta_{3} + 108 \beta_{5} + 13752 \beta_{6} - 52 \beta_{7} + 172 \beta_{8} + 464 \beta_{9} - 476 \beta_{11} + 1492 \beta_{12} + 12 \beta_{13} + 12 \beta_{14} - 12 \beta_{15} - 40 \beta_{16} + 92 \beta_{17} + 52 \beta_{18} - 56 \beta_{19} ) q^{32}$$ $$+ ( 11499543 + 263019 \beta_{1} + 53840 \beta_{2} - 14861 \beta_{3} + 873 \beta_{5} - 1087 \beta_{6} + 470 \beta_{7} + 158 \beta_{8} + 111 \beta_{9} - 108 \beta_{10} - 467 \beta_{11} - 470 \beta_{12} - 45 \beta_{13} - 75 \beta_{14} + 153 \beta_{15} - 9 \beta_{17} - 22 \beta_{18} + 9 \beta_{19} ) q^{33}$$ $$+ ( 11107956 - 54608 \beta_{1} - 3868 \beta_{2} - 169848 \beta_{3} - 1044 \beta_{4} + 80 \beta_{5} - 8 \beta_{6} - 488 \beta_{7} + 20 \beta_{8} + 488 \beta_{9} - 100 \beta_{10} - 206 \beta_{11} - 1904 \beta_{12} + 156 \beta_{13} - 4 \beta_{14} - 4 \beta_{15} - 4 \beta_{17} - 76 \beta_{18} + 80 \beta_{19} ) q^{34}$$ $$+ ( 1021 + 850793 \beta_{1} + 648 \beta_{2} - 26182 \beta_{3} + 1355 \beta_{5} - 648 \beta_{6} - 36 \beta_{7} + 2102 \beta_{8} + 120 \beta_{9} + 1454 \beta_{11} + 312 \beta_{12} + 96 \beta_{13} + 72 \beta_{14} - 96 \beta_{15} - 253 \beta_{16} - 24 \beta_{17} + 36 \beta_{18} - 120 \beta_{19} ) q^{35}$$ $$+ ( 14905089 + 111286 \beta_{1} + 5403 \beta_{2} - 55380 \beta_{3} - 2256 \beta_{4} + 110 \beta_{5} - 22018 \beta_{6} + 779 \beta_{7} - 133 \beta_{8} + 136 \beta_{9} - 240 \beta_{10} + 1867 \beta_{11} + 3061 \beta_{12} - 99 \beta_{13} - 23 \beta_{14} + 163 \beta_{15} - 166 \beta_{16} + 77 \beta_{17} - 51 \beta_{18} + 50 \beta_{19} ) q^{36}$$ $$+ ( -2249083 + 10996 \beta_{1} - 115099 \beta_{2} + 34526 \beta_{3} + 58 \beta_{5} - 37 \beta_{6} - 832 \beta_{7} + 172 \beta_{8} + 505 \beta_{9} - 274 \beta_{10} - 355 \beta_{11} + 832 \beta_{12} + 195 \beta_{13} + 34 \beta_{14} + 79 \beta_{15} - 26 \beta_{17} - 32 \beta_{18} + 58 \beta_{19} ) q^{37}$$ $$+ ( 24 + 141856 \beta_{1} - 5288 \beta_{2} + 436947 \beta_{3} + 676 \beta_{5} - 23718 \beta_{6} - 64 \beta_{7} + 194 \beta_{8} - 148 \beta_{9} - 2257 \beta_{11} - 5113 \beta_{12} + 174 \beta_{13} - 33 \beta_{14} - 174 \beta_{15} - 420 \beta_{16} - 46 \beta_{17} + 64 \beta_{18} - 128 \beta_{19} ) q^{38}$$ $$+ ( -27448 - 1992112 \beta_{1} - 139337 \beta_{2} - 41152 \beta_{3} - 870 \beta_{4} - 1738 \beta_{5} + 543 \beta_{6} + 1154 \beta_{7} - 37 \beta_{8} + 39 \beta_{9} - 582 \beta_{10} - 2927 \beta_{11} - 922 \beta_{12} - 144 \beta_{13} - 227 \beta_{14} - 50 \beta_{15} - 277 \beta_{16} + 11 \beta_{17} - 33 \beta_{18} + 155 \beta_{19} ) q^{39}$$ $$+ ( -21984696 + 234204 \beta_{1} + 200 \beta_{2} + 729816 \beta_{3} + 6068 \beta_{4} + 12 \beta_{5} + 48 \beta_{6} - 1334 \beta_{7} - 1374 \beta_{8} + 504 \beta_{9} - 712 \beta_{10} - 904 \beta_{11} + 9634 \beta_{12} - 66 \beta_{13} + 66 \beta_{14} - 90 \beta_{15} - 174 \beta_{17} + 162 \beta_{18} + 12 \beta_{19} ) q^{40}$$ $$+ ( 4512 - 2156910 \beta_{1} + 2026 \beta_{2} - 9500 \beta_{3} - 1292 \beta_{5} + 3358 \beta_{6} - 138 \beta_{7} + 8838 \beta_{8} + 1020 \beta_{9} + 4949 \beta_{11} + 138 \beta_{12} + 40 \beta_{13} + 348 \beta_{14} - 40 \beta_{15} + 236 \beta_{17} + 138 \beta_{18} - 108 \beta_{19} ) q^{41}$$ $$+ ( 14091001 - 611337 \beta_{1} - 5091 \beta_{2} + 438 \beta_{3} + 6891 \beta_{4} + 1576 \beta_{5} + 13544 \beta_{6} + 1728 \beta_{7} + 60 \beta_{8} + 144 \beta_{9} + 285 \beta_{10} + 7194 \beta_{11} - 13496 \beta_{12} - 72 \beta_{13} + 224 \beta_{14} + 608 \beta_{15} + 376 \beta_{16} + 16 \beta_{17} - 52 \beta_{18} - 128 \beta_{19} ) q^{42}$$ $$+ ( 47466 - 2135 \beta_{1} + 211017 \beta_{2} - 7760 \beta_{3} - 3526 \beta_{4} + 62 \beta_{5} - 1010 \beta_{6} - 1606 \beta_{7} + 15500 \beta_{8} - 514 \beta_{9} + 1356 \beta_{10} + 165 \beta_{11} + 1043 \beta_{12} + 288 \beta_{13} - 598 \beta_{14} + 164 \beta_{15} + 350 \beta_{17} - 412 \beta_{18} + 62 \beta_{19} ) q^{43}$$ $$+ ( -2694 - 704930 \beta_{1} + 15554 \beta_{2} - 1278420 \beta_{3} - 582 \beta_{5} + 2998 \beta_{6} + 215 \beta_{7} - 3841 \beta_{8} - 2032 \beta_{9} - 9249 \beta_{11} + 15497 \beta_{12} + 85 \beta_{13} - 135 \beta_{14} - 85 \beta_{15} + 330 \beta_{16} - 515 \beta_{17} - 215 \beta_{18} + 58 \beta_{19} ) q^{44}$$ $$+ ( 51069034 + 6278151 \beta_{1} + 288930 \beta_{2} - 60610 \beta_{3} - 1146 \beta_{5} + 13013 \beta_{6} + 1782 \beta_{7} - 1338 \beta_{8} - 722 \beta_{9} + 792 \beta_{10} - 11643 \beta_{11} - 1782 \beta_{12} + 78 \beta_{13} + 582 \beta_{14} - 870 \beta_{15} + 42 \beta_{17} - 6 \beta_{18} + 294 \beta_{19} ) q^{45}$$ $$+ ( -119643400 - 492544 \beta_{1} - 34956 \beta_{2} - 1541110 \beta_{3} - 6464 \beta_{4} - 512 \beta_{5} - 108 \beta_{6} - 2304 \beta_{7} + 1096 \beta_{8} - 2496 \beta_{9} + 256 \beta_{10} - 1626 \beta_{11} - 18418 \beta_{12} - 1120 \beta_{13} + 10 \beta_{14} - 96 \beta_{15} - 224 \beta_{17} + 736 \beta_{18} - 512 \beta_{19} ) q^{46}$$ $$+ ( -14554 + 9582606 \beta_{1} - 9012 \beta_{2} + 45292 \beta_{3} - 3758 \beta_{5} + 3656 \beta_{6} - 124 \beta_{7} - 30864 \beta_{8} - 568 \beta_{9} + 16514 \beta_{11} - 1828 \beta_{12} - 160 \beta_{13} - 1224 \beta_{14} + 160 \beta_{15} + 1498 \beta_{16} + 408 \beta_{17} + 124 \beta_{18} + 568 \beta_{19} ) q^{47}$$ $$+ ( 34460572 + 537750 \beta_{1} + 53636 \beta_{2} - 200824 \beta_{3} - 5442 \beta_{4} - 722 \beta_{5} + 47912 \beta_{6} + 2083 \beta_{7} + 1875 \beta_{8} - 1188 \beta_{9} + 2076 \beta_{10} + 20818 \beta_{11} + 25119 \beta_{12} + 341 \beta_{13} + 467 \beta_{14} - 343 \beta_{15} + 680 \beta_{16} - 797 \beta_{17} + 267 \beta_{18} - 230 \beta_{19} ) q^{48}$$ $$+ ( -50980915 + 27184 \beta_{1} - 271961 \beta_{2} + 80212 \beta_{3} - 195 \beta_{5} + 3588 \beta_{6} - 1984 \beta_{7} - 45000 \beta_{8} - 4122 \beta_{9} + 1092 \beta_{10} - 2192 \beta_{11} + 1984 \beta_{12} - 741 \beta_{13} + 1185 \beta_{14} - 351 \beta_{15} + 867 \beta_{17} - 672 \beta_{18} - 195 \beta_{19} ) q^{49}$$ $$+ ( 468 - 6040571 \beta_{1} - 37266 \beta_{2} + 3184374 \beta_{3} - 6300 \beta_{5} + 74696 \beta_{6} + 812 \beta_{7} - 2164 \beta_{8} + 3260 \beta_{9} - 28668 \beta_{11} - 36924 \beta_{12} - 1382 \beta_{13} + 486 \beta_{14} + 1382 \beta_{15} + 2196 \beta_{16} - 242 \beta_{17} - 812 \beta_{18} + 40 \beta_{19} ) q^{50}$$ $$+ ( -44939 - 14893905 \beta_{1} - 229088 \beta_{2} + 17294 \beta_{3} + 10548 \beta_{4} + 8535 \beta_{5} - 2486 \beta_{6} + 1576 \beta_{7} + 848 \beta_{8} - 198 \beta_{9} + 1260 \beta_{10} - 28540 \beta_{11} - 4894 \beta_{12} + 864 \beta_{13} - 1506 \beta_{14} - 444 \beta_{15} + 1749 \beta_{16} + 642 \beta_{17} - 122 \beta_{18} - 222 \beta_{19} ) q^{51}$$ $$+ ( -197953406 + 1056848 \beta_{1} + 41302 \beta_{2} + 3296160 \beta_{3} - 7136 \beta_{4} - 160 \beta_{5} - 1952 \beta_{6} - 1808 \beta_{7} + 23840 \beta_{8} + 1184 \beta_{9} + 3680 \beta_{10} - 536 \beta_{11} + 44464 \beta_{12} - 96 \beta_{13} - 544 \beta_{14} + 224 \beta_{15} - 640 \beta_{17} + 800 \beta_{18} - 160 \beta_{19} ) q^{52}$$ $$+ ( 15936 - 18260553 \beta_{1} + 14764 \beta_{2} - 80328 \beta_{3} + 8920 \beta_{5} - 11291 \beta_{6} - 172 \beta_{7} + 41780 \beta_{8} - 7672 \beta_{9} + 40374 \beta_{11} + 172 \beta_{12} + 496 \beta_{13} + 1032 \beta_{14} - 496 \beta_{15} - 152 \beta_{17} + 172 \beta_{18} + 1432 \beta_{19} ) q^{53}$$ $$+ ( -212475928 - 20876 \beta_{1} - 96660 \beta_{2} - 85739 \beta_{3} - 14424 \beta_{4} - 12556 \beta_{5} - 181530 \beta_{6} + 272 \beta_{7} + 1619 \beta_{8} - 444 \beta_{9} - 528 \beta_{10} + 52135 \beta_{11} - 48947 \beta_{12} - 342 \beta_{13} - 299 \beta_{14} - 1634 \beta_{15} - 1780 \beta_{16} - 754 \beta_{17} + 876 \beta_{18} - 1024 \beta_{19} ) q^{54}$$ $$+ ( 18150 + 69730 \beta_{1} + 28430 \beta_{2} + 216950 \beta_{3} + 22930 \beta_{4} + 370 \beta_{5} - 2110 \beta_{6} - 170 \beta_{7} + 31780 \beta_{8} + 850 \beta_{9} - 3660 \beta_{10} + 795 \beta_{11} + 660 \beta_{12} - 240 \beta_{13} - 1610 \beta_{14} - 980 \beta_{15} + 130 \beta_{17} - 500 \beta_{18} + 370 \beta_{19} ) q^{55}$$ $$+ ( 16920 - 1722972 \beta_{1} + 68832 \beta_{2} - 4874698 \beta_{3} + 974 \beta_{5} - 232552 \beta_{6} + 636 \beta_{7} + 34140 \beta_{8} + 1968 \beta_{9} - 69692 \beta_{11} + 60772 \beta_{12} - 1204 \beta_{13} + 1452 \beta_{14} + 1204 \beta_{15} - 1320 \beta_{16} - 68 \beta_{17} - 636 \beta_{18} + 360 \beta_{19} ) q^{56}$$ $$+ ( 71889345 + 37489209 \beta_{1} - 216807 \beta_{2} + 216213 \beta_{3} - 10518 \beta_{5} - 71829 \beta_{6} - 1230 \beta_{7} - 1854 \beta_{8} + 2211 \beta_{9} - 2736 \beta_{10} - 74517 \beta_{11} + 1230 \beta_{12} + 894 \beta_{13} + 1434 \beta_{14} + 1842 \beta_{15} + 1686 \beta_{17} - 18 \beta_{18} - 1590 \beta_{19} ) q^{57}$$ $$+ ( 483574445 - 1705503 \beta_{1} - 172923 \beta_{2} - 5317782 \beta_{3} + 33919 \beta_{4} + 480 \beta_{5} - 560 \beta_{6} + 2448 \beta_{7} + 3192 \beta_{8} + 4976 \beta_{9} + 177 \beta_{10} + 2732 \beta_{11} - 69792 \beta_{12} + 2984 \beta_{13} + 744 \beta_{14} + 2024 \beta_{15} - 24 \beta_{17} - 456 \beta_{18} + 480 \beta_{19} ) q^{58}$$ $$+ ( 3596 + 51970723 \beta_{1} + 1911 \beta_{2} - 336084 \beta_{3} - 6428 \beta_{5} + 656 \beta_{6} + 8 \beta_{7} + 7008 \beta_{8} - 112 \beta_{9} + 98692 \beta_{11} + 2711 \beta_{12} - 64 \beta_{13} - 144 \beta_{14} + 64 \beta_{15} - 5132 \beta_{16} + 48 \beta_{17} - 8 \beta_{18} + 112 \beta_{19} ) q^{59}$$ $$+ ( 284402990 + 3630138 \beta_{1} - 10602 \beta_{2} - 50844 \beta_{3} + 48984 \beta_{4} + 1198 \beta_{5} + 292394 \beta_{6} - 4587 \beta_{7} + 301 \beta_{8} + 5856 \beta_{9} - 9840 \beta_{10} + 112713 \beta_{11} + 78643 \beta_{12} - 513 \beta_{13} - 2509 \beta_{14} - 391 \beta_{15} - 938 \beta_{16} + 655 \beta_{17} + 419 \beta_{18} - 722 \beta_{19} ) q^{60}$$ $$+ ( 343849569 - 83452 \beta_{1} + 811797 \beta_{2} - 238074 \beta_{3} + 1894 \beta_{5} - 4633 \beta_{6} + 5440 \beta_{7} + 52540 \beta_{8} + 17797 \beta_{9} - 2026 \beta_{10} + 3105 \beta_{11} - 5440 \beta_{12} + 2907 \beta_{13} - 1538 \beta_{14} - 881 \beta_{15} - 2438 \beta_{17} + 544 \beta_{18} + 1894 \beta_{19} ) q^{61}$$ $$+ ( -2290 + 1998936 \beta_{1} - 69853 \beta_{2} + 6262612 \beta_{3} + 33062 \beta_{5} + 343670 \beta_{6} - 1216 \beta_{7} + 5478 \beta_{8} - 11278 \beta_{9} - 139825 \beta_{11} - 72131 \beta_{12} + 2669 \beta_{13} - 1218 \beta_{14} - 2669 \beta_{15} - 6182 \beta_{16} - 237 \beta_{17} + 1216 \beta_{18} + 640 \beta_{19} ) q^{62}$$ $$+ ( 205154 - 86547197 \beta_{1} + 1051578 \beta_{2} + 24535 \beta_{3} - 64419 \beta_{4} - 4846 \beta_{5} + 318 \beta_{6} - 8416 \beta_{7} - 3154 \beta_{8} - 114 \beta_{9} + 2532 \beta_{10} - 164240 \beta_{11} + 21040 \beta_{12} + 3034 \beta_{14} + 844 \beta_{15} - 6580 \beta_{16} - 730 \beta_{17} + 570 \beta_{18} - 730 \beta_{19} ) q^{63}$$ $$+ ( -72689552 + 1877768 \beta_{1} + 65392 \beta_{2} + 5869264 \beta_{3} - 54520 \beta_{4} - 392 \beta_{5} + 9920 \beta_{6} + 11012 \beta_{7} - 128156 \beta_{8} - 8816 \beta_{9} - 11952 \beta_{10} + 3608 \beta_{11} + 69108 \beta_{12} + 540 \beta_{13} + 4324 \beta_{14} + 1324 \beta_{15} + 2596 \beta_{17} - 2204 \beta_{18} - 392 \beta_{19} ) q^{64}$$ $$+ ( -49260 - 106360250 \beta_{1} - 50780 \beta_{2} - 412320 \beta_{3} - 3350 \beta_{5} + 127190 \beta_{6} + 1310 \beta_{7} - 136690 \beta_{8} + 30800 \beta_{9} + 203325 \beta_{11} - 1310 \beta_{12} - 3890 \beta_{13} - 3030 \beta_{14} + 3890 \beta_{15} + 1270 \beta_{17} - 1310 \beta_{18} - 3830 \beta_{19} ) q^{65}$$ $$+ ( -547999329 + 11585803 \beta_{1} + 265050 \beta_{2} + 272447 \beta_{3} - 47445 \beta_{4} + 54870 \beta_{5} - 238496 \beta_{6} - 12882 \beta_{7} - 1328 \beta_{8} + 5982 \beta_{9} - 1935 \beta_{10} + 206749 \beta_{11} - 69722 \beta_{12} + 2317 \beta_{13} - 441 \beta_{14} + 327 \beta_{15} + 4070 \beta_{16} + 915 \beta_{17} - 2052 \beta_{18} + 3876 \beta_{19} ) q^{66}$$ $$+ ( -432636 - 391477 \beta_{1} - 1867793 \beta_{2} - 1201422 \beta_{3} - 97496 \beta_{4} - 1140 \beta_{5} + 11244 \beta_{6} + 14244 \beta_{7} - 171432 \beta_{8} + 2124 \beta_{9} - 2952 \beta_{10} - 1518 \beta_{11} - 17741 \beta_{12} - 1632 \beta_{13} + 7332 \beta_{14} + 648 \beta_{15} - 2772 \beta_{17} + 3912 \beta_{18} - 1140 \beta_{19} ) q^{67}$$ $$+ ( -78392 + 9391512 \beta_{1} + 8888 \beta_{2} - 5426224 \beta_{3} + 2712 \beta_{5} - 158248 \beta_{6} - 3108 \beta_{7} - 182788 \beta_{8} + 14880 \beta_{9} - 252756 \beta_{11} + 64996 \beta_{12} + 1924 \beta_{13} - 7116 \beta_{14} - 1924 \beta_{15} + 1544 \beta_{16} + 4292 \beta_{17} + 3108 \beta_{18} - 1944 \beta_{19} ) q^{68}$$ $$+ ( -665933610 + 135346362 \beta_{1} - 2496572 \beta_{2} + 1311494 \beta_{3} + 36324 \beta_{5} + 344950 \beta_{6} - 17570 \beta_{7} + 9694 \beta_{8} - 3390 \beta_{9} + 2160 \beta_{10} - 298111 \beta_{11} + 17570 \beta_{12} - 720 \beta_{13} - 9156 \beta_{14} - 1440 \beta_{15} - 7380 \beta_{17} - 14 \beta_{18} + 4356 \beta_{19} ) q^{69}$$ $$+ ( 1759616192 - 1600612 \beta_{1} + 886780 \beta_{2} - 4967070 \beta_{3} + 1720 \beta_{4} + 3072 \beta_{5} + 2668 \beta_{6} + 21296 \beta_{7} - 19764 \beta_{8} + 7544 \beta_{9} - 2864 \beta_{10} + 8196 \beta_{11} - 77026 \beta_{12} - 1396 \beta_{13} - 2994 \beta_{14} - 7540 \beta_{15} + 1116 \beta_{17} - 4188 \beta_{18} + 3072 \beta_{19} ) q^{70}$$ $$+ ( 119030 + 180819464 \beta_{1} + 81868 \beta_{2} - 1312780 \beta_{3} + 53398 \beta_{5} - 33644 \beta_{6} + 1402 \beta_{7} + 255350 \beta_{8} + 5140 \beta_{9} + 344029 \beta_{11} + 11540 \beta_{12} + 1168 \beta_{13} + 11916 \beta_{14} - 1168 \beta_{15} + 8074 \beta_{16} - 3972 \beta_{17} - 1402 \beta_{18} - 5140 \beta_{19} ) q^{71}$$ $$+ ( 1521890600 + 14689734 \beta_{1} + 56040 \beta_{2} - 652511 \beta_{3} - 33372 \beta_{4} + 1785 \beta_{5} - 128912 \beta_{6} - 23166 \beta_{7} - 29190 \beta_{8} - 7912 \beta_{9} + 25560 \beta_{10} + 346776 \beta_{11} + 55674 \beta_{12} - 1626 \beta_{13} + 10074 \beta_{14} + 3822 \beta_{15} - 3456 \beta_{16} + 6186 \beta_{17} - 3846 \beta_{18} + 3612 \beta_{19} ) q^{72}$$ $$+ ( -620604720 - 273544 \beta_{1} + 2985056 \beta_{2} - 902484 \beta_{3} - 8062 \beta_{5} - 22006 \beta_{6} + 24768 \beta_{7} + 308216 \beta_{8} - 45350 \beta_{9} - 1236 \beta_{10} + 25582 \beta_{11} - 24768 \beta_{12} - 7444 \beta_{13} - 7606 \beta_{14} + 8680 \beta_{15} + 1886 \beta_{17} + 6176 \beta_{18} - 8062 \beta_{19} ) q^{73}$$ $$+ ( 1652 - 1842346 \beta_{1} + 574 \beta_{2} + 1098406 \beta_{3} - 110332 \beta_{5} - 345528 \beta_{6} - 3860 \beta_{7} + 5324 \beta_{8} + 220 \beta_{9} - 437708 \beta_{11} - 1724 \beta_{12} + 3626 \beta_{13} - 810 \beta_{14} - 3626 \beta_{15} + 4660 \beta_{16} + 4094 \beta_{17} + 3860 \beta_{18} + 296 \beta_{19} ) q^{74}$$ $$+ ( 673469 - 272954053 \beta_{1} + 3408412 \beta_{2} + 5000827 \beta_{3} + 239790 \beta_{4} - 59125 \beta_{5} + 22638 \beta_{6} - 25174 \beta_{7} - 11902 \beta_{8} + 2910 \beta_{9} - 18420 \beta_{10} - 512819 \beta_{11} - 2842 \beta_{12} - 9216 \beta_{13} + 11818 \beta_{14} + 3076 \beta_{15} + 12653 \beta_{16} - 5986 \beta_{17} + 84 \beta_{18} + 3230 \beta_{19} ) q^{75}$$ $$+ ( -3004179250 - 1654952 \beta_{1} + 291402 \beta_{2} - 5159048 \beta_{3} + 105238 \beta_{4} + 3106 \beta_{5} - 30092 \beta_{6} + 28755 \beta_{7} + 442429 \beta_{8} + 5360 \beta_{9} + 23456 \beta_{10} + 30101 \beta_{11} - 82561 \beta_{12} - 149 \beta_{13} - 19379 \beta_{14} - 6361 \beta_{15} + 2305 \beta_{17} - 5411 \beta_{18} + 3106 \beta_{19} ) q^{76}$$ $$+ ( -136080 - 267347602 \beta_{1} - 30788 \beta_{2} - 1116680 \beta_{3} - 85328 \beta_{5} - 817718 \beta_{6} + 124 \beta_{7} - 222308 \beta_{8} - 72824 \beta_{9} + 563826 \beta_{11} - 124 \beta_{12} + 11512 \beta_{13} - 11424 \beta_{14} - 11512 \beta_{15} - 11760 \beta_{17} - 124 \beta_{18} + 752 \beta_{19} ) q^{77}$$ $$+ ( -4052318260 + 5772 \beta_{1} - 1902038 \beta_{2} + 350786 \beta_{3} + 148344 \beta_{4} - 148868 \beta_{5} + 1033080 \beta_{6} - 27632 \beta_{7} - 17938 \beta_{8} - 28500 \beta_{9} + 10896 \beta_{10} + 563858 \beta_{11} + 120552 \beta_{12} - 2034 \beta_{13} + 914 \beta_{14} + 8330 \beta_{15} + 772 \beta_{16} + 7450 \beta_{17} - 2748 \beta_{18} + 1600 \beta_{19} ) q^{78}$$ $$+ ( -757566 + 643457 \beta_{1} - 3412554 \beta_{2} + 1962119 \beta_{3} + 297877 \beta_{4} - 3066 \beta_{5} + 12726 \beta_{6} + 28690 \beta_{7} - 189588 \beta_{8} - 11802 \beta_{9} + 44604 \beta_{10} + 13001 \beta_{11} + 13912 \beta_{12} + 4368 \beta_{13} + 10962 \beta_{14} + 10500 \beta_{15} + 1302 \beta_{17} + 1764 \beta_{18} - 3066 \beta_{19} ) q^{79}$$ $$+ ( 280016 - 17032672 \beta_{1} + 41248 \beta_{2} + 15588728 \beta_{3} - 14920 \beta_{5} + 1402896 \beta_{6} - 2776 \beta_{7} + 656232 \beta_{8} - 48160 \beta_{9} - 680136 \beta_{11} - 158568 \beta_{12} + 8936 \beta_{13} + 22632 \beta_{14} - 8936 \beta_{15} + 8272 \beta_{16} - 3384 \beta_{17} + 2776 \beta_{18} + 2800 \beta_{19} ) q^{80}$$ $$+ ( -2123845653 + 342236538 \beta_{1} - 3585447 \beta_{2} + 2437842 \beta_{3} + 16515 \beta_{5} - 1407192 \beta_{6} - 27000 \beta_{7} + 31464 \beta_{8} + 1266 \beta_{9} + 16632 \beta_{10} - 711954 \beta_{11} + 27000 \beta_{12} - 19017 \beta_{13} - 2817 \beta_{14} + 2385 \beta_{15} + 5661 \beta_{17} + 2520 \beta_{18} - 7677 \beta_{19} ) q^{81}$$ $$+ ( 4408466582 + 5388158 \beta_{1} - 2476506 \beta_{2} + 16784396 \beta_{3} - 173166 \beta_{4} - 6144 \beta_{5} + 3328 \beta_{6} + 24192 \beta_{7} - 16128 \beta_{8} - 51712 \beta_{9} + 6766 \beta_{10} + 13376 \beta_{11} + 208896 \beta_{12} - 11008 \beta_{13} - 384 \beta_{14} + 1280 \beta_{15} + 5376 \beta_{17} + 768 \beta_{18} - 6144 \beta_{19} ) q^{82}$$ $$+ ( -26079 + 400912732 \beta_{1} - 79025 \beta_{2} + 4749302 \beta_{3} - 72729 \beta_{5} + 29624 \beta_{6} + 2300 \beta_{7} - 52066 \beta_{8} - 5704 \beta_{9} + 762174 \beta_{11} - 74241 \beta_{12} - 5152 \beta_{13} - 1656 \beta_{14} + 5152 \beta_{15} + 8415 \beta_{16} + 552 \beta_{17} - 2300 \beta_{18} + 5704 \beta_{19} ) q^{83}$$ $$+ ( 6275029520 + 14330850 \beta_{1} - 726512 \beta_{2} + 512820 \beta_{3} - 244416 \beta_{4} - 9886 \beta_{5} - 1841966 \beta_{6} - 17731 \beta_{7} + 68661 \beta_{8} - 40056 \beta_{9} - 22848 \beta_{10} + 839057 \beta_{11} - 324813 \beta_{12} + 10435 \beta_{13} - 35081 \beta_{14} - 6851 \beta_{15} + 17158 \beta_{16} - 10045 \beta_{17} + 867 \beta_{18} + 4382 \beta_{19} ) q^{84}$$ $$+ ( 857196228 - 257808 \beta_{1} + 2471740 \beta_{2} - 726200 \beta_{3} + 9928 \beta_{5} + 16532 \beta_{6} + 11904 \beta_{7} - 237296 \beta_{8} + 60796 \beta_{9} + 16904 \beta_{10} - 13076 \beta_{11} - 11904 \beta_{12} + 1476 \beta_{13} + 2344 \beta_{14} - 18380 \beta_{15} - 6536 \beta_{17} - 3392 \beta_{18} + 9928 \beta_{19} ) q^{85}$$ $$+ ( 14784 - 9625536 \beta_{1} + 353852 \beta_{2} - 30783245 \beta_{3} + 238636 \beta_{5} - 2384382 \beta_{6} + 6592 \beta_{7} - 40406 \beta_{8} + 76804 \beta_{9} - 891373 \beta_{11} + 378259 \beta_{12} - 15894 \beta_{13} + 6711 \beta_{14} + 15894 \beta_{15} + 28884 \beta_{16} + 2710 \beta_{17} - 6592 \beta_{18} - 5248 \beta_{19} ) q^{86}$$ $$+ ( 137466 - 465634689 \beta_{1} + 555729 \beta_{2} - 1488063 \beta_{3} - 577635 \beta_{4} + 101732 \beta_{5} - 40943 \beta_{6} - 6612 \beta_{7} + 127555 \beta_{8} - 2967 \beta_{9} + 21918 \beta_{10} - 886170 \beta_{11} + 88766 \beta_{12} + 12096 \beta_{13} - 15965 \beta_{14} - 4790 \beta_{15} + 3815 \beta_{16} + 7757 \beta_{17} - 2417 \beta_{18} - 4339 \beta_{19} ) q^{87}$$ $$+ ( -8304716296 - 11873052 \beta_{1} - 5384 \beta_{2} - 37018072 \beta_{3} + 312540 \beta_{4} + 2532 \beta_{5} + 86960 \beta_{6} - 4642 \beta_{7} - 1281930 \beta_{8} + 53128 \beta_{9} - 15736 \beta_{10} - 60352 \beta_{11} - 479002 \beta_{12} + 1258 \beta_{13} + 48534 \beta_{14} - 3806 \beta_{15} - 13914 \beta_{17} + 11382 \beta_{18} + 2532 \beta_{19} ) q^{88}$$ $$+ ( 310590 - 470615554 \beta_{1} + 122507 \beta_{2} - 1893790 \beta_{3} + 133121 \beta_{5} + 2789892 \beta_{6} - 2704 \beta_{7} + 581792 \beta_{8} + 91286 \beta_{9} + 942642 \beta_{11} + 2704 \beta_{12} - 14047 \beta_{13} + 24597 \beta_{14} + 14047 \beta_{15} + 19455 \beta_{17} + 2704 \beta_{18} + 2305 \beta_{19} ) q^{89}$$ $$+ ( -12897582461 + 48942487 \beta_{1} + 6610533 \beta_{2} - 5974828 \beta_{3} + 176997 \beta_{4} + 259220 \beta_{5} + 1935712 \beta_{6} + 9284 \beta_{7} + 6032 \beta_{8} + 22820 \beta_{9} - 8637 \beta_{10} + 1041214 \beta_{11} + 638452 \beta_{12} - 10314 \beta_{13} - 1598 \beta_{14} - 19646 \beta_{15} - 30028 \beta_{16} - 9814 \beta_{17} + 12216 \beta_{18} - 21640 \beta_{19} ) q^{90}$$ $$+ ( 846894 - 3787504 \beta_{1} + 2467360 \beta_{2} - 11691890 \beta_{3} - 683338 \beta_{4} + 9354 \beta_{5} - 54918 \beta_{6} - 19810 \beta_{7} + 827844 \beta_{8} + 19914 \beta_{9} - 87804 \beta_{10} + 14311 \beta_{11} - 146322 \beta_{12} - 5280 \beta_{13} - 41490 \beta_{14} - 23988 \beta_{15} + 4074 \beta_{17} - 13428 \beta_{18} + 9354 \beta_{19} ) q^{91}$$ $$+ ( -770428 - 100588692 \beta_{1} - 1161836 \beta_{2} + 59571704 \beta_{3} + 17540 \beta_{5} + 1598172 \beta_{6} + 18118 \beta_{7} - 1676458 \beta_{8} - 8672 \beta_{9} - 1023402 \beta_{11} - 705030 \beta_{12} - 24270 \beta_{13} - 63078 \beta_{14} + 24270 \beta_{15} - 36188 \beta_{16} - 11966 \beta_{17} - 18118 \beta_{18} + 11396 \beta_{19} ) q^{92}$$ $$+ ( 5720508485 + 533293489 \beta_{1} + 5632705 \beta_{2} + 581558 \beta_{3} - 209314 \beta_{5} + 4138778 \beta_{6} + 37712 \beta_{7} + 316 \beta_{8} + 9777 \beta_{9} - 55986 \beta_{10} - 1041075 \beta_{11} - 37712 \beta_{12} + 58007 \beta_{13} + 40534 \beta_{14} - 2021 \beta_{15} + 10946 \beta_{17} - 4272 \beta_{18} + 3422 \beta_{19} ) q^{93}$$ $$+ ( 19599834272 + 20790600 \beta_{1} + 8242488 \beta_{2} + 64869524 \beta_{3} - 45168 \beta_{4} - 5120 \beta_{5} - 17704 \beta_{6} - 53600 \beta_{7} + 115912 \beta_{8} + 42256 \beta_{9} - 672 \beta_{10} - 6240 \beta_{11} + 877068 \beta_{12} + 36072 \beta_{13} + 18748 \beta_{14} + 46312 \beta_{15} - 8888 \beta_{17} + 14008 \beta_{18} - 5120 \beta_{19} ) q^{94}$$ $$+ ( -531104 + 639309198 \beta_{1} - 76832 \beta_{2} - 26240072 \beta_{3} - 99460 \beta_{5} + 98372 \beta_{6} - 12526 \beta_{7} - 1149278 \beta_{8} - 12220 \beta_{9} + 1120169 \beta_{11} + 265032 \beta_{12} + 6416 \beta_{13} - 55908 \beta_{14} - 6416 \beta_{15} - 65248 \beta_{16} + 18636 \beta_{17} + 12526 \beta_{18} + 12220 \beta_{19} ) q^{95}$$ $$+ ( 22343702024 + 34474056 \beta_{1} + 55168 \beta_{2} + 160508 \beta_{3} + 221832 \beta_{4} + 29700 \beta_{5} - 1175816 \beta_{6} + 64528 \beta_{7} - 32880 \beta_{8} + 160224 \beta_{9} - 66096 \beta_{10} + 1064348 \beta_{11} - 904696 \beta_{12} - 9432 \beta_{13} + 87024 \beta_{14} - 11232 \beta_{15} - 18216 \beta_{16} - 16128 \beta_{17} + 21784 \beta_{18} - 25920 \beta_{19} ) q^{96}$$ $$+ ( 3785463654 + 1017880 \beta_{1} - 10747697 \beta_{2} + 3226360 \beta_{3} + 11259 \beta_{5} + 108258 \beta_{6} - 71296 \beta_{7} - 1445280 \beta_{8} + 16548 \beta_{9} - 43032 \beta_{10} - 58598 \beta_{11} + 71296 \beta_{12} + 32775 \beta_{13} + 49047 \beta_{14} + 10257 \beta_{15} + 20421 \beta_{17} - 31680 \beta_{18} + 11259 \beta_{19} ) q^{97}$$ $$+ ( -38148 - 78531389 \beta_{1} + 1089546 \beta_{2} - 86287838 \beta_{3} - 284756 \beta_{5} + 41304 \beta_{6} + 10884 \beta_{7} + 18852 \beta_{8} - 123468 \beta_{9} - 1058996 \beta_{11} + 1085772 \beta_{12} + 7566 \beta_{13} - 8718 \beta_{14} - 7566 \beta_{15} - 103620 \beta_{16} - 29334 \beta_{17} - 10884 \beta_{18} + 3192 \beta_{19} ) q^{98}$$ $$+ ( -2732786 - 483094015 \beta_{1} - 13419189 \beta_{2} - 9275086 \beta_{3} + 924882 \beta_{4} + 186286 \beta_{5} - 54048 \beta_{6} + 90766 \beta_{7} - 355082 \beta_{8} - 11712 \beta_{9} + 64536 \beta_{10} - 892945 \beta_{11} - 363337 \beta_{12} + 36288 \beta_{13} - 57952 \beta_{14} - 14776 \beta_{15} - 86570 \beta_{16} + 26488 \beta_{17} + 6126 \beta_{18} - 9800 \beta_{19} ) q^{99}$$ $$+O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$20q$$ $$\mathstrut +\mathstrut 1976q^{4}$$ $$\mathstrut +\mathstrut 13128q^{6}$$ $$\mathstrut +\mathstrut 103620q^{9}$$ $$\mathstrut +\mathstrut O(q^{10})$$ $$20q$$ $$\mathstrut +\mathstrut 1976q^{4}$$ $$\mathstrut +\mathstrut 13128q^{6}$$ $$\mathstrut +\mathstrut 103620q^{9}$$ $$\mathstrut +\mathstrut 202000q^{10}$$ $$\mathstrut +\mathstrut 1064760q^{12}$$ $$\mathstrut +\mathstrut 1543864q^{13}$$ $$\mathstrut +\mathstrut 3482912q^{16}$$ $$\mathstrut +\mathstrut 2557392q^{18}$$ $$\mathstrut -\mathstrut 12211752q^{21}$$ $$\mathstrut -\mathstrut 5920464q^{22}$$ $$\mathstrut +\mathstrut 12133152q^{24}$$ $$\mathstrut -\mathstrut 141128700q^{25}$$ $$\mathstrut -\mathstrut 3604848q^{28}$$ $$\mathstrut +\mathstrut 73456080q^{30}$$ $$\mathstrut +\mathstrut 229769760q^{33}$$ $$\mathstrut +\mathstrut 222167104q^{34}$$ $$\mathstrut +\mathstrut 298087896q^{36}$$ $$\mathstrut -\mathstrut 44517800q^{37}$$ $$\mathstrut -\mathstrut 439643840q^{40}$$ $$\mathstrut +\mathstrut 281776944q^{42}$$ $$\mathstrut +\mathstrut 1020227520q^{45}$$ $$\mathstrut -\mathstrut 2392795680q^{46}$$ $$\mathstrut +\mathstrut 689090592q^{48}$$ $$\mathstrut -\mathstrut 1018138084q^{49}$$ $$\mathstrut -\mathstrut 3959246384q^{52}$$ $$\mathstrut -\mathstrut 4249352520q^{54}$$ $$\mathstrut +\mathstrut 1438636392q^{57}$$ $$\mathstrut +\mathstrut 9671853040q^{58}$$ $$\mathstrut +\mathstrut 5688375360q^{60}$$ $$\mathstrut +\mathstrut 6873199864q^{61}$$ $$\mathstrut -\mathstrut 1452752512q^{64}$$ $$\mathstrut -\mathstrut 10961242896q^{66}$$ $$\mathstrut -\mathstrut 13308470976q^{69}$$ $$\mathstrut +\mathstrut 35188514400q^{70}$$ $$\mathstrut +\mathstrut 30438127680q^{72}$$ $$\mathstrut -\mathstrut 12426469112q^{73}$$ $$\mathstrut -\mathstrut 60088673808q^{76}$$ $$\mathstrut -\mathstrut 81037845456q^{78}$$ $$\mathstrut -\mathstrut 42462874764q^{81}$$ $$\mathstrut +\mathstrut 88180337440q^{82}$$ $$\mathstrut +\mathstrut 125502443664q^{84}$$ $$\mathstrut +\mathstrut 17135502080q^{85}$$ $$\mathstrut -\mathstrut 