# Properties

 Label 24.8.f.c Level 24 Weight 8 Character orbit 24.f Analytic conductor 7.497 Analytic rank 0 Dimension 20 CM no Inner twists 4

# Related objects

## Newspace parameters

 Level: $$N$$ = $$24 = 2^{3} \cdot 3$$ Weight: $$k$$ = $$8$$ Character orbit: $$[\chi]$$ = 24.f (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$7.49724061162$$ Analytic rank: $$0$$ Dimension: $$20$$ Coefficient field: $$\mathbb{Q}[x]/(x^{20} - \cdots)$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{52}\cdot 3^{20}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{2} + ( 6 + \beta_{1} - \beta_{4} ) q^{3} + ( 27 + \beta_{3} + \beta_{4} ) q^{4} + ( -\beta_{1} + \beta_{9} ) q^{5} + ( -71 + 6 \beta_{1} + \beta_{2} + \beta_{3} + \beta_{6} ) q^{6} + ( -1 + 2 \beta_{3} + \beta_{4} + \beta_{7} ) q^{7} + ( 28 \beta_{1} + \beta_{2} - \beta_{4} - \beta_{5} - \beta_{6} - \beta_{10} + \beta_{13} ) q^{8} + ( 254 + 6 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} - 6 \beta_{4} + 2 \beta_{5} + \beta_{6} - \beta_{11} + \beta_{17} + \beta_{18} ) q^{9} +O(q^{10})$$ $$q + \beta_{1} q^{2} + ( 6 + \beta_{1} - \beta_{4} ) q^{3} + ( 27 + \beta_{3} + \beta_{4} ) q^{4} + ( -\beta_{1} + \beta_{9} ) q^{5} + ( -71 + 6 \beta_{1} + \beta_{2} + \beta_{3} + \beta_{6} ) q^{6} + ( -1 + 2 \beta_{3} + \beta_{4} + \beta_{7} ) q^{7} + ( 28 \beta_{1} + \beta_{2} - \beta_{4} - \beta_{5} - \beta_{6} - \beta_{10} + \beta_{13} ) q^{8} + ( 254 + 6 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} - 6 \beta_{4} + 2 \beta_{5} + \beta_{6} - \beta_{11} + \beta_{17} + \beta_{18} ) q^{9} + ( -168 - \beta_{3} - \beta_{4} - \beta_{7} - \beta_{8} - \beta_{15} - 2 \beta_{16} - \beta_{17} ) q^{10} + ( 60 \beta_{1} + 5 \beta_{2} - \beta_{3} + 8 \beta_{4} + \beta_{5} - \beta_{6} - \beta_{10} - \beta_{11} - 3 \beta_{18} ) q^{11} + ( -1339 - 52 \beta_{1} - 3 \beta_{2} + 13 \beta_{3} - 30 \beta_{4} - 2 \beta_{5} + \beta_{7} + 2 \beta_{8} + \beta_{9} - \beta_{10} + 3 \beta_{11} + 2 \beta_{13} + \beta_{14} - \beta_{17} - 2 \beta_{18} + \beta_{19} ) q^{12} + ( -3 + 11 \beta_{1} - 2 \beta_{2} + 11 \beta_{3} - \beta_{5} - 4 \beta_{6} - 8 \beta_{7} - 3 \beta_{8} - 2 \beta_{11} + 2 \beta_{14} + \beta_{16} + \beta_{17} - \beta_{18} ) q^{13} + ( -1 - 18 \beta_{1} + 25 \beta_{2} - 8 \beta_{3} + 34 \beta_{4} + 3 \beta_{5} - \beta_{6} - \beta_{7} - \beta_{8} + 6 \beta_{9} - \beta_{10} - 11 \beta_{11} - \beta_{12} + 4 \beta_{13} - \beta_{16} + \beta_{18} - 2 \beta_{19} ) q^{14} + ( 5 - 17 \beta_{1} - 3 \beta_{2} - 3 \beta_{3} - \beta_{4} - 3 \beta_{5} + \beta_{6} + 3 \beta_{7} - 2 \beta_{8} + 2 \beta_{9} - 3 \beta_{10} + 3 \beta_{11} - 4 \beta_{12} + 4 \beta_{13} + 2 \beta_{14} - 4 \beta_{15} - 2 \beta_{16} - 2 \beta_{17} - 2 \beta_{18} - 4 \beta_{19} ) q^{15} + ( -1689 + 40 \beta_{1} + 2 \beta_{2} + 32 \beta_{3} - 36 \beta_{4} + \beta_{5} - 9 \beta_{6} - 11 \beta_{7} + 3 \beta_{8} + 4 \beta_{10} - 15 \beta_{11} + 4 \beta_{12} - \beta_{14} + 2 \beta_{15} + 4 \beta_{16} + 7 \beta_{17} + \beta_{18} ) q^{16} + ( 80 \beta_{1} - 26 \beta_{2} + 28 \beta_{3} - 168 \beta_{4} - 9 \beta_{5} - 2 \beta_{6} - 2 \beta_{10} + 28 \beta_{11} - 5 \beta_{18} ) q^{17} + ( -597 + 202 \beta_{1} + 22 \beta_{2} - 15 \beta_{3} + 84 \beta_{4} - \beta_{5} + 5 \beta_{6} + 16 \beta_{7} + 11 \beta_{8} + 4 \beta_{9} + 2 \beta_{10} - 10 \beta_{11} + 2 \beta_{13} - 3 \beta_{14} - 5 \beta_{15} + 4 \beta_{16} - 7 \beta_{18} + 4 \beta_{19} ) q^{18} + ( 6722 - 84 \beta_{1} + 12 \beta_{2} - 61 \beta_{3} + 108 \beta_{4} + 6 \beta_{5} + 12 \beta_{6} - 16 \beta_{8} + 11 \beta_{11} - 10 \beta_{16} + 2 \beta_{17} + 6 \beta_{18} ) q^{19} + ( 7 - 192 \beta_{1} + 22 \beta_{2} - 16 \beta_{3} + 36 \beta_{4} + 9 \beta_{5} - \beta_{6} + 7 \beta_{7} + 5 \beta_{8} + 10 \beta_{9} - 10 \beta_{10} - \beta_{11} + 6 \beta_{12} - 6 \beta_{13} + \beta_{14} + 6 \beta_{16} - \beta_{17} + 15 \beta_{18} + 10 \beta_{19} ) q^{20} + ( -11 + 654 \beta_{1} - 20 \beta_{2} + 27 \beta_{3} + 40 \beta_{4} + 9 \beta_{5} + 14 \beta_{6} + 8 \beta_{7} - 3 \beta_{8} - 15 \beta_{9} - 2 \beta_{10} - 6 \beta_{11} - 8 \beta_{12} + 2 \beta_{14} + 8 \beta_{15} - 3 \beta_{16} + \beta_{17} + 9 \beta_{18} ) q^{21} + ( -8386 + 15 \beta_{1} + 8 \beta_{2} + 82 \beta_{3} - 19 \beta_{4} + 4 \beta_{5} + 6 \beta_{6} + 18 \beta_{7} + 2 \beta_{8} + 5 \beta_{10} - 18 \beta_{11} + 5 \beta_{12} - 13 \beta_{14} + 10 \beta_{15} - 19 \beta_{16} - 29 \beta_{17} + 4 \beta_{18} ) q^{22} + ( 4 + 550 \beta_{1} - 38 \beta_{2} + 34 \beta_{3} + 36 \beta_{4} + 24 \beta_{5} + 34 \beta_{6} + 4 \beta_{7} - 4 \beta_{8} - 28 \beta_{9} - 2 \beta_{10} + 22 \beta_{11} - 56 \beta_{13} + 4 \beta_{14} - 4 \beta_{17} - 6 \beta_{18} - 8 \beta_{19} ) q^{23} + ( -3711 - 1356 \beta_{1} - 25 \beta_{2} - 76 \beta_{3} - 51 \beta_{4} - 14 \beta_{5} + 56 \beta_{6} + 9 \beta_{7} + 23 \beta_{8} + 46 \beta_{9} + 9 \beta_{10} - 21 \beta_{11} + 4 \beta_{12} + 29 \beta_{13} - 5 \beta_{14} + 10 \beta_{15} - 16 \beta_{16} + 7 \beta_{17} + 31 \beta_{18} - 2 \beta_{19} ) q^{24} + ( 35657 - 148 \beta_{1} - 34 \beta_{2} - 126 \beta_{3} + 262 \beta_{4} - 17 \beta_{5} - 34 \beta_{6} - 24 \beta_{8} + 48 \beta_{11} + 44 \beta_{16} + 10 \beta_{17} - 17 \beta_{18} ) q^{25} + ( -2 + 48 \beta_{1} + 54 \beta_{2} - 8 \beta_{3} - 68 \beta_{4} - 62 \beta_{5} - 22 \beta_{6} - 2 \beta_{7} + 6 \beta_{8} - 36 \beta_{9} + 14 \beta_{10} + 10 \beta_{11} + 2 \beta_{12} - 4 \beta_{13} - 4 \beta_{14} + 2 \beta_{16} + 4 \beta_{17} + 18 \beta_{18} + 12 \beta_{19} ) q^{26} + ( 4296 - 2097 \beta_{1} - 231 \beta_{2} - 39 \beta_{3} - 261 \beta_{4} + 3 \beta_{5} + 15 \beta_{6} + 27 \beta_{10} + 57 \beta_{11} + 54 \beta_{16} + 42 \beta_{17} - 21 \beta_{18} ) q^{27} + ( -33487 + 32 \beta_{1} - 70 \beta_{2} + 32 \beta_{3} + 80 \beta_{4} - 35 \beta_{5} - 93 \beta_{6} - 17 \beta_{7} + 43 \beta_{8} + 10 \beta_{10} + 31 \beta_{11} + 10 \beta_{12} - 7 \beta_{14} - 20 \beta_{15} + 18 \beta_{16} - 29 \beta_{17} - 35 \beta_{18} ) q^{28} + ( -48 + 627 \beta_{1} - 4 \beta_{2} - 16 \beta_{3} + 96 \beta_{5} - 92 \beta_{6} - 48 \beta_{7} - 16 \beta_{8} - 19 \beta_{9} + 52 \beta_{10} - 64 \beta_{11} - 32 \beta_{12} - 96 \beta_{13} - 16 \beta_{14} - 32 \beta_{16} + 16 \beta_{17} + 40 \beta_{18} - 32 \beta_{19} ) q^{29} + ( -2081 - 14 \beta_{1} - 35 \beta_{2} - 78 \beta_{3} + 100 \beta_{4} - \beta_{5} - 61 \beta_{6} - 119 \beta_{7} + 21 \beta_{8} - 114 \beta_{9} + 59 \beta_{10} - 71 \beta_{11} - \beta_{12} + 48 \beta_{13} - 8 \beta_{14} - 14 \beta_{15} + 27 \beta_{16} - 30 \beta_{17} + 25 \beta_{18} + 6 \beta_{19} ) q^{30} + ( 131 - 120 \beta_{1} - 4 \beta_{2} - 442 \beta_{3} - 107 \beta_{4} - 2 \beta_{5} - 76 \beta_{6} + 61 \beta_{7} + 56 \beta_{8} + 8 \beta_{10} + 40 \beta_{11} + 8 \beta_{12} + 48 \beta_{14} + 40 \beta_{15} + 28 \beta_{16} + 40 \beta_{17} - 2 \beta_{18} ) q^{31} + ( 34 - 2132 \beta_{1} + 70 \beta_{2} - 8 \beta_{3} + 814 \beta_{4} - 112 \beta_{5} + 12 \beta_{6} + 34 \beta_{7} + 6 \beta_{8} - 148 \beta_{9} - 114 \beta_{10} + 10 \beta_{11} + 20 \beta_{12} - 38 \beta_{13} + 14 \beta_{14} + 20 \beta_{16} - 14 \beta_{17} - 86 \beta_{18} + 12 \beta_{19} ) q^{32} + ( 13423 - 642 \beta_{1} + 24 \beta_{2} - 490 \beta_{3} - 194 \beta_{4} + 61 \beta_{5} + 167 \beta_{6} - 120 \beta_{8} + 150 \beta_{10} - 41 \beta_{11} - 84 \beta_{16} - 67 \beta_{17} - 24 \beta_{18} ) q^{33} + ( 1190 + 120 \beta_{1} + 132 \beta_{2} + 86 \beta_{3} - 302 \beta_{4} + 66 \beta_{5} + 182 \beta_{6} - 136 \beta_{7} - 62 \beta_{8} - 10 \beta_{10} - 116 \beta_{11} - 10 \beta_{12} - 20 \beta_{14} + 38 \beta_{15} - 50 \beta_{16} + 14 \beta_{17} + 66 \beta_{18} ) q^{34} + ( -4066 \beta_{1} - 569 \beta_{2} + 216 \beta_{3} - 776 \beta_{4} - 69 \beta_{5} - 259 \beta_{6} - 259 \beta_{10} + 216 \beta_{11} + 7 \beta_{18} ) q^{35} + ( -20558 - 1436 \beta_{1} - 52 \beta_{2} + 95 \beta_{3} + 1413 \beta_{4} + 99 \beta_{5} + 49 \beta_{6} + 163 \beta_{7} + 59 \beta_{8} - 296 \beta_{9} + 8 \beta_{10} + 15 \beta_{11} + 30 \beta_{12} + 86 \beta_{13} + 9 \beta_{14} + 28 \beta_{15} - 50 \beta_{16} - 37 \beta_{17} + \beta_{18} - 8 \beta_{19} ) q^{36} + ( 255 - 151 \beta_{1} + 2 \beta_{2} - 175 \beta_{3} - 112 \beta_{4} + \beta_{5} + 284 \beta_{6} - 24 \beta_{7} - 17 \beta_{8} - 80 \beta_{10} + 122 \beta_{11} - 80 \beta_{12} - 42 \beta_{14} - 80 \beta_{15} - 61 \beta_{16} - 101 \beta_{17} + \beta_{18} ) q^{37} + ( -34 + 6771 \beta_{1} + 490 \beta_{2} - 218 \beta_{3} + 79 \beta_{4} + 186 \beta_{5} - 194 \beta_{6} - 34 \beta_{7} + 24 \beta_{8} + 368 \beta_{9} + 67 \beta_{10} - 146 \beta_{11} - 5 \beta_{12} - 104 \beta_{13} - 29 \beta_{14} - 5 \beta_{16} + 29 \beta_{17} + 66 \beta_{18} + 48 \beta_{19} ) q^{38} + ( -262 - 217 \beta_{1} - 123 \beta_{2} + 703 \beta_{3} + 226 \beta_{4} - 213 \beta_{5} - 135 \beta_{6} - 18 \beta_{7} - 58 \beta_{8} - 14 \beta_{9} - 39 \beta_{10} - 5 \beta_{11} + 100 \beta_{12} + 164 \beta_{13} + 66 \beta_{14} - 28 \beta_{15} + 90 \beta_{16} + 22 \beta_{17} - 124 \beta_{18} + 28 \beta_{19} ) q^{39} + ( -22018 + 772 \beta_{1} + 44 \beta_{2} - 104 \beta_{3} - 1996 \beta_{4} + 22 \beta_{5} + 18 \beta_{6} + 370 \beta_{7} + 46 \beta_{8} - 4 \beta_{10} - 170 \beta_{11} - 4 \beta_{12} + 38 \beta_{14} - 80 \beta_{15} + 52 \beta_{16} + 210 \beta_{17} + 22 \beta_{18} ) q^{40} + ( 20176 \beta_{1} + 1868 \beta_{2} - 294 \beta_{3} + 522 \beta_{4} - 40 \beta_{5} + 62 \beta_{6} + 62 \beta_{10} - 294 \beta_{11} - 10 \beta_{18} ) q^{41} + ( 86006 + 400 \beta_{1} + 30 \beta_{2} + 535 \beta_{3} + 307 \beta_{4} - 102 \beta_{5} + 2 \beta_{6} - 51 \beta_{7} - 27 \beta_{8} + 684 \beta_{9} + 198 \beta_{10} + 170 \beta_{11} - 30 \beta_{12} + 180 \beta_{13} - 36 \beta_{14} + 15 \beta_{15} + 51 \beta_{17} + 2 \beta_{18} - 36 \beta_{19} ) q^{42} + ( -115542 + 228 \beta_{1} + 284 \beta_{2} + 1733 \beta_{3} + 1120 \beta_{4} + 142 \beta_{5} + 284 \beta_{6} + 400 \beta_{8} - 99 \beta_{11} - 58 \beta_{16} + 226 \beta_{17} + 142 \beta_{18} ) q^{43} + ( 59 - 6460 \beta_{1} - 292 \beta_{2} + 776 \beta_{3} - 3262 \beta_{4} + 39 \beta_{5} + 117 \beta_{6} + 59 \beta_{7} - 55 \beta_{8} + 850 \beta_{9} - 396 \beta_{10} + 611 \beta_{11} + 2 \beta_{12} - 80 \beta_{13} + 57 \beta_{14} + 2 \beta_{16} - 57 \beta_{17} - 93 \beta_{18} - 110 \beta_{19} ) q^{44} + ( -1126 + 4837 \beta_{1} + 78 \beta_{2} + 2302 \beta_{3} + 840 \beta_{4} - 8 \beta_{5} - 138 \beta_{6} - 272 \beta_{7} - 286 \beta_{8} + 157 \beta_{9} + 74 \beta_{10} - 680 \beta_{11} - 72 \beta_{12} + 416 \beta_{13} - 156 \beta_{14} + 40 \beta_{15} - 122 \beta_{16} + 10 \beta_{17} + 192 \beta_{18} - 32 \beta_{19} ) q^{45} + ( 73144 + 1704 \beta_{1} - 240 \beta_{2} + 1140 \beta_{3} - 3124 \beta_{4} - 120 \beta_{5} + 136 \beta_{6} + 172 \beta_{7} - 492 \beta_{8} - 144 \beta_{10} - 256 \beta_{11} - 144 \beta_{12} + 56 \beta_{14} - 60 \beta_{15} - 24 \beta_{16} - 292 \beta_{17} - 120 \beta_{18} ) q^{46} + ( 296 - 9820 \beta_{1} + 248 \beta_{2} + 476 \beta_{3} - 216 \beta_{4} - 304 \beta_{5} + 896 \beta_{6} + 296 \beta_{7} + 24 \beta_{8} + 328 \beta_{9} - 328 \beta_{10} + 548 \beta_{11} + 160 \beta_{12} + 16 \beta_{13} + 136 \beta_{14} + 160 \beta_{16} - 136 \beta_{17} - 284 \beta_{18} + 48 \beta_{19} ) q^{47} + ( 93039 - 4692 \beta_{1} - 1116 \beta_{2} - 1704 \beta_{3} + 1742 \beta_{4} - 53 \beta_{5} - 63 \beta_{6} - 347 \beta_{7} + 219 \beta_{8} + 1008 \beta_{9} - 82 \beta_{10} + 149 \beta_{11} + 80 \beta_{12} - 30 \beta_{13} + 151 \beta_{14} - 50 \beta_{15} + 120 \beta_{16} + 139 \beta_{17} - 259 \beta_{18} + 48 \beta_{19} ) q^{48} + ( 123941 + 252 \beta_{1} - 322 \beta_{2} + 2170 \beta_{3} + 1694 \beta_{4} - 161 \beta_{5} - 322 \beta_{6} + 280 \beta_{8} + 168 \beta_{11} - 140 \beta_{16} - 462 \beta_{17} - 161 \beta_{18} ) q^{49} + ( -232 + 36849 \beta_{1} + 760 \beta_{2} + 164 \beta_{3} - 3050 \beta_{4} + 356 \beta_{5} - 412 \beta_{6} - 232 \beta_{7} - 68 \beta_{8} - 1544 \beta_{9} + 326 \beta_{10} - 40 \beta_{11} - 150 \beta_{12} - 260 \beta_{13} - 82 \beta_{14} - 150 \beta_{16} + 82 \beta_{17} + 16 \beta_{18} - 136 \beta_{19} ) q^{50} + ( -370632 - 4630 \beta_{1} + 383 \beta_{2} - 1514 \beta_{3} - 1444 \beta_{4} + 447 \beta_{5} + 105 \beta_{6} - 144 \beta_{8} - 111 \beta_{10} - 1058 \beta_{11} - 240 \beta_{16} - 24 \beta_{17} + 271 \beta_{18} ) q^{51} + ( -125454 - 1648 \beta_{1} + 260 \beta_{2} - 112 \beta_{3} + 4368 \beta_{4} + 130 \beta_{5} - 130 \beta_{6} - 402 \beta_{7} - 370 \beta_{8} + 60 \beta_{10} + 142 \beta_{11} + 60 \beta_{12} + 210 \beta_{14} + 160 \beta_{15} + 140 \beta_{16} + 510 \beta_{17} + 130 \beta_{18} ) q^{52} + ( -176 - 9953 \beta_{1} + 712 \beta_{2} - 888 \beta_{3} - 928 \beta_{4} + 160 \beta_{5} - 1192 \beta_{6} - 176 \beta_{7} + 176 \beta_{8} + 57 \beta_{9} + 392 \beta_{10} - 360 \beta_{11} + 32 \beta_{13} - 176 \beta_{14} + 176 \beta_{17} + 264 \beta_{18} + 352 \beta_{19} ) q^{53} + ( 285273 + 1905 \beta_{1} + 1203 \beta_{2} - 3123 \beta_{3} + 189 \beta_{4} + 228 \beta_{5} + 561 \beta_{6} - 102 \beta_{7} + 138 \beta_{8} - 1776 \beta_{9} + 237 \beta_{10} - 42 \beta_{11} - 171 \beta_{12} - 24 \beta_{13} - 117 \beta_{14} + 42 \beta_{15} - 291 \beta_{16} - 117 \beta_{17} - 132 \beta_{18} - 48 \beta_{19} ) q^{54} + ( 2374 - 1944 \beta_{1} - 444 \beta_{2} - 7368 \beta_{3} - 1806 \beta_{4} - 222 \beta_{5} - 1300 \beta_{6} + 426 \beta_{7} + 920 \beta_{8} + 312 \beta_{10} + 664 \beta_{11} + 312 \beta_{12} - 80 \beta_{14} - 232 \beta_{15} + 116 \beta_{16} + 136 \beta_{17} - 222 \beta_{18} ) q^{55} + ( 106 - 34452 \beta_{1} - 2384 \beta_{2} + 584 \beta_{3} + 892 \beta_{4} + 14 \beta_{5} + 186 \beta_{6} + 106 \beta_{7} - 50 \beta_{8} - 2180 \beta_{9} - 516 \beta_{10} + 434 \beta_{11} + 28 \beta_{12} - 220 \beta_{13} + 78 \beta_{14} + 28 \beta_{16} - 78 \beta_{17} + 114 \beta_{18} - 100 \beta_{19} ) q^{56} + ( -252617 + 2074 \beta_{1} - 314 \beta_{2} - 2376 \beta_{3} - 8188 \beta_{4} - 380 \beta_{5} - 211 \beta_{6} - 240 \beta_{8} + 342 \beta_{10} - 817 \beta_{11} + 456 \beta_{16} - 97 \beta_{17} + 71 \beta_{18} ) q^{57} + ( 72680 - 4512 \beta_{1} - 608 \beta_{2} + 275 \beta_{3} + 10291 \beta_{4} - 304 \beta_{5} - 16 \beta_{6} - 221 \beta_{7} - 1277 \beta_{8} - 160 \beta_{10} + 1296 \beta_{11} - 160 \beta_{12} - 112 \beta_{14} - 205 \beta_{15} + 134 \beta_{16} - 541 \beta_{17} - 304 \beta_{18} ) q^{58} + ( -45490 \beta_{1} - 2780 \beta_{2} - 699 \beta_{3} + 9032 \beta_{4} - 156 \beta_{5} - 484 \beta_{6} - 484 \beta_{10} - 699 \beta_{11} + 68 \beta_{18} ) q^{59} + ( 111183 - 2784 \beta_{1} - 3858 \beta_{2} + 496 \beta_{3} + 1088 \beta_{4} - 681 \beta_{5} - 23 \beta_{6} + 645 \beta_{7} + 1085 \beta_{8} - 2660 \beta_{9} - 282 \beta_{10} + 3365 \beta_{11} + 46 \beta_{12} - 316 \beta_{13} + 163 \beta_{14} - 140 \beta_{15} + 662 \beta_{16} + 497 \beta_{17} - 125 \beta_{18} + 28 \beta_{19} ) q^{60} + ( 1809 - 1817 \beta_{1} + 1134 \beta_{2} - 4945 \beta_{3} - 1840 \beta_{4} + 567 \beta_{5} + 1236 \beta_{6} - 1064 \beta_{7} + 449 \beta_{8} + 112 \beta_{10} - 378 \beta_{11} + 112 \beta_{12} - 438 \beta_{14} + 624 \beta_{15} - 163 \beta_{16} + 21 \beta_{17} + 567 \beta_{18} ) q^{61} + ( -389 - 6378 \beta_{1} - 1555 \beta_{2} - 1832 \beta_{3} + 15586 \beta_{4} - 89 \beta_{5} + 131 \beta_{6} - 389 \beta_{7} - 181 \beta_{8} + 2558 \beta_{9} + 1067 \beta_{10} - 2375 \beta_{11} - 285 \beta_{12} - 164 \beta_{13} - 104 \beta_{14} - 285 \beta_{16} + 104 \beta_{17} + 341 \beta_{18} - 362 \beta_{19} ) q^{62} + ( -2593 - 46442 \beta_{1} + 2694 \beta_{2} + 6520 \beta_{3} + 