Properties

Label 20.7.b.a
Level 20
Weight 7
Character orbit 20.b
Analytic conductor 4.601
Analytic rank 0
Dimension 12
CM No
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 20 = 2^{2} \cdot 5 \)
Weight: \( k \) = \( 7 \)
Character orbit: \([\chi]\) = 20.b (of order \(2\) and degree \(1\))

Newform invariants

Self dual: No
Analytic conductor: \(4.6010816724\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{33}\cdot 5^{7} \)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \( + ( -1 - \beta_{2} ) q^{2} \) \( + ( -\beta_{2} - \beta_{6} ) q^{3} \) \( + ( 13 + \beta_{2} + \beta_{3} ) q^{4} \) \( + ( -\beta_{1} - \beta_{2} ) q^{5} \) \( + ( -56 + \beta_{2} + \beta_{3} + 2 \beta_{6} + \beta_{10} ) q^{6} \) \( + ( 2 + 7 \beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} - 4 \beta_{6} + \beta_{10} ) q^{7} \) \( + ( 35 - \beta_{1} - 13 \beta_{2} + 3 \beta_{6} + \beta_{7} - 2 \beta_{11} ) q^{8} \) \( + ( -166 - 2 \beta_{1} - 19 \beta_{2} - \beta_{3} - 3 \beta_{5} + 4 \beta_{7} - \beta_{8} + 2 \beta_{9} + 2 \beta_{10} - \beta_{11} ) q^{9} \) \(+O(q^{10})\) \( q\) \( + ( -1 - \beta_{2} ) q^{2} \) \( + ( -\beta_{2} - \beta_{6} ) q^{3} \) \( + ( 13 + \beta_{2} + \beta_{3} ) q^{4} \) \( + ( -\beta_{1} - \beta_{2} ) q^{5} \) \( + ( -56 + \beta_{2} + \beta_{3} + 2 \beta_{6} + \beta_{10} ) q^{6} \) \( + ( 2 + 7 \beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} - 4 \beta_{6} + \beta_{10} ) q^{7} \) \( + ( 35 - \beta_{1} - 13 \beta_{2} + 3 \beta_{6} + \beta_{7} - 2 \beta_{11} ) q^{8} \) \( + ( -166 - 2 \beta_{1} - 19 \beta_{2} - \beta_{3} - 3 \beta_{5} + 4 \beta_{7} - \beta_{8} + 2 \beta_{9} + 2 \beta_{10} - \beta_{11} ) q^{9} \) \( + ( 62 + \beta_{1} - \beta_{2} - \beta_{6} - \beta_{7} - \beta_{8} - \beta_{9} ) q^{10} \) \( + ( -8 - 2 \beta_{1} - 29 \beta_{2} + 7 \beta_{3} + 3 \beta_{4} + 4 \beta_{5} + 17 \beta_{6} + 3 \beta_{7} - 2 \beta_{8} - \beta_{9} + 2 \beta_{11} ) q^{11} \) \( + ( -24 - \beta_{1} + 51 \beta_{2} - 3 \beta_{3} - 2 \beta_{4} - 4 \beta_{5} - 32 \beta_{6} - 5 \beta_{7} + \beta_{8} - 2 \beta_{10} - 5 \beta_{11} ) q^{12} \) \( + ( -419 + 2 \beta_{1} + 31 \beta_{2} + 13 \beta_{3} + 4 \beta_{4} + 3 \beta_{5} + \beta_{8} + 6 \beta_{9} + 2 \beta_{10} + 5 \beta_{11} ) q^{13} \) \( + ( 524 + 10 \beta_{1} - 11 \beta_{2} - 13 \beta_{3} - 8 \beta_{4} - 6 \beta_{5} - 16 \beta_{6} - 2 \beta_{7} + 2 \beta_{8} - 4 \beta_{9} + \beta_{10} - 2 \beta_{11} ) q^{14} \) \( + ( -3 - 2 \beta_{1} - 19 \beta_{2} - 5 \beta_{3} + 9 \beta_{6} - \beta_{7} - \beta_{8} - \beta_{9} + 5 \beta_{10} + 5 \beta_{11} ) q^{15} \) \( + ( 258 - 22 \beta_{1} - 24 \beta_{2} + 18 \beta_{3} - 8 \beta_{4} + 18 \beta_{5} + 40 \beta_{6} - 10 \beta_{7} + 6 \beta_{8} - 10 \beta_{9} - 14 \beta_{10} ) q^{16} \) \( + ( 561 - 12 \beta_{1} - 73 \beta_{2} - 29 \beta_{3} + 8 \beta_{4} + \beta_{5} + 8 \beta_{6} + 12 \beta_{7} + 11 \beta_{8} + 2 \beta_{9} - 6 \beta_{10} + 11 \beta_{11} ) q^{17} \) \( + ( 1301 + 62 \beta_{1} + 155 \beta_{2} + 8 \beta_{3} + 24 \beta_{5} + 58 \beta_{6} - 30 \beta_{7} + 10 \beta_{8} - 6 \beta_{9} - 8 \beta_{10} ) q^{18} \) \( + ( 34 + 6 \beta_{1} + 79 \beta_{2} - 19 \beta_{3} - 5 \beta_{4} - 26 \beta_{5} - 121 \beta_{6} - \beta_{7} - 20 \beta_{8} + 3 \beta_{9} - 12 \beta_{10} + 12 \beta_{11} ) q^{19} \) \( + ( -297 - 7 \beta_{1} - 60 \beta_{2} - 10 \beta_{4} - 10 \beta_{5} + 16 \beta_{6} + \beta_{7} - 9 \beta_{8} + 6 \beta_{9} - 5 \beta_{11} ) q^{20} \) \( + ( -2244 - 18 \beta_{1} + 208 \beta_{2} - 30 \beta_{3} + 8 \beta_{4} + 14 \beta_{5} - 8 \beta_{6} + 16 \beta_{7} + 10 \beta_{8} + 4 \beta_{9} - 4 \beta_{10} + 10 \beta_{11} ) q^{21} \) \( + ( -2188 + 108 \beta_{1} + 58 \beta_{2} + 50 \beta_{3} - 24 \beta_{4} + 12 \beta_{5} + 12 \beta_{6} + 12 \beta_{7} - 16 \beta_{8} + 8 \beta_{9} - 30 \beta_{10} - 24 \beta_{11} ) q^{22} \) \( + ( -40 + 36 \beta_{1} - 235 \beta_{2} - 89 \beta_{3} + 15 \beta_{4} + 13 \beta_{5} + 4 \beta_{6} - 14 \beta_{7} - 18 \beta_{8} + 18 \beta_{9} - 11 \beta_{10} - 22 \beta_{11} ) q^{23} \) \( + ( 2392 - 168 \beta_{1} + 10 \beta_{2} - 62 \beta_{3} - 14 \beta_{5} + 178 \beta_{6} + 10 \beta_{8} - 26 \beta_{9} + 26 \beta_{10} + 16 \beta_{11} ) q^{24} \) \( + 3125 q^{25} \) \( + ( -1488 + 146 \beta_{1} + 458 \beta_{2} - 24 \beta_{3} + 32 \beta_{4} - 56 \beta_{5} + 6 \beta_{6} + 46 \beta_{7} + 22 \beta_{8} - 10 \beta_{9} + 8 \beta_{10} - 16 \beta_{11} ) q^{26} \) \( + ( -4 - 68 \beta_{1} + 288 \beta_{2} + 120 \beta_{3} + 12 \beta_{4} + 18 \beta_{5} + 362 \beta_{6} + 22 \beta_{7} + 44 \beta_{8} - 34 \beta_{9} + 70 \beta_{10} + 36 \beta_{11} ) q^{27} \) \( + ( 1556 - 189 \beta_{1} - 633 \beta_{2} - 15 \beta_{3} + 22 \beta_{4} + 24 \beta_{5} - 220 \beta_{6} - 49 \beta_{7} + \beta_{8} + 36 \beta_{9} + 50 \beta_{10} + 23 \beta_{11} ) q^{28} \) \( + ( -6272 - 12 \beta_{1} - 378 \beta_{2} + 50 \beta_{3} - 32 \beta_{4} - 58 \beta_{5} - 8 \beta_{6} + 8 \beta_{7} - 46 \beta_{8} + 