166086469440q^{88}$$ $$\mathstrut -\mathstrut 257976145200q^{90}$$ $$\mathstrut +\mathstrut 114387515256q^{93}$$ $$\mathstrut +\mathstrut 391966360512q^{94}$$ $$\mathstrut +\mathstrut 446868262272q^{96}$$ $$\mathstrut +\mathstrut 75764383528q^{97}$$ $$\mathstrut +\mathstrut O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{20}\mathstrut -\mathstrut$$ $$247$$ $$x^{18}\mathstrut -\mathstrut$$ $$23916$$ $$x^{16}\mathstrut +\mathstrut$$ $$14713536$$ $$x^{14}\mathstrut -\mathstrut$$ $$45723119616$$ $$x^{12}\mathstrut +\mathstrut$$ $$40864324780032$$ $$x^{10}\mathstrut -\mathstrut$$ $$11986041468616704$$ $$x^{8}\mathstrut +\mathstrut$$ $$1011106494856298496$$ $$x^{6}\mathstrut -\mathstrut$$ $$430832354752771129344$$ $$x^{4}\mathstrut -\mathstrut$$ $$1166424521268802367782912$$ $$x^{2}\mathstrut +\mathstrut$$ $$1237940039285380274899124224$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$2 \nu$$ $$\beta_{2}$$ $$=$$ $$4 \nu^{2} - 99$$ $$\beta_{3}$$ $$=$$ $$($$$$46131569375$$ $$\nu^{19}\mathstrut -\mathstrut$$ $$239928354304$$ $$\nu^{18}\mathstrut -\mathstrut$$ $$22887903126825$$ $$\nu^{17}\mathstrut +\mathstrut$$ $$259755436976640$$ $$\nu^{16}\mathstrut +\mathstrut$$ $$19556588970264300$$ $$\nu^{15}\mathstrut +\mathstrut$$ $$53810623904876544$$ $$\nu^{14}\mathstrut -\mathstrut$$ $$6529423698712632000$$ $$\nu^{13}\mathstrut +\mathstrut$$ $$38232749261706461184$$ $$\nu^{12}\mathstrut +\mathstrut$$ $$897349730929067520000$$ $$\nu^{11}\mathstrut -\mathstrut$$ $$55782354449881500745728$$ $$\nu^{10}\mathstrut +\mathstrut$$ $$590330585908631489740800$$ $$\nu^{9}\mathstrut +\mathstrut$$ $$11627256762088427341479936$$ $$\nu^{8}\mathstrut -\mathstrut$$ $$1177886442391478235404697600$$ $$\nu^{7}\mathstrut +\mathstrut$$ $$10971145661586545313891483648$$ $$\nu^{6}\mathstrut +\mathstrut$$ $$400948207094769548080132915200$$ $$\nu^{5}\mathstrut -\mathstrut$$ $$1110769065760608792454216286208$$ $$\nu^{4}\mathstrut -\mathstrut$$ $$271316700506973624203911102464000$$ $$\nu^{3}\mathstrut +\mathstrut$$ $$2393219287980435280047879833518080$$ $$\nu^{2}\mathstrut +\mathstrut$$ $$72047149669348558893988904225996800$$ $$\nu\mathstrut -\mathstrut$$ $$1574683656749145806442103199856852992$$$$)/$$$$79\!\cdots\!00$$ $$\beta_{4}$$ $$=$$ $$($$$$-$$$$9226313875$$ $$\nu^{19}\mathstrut -\mathstrut$$ $$7553005215232$$ $$\nu^{18}\mathstrut +\mathstrut$$ $$4577580625365$$ $$\nu^{17}\mathstrut +\mathstrut$$ $$3427472345510400$$ $$\nu^{16}\mathstrut -\mathstrut$$ $$3911317794052860$$ $$\nu^{15}\mathstrut -\mathstrut$$ $$1191761025029265408$$ $$\nu^{14}\mathstrut +\mathstrut$$ $$1305884739742526400$$ $$\nu^{13}\mathstrut +\mathstrut$$ $$1656950916513972191232$$ $$\nu^{12}\mathstrut -\mathstrut$$ $$179469946185813504000$$ $$\nu^{11}\mathstrut +\mathstrut$$ $$23303313749598573428736$$ $$\nu^{10}\mathstrut -\mathstrut$$ $$118066117181726297948160$$ $$\nu^{9}\mathstrut -\mathstrut$$ $$26759412697756590504148992$$ $$\nu^{8}\mathstrut +\mathstrut$$ $$235577288478295647080939520$$ $$\nu^{7}\mathstrut +\mathstrut$$ $$53114190156844970000309551104$$ $$\nu^{6}\mathstrut -\mathstrut$$ $$80189641418953909616026583040$$ $$\nu^{5}\mathstrut -\mathstrut$$ $$102739084009715981330786291810304$$ $$\nu^{4}\mathstrut +\mathstrut$$ $$54263340101394724840782220492800$$ $$\nu^{3}\mathstrut +\mathstrut$$ $$44722723994641766129675232513884160$$ $$\nu^{2}\mathstrut -\mathstrut$$ $$15472806053487858216995241297182720$$ $$\nu\mathstrut -\mathstrut$$ $$11229575540777706173909771556755603456$$$$)/$$$$53\!\cdots\!80$$ $$\beta_{5}$$ $$=$$ $$($$$$46131569375$$ $$\nu^{19}\mathstrut -\mathstrut$$ $$239928354304$$ $$\nu^{18}\mathstrut -\mathstrut$$ $$22887903126825$$ $$\nu^{17}\mathstrut +\mathstrut$$ $$259755436976640$$ $$\nu^{16}\mathstrut +\mathstrut$$ $$19556588970264300$$ $$\nu^{15}\mathstrut +\mathstrut$$ $$53810623904876544$$ $$\nu^{14}\mathstrut -\mathstrut$$ $$6529423698712632000$$ $$\nu^{13}\mathstrut +\mathstrut$$ $$38232749261706461184$$ $$\nu^{12}\mathstrut +\mathstrut$$ $$897349730929067520000$$ $$\nu^{11}\mathstrut -\mathstrut$$ $$55782354449881500745728$$ $$\nu^{10}\mathstrut +\mathstrut$$ $$590330585908631489740800$$ $$\nu^{9}\mathstrut +\mathstrut$$ $$11627256762088427341479936$$ $$\nu^{8}\mathstrut -\mathstrut$$ $$1177886442391478235404697600$$ $$\nu^{7}\mathstrut +\mathstrut$$ $$10971145661586545313891483648$$ $$\nu^{6}\mathstrut +\mathstrut$$ $$400948207094769548080132915200$$ $$\nu^{5}\mathstrut -\mathstrut$$ $$1110769065760608792454216286208$$ $$\nu^{4}\mathstrut +\mathstrut$$ $$20996205691855955139745297937203200$$ $$\nu^{3}\mathstrut +\mathstrut$$ $$2393219287980435280047879833518080$$ $$\nu^{2}\mathstrut -\mathstrut$$ $$449007148943543195822766717245849600$$ $$\nu\mathstrut -\mathstrut$$ $$1574683656749145806442103199856852992$$$$)/$$$$26\!\cdots\!00$$ $$\beta_{6}$$ $$=$$ $$($$$$90366381$$ $$\nu^{19}\mathstrut +\mathstrut$$ $$142181548565$$ $$\nu^{17}\mathstrut +\mathstrut$$ $$46580767060484$$ $$\nu^{15}\mathstrut -\mathstrut$$ $$5363773315802176$$ $$\nu^{13}\mathstrut +\mathstrut$$ $$22436822557367816192$$ $$\nu^{11}\mathstrut +\mathstrut$$ $$5254313046384963485696$$ $$\nu^{9}\mathstrut -\mathstrut$$ $$240863080265417198927872$$ $$\nu^{7}\mathstrut +\mathstrut$$ $$3469516219226795629516685312$$ $$\nu^{5}\mathstrut -\mathstrut$$ $$662001721306173609090765291520$$ $$\nu^{3}\mathstrut +\mathstrut$$ $$76055560082787482507355134885888$$ $$\nu$$$$)/$$$$44\!\cdots\!00$$ $$\beta_{7}$$ $$=$$ $$($$$$-$$$$36763771475$$ $$\nu^{19}\mathstrut -\mathstrut$$ $$298065737291264$$ $$\nu^{18}\mathstrut +\mathstrut$$ $$158494922939925$$ $$\nu^{17}\mathstrut +\mathstrut$$ $$81842455582947840$$ $$\nu^{16}\mathstrut -\mathstrut$$ $$267699088226082300$$ $$\nu^{15}\mathstrut +\mathstrut$$ $$9099512565774895104$$ $$\nu^{14}\mathstrut +\mathstrut$$ $$93165443595733310400$$ $$\nu^{13}\mathstrut -\mathstrut$$ $$2673320262654211424256$$ $$\nu^{12}\mathstrut -\mathstrut$$ $$37405222621281219686400$$ $$\nu^{11}\mathstrut +\mathstrut$$ $$10891637640586969535741952$$ $$\nu^{10}\mathstrut +\mathstrut$$ $$15330136321678026237542400$$ $$\nu^{9}\mathstrut -\mathstrut$$ $$11301552649261375108335796224$$ $$\nu^{8}\mathstrut +\mathstrut$$ $$8565022094496776194306867200$$ $$\nu^{7}\mathstrut +\mathstrut$$ $$3904537820300378081238102048768$$ $$\nu^{6}\mathstrut -\mathstrut$$ $$4643127697977095954501940019200$$ $$\nu^{5}\mathstrut -\mathstrut$$ $$336971416754697943806994261475328$$ $$\nu^{4}\mathstrut +\mathstrut$$ $$3284581484215287182324236969574400$$ $$\nu^{3}\mathstrut +\mathstrut$$ $$147863901090117001159363811113697280$$ $$\nu^{2}\mathstrut -\mathstrut$$ $$1593247699565736691208787584037683200$$ $$\nu\mathstrut +\mathstrut$$ $$239853310276670848164756266046275452928$$$$)/$$$$26\!\cdots\!00$$ $$\beta_{8}$$ $$=$$ $$($$$$-$$$$468610067$$ $$\nu^{19}\mathstrut -\mathstrut$$ $$22448057600$$ $$\nu^{18}\mathstrut +\mathstrut$$ $$507334837845$$ $$\nu^{17}\mathstrut +\mathstrut$$ $$40351311686400$$ $$\nu^{16}\mathstrut +\mathstrut$$ $$105098874814212$$ $$\nu^{15}\mathstrut -\mathstrut$$ $$14078480870016000$$ $$\nu^{14}\mathstrut +\mathstrut$$ $$74673338401770432$$ $$\nu^{13}\mathstrut +\mathstrut$$ $$5872322256906240000$$ $$\nu^{12}\mathstrut -\mathstrut$$ $$108949911034924806144$$ $$\nu^{11}\mathstrut -\mathstrut$$ $$2528915718054582681600$$ $$\nu^{10}\mathstrut +\mathstrut$$ $$22709485863453959651328$$ $$\nu^{9}\mathstrut -\mathstrut$$ $$1220608474317109120204800$$ $$\nu^{8}\mathstrut +\mathstrut$$ $$21428018870286221316194304$$ $$\nu^{7}\mathstrut +\mathstrut$$ $$692000542347252213586329600$$ $$\nu^{6}\mathstrut -\mathstrut$$ $$2169470831563689047762141184$$ $$\nu^{5}\mathstrut -\mathstrut$$ $$491097124696682718509924352000$$ $$\nu^{4}\mathstrut +\mathstrut$$ $$4674256421836787656343515299840$$ $$\nu^{3}\mathstrut +\mathstrut$$ $$245812780064377887395567527526400$$ $$\nu^{2}\mathstrut -\mathstrut$$ $$3075554017088175403207232812220416$$ $$\nu\mathstrut -\mathstrut$$ $$106430100683834190110937046004531200$$$$)/$$$$77\!