105 \beta_{4} + 82 \beta_{5} + 1206 \beta_{6} + 1585 \beta_{7} - 292 \beta_{8} - 188 \beta_{9} - 226 \beta_{10} - 98 \beta_{11} + 120 \beta_{12} - 184 \beta_{13} + 36 \beta_{14} + 280 \beta_{15} + 76 \beta_{16} - 68 \beta_{17} - 12 \beta_{18} - 8 \beta_{19} ) q^{63} + ( -204490 + 8040 \beta_{1} + 68 \beta_{2} - 128 \beta_{3} - 17968 \beta_{4} + 34 \beta_{5} - 866 \beta_{6} - 62 \beta_{7} - 882 \beta_{8} + 256 \beta_{10} - 2622 \beta_{11} + 256 \beta_{12} + 166 \beta_{14} + 572 \beta_{15} + 16 \beta_{16} - 66 \beta_{17} + 34 \beta_{18} ) q^{64} + ( 27472 \beta_{1} + 2568 \beta_{2} - 222 \beta_{3} - 3030 \beta_{4} - 26 \beta_{5} + 1210 \beta_{6} + 1210 \beta_{10} - 222 \beta_{11} + 540 \beta_{18} ) q^{65} + ( 54681 + 9991 \beta_{1} + 3530 \beta_{2} - 1393 \beta_{3} + 6662 \beta_{4} + 1049 \beta_{5} + 643 \beta_{6} + 600 \beta_{7} + 949 \beta_{8} + 2756 \beta_{9} + 24 \beta_{10} + 2850 \beta_{11} - 298 \beta_{12} - 926 \beta_{13} + 33 \beta_{14} + 25 \beta_{15} - 558 \beta_{16} - 738 \beta_{17} + 659 \beta_{18} + 68 \beta_{19} ) q^{66} + ( 179508 - 3196 \beta_{1} - 640 \beta_{2} + 8119 \beta_{3} + 16590 \beta_{4} - 320 \beta_{5} - 640 \beta_{6} + 1440 \beta_{8} + 1479 \beta_{11} + 600 \beta_{16} - 40 \beta_{17} - 320 \beta_{18} ) q^{67} + ( 252 + 9104 \beta_{1} - 300 \beta_{2} + 1888 \beta_{3} - 14164 \beta_{4} - 896 \beta_{5} + 40 \beta_{6} + 252 \beta_{7} + 228 \beta_{8} + 2696 \beta_{9} - 68 \beta_{10} + 2572 \beta_{11} + 240 \beta_{12} - 52 \beta_{13} + 12 \beta_{14} + 240 \beta_{16} - 12 \beta_{17} - 788 \beta_{18} + 456 \beta_{19} ) q^{68} + ( -3302 + 63538 \beta_{1} - 3926 \beta_{2} + 10406 \beta_{3} + 5704 \beta_{4} + 400 \beta_{5} - 102 \beta_{6} - 432 \beta_{7} - 1150 \beta_{8} - 536 \beta_{9} + 30 \beta_{10} - 16 \beta_{11} - 200 \beta_{12} - 1504 \beta_{13} + 4 \beta_{14} - 632 \beta_{15} - 10 \beta_{16} - 278 \beta_{17} - 264 \beta_{18} + 352 \beta_{19} ) q^{69} + ( 585418 + 13606 \beta_{1} + 836 \beta_{2} - 3466 \beta_{3} - 33488 \beta_{4} + 418 \beta_{5} + 250 \beta_{6} - 1136 \beta_{7} - 1450 \beta_{8} + 364 \beta_{10} - 4700 \beta_{11} + 364 \beta_{12} - 506 \beta_{14} + 110 \beta_{15} + 420 \beta_{16} + 1004 \beta_{17} + 418 \beta_{18} ) q^{70} + ( 172 + 88698 \beta_{1} - 4662 \beta_{2} + 110 \beta_{3} + 4748 \beta_{4} - 472 \beta_{5} + 426 \beta_{6} + 172 \beta_{7} + 20 \beta_{8} - 1620 \beta_{9} - 258 \beta_{10} + 170 \beta_{11} + 96 \beta_{12} + 600 \beta_{13} + 76 \beta_{14} + 96 \beta_{16} - 76 \beta_{17} - 162 \beta_{18} + 40 \beta_{19} ) q^{71} + ( 349624 - 23232 \beta_{1} - 2045 \beta_{2} - 2608 \beta_{3} + 6237 \beta_{4} + 337 \beta_{5} - 919 \beta_{6} + 1068 \beta_{7} + 1956 \beta_{8} + 3948 \beta_{9} - 771 \beta_{10} - 1700 \beta_{11} - 180 \beta_{12} + 3 \beta_{13} - 252 \beta_{14} + 60 \beta_{15} + 492 \beta_{16} - 112 \beta_{17} + 116 \beta_{18} - 372 \beta_{19} ) q^{72} + ( 993594 - 8928 \beta_{1} + 500 \beta_{2} + 4512 \beta_{3} + 23640 \beta_{4} + 250 \beta_{5} + 500 \beta_{6} + 280 \beta_{8} + 2052 \beta_{11} - 780 \beta_{16} - 280 \beta_{17} + 250 \beta_{18} ) q^{73} + ( 314 + 13008 \beta_{1} - 4894 \beta_{2} + 2984 \beta_{3} - 28700 \beta_{4} + 726 \beta_{5} + 814 \beta_{6} + 314 \beta_{7} + 306 \beta_{8} - 2476 \beta_{9} + 778 \beta_{10} + 3902 \beta_{11} + 310 \beta_{12} + 996 \beta_{13} + 4 \beta_{14} + 310 \beta_{16} - 4 \beta_{17} + 150 \beta_{18} + 612 \beta_{19} ) q^{74} + ( -426388 + 77153 \beta_{1} + 2819 \beta_{2} - 7794 \beta_{3} - 39677 \beta_{4} - 1771 \beta_{5} - 383 \beta_{6} - 1776 \beta_{8} - 1251 \beta_{10} + 1222 \beta_{11} - 366 \beta_{16} + 502 \beta_{17} - 263 \beta_{18} ) q^{75} + ( -591659 - 9556 \beta_{1} + 534 \beta_{2} + 6094 \beta_{3} + 23850 \beta_{4} + 267 \beta_{5} + 797 \beta_{6} + 2341 \beta_{7} - 943 \beta_{8} + 18 \beta_{10} + 1929 \beta_{11} + 18 \beta_{12} - 317 \beta_{14} - 560 \beta_{15} - 942 \beta_{16} - 147 \beta_{17} + 267 \beta_{18} ) q^{76} + ( -560 - 85218 \beta_{1} + 5016 \beta_{2} + 744 \beta_{3} - 4128 \beta_{4} - 352 \beta_{5} + 392 \beta_{6} - 560 \beta_{7} - 720 \beta_{8} + 410 \beta_{9} - 328 \beta_{10} - 1416 \beta_{11} - 640 \beta_{12} + 1824 \beta_{13} + 80 \beta_{14} - 640 \beta_{16} - 80 \beta_{17} + 200 \beta_{18} - 1440 \beta_{19} ) q^{77} + ( -63540 - 820 \beta_{1} + 8440 \beta_{2} - 2390 \beta_{3} - 1358 \beta_{4} - 464 \beta_{5} - 1416 \beta_{6} - 90 \beta_{7} + 2146 \beta_{8} - 3328 \beta_{9} - 1080 \beta_{10} - 4968 \beta_{11} + 260 \beta_{12} - 572 \beta_{13} + 224 \beta_{14} + 290 \beta_{15} + 64 \beta_{16} + 618 \beta_{17} + 828 \beta_{18} + 32 \beta_{19} ) q^{78} + ( 5795 - 1840 \beta_{1} - 356 \beta_{2} - 11906 \beta_{3} - 4723 \beta_{4} - 178 \beta_{5} + 36 \beta_{6} - 2043 \beta_{7} + 528 \beta_{8} - 408 \beta_{10} + 1352 \beta_{11} - 408 \beta_{12} + 832 \beta_{14} + 136 \beta_{15} + 212 \beta_{16} + 144 \beta_{17} - 178 \beta_{18} ) q^{79} + ( 164 - 49592 \beta_{1} + 15600 \beta_{2} - 9168 \beta_{3} + 47944 \beta_{4} + 1516 \beta_{5} + 1540 \beta_{6} + 164 \beta_{7} + 76 \beta_{8} - 2088 \beta_{9} + 1144 \beta_{10} - 8940 \beta_{11} + 120 \beta_{12} + 1800 \beta_{13} + 44 \beta_{14} + 120 \beta_{16} - 44 \beta_{17} + 20 \beta_{18} + 152 \beta_{19} ) q^{80} + ( -334575 - 28512 \beta_{1} - 2178 \beta_{2} - 14886 \beta_{3} - 8802 \beta_{4} + 117 \beta_{5} - 1764 \beta_{6} - 2592 \beta_{8} - 2268 \beta_{10} - 1422 \beta_{11} + 504 \beta_{17} - 1413 \beta_{18} ) q^{81} + ( -2645168 - 4132 \beta_{1} - 1136 \beta_{2} + 24808 \beta_{3} + 36260 \beta_{4} - 568 \beta_{5} - 2304 \beta_{6} + 208 \beta_{7} + 8 \beta_{8} + 220 \beta_{10} + 1480 \beta_{11} + 220 \beta_{12} + 508 \beta_{14} + 120 \beta_{15} + 140 \beta_{16} - 388 \beta_{17} - 568 \beta_{18} ) q^{82} + ( -87032 \beta_{1} - 8421 \beta_{2} + 1999 \beta_{3} - 16096 \beta_{4} + 3055 \beta_{5} + 2265 \beta_{6} + 2265 \beta_{10} + 1999 \beta_{11} + 75 \beta_{18} ) q^{83} + ( -326847 + 72080 \beta_{1} + 218 \beta_{2} - 1888 \beta_{3} + 33028 \beta_{4} + 467 \beta_{5} - 1259 \beta_{6} - 3987 \beta_{7} + 667 \beta_{8} - 886 \beta_{9} - 510 \beta_{10} + 3525 \beta_{11} - 190 \beta_{12} + 106 \beta_{13} - 261 \beta_{14} - 272 \beta_{15} - 1710 \beta_{16} - 507 \beta_{17} + 137 \beta_{18} - 118 \beta_{19} ) q^{84} + ( 10008 - 7384 \beta_{1} - 2832 \beta_{2} - 22488 \beta_{3} - 5440 \beta_{4} - 1416 \beta_{5} - 1792 \beta_{6} + 4704 \beta_{7} + 2520 \beta_{8} - 832 \beta_{10} + 5488 \beta_{11} - 832 \beta_{12} + 1456 \beta_{14} - 1600 \beta_{15} + 312 \beta_{16} - 296 \beta_{17} - 1416 \beta_{18} ) q^{85} + ( 1114 - 139087 \beta_{1} - 12146 \beta_{2} - 4670 \beta_{3} + 49029 \beta_{4} - 3746 \beta_{5} + 202 \beta_{6} + 1114 \beta_{7} + 568 \beta_{8} + 2992 \beta_{9} - 2255 \beta_{10} - 2966 \beta_{11} + 841 \beta_{12} + 3768 \beta_{13} + 273 \beta_{14} + 841 \beta_{16} - 273 \beta_{17} - 1786 \beta_{18} + 1136 \beta_{19} ) q^{86} + ( -10503 - 160573 \beta_{1} + 8461 \beta_{2} + 26825 \beta_{3} - 3213 \beta_{4} + 1633 \beta_{5} - 2711 \beta_{6} - 4905 \beta_{7} - 3906 \beta_{8} + 2778 \beta_{9} + 885 \beta_{10} - 3841 \beta_{11} - 660 \beta_{12} - 2508 \beta_{13} + 426 \beta_{14} - 180 \beta_{15} - 210 \beta_{16} + 494 \beta_{17} + 286 \beta_{18} - 372 \beta_{19} ) q^{87} + ( -674684 + 36628 \beta_{1} - 704 \beta_{2} + 1104 \beta_{3} - 81612 \beta_{4} - 352 \beta_{5} + 1816 \beta_{6} - 3012 \beta_{7} - 2180 \beta_{8} - 580 \beta_{10} - 8600 \beta_{11} - 580 \beta_{12} - 780 \beta_{14} - 1460 \beta_{15} - 1452 \beta_{16} - 2056 \beta_{17} - 352 \beta_{18} ) q^{88} + ( 47840 \beta_{1} + 350 \beta_{2} + 2478 \beta_{3} - 16494 \beta_{4} + 1717 \beta_{5} - 712 \beta_{6} - 712 \beta_{10} + 2478 \beta_{11} - 3201 \beta_{18} ) q^{89} + ( 520948 + 4832 \beta_{1} + 13188 \beta_{2} + 3299 \beta_{3} - 2589 \beta_{4} - 2212 \beta_{5} - 4020 \beta_{6} - 2785 \beta_{7} + 3103 \beta_{8} + 680 \beta_{9} - 2300 \beta_{10} + 10388 \beta_{11} + 1428 \beta_{12} + 1456 \beta_{13} - 504 \beta_{14} + 131 \beta_{15} + 1370 \beta_{16} + 1211 \beta_{17} - 1740 \beta_{18} + 392 \beta_{19} ) q^{90} + ( 3392938 - 10576 \beta_{1} + 1940 \beta_{2} - 2818 \beta_{3} + 19766 \beta_{4} + 970 \beta_{5} + 1940 \beta_{6} - 528 \beta_{8} + 1542 \beta_{11} - 110 \beta_{16} + 1830 \beta_{17} + 970 \beta_{18} ) q^{91} + ( -524 + 109280 \beta_{1} + 24728 \beta_{2} + 4864 \beta_{3} - 75040 \beta_{4} - 292 \beta_{5} + 2148 \beta_{6} - 524 \beta_{7} - 356 \beta_{8} + 568 \beta_{9} + 2904 \beta_{10} + 3796 \beta_{11} - 440 \beta_{12} + 2664 \beta_{13} - 84 \beta_{14} - 440 \beta_{16} + 84 \beta_{17} + 2196 \beta_{18} - 712 \beta_{19} ) q^{92} + ( -12031 + 247834 \beta_{1} - 12170 \beta_{2} + 26111 \beta_{3} + 22512 \beta_{4} + 1167 \beta_{5} + 1196 \beta_{6} + 3128 \beta_{7} - 1967 \beta_{8} + 77 \beta_{9} + 184 \beta_{10} - 2698 \beta_{11} + 720 \beta_{12} - 1664 \beta_{13} + 746 \beta_{14} + 1680 \beta_{15} + 381 \beta_{16} + 837 \beta_{17} - 17 \beta_{18} - 1408 \beta_{19} ) q^{93} + ( -1234624 + 45136 \beta_{1} + 520 \beta_{3} - 102216 \beta_{4} + 544 \beta_{6} + 1128 \beta_{7} + 1912 \beta_{8} - 576 \beta_{10} - 9904 \beta_{11} - 576 \beta_{12} + 1184 \beta_{14} - 120 \beta_{15} - 1616 \beta_{16} - 888 \beta_{17} ) q^{94} + ( -2444 + 357998 \beta_{1} - 17022 \beta_{2} - 3462 \beta_{3} + 16980 \beta_{4} + 2360 \beta_{5} - 6838 \beta_{6} - 2444 \beta_{7} - 372 \beta_{8} + 4884 \beta_{9} + 2486 \beta_{10} - 4578 \beta_{11} - 1408 \beta_{12} + 168 \beta_{13} - 1036 \beta_{14} - 1408 \beta_{16} + 1036 \beta_{17} + 2258 \beta_{18} - 744 \beta_{19} ) q^{95} + ( 1794408 + 80428 \beta_{1} - 14498 \beta_{2} - 9944 \beta_{3} + 11234 \beta_{4} - 1338 \beta_{5} + 1214 \beta_{6} + 1860 \beta_{7} + 292 \beta_{8} - 4468 \beta_{9} + 2082 \beta_{10} - 5628 \beta_{11} + 572 \beta_{12} - 1682 \beta_{13} - 316 \beta_{14} - 412 \beta_{15} - 2948 \beta_{16} - 2264 \beta_{17} + 1108 \beta_{18} + 1196 \beta_{19} ) q^{96} + ( 270604 + 348 \beta_{1} + 946 \beta_{2} + 12858 \beta_{3} + 16150 \beta_{4} + 473 \beta_{5} + 946 \beta_{6} + 4080 \beta_{8} + 460 \beta_{11} + 3528 \beta_{16} + 4474 \beta_{17} + 473 \beta_{18} ) q^{97} + ( 504 + 140517 \beta_{1} - 22344 \beta_{2} + 8708 \beta_{3} - 24570 \beta_{4} - 3388 \beta_{5} + 1092 \beta_{6} + 504 \beta_{7} - 644 \beta_{8} + 3192 \beta_{9} - 4074 \beta_{10} + 6776 \beta_{11} - 70 \beta_{12} + 1596 \beta_{13} + 574 \beta_{14} - 70 \beta_{16} - 574 \beta_{17} + 1904 \beta_{18} - 1288 \beta_{19} ) q^{98} + ( 515846 + 367902 \beta_{1} + 28856 \beta_{2} - 18653 \beta_{3} - 16290 \beta_{4} + 1550 \beta_{5} - 4076 \beta_{6} - 3168 \beta_{8} + 1296 \beta_{10} - 2605 \beta_{11} + 2790 \beta_{16} - 2582 \beta_{17} - 1214 \beta_{18} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$20q + 120q^{3} + 548q^{4} - 1416q^{6} + 5076q^{9} + O(q^{10})$$ $$20q + 120q^{3} + 548q^{4} - 1416q^{6} + 5076q^{9} - 3384q^{10} - 26700q^{12} - 33304q^{16} - 11880q^{18} + 133744q^{19} - 166880q^{22} - 74424q^{24} + 711596q^{25} + 85320q^{27} - 669360q^{28} - 41904q^{30} + 263640q^{33} + 24112q^{34} - 409428q^{36} - 437088q^{40} + 1724040q^{42} - 2292080q^{43} + 1466400q^{46} + 1846200q^{48} + 2495228q^{49} - 7417536q^{51} - 2511360q^{52} + 5680584q^{54} - 5067120q^{57} + 1427880q^{58} + 2211120q^{60} - 4070608q^{64} + 1062024q^{66} + 3654640q^{67} + 11712816q^{70} + 7004880q^{72} + 19892680q^{73} - 8612088q^{75} - 11799464q^{76} - 1226160q^{78} - 6817932q^{81} - 52710080q^{82} - 6585408q^{84} - 13470800q^{88} + 10393704q^{90} + 67826976q^{91} - 24601920q^{94} + 35857680q^{96} + 5561800q^{97} + 10152864q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{20} - 274 x^{18} + 45864 x^{16} - 5031360 x^{14} + 776389632 x^{12} - 102828146688 x^{10} + 12720367730688 x^{8} - 1350595415900160 x^{6} + 201712005185273856 x^{4} - 19743780766392254464 x^{2} + 1180591620717411303424$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{19} - 274 \nu^{17} + 45864 \nu^{15} - 5031360 \nu^{13} + 776389632 \nu^{11} - 102828146688 \nu^{9} + 12720367730688 \nu^{7} - 1350595415900160 \nu^{5} + 201712005185273856 \nu^{3} - 19743780766392254464 \nu$$$$)/ 9223372036854775808$$ $$\beta_{2}$$ $$=$$ $$($$$$-67470075 \nu^{19} + 620241440 \nu^{18} + 7481718950 \nu^{17} - 87283107392 \nu^{16} - 603720648760 \nu^{15} + 11830702224640 \nu^{14} + 74995432727360 \nu^{13} - 675299382278144 \nu^{12} - 17942103215057920 \nu^{11} + 258163694781890560 \nu^{10} + 1141389515184865280 \nu^{9} - 20897276187801288704 \nu^{8} - 97937657864629780480 \nu^{7} + 2287807466056764620800 \nu^{6} + 31353350171691088609280 \nu^{5} - 318723544443333235441664 \nu^{4} - 3401840897950015773736960 \nu^{3} + 64586383234089189722030080 \nu^{2} + 949984376297812261779537920 \nu - 2270017094097738660076060672$$$$)/$$$$64\!\cdots\!40$$ $$\beta_{3}$$ $$=$$ $$($$$$176868855 \nu^{19} - 3029440960 \nu^{18} - 26451903070 \nu^{17} + 664740728704 \nu^{16} + 3130459423640 \nu^{15} - 105710177830400 \nu^{14} - 360949274643520 \nu^{13} + 10351490889945088 \nu^{12} + 68437218275302400 \nu^{11} - 1905255894958407680 \nu^{10} - 6594230058547609600 \nu^{9} + 225749796244525416448 \nu^{8} + 729223861800796160000 \nu^{7} - 27331819564499090800640 \nu^{6} - 119335417111012295311360 \nu^{5} + 3053605670972218176176128 \nu^{4} + 15261204956271849399910400 \nu^{3} - 490027087977375941148016640 \nu^{2} - 665271670662183154226298880 \nu + 36380313912834696188688072704$$$$)/$$$$12\!\cdots\!80$$ $$\beta_{4}$$ $$=$$ $$($$$$-176868855 \nu^{19} + 1240482880 \nu^{18} + 26451903070 \nu^{17} - 174566214784 \nu^{16} - 3130459423640 \nu^{15} + 23661404449280 \nu^{14} + 360949274643520 \nu^{13} - 1350598764556288 \nu^{12} - 68437218275302400 \nu^{11} + 516327389563781120 \nu^{10} + 6594230058547609600 \nu^{9} - 41794552375602577408 \nu^{8} - 729223861800796160000 \nu^{7} + 4575614932113529241600 \nu^{6} + 119335417111012295311360 \nu^{5} - 637447088886666470883328 \nu^{4} - 15261204956271849399910400 \nu^{3} + 129172766468178379444060160 \nu^{2} + 665271670662183154226298880 \nu - 4540034188195477320152121344$$$$)/$$$$12\!\cdots\!80$$ $$\beta_{5}$$ $$=$$ $$($$$$-343887579 \nu^{19} + 2480965760 \nu^{18} - 309649627546 \nu^{17} - 349132429568 \nu^{16} + 42654920740040 \nu^{15} + 47322808898560 \nu^{14} - 5019207879120064 \nu^{13} - 2701197529112576 \nu^{12} + 249849354898002944 \nu^{11} + 1032654779127562240 \nu^{10} - 158818492905752952832 \nu^{9} - 83589104751205154816 \nu^{8} + 6045239330621672652800 \nu^{7} + 9151229864227058483200 \nu^{6} - 1453505891903586942582784 \nu^{5} - 1274894177773332941766656 \nu^{4} + 117417405794707067170193408 \nu^{3} + 258345532936356758888120320 \nu^{2} - 33849331021006522850901753856 \nu - 9080068376390954640304242688$$$$)/$$$$12\!\cdots\!80$$ $$\beta_{6}$$ $$=$$ $$($$$$112575697 \nu^{19} - 5659803360 \nu^{18} + 8849711182 \nu^{17} + 846460898240 \nu^{16} - 422947105880 \nu^{15} - 100174701556480 \nu^{14} + 190756083519040 \nu^{13} + 11550376788592640 \nu^{12} + 44183443500172288 \nu^{11} - 2189990984809676800 \nu^{10} + 4946526181421154304 \nu^{9} + 211015361873523507200 \nu^{8} + 94578587359227412480 \nu^{7} - 23335163577625477120000 \nu^{6} - 5624542482332458680320 \nu^{5} + 3818733347552393449963520 \nu^{4} + 2309562732626491406811136 \nu^{3} - 488358558600699180797132800 \nu^{2} + 1395226585471861890264072192 \nu + 24446939830637880620582174720$$$$)/$$$$32\!