4 \beta_{9} + 36 \beta_{10} - 54 \beta_{11} ) q^{29} \) \( + ( -1056 + 86 \beta_{1} - 53 \beta_{2} - 15 \beta_{3} - 50 \beta_{5} - 292 \beta_{6} - 22 \beta_{7} - 22 \beta_{8} + 28 \beta_{9} + 15 \beta_{10} - 10 \beta_{11} ) q^{30} \) \( + ( 42 - 32 \beta_{1} + 80 \beta_{2} + 244 \beta_{3} - 26 \beta_{4} + 20 \beta_{5} - 336 \beta_{6} + 40 \beta_{7} + 58 \beta_{8} - 16 \beta_{9} - 58 \beta_{10} - 50 \beta_{11} ) q^{31} \) \( + ( -4988 - 352 \beta_{1} - 196 \beta_{2} + 104 \beta_{3} + 40 \beta_{4} + 36 \beta_{5} - 224 \beta_{6} + 72 \beta_{7} - 64 \beta_{8} + 12 \beta_{9} - 4 \beta_{10} - 4 \beta_{11} ) q^{32} \) \( + ( 9281 + 208 \beta_{1} + \beta_{2} + 401 \beta_{3} - 32 \beta_{4} - 13 \beta_{5} - 8 \beta_{6} - 148 \beta_{7} - 79 \beta_{8} + 22 \beta_{9} + 54 \beta_{10} - 39 \beta_{11} ) q^{33} \) \( + ( 3926 + 296 \beta_{1} - 634 \beta_{2} + 72 \beta_{3} + 64 \beta_{4} + 120 \beta_{5} - 496 \beta_{6} + 56 \beta_{7} + 32 \beta_{8} - 16 \beta_{9} + 24 \beta_{10} + 96 \beta_{11} ) q^{34} \) \( + ( -80 + 30 \beta_{1} - 570 \beta_{2} - 175 \beta_{3} + 5 \beta_{4} + 30 \beta_{5} + 10 \beta_{6} - 45 \beta_{7} + 10 \beta_{8} + 15 \beta_{9} + 50 \beta_{10} - 10 \beta_{11} ) q^{35} \) \( + ( -10651 - 594 \beta_{1} - 1161 \beta_{2} - 33 \beta_{3} + 20 \beta_{4} - 204 \beta_{5} - 416 \beta_{6} + 222 \beta_{7} + 50 \beta_{8} + 20 \beta_{9} - 32 \beta_{10} + 10 \beta_{11} ) q^{36} \) \( + ( 4838 - 130 \beta_{1} - 2152 \beta_{2} - 294 \beta_{3} - 32 \beta_{4} - 50 \beta_{5} + 128 \beta_{6} - 72 \beta_{7} + 58 \beta_{8} - 148 \beta_{9} - 116 \beta_{10} - 6 \beta_{11} ) q^{37} \) \( + ( 6052 + 468 \beta_{1} - 124 \beta_{2} - 188 \beta_{3} + 40 \beta_{4} - 92 \beta_{5} + 1176 \beta_{6} + 36 \beta_{7} - 136 \beta_{8} + 136 \beta_{9} + 188 \beta_{10} + 48 \beta_{11} ) q^{38} \) \( + ( 24 + 36 \beta_{1} + 784 \beta_{2} + 212 \beta_{3} - 44 \beta_{4} + 42 \beta_{5} + 338 \beta_{6} + 26 \beta_{7} + 64 \beta_{8} + 18 \beta_{9} - 150 \beta_{10} - 144 \beta_{11} ) q^{39} \) \( + ( 4639 + 23 \beta_{1} + 249 \beta_{2} + 10 \beta_{3} + 40 \beta_{4} - 10 \beta_{5} + 533 \beta_{6} - 87 \beta_{7} - 22 \beta_{8} + 18 \beta_{9} - 10 \beta_{10} + 10 \beta_{11} ) q^{40} \) \( + ( -539 + 26 \beta_{1} + 4119 \beta_{2} - 595 \beta_{3} - 80 \beta_{4} + 183 \beta_{5} - 240 \beta_{6} + 92 \beta_{7} + 29 \beta_{8} - 170 \beta_{9} - 90 \beta_{10} - 99 \beta_{11} ) q^{41} \) \( + ( -11118 + 362 \beta_{1} + 2380 \beta_{2} - 176 \beta_{3} + 64 \beta_{4} + 144 \beta_{5} - 442 \beta_{6} + 22 \beta_{7} + 38 \beta_{8} - 26 \beta_{9} + 16 \beta_{10} + 96 \beta_{11} ) q^{42} \) \( + ( 40 + 192 \beta_{1} - 637 \beta_{2} - 1050 \beta_{3} - 178 \beta_{4} - 154 \beta_{5} - 307 \beta_{6} - 288 \beta_{7} - 76 \beta_{8} + 96 \beta_{9} + 110 \beta_{10} + 76 \beta_{11} ) q^{43} \) \( + ( -18456 - 354 \beta_{1} + 2806 \beta_{2} + 250 \beta_{3} + 92 \beta_{4} + 288 \beta_{5} + 2360 \beta_{6} + 86 \beta_{7} + 90 \beta_{8} + 120 \beta_{9} - 92 \beta_{10} + 6 \beta_{11} ) q^{44} \) \( + ( 6735 + 101 \beta_{1} + 2786 \beta_{2} + 365 \beta_{3} + 20 \beta_{4} + 195 \beta_{5} - 120 \beta_{6} - 160 \beta_{7} - 15 \beta_{8} + 30 \beta_{9} + 10 \beta_{10} + 45 \beta_{11} ) q^{45} \) \( + ( -12380 + 74 \beta_{1} + 643 \beta_{2} + 249 \beta_{3} - 120 \beta_{4} - 182 \beta_{5} + 1876 \beta_{6} + 46 \beta_{7} + 74 \beta_{8} - 148 \beta_{9} - 209 \beta_{10} + 214 \beta_{11} ) q^{46} \) \( + ( 1072 + 52 \beta_{1} + 6287 \beta_{2} + 243 \beta_{3} - 61 \beta_{4} - 495 \beta_{5} - 518 \beta_{6} + 42 \beta_{7} + 14 \beta_{8} + 26 \beta_{9} - 223 \beta_{10} - 134 \beta_{11} ) q^{47} \) \( + ( 24660 + 128 \beta_{1} - 3304 \beta_{2} - 356 \beta_{3} - 104 \beta_{4} - 24 \beta_{5} - 2932 \beta_{6} - 344 \beta_{7} - 244 \beta_{8} - 40 \beta_{9} - 64 \beta_{10} - 12 \beta_{11} ) q^{48} \) \( + ( 5388 + 494 \beta_{1} - 1417 \beta_{2} - 103 \beta_{3} + 144 \beta_{4} + 155 \beta_{5} + 256 \beta_{6} - 212 \beta_{7} + 233 \beta_{8} - 82 \beta_{9} - 226 \beta_{10} + 281 \beta_{11} ) q^{49} \) \( + ( -3125 - 3125 \beta_{2} ) q^{50} \) \( + ( -1506 - 344 \beta_{1} - 7840 \beta_{2} + 1030 \beta_{3} + 60 \beta_{4} + 786 \beta_{5} + 1026 \beta_{6} + 284 \beta_{7} - 46 \beta_{8} - 172 \beta_{9} + 8 \beta_{10} + 278 \beta_{11} ) q^{51} \) \( + ( 46010 + 2 \beta_{1} + 980 \beta_{2} - 340 \beta_{3} - 244 \beta_{4} + 556 \beta_{5} - 544 \beta_{6} + 82 \beta_{7} + 350 \beta_{8} - 52 \beta_{9} - 160 \beta_{10} + 6 \beta_{11} ) q^{52} \) \( + ( 26291 - 654 \beta_{1} - 4123 \beta_{2} + 355 \beta_{3} + 20 \beta_{4} - 267 \beta_{5} + 144 \beta_{6} + 88 \beta_{7} - 89 \beta_{8} + 170 \beta_{9} + 150 \beta_{10} - 21 \beta_{11} ) q^{53} \) \( + ( 13360 - 12 \beta_{1} - 1038 \beta_{2} - 330 \beta_{3} - 96 \beta_{4} + 84 \beta_{5} - 5992 \beta_{6} - 212 \beta_{7} + 12 \beta_{8} + 56 \beta_{9} - 182 \beta_{10} - 476 \beta_{11} ) q^{54} \) \( + ( -853 + 98 \beta_{1} - 5064 \beta_{2} - 80 \beta_{3} - 