Properties

 Label 20.9.d.c Level 20 Weight 9 Character orbit 20.d Analytic conductor 8.148 Analytic rank 0 Dimension 20 CM No Inner twists 4

Related objects

Newspace parameters

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

Newform invariants

 Self dual: No Analytic conductor: $$8.14757220122$$ Analytic rank: $$0$$ Dimension: $$20$$ Coefficient field: $$\mathbb{Q}[x]/(x^{20} + \cdots)$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{74}\cdot 3^{4}\cdot 5^{12}$$ 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.

 $$f(q)$$ $$=$$ $$q$$ $$+ \beta_{1} q^{2}$$ $$+ ( \beta_{1} + \beta_{6} ) q^{3}$$ $$+ ( -38 + \beta_{2} ) q^{4}$$ $$+ ( -71 + \beta_{1} + \beta_{7} ) q^{5}$$ $$+ ( 170 + \beta_{2} + \beta_{8} ) q^{6}$$ $$+ ( 3 \beta_{1} + 3 \beta_{6} - \beta_{13} ) q^{7}$$ $$+ ( -38 \beta_{1} + \beta_{3} ) q^{8}$$ $$+ ( 130 - \beta_{1} - 5 \beta_{2} - 3 \beta_{7} + 2 \beta_{8} - \beta_{9} + \beta_{11} ) q^{9}$$ $$+O(q^{10})$$ $$q$$ $$+ \beta_{1} q^{2}$$ $$+ ( \beta_{1} + \beta_{6} ) q^{3}$$ $$+ ( -38 + \beta_{2} ) q^{4}$$ $$+ ( -71 + \beta_{1} + \beta_{7} ) q^{5}$$ $$+ ( 170 + \beta_{2} + \beta_{8} ) q^{6}$$ $$+ ( 3 \beta_{1} + 3 \beta_{6} - \beta_{13} ) q^{7}$$ $$+ ( -38 \beta_{1} + \beta_{3} ) q^{8}$$ $$+ ( 130 - \beta_{1} - 5 \beta_{2} - 3 \beta_{7} + 2 \beta_{8} - \beta_{9} + \beta_{11} ) q^{9}$$ $$+ ( -208 - 68 \beta_{1} - \beta_{4} + 7 \beta_{6} - \beta_{7} + \beta_{9} + \beta_{15} ) q^{10}$$ $$+ ( 1 - 2 \beta_{2} - \beta_{8} + \beta_{12} ) q^{11}$$ $$+ ( 139 \beta_{1} + \beta_{3} + \beta_{5} - 96 \beta_{6} + 2 \beta_{7} - \beta_{13} - \beta_{18} - \beta_{19} ) q^{12}$$ $$+ ( 88 \beta_{1} - \beta_{2} - \beta_{5} - 2 \beta_{6} + 4 \beta_{7} + \beta_{9} - \beta_{10} + \beta_{11} - \beta_{13} + \beta_{15} + \beta_{17} + \beta_{18} - \beta_{19} ) q^{13}$$ $$+ ( 442 - 3 \beta_{1} + 2 \beta_{2} - 2 \beta_{4} - 7 \beta_{7} + 3 \beta_{8} - \beta_{9} + 2 \beta_{11} + \beta_{12} - \beta_{15} + \beta_{16} + \beta_{18} ) q^{14}$$ $$+ ( -8 + 55 \beta_{1} + 19 \beta_{2} + 5 \beta_{3} + 7 \beta_{5} - 216 \beta_{6} - \beta_{7} + 7 \beta_{8} - 3 \beta_{13} + \beta_{15} + \beta_{16} + \beta_{17} - \beta_{18} + \beta_{19} ) q^{15}$$ $$+ ( -2943 + \beta_{1} - 40 \beta_{2} - \beta_{3} - 3 \beta_{4} - \beta_{5} + \beta_{6} - \beta_{7} + \beta_{8} + 3 \beta_{9} - \beta_{11} + 3 \beta_{12} + \beta_{13} + \beta_{14} - 2 \beta_{15} - \beta_{16} - \beta_{17} + \beta_{18} - \beta_{19} ) q^{16}$$ $$+ ( -845 \beta_{1} - 3 \beta_{2} + 8 \beta_{3} + 17 \beta_{5} + 26 \beta_{6} - 13 \beta_{7} + 3 \beta_{9} - 3 \beta_{10} + 2 \beta_{11} - 7 \beta_{13} + 7 \beta_{15} - \beta_{17} + 7 \beta_{18} + \beta_{19} ) q^{17}$$ $$+ ( 293 \beta_{1} + \beta_{2} - 5 \beta_{3} - 32 \beta_{5} + 445 \beta_{6} + 29 \beta_{7} - \beta_{9} + \beta_{10} + 2 \beta_{11} - 3 \beta_{13} - \beta_{14} - 5 \beta_{15} + \beta_{17} - 4 \beta_{18} - 3 \beta_{19} ) q^{18}$$ $$+ ( -19 + 4 \beta_{1} + 47 \beta_{2} + 2 \beta_{3} + 2 \beta_{5} - 2 \beta_{6} - 8 \beta_{7} + 21 \beta_{8} + \beta_{9} - \beta_{10} - 3 \beta_{12} - 2 \beta_{13} - 2 \beta_{14} + 8 \beta_{15} - 2 \beta_{16} + 2 \beta_{17} - 6 \beta_{18} + 2 \beta_{19} ) q^{19}$$ $$+ ( -179 - 275 \beta_{1} - 70 \beta_{2} - 9 \beta_{4} - 3 \beta_{5} - 169 \beta_{6} - 37 \beta_{7} + 7 \beta_{8} - 10 \beta_{9} + 9 \beta_{10} + \beta_{11} - 5 \beta_{12} - 10 \beta_{13} + 3 \beta_{14} - 2 \beta_{15} - \beta_{16} - 3 \beta_{17} - 8 \beta_{18} + 2 \beta_{19} ) q^{20}$$ $$+ ( 20468 - 7 \beta_{1} + 112 \beta_{2} + 28 \beta_{4} + 21 \beta_{7} + 28 \beta_{8} - 28 \beta_{9} + 7 \beta_{11} ) q^{21}$$ $$+ ( 83 \beta_{1} + 5 \beta_{2} - 6 \beta_{3} + \beta_{5} + 248 \beta_{6} + 38 \beta_{7} - 5 \beta_{9} + 5 \beta_{10} + 4 \beta_{11} + 26 \beta_{13} - 4 \beta_{14} - 6 \beta_{15} - 12 \beta_{17} - 10 \beta_{18} - 4 \beta_{19} ) q^{22}$$ $$+ ( 375 \beta_{1} - \beta_{2} + 12 \beta_{3} - 4 \beta_{5} + 839 \beta_{6} + 14 \beta_{7} + \beta_{9} - \beta_{10} - \beta_{13} - 2 \beta_{14} - 14 \beta_{15} + 12 \beta_{17} + 16 \beta_{18} + 12 \beta_{19} ) q^{23}$$ $$+ ( -7897 + 51 \beta_{1} + 148 \beta_{2} - 3 \beta_{3} - 21 \beta_{4} - 3 \beta_{5} + 3 \beta_{6} + 205 \beta_{7} - 105 \beta_{8} + 37 \beta_{9} - 55 \beta_{11} - 3 \beta_{12} + 3 \beta_{13} + 3 \beta_{14} - 10 \beta_{15} + \beta_{16} - 3 \beta_{17} + 7 \beta_{18} - 3 \beta_{19} ) q^{24}$$ $$+ ( -32918 - 754 \beta_{1} + 114 \beta_{2} - 40 \beta_{3} + 36 \beta_{4} + 9 \beta_{5} + 10 \beta_{6} - 52 \beta_{7} - 46 \beta_{8} - 11 \beta_{10} + \beta_{11} - 3 \beta_{13} + 8 \beta_{14} + 19 \beta_{15} + 3 \beta_{17} + 11 \beta_{18} + 13 \beta_{19} ) q^{25}$$ $$+ ( -22052 + 3 \beta_{1} + 38 \beta_{2} - 3 \beta_{3} - 22 \beta_{4} - 3 \beta_{5} + 3 \beta_{6} - 123 \beta_{7} + 20 \beta_{8} + 72 \beta_{9} - 6 \beta_{10} - 6 \beta_{11} - 16 \beta_{12} + 3 \beta_{13} + 3 \beta_{14} - 9 \beta_{15} - 3 \beta_{17} + 6 \beta_{18} - 3 \beta_{19} ) q^{26}$$ $$+ ( 6834 \beta_{1} - 6 \beta_{2} - 72 \beta_{3} + 132 \beta_{5} - 1242 \beta_{6} + 12 \beta_{7} + 6 \beta_{9} - 6 \beta_{10} - 42 \beta_{13} - 12 \beta_{14} - 12 \beta_{15} + 24 \beta_{18} ) q^{27}$$ $$+ ( 639 \beta_{1} + 14 \beta_{2} - 21 \beta_{3} - 147 \beta_{5} + 744 \beta_{6} - 182 \beta_{7} - 