# 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

# Related objects

## 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: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

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

## Hecke kernels

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