5 \beta_{4} + 395 \beta_{5} - 111 \beta_{6} + 9 \beta_{7} - 121 \beta_{8} + 49 \beta_{9} - 170 \beta_{10} - 35 \beta_{11} ) q^{55} \) \( + ( -20740 + 564 \beta_{1} - 1410 \beta_{2} + 194 \beta_{3} + 80 \beta_{4} - 966 \beta_{5} - 898 \beta_{6} - 140 \beta_{7} - 86 \beta_{8} - 322 \beta_{9} + 274 \beta_{10} - 32 \beta_{11} ) q^{56} \) \( + ( -103305 + 668 \beta_{1} - 14753 \beta_{2} + 419 \beta_{3} + 72 \beta_{4} - 1215 \beta_{5} + 648 \beta_{6} + 716 \beta_{7} - 245 \beta_{8} + 578 \beta_{9} + 506 \beta_{10} - 117 \beta_{11} ) q^{57} \) \( + ( 29706 - 476 \beta_{1} + 6046 \beta_{2} + 176 \beta_{3} - 256 \beta_{4} + 16 \beta_{5} + 2092 \beta_{6} - 612 \beta_{7} - 52 \beta_{8} - 20 \beta_{9} - 176 \beta_{10} - 256 \beta_{11} ) q^{58} \) \( + ( 1054 - 294 \beta_{1} + 6021 \beta_{2} - 161 \beta_{3} + 465 \beta_{4} - 522 \beta_{5} + 593 \beta_{6} + 49 \beta_{7} + 108 \beta_{8} - 147 \beta_{9} + 808 \beta_{10} + 284 \beta_{11} ) q^{59} \) \( + ( 12988 + 77 \beta_{1} + 1049 \beta_{2} - 145 \beta_{3} + 10 \beta_{4} - 440 \beta_{5} + 2076 \beta_{6} + \beta_{7} + 111 \beta_{8} - 4 \beta_{9} + 270 \beta_{10} + 25 \beta_{11} ) q^{60} \) \( + ( 8060 - 1398 \beta_{1} + 22374 \beta_{2} - 4 \beta_{3} + 248 \beta_{4} + 1276 \beta_{5} - 1168 \beta_{6} + 328 \beta_{7} + 148 \beta_{8} + 312 \beta_{9} + 64 \beta_{10} + 284 \beta_{11} ) q^{61} \) \( + ( -408 - 1812 \beta_{1} + 494 \beta_{2} + 594 \beta_{3} + 208 \beta_{4} + 1180 \beta_{5} + 1376 \beta_{6} + 260 \beta_{7} + 300 \beta_{8} - 264 \beta_{9} + 246 \beta_{10} - 220 \beta_{11} ) q^{62} \) \( + ( 64 - 228 \beta_{1} + 1355 \beta_{2} + 305 \beta_{3} + 737 \beta_{4} - 21 \beta_{5} + 1236 \beta_{6} + 302 \beta_{7} - 166 \beta_{8} - 114 \beta_{9} + 475 \beta_{10} + 206 \beta_{11} ) q^{63} \) \( + ( -45208 + 2336 \beta_{1} + 6272 \beta_{2} + 184 \beta_{3} - 432 \beta_{4} + 368 \beta_{5} + 3816 \beta_{6} - 368 \beta_{7} - 504 \beta_{8} - 304 \beta_{9} - 192 \beta_{10} - 328 \beta_{11} ) q^{64} \) \( + ( -9415 + 144 \beta_{1} + 7579 \beta_{2} - 85 \beta_{3} - 80 \beta_{4} + 145 \beta_{5} - 520 \beta_{6} + 340 \beta_{7} - 165 \beta_{8} + 130 \beta_{9} + 210 \beta_{10} - 205 \beta_{11} ) q^{65} \) \( + ( -6876 - 1884 \beta_{1} - 8960 \beta_{2} + 56 \beta_{3} - 256 \beta_{4} - 1880 \beta_{5} + 4004 \beta_{6} + 316 \beta_{7} - 108 \beta_{8} + 164 \beta_{9} - 56 \beta_{10} - 1024 \beta_{11} ) q^{66} \) \( + ( 2348 + 312 \beta_{1} + 13341 \beta_{2} - 290 \beta_{3} + 42 \beta_{4} - 1358 \beta_{5} - 1237 \beta_{6} + 92 \beta_{7} - 616 \beta_{8} + 156 \beta_{9} - 610 \beta_{10} + 56 \beta_{11} ) q^{67} \) \( + ( -139718 + 1864 \beta_{1} - 2198 \beta_{2} + 1322 \beta_{3} - 16 \beta_{4} + 144 \beta_{5} - 2816 \beta_{6} + 712 \beta_{7} + 248 \beta_{8} + 528 \beta_{9} + 608 \beta_{10} - 264 \beta_{11} ) q^{68} \) \( + ( 892 - 3174 \beta_{1} - 23048 \beta_{2} - 1282 \beta_{3} + 32 \beta_{4} - 1494 \beta_{5} + 888 \beta_{6} + 1176 \beta_{7} - 34 \beta_{8} + 300 \beta_{9} + 268 \beta_{10} - 170 \beta_{11} ) q^{69} \) \( + ( -29820 - 480 \beta_{1} + 275 \beta_{2} + 295 \beta_{3} - 40 \beta_{4} - 640 \beta_{5} - 1910 \beta_{6} - 280 \beta_{7} + 180 \beta_{8} - 160 \beta_{9} - 45 \beta_{10} + 260 \beta_{11} ) q^{70} \) \( + ( -2950 + 580 \beta_{1} - 25946 \beta_{2} - 2178 \beta_{3} + 56 \beta_{4} + 1396 \beta_{5} - 7518 \beta_{6} - 558 \beta_{7} + 110 \beta_{8} + 290 \beta_{9} + 302 \beta_{10} - 422 \beta_{11} ) q^{71} \) \( + ( 193503 + 2899 \beta_{1} + 8283 \beta_{2} + 492 \beta_{3} + 304 \beta_{4} + 2388 \beta_{5} + 211 \beta_{6} - 851 \beta_{7} - 84 \beta_{8} - 612 \beta_{9} + 468 \beta_{10} + 302 \beta_{11} ) q^{72} \) \( + ( -34911 + 2444 \beta_{1} + 2631 \beta_{2} - 1605 \beta_{3} + 320 \beta_{4} + 529 \beta_{5} + 344 \beta_{6} - 28 \beta_{7} + 699 \beta_{8} - 350 \beta_{9} - 670 \beta_{10} + 611 \beta_{11} ) q^{73} \) \( + ( 131784 - 2902 \beta_{1} - 6878 \beta_{2} + 1968 \beta_{3} - 256 \beta_{4} + 656 \beta_{5} - 3930 \beta_{6} + 278 \beta_{7} - 634 \beta_{8} + 166 \beta_{9} + 208 \beta_{10} + 640 \beta_{11} ) q^{74} \) \( + ( -3125 \beta_{2} - 3125 \beta_{6} ) q^{75} \) \( + ( 14760 + 4420 \beta_{1} - 6812 \beta_{2} - 692 \beta_{3} - 152 \beta_{4} - 1752 \beta_{5} + 136 \beta_{6} - 748 \beta_{7} + 452 \beta_{8} + 584 \beta_{9} - 1312 \beta_{10} - 92 \beta_{11} ) q^{76} \) \( + ( -757 - 2288 \beta_{1} - 34605 \beta_{2} + 1587 \beta_{3} - 204 \beta_{4} - 1019 \beta_{5} + 1904 \beta_{6} - 1928 \beta_{7} + 23 \beta_{8} - 726 \beta_{9} - 522 \beta_{10} + 91 \beta_{11} ) q^{77} \) \( + ( 38960 - 3540 \beta_{1} + 1058 \beta_{2} + 382 \beta_{3} + 352 \beta_{4} + 1708 \beta_{5} + 5456 \beta_{6} + 564 \beta_{7} + 692 \beta_{8} - 728 \beta_{9} - 822 \beta_{10} + 252 \beta_{11} ) q^{78} \) \( + ( 906 - 216 \beta_{1} + 9986 \beta_{2} - 6 \beta_{3} - 1300 \beta_{4} - 794 \beta_{5} + 5096 \beta_{6} - 244 \beta_{7} - 330 \beta_{8} - 108 \beta_{9} - 488 \beta_{10} + 898 \beta_{11} ) q^{79} \) \( + ( 60414 + 10 \beta_{1} - 5184 \beta_{2} - 410 \beta_{3} - 160 \beta_{4} - 1110 \beta_{5} + 668 \beta_{6} + 558 \beta_{7} - 102 \beta_{8} - 162 \beta_{9} - 590 \beta_{10} - 20 \beta_{11} ) q^{80} \) \( + ( 199156 + 5698 \beta_{1} + 59559 \beta_{2} + 2517 \beta_{3} - 16 \beta_{4} + 3087 \beta_{5} - 2784 \beta_{6} - 1124 \beta_{7} - 251 \beta_{8} + 118 \beta_{9} + 134 \beta_{10} + 85 \beta_{11} ) q^{81} \) \( + ( -262680 - 2758 \beta_{1} + 3002 \beta_{2} - 3368 \beta_{3} - 640 \beta_{4} + 2632 \beta_{5} - 2706 \beta_{6} - 1114 \beta_{7} - 290 \beta_{8} + 366 \beta_{9} - 152 \beta_{10} + 1088 \beta_{11} ) q^{82} \) \( + ( 396 - 692 \beta_{1} + 9521 \beta_{2} + 5860 \beta_{3} - 1272 \beta_{4} + 726 \beta_{5} + 3015 \beta_{6} + 798 \beta_{7} + 1396 \beta_{8} - 346 \beta_{9} - 1830 \beta_{10} - 1156 \beta_{11} ) q^{83} \) \( + ( -184680 + 1498 \beta_{1} + 12994 \beta_{2} - 1830 \beta_{3} - 36 \beta_{4} - 100 \beta_{5} - 3936 \beta_{6} + 682 \beta_{7} + 262 \beta_{8} + 572 \beta_{9} + 576 \beta_{10} + 238 \beta_{11} ) q^{84} \) \( + ( 34365 - 1254 \beta_{1} + 18011 \beta_{2} - 1615 \beta_{3} - 20 \beta_{4} + 1255 \beta_{5} - 880 \beta_{6} - 40 \beta_{7} + 365 \beta_{8} - 530 \beta_{9} - 510 \beta_{10} + 105 \beta_{11} ) q^{85} \) \( + ( -5504 - 692 \beta_{1} - 2639 \beta_{2} - 2091 \beta_{3} + 1424 \beta_{4} - 3956 \beta_{5} - 3370 \beta_{6} - 1372 \beta_{7} - 388 \beta_{8} + 936 \beta_{9} + 1689 \beta_{10} + 2084 \beta_{11} ) q^{86} \) \( + ( 4232 - 24 \beta_{1} + 35810 \beta_{2} - 2208 \beta_{3} + 104 \beta_{4} - 2148 \beta_{5} + 12686 \beta_{6} - 684 \beta_{7} + 632 \beta_{8} - 12 \beta_{9} + 1508 \beta_{10} + 88 \beta_{11} ) q^{87} \) \( + ( -20488 + 1800 \beta_{1} + 20140 \beta_{2} - 1548 \beta_{3} + 736 \beta_{4} + 900 \beta_{5} - 1844 \beta_{6} + 968 \beta_{7} + 644 \beta_{8} - 1236 \beta_{9} - 2892 \beta_{10} + 224 \beta_{11} ) q^{88} \) \( + ( 18716 + 8160 \beta_{1} - 11702 \beta_{2} + 2538 \beta_{3} - 1008 \beta_{4} - 882 \beta_{5} + 768 \beta_{6} - 2456 \beta_{7} - 838 \beta_{8} - 1140 \beta_{9} - 132 \beta_{10} - 1046 \beta_{11} ) q^{89} \) \( + ( -179892 - 801 \beta_{1} - 4429 \beta_{2} - 2200 \beta_{3} + 160 \beta_{4} - 2040 \beta_{5} + 1081 \beta_{6} + 1121 \beta_{7} - 119 \beta_{8} + 41 \beta_{9} + 200 \beta_{10} - 720 \beta_{11} ) q^{90} \) \( + ( -10118 + 1508 \beta_{1} - 51850 \beta_{2} - 3808 \beta_{3} - 438 \beta_{4} + 4774 \beta_{5} + 8896 \beta_{6} - 974 \beta_{7} - 350 \beta_{8} + 754 \beta_{9} - 752 \beta_{10} - 938 \beta_{11} ) q^{91} \) \( + ( 344228 + 373 \beta_{1} + 6001 \beta_{2} - 473 \beta_{3} + 1402 \beta_{4} + 1568 \beta_{5} - 8732 \beta_{6} + 1481 \beta_{7} - 1073 \beta_{8} + 1892 \beta_{9} - 42 \beta_{10} - 47 \beta_{11} ) q^{92} \) \( + ( -303252 - 8476 \beta_{1} - 13452 \beta_{2} - 3472 \beta_{3} - 536 \beta_{4} - 824 \beta_{5} - 64 \beta_{6} + 1096 \beta_{7} - 72 \beta_{8} - 704 \beta_{9} - 168 \beta_{10} - 832 \beta_{11} ) q^{93} \) \( + ( 391076 - 2622 \beta_{1} - 5503 \beta_{2} - 6529 \beta_{3} + 488 \beta_{4} + 1922 \beta_{5} + 12472 \beta_{6} + 854 \beta_{7} + 386 \beta_{8} - 452 \beta_{9} + 573 \beta_{10} + 350 \beta_{11} ) q^{94} \) \( + ( -3421 - 1114 \beta_{1} - 18348 \beta_{2} + 3440 \beta_{3} - 485 \beta_{4} + 2065 \beta_{5} + 1523 \beta_{6} + 563 \beta_{7} + 703 \beta_{8} - 557 \beta_{9} + 60 \beta_{10} + 405 \beta_{11} ) q^{95} \) \( + ( -224696 - 1640 \beta_{1} - 21240 \beta_{2} + 2224 \beta_{3} - 512 \beta_{4} - 3744 \beta_{5} + 14776 \beta_{6} + 104 \beta_{7} - 1728 \beta_{8} + 992 \beta_{9} + 3168 \beta_{10} + 400 \beta_{11} ) q^{96} \) \( + ( 183847 + 8552 \beta_{1} - 62683 \beta_{2} - 1955 \beta_{3} - 160 \beta_{4} - 4137 \beta_{5} + 3576 \beta_{6} + 956 \beta_{7} - 3 \beta_{8} - 130 \beta_{9} + 30 \beta_{10} - 347 \beta_{11} ) q^{97} \) \( + ( 89845 - 650 \beta_{1} - 6621 \beta_{2} + 2424 \beta_{3} + 1152 \beta_{4} - 1368 \beta_{5} - 10606 \beta_{6} + 4330 \beta_{7} + 194 \beta_{8} + 658 \beta_{9} + 1096 \beta_{10} + 960 \beta_{11} ) q^{98} \) \( + ( 7670 + 2030 \beta_{1} + 29321 \beta_{2} - 2563 \beta_{3} + 1343 \beta_{4} - 3534 \beta_{5} - 19211 \beta_{6} - 269 \beta_{7} - 144 \beta_{8} + 1015 \beta_{9} - 1164 \beta_{10} - 2632 \beta_{11} ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(12q \) \(\mathstrut -\mathstrut 10q^{2} \) \(\mathstrut +\mathstrut 156q^{4} \) \(\mathstrut -\mathstrut 672q^{6} \) \(\mathstrut +\mathstrut 440q^{8} \) \(\mathstrut -\mathstrut 1996q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(12q \) \(\mathstrut -\mathstrut 10q^{2} \) \(\mathstrut +\mathstrut 156q^{4} \) \(\mathstrut -\mathstrut 672q^{6} \) \(\mathstrut +\mathstrut 440q^{8} \) \(\mathstrut -\mathstrut 1996q^{9} \) \(\mathstrut +\mathstrut 750q^{10} \) \(\mathstrut -\mathstrut 440q^{12} \) \(\mathstrut -\mathstrut 5040q^{13} \) \(\mathstrut +\mathstrut 6248q^{14} \) \(\mathstrut +\mathstrut 3312q^{16} \) \(\mathstrut +\mathstrut 6840q^{17} \) \(\mathstrut +\mathstrut 15790q^{18} \) \(\mathstrut -\mathstrut 3500q^{20} \) \(\mathstrut -\mathstrut 27464q^{21} \) \(\mathstrut -\mathstrut 26160q^{22} \) \(\mathstrut +\mathstrut 28528q^{24} \) \(\mathstrut +\mathstrut 37500q^{25} \) \(\mathstrut -\mathstrut 18684q^{26} \) \(\mathstrut +\mathstrut 19320q^{28} \) \(\mathstrut -\mathstrut 74968q^{29} \) \(\mathstrut -\mathstrut 13000q^{30} \) \(\mathstrut -\mathstrut 60800q^{32} \) \(\mathstrut +\mathstrut 112880q^{33} \) \(\mathstrut +\mathstrut 48204q^{34} \) \(\mathstrut -\mathstrut 128580q^{36} \) \(\mathstrut +\mathstrut 62640q^{37} \) \(\mathstrut +\mathstrut 74800q^{38} \) \(\mathstrut +\mathstrut 57000q^{40} \) \(\mathstrut -\mathstrut 16976q^{41} \) \(\mathstrut -\mathstrut 138360q^{42} \) \(\mathstrut -\mathstrut 222160q^{44} \) \(\mathstrut +\mathstrut 77000q^{45} \) \(\mathstrut -\mathstrut 144792q^{46} \) \(\mathstrut +\mathstrut 297600q^{48} \) \(\mathstrut +\mathstrut 72564q^{49} \) \(\mathstrut -\mathstrut 31250q^{50} \) \(\mathstrut +\mathstrut 548280q^{52} \) \(\mathstrut +\mathstrut 322160q^{53} \) \(\mathstrut +\mathstrut 150416q^{54} \) \(\mathstrut -\mathstrut 246512q^{56} \) \(\mathstrut -\mathstrut 1213440q^{57} \) \(\mathstrut +\mathstrut 350700q^{58} \) \(\mathstrut +\mathstrut 157000q^{60} \) \(\mathstrut +\mathstrut 46464q^{61} \) \(\mathstrut -\mathstrut 7120q^{62} \) \(\mathstrut -\mathstrut 542784q^{64} \) \(\mathstrut -\mathstrut 133000q^{65} \) \(\mathstrut -\mathstrut 65200q^{66} \) \(\mathstrut -\mathstrut 1678280q^{68} \) \(\mathstrut +\mathstrut 41256q^{69} \) \(\mathstrut -\mathstrut 360000q^{70} \) \(\mathstrut +\mathstrut 2317560q^{72} \) \(\mathstrut -\mathstrut 415080q^{73} \) \(\mathstrut +\mathstrut 1581924q^{74} \) \(\mathstrut +\mathstrut 208320q^{76} \) \(\mathstrut +\mathstrut 75600q^{77} \) \(\mathstrut +\mathstrut 473200q^{78} \) \(\mathstrut +\mathstrut 734000q^{80} \) \(\mathstrut +\mathstrut 2287428q^{81} \) \(\mathstrut -\mathstrut 3169500q^{82} \) \(\mathstrut -\mathstrut 2256224q^{84} \) \(\mathstrut +\mathstrut 372000q^{85} \) \(\mathstrut -\mathstrut 62512q^{86} \) \(\mathstrut -\mathstrut 278880q^{88} \) \(\mathstrut +\mathstrut 278392q^{89} \) \(\mathstrut -\mathstrut 2162250q^{90} \) \(\mathstrut +\mathstrut 4095720q^{92} \) \(\mathstrut -\mathstrut 3646000q^{93} \) \(\mathstrut +\mathstrut 4706568q^{94} \) \(\mathstrut -\mathstrut 2641152q^{96} \) \(\mathstrut +\mathstrut 2344680q^{97} \) \(\mathstrut +\mathstrut 1050270q^{98} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{12}\mathstrut -\mathstrut \) \(x^{11}\mathstrut -\mathstrut \) \(18\) \(x^{10}\mathstrut -\mathstrut \) \(30\) \(x^{9}\mathstrut +\mathstrut \) \(174\) \(x^{8}\mathstrut +\mathstrut \) \(1853\) \(x^{7}\mathstrut +\mathstrut \) \(10388\) \(x^{6}\mathstrut -\mathstrut \) \(17262\) \(x^{5}\mathstrut -\mathstrut \) \(88626\) \(x^{4}\mathstrut -\mathstrut \) \(240665\) \(x^{3}\mathstrut -\mathstrut \) \(173034\) \(x^{2}\mathstrut +\mathstrut \) \(3703420\) \(x\mathstrut +\mathstrut \) \(7603655\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\(-\)\(229430991652916353\) \(\nu^{11}\mathstrut +\mathstrut \) \(4602523686712292111\) \(\nu^{10}\mathstrut +\mathstrut \) \(17847734682372217233\) \(\nu^{9}\mathstrut -\mathstrut \) \(138295517074660635728\) \(\nu^{8}\mathstrut -\mathstrut \) \(918865866065544137519\) \(\nu^{7}\mathstrut +\mathstrut \) \(2725792408921615354870\) \(\nu^{6}\mathstrut +\mathstrut \) \(13439762669135733009606\) \(\nu^{5}\mathstrut +\mathstrut \) \(40515061354013798218710\) \(\nu^{4}\mathstrut -\mathstrut \) \(39943189723075924235172\) \(\nu^{3}\mathstrut -\mathstrut \) \(989486951546215591280163\) \(\nu^{2}\mathstrut -\mathstrut \) \(2686227666308942885313620\) \(\nu\mathstrut +\mathstrut \) \(6739115216764128780711255\)\()/\)\(18\!\cdots\!60\)
\(\beta_{2}\)\(=\)\((\)\(304451778789345153\) \(\nu^{11}\mathstrut +\mathstrut \) \(99751686729570289\) \(\nu^{10}\mathstrut -\mathstrut \) \(5223693543186182033\) \(\nu^{9}\mathstrut -\mathstrut \) \(19062481820272023472\) \(\nu^{8}\mathstrut +\mathstrut \) \(17616497214921679919\) \(\nu^{7}\mathstrut +\mathstrut \) \(680769074275008357130\) \(\nu^{6}\mathstrut +\mathstrut \) \(4622670121127939604794\) \(\nu^{5}\mathstrut -\mathstrut \) \(1095986237557791082710\) \(\nu^{4}\mathstrut -\mathstrut \) \(36737530065508947809628\) \(\nu^{3}\mathstrut -\mathstrut \) \(145173204708031231139037\) \(\nu^{2}\mathstrut -\mathstrut \) \(209522322886653443646380\) \(\nu\mathstrut +\mathstrut \) \(1300181042601519293992745\)\()/\)\(18\!\cdots\!60\)