\cdots\!00$$ $$\beta_{9}$$ $$=$$ $$($$$$-$$$$598262172925$$ $$\nu^{19}\mathstrut -\mathstrut$$ $$78117971797504$$ $$\nu^{18}\mathstrut +\mathstrut$$ $$245061855234075$$ $$\nu^{17}\mathstrut -\mathstrut$$ $$61662761079068160$$ $$\nu^{16}\mathstrut +\mathstrut$$ $$21718767005754300$$ $$\nu^{15}\mathstrut +\mathstrut$$ $$51272586309057644544$$ $$\nu^{14}\mathstrut -\mathstrut$$ $$1680353395577080920000$$ $$\nu^{13}\mathstrut +\mathstrut$$ $$3684971218613189443584$$ $$\nu^{12}\mathstrut -\mathstrut$$ $$343690054636713113702400$$ $$\nu^{11}\mathstrut -\mathstrut$$ $$882593333782537492758528$$ $$\nu^{10}\mathstrut -\mathstrut$$ $$198864629395050739138560000$$ $$\nu^{9}\mathstrut -\mathstrut$$ $$1915429115930917298308644864$$ $$\nu^{8}\mathstrut -\mathstrut$$ $$118575635726017791188061388800$$ $$\nu^{7}\mathstrut -\mathstrut$$ $$4924509392021741116805666045952$$ $$\nu^{6}\mathstrut +\mathstrut$$ $$128347953685032807704315088076800$$ $$\nu^{5}\mathstrut +\mathstrut$$ $$3652037944105187095153747156795392$$ $$\nu^{4}\mathstrut -\mathstrut$$ $$31630217668136319022817769160704000$$ $$\nu^{3}\mathstrut -\mathstrut$$ $$64861270229500117352856674862366720$$ $$\nu^{2}\mathstrut +\mathstrut$$ $$2959773742109028651297735524220928000$$ $$\nu\mathstrut +\mathstrut$$ $$82100011620472339439472437240012996608$$$$)/$$$$79\!\cdots\!00$$ $$\beta_{10}$$ $$=$$ $$($$$$876499818125$$ $$\nu^{19}\mathstrut -\mathstrut$$ $$704858789234176$$ $$\nu^{18}\mathstrut -\mathstrut$$ $$434870159409675$$ $$\nu^{17}\mathstrut -\mathstrut$$ $$116589674246960640$$ $$\nu^{16}\mathstrut +\mathstrut$$ $$371575190435021700$$ $$\nu^{15}\mathstrut +\mathstrut$$ $$7162632924105639936$$ $$\nu^{14}\mathstrut -\mathstrut$$ $$124059050275540008000$$ $$\nu^{13}\mathstrut -\mathstrut$$ $$68995303632294469730304$$ $$\nu^{12}\mathstrut +\mathstrut$$ $$17049644887652282880000$$ $$\nu^{11}\mathstrut -\mathstrut$$ $$6443033077526701887455232$$ $$\nu^{10}\mathstrut +\mathstrut$$ $$11216281132263998305075200$$ $$\nu^{9}\mathstrut -\mathstrut$$ $$3147449340918542980733730816$$ $$\nu^{8}\mathstrut -\mathstrut$$ $$22379842405438086472689254400$$ $$\nu^{7}\mathstrut -\mathstrut$$ $$11001634944582836677350877298688$$ $$\nu^{6}\mathstrut +\mathstrut$$ $$7618015934800621413522525388800$$ $$\nu^{5}\mathstrut +\mathstrut$$ $$655094595948353532115823307522048$$ $$\nu^{4}\mathstrut -\mathstrut$$ $$5155017309632498859874310946816000$$ $$\nu^{3}\mathstrut +\mathstrut$$ $$76764242669034258133694058457989120$$ $$\nu^{2}\mathstrut +\mathstrut$$ $$1464599694483255798423560620972441600$$ $$\nu\mathstrut +\mathstrut$$ $$280102002702068699742263678893557809152$$$$)/$$$$79\!\cdots\!00$$ $$\beta_{11}$$ $$=$$ $$($$$$-$$$$1642708321825$$ $$\nu^{19}\mathstrut -\mathstrut$$ $$239928354304$$ $$\nu^{18}\mathstrut +\mathstrut$$ $$394255549999575$$ $$\nu^{17}\mathstrut +\mathstrut$$ $$259755436976640$$ $$\nu^{16}\mathstrut +\mathstrut$$ $$59946883808203500$$ $$\nu^{15}\mathstrut +\mathstrut$$ $$53810623904876544$$ $$\nu^{14}\mathstrut -\mathstrut$$ $$31378230236119915200$$ $$\nu^{13}\mathstrut +\mathstrut$$ $$38232749261706461184$$ $$\nu^{12}\mathstrut +\mathstrut$$ $$78116378088539093299200$$ $$\nu^{11}\mathstrut -\mathstrut$$ $$55782354449881500745728$$ $$\nu^{10}\mathstrut -\mathstrut$$ $$68422971229562075322777600$$ $$\nu^{9}\mathstrut +\mathstrut$$ $$11627256762088427341479936$$ $$\nu^{8}\mathstrut +\mathstrut$$ $$19064618527385844362457907200$$ $$\nu^{7}\mathstrut +\mathstrut$$ $$10971145661586545313891483648$$ $$\nu^{6}\mathstrut -\mathstrut$$ $$1306648775669954963539230720000$$ $$\nu^{5}\mathstrut -\mathstrut$$ $$1110769065760608792454216286208$$ $$\nu^{4}\mathstrut +\mathstrut$$ $$456290166619136172775910984908800$$ $$\nu^{3}\mathstrut +\mathstrut$$ $$2393219287980435280047879833518080$$ $$\nu^{2}\mathstrut +\mathstrut$$ $$1850543709730698476779241510168166400$$ $$\nu\mathstrut -\mathstrut$$ $$1574683656749145806442103199856852992$$$$)/$$$$39\!\cdots\!00$$ $$\beta_{12}$$ $$=$$ $$($$$$-$$$$599710401875$$ $$\nu^{19}\mathstrut -\mathstrut$$ $$3358996960256$$ $$\nu^{18}\mathstrut +\mathstrut$$ $$297542740648725$$ $$\nu^{17}\mathstrut +\mathstrut$$ $$3636576117672960$$ $$\nu^{16}\mathstrut -\mathstrut$$ $$254235656613435900$$ $$\nu^{15}\mathstrut +\mathstrut$$ $$753348734668271616$$ $$\nu^{14}\mathstrut +\mathstrut$$ $$84882508083264216000$$ $$\nu^{13}\mathstrut +\mathstrut$$ $$535258489663890456576$$ $$\nu^{12}\mathstrut -\mathstrut$$ $$11665546502077877760000$$ $$\nu^{11}\mathstrut -\mathstrut$$ $$780952962298341010440192$$ $$\nu^{10}\mathstrut -\mathstrut$$ $$7674297616812209366630400$$ $$\nu^{9}\mathstrut +\mathstrut$$ $$162781594669237982780719104$$ $$\nu^{8}\mathstrut +\mathstrut$$ $$15312523751089217060261068800$$ $$\nu^{7}\mathstrut +\mathstrut$$ $$153596039262211634394480771072$$ $$\nu^{6}\mathstrut -\mathstrut$$ $$5212326692232004125041727897600$$ $$\nu^{5}\mathstrut -\mathstrut$$ $$15550766920648523094359028006912$$ $$\nu^{4}\mathstrut +\mathstrut$$ $$3527117106590657114650844332032000$$ $$\nu^{3}\mathstrut +\mathstrut$$ $$28188189433635361729683015409336320$$ $$\nu^{2}\mathstrut -\mathstrut$$ $$1003073953177665418009197033186918400$$ $$\nu\mathstrut -\mathstrut$$ $$21913978399685295668462509067063001088$$$$)/$$$$13\!\cdots\!00$$ $$\beta_{13}$$ $$=$$ $$($$$$-$$$$2059342161337$$ $$\nu^{19}\mathstrut -\mathstrut$$ $$122888043929344$$ $$\nu^{18}\mathstrut -\mathstrut$$ $$7256757377112705$$ $$\nu^{17}\mathstrut -\mathstrut$$ $$74548276279023360$$ $$\nu^{16}\mathstrut +\mathstrut$$ $$2380369484014821132$$ $$\nu^{15}\mathstrut -\mathstrut$$ $$31596501599966063616$$ $$\nu^{14}\mathstrut +\mathstrut$$ $$965034429295281029952$$ $$\nu^{13}\mathstrut +\mathstrut$$ $$49873315037495650074624$$ $$\nu^{12}\mathstrut +\mathstrut$$ $$389213420540595849916416$$ $$\nu^{11}\mathstrut -\mathstrut$$ $$22717862552347329149534208$$ $$\nu^{10}\mathstrut +\mathstrut$$ $$164586163159111090918391808$$ $$\nu^{9}\mathstrut -\mathstrut$$ $$8028302850833161666852552704$$ $$\nu^{8}\mathstrut -\mathstrut$$ $$187451746606807105715155501056$$ $$\nu^{7}\mathstrut +\mathstrut$$ $$521848681215751088383331401728$$ $$\nu^{6}\mathstrut +\mathstrut$$ $$44110920201420182273173462450176$$ $$\nu^{5}\mathstrut -\mathstrut$$ $$1267254250130615885517071201599488$$ $$\nu^{4}\mathstrut +\mathstrut$$ $$2089598226915952701036641585725440$$ $$\nu^{3}\mathstrut +\mathstrut$$ $$1438468950404765511202498888681390080$$ $$\nu^{2}\mathstrut +\mathstrut$$ $$10846674409688296363107785263990964224$$ $$\nu\mathstrut -\mathstrut$$ $$314902340819552224520706177538470182912$$$$)/$$$$19\!\cdots\!00$$ $$\beta_{14}$$ $$=$$ $$($$$$-$$$$16628286142183$$ $$\nu^{19}\mathstrut +\mathstrut$$ $$607974004777472$$ $$\nu^{18}\mathstrut +\mathstrut$$ $$14622089403463905$$ $$\nu^{17}\mathstrut -\mathstrut$$ $$1085482806128709120$$ $$\nu^{16}\mathstrut +\mathstrut$$ $$3143284153218307188$$ $$\nu^{15}\mathstrut +\mathstrut$$ $$372511617855396292608$$ $$\nu^{14}\mathstrut +\mathstrut$$ $$2038629172992427263168$$ $$\nu^{13}\mathstrut -\mathstrut$$ $$157988715986125111590912$$ $$\nu^{12}\mathstrut -\mathstrut$$ $$3039467781032383144390656$$ $$\nu^{11}\mathstrut +\mathstrut$$ $$69728493318830113846984704$$ $$\nu^{10}\mathstrut +\mathstrut$$ $$500867039793689454021967872$$ $$\nu^{9}\mathstrut +\mathstrut$$ $$31997507979448910840648957952$$ $$\nu^{8}\mathstrut +\mathstrut$$ $$642770852140160869081175556096$$ $$\nu^{7}\mathstrut -\mathstrut$$ $$18895581702901464383019773067264$$ $$\nu^{6}\mathstrut -\mathstrut$$ $$73573187175405375995390843682816$$ $$\nu^{5}\mathstrut +\mathstrut$$ $$13122732917752186875683757247954944$$ $$\nu^{4}\mathstrut +\mathstrut$$ $$134850384290315088153197380223631360$$ $$\nu^{3}\mathstrut -\mathstrut$$ $$6870736870936966343083115380620656640$$ $$\nu^{2}\mathstrut -\mathstrut$$ $$84055983102218869175090724014007517184$$ $$\nu\mathstrut +\mathstrut$$ $$2906737591377564430123691879676988358656$$$$)/$$$$79\!\cdots\!00$$ $$\beta_{15}$$ $$=$$ $$($$$$19147518521161$$ $$\nu^{19}\mathstrut +\mathstrut$$ $$1348047059809792$$ $$\nu^{18}\mathstrut +\mathstrut$$ $$21463383910277265$$ $$\nu^{17}\mathstrut -\mathstrut$$ $$685295882504133120$$ $$\nu^{16}\mathstrut -\mathstrut$$ $$10402180382210357196$$ $$\nu^{15}\mathstrut +\mathstrut$$ $$265514517916036773888$$ $$\nu^{14}\mathstrut -\mathstrut$$ $$3200878256230621083456$$ $$\nu^{13}\mathstrut +\mathstrut$$ $$11295489965198363885568$$ $$\nu^{12}\mathstrut +\mathstrut$$ $$30898934433934614269952$$ $$\nu^{11}\mathstrut -\mathstrut$$ $$80670937341667042872262656$$ $$\nu^{10}\mathstrut -\mathstrut$$ $$578157123161926569399681024$$ $$\nu^{9}\mathstrut -\mathstrut$$ $$6432831683083585429048393728$$ $$\nu^{8}\mathstrut +\mathstrut$$ $$538247396593968986692169760768$$ $$\nu^{7}\mathstrut -\mathstrut$$ $$20777602669647249706131866517504$$ $$\nu^{6}\mathstrut -\mathstrut$$ $$259714444488182396885307362377728$$ $$\nu^{5}\mathstrut -\mathstrut$$ $$5614499082890471198360357742575616$$ $$\nu^{4}\mathstrut -\mathstrut$$ $$42769306363831179200323345365073920$$ $$\nu^{3}\mathstrut -\mathstrut$$ $$1390960470315401954509993751598858240$$ $$\nu^{2}\mathstrut -\mathstrut$$ $$10262883303858041422809346997928067072$$ $$\nu\mathstrut +\mathstrut$$ $$235306592015734941495005596577762902016$$$$)/$$$$79\!