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

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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:

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 \(S_{11}^{\mathrm{new}}(20, [\chi])\).