14 \beta_{9} + 14 \beta_{10} - 56 \beta_{11} + 221 \beta_{13} - 28 \beta_{15} - 35 \beta_{18} - 7 \beta_{19} ) q^{28}$$ $$+ ( 33232 - 8 \beta_{1} - 602 \beta_{2} + 100 \beta_{4} + 4 \beta_{7} - 56 \beta_{8} - 70 \beta_{9} - 48 \beta_{10} + 8 \beta_{11} ) q^{29}$$ $$+ ( 33374 - 658 \beta_{1} + 89 \beta_{2} - 10 \beta_{3} - 26 \beta_{4} - 59 \beta_{5} - 1948 \beta_{6} + 213 \beta_{7} - 229 \beta_{8} + 28 \beta_{9} + 53 \beta_{10} + 22 \beta_{11} + 15 \beta_{12} + 182 \beta_{13} + 16 \beta_{14} - 25 \beta_{15} - \beta_{16} - 27 \beta_{18} ) q^{30}$$ $$+ ( 318 - 20 \beta_{1} - 733 \beta_{2} + 14 \beta_{3} - 32 \beta_{4} + 14 \beta_{5} - 14 \beta_{6} + 56 \beta_{7} - 576 \beta_{8} - 57 \beta_{9} + 57 \beta_{10} + 32 \beta_{11} - 18 \beta_{12} - 14 \beta_{13} - 14 \beta_{14} + 40 \beta_{15} + 2 \beta_{16} + 14 \beta_{17} - 26 \beta_{18} + 14 \beta_{19} ) q^{31}$$ $$+ ( -2916 \beta_{1} - 24 \beta_{3} + 28 \beta_{5} + 280 \beta_{6} - 528 \beta_{7} - 112 \beta_{11} - 108 \beta_{13} - 24 \beta_{14} - 8 \beta_{15} - 40 \beta_{17} - 4 \beta_{18} - 28 \beta_{19} ) q^{32}$$ $$+ ( -6167 \beta_{1} - 19 \beta_{2} - 72 \beta_{3} + 89 \beta_{5} + 154 \beta_{6} - 87 \beta_{7} + 19 \beta_{9} - 19 \beta_{10} + 16 \beta_{11} - 23 \beta_{13} + 16 \beta_{14} + 55 \beta_{15} + 15 \beta_{17} + 39 \beta_{18} + 17 \beta_{19} ) q^{33}$$ $$+ ( 217476 - 250 \beta_{1} - 752 \beta_{2} - 10 \beta_{3} - 72 \beta_{4} - 10 \beta_{5} + 10 \beta_{6} - 1006 \beta_{7} + 44 \beta_{8} - 160 \beta_{9} - 20 \beta_{10} + 256 \beta_{11} + 10 \beta_{13} + 10 \beta_{14} - 14 \beta_{15} - 16 \beta_{16} - 10 \beta_{17} + 4 \beta_{18} - 10 \beta_{19} ) q^{34}$$ $$+ ( 453 - 8987 \beta_{1} - 1187 \beta_{2} + 140 \beta_{3} + 32 \beta_{4} - 140 \beta_{5} + 653 \beta_{6} - 98 \beta_{7} + 715 \beta_{8} + 121 \beta_{9} + 7 \beta_{10} - 32 \beta_{11} + 5 \beta_{12} + 4 \beta_{13} + 14 \beta_{14} + 2 \beta_{15} - 16 \beta_{16} + 28 \beta_{17} - 16 \beta_{18} + 28 \beta_{19} ) q^{35}$$ $$+ ( -359632 + 214 \beta_{1} + 35 \beta_{2} + 6 \beta_{3} - 6 \beta_{4} + 6 \beta_{5} - 6 \beta_{6} + 690 \beta_{7} + 490 \beta_{8} + 178 \beta_{9} - 48 \beta_{10} - 226 \beta_{11} - 6 \beta_{12} - 6 \beta_{13} - 6 \beta_{14} + 18 \beta_{16} + 6 \beta_{17} + 6 \beta_{18} + 6 \beta_{19} ) q^{36}$$ $$+ ( -3737 \beta_{1} - 34 \beta_{2} + 272 \beta_{3} + 134 \beta_{5} + 188 \beta_{6} + 67 \beta_{7} + 34 \beta_{9} - 34 \beta_{10} + 25 \beta_{11} - 90 \beta_{13} - 32 \beta_{14} + 26 \beta_{15} - 22 \beta_{17} + 58 \beta_{18} - 42 \beta_{19} ) q^{37}$$ $$+ ( -1479 \beta_{1} + 23 \beta_{2} + 86 \beta_{3} - 45 \beta_{5} - 5440 \beta_{6} + 1194 \beta_{7} - 23 \beta_{9} + 23 \beta_{10} + 252 \beta_{11} - 618 \beta_{13} + 28 \beta_{14} + 22 \beta_{15} - 12 \beta_{17} - 46 \beta_{18} + 28 \beta_{19} ) q^{38}$$ $$+ ( -2282 + 5620 \beta_{2} + 32 \beta_{4} - 64 \beta_{7} + 714 \beta_{8} - 64 \beta_{9} - 192 \beta_{10} - 32 \beta_{11} + 54 \beta_{12} - 32 \beta_{15} + 32 \beta_{16} + 32 \beta_{18} ) q^{39}$$ $$+ ( -47105 + 3265 \beta_{1} - 264 \beta_{2} - 20 \beta_{3} - 13 \beta_{4} - 123 \beta_{5} + 10123 \beta_{6} + 237 \beta_{7} - 273 \beta_{8} - 95 \beta_{9} + 148 \beta_{10} - 223 \beta_{11} + 45 \beta_{12} - 377 \beta_{13} + 11 \beta_{14} - 10 \beta_{15} + 17 \beta_{16} + 53 \beta_{17} + 35 \beta_{18} - 7 \beta_{19} ) q^{40}$$ $$+ ( 352371 - 133 \beta_{1} + 2949 \beta_{2} - 224 \beta_{4} + 133 \beta_{7} + 1022 \beta_{8} - 159 \beta_{9} + 240 \beta_{10} + 133 \beta_{11} ) q^{41}$$ $$+ ( 23807 \beta_{1} - 70 \beta_{2} + 91 \beta_{3} + 343 \beta_{5} + 8827 \beta_{6} + 2429 \beta_{7} + 70 \beta_{9} - 70 \beta_{10} + 490 \beta_{11} + 133 \beta_{13} - 35 \beta_{14} + 35 \beta_{15} + 35 \beta_{17} + 98 \beta_{18} - 77 \beta_{19} ) q^{42}$$ $$+ ( 14513 \beta_{1} + 24 \beta_{2} + 176 \beta_{3} + 532 \beta_{5} - 11379 \beta_{6} - 104 \beta_{7} - 24 \beta_{9} + 24 \beta_{10} - 82 \beta_{13} + 48 \beta_{14} + 104 \beta_{15} - 56 \beta_{17} - 152 \beta_{18} - 56 \beta_{19} ) q^{43}$$ $$+ ( 133176 + 448 \beta_{1} + 592 \beta_{2} + 24 \beta_{3} - 8 \beta_{4} + 24 \beta_{5} - 24 \beta_{6} + 2576 \beta_{7} + 40 \beta_{8} - 480 \beta_{9} - 192 \beta_{10} - 408 \beta_{11} + 16 \beta_{12} - 24 \beta_{13} - 24 \beta_{14} + 88 \beta_{15} - 16 \beta_{16} + 24 \beta_{17} - 64 \beta_{18} + 24 \beta_{19} ) q^{44}$$ $$+ ( -937923 + 44987 \beta_{1} + 3577 \beta_{2} - 80 \beta_{3} - 132 \beta_{4} - 713 \beta_{5} - 1250 \beta_{6} - 37 \beta_{7} - 648 \beta_{8} + 455 \beta_{9} + 127 \beta_{10} - 37 \beta_{11} + 111 \beta_{13} - 16 \beta_{14} - 143 \beta_{15} + 49 \beta_{17} - 127 \beta_{18} - 81 \beta_{19} ) q^{45}$$ $$+ ( 27906 - 581 \beta_{1} + 72 \beta_{2} + 16 \beta_{3} + 362 \beta_{4} + 16 \beta_{5} - 16 \beta_{6} - 3841 \beta_{7} + 1143 \beta_{8} + 289 \beta_{9} - 128 \beta_{10} + 598 \beta_{11} + 159 \beta_{12} - 16 \beta_{13} - 16 \beta_{14} + 49 \beta_{15} - \beta_{16} + 16 \beta_{17} - 33 \beta_{18} + 16 \beta_{19} ) q^{46}$$ $$+ ( -32457 \beta_{1} - 5 \beta_{2} - 212 \beta_{3} - 952 \beta_{5} + 14363 \beta_{6} - 66 \beta_{7} + 5 \beta_{9} - 5 \beta_{10} + 73 \beta_{13} - 10 \beta_{14} + 66 \beta_{15} - 76 \beta_{17} - 56 \beta_{18} - 76 \beta_{19} ) q^{47}$$ $$+ ( -10188 \beta_{1} - 104 \beta_{2} + 140 \beta_{3} + 220 \beta_{5} - 4376 \beta_{6} - 4448 \beta_{7} + 104 \beta_{9} - 104 \beta_{10} - 688 \beta_{11} + 316 \beta_{13} + 56 \beta_{14} + 312 \beta_{15} + 8 \beta_{17} + 244 \beta_{18} + 92 \beta_{19} ) q^{48}$$ $$+ ( 591934 + 287 \beta_{1} - 15001 \beta_{2} - 616 \beta_{4} + 21 \beta_{7} - 798 \beta_{8} + 259 \beta_{9} - 448 \beta_{10} - 287 \beta_{11} ) q^{49}$$ $$+ ( 196276 - 37341 \beta_{1} - 1125 \beta_{2} + 135 \beta_{3} + 30 \beta_{4} + 798 \beta_{5} - 11113 \beta_{6} - 561 \beta_{7} + 28 \beta_{8} - 241 \beta_{9} + 391 \beta_{10} + 1334 \beta_{11} - 160 \beta_{12} + 225 \beta_{13} - 73 \beta_{14} + 141 \beta_{15} + 16 \beta_{16} + 73 \beta_{17} + 168 \beta_{18} + 93 \beta_{19} ) q^{50}$$ $$+ ( 5178 - 220 \beta_{1} - 12459 \beta_{2} - 126 \beta_{3} - 128 \beta_{4} - 126 \beta_{5} + 126 \beta_{6} + 728 \beta_{7} - 4376 \beta_{8} - 191 \beta_{9} + 447 \beta_{10} + 128 \beta_{11} + 138 \beta_{12} + 126 \beta_{13} + 126 \beta_{14} - 344 \beta_{15} - 34 \beta_{16} - 126 \beta_{17} + 218 \beta_{18} - 126 \beta_{19} ) q^{51}$$ $$+ ( -40592 \beta_{1} - 30 \beta_{2} + 174 \beta_{3} + 1016 \beta_{5} - 51872 \beta_{6} - 5048 \beta_{7} + 30 \beta_{9} - 30 \beta_{10} - 984 \beta_{11} - 402 \beta_{13} + 56 \beta_{14} - 28 \beta_{15} + 200 \beta_{17} + 166 \beta_{18} + 162 \beta_{19} ) q^{52}$$ $$+ ( -93116 \beta_{1} + 265 \beta_{2} + 112 \beta_{3} + 1121 \beta_{5} + 2514 \beta_{6} + 472 \beta_{7} - 265 \beta_{9} + 265 \beta_{10} - 165 \beta_{11} + 369 \beta_{13} - 64 \beta_{14} - 497 \beta_{15} - 161 \beta_{17} - 433 \beta_{18} + 33 \beta_{19} ) q^{53}$$ $$+ ( 1851972 - 642 \beta_{1} + 8472 \beta_{2} + 96 \beta_{3} + 516 \beta_{4} + 96 \beta_{5} - 96 \beta_{6} - 3354 \beta_{7} - 1314 \beta_{8} - 678 \beta_{9} - 768 \beta_{10} + 636 \beta_{11} - 90 \beta_{12} - 96 \beta_{13} - 96 \beta_{14} + 186 \beta_{15} + 102 \beta_{16} + 96 \beta_{17} - 90 \beta_{18} + 96 \beta_{19} ) q^{54}$$ $$+ ( 5646 - 58416 \beta_{1} - 14522 \beta_{2} - 665 \beta_{3} + 96 \beta_{4} - 1443 \beta_{5} + 6463 \beta_{6} - 205 \beta_{7} + 2187 \beta_{8} + 723 \beta_{9} + 301 \beta_{10} - 96 \beta_{11} - 90 \beta_{12} + 514 \beta_{13} - 38 \beta_{14} - 83 \beta_{15} + 103 \beta_{16} - 165 \beta_{17} + 121 \beta_{18} - 165 \beta_{19} ) q^{55}$$ $$+ ( -1752559 + 1685 \beta_{1} + 188 \beta_{2} - 21 \beta_{3} + 1037 \beta_{4} - 21 \beta_{5} + 21 \beta_{6} + 6923 \beta_{7} + 737 \beta_{8} + 403 \beta_{9} - 1569 \beta_{11} - 165 \beta_{12} + 21 \beta_{13} + 21 \beta_{14} + 74 \beta_{15} - 137 \beta_{16} - 21 \beta_{17} - 95 \beta_{18} - 21 \beta_{19} ) q^{56}$$ $$+ ( 139367 \beta_{1} + 379 \beta_{2} - 1256 \beta_{3} - 2537 \beta_{5} - 4298 \beta_{6} - 121 \beta_{7} - 379 \beta_{9} + 379 \beta_{10} - 400 \beta_{11} + 807 \beta_{13} + 80 \beta_{14} - 647 \beta_{15} + 49 \beta_{17} - 727 \beta_{18} + 111 \beta_{19} ) q^{57}$$ $$+ ( 28396 \beta_{1} - 206 \beta_{2} - 760 \beta_{3} - 2362 \beta_{5} - 19804 \beta_{6} + 6840 \beta_{7} + 206 \beta_{9} - 206 \beta_{10} + 1472 \beta_{11} + 648 \beta_{13} - 108 \beta_{14} + 88 \beta_{15} + 108 \beta_{17} + 540 \beta_{18} + 20 \beta_{19} ) q^{58}$$ $$+ ( -14165 + 156 \beta_{1} + 35305 \beta_{2} - 114 \beta_{3} + 64 \beta_{4} - 114 \beta_{5} + 114 \beta_{6} - 56 \beta_{7} + 547 \beta_{8} - 953 \beta_{9} - 1095 \beta_{10} - 64 \beta_{11} - 421 \beta_{12} + 114 \beta_{13} + 114 \beta_{14} - 136 \beta_{15} - 206 \beta_{16} - 114 \beta_{17} + 22 \beta_{18} - 114 \beta_{19} ) q^{59}$$ $$+ ( -1136176 + 47199 \beta_{1} - 1454 \beta_{2} + 215 \beta_{3} - 80 \beta_{4} + 3861 \beta_{5} + 43288 \beta_{6} + 1090 \beta_{7} - 2144 \beta_{8} + 1174 \beta_{9} + 866 \beta_{10} - 1976 \beta_{11} - 120 \beta_{12} + 433 \beta_{13} - 128 \beta_{14} + 180 \beta_{15} - 120 \beta_{16} - 128 \beta_{17} - 71 \beta_{18} + 117 \beta_{19} ) q^{60}$$ $$+ ( -890108 + 577 \beta_{1} + 17610 \beta_{2} - 616 \beta_{4} + 1233 \beta_{7} - 1036 \beta_{8} + 1482 \beta_{9} + 656 \beta_{10} - 577 \beta_{11} ) q^{61}$$ $$+ ( 47912 \beta_{1} + 208 \beta_{2} - 696 \beta_{3} + 808 \beta_{5} + 141544 \beta_{6} + 5400 \beta_{7} - 208 \beta_{9} + 208 \beta_{10} + 1264 \beta_{11} - 344 \beta_{13} + 248 \beta_{14} + 8 \beta_{15} + 72 \beta_{17} + 32 \beta_{18} + 696 \beta_{19} ) q^{62}$$ $$+ ( 246933 \beta_{1} - 105 \beta_{2} - 1428 \beta_{3} + 2940 \beta_{5} + 69189 \beta_{6} + 126 \beta_{7} + 105 \beta_{9} - 105 \beta_{10} + 1185 \beta_{13} - 210 \beta_{14} - 126 \beta_{15} - 84 \beta_{17} + 336 \beta_{18} - 84 \beta_{19} ) q^{63}$$ $$+ ( 1010200 + 232 \beta_{1} - 112 \beta_{2} - 72 \beta_{3} + 888 \beta_{4} - 72 \beta_{5} + 72 \beta_{6} + 4952 \beta_{7} - 168 \beta_{8} - 3032 \beta_{9} - 1664 \beta_{10} - 408 \beta_{11} + 72 \beta_{12} + 72 \beta_{13} + 72 \beta_{14} - 320 \beta_{15} + 104 \beta_{16} - 72 \beta_{17} + 248 \beta_{18} - 72 \beta_{19} ) q^{64}$$ $$+ ( -2223027 + 151863 \beta_{1} + 32023 \beta_{2} + 1360 \beta_{3} + 172 \beta_{4} - 1562 \beta_{5} - 3860 \beta_{6} + 397 \beta_{7} - 942 \beta_{8} + 1355 \beta_{9} + 998 \beta_{10} - 1043 \beta_{11} - 406 \beta_{13} - 104 \beta_{14} + 198 \beta_{15} - 434 \beta_{17} + 302 \beta_{18} + 226 \beta_{19} ) q^{65}$$ $$+ ( 1597892 - 1872 \beta_{1} - 6580 \beta_{2} + 36 \beta_{3} - 212 \beta_{4} + 36 \beta_{5} - 36 \beta_{6} - 7640 \beta_{7} + 300 \beta_{8} - 80 \beta_{9} + 840 \beta_{10} + 1892 \beta_{11} - 480 \beta_{12} - 36 \beta_{13} - 36 \beta_{14} + 92 \beta_{15} + 16 \beta_{16} + 36 \beta_{17} - 56 \beta_{18} + 36 \beta_{19} ) q^{66}$$ $$+ ( -230967 \beta_{1} + 204 \beta_{2} + 3072 \beta_{3} - 2028 \beta_{5} - 75531 \beta_{6} - 96 \beta_{7} - 204 \beta_{9} + 204 \beta_{10} + 1778 \beta_{13} + 408 \beta_{14} + 96 \beta_{15} + 312 \beta_{17} - 504 \beta_{18} + 312 \beta_{19} ) q^{67}$$ $$+ ( 254704 \beta_{1} + 284 \beta_{2} - 316 \beta_{3} - 9760 \beta_{5} + 104512 \beta_{6} - 3440 \beta_{7} - 284 \beta_{9} + 284 \beta_{10} - 1488 \beta_{11} - 3660 \beta_{13} - 368 \beta_{14} - 1416 \beta_{15} + 112 \beta_{17} - 444 \beta_{18} - 244 \beta_{19} ) q^{68}$$ $$+ ( 5608826 - 1275 \beta_{1} - 53250 \beta_{2} + 596 \beta_{4} - 3531 \beta_{7} + 2248 \beta_{8} - 3198 \beta_{9} - 1776 \beta_{10} + 1275 \beta_{11} ) q^{69}$$ $$+ ( -2353614 - 54039 \beta_{1} - 11684 \beta_{2} - 910 \beta_{3} - 76 \beta_{4} + 6965 \beta_{5} - 161144 \beta_{6} - 896 \beta_{7} + 1325 \beta_{8} + 1677 \beta_{9} + 889 \beta_{10} + 1056 \beta_{11} + 570 \beta_{12} - 3662 \beta_{13} + 28 \beta_{14} - 136 \beta_{15} - 102 \beta_{16} - 364 \beta_{17} - 312 \beta_{18} - 644 \beta_{19} ) q^{70}$$ $$+ ( 4124 + 372 \beta_{1} - 11647 \beta_{2} + 354 \beta_{3} + 256 \beta_{4} + 354 \beta_{5} - 354 \beta_{6} - 1592 \beta_{7} + 10230 \beta_{8} + 1585 \beta_{9} + 207 \beta_{10} - 256 \beta_{11} - 564 \beta_{12} - 354 \beta_{13} - 354 \beta_{14} + 824 \beta_{15} + 238 \beta_{16} + 354 \beta_{17} - 470 \beta_{18} + 354 \beta_{19} ) q^{71}$$ $$+ ( -431690 \beta_{1} - 88 \beta_{2} - 481 \beta_{3} + 1088 \beta_{5} - 212272 \beta_{6} - 4496 \beta_{7} + 88 \beta_{9} - 88 \beta_{10} - 512 \beta_{11} + 3096 \beta_{13} + 400 \beta_{14} + 992 \beta_{15} - 16 \beta_{17} - 344 \beta_{18} - 120 \beta_{19} ) q^{72}$$ $$+ ( -849651 \beta_{1} - 987 \beta_{2} + 1528 \beta_{3} + 12273 \beta_{5} + 23882 \beta_{6} - 1075 \beta_{7} + 987 \beta_{9} - 987 \beta_{10} + 156 \beta_{11} - 1391 \beta_{13} - 144 \beta_{14} + 1103 \beta_{15} + 583 \beta_{17} + 1247 \beta_{18} - 871 \beta_{19} ) q^{73}$$ $$+ ( 956728 - 181 \beta_{1} - 2054 \beta_{2} - 303 \beta_{3} - 1266 \beta_{4} - 303 \beta_{5} + 303 \beta_{6} - 3771 \beta_{7} + 584 \beta_{8} - 664 \beta_{9} - 2142 \beta_{10} + 166 \beta_{11} + 576 \beta_{12} + 303 \beta_{13} + 303 \beta_{14} - 621 \beta_{15} - 288 \beta_{16} - 303 \beta_{17} + 318 \beta_{18} - 303 \beta_{19} ) q^{74}$$ $$+ ( 23602 - 337283 \beta_{1} - 