Properties

Label 20.9.b.a
Level 20
Weight 9
Character orbit 20.b
Analytic conductor 8.148
Analytic rank 0
Dimension 16
CM No
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 20 = 2^{2} \cdot 5 \)
Weight: \( k \) = \( 9 \)
Character orbit: \([\chi]\) = 20.b (of order \(2\) and degree \(1\))

Newform invariants

Self dual: No
Analytic conductor: \(8.14757220122\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{58}\cdot 5^{16} \)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \( -\beta_{1} q^{2} \) \( + ( \beta_{1} + \beta_{3} ) q^{3} \) \( + ( -3 + \beta_{2} ) q^{4} \) \( + ( -1 - 2 \beta_{1} + \beta_{4} ) q^{5} \) \( + ( 271 - \beta_{2} + 5 \beta_{3} - \beta_{4} + \beta_{5} ) q^{6} \) \( + ( 7 + 17 \beta_{1} - \beta_{2} - 3 \beta_{3} + \beta_{12} ) q^{7} \) \( + ( -885 + 3 \beta_{1} + \beta_{2} - 2 \beta_{3} + \beta_{5} + \beta_{8} ) q^{8} \) \( + ( -2374 + 146 \beta_{1} - 6 \beta_{2} + 2 \beta_{4} - 2 \beta_{5} + 4 \beta_{6} - \beta_{11} - \beta_{14} ) q^{9} \) \(+O(q^{10})\) \( q\) \( -\beta_{1} q^{2} \) \( + ( \beta_{1} + \beta_{3} ) q^{3} \) \( + ( -3 + \beta_{2} ) q^{4} \) \( + ( -1 - 2 \beta_{1} + \beta_{4} ) q^{5} \) \( + ( 271 - \beta_{2} + 5 \beta_{3} - \beta_{4} + \beta_{5} ) q^{6} \) \( + ( 7 + 17 \beta_{1} - \beta_{2} - 3 \beta_{3} + \beta_{12} ) q^{7} \) \( + ( -885 + 3 \beta_{1} + \beta_{2} - 2 \beta_{3} + \beta_{5} + \beta_{8} ) q^{8} \) \( + ( -2374 + 146 \beta_{1} - 6 \beta_{2} + 2 \beta_{4} - 2 \beta_{5} + 4 \beta_{6} - \beta_{11} - \beta_{14} ) q^{9} \) \( + ( 546 + \beta_{1} + 2 \beta_{2} + 4 \beta_{3} - \beta_{10} - \beta_{14} ) q^{10} \) \( + ( -69 - 106 \beta_{1} - 22 \beta_{2} + 58 \beta_{3} + 2 \beta_{4} - 6 \beta_{5} + 7 \beta_{6} + 3 \beta_{7} - \beta_{8} + \beta_{11} + 3 \beta_{12} + \beta_{13} ) q^{11} \) \( + ( -4124 - 288 \beta_{1} + 48 \beta_{3} - 22 \beta_{4} + \beta_{5} - 3 \beta_{6} + 3 \beta_{7} - 3 \beta_{8} - \beta_{9} + 2 \beta_{10} + \beta_{12} + 3 \beta_{13} - \beta_{14} ) q^{12} \) \( + ( 3116 - 248 \beta_{1} + 5 \beta_{2} + 4 \beta_{3} + 7 \beta_{4} - 2 \beta_{5} - 3 \beta_{6} + 2 \beta_{7} - 2 \beta_{9} - 4 \beta_{10} - \beta_{11} - \beta_{14} + 2 \beta_{15} ) q^{13} \) \( + ( 4253 - 3 \beta_{1} - 20 \beta_{2} + 41 \beta_{3} + 16 \beta_{4} - 6 \beta_{5} - \beta_{6} + 6 \beta_{7} - 4 \beta_{9} + \beta_{10} + 4 \beta_{12} - 4 \beta_{13} ) q^{14} \) \( + ( -60 - 189 \beta_{1} + 23 \beta_{2} - 14 \beta_{3} - 6 \beta_{4} + \beta_{10} + 5 \beta_{13} + \beta_{14} ) q^{15} \) \( + ( -4782 + 883 \beta_{1} - 19 \beta_{2} + 10 \beta_{3} + 23 \beta_{4} - 6 \beta_{5} + 25 \beta_{6} - 6 \beta_{7} - \beta_{8} - 3 \beta_{9} - \beta_{10} + \beta_{11} + 6 \beta_{12} + 10 \beta_{13} - \beta_{14} - \beta_{15} ) q^{16} \) \( + ( 1407 - 930 \beta_{1} + 42 \beta_{2} - 92 \beta_{4} + 18 \beta_{5} - 28 \beta_{6} + 8 \beta_{7} - 4 \beta_{8} + 4 \beta_{9} - \beta_{11} - 9 \beta_{14} + 4 \beta_{15} ) q^{17} \) \( + ( -37828 + 2315 \beta_{1} - 176 \beta_{2} + 168 \beta_{3} + 52 \beta_{4} - 16 \beta_{5} + 52 \beta_{6} - 8 \beta_{7} + 4 \beta_{8} + 4 \beta_{9} + 14 \beta_{10} + 4 \beta_{11} + 8 \beta_{12} - 8 \beta_{13} - 10 \beta_{14} - 4 \beta_{15} ) q^{18} \) \( + ( -241 - 720 \beta_{1} + 26 \beta_{2} - 74 \beta_{3} - 6 \beta_{4} + 22 \beta_{5} - 7 \beta_{6} + 13 \beta_{7} + 17 \beta_{8} - 2 \beta_{10} - 17 \beta_{11} - 5 \beta_{12} - 11 \beta_{13} + 14 \beta_{14} ) q^{19} \) \( + ( 10499 - 545 \beta_{1} - 14 \beta_{2} + 202 \beta_{3} - 11 \beta_{4} - 5 \beta_{5} + 10 \beta_{6} + 5 \beta_{7} - 3 \beta_{10} + 5 \beta_{11} + 15 \beta_{12} + 5 \beta_{13} + 2 \beta_{14} - 5 \beta_{15} ) q^{20} \) \( + ( 26298 + 1096 \beta_{1} + 32 \beta_{2} - 64 \beta_{3} + 166 \beta_{4} - 32 \beta_{5} + 76 \beta_{6} - 16 \beta_{7} + 20 \beta_{8} + 20 \beta_{9} - 56 \beta_{10} + 6 \beta_{11} - 18 \beta_{14} + 4 \beta_{15} ) q^{21} \) \( + ( -24150 + 42 \beta_{1} + 132 \beta_{2} - 286 \beta_{3} + 64 \beta_{4} + 104 \beta_{5} - 126 \beta_{6} - 24 \beta_{7} - 8 \beta_{8} - 12 \beta_{9} - 2 \beta_{10} - 8 \beta_{11} + 4 \beta_{12} - 20 \beta_{13} + 4 \beta_{14} - 8 \beta_{15} ) q^{22} \) \( + ( -711 - 1945 \beta_{1} - 13 \beta_{2} + 33 \beta_{3} - 28 \beta_{4} + 12 \beta_{6} - 28 \beta_{7} - 28 \beta_{8} + 16 \beta_{9} + 10 \beta_{10} - 4 \beta_{11} - 17 \beta_{12} + 6 \beta_{13} + 34 \beta_{14} + 16 \beta_{15} ) q^{23} \) \( + ( 12264 + 4133 \beta_{1} + 423 \beta_{2} - 706 \beta_{3} - 369 \beta_{4} + 80 \beta_{5} - 115 \beta_{6} - 18 \beta_{7} + 21 \beta_{8} - 11 \beta_{9} + 71 \beta_{10} - 11 \beta_{11} - 54 \beta_{12} - 10 \beta_{13} + 7 \beta_{14} - 21 \beta_{15} ) q^{24} \) \( + 78125 q^{25} \) \( + ( 63364 - 3118 \beta_{1} + 88 \beta_{2} + 728 \beta_{3} + 500 \beta_{4} + 56 \beta_{5} - 44 \beta_{6} - 24 \beta_{7} - 20 \beta_{8} - 4 \beta_{9} + 2 \beta_{10} + 76 \beta_{11} - 88 \beta_{12} + 24 \beta_{13} + 34 \beta_{14} - 12 \beta_{15} ) q^{26} \) \( + ( 1960 + 2306 \beta_{1} + 46 \beta_{2} - 2846 \beta_{3} + 204 \beta_{4} - 228 \beta_{5} - 50 \beta_{6} - 2 \beta_{7} + 22 \beta_{8} + 32 \beta_{9} + 96 \beta_{10} + 10 \beta_{11} - 28 \beta_{12} - 46 \beta_{13} + 72 \beta_{14} + 32 \beta_{15} ) q^{27} \) \( + ( 79496 - 4258 \beta_{1} + 158 \beta_{2} - 540 \beta_{3} - 804 \beta_{4} + 173 \beta_{5} - 273 \beta_{6} - 109 \beta_{7} + 23 \beta_{8} + \beta_{9} + 84 \beta_{10} + 118 \beta_{11} - 31 \beta_{12} - 13 \beta_{13} + 45 \beta_{14} + 10 \beta_{15} ) q^{28} \) \( + ( 168534 - 12320 \beta_{1} + 938 \beta_{2} - 72 \beta_{3} + 22 \beta_{4} - 328 \beta_{5} - 126 \beta_{6} + 124 \beta_{7} - 140 \beta_{8} - 24 \beta_{9} - 192 \beta_{10} - 102 \beta_{11} - 62 \beta_{14} - 16 \beta_{15} ) q^{29} \) \( + ( -50405 + 19 \beta_{1} + 152 \beta_{2} + 119 \beta_{3} + 216 \beta_{4} - 10 \beta_{5} + 85 \beta_{6} - 10 \beta_{7} + 20 \beta_{8} - 20 \beta_{9} - 21 \beta_{10} - 60 \beta_{11} - 80 \beta_{12} - 16 \beta_{14} - 20 \beta_{15} ) q^{30} \) \( + ( -194 - 74 \beta_{1} - 1138 \beta_{2} - 188 \beta_{3} + 144 \beta_{4} - 36 \beta_{5} + 34 \beta_{6} + 26 \beta_{7} - 126 \beta_{8} + 32 \beta_{9} - 38 \beta_{10} - 2 \beta_{11} - 18 \beta_{12} + 48 \beta_{13} + 74 \beta_{14} + 32 \beta_{15} ) q^{31} \) \( + ( -273708 + 3998 \beta_{1} - 1054 \beta_{2} + 3732 \beta_{3} - 754 \beta_{4} - 104 \beta_{5} + 342 \beta_{6} + 56 \beta_{7} - 70 \beta_{8} - 50 \beta_{9} - 178 \beta_{10} - 78 \beta_{11} - 184 \beta_{12} - 56 \beta_{13} - 46 \beta_{14} - 50 \beta_{15} ) q^{32} \) \( + ( -348187 - 5850 \beta_{1} - 1668 \beta_{2} + 904 \beta_{3} + 58 \beta_{4} - 82 \beta_{5} + 222 \beta_{6} - 164 \beta_{7} + 196 \beta_{8} + 88 \beta_{9} - 104 \beta_{10} + 17 \beta_{11} - 103 \beta_{14} + 32 \beta_{15} ) q^{33} \) \( + ( 238912 - 502 \beta_{1} + 1156 \beta_{2} - 5520 \beta_{3} + 1316 \beta_{4} + 48 \beta_{5} - 284 \beta_{6} + 88 \beta_{7} + 52 \beta_{8} - 12 \beta_{9} + 284 \beta_{10} + 52 \beta_{11} + 168 \beta_{12} + 88 \beta_{13} + 36 \beta_{14} - 52 \beta_{15} ) q^{34} \) \( + ( 