\cdots\!20$$ $$\beta_{7}$$ $$=$$ $$($$$$-176868855 \nu^{19} + 46663936192 \nu^{18} + 26451903070 \nu^{17} - 7650400378240 \nu^{16} - 3130459423640 \nu^{15} + 883852122037760 \nu^{14} + 360949274643520 \nu^{13} - 98847944326402048 \nu^{12} - 68437218275302400 \nu^{11} + 20665734539185946624 \nu^{10} + 6594230058547609600 \nu^{9} - 1762765790002246844416 \nu^{8} - 729223861800796160000 \nu^{7} + 198617799574511345991680 \nu^{6} + 119335417111012295311360 \nu^{5} - 27910231241182783039602688 \nu^{4} - 15261204956271849399910400 \nu^{3} + 4198176475754643076407099392 \nu^{2} + 665271670662183154226298880 \nu - 179938669808013424602147454976$$$$)/$$$$12\!\cdots\!80$$ $$\beta_{8}$$ $$=$$ $$($$$$-367713945 \nu^{19} - 25214600320 \nu^{18} + 56733294530 \nu^{17} + 4263582978304 \nu^{16} - 6901924889320 \nu^{15} - 624728791331840 \nu^{14} + 792218019016640 \nu^{13} + 48612256609067008 \nu^{12} - 147725440499000320 \nu^{11} - 12428023749204377600 \nu^{10} + 14625610459821178880 \nu^{9} + 1220578569616409755648 \nu^{8} - 1636230572292104519680 \nu^{7} - 141478452810973152542720 \nu^{6} + 257547073144567963320320 \nu^{5} + 17447629168443318650011648 \nu^{4} - 33341584299334304750632960 \nu^{3} - 2118063107382358695697448960 \nu^{2} + 1606487061103944559021260800 \nu + 139801622867552872063758761984$$$$)/$$$$12\!\cdots\!80$$ $$\beta_{9}$$ $$=$$ $$($$$$-8511439 \nu^{19} + 3020598158 \nu^{17} - 288761902424 \nu^{15} + 33183569749568 \nu^{13} - 5339962910997504 \nu^{11} + 935646678877667328 \nu^{9} - 71620747207297204224 \nu^{7} + 11288455140532434239488 \nu^{5} - 1510368000915481884098560 \nu^{3} + 147156128792375308928942080 \nu$$$$)/$$$$11\!\cdots\!40$$ $$\beta_{10}$$ $$=$$ $$($$$$-23408920 \nu^{19} + 373120255 \nu^{18} + 6526306416 \nu^{17} - 55631403246 \nu^{16} - 919469947200 \nu^{15} + 6630628291800 \nu^{14} + 61742774618112 \nu^{13} - 743001654983232 \nu^{12} - 13029995526205440 \nu^{11} + 144942052012538880 \nu^{10} + 2064258185792520192 \nu^{9} - 13841499997964009472 \nu^{8} - 168257420302913372160 \nu^{7} + 1529941706915866214400 \nu^{6} + 19185863009259038441472 \nu^{5} - 248630944985878754230272 \nu^{4} - 4461287883387838031462400 \nu^{3} + 32540734388608985978634240 \nu^{2} + 365049355560191487970902016 \nu - 1598871773605421871913762816$$$$)/$$$$20\!\cdots\!20$$ $$\beta_{11}$$ $$=$$ $$($$$$183080515 \nu^{19} + 1439256000 \nu^{18} - 28153897910 \nu^{17} - 229030049664 \nu^{16} + 3415350997880 \nu^{15} + 32777934824960 \nu^{14} - 392202372301120 \nu^{13} - 2350697889599488 \nu^{12} + 73259886696811520 \nu^{11} + 670652779052072960 \nu^{10} - 7232963544203591680 \nu^{9} - 62234023916594003968 \nu^{8} + 808238461218801582080 \nu^{7} + 7104082113489702748160 \nu^{6} - 127724856632142683176960 \nu^{5} - 905909153562838882582528 \nu^{4} + 16514171350401007600271360 \nu^{3} + 169267691080311441855610880 \nu^{2} - 787913323897551265590149120 \nu - 8077843046488723861100560384$$$$)/$$$$14\!\cdots\!20$$ $$\beta_{12}$$ $$=$$ $$($$$$585029869 \nu^{19} + 38141204352 \nu^{18} - 50164026794 \nu^{17} - 8177313523456 \nu^{16} + 18248082775048 \nu^{15} + 738475155270656 \nu^{14} + 418399062453568 \nu^{13} - 104986031637864448 \nu^{12} + 344184310189487104 \nu^{11} + 19945347754623893504 \nu^{10} - 3944833581139165184 \nu^{9} - 2233943687850102882304 \nu^{8} + 2660188554032261890048 \nu^{7} + 196956371065039367438336 \nu^{6} - 408230192328260999708672 \nu^{5} - 41433944485106499041886208 \nu^{4} + 88977523864899086712832000 \nu^{3} + 4717790095984032362446979072 \nu^{2} - 635060460071117926862684160 \nu - 289052317905395063160599216128$$$$)/$$$$64\!\cdots\!40$$ $$\beta_{13}$$ $$=$$ $$($$$$334078239 \nu^{19} + 620241440 \nu^{18} - 93024369454 \nu^{17} - 87283107392 \nu^{16} + 12523197688280 \nu^{15} + 11830702224640 \nu^{14} - 1148339020165696 \nu^{13} - 675299382278144 \nu^{12} + 179557197105314816 \nu^{11} + 258163694781890560 \nu^{10} - 29727424813540507648 \nu^{9} - 20897276187801288704 \nu^{8} + 2551673299540744601600 \nu^{7} + 2287807466056764620800 \nu^{6} - 320373449078837667168256 \nu^{5} - 318723544443333235441664 \nu^{4} + 36154051611926337758953472 \nu^{3} + 64586383234089189722030080 \nu^{2} - 4638794152966537195137531904 \nu - 2270017094097738660076060672$$$$)/$$$$21\!\cdots\!80$$ $$\beta_{14}$$ $$=$$ $$($$$$-3421024153 \nu^{19} + 81935374912 \nu^{18} + 583128146370 \nu^{17} - 27895536621696 \nu^{16} - 59736877517544 \nu^{15} + 3133683366812160 \nu^{14} + 7945275676684224 \nu^{13} - 382123180007288832 \nu^{12} - 1374520280345158656 \nu^{11} + 46709207148971753472 \nu^{10} + 170967159693274841088 \nu^{9} - 9424318485405332668416 \nu^{8} - 15844327075257866256384 \nu^{7} + 636726342792302611660800 \nu^{6} + 1982837387739580537503744 \nu^{5} - 103328952097634195154665472 \nu^{4} - 323274210077250614395404288 \nu^{3} + 13460550666279957019943239680 \nu^{2} + 20294203798779213357858684928 \nu - 1593744964547285331425477263360$$$$)/$$$$12\!\cdots\!80$$ $$\beta_{15}$$ $$=$$ $$($$$$1012048949 \nu^{19} - 31317375584 \nu^{18} - 143545301178 \nu^{17} + 5528689605312 \nu^{16} + 15293887129416 \nu^{15} - 536246907191040 \nu^{14} - 1885800154312896 \nu^{13} + 60940730662103040 \nu^{12} + 404922818155287552 \nu^{11} - 14262186971055095808 \nu^{10} - 39443821256534458368 \nu^{9} + 1580926172207152889856 \nu^{8} + 4237966321623893016576 \nu^{7} - 159927259539857033134080 \nu^{6} - 531167743694980666884096 \nu^{5} + 22094602036680759265198080 \nu^{4} + 86943416491007107355639808 \nu^{3} - 3256216571989507680201867264 \nu^{2} - 3844642281888487237757173760 \nu + 172806517683863918176228081664$$$$)/$$$$32\!\cdots\!20$$ $$\beta_{16}$$ $$=$$ $$($$$$-3680481851 \nu^{19} - 32083178880 \nu^{18} + 517447910182 \nu^{17} + 19692744459008 \nu^{16} - 54273623628344 \nu^{15} - 1894636902341632 \nu^{14} + 6750982598234944 \nu^{13} + 234806482825355264 \nu^{12} - 1471965832122149888 \nu^{11} - 30192830511859499008 \nu^{10} + 143149674566316654592 \nu^{9} + 6295934218534802948096 \nu^{8} - 15315634714203467546624 \nu^{7} - 389781237566583479468032 \nu^{6} + 1867123901635354704216064 \nu^{5} + 69477314308896695678664704 \nu^{4} - 314432081664694124671926272 \nu^{3} - 9323183097206738444666011648 \nu^{2} + 13772082066450004392007434240 \nu + 1092241701559431861094861242368$$$$)/$$$$12\!\cdots\!80$$ $$\beta_{17}$$ $$=$$ $$($$$$1928675353 \nu^{19} + 20259606880 \nu^{18} - 271949906626 \nu^{17} - 6438475962560 \nu^{16} + 28702041525992 \nu^{15} + 675319926019840 \nu^{14} - 3555965936439232 \nu^{13} - 83091530006128640 \nu^{12} + 770201525198726144 \nu^{11} + 14612706781429596160 \nu^{10} - 74871952312432132096 \nu^{9} - 2180570621320217231360 \nu^{8} + 8022429288002131853312 \nu^{7} + 144769928176286983782400 \nu^{6} - 993229659373183499763712 \nu^{5} - 23907265253581614428979200 \nu^{4} + 164846643310482987035918336 \nu^{3} + 3485923966144483266584903680 \nu^{2} - 7218676868556093773116866560 \nu - 324969136122728633951531827200$$$$)/$$$$64\!\cdots\!40$$ $$\beta_{18}$$ $$=$$ $$($$$$-6530683329 \nu^{19} + 2480965760 \nu^{18} + 1069791510674 \nu^{17} - 349132429568 \nu^{16} - 124056927844648 \nu^{15} + 47322808898560 \nu^{14} + 14248169149748672 \nu^{13} - 2701197529112576 \nu^{12} - 3021452828838052864 \nu^{11} + 1032654779127562240 \nu^{10} + 355666092804088070144 \nu^{9} - 83589104751205154816 \nu^{8} - 31954912260877082165248 \nu^{7} + 9151229864227058483200 \nu^{6} + 4315819175989944403361792 \nu^{5} - 1274894177773332941766656 \nu^{4} - 648306780108513664470876160 \nu^{3} + 258345532936356758888120320 \nu^{2} + 41336340061848975538053447680 \nu - 9080068376390954640304242688$$$$)/$$$$12\!