\(\beta_{3}\)\(=\)\((\)\(492655598067061225\) \(\nu^{11}\mathstrut -\mathstrut \) \(779141392859147189\) \(\nu^{10}\mathstrut -\mathstrut \) \(11590855480776886617\) \(\nu^{9}\mathstrut -\mathstrut \) \(15167230181595406146\) \(\nu^{8}\mathstrut +\mathstrut \) \(177612753020868385241\) \(\nu^{7}\mathstrut +\mathstrut \) \(1250207873560385509334\) \(\nu^{6}\mathstrut +\mathstrut \) \(3606428560613602341218\) \(\nu^{5}\mathstrut -\mathstrut \) \(25636663894693266283466\) \(\nu^{4}\mathstrut -\mathstrut \) \(73586804181633597066468\) \(\nu^{3}\mathstrut -\mathstrut \) \(25838522809951981526101\) \(\nu^{2}\mathstrut +\mathstrut \) \(678254905293605620659450\) \(\nu\mathstrut +\mathstrut \) \(3109235208383557991128915\)\()/\)\(36\!\cdots\!12\)
\(\beta_{4}\)\(=\)\((\)\(-\)\(14355939228549118447\) \(\nu^{11}\mathstrut +\mathstrut \) \(220815533436541303524\) \(\nu^{10}\mathstrut -\mathstrut \) \(1119834385218488263553\) \(\nu^{9}\mathstrut -\mathstrut \) \(1770674455168369851267\) \(\nu^{8}\mathstrut +\mathstrut \) \(17090617648813939138994\) \(\nu^{7}\mathstrut +\mathstrut \) \(26233100033248331622580\) \(\nu^{6}\mathstrut -\mathstrut \) \(230852553891625044928146\) \(\nu^{5}\mathstrut +\mathstrut \) \(1028930075717030293596900\) \(\nu^{4}\mathstrut -\mathstrut \) \(11935906461210190704966968\) \(\nu^{3}\mathstrut +\mathstrut \) \(15172700604347656856708923\) \(\nu^{2}\mathstrut +\mathstrut \) \(48405256090570333502776245\) \(\nu\mathstrut -\mathstrut \) \(61319593906348688392212700\)\()/\)\(99\!\cdots\!80\)
\(\beta_{5}\)\(=\)\((\)\(19925989943420904579\) \(\nu^{11}\mathstrut -\mathstrut \) \(104574008316173903958\) \(\nu^{10}\mathstrut -\mathstrut \) \(45771293527532439539\) \(\nu^{9}\mathstrut +\mathstrut \) \(197240137050733042169\) \(\nu^{8}\mathstrut +\mathstrut \) \(849801212580941543332\) \(\nu^{7}\mathstrut +\mathstrut \) \(27024367443938499230800\) \(\nu^{6}\mathstrut +\mathstrut \) \(126331078398491499456642\) \(\nu^{5}\mathstrut -\mathstrut \) \(913145548920149025183120\) \(\nu^{4}\mathstrut +\mathstrut \) \(652625494895705364718096\) \(\nu^{3}\mathstrut -\mathstrut \) \(6609002444269178712358751\) \(\nu^{2}\mathstrut -\mathstrut \) \(29119967657986025139013415\) \(\nu\mathstrut +\mathstrut \) \(84029788076530736579963210\)\()/\)\(99\!\cdots\!80\)
\(\beta_{6}\)\(=\)\((\)\(9874148449089948091\) \(\nu^{11}\mathstrut -\mathstrut \) \(47792045919860539935\) \(\nu^{10}\mathstrut -\mathstrut \) \(70559544720015714795\) \(\nu^{9}\mathstrut +\mathstrut \) \(174643737250563046106\) \(\nu^{8}\mathstrut +\mathstrut \) \(1725148477992536125099\) \(\nu^{7}\mathstrut +\mathstrut \) \(10506384299449608662882\) \(\nu^{6}\mathstrut +\mathstrut \) \(55610148790149750133222\) \(\nu^{5}\mathstrut -\mathstrut \) \(427993786499911089266622\) \(\nu^{4}\mathstrut +\mathstrut \) \(126548952988250590182772\) \(\nu^{3}\mathstrut -\mathstrut \) \(2004142628774698084494911\) \(\nu^{2}\mathstrut +\mathstrut \) \(3828086125189702420446030\) \(\nu\mathstrut +\mathstrut \) \(30074924445179950384845305\)\()/\)\(39\!\cdots\!32\)
\(\beta_{7}\)\(=\)\((\)\(-\)\(55090089075737787399\) \(\nu^{11}\mathstrut +\mathstrut \) \(209217448185747312223\) \(\nu^{10}\mathstrut +\mathstrut \) \(1496137379791207598519\) \(\nu^{9}\mathstrut -\mathstrut \) \(8276966371419389315174\) \(\nu^{8}\mathstrut -\mathstrut \) \(13968591595376811661907\) \(\nu^{7}\mathstrut +\mathstrut \) \(19104380607229813383310\) \(\nu^{6}\mathstrut -\mathstrut \) \(158572037641265325867982\) \(\nu^{5}\mathstrut +\mathstrut \) \(1523919404549890265786990\) \(\nu^{4}\mathstrut +\mathstrut \) \(2102484689332632229948204\) \(\nu^{3}\mathstrut -\mathstrut \) \(55332571667484071034838149\) \(\nu^{2}\mathstrut +\mathstrut \) \(38470498365793718360251590\) \(\nu\mathstrut -\mathstrut \) \(99074941301068926035712185\)\()/\)\(19\!\cdots\!60\)
\(\beta_{8}\)\(=\)\((\)\(134515215159071431473\) \(\nu^{11}\mathstrut -\mathstrut \) \(510265244715824507701\) \(\nu^{10}\mathstrut -\mathstrut \) \(921891582565766356193\) \(\nu^{9}\mathstrut +\mathstrut \) \(8810258529214383328198\) \(\nu^{8}\mathstrut +\mathstrut \) \(4157346390418437180169\) \(\nu^{7}\mathstrut -\mathstrut \) \(48768190632458189959690\) \(\nu^{6}\mathstrut +\mathstrut \) \(998320389519415274133714\) \(\nu^{5}\mathstrut -\mathstrut \) \(1605255921159998531890730\) \(\nu^{4}\mathstrut +\mathstrut \) \(11226666748284266790323612\) \(\nu^{3}\mathstrut +\mathstrut \) \(11434172517624091124684483\) \(\nu^{2}\mathstrut -\mathstrut \) \(118159837026695347333866430\) \(\nu\mathstrut -\mathstrut \) \(186483903294652362770468525\)\()/\)\(19\!\cdots\!60\)
\(\beta_{9}\)\(=\)\((\)\(-\)\(6831406141799545928\) \(\nu^{11}\mathstrut +\mathstrut \) \(11898226767942619203\) \(\nu^{10}\mathstrut +\mathstrut \) \(101418314090124938740\) \(\nu^{9}\mathstrut +\mathstrut \) \(287167625737812918837\) \(\nu^{8}\mathstrut -\mathstrut \) \(1710305673626986297485\) \(\nu^{7}\mathstrut -\mathstrut \) \(15404378326265046925966\) \(\nu^{6}\mathstrut -\mathstrut \) \(50931999920944157544184\) \(\nu^{5}\mathstrut +\mathstrut \) \(193574586311925075023118\) \(\nu^{4}\mathstrut +\mathstrut \) \(386157383108126614531156\) \(\nu^{3}\mathstrut +\mathstrut \) \(2274300684284867393775964\) \(\nu^{2}\mathstrut -\mathstrut \) \(5478452150301990459663355\) \(\nu\mathstrut -\mathstrut \) \(29897302327886864290889929\)\()/\)\(49\!\cdots\!04\)