2145 + 3015 \beta_{1} + 1120 \beta_{2} - 2105 \beta_{3} + 90 \beta_{4} - 230 \beta_{5} - 85 \beta_{6} + 15 \beta_{7} + 155 \beta_{8} + 140 \beta_{10} + 5 \beta_{11} + 5 \beta_{12} - 55 \beta_{13} + 20 \beta_{14} ) q^{35} \) \( + ( -326227 + 37658 \beta_{1} - 1725 \beta_{2} - 3780 \beta_{3} - 2514 \beta_{4} - 38 \beta_{5} + 340 \beta_{6} + 70 \beta_{7} - 104 \beta_{8} + 40 \beta_{9} - 34 \beta_{10} - 2 \beta_{11} + 466 \beta_{12} - 122 \beta_{13} - 140 \beta_{14} + 2 \beta_{15} ) q^{36} \) \( + ( 576704 + 35152 \beta_{1} + 340 \beta_{2} - 1216 \beta_{3} - 310 \beta_{4} + 692 \beta_{5} + 272 \beta_{6} - 48 \beta_{7} + 128 \beta_{8} + 48 \beta_{9} + 208 \beta_{10} + 162 \beta_{11} + 34 \beta_{14} + 80 \beta_{15} ) q^{37} \) \( + ( -195640 - 266 \beta_{1} - 202 \beta_{2} + 4052 \beta_{3} + 2994 \beta_{4} - 74 \beta_{5} + 150 \beta_{6} - 352 \beta_{7} + 160 \beta_{8} - 4 \beta_{9} + 362 \beta_{10} + 64 \beta_{11} + 20 \beta_{12} + 156 \beta_{13} - 36 \beta_{14} + 112 \beta_{15} ) q^{38} \) \( + ( -13232 - 25534 \beta_{1} - 310 \beta_{2} + 9110 \beta_{3} - 948 \beta_{4} + 1220 \beta_{5} + 198 \beta_{6} + 254 \beta_{7} + 86 \beta_{8} - 128 \beta_{9} - 452 \beta_{10} - 278 \beta_{11} + 92 \beta_{12} + 150 \beta_{13} - 68 \beta_{14} - 128 \beta_{15} ) q^{39} \) \( + ( 19191 - 10786 \beta_{1} + 626 \beta_{2} + 3232 \beta_{3} - 1163 \beta_{4} + 185 \beta_{5} - 245 \beta_{6} - 10 \beta_{7} + 40 \beta_{8} - 85 \beta_{9} - 83 \beta_{10} - 145 \beta_{11} + 130 \beta_{12} - 50 \beta_{13} + 17 \beta_{14} - 15 \beta_{15} ) q^{40} \) \( + ( -541947 - 20154 \beta_{1} + 3702 \beta_{2} - 1280 \beta_{3} + 46 \beta_{4} + 194 \beta_{5} - 908 \beta_{6} + 448 \beta_{7} - 392 \beta_{8} - 168 \beta_{9} - 48 \beta_{10} - 75 \beta_{11} + 37 \beta_{14} + 56 \beta_{15} ) q^{41} \) \( + ( -271756 - 26560 \beta_{1} - 3316 \beta_{2} + 12040 \beta_{3} + 7960 \beta_{4} - 512 \beta_{5} + 1112 \beta_{6} + 400 \beta_{7} - 296 \beta_{8} - 40 \beta_{9} - 1010 \beta_{10} - 168 \beta_{11} + 816 \beta_{12} + 208 \beta_{13} - 98 \beta_{14} - 88 \beta_{15} ) q^{42} \) \( + ( -14350 - 24351 \beta_{1} + 3402 \beta_{2} + 16101 \beta_{3} - 1320 \beta_{4} + 272 \beta_{5} + 644 \beta_{6} - 324 \beta_{7} + 140 \beta_{8} - 64 \beta_{9} + 52 \beta_{10} + 84 \beta_{11} + 310 \beta_{12} - 112 \beta_{13} - 276 \beta_{14} - 64 \beta_{15} ) q^{43} \) \( + ( 1063896 + 23820 \beta_{1} + 420 \beta_{2} + 11560 \beta_{3} - 6712 \beta_{4} - 926 \beta_{5} + 70 \beta_{6} + 478 \beta_{7} + 86 \beta_{8} + 10 \beta_{9} + 184 \beta_{10} + 220 \beta_{11} + 10 \beta_{12} + 62 \beta_{13} - 30 \beta_{14} + 164 \beta_{15} ) q^{44} \) \( + ( 94319 - 9342 \beta_{1} - 2885 \beta_{2} + 1660 \beta_{3} - 2194 \beta_{4} + 650 \beta_{5} - 465 \beta_{6} - 130 \beta_{7} + 140 \beta_{8} + 30 \beta_{9} + 620 \beta_{10} + 55 \beta_{11} + 15 \beta_{14} + 10 \beta_{15} ) q^{45} \) \( + ( -498969 + 717 \beta_{1} + 1102 \beta_{2} - 1817 \beta_{3} + 8234 \beta_{4} + 96 \beta_{5} - 25 \beta_{6} + 686 \beta_{7} - 376 \beta_{8} + 92 \beta_{9} - 359 \beta_{10} - 280 \beta_{11} - 836 \beta_{12} - 284 \beta_{13} - 112 \beta_{14} + 184 \beta_{15} ) q^{46} \) \( + ( 10925 + 13341 \beta_{1} - 169 \beta_{2} - 16505 \beta_{3} + 1068 \beta_{4} - 2264 \beta_{5} + 452 \beta_{6} + 132 \beta_{7} - 236 \beta_{8} + 506 \beta_{10} + 684 \beta_{11} + 251 \beta_{12} + 606 \beta_{13} - 310 \beta_{14} ) q^{47} \) \( + ( -165204 - 11416 \beta_{1} - 1288 \beta_{2} - 15344 \beta_{3} - 12548 \beta_{4} - 824 \beta_{5} + 1352 \beta_{6} - 628 \beta_{7} + 884 \beta_{8} + 248 \beta_{9} + 1164 \beta_{10} - 124 \beta_{11} + 20 \beta_{12} + 156 \beta_{13} + 64 \beta_{14} + 124 \beta_{15} ) q^{48} \) \( + ( -2026332 + 67486 \beta_{1} - 1070 \beta_{2} - 784 \beta_{3} + 5414 \beta_{4} - 1822 \beta_{5} + 2128 \beta_{6} - 248 \beta_{7} - 64 \beta_{8} - 200 \beta_{9} - 256 \beta_{10} - 247 \beta_{11} + 265 \beta_{14} - 312 \beta_{15} ) q^{49} \) \( -78125 \beta_{1} q^{50} \) \( + ( 59808 + 163938 \beta_{1} - 14188 \beta_{2} - 5312 \beta_{3} + 3672 \beta_{4} + 1236 \beta_{5} - 1986 \beta_{6} + 534 \beta_{7} - 498 \beta_{8} - 192 \beta_{9} - 1166 \beta_{10} - 78 \beta_{11} - 108 \beta_{12} - 724 \beta_{13} - 590 \beta_{14} - 192 \beta_{15} ) q^{51} \) \( + ( 400402 - 64050 \beta_{1} + 5840 \beta_{2} - 3660 \beta_{3} - 12598 \beta_{4} + 1438 \beta_{5} - 2292 \beta_{6} + 162 \beta_{7} - 712 \beta_{8} + 168 \beta_{9} - 2214 \beta_{10} - 550 \beta_{11} - 794 \beta_{12} + 482 \beta_{13} - 180 \beta_{14} + 38 \beta_{15} ) q^{52} \) \( + ( 121406 - 63452 \beta_{1} - 17487 \beta_{2} + 11108 \beta_{3} - 2817 \beta_{4} - 830 \beta_{5} - 403 \beta_{6} - 1054 \beta_{7} + 728 \beta_{8} - 618 \beta_{9} + 1788 \beta_{10} - 147 \beta_{11} + 797 \beta_{14} - 326 \beta_{15} ) q^{53} \) \( + ( 580634 + 7498 \beta_{1} - 960 \beta_{2} - 69742 \beta_{3} + 19568 \beta_{4} - 500 \beta_{5} - 490 \beta_{6} - 732 \beta_{7} + 336 \beta_{8} + 464 \beta_{9} - 22 \beta_{10} + 208 \beta_{11} - 576 \beta_{12} - 736 \beta_{13} + 168 \beta_{14} + 528 \beta_{15} ) q^{54} \) \( + ( 15285 + 29442 \beta_{1} + 8826 \beta_{2} - 4583 \beta_{3} - 642 \beta_{4} - 420 \beta_{5} - 1010 \beta_{6} - 930 \beta_{7} + 230 \beta_{8} + 80 \beta_{9} + 717 \beta_{10} + 250 \beta_{11} - 45 \beta_{12} + 195 \beta_{13} - 43 \beta_{14} + 80 \beta_{15} ) q^{55} \) \( + ( 2132316 - 72795 \beta_{1} + 9831 \beta_{2} - 48978 \beta_{3} - 15581 \beta_{4} + 36 \beta_{5} - 767 \beta_{6} + 1678 \beta_{7} - 1323 \beta_{8} - 23 \beta_{9} - 1253 \beta_{10} + 113 \beta_{11} - 566 \beta_{12} + 534 \beta_{13} + 1299 \beta_{14} + 239 \beta_{15} ) q^{56} \) \( + ( 808901 + 141150 \beta_{1} + 11850 \beta_{2} - 9856 \beta_{3} - 4628 \beta_{4} + 3458 \beta_{5} - 844 \beta_{6} + 360 \beta_{7} - 612 \beta_{8} + 196 \beta_{9} + 1568 \beta_{10} + 919 \beta_{11} + 975 \beta_{14} - 252 \beta_{15} ) q^{57} \) \( + ( 3172936 - 168170 \beta_{1} + 8208 \beta_{2} + 62256 \beta_{3} + 12040 \beta_{4} + 752 \beta_{5} - 3448 \beta_{6} - 1328 \beta_{7} + 2680 \beta_{8} + 472 \beta_{9} + 1636 \beta_{10} + 312 \beta_{11} - 176 \beta_{12} - 1232 \beta_{13} + 804 \beta_{14} - 184 \beta_{15} ) q^{58} \) \( + ( -11875 - 6636 \beta_{1} - 12950 \beta_{2} + 18998 \beta_{3} + 582 \beta_{4} + 666 \beta_{5} + 1559 \beta_{6} - 781 \beta_{7} - 1681 \beta_{8} + 64 \beta_{9} - 1046 \beta_{10} + 593 \beta_{11} - 839 \beta_{12} - 861 \beta_{13} - 854 \beta_{14} + 64 \beta_{15} ) q^{59} \) \( + ( -1507840 + 45106 \beta_{1} - 2182 \beta_{2} + 34236 \beta_{3} - 4796 \beta_{4} - 885 \beta_{5} + 3145 \beta_{6} - 1115 \beta_{7} + 625 \beta_{8} + 375 \beta_{9} + 396 \beta_{10} + 10 \beta_{11} - 345 \beta_{12} + 165 \beta_{13} - 469 \beta_{14} - 10 \beta_{15} ) q^{60} \) \( + ( 530276 - 6864 \beta_{1} + 21846 \beta_{2} - 11256 \beta_{3} + 9768 \beta_{4} - 744 \beta_{5} - 910 \beta_{6} + 820 \beta_{7} - 1560 \beta_{8} + 532 \beta_{9} - 1352 \beta_{10} + 368 \beta_{11} + 576 \beta_{14} - 740 \beta_{15} ) q^{61} \) \( + ( 73854 - 1266 \beta_{1} - 3208 \beta_{2} + 5462 \beta_{3} + 22224 \beta_{4} - 268 \beta_{5} - 4390 \beta_{6} + 1156 \beta_{7} - 1704 \beta_{8} - 858 \beta_{10} - 968 \beta_{11} - 2136 \beta_{12} - 680 \beta_{13} - 296 \beta_{14} + 200 \beta_{15} ) q^{62} \) \( + ( -55985 - 204455 \beta_{1} + 66389 \beta_{2} - 8793 \beta_{3} - 11604 \beta_{4} + 4576 \beta_{5} - 3652 \beta_{6} - 3180 \beta_{7} + 4212 \beta_{8} - 336 \beta_{9} + 2478 \beta_{10} - 1172 \beta_{11} - 2503 \beta_{12} + 810 \beta_{13} + 198 \beta_{14} - 336 \beta_{15} ) q^{63} \) \( + ( -2909816 + 257304 \beta_{1} - 8840 \beta_{2} + 102864 \beta_{3} - 17872 \beta_{4} + 2096 \beta_{5} - 1688 \beta_{6} - 2088 \beta_{7} - 432 \beta_{8} + 1144 \beta_{9} + 1232 \beta_{10} + 704 \beta_{11} + 72 \beta_{12} + 216 \beta_{13} + 1176 \beta_{14} + 192 \beta_{15} ) q^{64} \) \( + ( 496517 - 161726 \beta_{1} - 15180 \beta_{2} + 11000 \beta_{3} + 4558 \beta_{4} + 130 \beta_{5} - 1190 \beta_{6} - 1340 \beta_{7} + 1540 \beta_{8} - 80 \beta_{9} + 120 \beta_{10} + 515 \beta_{11} + 395 \beta_{14} + 200 \beta_{15} ) q^{65} \) \( + ( 1324504 + 353316 \beta_{1} - 3756 \beta_{2} + 21248 \beta_{3} + 28700 \beta_{4} - 3872 \beta_{5} + 11804 \beta_{6} + 1800 \beta_{7} - 3108 \beta_{8} - 196 \beta_{9} - 3192 \beta_{10} - 356 \beta_{11} + 3608 \beta_{12} + 1128 \beta_{13} - 272 \beta_{14} - 540 \beta_{15} ) q^{66} \) \( + ( -64010 - 192361 \beta_{1} - 16838 \beta_{2} - 32273 \beta_{3} - 528 \beta_{4} + 3920 \beta_{5} + 2508 \beta_{6} + 916 \beta_{7} + 36 \beta_{8} - 32 \beta_{9} - 1968 \beta_{10} - 1412 \beta_{11} - 974 \beta_{12} - 3052 \beta_{13} + 344 \beta_{14} - 32 \beta_{15} ) q^{67} \) \( + ( 967290 - 228616 \beta_{1} + 12506 \beta_{2} - 110448 \beta_{3} - 22216 \beta_{4} - 3184 \beta_{5} - 1208 \beta_{6} - 1776 \beta_{7} + 3256 \beta_{8} - 1016 \beta_{9} + 1016 \beta_{10} - 1688 \beta_{11} + 560 \beta_{12} - 688 \beta_{13} - 2280 \beta_{14} - 360 \beta_{15} ) q^{68} \) \( + ( 473456 - 21884 \beta_{1} + 11890 \beta_{2} - 7176 \beta_{3} - 21392 \beta_{4} + 7824 \beta_{5} - 4526 \beta_{6} + 380 \beta_{7} + 980 \beta_{8} + 712 \beta_{9} - 32 \beta_{10} + 2022 \beta_{11} - 50 \beta_{14} + 1360 \beta_{15} ) q^{69} \) \( + ( 685005 + 9100 \beta_{1} + 685 \beta_{2} - 73465 \beta_{3} + 2545 \beta_{4} + 715 \beta_{5} + 3480 \beta_{6} - 2620 \beta_{7} + 2280 \beta_{8} + 220 \beta_{9} + 1000 \beta_{10} + 680 \beta_{11} + 940 \beta_{12} - 60 \beta_{13} + 340 \beta_{14} + 200 \beta_{15} ) q^{70} \) \( + ( 152856 + 477374 \beta_{1} - 58062 \beta_{2} + 25412 \beta_{3} + 14972 \beta_{4} - 4424 \beta_{5} - 3204 \beta_{6} + 5724 \beta_{7} - 3092 \beta_{8} - 96 \beta_{9} - 2278 \beta_{10} + 468 \beta_{11} + 1352 \beta_{12} + 3750 \beta_{13} + 746 \beta_{14} - 96 \beta_{15} ) q^{71} \) \( + ( -2512153 + 325521 \beta_{1} - 35381 \beta_{2} - 38750 \beta_{3} + 5094 \beta_{4} - 5491 \beta_{5} - 5958 \beta_{6} + 3732 \beta_{7} - 801 \beta_{8} - 1094 \beta_{9} + 4726 \beta_{10} + 1362 \beta_{11} + 6268 \beta_{12} - 2204 \beta_{13} + 2126 \beta_{14} - 18 \beta_{15} ) q^{72} \) \( + ( 3791573 - 147950 \beta_{1} - 46920 \beta_{2} + 27720 \beta_{3} - 13650 \beta_{4} - 318 \beta_{5} + 806 \beta_{6} - 2148 \beta_{7} + 2356 \beta_{8} + 712 \beta_{9} + 3224 \beta_{10} - 1261 \beta_{11} - 2181 \beta_{14} + 208 \beta_{15} ) q^{73} \) \( + ( -8812236 - 579970 \beta_{1} - 27188 \beta_{2} - 132984 \beta_{3} - 744 \beta_{4} + 1120 \beta_{5} - 2280 \beta_{6} + 2704 \beta_{7} - 3144 \beta_{8} - 968 \beta_{9} - 1106 \beta_{10} + 696 \beta_{11} + 496 \beta_{12} + 3088 \beta_{13} - 1122 \beta_{14} - 184 \beta_{15} ) q^{74} \) \( + ( 78125 \beta_{1} + 78125 \beta_{3} ) q^{75} \) \( + ( 266320 + 205972 \beta_{1} + 4012 \beta_{2} - 78280 \beta_{3} - 1884 \beta_{4} + 3116 \beta_{5} + 18128 \beta_{6} + 2732 \beta_{7} - 2784 \beta_{8} - 376 \beta_{9} - 5628 \beta_{10} + 420 \beta_{11} - 5148 \beta_{12} - 628 \beta_{13} - 4112 \beta_{14} - 1060 \beta_{15} ) q^{76} \) \( + ( -9490876 + 820064 \beta_{1} + 31769 \beta_{2} - 30908 \beta_{3} - 439 \beta_{4} - 16766 \beta_{5} + 20885 \beta_{6} + 4242 \beta_{7} - 3480 \beta_{8} - 1738 \beta_{9} - 9540 \beta_{10} - 5627 \beta_{11} - 4651 \beta_{14} + 762 \beta_{15} ) q^{77} \) \( + ( -6645078 - 36482 \beta_{1} + 4668 \beta_{2} + 350922 \beta_{3} - 26292 \beta_{4} - 584 \beta_{5} + 1954 \beta_{6} - 1940 \beta_{7} + 352 \beta_{8} - 2592 \beta_{9} + 4702 \beta_{10} - 1376 \beta_{11} + 864 \beta_{12} + 4928 \beta_{13} - 1560 \beta_{14} - 1024 \beta_{15} ) q^{78} \) \( + ( -60552 - 316628 \beta_{1} + 9756 \beta_{2} - 150482 \beta_{3} + 408 \beta_{4} - 13860 \beta_{5} + 6010 \beta_{6} + 34 \beta_{7} - 566 \beta_{8} + 1664 \beta_{9} + 5390 \beta_{10} + 1654 \beta_{11} + 4996 \beta_{12} + 2652 \beta_{13} + 3758 \beta_{14} + 1664 \beta_{15} ) q^{79} \) \( + ( 2298854 - 26173 \beta_{1} + 4237 \beta_{2} + 113354 \beta_{3} - 7533 \beta_{4} + 1010 \beta_{5} + 3865 \beta_{6} - 290 \beta_{7} + 1395 \beta_{8} - 515 \beta_{9} + 3499 \beta_{10} + 1525 \beta_{11} - 1310 \beta_{12} - 130 \beta_{13} + 1479 \beta_{14} + 395 \beta_{15} ) q^{80} \) \( + ( 8291112 - 1316114 \beta_{1} + 39466 \beta_{2} + 10000 \beta_{3} + 52146 \beta_{4} - 2726 \beta_{5} - 20752 \beta_{6} + 5864 \beta_{7} - 3472 \beta_{8} - 1080 \beta_{9} - 9040 \beta_{10} - 1627 \beta_{11} - 2939 \beta_{14} + 2392 \beta_{15} ) q^{81} \) \( + ( 5482548 + 545618 \beta_{1} + 31216 \beta_{2} - 45320 \beta_{3} - 9956 \beta_{4} + 9104 \beta_{5} - 25444 \beta_{6} - 2840 \beta_{7} + 5452 \beta_{8} + 76 \beta_{9} + 7146 \beta_{10} + 3660 \beta_{11} - 7016 \beta_{12} - 152 \beta_{13} + 1634 \beta_{14} - 76 \beta_{15} ) q^{82} \) \( + ( -125188 - 289695 \beta_{1} + 9370 \beta_{2} + 36345 \beta_{3} + 1212 \beta_{4} - 21924 \beta_{5} + 15390 \beta_{6} + 2862 \beta_{7} + 3494 \beta_{8} + 960 \beta_{9} + 7224 \beta_{10} + 2554 \beta_{11} + 11664 \beta_{12} - 2534 \beta_{13} + 1728 \beta_{14} + 960 \beta_{15} ) q^{83} \) \( + ( 4237352 + 226998 \beta_{1} - 10298 \beta_{2} + 514916 \beta_{3} + 53106 \beta_{4} + 4894 \beta_{5} - 12748 \beta_{6} + 2338 \beta_{7} - 2096 \beta_{8} - 4048 \beta_{9} - 11070 \beta_{10} - 2766 \beta_{11} + 2918 \beta_{12} - 2910 \beta_{13} - 6332 \beta_{14} - 1330 \beta_{15} ) q^{84} \) \( + ( -6864098 - 429156 \beta_{1} - 28145 \beta_{2} + 25820 \beta_{3} + 4973 \beta_{4} - 7450 \beta_{5} - 805 \beta_{6} - 1010 \beta_{7} + 280 \beta_{8} + 810 \beta_{9} - 1260 \beta_{10} - 2265 \beta_{11} - 2345 \beta_{14} - 730 \beta_{15} ) q^{85} \) \( + ( -6372981 - 898 \beta_{1} + 35365 \beta_{2} + 106965 \beta_{3} - 89603 \beta_{4} + 13967 \beta_{5} + 12394 \beta_{6} + 4868 \beta_{7} + 1872 \beta_{8} - 712 \beta_{9} - 618 \beta_{10} + 2448 \beta_{11} + 6264 \beta_{12} - 376 \beta_{13} + 784 \beta_{14} - 656 \beta_{15} ) q^{86} \) \( + ( -893952 - 2150438 \beta_{1} - 34836 \beta_{2} + 210218 \beta_{3} - 36480 \beta_{4} + 14560 \beta_{5} + 35380 \beta_{6} + 4988 \beta_{7} - 6052 \beta_{8} - 176 \beta_{9} - 8384 \beta_{10} - 3516 \beta_{11} + 1176 \beta_{12} + 5964 \beta_{13} + 2632 \beta_{14} - 176 \beta_{15} ) q^{87} \) \( + ( 2505600 - 1029110 \beta_{1} - 11826 \beta_{2} - 272900 \beta_{3} + 37998 \beta_{4} + 27488 \beta_{5} - 16518 \beta_{6} - 7316 \beta_{7} + 1146 \beta_{8} - 182 \beta_{9} + 8094 \beta_{10} + 3034 \beta_{11} - 2044 \beta_{12} + 956 \beta_{13} + 9422 \beta_{14} - 1562 \beta_{15} ) q^{88} \) \( + ( 7330548 + 1785500 \beta_{1} - 9080 \beta_{2} - 40080 \beta_{3} + 20260 \beta_{4} + 1204 \beta_{5} + 32044 \beta_{6} - 888 \beta_{7} + 960 \beta_{8} + 1976 \beta_{9} + 4416 \beta_{10} - 2222 \beta_{11} - 4270 \beta_{14} + 72 \beta_{15} ) q^{89} \) \( + ( 2097966 - 77979 \beta_{1} + 26162 \beta_{2} - 198036 \beta_{3} - 44220 \beta_{4} - 1720 \beta_{5} + 6180 \beta_{6} + 1000 \beta_{7} - 2420 \beta_{8} - 260 \beta_{9} + 6269 \beta_{10} - 340 \beta_{11} + 840 \beta_{12} + 760 \beta_{13} - 851 \beta_{14} + 20 \beta_{15} ) q^{90} \) \( + ( 1125046 + 2875290 \beta_{1} - 33848 \beta_{2} - 129876 \beta_{3} + 45372 \beta_{4} + 8192 \beta_{5} - 54448 \beta_{6} - 7600 \beta_{7} - 432 \beta_{8} + 2240 \beta_{9} + 1190 \beta_{10} - 2832 \beta_{11} - 4558 \beta_{12} - 7474 \beta_{13} + 4422 \beta_{14} + 2240 \beta_{15} ) q^{91} \) \( + ( -673008 + 498078 \beta_{1} - 6578 \beta_{2} - 80508 \beta_{3} + 98552 \beta_{4} + 8483 \beta_{5} - 28947 \beta_{6} - 6443 \beta_{7} + 2357 \beta_{8} + 4299 \beta_{9} + 1920 \beta_{10} + 5462 \beta_{11} + 183 \beta_{12} + 4469 \beta_{13} - 8201 \beta_{14} + 554 \beta_{15} ) q^{92} \) \( + ( 6702206 + 247964 \beta_{1} - 9290 \beta_{2} - 5880 \beta_{3} - 52698 \beta_{4} + 12232 \beta_{5} + 3370 \beta_{6} - 4940 \beta_{7} + 7264 \beta_{8} + 3500 \beta_{9} - 2744 \beta_{10} + 5884 \beta_{11} + 60 \beta_{14} + 2324 \beta_{15} ) q^{93} \) \( + ( 3684961 + 42385 \beta_{1} + 28020 \beta_{2} - 501779 \beta_{3} - 43264 \beta_{4} - 1134 \beta_{5} - 9933 \beta_{6} - 3514 \beta_{7} + 1384 \beta_{8} - 692 \beta_{9} - 9907 \beta_{10} - 4536 \beta_{11} - 3220 \beta_{12} - 6476 \beta_{13} + 1808 \beta_{14} - 5160 \beta_{15} ) q^{94} \) \( + ( 364545 + 952736 \beta_{1} + 5128 \beta_{2} - 16949 \beta_{3} + 6794 \beta_{4} + 15680 \beta_{5} - 22020 \beta_{6} - 1100 \beta_{7} + 3060 \beta_{8} - 880 \beta_{9} - 2919 \beta_{10} - 3860 \beta_{11} + 1635 \beta_{12} - 1695 \beta_{13} + 81 \beta_{14} - 880 \beta_{15} ) q^{95} \) \( + ( -28124328 + 194984 \beta_{1} + 12200 \beta_{2} - 451504 \beta_{3} + 80448 \beta_{4} - 19000 \beta_{5} + 58016 \beta_{6} - 3200 \beta_{7} + 136 \beta_{8} + 80 \beta_{9} + 12864 \beta_{10} - 3888 \beta_{11} + 672 \beta_{12} - 2464 \beta_{13} + 6096 \beta_{14} - 592 \beta_{15} ) q^{96} \) \( + ( 11233579 + 1191886 \beta_{1} + 89764 \beta_{2} - 78328 \beta_{3} + 51266 \beta_{4} - 7546 \beta_{5} + 24846 \beta_{6} - 3556 \beta_{7} + 452 \beta_{8} + 3224 \beta_{9} - 13032 \beta_{10} + 6013 \beta_{11} + 5893 \beta_{14} - 3104 \beta_{15} ) q^{97} \) \( + ( -17267284 + 1964755 \beta_{1} - 86664 \beta_{2} + 423464 \beta_{3} - 56276 \beta_{4} - 4944 \beta_{5} + 16556 \beta_{6} - 7672 \beta_{7} + 3708 \beta_{8} + 2236 \beta_{9} - 10946 \beta_{10} - 4100 \beta_{11} - 5576 \beta_{12} - 9016 \beta_{13} - 3338 \beta_{14} + 2308 \beta_{15} ) q^{98} \) \( + ( -445413 - 1506394 \beta_{1} + 118378 \beta_{2} - 242704 \beta_{3} - 13614 \beta_{4} - 9482 \beta_{5} + 10681 \beta_{6} + 4973 \beta_{7} + 14129 \beta_{8} - 1088 \beta_{9} + 7530 \beta_{10} - 2481 \beta_{11} - 20865 \beta_{12} - 4271 \beta_{13} + 1626 \beta_{14} - 1088 \beta_{15} ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(16q \) \(\mathstrut +\mathstrut 6q^{2} \) \(\mathstrut -\mathstrut 52q^{4} \) \(\mathstrut +\mathstrut 4368q^{6} \) \(\mathstrut -\mathstrut 14184q^{8} \) \(\mathstrut -\mathstrut 38800q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(16q \) \(\mathstrut +\mathstrut 6q^{2} \) \(\mathstrut -\mathstrut 52q^{4} \) \(\mathstrut +\mathstrut 4368q^{6} \) \(\mathstrut -\mathstrut 14184q^{8} \) \(\mathstrut -\mathstrut 38800q^{9} \) \(\mathstrut +\mathstrut 8750q^{10} \) \(\mathstrut -\mathstrut 64040q^{12} \) \(\mathstrut +\mathstrut 51392q^{13} \) \(\mathstrut +\mathstrut 68472q^{14} \) \(\mathstrut -\mathstrut 81424q^{16} \) \(\mathstrut +\mathstrut 27552q^{17} \) \(\mathstrut -\mathstrut 616994q^{18} \) \(\mathstrut +\mathstrut 172500q^{20} \) \(\mathstrut +\mathstrut 414496q^{21} \) \(\mathstrut -\mathstrut 389120q^{22} \) \(\mathstrut +\mathstrut 163792q^{24} \) \(\mathstrut +\mathstrut 1250000q^{25} \) \(\mathstrut +\mathstrut 1037124q^{26} \) \(\mathstrut +\mathstrut 1288520q^{28} \) \(\mathstrut +\mathstrut 2764896q^{29} \) \(\mathstrut -\mathstrut 805000q^{30} \) \(\mathstrut -\mathstrut 4379904q^{32} \) \(\mathstrut -\mathstrut 5521600q^{33} \) \(\mathstrut +\mathstrut 3793964q^{34} \) \(\mathstrut -\mathstrut 5468916q^{36} \) \(\mathstrut +\mathstrut 9009472q^{37} \) \(\mathstrut -\mathstrut 3087360q^{38} \) \(\mathstrut +\mathstrut 385000q^{40} \) \(\mathstrut -\mathstrut 8576448q^{41} \) \(\mathstrut -\mathstrut 4067400q^{42} \) \(\mathstrut +\mathstrut 16921200q^{44} \) \(\mathstrut +\mathstrut 1580000q^{45} \) \(\mathstrut -\mathstrut 7974152q^{46} \) \(\mathstrut -\mathstrut 2696640q^{48} \) \(\mathstrut -\mathstrut 32803600q^{49} \) \(\mathstrut +\mathstrut 468750q^{50} \) \(\mathstrut +\mathstrut 6679352q^{52} \) \(\mathstrut +\mathstrut 2452032q^{53} \) \(\mathstrut +\mathstrut 8898704q^{54} \) \(\mathstrut +\mathstrut 34134768q^{56} \) \(\mathstrut +\mathstrut 11957760q^{57} \) \(\mathstrut +\mathstrut 52156572q^{58} \) \(\mathstrut -\mathstrut 24185000q^{60} \) \(\mathstrut +\mathstrut 8371712q^{61} \) \(\mathstrut +\mathstrut 1290000q^{62} \) \(\mathstrut -\mathstrut 47543872q^{64} \) \(\mathstrut +\mathstrut 9060000q^{65} \) \(\mathstrut +\mathstrut 19358000q^{66} \) \(\mathstrut +\mathstrut 16095192q^{68} \) \(\mathstrut +\mathstrut 7527264q^{69} \) \(\mathstrut +\mathstrut 10500000q^{70} \) \(\mathstrut -\mathstrut 42242664q^{72} \) \(\mathstrut +\mathstrut 61907232q^{73} \) \(\mathstrut -\mathstrut 138210876q^{74} \) \(\mathstrut +\mathstrut 2570400q^{76} \) \(\mathstrut -\mathstrut 156997440q^{77} \) \(\mathstrut -\mathstrut 104032400q^{78} \) \(\mathstrut +\mathstrut 37590000q^{80} \) \(\mathstrut +\mathstrut 140586672q^{81} \) \(\mathstrut +\mathstrut 83921012q^{82} \) \(\mathstrut +\mathstrut 69761824q^{84} \) \(\mathstrut -\mathstrut 106960000q^{85} \) \(\mathstrut -\mathstrut 101724672q^{86} \) \(\mathstrut +\mathstrut 44728480q^{88} \) \(\mathstrut +\mathstrut 106647456q^{89} \) \(\mathstrut +\mathstrut 32613750q^{90} \) \(\mathstrut -\mathstrut 13876200q^{92} \) \(\mathstrut +\mathstrut 105563840q^{93} \) \(\mathstrut +\mathstrut 55264632q^{94} \) \(\mathstrut -\mathstrut 453389952q^{96} \) \(\mathstrut +\mathstrut 171851232q^{97} \) \(\mathstrut -\mathstrut 285387714q^{98} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{16}\mathstrut -\mathstrut \) \(5\) \(x^{15}\mathstrut +\mathstrut \) \(26\) \(x^{14}\mathstrut -\mathstrut \) \(834\) \(x^{13}\mathstrut +\mathstrut \) \(4390\) \(x^{12}\mathstrut -\mathstrut \) \(61783\) \(x^{11}\mathstrut +\mathstrut \) \(466168\) \(x^{10}\mathstrut -\mathstrut \) \(1105435\) \(x^{9}\mathstrut +\mathstrut \) \(46850799\) \(x^{8}\mathstrut -\mathstrut \) \(116275535\) \(x^{7}\mathstrut +\mathstrut \) \(626274432\) \(x^{6}\mathstrut -\mathstrut \) \(8558999923\) \(x^{5}\mathstrut +\mathstrut \) \(34408048994\) \(x^{4}\mathstrut -\mathstrut \) \(448299413930\) \(x^{3}\mathstrut +\mathstrut \) \(1712647133330\) \(x^{2}\mathstrut +\mathstrut \) \(15986651928135\) \(x\mathstrut +\mathstrut \) \(206161212459445\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\(-\)\(71\!