\cdots\!80$$ $$\beta_{19}$$ $$=$$ $$($$$$-9234345027 \nu^{19} - 43570858688 \nu^{18} + 2725068754870 \nu^{17} + 1894292160896 \nu^{16} - 298237616956280 \nu^{15} - 197590658444800 \nu^{14} + 34280229780734272 \nu^{13} + 34980864241553408 \nu^{12} - 6017159139631385600 \nu^{11} - 14407308683941445632 \nu^{10} + 783386341516353667072 \nu^{9} - 95332354979579035648 \nu^{8} - 58892237181535492505600 \nu^{7} - 94511229670833006837760 \nu^{6} + 9878014955444494592376832 \nu^{5} + 19694232317919543016030208 \nu^{4} - 1475153371177541697586331648 \nu^{3} - 2477159695873761139411124224 \nu^{2} + 117742728864315430521234522112 \nu - 159793821601400915737626804224$$$$)/$$$$12\!\cdots\!80$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$-\beta_{4} + \beta_{2} - 3 \beta_{1}$$$$)/16$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{11} + \beta_{8} + 5 \beta_{4} - 2 \beta_{3} - 3 \beta_{1} + 220$$$$)/8$$ $$\nu^{3}$$ $$=$$ $$($$$$-4 \beta_{13} + 2 \beta_{11} - 3 \beta_{10} - 3 \beta_{6} - 7 \beta_{4} + 2 \beta_{3} + 6 \beta_{2} + 57 \beta_{1}$$$$)/4$$ $$\nu^{4}$$ $$=$$ $$($$$$\beta_{18} + 2 \beta_{17} + 2 \beta_{16} - 4 \beta_{15} + 2 \beta_{14} - 8 \beta_{12} + 7 \beta_{11} - 8 \beta_{10} + 15 \beta_{8} + 22 \beta_{7} + 24 \beta_{6} + \beta_{5} - 87 \beta_{4} - 8 \beta_{3} + 2 \beta_{2} + 21 \beta_{1} - 3326$$$$)/2$$ $$\nu^{5}$$ $$=$$ $$($$$$-24 \beta_{19} - 48 \beta_{18} + 28 \beta_{17} - 40 \beta_{16} - 28 \beta_{14} + 76 \beta_{13} - 40 \beta_{12} - 270 \beta_{11} - 111 \beta_{10} + 296 \beta_{9} - 12 \beta_{8} - 68 \beta_{7} - 363 \beta_{6} - 164 \beta_{5} + 1902 \beta_{4} - 234 \beta_{3} - 115 \beta_{2} + 2476 \beta_{1} - 68$$$$)/2$$ $$\nu^{6}$$ $$=$$ $$-323 \beta_{18} - 902 \beta_{17} - 406 \beta_{16} - 572 \beta_{15} - 166 \beta_{14} - 256 \beta_{12} - 732 \beta_{11} - 256 \beta_{10} - 324 \beta_{8} + 62 \beta_{7} + 288 \beta_{6} - 323 \beta_{5} - 20534 \beta_{4} - 7898 \beta_{3} - 646 \beta_{2} + 5326 \beta_{1} - 200026$$ $$\nu^{7}$$ $$=$$ $$3848 \beta_{19} + 656 \beta_{18} + 1324 \beta_{17} + 600 \beta_{16} - 1324 \beta_{14} + 5940 \beta_{13} + 600 \beta_{12} - 6002 \beta_{11} + 4551 \beta_{10} - 29112 \beta_{9} + 1924 \beta_{8} - 724 \beta_{7} - 7365 \beta_{6} - 11572 \beta_{5} + 32400 \beta_{4} - 11774 \beta_{3} + 7837 \beta_{2} + 1165190 \beta_{1} - 724$$ $$\nu^{8}$$ $$=$$ $$17782 \beta_{18} - 99860 \beta_{17} - 79604 \beta_{16} + 27672 \beta_{15} - 43924 \beta_{14} + 15776 \beta_{12} - 11852 \beta_{11} + 15776 \beta_{10} + 25732 \beta_{8} + 162788 \beta_{7} + 32160 \beta_{6} + 17782 \beta_{5} - 505512 \beta_{4} - 464228 \beta_{3} + 35564 \beta_{2} - 132688 \beta_{1} - 151943260$$ $$\nu^{9}$$ $$=$$ $$-243600 \beta_{19} + 1236960 \beta_{18} - 25880 \beta_{17} - 95920 \beta_{16} + 25880 \beta_{14} + 232440 \beta_{13} - 95920 \beta_{12} + 1090292 \beta_{11} + 146394 \beta_{10} - 218640 \beta_{9} - 121800 \beta_{8} - 70040 \beta_{7} + 379314 \beta_{6} - 992472 \beta_{5} - 1338440 \beta_{4} + 1455692 \beta_{3} - 9416986 \beta_{2} + 84517804 \beta_{1} - 70040$$ $$\nu^{10}$$ $$=$$ $$9852580 \beta_{18} + 15045448 \beta_{17} + 1656328 \beta_{16} + 2495376 \beta_{15} - 5780152 \beta_{14} + 1959488 \beta_{12} - 18846552 \beta_{11} + 1959488 \beta_{10} - 12358008 \beta_{8} + 8058264 \beta_{7} + 19606848 \beta_{6} + 9852580 \beta_{5} - 56611328 \beta_{4} - 10136312 \beta_{3} + 19705160 \beta_{2} + 7504400 \beta_{1} - 10579846824$$ $$\nu^{11}$$ $$=$$ $$-67022432 \beta_{19} - 65536704 \beta_{18} - 29816336 \beta_{17} - 3694880 \beta_{16} + 29816336 \beta_{14} + 70316496 \beta_{13} - 3694880 \beta_{12} - 256303432 \beta_{11} + 139220764 \beta_{10} + 158048928 \beta_{9} - 33511216 \beta_{8} + 26121456 \beta_{7} + 407567788 \beta_{6} - 34167440 \beta_{5} + 2579892144 \beta_{4} - 155769784 \beta_{3} - 419012092 \beta_{2} + 2325349544 \beta_{1} + 26121456$$ $$\nu^{12}$$ $$=$$ $$73104344 \beta_{18} - 637727824 \beta_{17} - 805190608 \beta_{16} - 392236192 \beta_{15} - 403831888 \beta_{14} + 32849792 \beta_{12} + 2103356592 \beta_{11} + 32849792 \beta_{10} - 3179780112 \beta_{8} - 2831905392 \beta_{7} + 451491200 \beta_{6} + 73104344 \beta_{5} + 38826374592 \beta_{4} + 10923094192 \beta_{3} + 146208688 \beta_{2} - 12028618720 \beta_{1} - 469614905072$$ $$\nu^{13}$$ $$=$$ $$6520396224 \beta_{19} - 9033642112 \beta_{18} - 5922604128 \beta_{17} + 9182802240 \beta_{16} + 5922604128 \beta_{14} + 11723660512 \beta_{13} + 9182802240 \beta_{12} + 393894736 \beta_{11} - 9900479960 \beta_{10} - 61949509696 \beta_{9} + 3260198112 \beta_{8} + 15105406368 \beta_{7} + 43402957192 \beta_{6} - 5076305760 \beta_{5} + 182632406432 \beta_{4} - 9386699600 \beta_{3} - 31780106472 \beta_{2} - 1505666642064 \beta_{1} + 15105406368$$ $$\nu^{14}$$ $$=$$ $$24231938576 \beta_{18} - 31505920992 \beta_{17} + 10848194336 \beta_{16} + 149303277632 \beta_{15} + 47050722336 \beta_{14} + 11434562816 \beta_{12} + 319804087584 \beta_{11} + 11434562816 \beta_{10} - 284875724640 \beta_{8} - 35929275552 \beta_{7} - 32890533632 \beta_{6} + 24231938576 \beta_{5} + 4074443401600 \beta_{4} + 99304154144 \beta_{3} + 48463877152 \beta_{2} - 1809010635840 \beta_{1} + 175411632106592$$ $$\nu^{15}$$ $$=$$ $$-174302113152 \beta_{19} + 822023900416 \beta_{18} - 454890849856 \beta_{17} + 367739793280 \beta_{16} + 454890849856 \beta_{14} + 2452621046080 \beta_{13} + 367739793280 \beta_{12} + 2110183606752 \beta_{11} - 2563869132816 \beta_{10} + 7412601157248 \beta_{9} - 87151056576 \beta_{8} + 822630643136 \beta_{7} + 1530148515888 \beta_{6} + 1301414176704 \beta_{5} - 27150698605632 \beta_{4} + 2371636776480 \beta_{3} + 9454666023312 \beta_{2} - 66792897881952 \beta_{1} + 822630643136$$ $$\nu^{16}$$ $$=$$ $$-4918911747488 \beta_{18} + 37587778847936 \beta_{17} + 37978188674752 \beta_{16} + 7796442409344 \beta_{15} + 11041562690752 \beta_{14} + 6202293386752 \beta_{12} + 20539589531328 \beta_{11} + 6202293386752 \beta_{10} + 12537051311040 \beta_{8} - 50975189178304 \beta_{7} - 39486266345984 \beta_{6} - 4918911747488 \beta_{5} + 92686337729792 \beta_{4} - 198921912353600 \beta_{3} - 9837823494976 \beta_{2} - 71509713089920 \beta_{1} + 10317444868410944$$ $$\nu^{17}$$ $$=$$ $$53431420690176 \beta_{19} - 277804069933568 \beta_{18} - 24870215769472 \beta_{17} + 51585926114560 \beta_{16} + 24870215769472 \beta_{14} - 352572444762240 \beta_{13} + 51585926114560 \beta_{12} + 199451294969664 \beta_{11} - 63558740850272 \beta_{10} + 961108482782976 \beta_{9} + 26715710345088 \beta_{8} + 76456141884032 \beta_{7} + 160273201074976 \beta_{6} + 478236338747008 \beta_{5} - 880956396210560 \beta_{4} + 119304163934400 \beta_{3} + 770318992803680 \beta_{2} + 19483374884632512 \beta_{1} + 76456141884032$$ $$\nu^{18}$$ $$=$$ $$-3128315548515776 \beta_{18} - 589780755127168 \beta_{17} + 2900855333605504 \beta_{16} - 1201965861172992 \beta_{15} + 2268833268294784 \beta_{14} - 704804121168896 \beta_{12} + 3781037944914048 \beta_{11} - 704804121168896 \beta_{10} + 1779541099585152 \beta_{8} - 2016954556905088 \beta_{7} - 6411052001819648 \beta_{6} - 3128315548515776 \beta_{5} + 30168371570159104 \beta_{4} + 29466442077918336 \beta_{3} - 6256631097031552 \beta_{2} + 5458667128135424 \beta_{1} - 2441351611176587904$$ $$\nu^{19}$$ $$=$$ $$17272326896835072 \beta_{19} - 21953654291803136 \beta_{18} + 6804391957020416 \beta_{17} + 1831771491397120 \beta_{16} - 6804391957020416 \beta_{14} - 3321882336228096 \beta_{13} + 1831771491397120 \beta_{12} + 64115316440535936 \beta_{11} - 24239249195387456 \beta_{10} + 36695471591680512 \beta_{9} + 8636163448417536 \beta_{8} - 4972620465623296 \beta_{7} - 85478776808571200 \beta_{6} + 6732436311652096 \beta_{5} - 226597806998743296 \beta_{4} + 38206826095283328 \beta_{3} - 271403597017403328 \beta_{2} - 2790875940615743872 \beta_{1} - 4972620465623296$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/24\mathbb{Z}\right)^\times$$.