\(\beta_{10}\)\(=\)\((\)\(72447598830897678264\) \(\nu^{11}\mathstrut -\mathstrut \) \(216686329311830291793\) \(\nu^{10}\mathstrut -\mathstrut \) \(300849134817389066624\) \(\nu^{9}\mathstrut -\mathstrut \) \(1733497049912086386311\) \(\nu^{8}\mathstrut +\mathstrut \) \(5953630367627150492687\) \(\nu^{7}\mathstrut +\mathstrut \) \(113738322440727101508130\) \(\nu^{6}\mathstrut +\mathstrut \) \(645458824689634222179772\) \(\nu^{5}\mathstrut -\mathstrut \) \(1858658847327154682250870\) \(\nu^{4}\mathstrut +\mathstrut \) \(2191471038750030876402836\) \(\nu^{3}\mathstrut -\mathstrut \) \(21530186387644625286197496\) \(\nu^{2}\mathstrut +\mathstrut \) \(15576577817869413483925685\) \(\nu\mathstrut +\mathstrut \) \(226301084863597549489901295\)\()/\)\(49\!\cdots\!40\)
\(\beta_{11}\)\(=\)\((\)\(324339192997431574583\) \(\nu^{11}\mathstrut -\mathstrut \) \(691817491392763120011\) \(\nu^{10}\mathstrut -\mathstrut \) \(5443100397964030390503\) \(\nu^{9}\mathstrut -\mathstrut \) \(5225807092980164314622\) \(\nu^{8}\mathstrut +\mathstrut \) \(85844468308166976106119\) \(\nu^{7}\mathstrut +\mathstrut \) \(519884663791896951984970\) \(\nu^{6}\mathstrut +\mathstrut \) \(2277737504189018698565534\) \(\nu^{5}\mathstrut -\mathstrut \) \(9583646082882920221111510\) \(\nu^{4}\mathstrut -\mathstrut \) \(12598447869296603382631228\) \(\nu^{3}\mathstrut -\mathstrut \) \(47648273975845351042775307\) \(\nu^{2}\mathstrut +\mathstrut \) \(83254648987909539614052870\) \(\nu\mathstrut +\mathstrut \) \(721530890819273355936635885\)\()/\)\(19\!\cdots\!60\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(25\) \(\beta_{6}\mathstrut -\mathstrut \) \(25\) \(\beta_{5}\mathstrut -\mathstrut \) \(59\) \(\beta_{2}\mathstrut +\mathstrut \) \(16\) \(\beta_{1}\mathstrut +\mathstrut \) \(50\)\()/800\)
\(\nu^{2}\)\(=\)\((\)\(20\) \(\beta_{11}\mathstrut +\mathstrut \) \(35\) \(\beta_{10}\mathstrut +\mathstrut \) \(39\) \(\beta_{9}\mathstrut -\mathstrut \) \(31\) \(\beta_{8}\mathstrut -\mathstrut \) \(26\) \(\beta_{7}\mathstrut +\mathstrut \) \(14\) \(\beta_{6}\mathstrut -\mathstrut \) \(30\) \(\beta_{5}\mathstrut +\mathstrut \) \(20\) \(\beta_{4}\mathstrut -\mathstrut \) \(35\) \(\beta_{3}\mathstrut -\mathstrut \) \(326\) \(\beta_{2}\mathstrut +\mathstrut \) \(46\) \(\beta_{1}\mathstrut +\mathstrut \) \(2402\)\()/800\)
\(\nu^{3}\)\(=\)\((\)\(130\) \(\beta_{11}\mathstrut -\mathstrut \) \(30\) \(\beta_{10}\mathstrut +\mathstrut \) \(8\) \(\beta_{9}\mathstrut -\mathstrut \) \(112\) \(\beta_{8}\mathstrut -\mathstrut \) \(82\) \(\beta_{7}\mathstrut +\mathstrut \) \(3\) \(\beta_{6}\mathstrut -\mathstrut \) \(255\) \(\beta_{5}\mathstrut -\mathstrut \) \(130\) \(\beta_{4}\mathstrut -\mathstrut \) \(470\) \(\beta_{3}\mathstrut -\mathstrut \) \(435\) \(\beta_{2}\mathstrut +\mathstrut \) \(104\) \(\beta_{1}\mathstrut +\mathstrut \) \(9644\)\()/800\)
\(\nu^{4}\)\(=\)\((\)\(75\) \(\beta_{11}\mathstrut +\mathstrut \) \(190\) \(\beta_{10}\mathstrut -\mathstrut \) \(16\) \(\beta_{9}\mathstrut -\mathstrut \) \(661\) \(\beta_{8}\mathstrut -\mathstrut \) \(706\) \(\beta_{7}\mathstrut +\mathstrut \) \(1719\) \(\beta_{6}\mathstrut -\mathstrut \) \(880\) \(\beta_{5}\mathstrut +\mathstrut \) \(170\) \(\beta_{4}\mathstrut -\mathstrut \) \(4015\) \(\beta_{3}\mathstrut +\mathstrut \) \(9062\) \(\beta_{2}\mathstrut +\mathstrut \) \(84\) \(\beta_{1}\mathstrut +\mathstrut \) \(11297\)\()/800\)
\(\nu^{5}\)\(=\)\((\)\(25\) \(\beta_{11}\mathstrut +\mathstrut \) \(35\) \(\beta_{10}\mathstrut +\mathstrut \) \(81\) \(\beta_{9}\mathstrut -\mathstrut \) \(344\) \(\beta_{8}\mathstrut -\mathstrut \) \(264\) \(\beta_{7}\mathstrut +\mathstrut \) \(261\) \(\beta_{6}\mathstrut -\mathstrut \) \(360\) \(\beta_{5}\mathstrut -\mathstrut \) \(250\) \(\beta_{4}\mathstrut -\mathstrut \) \(510\) \(\beta_{3}\mathstrut +\mathstrut \) \(12322\) \(\beta_{2}\mathstrut -\mathstrut \) \(970\) \(\beta_{1}\mathstrut -\mathstrut \) \(34887\)\()/80\)
\(\nu^{6}\)\(=\)\((\)\(-\)\(6110\) \(\beta_{11}\mathstrut -\mathstrut \) \(535\) \(\beta_{10}\mathstrut -\mathstrut \) \(12585\) \(\beta_{9}\mathstrut -\mathstrut \) \(14935\) \(\beta_{8}\mathstrut -\mathstrut \) \(22360\) \(\beta_{7}\mathstrut -\mathstrut \) \(15530\) \(\beta_{6}\mathstrut +\mathstrut \) \(8550\) \(\beta_{5}\mathstrut -\mathstrut \) \(9850\) \(\beta_{4}\mathstrut -\mathstrut \) \(44165\) \(\beta_{3}\mathstrut +\mathstrut \) \(223164\) \(\beta_{2}\mathstrut -\mathstrut \) \(26746\) \(\beta_{1}\mathstrut -\mathstrut \) \(4568500\)\()/800\)