\cdots\!16\) \(\nu^{15}\mathstrut -\mathstrut \) \(16\!\cdots\!44\) \(\nu^{14}\mathstrut +\mathstrut \) \(46\!\cdots\!50\) \(\nu^{13}\mathstrut +\mathstrut \) \(49\!\cdots\!44\) \(\nu^{12}\mathstrut +\mathstrut \) \(28\!\cdots\!42\) \(\nu^{11}\mathstrut +\mathstrut \) \(19\!\cdots\!52\) \(\nu^{10}\mathstrut -\mathstrut \) \(26\!\cdots\!96\) \(\nu^{9}\mathstrut -\mathstrut \) \(13\!\cdots\!50\) \(\nu^{8}\mathstrut -\mathstrut \) \(25\!\cdots\!14\) \(\nu^{7}\mathstrut +\mathstrut \) \(13\!\cdots\!08\) \(\nu^{6}\mathstrut +\mathstrut \) \(22\!\cdots\!64\) \(\nu^{5}\mathstrut +\mathstrut \) \(69\!\cdots\!80\) \(\nu^{4}\mathstrut -\mathstrut \) \(37\!\cdots\!50\) \(\nu^{3}\mathstrut +\mathstrut \) \(83\!\cdots\!60\) \(\nu^{2}\mathstrut +\mathstrut \) \(28\!\cdots\!00\) \(\nu\mathstrut -\mathstrut \) \(32\!\cdots\!90\)\()/\)\(13\!\cdots\!65\)
\(\beta_{2}\)\(=\)\((\)\(-\)\(19\!\cdots\!64\) \(\nu^{15}\mathstrut +\mathstrut \) \(26\!\cdots\!52\) \(\nu^{14}\mathstrut -\mathstrut \) \(48\!\cdots\!00\) \(\nu^{13}\mathstrut +\mathstrut \) \(11\!\cdots\!40\) \(\nu^{12}\mathstrut -\mathstrut \) \(97\!\cdots\!88\) \(\nu^{11}\mathstrut +\mathstrut \) \(23\!\cdots\!64\) \(\nu^{10}\mathstrut -\mathstrut \) \(80\!\cdots\!48\) \(\nu^{9}\mathstrut +\mathstrut \) \(33\!\cdots\!80\) \(\nu^{8}\mathstrut -\mathstrut \) \(22\!\cdots\!80\) \(\nu^{7}\mathstrut +\mathstrut \) \(26\!\cdots\!88\) \(\nu^{6}\mathstrut +\mathstrut \) \(28\!\cdots\!20\) \(\nu^{5}\mathstrut +\mathstrut \) \(68\!\cdots\!56\) \(\nu^{4}\mathstrut -\mathstrut \) \(93\!\cdots\!80\) \(\nu^{3}\mathstrut +\mathstrut \) \(62\!\cdots\!40\) \(\nu^{2}\mathstrut -\mathstrut \) \(84\!\cdots\!60\) \(\nu\mathstrut +\mathstrut \) \(74\!\cdots\!15\)\()/\)\(13\!\cdots\!65\)
\(\beta_{3}\)\(=\)\((\)\(48\!\cdots\!09\) \(\nu^{15}\mathstrut -\mathstrut \) \(43\!\cdots\!83\) \(\nu^{14}\mathstrut +\mathstrut \) \(29\!\cdots\!19\) \(\nu^{13}\mathstrut -\mathstrut \) \(44\!\cdots\!36\) \(\nu^{12}\mathstrut +\mathstrut \) \(19\!\cdots\!03\) \(\nu^{11}\mathstrut -\mathstrut \) \(28\!\cdots\!50\) \(\nu^{10}\mathstrut +\mathstrut \) \(25\!\cdots\!82\) \(\nu^{9}\mathstrut +\mathstrut \) \(55\!\cdots\!81\) \(\nu^{8}\mathstrut +\mathstrut \) \(91\!\cdots\!91\) \(\nu^{7}\mathstrut +\mathstrut \) \(41\!\cdots\!32\) \(\nu^{6}\mathstrut -\mathstrut \) \(65\!\cdots\!24\) \(\nu^{5}\mathstrut +\mathstrut \) \(13\!\cdots\!61\) \(\nu^{4}\mathstrut -\mathstrut \) \(51\!\cdots\!80\) \(\nu^{3}\mathstrut +\mathstrut \) \(16\!\cdots\!25\) \(\nu^{2}\mathstrut +\mathstrut \) \(44\!\cdots\!05\) \(\nu\mathstrut +\mathstrut \) \(18\!\cdots\!35\)\()/\)\(21\!\cdots\!80\)
\(\beta_{4}\)\(=\)\((\)\(87\!\cdots\!68\) \(\nu^{15}\mathstrut +\mathstrut \) \(20\!\cdots\!12\) \(\nu^{14}\mathstrut -\mathstrut \) \(56\!\cdots\!50\) \(\nu^{13}\mathstrut -\mathstrut \) \(60\!\cdots\!12\) \(\nu^{12}\mathstrut -\mathstrut \) \(34\!\cdots\!66\) \(\nu^{11}\mathstrut -\mathstrut \) \(24\!\cdots\!96\) \(\nu^{10}\mathstrut +\mathstrut \) \(32\!\cdots\!08\) \(\nu^{9}\mathstrut +\mathstrut \) \(16\!\cdots\!50\) \(\nu^{8}\mathstrut +\mathstrut \) \(31\!\cdots\!22\) \(\nu^{7}\mathstrut -\mathstrut \) \(16\!\cdots\!84\) \(\nu^{6}\mathstrut -\mathstrut \) \(27\!\cdots\!72\) \(\nu^{5}\mathstrut -\mathstrut \) \(84\!\cdots\!40\) \(\nu^{4}\mathstrut +\mathstrut \) \(46\!\cdots\!50\) \(\nu^{3}\mathstrut -\mathstrut \) \(10\!\cdots\!80\) \(\nu^{2}\mathstrut -\mathstrut \) \(23\!\cdots\!50\) \(\nu\mathstrut +\mathstrut \) \(22\!\cdots\!10\)\()/\)\(13\!\cdots\!65\)
\(\beta_{5}\)\(=\)\((\)\(43\!\cdots\!91\) \(\nu^{15}\mathstrut -\mathstrut \) \(20\!\cdots\!25\) \(\nu^{14}\mathstrut -\mathstrut \) \(15\!\cdots\!27\) \(\nu^{13}\mathstrut +\mathstrut \) \(96\!\cdots\!68\) \(\nu^{12}\mathstrut -\mathstrut \) \(79\!\cdots\!83\) \(\nu^{11}\mathstrut +\mathstrut \) \(36\!\cdots\!22\) \(\nu^{10}\mathstrut -\mathstrut \) \(27\!\cdots\!30\) \(\nu^{9}\mathstrut +\mathstrut \) \(31\!\cdots\!47\) \(\nu^{8}\mathstrut -\mathstrut \) \(18\!\cdots\!23\) \(\nu^{7}\mathstrut +\mathstrut \) \(18\!\cdots\!88\) \(\nu^{6}\mathstrut -\mathstrut \) \(11\!\cdots\!68\) \(\nu^{5}\mathstrut +\mathstrut \) \(14\!\cdots\!55\) \(\nu^{4}\mathstrut -\mathstrut \) \(61\!\cdots\!80\) \(\nu^{3}\mathstrut +\mathstrut \) \(11\!\cdots\!15\) \(\nu^{2}\mathstrut -\mathstrut \) \(25\!\cdots\!05\) \(\nu\mathstrut +\mathstrut \) \(14\!\cdots\!85\)\()/\)\(21\!\cdots\!80\)
\(\beta_{6}\)\(=\)\((\)\(-\)\(25\!\cdots\!87\) \(\nu^{15}\mathstrut +\mathstrut \) \(10\!\cdots\!89\) \(\nu^{14}\mathstrut -\mathstrut \) \(74\!\cdots\!09\) \(\nu^{13}\mathstrut +\mathstrut \) \(21\!\cdots\!52\) \(\nu^{12}\mathstrut -\mathstrut \) \(12\!\cdots\!93\) \(\nu^{11}\mathstrut +\mathstrut \) \(15\!\cdots\!42\) \(\nu^{10}\mathstrut -\mathstrut \) \(10\!\cdots\!14\) \(\nu^{9}\mathstrut +\mathstrut \) \(36\!\cdots\!09\) \(\nu^{8}\mathstrut -\mathstrut \) \(13\!\cdots\!97\) \(\nu^{7}\mathstrut +\mathstrut \) \(31\!\cdots\!68\) \(\nu^{6}\mathstrut -\mathstrut \) \(22\!\cdots\!80\) \(\nu^{5}\mathstrut +\mathstrut \) \(25\!\cdots\!85\) \(\nu^{4}\mathstrut -\mathstrut \) \(14\!\cdots\!20\) \(\nu^{3}\mathstrut +\mathstrut \) \(10\!\cdots\!25\) \(\nu^{2}\mathstrut -\mathstrut \) \(44\!\cdots\!75\) \(\nu\mathstrut -\mathstrut \) \(35\!\cdots\!85\)\()/\)\(10\!\cdots\!40\)
\(\beta_{7}\)\(=\)\((\)\(-\)\(53\!\cdots\!87\) \(\nu^{15}\mathstrut +\mathstrut \) \(84\!\cdots\!57\) \(\nu^{14}\mathstrut -\mathstrut \) \(16\!\cdots\!65\) \(\nu^{13}\mathstrut +\mathstrut \) \(11\!\cdots\!48\) \(\nu^{12}\mathstrut -\mathstrut \) \(39\!\cdots\!41\) \(\nu^{11}\mathstrut +\mathstrut \) \(12\!\cdots\!54\) \(\nu^{10}\mathstrut -\mathstrut \) \(78\!\cdots\!82\) \(\nu^{9}\mathstrut +\mathstrut \) \(91\!\cdots\!45\) \(\nu^{8}\mathstrut -\mathstrut \) \(70\!\cdots\!93\) \(\nu^{7}\mathstrut +\mathstrut \) \(82\!\cdots\!16\) \(\nu^{6}\mathstrut -\mathstrut \) \(51\!\cdots\!92\) \(\nu^{5}\mathstrut +\mathstrut \) \(39\!\cdots\!05\) \(\nu^{4}\mathstrut -\mathstrut \) \(12\!\cdots\!80\) \(\nu^{3}\mathstrut +\mathstrut \) \(19\!\cdots\!25\) \(\nu^{2}\mathstrut -\mathstrut \) \(30\!\cdots\!75\) \(\nu\mathstrut +\mathstrut \) \(37\!\cdots\!35\)\()/\)\(21\!\cdots\!80\)