\cdots\!00$$ $$\beta_{16}$$ $$=$$ $$($$$$-$$$$67315768751491$$ $$\nu^{19}\mathstrut -\mathstrut$$ $$49572547279360$$ $$\nu^{18}\mathstrut +\mathstrut$$ $$57887903279997285$$ $$\nu^{17}\mathstrut +\mathstrut$$ $$86535817888396800$$ $$\nu^{16}\mathstrut -\mathstrut$$ $$23116739464315131324$$ $$\nu^{15}\mathstrut -\mathstrut$$ $$28025569463219619840$$ $$\nu^{14}\mathstrut +\mathstrut$$ $$17156097341881911039936$$ $$\nu^{13}\mathstrut +\mathstrut$$ $$12600007221069576437760$$ $$\nu^{12}\mathstrut -\mathstrut$$ $$3943619450344111480614912$$ $$\nu^{11}\mathstrut -\mathstrut$$ $$6015954707324007843102720$$ $$\nu^{10}\mathstrut -\mathstrut$$ $$2111080019336329260728057856$$ $$\nu^{9}\mathstrut -\mathstrut$$ $$2325397303970113068057231360$$ $$\nu^{8}\mathstrut +\mathstrut$$ $$947536734565477230709196193792$$ $$\nu^{7}\mathstrut +\mathstrut$$ $$1581784295650970713133175275520$$ $$\nu^{6}\mathstrut -\mathstrut$$ $$939246874081829721381436372549632$$ $$\nu^{5}\mathstrut -\mathstrut$$ $$1022428447365215339395138317189120$$ $$\nu^{4}\mathstrut +\mathstrut$$ $$573574010319336994430638717057105920$$ $$\nu^{3}\mathstrut +\mathstrut$$ $$539322862891552442586840493876838400$$ $$\nu^{2}\mathstrut -\mathstrut$$ $$188977894050805310016426886574341357568$$ $$\nu\mathstrut -\mathstrut$$ $$241581125730832472345544137261742817280$$$$)/$$$$79\!\cdots\!00$$ $$\beta_{17}$$ $$=$$ $$($$$$77752598588255$$ $$\nu^{19}\mathstrut -\mathstrut$$ $$1513610837722624$$ $$\nu^{18}\mathstrut -\mathstrut$$ $$33558837933971625$$ $$\nu^{17}\mathstrut +\mathstrut$$ $$1010192911603223040$$ $$\nu^{16}\mathstrut +\mathstrut$$ $$15080693918489719020$$ $$\nu^{15}\mathstrut -\mathstrut$$ $$784958016157527668736$$ $$\nu^{14}\mathstrut +\mathstrut$$ $$303187723758517273920$$ $$\nu^{13}\mathstrut +\mathstrut$$ $$202932145804771167535104$$ $$\nu^{12}\mathstrut -\mathstrut$$ $$3236441107805450132951040$$ $$\nu^{11}\mathstrut -\mathstrut$$ $$21558258125752034779987968$$ $$\nu^{10}\mathstrut -\mathstrut$$ $$742890654813919552623083520$$ $$\nu^{9}\mathstrut -\mathstrut$$ $$21454678355626107336218640384$$ $$\nu^{8}\mathstrut -\mathstrut$$ $$450042237173729535477531279360$$ $$\nu^{7}\mathstrut +\mathstrut$$ $$56883127992053949267051427135488$$ $$\nu^{6}\mathstrut +\mathstrut$$ $$1139710695646396110383833531023360$$ $$\nu^{5}\mathstrut -\mathstrut$$ $$24406261697714745713004469549006848$$ $$\nu^{4}\mathstrut -\mathstrut$$ $$165910551883710751944020159417548800$$ $$\nu^{3}\mathstrut +\mathstrut$$ $$9219747134140788721780149095529185280$$ $$\nu^{2}\mathstrut +\mathstrut$$ $$20995905308932208702965773590006333440$$ $$\nu\mathstrut -\mathstrut$$ $$2676074463828503951306550662363088945152$$$$)/$$$$79\!\cdots\!00$$ $$\beta_{18}$$ $$=$$ $$($$$$20861361537311$$ $$\nu^{19}\mathstrut +\mathstrut$$ $$664254202777088$$ $$\nu^{18}\mathstrut -\mathstrut$$ $$9712105016906985$$ $$\nu^{17}\mathstrut -\mathstrut$$ $$378508764894036480$$ $$\nu^{16}\mathstrut +\mathstrut$$ $$3659925436052685804$$ $$\nu^{15}\mathstrut +\mathstrut$$ $$333973837738325317632$$ $$\nu^{14}\mathstrut +\mathstrut$$ $$1116236301105943450944$$ $$\nu^{13}\mathstrut -\mathstrut$$ $$112963646743620701749248$$ $$\nu^{12}\mathstrut -\mathstrut$$ $$265637681025076670926848$$ $$\nu^{11}\mathstrut +\mathstrut$$ $$21419056014545158894780416$$ $$\nu^{10}\mathstrut -\mathstrut$$ $$100267392562841143370317824$$ $$\nu^{9}\mathstrut +\mathstrut$$ $$6227434081898698178253815808$$ $$\nu^{8}\mathstrut -\mathstrut$$ $$75062984890221276515933356032$$ $$\nu^{7}\mathstrut -\mathstrut$$ $$18501514133614160826699985453056$$ $$\nu^{6}\mathstrut +\mathstrut$$ $$197189628916290134997291679875072$$ $$\nu^{5}\mathstrut +\mathstrut$$ $$6833679502713532408439517663461376$$ $$\nu^{4}\mathstrut -\mathstrut$$ $$30265381768839622117935619265003520$$ $$\nu^{3}\mathstrut -\mathstrut$$ $$4540413934350005252197418343868661760$$ $$\nu^{2}\mathstrut +\mathstrut$$ $$11057913549659291865695803256684412928$$ $$\nu\mathstrut +\mathstrut$$ $$1453673518702250559917566267443562676224$$$$)/$$$$19\!\cdots\!00$$ $$\beta_{19}$$ $$=$$ $$($$$$52728422385041$$ $$\nu^{19}\mathstrut -\mathstrut$$ $$64528628019712$$ $$\nu^{18}\mathstrut +\mathstrut$$ $$8735714394333465$$ $$\nu^{17}\mathstrut -\mathstrut$$ $$288344739388654080$$ $$\nu^{16}\mathstrut -\mathstrut$$ $$2076715176723154476$$ $$\nu^{15}\mathstrut +\mathstrut$$ $$160264440885605701632$$ $$\nu^{14}\mathstrut +\mathstrut$$ $$3411816834702714588864$$ $$\nu^{13}\mathstrut -\mathstrut$$ $$16644856777799873691648$$ $$\nu^{12}\mathstrut -\mathstrut$$ $$1144271951083651373887488$$ $$\nu^{11}\mathstrut +\mathstrut$$ $$8537469759136614177570816$$ $$\nu^{10}\mathstrut -\mathstrut$$ $$211428997219881600332857344$$ $$\nu^{9}\mathstrut +\mathstrut$$ $$1180381335703132787083051008$$ $$\nu^{8}\mathstrut +\mathstrut$$ $$946834725462405301524662059008$$ $$\nu^{7}\mathstrut -\mathstrut$$ $$12672481859836240755147046649856$$ $$\nu^{6}\mathstrut -\mathstrut$$ $$164157668945818972363059225427968$$ $$\nu^{5}\mathstrut +\mathstrut$$ $$9314498941902225996531690243096576$$ $$\nu^{4}\mathstrut -\mathstrut$$ $$6040904779116899729561630980177920$$ $$\nu^{3}\mathstrut -\mathstrut$$ $$1197981035491800412489757689758351360$$ $$\nu^{2}\mathstrut -\mathstrut$$ $$31184387169193157452512097371561132032$$ $$\nu\mathstrut +\mathstrut$$ $$600134170201916187129337659532534349824$$$$)/$$$$39\!\cdots\!00$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$$$/2$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{2}\mathstrut +\mathstrut$$ $$99$$$$)/4$$ $$\nu^{3}$$ $$=$$ $$($$$$\beta_{5}\mathstrut -\mathstrut$$ $$3$$ $$\beta_{3}\mathstrut +\mathstrut$$ $$98$$ $$\beta_{1}$$$$)/8$$ $$\nu^{4}$$ $$=$$ $$($$$$2$$ $$\beta_{19}\mathstrut -\mathstrut$$ $$\beta_{18}\mathstrut -\mathstrut$$ $$\beta_{17}\mathstrut -\mathstrut$$ $$3$$ $$\beta_{15}\mathstrut -\mathstrut$$ $$\beta_{14}\mathstrut +\mathstrut$$ $$\beta_{13}\mathstrut +\mathstrut$$ $$3$$ $$\beta_{12}\mathstrut -\mathstrut$$ $$2$$ $$\beta_{11}\mathstrut -\mathstrut$$ $$4$$ $$\beta_{10}\mathstrut +\mathstrut$$ $$12$$ $$\beta_{9}\mathstrut -\mathstrut$$ $$\beta_{8}\mathstrut -\mathstrut$$ $$\beta_{7}\mathstrut +\mathstrut$$ $$2$$ $$\beta_{5}\mathstrut -\mathstrut$$ $$34$$ $$\beta_{4}\mathstrut +\mathstrut$$ $$188$$ $$\beta_{3}\mathstrut +\mathstrut$$ $$84$$ $$\beta_{2}\mathstrut +\mathstrut$$ $$54$$ $$\beta_{1}\mathstrut +\mathstrut$$ $$174164$$$$)/16$$ $$\nu^{5}$$ $$=$$ $$($$$$-$$$$14$$ $$\beta_{19}\mathstrut +\mathstrut$$ $$13$$ $$\beta_{18}\mathstrut +\mathstrut$$ $$23$$ $$\beta_{17}\mathstrut -\mathstrut$$ $$10$$ $$\beta_{16}\mathstrut -\mathstrut$$ $$3$$ $$\beta_{15}\mathstrut +\mathstrut$$ $$3$$ $$\beta_{14}\mathstrut +\mathstrut$$ $$3$$ $$\beta_{13}\mathstrut +\mathstrut$$ $$373$$ $$\beta_{12}\mathstrut -\mathstrut$$ $$119$$ $$\beta_{11}\mathstrut +\mathstrut$$ $$116$$ $$\beta_{9}\mathstrut +\mathstrut$$ $$43$$ $$\beta_{8}\mathstrut -\mathstrut$$ $$13$$ $$\beta_{7}\mathstrut +\mathstrut$$ $$3438$$ $$\beta_{6}\mathstrut +\mathstrut$$ $$27$$ $$\beta_{5}\mathstrut -\mathstrut$$ $$31549$$ $$\beta_{3}\mathstrut +\mathstrut$$ $$324$$ $$\beta_{2}\mathstrut +\mathstrut$$ $$33168$$ $$\beta_{1}\mathstrut +\mathstrut$$ $$70$$$$)/8$$ $$\nu^{6}$$ $$=$$ $$($$$$-$$$$98$$ $$\beta_{19}\mathstrut -\mathstrut$$ $$551$$ $$\beta_{18}\mathstrut +\mathstrut$$ $$649$$ $$\beta_{17}\mathstrut +\mathstrut$$ $$331$$ $$\beta_{15}\mathstrut +\mathstrut$$ $$1081$$ $$\beta_{14}\mathstrut +\mathstrut$$ $$135$$ $$\beta_{13}\mathstrut +\mathstrut$$ $$17277$$ $$\beta_{12}\mathstrut +\mathstrut$$ $$902$$ $$\beta_{11}\mathstrut -\mathstrut$$ $$2988$$ $$\beta_{10}\mathstrut -\mathstrut$$ $$2204$$ $$\beta_{9}\mathstrut -\mathstrut$$ $$32039$$ $$\beta_{8}\mathstrut +\mathstrut$$ $$2753$$ $$\beta_{7}\mathstrut +\mathstrut$$ $$2480$$ $$\beta_{6}\mathstrut -\mathstrut$$ $$98$$ $$\beta_{5}\mathstrut -\mathstrut$$ $$13630$$ $$\beta_{4}\mathstrut +\mathstrut$$ $$1467316$$ $$\beta_{3}\mathstrut +\mathstrut$$ $$16348$$ $$\beta_{2}\mathstrut +\mathstrut$$ $$469442$$ $$\beta_{1}\mathstrut -\mathstrut$$ $$18172388$$$$)/16$$ $$\nu^{7}$$ $$=$$ $$($$$$9982$$ $$\beta_{19}\mathstrut +\mathstrut$$ $$1291$$ $$\beta_{18}\mathstrut +\mathstrut$$ $$2209$$ $$\beta_{17}\mathstrut -\mathstrut$$ $$21766$$ $$\beta_{16}\mathstrut -\mathstrut$$ $$373$$ $$\beta_{15}\mathstrut +\mathstrut$$ $$10197$$ $$\beta_{14}\mathstrut +\mathstrut$$ $$373$$ $$\beta_{13}\mathstrut +\mathstrut$$ $$193315$$ $$\beta_{12}\mathstrut +\mathstrut$$ $$294655$$ $$\beta_{11}\mathstrut -\mathstrut$$ $$34420$$ $$\beta_{9}\mathstrut +\mathstrut$$ $$394621$$ $$\beta_{8}\mathstrut -\mathstrut$$ $$1291$$ $$\beta_{7}\mathstrut -\mathstrut$$ $$250334$$ $$\beta_{6}\mathstrut -\mathstrut$$ $$4199$$ $$\beta_{5}\mathstrut -\mathstrut$$ $$16136303$$ $$\beta_{3}\mathstrut +\mathstrut$$ $$303532$$ $$\beta_{2}\mathstrut -\mathstrut$$ $$9707064$$ $$\beta_{1}\mathstrut +\mathstrut$$ $$169258$$$$)/8$$ $$\nu^{8}$$ $$=$$ $$($$$$-$$$$17846$$ $$\beta_{19}\mathstrut -\mathstrut$$ $$56333$$ $$\beta_{18}\mathstrut +\mathstrut$$ $$74179$$ $$\beta_{17}\mathstrut -\mathstrut$$ $$518631$$ $$\beta_{15}\mathstrut +\mathstrut$$ $$323731$$ $$\beta_{14}\mathstrut -\mathstrut$$ $$554323$$ $$\beta_{13}\mathstrut +\mathstrut$$ $$9963231$$ $$\beta_{12}\mathstrut -\mathstrut$$ $$1762190$$ $$\beta_{11}\mathstrut +\mathstrut$$ $$566620$$ $$\beta_{10}\mathstrut -\mathstrut$$ $$848244$$ $$\beta_{9}\mathstrut -\mathstrut$$ $$15937805$$ $$\beta_{8}\mathstrut -\mathstrut$$ $$1895989$$ $$\beta_{7}\mathstrut +\mathstrut$$ $$1240272$$ $$\beta_{6}\mathstrut -\mathstrut$$ $$17846$$ $$\beta_{5}\mathstrut +\mathstrut$$ $$7629014$$ $$\beta_{4}\mathstrut +\mathstrut$$ $$728802300$$ $$\beta_{3}\mathstrut -\mathstrut$$ $$26383436$$ $$\beta_{2}\mathstrut +\mathstrut$$ $$234490102$$ $$\beta_{1}\mathstrut +\mathstrut$$ $$286478617972$$$$)/16$$ $$\nu^{9}$$ $$=$$ $$($$$$-$$$$1926358$$ $$\beta_{19}\mathstrut -\mathstrut$$ $$1083367$$ $$\beta_{18}\mathstrut -\mathstrut$$ $$4659189$$ $$\beta_{17}\mathstrut -\mathstrut$$ $$6790914$$ $$\beta_{16}\mathstrut -\mathstrut$$ $$2492455$$ $$\beta_{15}\mathstrut +\mathstrut$$ $$5772999$$ $$\beta_{14}\mathstrut +\mathstrut$$ $$2492455$$ $$\beta_{13}\mathstrut +\mathstrut$$ $$64467233$$ $$\beta_{12}\mathstrut -\mathstrut$$ $$234549547$$ $$\beta_{11}\mathstrut -\mathstrut$$ $$13749500$$ $$\beta_{9}\mathstrut +\mathstrut$$ $$159973567$$ $$\beta_{8}\mathstrut +\mathstrut$$ $$1083367$$ $$\beta_{7}\mathstrut +\mathstrut$$ $$618883894$$ $$\beta_{6}\mathstrut -\mathstrut$$ $$12309437$$ $$\beta_{5}\mathstrut -\mathstrut$$ $$4443663765$$ $$\beta_{3}\mathstrut +\mathstrut$$ $$109309348$$ $$\beta_{2}\mathstrut +\mathstrut$$ $$70170590008$$ $$\beta_{1}\mathstrut +\mathstrut$$ $$67880718$$$$)/8$$ $$\nu^{10}$$ $$=$$ $$($$$$-$$$$11602802$$ $$\beta_{19}\mathstrut +\mathstrut$$ $$33234625$$ $$\beta_{18}\mathstrut -\mathstrut$$ $$21631823$$ $$\beta_{17}\mathstrut -\mathstrut$$ $$219962109$$ $$\beta_{15}\mathstrut +\mathstrut$$ $$92413345$$ $$\beta_{14}\mathstrut -\mathstrut$$ $$243167713$$ $$\beta_{13}\mathstrut -\mathstrut$$ $$5805802843$$ $$\beta_{12}\mathstrut -\mathstrut$$ $$185638394$$ $$\beta_{11}\mathstrut -\mathstrut$$ $$588698060$$ $$\beta_{10}\mathstrut -\mathstrut$$ $$224683452$$ $$\beta_{9}\mathstrut -\mathstrut$$ $$5020829759$$ $$\beta_{8}\mathstrut +\mathstrut$$ $$286708745$$ $$\beta_{7}\mathstrut +\mathstrut$$ $$383156976$$ $$\beta_{6}\mathstrut -\mathstrut$$ $$11602802$$ $$\beta_{5}\mathstrut +\mathstrut$$ $$4497115090$$ $$\beta_{4}\mathstrut -\mathstrut$$ $$413099186412$$ $$\beta_{3}\mathstrut +\mathstrut$$ $$82294907004$$ $$\beta_{2}\mathstrut -\mathstrut$$ $$132261435214$$ $$\beta_{1}\mathstrut -\mathstrut$$ $$241062032403524$$$$)/16$$ $$\nu^{11}$$ $$=$$ $$($$$$1120245038$$ $$\beta_{19}\mathstrut -\mathstrut$$ $$245539837$$ $$\beta_{18}\mathstrut -\mathstrut$$ $$1709030327$$ $$\beta_{17}\mathstrut -\mathstrut$$ $$1448893430$$ $$\beta_{16}\mathstrut -\mathstrut$$ $$1217950653$$ $$\beta_{15}\mathstrut -\mathstrut$$ $$4766457699$$ $$\beta_{14}\mathstrut +\mathstrut$$ $$1217950653$$ $$\beta_{13}\mathstrut +\mathstrut$$ $$10371444443$$ $$\beta_{12}\mathstrut +\mathstrut$$ $$44631739735$$ $$\beta_{11}\mathstrut -\mathstrut$$ $$13261923092$$ $$\beta_{9}\mathstrut -\mathstrut$$ $$105497068603$$ $$\beta_{8}\mathstrut +\mathstrut$$ $$245539837$$ $$\beta_{7}\mathstrut +\mathstrut$$ $$424211272434$$ $$\beta_{6}\mathstrut +\mathstrut$$ $$17782539249$$ $$\beta_{5}\mathstrut -\mathstrut$$ $$1160922416343$$ $$\beta_{3}\mathstrut -\mathstrut$$ $$13149897012$$ $$\beta_{2}\mathstrut -\mathstrut$$ $$60666986586264$$ $$\beta_{1}\mathstrut -\mathstrut$$ $$54490268230$$$$)/8$$ $$\nu^{12}$$ $$=$$ $$($$$$84428086202$$ $$\beta_{19}\mathstrut -\mathstrut$$ $$26861954821$$ $$\beta_{18}\mathstrut -\mathstrut$$ $$57566131381$$ $$\beta_{17}\mathstrut -\mathstrut$$ $$37109285071$$ $$\beta_{15}\mathstrut +\mathstrut$$ $$26989162139$$ $$\beta_{14}\mathstrut +\mathstrut$$ $$131746887333$$ $$\beta_{13}\mathstrut -\mathstrut$$ $$482249771385$$ $$\beta_{12}\mathstrut -\mathstrut$$ $$93662384798$$ $$\beta_{11}\mathstrut -\mathstrut$$ $$535597147140$$ $$\beta_{10}\mathstrut +\mathstrut$$ $$642157626284$$ $$\beta_{9}\mathstrut -\mathstrut$$ $$767340864517$$ $$\beta_{8}\mathstrut -\mathstrut$$ $$85989829997$$ $$\beta_{7}\mathstrut +\mathstrut$$ $$33521477968$$ $$\beta_{6}\mathstrut +\mathstrut$$ $$84428086202$$ $$\beta_{5}\mathstrut +\mathstrut$$ $$3034800033382$$ $$\beta_{4}\mathstrut -\mathstrut$$ $$23967212175460$$ $$\beta_{3}\mathstrut -\mathstrut$$ $$61505623044460$$ $$\beta_{2}\mathstrut -\mathstrut$$ $$7844214361082$$ $$\beta_{1}\mathstrut +\mathstrut$$ $$54439264050892372$$$$)/16$$ $$\nu^{13}$$ $$=$$ $$($$$$242499472538$$ $$\beta_{19}\mathstrut +\mathstrut$$ $$1062225369137$$ $$\beta_{18}\mathstrut +\mathstrut$$ $$348130941779$$ $$\beta_{17}\mathstrut +\mathstrut$$ $$1932600731470$$ $$\beta_{16}\mathstrut -\mathstrut$$ $$1776319796495$$ $$\beta_{15}\mathstrut -\mathstrut$$ $$768644733777$$ $$\beta_{14}\mathstrut +\mathstrut$$ $$1776319796495$$ $$\beta_{13}\mathstrut +\mathstrut$$ $$5048915033081$$ $$\beta_{12}\mathstrut -\mathstrut$$ $$11434652830579$$ $$\beta_{11}\mathstrut -\mathstrut$$ $$5254564454300$$ $$\beta_{9}\mathstrut -\mathstrut$$ $$8065373874457$$ $$\beta_{8}\mathstrut -\mathstrut$$ $$1062225369137$$ $$\beta_{7}\mathstrut +\mathstrut$$ $$128517737030342$$ $$\beta_{6}\mathstrut -\mathstrut$$ $$17255167411189$$ $$\beta_{5}\mathstrut -\mathstrut$$ $$437020238464525$$ $$\beta_{3}\mathstrut +\mathstrut$$ $$5619228234500$$ $$\beta_{2}\mathstrut +\mathstrut$$ $$13463303850048248$$ $$\beta_{1}\mathstrut -\mathstrut$$ $$6422437115170$$$$)/8$$ $$\nu^{14}$$ $$=$$ $$($$$$-$$$$28262175402274$$ $$\beta_{19}\mathstrut +\mathstrut$$ $$60137703195737$$ $$\beta_{18}\mathstrut -\mathstrut$$ $$31875527793463$$ $$\beta_{17}\mathstrut +\mathstrut$$ $$42970685388747$$ $$\beta_{15}\mathstrut +\mathstrut$$ $$42452345557049$$ $$\beta_{14}\mathstrut -\mathstrut$$ $$13553665415801$$ $$\beta_{13}\mathstrut +\mathstrut$$ $$1576445248230813$$ $$\beta_{12}\mathstrut +\mathstrut$$ $$75007685261462$$ $$\beta_{11}\mathstrut -\mathstrut$$ $$97635594443500$$ $$\beta_{10}\mathstrut -\mathstrut$$ $$30975783644508$$ $$\beta_{9}\mathstrut -\mathstrut$$ $$357875525388967$$ $$\beta_{8}\mathstrut +\mathstrut$$ $$209116394055265$$ $$\beta_{7}\mathstrut +\mathstrut$$ $$10058477931888$$ $$\beta_{6}\mathstrut -\mathstrut$$ $$28262175402274$$ $$\beta_{5}\mathstrut +\mathstrut$$ $$1151539460476546$$ $$\beta_{4}\mathstrut +\mathstrut$$ $$134637585821412084$$ $$\beta_{3}\mathstrut +\mathstrut$$ $$12197447742988828$$ $$\beta_{2}\mathstrut +\mathstrut$$ $$43186113798062690$$ $$\beta_{1}\mathstrut -\mathstrut$$ $$11063500004389784740$$$$)/16$$ $$\nu^{15}$$ $$=$$ $$($$$$334300856711806$$ $$\beta_{19}\mathstrut +\mathstrut$$ $$46520424696523$$ $$\beta_{18}\mathstrut -\mathstrut$$ $$285404129197407$$ $$\beta_{17}\mathstrut +\mathstrut$$ $$307896550139898$$ $$\beta_{16}\mathstrut -\mathstrut$$ $$378444978590453$$ $$\beta_{15}\mathstrut +\mathstrut$$ $$1168715433262293$$ $$\beta_{14}\mathstrut +\mathstrut$$ $$378444978590453$$ $$\beta_{13}\mathstrut -\mathstrut$$ $$18276927096838685$$ $$\beta_{12}\mathstrut +\mathstrut$$ $$26793702635271679$$ $$\beta_{11}\mathstrut -\mathstrut$$ $$3328750750210804$$ $$\beta_{9}\mathstrut +\mathstrut$$ $$34855446163866557$$ $$\beta_{8}\mathstrut -\mathstrut$$ $$46520424696523$$ $$\beta_{7}\mathstrut -\mathstrut$$ $$512549952323422$$ $$\beta_{6}\mathstrut +\mathstrut$$ $$1792618081606921$$ $$\beta_{5}\mathstrut +\mathstrut$$ $$1526181565647949185$$ $$\beta_{3}\mathstrut -\mathstrut$$ $$7337976572186644$$ $$\beta_{2}\mathstrut -\mathstrut$$ $$2277678734468607192$$ $$\beta_{1}\mathstrut +\mathstrut$$ $$14571372061217322$$$$)/8$$ $$\nu^{16}$$ $$=$$ $$($$$$-$$$$148043133669846$$ $$\beta_{19}\mathstrut +\mathstrut$$ $$26180663941687299$$ $$\beta_{18}\mathstrut -\mathstrut$$ $$26032620808017453$$ $$\beta_{17}\mathstrut -\mathstrut$$ $$46724170236825015$$ $$\beta_{15}\mathstrut -\mathstrut$$ $$67615707262618461$$ $$\beta_{14}\mathstrut -\mathstrut$$ $$47020256504164707$$ $$\beta_{13}\mathstrut +\mathstrut$$ $$145184776446505135$$ $$\beta_{12}\mathstrut -\mathstrut$$ $$5292671875455086$$ $$\beta_{11}\mathstrut -\mathstrut$$ $$18269425041927780$$ $$\beta_{10}\mathstrut +\mathstrut$$ $$30485433385208268$$ $$\beta_{9}\mathstrut +\mathstrut$$ $$1566991390917243651$$ $$\beta_{8}\mathstrut -\mathstrut$$ $$26187753860809765$$ $$\beta_{7}\mathstrut -\mathstrut$$ $$114539865096429360$$ $$\beta_{6}\mathstrut -\mathstrut$$ $$148043133669846$$ $$\beta_{5}\mathstrut +\mathstrut$$ $$176630166733831926$$ $$\beta_{4}\mathstrut +\mathstrut$$ $$12340419527251115068$$ $$\beta_{3}\mathstrut -\mathstrut$$ $$3004769573127831948$$ $$\beta_{2}\mathstrut +\mathstrut$$ $$3971324987239654294$$ $$\beta_{1}\mathstrut +\mathstrut$$ $$23159979889812405966900$$$$)/16$$ $$\nu^{17}$$ $$=$$ $$($$$$-$$$$15441452171127798$$ $$\beta_{19}\mathstrut -\mathstrut$$ $$194195113220287671$$ $$\beta_{18}\mathstrut -\mathstrut$$ $$64646036688224613$$ $$\beta_{17}\mathstrut +\mathstrut$$ $$602679577972900254$$ $$\beta_{16}\mathstrut +\mathstrut$$ $$323744189752350729$$ $$\beta_{15}\mathstrut +\mathstrut$$ $$223645379953401495$$ $$\beta_{14}\mathstrut -\mathstrut$$ $$323744189752350729$$ $$\beta_{13}\mathstrut -\mathstrut$$ $$5315732609610095151$$ $$\beta_{12}\mathstrut -\mathstrut$$ $$2074539421055336891$$ $$\beta_{11}\mathstrut +\mathstrut$$ $$839060267876889540$$ $$\beta_{9}\mathstrut +\mathstrut$$ $$1960838627298648591$$ $$\beta_{8}\mathstrut +\mathstrut$$ $$194195113220287671$$ $$\beta_{7}\mathstrut +\mathstrut$$ $$79322729055914048598$$ $$\beta_{6}\mathstrut -\mathstrut$$ $$733801132566146925$$ $$\beta_{5}\mathstrut +\mathstrut$$ $$458073699678311505979$$ $$\beta_{3}\mathstrut -\mathstrut$$ $$4718567291189623452$$ $$\beta_{2}\mathstrut +\mathstrut$$ $$5931306687337437075000$$ $$\beta_{1}\mathstrut +\mathstrut$$ $$1273401624244511406$$$$)/8$$ $$\nu^{18}$$ $$=$$ $$($$$$-$$$$4952817144020814162$$ $$\beta_{19}\mathstrut +\mathstrut$$ $$6528510030265107249$$ $$\beta_{18}\mathstrut -\mathstrut$$ $$1575692886244293087$$ $$\beta_{17}\mathstrut +\mathstrut$$ $$8169676679967086163$$ $$\beta_{15}\mathstrut -\mathstrut$$ $$11118500381650297839$$ $$\beta_{14}\mathstrut -\mathstrut$$ $$1735957608074542161$$ $$\beta_{13}\mathstrut -\mathstrut$$ $$240318150030246699819$$ $$\beta_{12}\mathstrut +\mathstrut$$ $$4434082807554055590$$ $$\beta_{11}\mathstrut -\mathstrut$$ $$80808594406875614988$$ $$\beta_{10}\mathstrut -\mathstrut$$ $$16820147525424919548$$ $$\beta_{9}\mathstrut +\mathstrut$$ $$428488183966032825393$$ $$\beta_{8}\mathstrut -\mathstrut$$ $$23179678477861549575$$ $$\beta_{7}\mathstrut -\mathstrut$$ $$31982370329511974928$$ $$\beta_{6}\mathstrut -\mathstrut$$ $$4952817144020814162$$ $$\beta_{5}\mathstrut -\mathstrut$$ $$208609269676507668558$$ $$\beta_{4}\mathstrut -\mathstrut$$ $$18749298330204427635756$$ $$\beta_{3}\mathstrut +\mathstrut$$ $$6244323012608557513404$$ $$\beta_{2}\mathstrut -\mathstrut$$ $$6009899696814312882030$$ $$\beta_{1}\mathstrut -\mathstrut$$ $$1500963881845785618019844$$$$)/16$$ $$\nu^{19}$$ $$=$$ $$($$$$264353649577719578958$$ $$\beta_{19}\mathstrut -\mathstrut$$ $$27108892988102479533$$ $$\beta_{18}\mathstrut +\mathstrut$$ $$87558597300783467577$$ $$\beta_{17}\mathstrut +\mathstrut$$ $$88271193197533497450$$ $$\beta_{16}\mathstrut +\mathstrut$$ $$141776383276988426643$$ $$\beta_{15}\mathstrut -\mathstrut$$ $$240157787123835970227$$ $$\beta_{14}\mathstrut -\mathstrut$$ $$141776383276988426643$$ $$\beta_{13}\mathstrut -\mathstrut$$ $$2044640604321636379989$$ $$\beta_{12}\mathstrut -\mathstrut$$ $$3315354838169745749145$$ $$\beta_{11}\mathstrut -\mathstrut$$ $$369409656577562633172$$ $$\beta_{9}\mathstrut -\mathstrut$$ $$5237800633619283587595$$ $$\beta_{8}\mathstrut +\mathstrut$$ $$27108892988102479533$$ $$\beta_{7}\mathstrut +\mathstrut$$ $$5318798053804332586194$$ $$\beta_{6}\mathstrut +\mathstrut$$ $$1784800993036943086689$$ $$\beta_{5}\mathstrut +\mathstrut$$ $$159556236651584802618393$$ $$\beta_{3}\mathstrut -\mathstrut$$ $$3181218263009358603252$$ $$\beta_{2}\mathstrut -\mathstrut$$ $$324203861615639326858520$$ $$\beta_{1}\mathstrut -\mathstrut$$ $$2549892911480535900390$$$$)/8$$

## Character Values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/12\mathbb{Z}\right)^\times$$.

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
11.1
 −22.4409 − 2.89952i −22.4409 + 2.89952i −20.8511 − 8.78821i −20.8511 + 8.78821i −16.0196 − 15.9804i −16.0196 + 15.9804i −12.1023 − 19.1190i −12.1023 + 19.1190i −0.544378 − 22.6209i −0.544378 + 22.6209i 0.544378 − 22.6209i 0.544378 + 22.6209i 12.1023 − 19.1190i 12.1023 + 19.1190i 16.0196 − 15.9804i 16.0196 + 15.9804i 20.8511 − 8.78821i 20.8511 + 8.78821i 22.4409 − 2.89952i 22.4409 + 2.89952i
−44.8817 5.79904i −353.249 228.828i 1980.74 + 520.542i 12652.7i 14527.4 + 12318.7i 32182.6i −85880.5 34849.2i 72422.0 + 161667.i −73373.6 + 567877.i
11.2 −44.8817 + 5.79904i −353.249 + 228.828i 1980.74 520.542i 12652.7i 14527.4 12318.7i 32182.6i −85880.5 + 34849.2i 72422.0 161667.i −73373.6 567877.i
11.3 −41.7022 17.5764i 364.303 210.784i 1430.14 + 1465.95i 505.366i −18897.1 + 2387.01i 77554.9i −33873.8 86269.9i 88286.9 153579.i −8882.52 + 21074.9i
11.4 −41.7022 + 17.5764i 364.303 + 210.784i 1430.14 1465.95i 505.366i −18897.1 2387.01i 77554.9i −33873.8 + 86269.9i 88286.9 + 153579.i −8882.52 21074.9i
11.5 −32.0392 31.9607i 70.6484 + 414.917i 5.02409 + 2047.99i 1842.04i 10997.5 15551.6i 16302.4i 65294.4 65776.7i −167165. + 58626.4i −58872.8 + 59017.4i
11.6 −32.0392 + 31.9607i 70.6484 414.917i 5.02409 2047.99i 1842.04i 10997.5 + 15551.6i 16302.4i 65294.4 + 65776.7i −167165. 58626.4i −58872.8 59017.4i
11.7 −24.2046 38.2379i −183.786 378.642i −876.277 + 1851.07i 9922.43i −10030.0 + 16192.5i 35930.2i 91990.8 11297.3i −109593. + 139178.i 379413. 240168.i
11.8 −24.2046 + 38.2379i −183.786 + 378.642i −876.277 1851.07i 9922.43i −10030.0 16192.5i 35930.2i 91990.8 + 11297.3i −109593. 139178.i 379413. + 240168.i
11.9 −1.08876 45.2417i −399.437 + 132.653i −2045.63 + 98.5144i 4150.68i 6436.35 + 17926.8i 39165.5i 6684.15 + 92440.6i 141953. 105973.i −187784. + 4519.08i
11.10 −1.08876 + 45.2417i −399.437 132.653i −2045.63 98.5144i 4150.68i 6436.35 17926.8i 39165.5i 6684.15 92440.6i 141953. + 105973.i −187784. 4519.08i
11.11 1.08876 45.2417i 399.437 132.653i −2045.63 98.5144i 4150.68i −5566.58 18215.7i 39165.5i −6684.15 + 92440.6i 141953. 105973.i −187784. 4519.08i
11.12 1.08876 + 45.2417i 399.437 + 132.653i −2045.63 + 98.5144i 4150.68i −5566.58 + 18215.7i 39165.5i −6684.15 92440.6i 141953. + 105973.i −187784. + 4519.08i
11.13 24.2046 38.2379i 183.786 + 378.642i −876.277 1851.07i 9922.43i 18926.9 + 2137.29i 35930.2i −91990.8 11297.3i −109593. + 139178.i 379413. + 240168.i
11.14 24.2046 + 38.2379i 183.786 378.642i −876.277 + 1851.07i 9922.43i 18926.9 2137.29i 35930.2i −91990.8 + 11297.3i −109593. 139178.i 379413. 240168.i
11.15 32.0392 31.9607i −70.6484 414.917i 5.02409 2047.99i 1842.04i −15524.6 11035.6i 16302.4i −65294.4 65776.7i −167165. + 58626.4i −58872.8 59017.4i
11.16 32.0392 + 31.9607i −70.6484 + 414.917i 5.02409 + 2047.99i 1842.04i −15524.6 + 11035.6i 16302.4i −65294.4 + 65776.7i −167165. 58626.4i −58872.8 + 59017.4i
11.17 41.7022 17.5764i −364.303 + 210.784i 1430.14 1465.95i 505.366i −11487.4 + 15193.3i 77554.9i 33873.8 86269.9i 88286.9 153579.i −8882.52 21074.9i
11.18 41.7022 + 17.5764i −364.303 210.784i 1430.14 + 1465.95i 505.366i −11487.4 15193.3i 77554.9i 33873.8 + 86269.9i 88286.9 + 153579.i −8882.52 + 21074.9i
11.19 44.8817 5.79904i 353.249 + 228.828i 1980.74 520.542i 12652.7i 17181.4 + 8221.72i 32182.6i 85880.5 34849.2i 72422.0 + 161667.i −73373.6 567877.i
11.20 44.8817 + 5.79904i 353.249 228.828i 1980.74 + 520.542i 12652.7i 17181.4 8221.72i 32182.6i 85880.5 + 34849.2i 72422.0 161667.i −73373.6 + 567877.i
 $$n$$: 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
3.b Odd 1 yes
4.b Odd 1 yes
12.b Even 1 yes

## Hecke kernels

There are no other newforms in $$S_{12}^{\mathrm{new}}(12, [\chi])$$.