\(\nu^{7}\)\(=\)\((\)\(-\)\(52365\) \(\beta_{11}\mathstrut +\mathstrut \) \(4835\) \(\beta_{10}\mathstrut -\mathstrut \) \(49969\) \(\beta_{9}\mathstrut -\mathstrut \) \(24294\) \(\beta_{8}\mathstrut -\mathstrut \) \(27494\) \(\beta_{7}\mathstrut -\mathstrut \) \(252199\) \(\beta_{6}\mathstrut +\mathstrut \) \(162800\) \(\beta_{5}\mathstrut -\mathstrut \) \(40500\) \(\beta_{4}\mathstrut -\mathstrut \) \(18410\) \(\beta_{3}\mathstrut +\mathstrut \) \(2286916\) \(\beta_{2}\mathstrut -\mathstrut \) \(523966\) \(\beta_{1}\mathstrut -\mathstrut \) \(9376277\)\()/800\)
\(\nu^{8}\)\(=\)\((\)\(-\)\(493935\) \(\beta_{11}\mathstrut -\mathstrut \) \(237015\) \(\beta_{10}\mathstrut -\mathstrut \) \(582083\) \(\beta_{9}\mathstrut +\mathstrut \) \(252982\) \(\beta_{8}\mathstrut -\mathstrut \) \(150178\) \(\beta_{7}\mathstrut -\mathstrut \) \(1243298\) \(\beta_{6}\mathstrut +\mathstrut \) \(750485\) \(\beta_{5}\mathstrut -\mathstrut \) \(320140\) \(\beta_{4}\mathstrut +\mathstrut \) \(856090\) \(\beta_{3}\mathstrut +\mathstrut \) \(6768875\) \(\beta_{2}\mathstrut -\mathstrut \) \(1562394\) \(\beta_{1}\mathstrut -\mathstrut \) \(99485809\)\()/800\)
\(\nu^{9}\)\(=\)\((\)\(-\)\(2625715\) \(\beta_{11}\mathstrut +\mathstrut \) \(603075\) \(\beta_{10}\mathstrut -\mathstrut \) \(1436079\) \(\beta_{9}\mathstrut +\mathstrut \) \(1908076\) \(\beta_{8}\mathstrut +\mathstrut \) \(931856\) \(\beta_{7}\mathstrut -\mathstrut \) \(9032474\) \(\beta_{6}\mathstrut +\mathstrut \) \(6353195\) \(\beta_{5}\mathstrut +\mathstrut \) \(324470\) \(\beta_{4}\mathstrut +\mathstrut \) \(8804550\) \(\beta_{3}\mathstrut +\mathstrut \) \(6260723\) \(\beta_{2}\mathstrut -\mathstrut \) \(8575874\) \(\beta_{1}\mathstrut -\mathstrut \) \(275264777\)\()/800\)
\(\nu^{10}\)\(=\)\((\)\(-\)\(459070\) \(\beta_{11}\mathstrut -\mathstrut \) \(165605\) \(\beta_{10}\mathstrut -\mathstrut \) \(460161\) \(\beta_{9}\mathstrut +\mathstrut \) \(868579\) \(\beta_{8}\mathstrut +\mathstrut \) \(536734\) \(\beta_{7}\mathstrut -\mathstrut \) \(2864076\) \(\beta_{6}\mathstrut +\mathstrut \) \(1287630\) \(\beta_{5}\mathstrut -\mathstrut \) \(100620\) \(\beta_{4}\mathstrut +\mathstrut \) \(3313535\) \(\beta_{3}\mathstrut -\mathstrut \) \(6375902\) \(\beta_{2}\mathstrut -\mathstrut \) \(1046790\) \(\beta_{1}\mathstrut -\mathstrut \) \(24994968\)\()/40\)
\(\nu^{11}\)\(=\)\((\)\(-\)\(34579650\) \(\beta_{11}\mathstrut +\mathstrut \) \(5947895\) \(\beta_{10}\mathstrut -\mathstrut \) \(19832475\) \(\beta_{9}\mathstrut +\mathstrut \) \(99242815\) \(\beta_{8}\mathstrut +\mathstrut \) \(69907980\) \(\beta_{7}\mathstrut -\mathstrut \) \(173934145\) \(\beta_{6}\mathstrut +\mathstrut \) \(108232785\) \(\beta_{5}\mathstrut +\mathstrut \) \(42971110\) \(\beta_{4}\mathstrut +\mathstrut \) \(272354705\) \(\beta_{3}\mathstrut -\mathstrut \) \(1668083059\) \(\beta_{2}\mathstrut +\mathstrut \) \(45093586\) \(\beta_{1}\mathstrut +\mathstrut \) \(4234554190\)\()/800\)

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
−2.35434 + 0.469325i
−2.35434 0.469325i
−4.07586 + 2.01083i
−4.07586 2.01083i
−0.552397 + 3.35995i
−0.552397 3.35995i
−0.601180 + 3.99996i
−0.601180 3.99996i
4.83379 + 2.37885i
4.83379 2.37885i
3.24999 + 1.01902i
3.24999 1.01902i
−7.94474 0.938649i 50.7798i 62.2379 + 14.9147i −55.9017 −47.6645 + 403.433i 91.7717i −480.464 176.913i −1849.59 444.125 + 52.4721i
11.2 −7.94474 + 0.938649i 50.7798i 62.2379 14.9147i −55.9017 −47.6645 403.433i 91.7717i −480.464 + 176.913i −1849.59 444.125 52.4721i
11.3 −6.91565 4.02166i 5.41240i 31.6525 + 55.6248i 55.9017 21.7668 37.4303i 454.127i 4.80679 511.977i 699.706 −386.597 224.818i
11.4 −6.91565 + 4.02166i 5.41240i 31.6525 55.6248i 55.9017 21.7668 + 37.4303i 454.127i 4.80679 + 511.977i 699.706 −386.597 + 224.818i
11.5 −4.34086 6.71989i 5.31460i −26.3138 + 58.3402i −55.9017 35.7136 23.0700i 502.023i 506.265 76.4207i 700.755 242.662 + 375.653i
11.6 −4.34086 + 6.71989i 5.31460i −26.3138 58.3402i −55.9017 35.7136 + 23.0700i 502.023i 506.265 + 76.4207i 700.755 242.662 375.653i
11.7 0.0337085 7.99993i 38.3717i −63.9977 0.539332i 55.9017 −306.971 1.29345i 111.263i −6.47188 + 511.959i −743.391 1.88436 447.210i
11.8 0.0337085 + 7.99993i 38.3717i −63.9977 + 0.539332i 55.9017 −306.971 + 1.29345i 111.263i −6.47188 511.959i −743.391 1.88436 + 447.210i
11.9 6.43150 4.75771i 20.5795i 18.7284 61.1984i −55.9017 −97.9113 132.357i 24.2619i −170.712 482.702i 305.483 −359.532 + 265.964i
11.10 6.43150 + 4.75771i 20.5795i 18.7284 + 61.1984i −55.9017 −97.9113 + 132.357i 24.2619i −170.712 + 482.702i 305.483 −359.532 265.964i
11.11 7.73604 2.03804i 28.9821i 55.6928 31.5327i 55.9017 59.0666 + 224.207i 435.848i 366.577 357.443i −110.960 432.458 113.930i
11.12 7.73604 + 2.03804i 28.9821i 55.6928 + 31.5327i 55.9017 59.0666 224.207i 435.848i 366.577 + 357.443i −110.960 432.458 + 113.930i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 11.12
Significant digits:
Format:

Inner twists

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

Hecke kernels

There are no other newforms in \(S_{7}^{\mathrm{new}}(20, [\chi])\).