\(\beta_{8}\)\(=\)\((\)\(-\)\(11\!\cdots\!73\) \(\nu^{15}\mathstrut +\mathstrut \) \(13\!\cdots\!11\) \(\nu^{14}\mathstrut +\mathstrut \) \(41\!\cdots\!21\) \(\nu^{13}\mathstrut -\mathstrut \) \(12\!\cdots\!76\) \(\nu^{12}\mathstrut +\mathstrut \) \(56\!\cdots\!37\) \(\nu^{11}\mathstrut -\mathstrut \) \(13\!\cdots\!94\) \(\nu^{10}\mathstrut +\mathstrut \) \(27\!\cdots\!62\) \(\nu^{9}\mathstrut -\mathstrut \) \(61\!\cdots\!81\) \(\nu^{8}\mathstrut +\mathstrut \) \(18\!\cdots\!01\) \(\nu^{7}\mathstrut -\mathstrut \) \(24\!\cdots\!92\) \(\nu^{6}\mathstrut +\mathstrut \) \(14\!\cdots\!92\) \(\nu^{5}\mathstrut -\mathstrut \) \(12\!\cdots\!73\) \(\nu^{4}\mathstrut -\mathstrut \) \(11\!\cdots\!60\) \(\nu^{3}\mathstrut +\mathstrut \) \(71\!\cdots\!75\) \(\nu^{2}\mathstrut -\mathstrut \) \(26\!\cdots\!65\) \(\nu\mathstrut -\mathstrut \) \(97\!\cdots\!15\)\()/\)\(21\!\cdots\!80\)
\(\beta_{9}\)\(=\)\((\)\(-\)\(13\!\cdots\!67\) \(\nu^{15}\mathstrut +\mathstrut \) \(32\!\cdots\!53\) \(\nu^{14}\mathstrut -\mathstrut \) \(64\!\cdots\!49\) \(\nu^{13}\mathstrut +\mathstrut \) \(72\!\cdots\!04\) \(\nu^{12}\mathstrut -\mathstrut \) \(40\!\cdots\!41\) \(\nu^{11}\mathstrut +\mathstrut \) \(37\!\cdots\!10\) \(\nu^{10}\mathstrut -\mathstrut \) \(51\!\cdots\!46\) \(\nu^{9}\mathstrut +\mathstrut \) \(34\!\cdots\!29\) \(\nu^{8}\mathstrut -\mathstrut \) \(23\!\cdots\!69\) \(\nu^{7}\mathstrut +\mathstrut \) \(19\!\cdots\!36\) \(\nu^{6}\mathstrut -\mathstrut \) \(12\!\cdots\!28\) \(\nu^{5}\mathstrut +\mathstrut \) \(81\!\cdots\!53\) \(\nu^{4}\mathstrut -\mathstrut \) \(17\!\cdots\!00\) \(\nu^{3}\mathstrut -\mathstrut \) \(10\!\cdots\!35\) \(\nu^{2}\mathstrut -\mathstrut \) \(28\!\cdots\!75\) \(\nu\mathstrut +\mathstrut \) \(14\!\cdots\!15\)\()/\)\(21\!\cdots\!80\)
\(\beta_{10}\)\(=\)\((\)\(-\)\(14\!\cdots\!15\) \(\nu^{15}\mathstrut +\mathstrut \) \(35\!\cdots\!25\) \(\nu^{14}\mathstrut -\mathstrut \) \(36\!\cdots\!21\) \(\nu^{13}\mathstrut +\mathstrut \) \(32\!\cdots\!32\) \(\nu^{12}\mathstrut -\mathstrut \) \(38\!\cdots\!17\) \(\nu^{11}\mathstrut +\mathstrut \) \(35\!\cdots\!26\) \(\nu^{10}\mathstrut -\mathstrut \) \(30\!\cdots\!94\) \(\nu^{9}\mathstrut +\mathstrut \) \(25\!\cdots\!01\) \(\nu^{8}\mathstrut -\mathstrut \) \(17\!\cdots\!57\) \(\nu^{7}\mathstrut +\mathstrut \) \(14\!\cdots\!52\) \(\nu^{6}\mathstrut -\mathstrut \) \(83\!\cdots\!16\) \(\nu^{5}\mathstrut +\mathstrut \) \(31\!\cdots\!89\) \(\nu^{4}\mathstrut -\mathstrut \) \(20\!\cdots\!60\) \(\nu^{3}\mathstrut +\mathstrut \) \(11\!\cdots\!85\) \(\nu^{2}\mathstrut -\mathstrut \) \(75\!\cdots\!75\) \(\nu\mathstrut +\mathstrut \) \(25\!\cdots\!75\)\()/\)\(21\!\cdots\!80\)
\(\beta_{11}\)\(=\)\((\)\(-\)\(17\!\cdots\!97\) \(\nu^{15}\mathstrut -\mathstrut \) \(16\!\cdots\!33\) \(\nu^{14}\mathstrut +\mathstrut \) \(39\!\cdots\!09\) \(\nu^{13}\mathstrut -\mathstrut \) \(19\!\cdots\!20\) \(\nu^{12}\mathstrut +\mathstrut \) \(23\!\cdots\!37\) \(\nu^{11}\mathstrut -\mathstrut \) \(30\!\cdots\!70\) \(\nu^{10}\mathstrut +\mathstrut \) \(24\!\cdots\!14\) \(\nu^{9}\mathstrut -\mathstrut \) \(22\!\cdots\!89\) \(\nu^{8}\mathstrut +\mathstrut \) \(14\!\cdots\!85\) \(\nu^{7}\mathstrut -\mathstrut \) \(13\!\cdots\!72\) \(\nu^{6}\mathstrut +\mathstrut \) \(12\!\cdots\!52\) \(\nu^{5}\mathstrut -\mathstrut \) \(51\!\cdots\!81\) \(\nu^{4}\mathstrut +\mathstrut \) \(58\!\cdots\!00\) \(\nu^{3}\mathstrut -\mathstrut \) \(21\!\cdots\!25\) \(\nu^{2}\mathstrut +\mathstrut \) \(10\!\cdots\!95\) \(\nu\mathstrut -\mathstrut \) \(77\!\cdots\!95\)\()/\)\(21\!\cdots\!80\)
\(\beta_{12}\)\(=\)\((\)\(-\)\(30\!\cdots\!06\) \(\nu^{15}\mathstrut +\mathstrut \) \(22\!\cdots\!77\) \(\nu^{14}\mathstrut -\mathstrut \) \(48\!\cdots\!66\) \(\nu^{13}\mathstrut +\mathstrut \) \(10\!\cdots\!09\) \(\nu^{12}\mathstrut +\mathstrut \) \(47\!\cdots\!93\) \(\nu^{11}\mathstrut +\mathstrut \) \(80\!\cdots\!70\) \(\nu^{10}\mathstrut -\mathstrut \) \(73\!\cdots\!88\) \(\nu^{9}\mathstrut -\mathstrut \) \(11\!\cdots\!04\) \(\nu^{8}\mathstrut -\mathstrut \) \(41\!\cdots\!09\) \(\nu^{7}\mathstrut -\mathstrut \) \(10\!\cdots\!68\) \(\nu^{6}\mathstrut +\mathstrut \) \(50\!\cdots\!56\) \(\nu^{5}\mathstrut -\mathstrut \) \(10\!\cdots\!54\) \(\nu^{4}\mathstrut +\mathstrut \) \(32\!\cdots\!15\) \(\nu^{3}\mathstrut +\mathstrut \) \(72\!\cdots\!55\) \(\nu^{2}\mathstrut -\mathstrut \) \(91\!\cdots\!60\) \(\nu\mathstrut -\mathstrut \) \(14\!\cdots\!85\)\()/\)\(36\!\cdots\!80\)
\(\beta_{13}\)\(=\)\((\)\(31\!\cdots\!53\) \(\nu^{15}\mathstrut -\mathstrut \) \(20\!\cdots\!81\) \(\nu^{14}\mathstrut -\mathstrut \) \(61\!\cdots\!37\) \(\nu^{13}\mathstrut -\mathstrut \) \(80\!\cdots\!02\) \(\nu^{12}\mathstrut +\mathstrut \) \(76\!\cdots\!21\) \(\nu^{11}\mathstrut -\mathstrut \) \(44\!\cdots\!30\) \(\nu^{10}\mathstrut -\mathstrut \) \(11\!\cdots\!86\) \(\nu^{9}\mathstrut +\mathstrut \) \(81\!\cdots\!37\) \(\nu^{8}\mathstrut +\mathstrut \) \(16\!\cdots\!37\) \(\nu^{7}\mathstrut +\mathstrut \) \(26\!\cdots\!24\) \(\nu^{6}\mathstrut -\mathstrut \) \(63\!\cdots\!28\) \(\nu^{5}\mathstrut +\mathstrut \) \(52\!\cdots\!37\) \(\nu^{4}\mathstrut -\mathstrut \) \(15\!\cdots\!10\) \(\nu^{3}\mathstrut +\mathstrut \) \(52\!\cdots\!75\) \(\nu^{2}\mathstrut -\mathstrut \) \(75\!\cdots\!15\) \(\nu\mathstrut +\mathstrut \) \(10\!\cdots\!65\)\()/\)\(21\!\cdots\!80\)
\(\beta_{14}\)\(=\)\((\)\(34\!\cdots\!43\) \(\nu^{15}\mathstrut -\mathstrut \) \(61\!\cdots\!37\) \(\nu^{14}\mathstrut +\mathstrut \) \(42\!\cdots\!97\) \(\nu^{13}\mathstrut -\mathstrut \) \(45\!\cdots\!08\) \(\nu^{12}\mathstrut +\mathstrut \) \(48\!\cdots\!41\) \(\nu^{11}\mathstrut -\mathstrut \) \(38\!\cdots\!66\) \(\nu^{10}\mathstrut +\mathstrut \) \(39\!\cdots\!78\) \(\nu^{9}\mathstrut -\mathstrut \) \(28\!\cdots\!57\) \(\nu^{8}\mathstrut +\mathstrut \) \(20\!\cdots\!53\) \(\nu^{7}\mathstrut -\mathstrut \) \(16\!\cdots\!36\) \(\nu^{6}\mathstrut +\mathstrut \) \(78\!\cdots\!48\) \(\nu^{5}\mathstrut -\mathstrut \) \(37\!\cdots\!01\) \(\nu^{4}\mathstrut +\mathstrut \) \(26\!\cdots\!20\) \(\nu^{3}\mathstrut -\mathstrut \) \(16\!\cdots\!05\) \(\nu^{2}\mathstrut +\mathstrut \) \(10\!\cdots\!55\) \(\nu\mathstrut -\mathstrut \) \(78\!\cdots\!75\)\()/\)\(21\!\cdots\!80\)
\(\beta_{15}\)\(=\)\((\)\(47\!\cdots\!67\) \(\nu^{15}\mathstrut -\mathstrut \) \(59\!\cdots\!25\) \(\nu^{14}\mathstrut +\mathstrut \) \(43\!\cdots\!29\) \(\nu^{13}\mathstrut -\mathstrut \) \(50\!\cdots\!80\) \(\nu^{12}\mathstrut +\mathstrut \) \(46\!\cdots\!05\) \(\nu^{11}\mathstrut -\mathstrut \) \(42\!\cdots\!94\) \(\nu^{10}\mathstrut +\mathstrut \) \(50\!\cdots\!82\) \(\nu^{9}\mathstrut -\mathstrut \) \(26\!\cdots\!09\) \(\nu^{8}\mathstrut +\mathstrut \) \(19\!\cdots\!25\) \(\nu^{7}\mathstrut -\mathstrut \) \(21\!\cdots\!40\) \(\nu^{6}\mathstrut +\mathstrut \) \(11\!\cdots\!92\) \(\nu^{5}\mathstrut -\mathstrut \) \(55\!\cdots\!17\) \(\nu^{4}\mathstrut +\mathstrut \) \(35\!\cdots\!00\) \(\nu^{3}\mathstrut -\mathstrut \) \(73\!\cdots\!05\) \(\nu^{2}\mathstrut +\mathstrut \) \(36\!\cdots\!55\) \(\nu\mathstrut -\mathstrut \) \(12\!\cdots\!95\)\()/\)\(21\!\cdots\!80\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{4}\mathstrut +\mathstrut \) \(123\) \(\beta_{1}\mathstrut +\mathstrut \) \(124\)\()/250\)