 $$n$$ $$7$$ $$13$$ $$17$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$-1$$

## 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
 10.9006 − 3.02933i 10.9006 + 3.02933i 10.4702 − 4.28654i 10.4702 + 4.28654i 10.0085 − 5.27546i 10.0085 + 5.27546i 6.74480 − 9.08337i 6.74480 + 9.08337i 3.79334 − 10.6588i 3.79334 + 10.6588i −3.79334 − 10.6588i −3.79334 + 10.6588i −6.74480 − 9.08337i −6.74480 + 9.08337i −10.0085 − 5.27546i −10.0085 + 5.27546i −10.4702 − 4.28654i −10.4702 + 4.28654i −10.9006 − 3.02933i −10.9006 + 3.02933i
−10.9006 3.02933i 35.1794 + 30.8125i 109.646 + 66.0430i −65.0827 −290.135 442.445i 90.3177i −995.145 1052.06i 288.180 + 2167.93i 709.441 + 197.157i
11.2 −10.9006 + 3.02933i 35.1794 30.8125i 109.646 66.0430i −65.0827 −290.135 + 442.445i 90.3177i −995.145 + 1052.06i 288.180 2167.93i 709.441 197.157i
11.3 −10.4702 4.28654i −24.1914 40.0222i 91.2511 + 89.7621i 477.520 81.7320 + 522.739i 632.635i −570.650 1330.98i −1016.56 + 1936.38i −4999.74 2046.91i
11.4 −10.4702 + 4.28654i −24.1914 + 40.0222i 91.2511 89.7621i 477.520 81.7320 522.739i 632.635i −570.650 + 1330.98i −1016.56 1936.38i −4999.74 + 2046.91i
11.5 −10.0085 5.27546i −43.1368 + 18.0615i 72.3391 + 105.599i −277.669 527.016 + 46.7976i 1066.27i −166.924 1438.50i 1534.56 1558.23i 2779.04 + 1464.83i
11.6 −10.0085 + 5.27546i −43.1368 18.0615i 72.3391 105.599i −277.669 527.016 46.7976i 1066.27i −166.924 + 1438.50i 1534.56 + 1558.23i 2779.04 1464.83i
11.7 −6.74480 9.08337i 15.4671 44.1335i −37.0154 + 122.531i −320.900 −505.204 + 157.178i 37.9939i 1362.66 490.223i −1708.54 1365.24i 2164.40 + 2914.85i
11.8 −6.74480 + 9.08337i 15.4671 + 44.1335i −37.0154 122.531i −320.900 −505.204 157.178i 37.9939i 1362.66 + 490.223i −1708.54 + 1365.24i 2164.40 2914.85i
11.9 −3.79334 10.6588i 46.6816 + 2.79715i −99.2212 + 80.8651i 395.205 −147.265 508.182i 1395.40i 1238.31 + 750.833i 2171.35 + 261.151i −1499.14 4212.42i
11.10 −3.79334 + 10.6588i 46.6816 2.79715i −99.2212 80.8651i 395.205 −147.265 + 508.182i 1395.40i 1238.31 750.833i 2171.35 261.151i −1499.14 + 4212.42i
11.11 3.79334 10.6588i 46.6816 + 2.79715i −99.2212 80.8651i −395.205 206.894 486.961i 1395.40i −1238.31 + 750.833i 2171.35 + 261.151i −1499.14 + 4212.42i
11.12 3.79334 + 10.6588i 46.6816 2.79715i −99.2212 + 80.8651i −395.205 206.894 + 486.961i 1395.40i −1238.31 750.833i 2171.35 261.151i −1499.14 4212.42i
11.13 6.74480 9.08337i 15.4671 44.1335i −37.0154 122.531i 320.900 −296.559 438.165i 37.9939i −1362.66 490.223i −1708.54 1365.24i 2164.40 2914.85i
11.14 6.74480 + 9.08337i 15.4671 + 44.1335i −37.0154 + 122.531i 320.900 −296.559 + 438.165i 37.9939i −1362.66 + 490.223i −1708.54 + 1365.24i 2164.40 + 2914.85i
11.15 10.0085 5.27546i −43.1368 + 18.0615i 72.3391 105.599i 277.669 −336.450 + 408.335i 1066.27i 166.924 1438.50i 1534.56 1558.23i 2779.04 1464.83i
11.16 10.0085 + 5.27546i −43.1368 18.0615i 72.3391 + 105.599i 277.669 −336.450 408.335i 1066.27i 166.924 + 1438.50i 1534.56 + 1558.23i 2779.04 + 1464.83i
11.17 10.4702 4.28654i −24.1914 40.0222i 91.2511 89.7621i −477.520 −424.846 315.344i 632.635i 570.650 1330.98i −1016.56 + 1936.38i −4999.74 + 2046.91i
11.18 10.4702 + 4.28654i −24.1914 + 40.0222i 91.2511 + 89.7621i −477.520 −424.846 + 315.344i 632.635i 570.650 + 1330.98i −1016.56 1936.38i −4999.74 2046.91i
11.19 10.9006 3.02933i 35.1794 + 30.8125i 109.646 66.0430i 65.0827 476.818 + 229.305i 90.3177i 995.145 1052.06i 288.180 + 2167.93i 709.441 197.157i
11.20 10.9006 + 3.02933i 35.1794 30.8125i 109.646 + 66.0430i 65.0827 476.818 229.305i 90.3177i 995.145 + 1052.06i 288.180 2167.93i 709.441 + 197.157i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 11.20 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
8.d odd 2 1 inner
24.f even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 24.8.f.c 20
3.b odd 2 1 inner 24.8.f.c 20
4.b odd 2 1 96.8.f.c 20
8.b even 2 1 96.8.f.c 20
8.d odd 2 1 inner 24.8.f.c 20
12.b even 2 1 96.8.f.c 20
24.f even 2 1 inner 24.8.f.c 20
24.h odd 2 1 96.8.f.c 20

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
24.8.f.c 20 1.a even 1 1 trivial
24.8.f.c 20 3.b odd 2 1 inner
24.8.f.c 20 8.d odd 2 1 inner
24.8.f.c 20 24.f even 2 1 inner
96.8.f.c 20 4.b odd 2 1
96.8.f.c 20 8.b even 2 1
96.8.f.c 20 12.b even 2 1
96.8.f.c 20 24.h odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{10} - 568524 T_{5}^{8} + 115131638832 T_{5}^{6} -$$$$99\!\cdots\!20$$$$T_{5}^{4} +$$$$32\!\cdots\!00$$$$T_{5}^{2} -$$$$11\!\cdots\!00$$ acting on $$S_{8}^{\mathrm{new}}(24, [\chi])$$.

