# 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

# Related objects

## 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: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

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

## Hecke kernels

There are no other newforms in $$S_{9}^{\mathrm{new}}(20, [\chi])$$.