\(\nu^{2}\)\(=\)\((\)\(2\) \(\beta_{14}\mathstrut +\mathstrut \) \(2\) \(\beta_{10}\mathstrut +\mathstrut \) \(2\) \(\beta_{4}\mathstrut -\mathstrut \) \(8\) \(\beta_{3}\mathstrut +\mathstrut \) \(121\) \(\beta_{2}\mathstrut +\mathstrut \) \(244\) \(\beta_{1}\mathstrut -\mathstrut \) \(719\)\()/500\)
\(\nu^{3}\)\(=\)\((\)\(-\)\(15\) \(\beta_{15}\mathstrut +\mathstrut \) \(12\) \(\beta_{14}\mathstrut +\mathstrut \) \(15\) \(\beta_{13}\mathstrut +\mathstrut \) \(45\) \(\beta_{12}\mathstrut +\mathstrut \) \(15\) \(\beta_{11}\mathstrut -\mathstrut \) \(3\) \(\beta_{10}\mathstrut -\mathstrut \) \(125\) \(\beta_{8}\mathstrut +\mathstrut \) \(15\) \(\beta_{7}\mathstrut +\mathstrut \) \(30\) \(\beta_{6}\mathstrut -\mathstrut \) \(140\) \(\beta_{5}\mathstrut -\mathstrut \) \(25\) \(\beta_{4}\mathstrut +\mathstrut \) \(832\) \(\beta_{3}\mathstrut +\mathstrut \) \(196\) \(\beta_{2}\mathstrut +\mathstrut \) \(218\) \(\beta_{1}\mathstrut +\mathstrut \) \(139713\)\()/1000\)
\(\nu^{4}\)\(=\)\((\)\(-\)\(125\) \(\beta_{15}\mathstrut -\mathstrut \) \(137\) \(\beta_{14}\mathstrut +\mathstrut \) \(1510\) \(\beta_{13}\mathstrut +\mathstrut \) \(410\) \(\beta_{12}\mathstrut +\mathstrut \) \(765\) \(\beta_{11}\mathstrut +\mathstrut \) \(203\) \(\beta_{10}\mathstrut -\mathstrut \) \(35\) \(\beta_{9}\mathstrut -\mathstrut \) \(785\) \(\beta_{8}\mathstrut -\mathstrut \) \(650\) \(\beta_{7}\mathstrut +\mathstrut \) \(4225\) \(\beta_{6}\mathstrut -\mathstrut \) \(2050\) \(\beta_{5}\mathstrut +\mathstrut \) \(7419\) \(\beta_{4}\mathstrut -\mathstrut \) \(8382\) \(\beta_{3}\mathstrut -\mathstrut \) \(1111\) \(\beta_{2}\mathstrut +\mathstrut \) \(153399\) \(\beta_{1}\mathstrut -\mathstrut \) \(130022\)\()/2000\)
\(\nu^{5}\)\(=\)\((\)\(375\) \(\beta_{15}\mathstrut +\mathstrut \) \(623\) \(\beta_{14}\mathstrut +\mathstrut \) \(700\) \(\beta_{13}\mathstrut +\mathstrut \) \(940\) \(\beta_{12}\mathstrut +\mathstrut \) \(1065\) \(\beta_{11}\mathstrut +\mathstrut \) \(2033\) \(\beta_{10}\mathstrut +\mathstrut \) \(175\) \(\beta_{9}\mathstrut +\mathstrut \) \(340\) \(\beta_{8}\mathstrut -\mathstrut \) \(580\) \(\beta_{7}\mathstrut -\mathstrut \) \(105\) \(\beta_{6}\mathstrut +\mathstrut \) \(135\) \(\beta_{5}\mathstrut +\mathstrut \) \(4671\) \(\beta_{4}\mathstrut +\mathstrut \) \(3628\) \(\beta_{3}\mathstrut +\mathstrut \) \(6984\) \(\beta_{2}\mathstrut +\mathstrut \) \(5280\) \(\beta_{1}\mathstrut +\mathstrut \) \(2485451\)\()/200\)
\(\nu^{6}\)\(=\)\((\)\(4755\) \(\beta_{15}\mathstrut +\mathstrut \) \(20720\) \(\beta_{14}\mathstrut +\mathstrut \) \(14515\) \(\beta_{13}\mathstrut +\mathstrut \) \(49705\) \(\beta_{12}\mathstrut +\mathstrut \) \(19165\) \(\beta_{11}\mathstrut +\mathstrut \) \(51255\) \(\beta_{10}\mathstrut +\mathstrut \) \(16350\) \(\beta_{9}\mathstrut +\mathstrut \) \(7520\) \(\beta_{8}\mathstrut -\mathstrut \) \(29545\) \(\beta_{7}\mathstrut +\mathstrut \) \(10550\) \(\beta_{6}\mathstrut -\mathstrut \) \(102715\) \(\beta_{5}\mathstrut +\mathstrut \) \(103979\) \(\beta_{4}\mathstrut +\mathstrut \) \(2297330\) \(\beta_{3}\mathstrut -\mathstrut \) \(165515\) \(\beta_{2}\mathstrut +\mathstrut \) \(6731047\) \(\beta_{1}\mathstrut +\mathstrut \) \(34662356\)\()/1000\)
\(\nu^{7}\)\(=\)\((\)\(-\)\(13715\) \(\beta_{15}\mathstrut +\mathstrut \) \(197313\) \(\beta_{14}\mathstrut +\mathstrut \) \(797250\) \(\beta_{13}\mathstrut +\mathstrut \) \(195550\) \(\beta_{12}\mathstrut +\mathstrut \) \(510675\) \(\beta_{11}\mathstrut +\mathstrut \) \(746193\) \(\beta_{10}\mathstrut +\mathstrut \) \(156315\) \(\beta_{9}\mathstrut +\mathstrut \) \(77635\) \(\beta_{8}\mathstrut -\mathstrut \) \(324150\) \(\beta_{7}\mathstrut -\mathstrut \) \(1085085\) \(\beta_{6}\mathstrut -\mathstrut \) \(2827760\) \(\beta_{5}\mathstrut -\mathstrut \) \(871527\) \(\beta_{4}\mathstrut +\mathstrut \) \(8256458\) \(\beta_{3}\mathstrut +\mathstrut \) \(6499549\) \(\beta_{2}\mathstrut +\mathstrut \) \(47408331\) \(\beta_{1}\mathstrut -\mathstrut \) \(1215185396\)\()/2000\)
\(\nu^{8}\)\(=\)\((\)\(899210\) \(\beta_{15}\mathstrut -\mathstrut \) \(681918\) \(\beta_{14}\mathstrut +\mathstrut \) \(3012380\) \(\beta_{13}\mathstrut +\mathstrut \) \(3162500\) \(\beta_{12}\mathstrut +\mathstrut \) \(3531030\) \(\beta_{11}\mathstrut +\mathstrut \) \(3259662\) \(\beta_{10}\mathstrut +\mathstrut \) \(401630\) \(\beta_{9}\mathstrut -\mathstrut \) \(2892685\) \(\beta_{8}\mathstrut +\mathstrut \) \(1178020\) \(\beta_{7}\mathstrut -\mathstrut \) \(4854390\) \(\beta_{6}\mathstrut -\mathstrut \) \(7425975\) \(\beta_{5}\mathstrut -\mathstrut \) \(7030590\) \(\beta_{4}\mathstrut +\mathstrut \) \(101759282\) \(\beta_{3}\mathstrut +\mathstrut \) \(8074031\) \(\beta_{2}\mathstrut -\mathstrut \) \(303336367\) \(\beta_{1}\mathstrut -\mathstrut \) \(6968504837\)\()/1000\)
\(\nu^{9}\)\(=\)\((\)\(11919945\) \(\beta_{15}\mathstrut -\mathstrut \) \(11166951\) \(\beta_{14}\mathstrut +\mathstrut \) \(23725000\) \(\beta_{13}\mathstrut +\mathstrut \) \(20836320\) \(\beta_{12}\mathstrut +\mathstrut \) \(13367815\) \(\beta_{11}\mathstrut +\mathstrut \) \(11729589\) \(\beta_{10}\mathstrut +\mathstrut \) \(8344965\) \(\beta_{9}\mathstrut -\mathstrut \) \(5890065\) \(\beta_{8}\mathstrut -\mathstrut \) \(13216860\) \(\beta_{7}\mathstrut +\mathstrut \) \(80779795\) \(\beta_{6}\mathstrut -\mathstrut \) \(121982750\) \(\beta_{5}\mathstrut +\mathstrut \) \(34446227\) \(\beta_{4}\mathstrut +\mathstrut \) \(1251480074\) \(\beta_{3}\mathstrut -\mathstrut \) \(264398853\) \(\beta_{2}\mathstrut -\mathstrut \) \(3955718533\) \(\beta_{1}\mathstrut -\mathstrut \) \(69182012056\)\()/1000\)
\(\nu^{10}\)\(=\)\((\)\(22322485\) \(\beta_{15}\mathstrut -\mathstrut \) \(5897839\) \(\beta_{14}\mathstrut +\mathstrut \) \(52435630\) \(\beta_{13}\mathstrut +\mathstrut \) \(32305090\) \(\beta_{12}\mathstrut +\mathstrut \) \(20782235\) \(\beta_{11}\mathstrut +\mathstrut \) \(62826701\) \(\beta_{10}\mathstrut +\mathstrut \) \(4570175\) \(\beta_{9}\mathstrut +\mathstrut \) \(34275270\) \(\beta_{8}\mathstrut -\mathstrut \) \(21348050\) \(\beta_{7}\mathstrut -\mathstrut \) \(81394685\) \(\beta_{6}\mathstrut -\mathstrut \) \(171497985\) \(\beta_{5}\mathstrut -\mathstrut \) \(406004323\) \(\beta_{4}\mathstrut +\mathstrut \) \(814425216\) \(\beta_{3}\mathstrut -\mathstrut \) \(419773762\) \(\beta_{2}\mathstrut -\mathstrut \) \(7846871800\) \(\beta_{1}\mathstrut -\mathstrut \) \(28755514013\)\()/200\)