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$1 - 274 T^{2} + 45864 T^{4} - 5031360 T^{6} + 776389632 T^{8} - 102828146688 T^{10} + 12720367730688 T^{12} - 1350595415900160 T^{14} + 201712005185273856 T^{16} - 19743780766392254464 T^{18} +$$$$11\!\cdots\!24$$$$T^{20}$$
$3$ $$( 1 - 60 T + 531 T^{2} + 25920 T^{3} + 766422 T^{4} - 160875720 T^{5} + 1676164914 T^{6} + 123974556480 T^{7} + 5554447550793 T^{8} - 1372607547297660 T^{9} + 50031545098999707 T^{10} )^{2}$$
$5$ $$( 1 + 212726 T^{2} + 34462341957 T^{4} + 4087236092863080 T^{6} +$$$$39\!\cdots\!50$$$$T^{8} +$$$$34\!\cdots\!00$$$$T^{10} +$$$$24\!\cdots\!50$$$$T^{12} +$$$$15\!\cdots\!00$$$$T^{14} +$$$$78\!\cdots\!25$$$$T^{16} +$$$$29\!\cdots\!50$$$$T^{18} +$$$$84\!\cdots\!25$$$$T^{20} )^{2}$$
$7$ $$( 1 - 4741522 T^{2} + 10982550245901 T^{4} - 16959553029024120120 T^{6} +$$$$19\!\cdots\!94$$$$T^{8} -$$$$18\!\cdots\!80$$$$T^{10} +$$$$13\!\cdots\!06$$$$T^{12} -$$$$78\!\cdots\!20$$$$T^{14} +$$$$34\!\cdots\!49$$$$T^{16} -$$$$10\!\cdots\!22$$$$T^{18} +$$$$14\!\cdots\!49$$$$T^{20} )^{2}$$
$11$ $$( 1 - 64937722 T^{2} + 3455038618503141 T^{4} -$$$$11\!\cdots\!40$$$$T^{6} +$$$$33\!\cdots\!62$$$$T^{8} -$$$$69\!\cdots\!64$$$$T^{10} +$$$$12\!\cdots\!42$$$$T^{12} -$$$$16\!\cdots\!40$$$$T^{14} +$$$$18\!\cdots\!61$$$$T^{16} -$$$$13\!\cdots\!42$$$$T^{18} +$$$$78\!\cdots\!01$$$$T^{20} )^{2}$$
$13$ $$( 1 - 352407394 T^{2} + 62269551776806821 T^{4} -$$$$72\!\cdots\!68$$$$T^{6} +$$$$63\!\cdots\!46$$$$T^{8} -$$$$44\!\cdots\!72$$$$T^{10} +$$$$25\!\cdots\!94$$$$T^{12} -$$$$11\!\cdots\!28$$$$T^{14} +$$$$38\!\cdots\!49$$$$T^{16} -$$$$84\!\cdots\!54$$$$T^{18} +$$$$94\!\cdots\!49$$$$T^{20} )^{2}$$
$17$ $$( 1 - 2730456490 T^{2} + 3432956136635397261 T^{4} -$$$$26\!\cdots\!12$$$$T^{6} +$$$$15\!\cdots\!94$$$$T^{8} -$$$$68\!\cdots\!00$$$$T^{10} +$$$$25\!\cdots\!26$$$$T^{12} -$$$$76\!\cdots\!92$$$$T^{14} +$$$$16\!\cdots\!29$$$$T^{16} -$$$$21\!\cdots\!90$$$$T^{18} +$$$$13\!\cdots\!49$$$$T^{20} )^{2}$$
$19$ $$( 1 - 33436 T + 3303470211 T^{2} - 96285443658240 T^{3} + 5143998080498378646 T^{4} -$$$$12\!\cdots\!92$$$$T^{5} +$$$$45\!\cdots\!94$$$$T^{6} -$$$$76\!\cdots\!40$$$$T^{7} +$$$$23\!\cdots\!09$$$$T^{8} -$$$$21\!\cdots\!76$$$$T^{9} +$$$$57\!\cdots\!99$$$$T^{10} )^{4}$$
$23$ $$( 1 + 14309806406 T^{2} + 96167780623641456861 T^{4} +$$$$42\!\cdots\!20$$$$T^{6} +$$$$14\!\cdots\!66$$$$T^{8} +$$$$48\!\cdots\!52$$$$T^{10} +$$$$17\!\cdots\!94$$$$T^{12} +$$$$57\!\cdots\!20$$$$T^{14} +$$$$14\!\cdots\!69$$$$T^{16} +$$$$25\!\cdots\!66$$$$T^{18} +$$$$20\!\cdots\!49$$$$T^{20} )^{2}$$
$29$ $$( 1 + 73882100390 T^{2} +$$$$28\!\cdots\!49$$$$T^{4} +$$$$79\!\cdots\!96$$$$T^{6} +$$$$17\!\cdots\!74$$$$T^{8} +$$$$33\!\cdots\!40$$$$T^{10} +$$$$53\!\cdots\!94$$$$T^{12} +$$$$70\!\cdots\!56$$$$T^{14} +$$$$74\!\cdots\!09$$$$T^{16} +$$$$57\!\cdots\!90$$$$T^{18} +$$$$23\!\cdots\!01$$$$T^{20} )^{2}$$
$31$ $$( 1 - 102708614914 T^{2} +$$$$77\!\cdots\!29$$$$T^{4} -$$$$38\!\cdots\!20$$$$T^{6} +$$$$15\!\cdots\!74$$$$T^{8} -$$$$46\!\cdots\!60$$$$T^{10} +$$$$11\!\cdots\!54$$$$T^{12} -$$$$21\!\cdots\!20$$$$T^{14} +$$$$33\!\cdots\!69$$$$T^{16} -$$$$33\!\cdots\!34$$$$T^{18} +$$$$24\!\cdots\!01$$$$T^{20} )^{2}$$
$37$ $$( 1 - 483705831538 T^{2} +$$$$12\!\cdots\!41$$$$T^{4} -$$$$20\!\cdots\!60$$$$T^{6} +$$$$27\!\cdots\!66$$$$T^{8} -$$$$29\!\cdots\!00$$$$T^{10} +$$$$24\!\cdots\!74$$$$T^{12} -$$$$16\!\cdots\!60$$$$T^{14} +$$$$88\!\cdots\!29$$$$T^{16} -$$$$31\!\cdots\!58$$$$T^{18} +$$$$59\!\cdots\!49$$$$T^{20} )^{2}$$
$41$ $$( 1 - 1090625824570 T^{2} +$$$$56\!\cdots\!21$$$$T^{4} -$$$$19\!\cdots\!36$$$$T^{6} +$$$$47\!\cdots\!34$$$$T^{8} -$$$$98\!\cdots\!00$$$$T^{10} +$$$$18\!\cdots\!74$$$$T^{12} -$$$$27\!\cdots\!56$$$$T^{14} +$$$$31\!\cdots\!01$$$$T^{16} -$$$$22\!\cdots\!70$$$$T^{18} +$$$$78\!\cdots\!01$$$$T^{20} )^{2}$$
$43$ $$( 1 + 573020 T + 765614977707 T^{2} + 388334673397088640 T^{3} +$$$$29\!\cdots\!86$$$$T^{4} +$$$$14\!\cdots\!20$$$$T^{5} +$$$$80\!\cdots\!02$$$$T^{6} +$$$$28\!\cdots\!60$$$$T^{7} +$$$$15\!\cdots\!01$$$$T^{8} +$$$$31\!\cdots\!20$$$$T^{9} +$$$$14\!\cdots\!07$$$$T^{10} )^{4}$$
$47$ $$( 1 + 2220997832726 T^{2} +$$$$28\!\cdots\!69$$$$T^{4} +$$$$26\!\cdots\!20$$$$T^{6} +$$$$19\!\cdots\!22$$$$T^{8} +$$$$10\!\cdots\!72$$$$T^{10} +$$$$49\!\cdots\!18$$$$T^{12} +$$$$17\!\cdots\!20$$$$T^{14} +$$$$48\!\cdots\!21$$$$T^{16} +$$$$96\!\cdots\!46$$$$T^{18} +$$$$11\!\cdots\!49$$$$T^{20} )^{2}$$
$53$ $$( 1 + 3461178859478 T^{2} +$$$$63\!\cdots\!13$$$$T^{4} +$$$$75\!\cdots\!12$$$$T^{6} +$$$$64\!\cdots\!46$$$$T^{8} +$$$$53\!\cdots\!64$$$$T^{10} +$$$$88\!\cdots\!74$$$$T^{12} +$$$$14\!\cdots\!32$$$$T^{14} +$$$$16\!\cdots\!17$$$$T^{16} +$$$$12\!\cdots\!38$$$$T^{18} +$$$$50\!\cdots\!49$$$$T^{20} )^{2}$$
$59$ $$( 1 - 19732197121882 T^{2} +$$$$18\!\cdots\!29$$$$T^{4} -$$$$10\!\cdots\!80$$$$T^{6} +$$$$43\!\cdots\!06$$$$T^{8} -$$$$12\!\cdots\!44$$$$T^{10} +$$$$26\!\cdots\!66$$$$T^{12} -$$$$41\!\cdots\!80$$$$T^{14} +$$$$43\!\cdots\!49$$$$T^{16} -$$$$29\!\cdots\!62$$$$T^{18} +$$$$91\!\cdots\!01$$$$T^{20} )^{2}$$
$61$ $$( 1 - 8130200545282 T^{2} +$$$$45\!\cdots\!25$$$$T^{4} -$$$$16\!\cdots\!48$$$$T^{6} +$$$$61\!\cdots\!70$$$$T^{8} -$$$$19\!\cdots\!52$$$$T^{10} +$$$$60\!\cdots\!70$$$$T^{12} -$$$$16\!\cdots\!88$$$$T^{14} +$$$$43\!\cdots\!25$$$$T^{16} -$$$$77\!\cdots\!02$$$$T^{18} +$$$$93\!\cdots\!01$$$$T^{20} )^{2}$$
$67$ $$( 1 - 913660 T + 20744505994131 T^{2} - 20152579863740122560 T^{3} +$$$$20\!\cdots\!10$$$$T^{4} -$$$$17\!\cdots\!20$$$$T^{5} +$$$$12\!\cdots\!30$$$$T^{6} -$$$$74\!\cdots\!40$$$$T^{7} +$$$$46\!\cdots\!77$$$$T^{8} -$$$$12\!\cdots\!60$$$$T^{9} +$$$$81\!\cdots\!43$$$$T^{10} )^{4}$$
$71$ $$( 1 + 78176481465446 T^{2} +$$$$28\!\cdots\!33$$$$T^{4} +$$$$63\!\cdots\!56$$$$T^{6} +$$$$95\!\cdots\!42$$$$T^{8} +$$$$10\!\cdots\!48$$$$T^{10} +$$$$78\!\cdots\!02$$$$T^{12} +$$$$43\!\cdots\!16$$$$T^{14} +$$$$16\!\cdots\!53$$$$T^{16} +$$$$36\!\cdots\!66$$$$T^{18} +$$$$38\!\cdots\!01$$$$T^{20} )^{2}$$
$73$ $$( 1 - 4973170 T + 56350864331589 T^{2} -$$$$20\!\cdots\!80$$$$T^{3} +$$$$12\!\cdots\!10$$$$T^{4} -$$$$33\!\cdots\!20$$$$T^{5} +$$$$13\!\cdots\!70$$$$T^{6} -$$$$25\!\cdots\!20$$$$T^{7} +$$$$75\!\cdots\!97$$$$T^{8} -$$$$74\!\cdots\!70$$$$T^{9} +$$$$16\!\cdots\!57$$$$T^{10} )^{4}$$
$79$ $$( 1 - 126095156012386 T^{2} +$$$$68\!\cdots\!69$$$$T^{4} -$$$$20\!\cdots\!20$$$$T^{6} +$$$$42\!\cdots\!54$$$$T^{8} -$$$$77\!\cdots\!60$$$$T^{10} +$$$$15\!\cdots\!74$$$$T^{12} -$$$$28\!\cdots\!20$$$$T^{14} +$$$$34\!\cdots\!29$$$$T^{16} -$$$$23\!\cdots\!06$$$$T^{18} +$$$$68\!\cdots\!01$$$$T^{20} )^{2}$$
$83$ $$( 1 - 179430499509514 T^{2} +$$$$15\!\cdots\!93$$$$T^{4} -$$$$80\!\cdots\!80$$$$T^{6} +$$$$31\!\cdots\!46$$$$T^{8} -$$$$93\!\cdots\!16$$$$T^{10} +$$$$22\!\cdots\!34$$$$T^{12} -$$$$43\!\cdots\!80$$$$T^{14} +$$$$60\!\cdots\!77$$$$T^{16} -$$$$52\!\cdots\!34$$$$T^{18} +$$$$21\!\cdots\!49$$$$T^{20} )^{2}$$
$89$ $$( 1 - 287003084492698 T^{2} +$$$$42\!\cdots\!01$$$$T^{4} -$$$$40\!\cdots\!00$$$$T^{6} +$$$$27\!\cdots\!22$$$$T^{8} -$$$$13\!\cdots\!76$$$$T^{10} +$$$$53\!\cdots\!02$$$$T^{12} -$$$$15\!\cdots\!00$$$$T^{14} +$$$$31\!\cdots\!21$$$$T^{16} -$$$$42\!\cdots\!78$$$$T^{18} +$$$$28\!\cdots\!01$$$$T^{20} )^{2}$$
$97$ $$( 1 - 1390450 T + 285922062427293 T^{2} +$$$$26\!\cdots\!80$$$$T^{3} +$$$$35\!\cdots\!78$$$$T^{4} +$$$$61\!\cdots\!00$$$$T^{5} +$$$$28\!\cdots\!14$$$$T^{6} +$$$$17\!\cdots\!20$$$$T^{7} +$$$$15\!\cdots\!21$$$$T^{8} -$$$$59\!\cdots\!50$$$$T^{9} +$$$$34\!\cdots\!93$$$$T^{10} )^{4}$$