\(\nu^{11}\)\(=\)\((\)\(1147601620\) \(\beta_{15}\mathstrut -\mathstrut \) \(1613309315\) \(\beta_{14}\mathstrut +\mathstrut \) \(1334290815\) \(\beta_{13}\mathstrut +\mathstrut \) \(2779943605\) \(\beta_{12}\mathstrut +\mathstrut \) \(597729340\) \(\beta_{11}\mathstrut +\mathstrut \) \(276428840\) \(\beta_{10}\mathstrut +\mathstrut \) \(527844335\) \(\beta_{9}\mathstrut +\mathstrut \) \(1022148260\) \(\beta_{8}\mathstrut +\mathstrut \) \(547829055\) \(\beta_{7}\mathstrut -\mathstrut \) \(6861598315\) \(\beta_{6}\mathstrut -\mathstrut \) \(3958759380\) \(\beta_{5}\mathstrut -\mathstrut \) \(16915108746\) \(\beta_{4}\mathstrut +\mathstrut \) \(91299153710\) \(\beta_{3}\mathstrut -\mathstrut \) \(23341571745\) \(\beta_{2}\mathstrut -\mathstrut \) \(115916206883\) \(\beta_{1}\mathstrut -\mathstrut \) \(6866702025759\)\()/1000\)
\(\nu^{12}\)\(=\)\((\)\(22323988735\) \(\beta_{15}\mathstrut -\mathstrut \) \(29642977867\) \(\beta_{14}\mathstrut +\mathstrut \) \(6550653720\) \(\beta_{13}\mathstrut -\mathstrut \) \(3130372960\) \(\beta_{12}\mathstrut -\mathstrut \) \(8186000255\) \(\beta_{11}\mathstrut -\mathstrut \) \(15676601377\) \(\beta_{10}\mathstrut +\mathstrut \) \(16554126755\) \(\beta_{9}\mathstrut +\mathstrut \) \(24857122255\) \(\beta_{8}\mathstrut -\mathstrut \) \(10201972160\) \(\beta_{7}\mathstrut -\mathstrut \) \(60195525145\) \(\beta_{6}\mathstrut -\mathstrut \) \(104609579490\) \(\beta_{5}\mathstrut -\mathstrut \) \(453720009777\) \(\beta_{4}\mathstrut +\mathstrut \) \(886749358318\) \(\beta_{3}\mathstrut -\mathstrut \) \(116911099441\) \(\beta_{2}\mathstrut -\mathstrut \) \(8275880420599\) \(\beta_{1}\mathstrut -\mathstrut \) \(199006383104476\)\()/2000\)
\(\nu^{13}\)\(=\)\((\)\(136002850625\) \(\beta_{15}\mathstrut -\mathstrut \) \(360807053871\) \(\beta_{14}\mathstrut +\mathstrut \) \(26931144990\) \(\beta_{13}\mathstrut -\mathstrut \) \(66284344590\) \(\beta_{12}\mathstrut -\mathstrut \) \(106032668385\) \(\beta_{11}\mathstrut -\mathstrut \) \(195212713531\) \(\beta_{10}\mathstrut -\mathstrut \) \(12806962485\) \(\beta_{9}\mathstrut +\mathstrut \) \(73949213565\) \(\beta_{8}\mathstrut -\mathstrut \) \(1701046250\) \(\beta_{7}\mathstrut -\mathstrut \) \(1241535626125\) \(\beta_{6}\mathstrut -\mathstrut \) \(485466541850\) \(\beta_{5}\mathstrut -\mathstrut \) \(6690853570395\) \(\beta_{4}\mathstrut +\mathstrut \) \(5429051102294\) \(\beta_{3}\mathstrut -\mathstrut \) \(3742280170593\) \(\beta_{2}\mathstrut -\mathstrut \) \(114801351523539\) \(\beta_{1}\mathstrut -\mathstrut \) \(1171110381859314\)\()/2000\)
\(\nu^{14}\)\(=\)\((\)\(690002435965\) \(\beta_{15}\mathstrut -\mathstrut \) \(2595594352266\) \(\beta_{14}\mathstrut -\mathstrut \) \(195256153965\) \(\beta_{13}\mathstrut -\mathstrut \) \(89071665735\) \(\beta_{12}\mathstrut -\mathstrut \) \(1098403501325\) \(\beta_{11}\mathstrut -\mathstrut \) \(2531419212461\) \(\beta_{10}\mathstrut -\mathstrut \) \(101799078160\) \(\beta_{9}\mathstrut +\mathstrut \) \(840254505090\) \(\beta_{8}\mathstrut +\mathstrut \) \(76419256635\) \(\beta_{7}\mathstrut -\mathstrut \) \(3292387382830\) \(\beta_{6}\mathstrut -\mathstrut \) \(286574578085\) \(\beta_{5}\mathstrut -\mathstrut \) \(14155813810313\) \(\beta_{4}\mathstrut +\mathstrut \) \(12156404375374\) \(\beta_{3}\mathstrut -\mathstrut \) \(28736956586723\) \(\beta_{2}\mathstrut -\mathstrut \) \(379192722091313\) \(\beta_{1}\mathstrut -\mathstrut \) \(7324070154746756\)\()/1000\)
\(\nu^{15}\)\(=\)\((\)\(3179941561745\) \(\beta_{15}\mathstrut -\mathstrut \) \(5554539006727\) \(\beta_{14}\mathstrut -\mathstrut \) \(3785987666310\) \(\beta_{13}\mathstrut -\mathstrut \) \(4667770832730\) \(\beta_{12}\mathstrut -\mathstrut \) \(6463889340145\) \(\beta_{11}\mathstrut -\mathstrut \) \(6287667525587\) \(\beta_{10}\mathstrut -\mathstrut \) \(20523475865\) \(\beta_{9}\mathstrut +\mathstrut \) \(7089821780375\) \(\beta_{8}\mathstrut -\mathstrut \) \(1170312522150\) \(\beta_{7}\mathstrut -\mathstrut \) \(18265038244245\) \(\beta_{6}\mathstrut +\mathstrut \) \(9710697378580\) \(\beta_{5}\mathstrut -\mathstrut \) \(70493748401839\) \(\beta_{4}\mathstrut -\mathstrut \) \(93747914458022\) \(\beta_{3}\mathstrut -\mathstrut \) \(63805762802491\) \(\beta_{2}\mathstrut -\mathstrut \) \(1823470049706365\) \(\beta_{1}\mathstrut -\mathstrut \) \(22285706746335484\)\()/400\)

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
9.29581 + 2.24761i
9.29581 2.24761i
6.86496 + 2.82928i
6.86496 2.82928i
3.05707 + 7.10588i
3.05707 7.10588i
2.32463 + 7.96873i
2.32463 7.96873i
−2.93811 + 7.65619i
−2.93811 7.65619i
−6.29294 + 5.63875i
−6.29294 5.63875i
−4.16577 + 5.52698i
−4.16577 5.52698i
−5.64565 + 3.35245i
−5.64565 3.35245i
−15.3556 4.49522i 98.1237i 215.586 + 138.053i 279.508 441.088 1506.74i 820.952i −2689.86 3088.99i −3067.27 −4292.01 1256.45i
11.2 −15.3556 + 4.49522i 98.1237i 215.586 138.053i 279.508 441.088 + 1506.74i 820.952i −2689.86 + 3088.99i −3067.27 −4292.01 + 1256.45i
11.3 −14.9660 5.65855i 25.1248i 191.962 + 169.372i −279.508 −142.170 + 376.017i 2973.76i −1914.50 3621.04i 5929.74 4183.12 + 1581.61i
11.4 −14.9660 + 5.65855i 25.1248i 191.962 169.372i −279.508 −142.170 376.017i 2973.76i −1914.50 + 3621.04i 5929.74 4183.12 1581.61i
11.5 −7.35022 14.2118i 110.171i −147.949 + 208.919i −279.508 1565.72 809.778i 3540.70i 4056.56 + 567.011i −5576.56 2054.45 + 3972.31i
11.6 −7.35022 + 14.2118i 110.171i −147.949 208.919i −279.508 1565.72 + 809.778i 3540.70i 4056.56 567.011i −5576.56 2054.45 3972.31i
11.7 −1.41320 15.9375i 39.9624i −252.006 + 45.0455i 279.508 636.899 56.4746i 2633.20i 1074.04 + 3952.68i 4964.01 −395.000 4454.66i
11.8 −1.41320 + 15.9375i 39.9624i −252.006 45.0455i 279.508 636.899 + 56.4746i 2633.20i 1074.04 3952.68i 4964.01 −395.000 + 4454.66i
11.9 4.64016 15.3124i 75.7492i −212.938 142.104i −279.508 −1159.90 351.488i 210.345i −3164.01 + 2601.20i 823.060 −1296.96 + 4279.94i
11.10 4.64016 + 15.3124i 75.7492i −212.938 + 142.104i −279.508 −1159.90 + 351.488i 210.345i −3164.01 2601.20i 823.060 −1296.96 4279.94i
11.11 11.3498 11.2775i 137.297i 1.63618 255.995i −279.508 1548.37 + 1558.29i 3940.57i −2868.41 2923.94i −12289.4 −3172.37 + 3152.16i
11.12 11.3498 + 11.2775i 137.297i 1.63618 + 255.995i −279.508 1548.37 1558.29i 3940.57i −2868.41 + 2923.94i −12289.4 −3172.37 3152.16i
11.13 11.5676 11.0540i 27.2434i 11.6196 255.736i 279.508 301.148 + 315.141i 3325.58i −2692.49 3086.70i 5818.80 3233.25 3089.68i
11.14 11.5676 + 11.0540i 27.2434i 11.6196 + 255.736i 279.508 301.148 315.141i 3325.58i −2692.49 + 3086.70i 5818.80 3233.25 + 3089.68i
11.15 14.5274 6.70489i 150.211i 166.089 194.809i 279.508 −1007.15 2182.17i 2626.96i 1106.66 3943.67i −16002.3 4060.52 1874.07i
11.16 14.5274 + 6.70489i 150.211i 166.089 + 194.809i 279.508 −1007.15 + 2182.17i 2626.96i 1106.66 + 3943.67i −16002.3 4060.52 + 1874.07i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 11.16
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 \(S_{9}^{\mathrm{new}}(20, [\chi])\).