57503 \beta_{2} - 460 \beta_{3} - 352 \beta_{4} - 4880 \beta_{5} - 85903 \beta_{6} + 738 \beta_{7} - 9090 \beta_{8} + 539 \beta_{9} + 1893 \beta_{10} + 352 \beta_{11} + 690 \beta_{12} - 3894 \beta_{13} - 54 \beta_{14} + 318 \beta_{15} - 304 \beta_{16} - 68 \beta_{17} - 264 \beta_{18} - 68 \beta_{19} ) q^{75}$$ $$+ ( -2746824 - 1040 \beta_{1} - 4944 \beta_{2} + 184 \beta_{3} - 4552 \beta_{4} + 184 \beta_{5} - 184 \beta_{6} - 704 \beta_{7} - 5944 \beta_{8} + 1200 \beta_{9} + 2368 \beta_{10} + 680 \beta_{11} + 960 \beta_{12} - 184 \beta_{13} - 184 \beta_{14} + 8 \beta_{15} + 544 \beta_{16} + 184 \beta_{17} + 176 \beta_{18} + 184 \beta_{19} ) q^{76}$$ $$+ ( 323435 \beta_{1} - 973 \beta_{2} - 1232 \beta_{3} - 3605 \beta_{5} - 8666 \beta_{6} + 2443 \beta_{7} + 973 \beta_{9} - 973 \beta_{10} + 1904 \beta_{11} - 1477 \beta_{13} + 448 \beta_{14} + 2373 \beta_{15} + 469 \beta_{17} + 1925 \beta_{18} + 427 \beta_{19} ) q^{77}$$ $$+ ( -70462 \beta_{1} + 334 \beta_{2} + 4284 \beta_{3} - 18698 \beta_{5} - 214576 \beta_{6} - 1020 \beta_{7} - 334 \beta_{9} + 334 \beta_{10} - 808 \beta_{11} + 7676 \beta_{13} - 88 \beta_{14} - 580 \beta_{15} - 264 \beta_{17} - 1308 \beta_{18} - 728 \beta_{19} ) q^{78}$$ $$+ ( -31830 - 564 \beta_{1} + 80683 \beta_{2} + 462 \beta_{3} - 416 \beta_{4} + 462 \beta_{5} - 462 \beta_{6} + 472 \beta_{7} - 6156 \beta_{8} - 3033 \beta_{9} - 1831 \beta_{10} + 416 \beta_{11} + 1818 \beta_{12} - 462 \beta_{13} - 462 \beta_{14} + 776 \beta_{15} + 610 \beta_{16} + 462 \beta_{17} - 314 \beta_{18} + 462 \beta_{19} ) q^{79}$$ $$+ ( 397431 + 41483 \beta_{1} + 848 \beta_{2} - 835 \beta_{3} + 795 \beta_{4} + 9973 \beta_{5} + 278639 \beta_{6} - 4519 \beta_{7} + 11063 \beta_{8} + 1677 \beta_{9} + 88 \beta_{10} + 1817 \beta_{11} + 5 \beta_{12} + 5219 \beta_{13} + 191 \beta_{14} - 390 \beta_{15} + 425 \beta_{16} - 639 \beta_{17} + 587 \beta_{18} - 779 \beta_{19} ) q^{80}$$ $$+ ( -7708326 - 1893 \beta_{1} - 7143 \beta_{2} + 4104 \beta_{4} - 6867 \beta_{7} - 1506 \beta_{8} - 2595 \beta_{9} - 1296 \beta_{10} + 1893 \beta_{11} ) q^{81}$$ $$+ ( 450926 \beta_{1} + 315 \beta_{2} + 3403 \beta_{3} + 7606 \beta_{5} + 298873 \beta_{6} - 4403 \beta_{7} - 315 \beta_{9} + 315 \beta_{10} - 1806 \beta_{11} - 2499 \beta_{13} + 91 \beta_{14} - 357 \beta_{15} - 91 \beta_{17} - 2408 \beta_{18} - 1687 \beta_{19} ) q^{82}$$ $$+ ( 1549713 \beta_{1} + 36 \beta_{2} + 2240 \beta_{3} + 30572 \beta_{5} - 31579 \beta_{6} + 832 \beta_{7} - 36 \beta_{9} + 36 \beta_{10} - 5598 \beta_{13} + 72 \beta_{14} - 832 \beta_{15} + 904 \beta_{17} + 760 \beta_{18} + 904 \beta_{19} ) q^{83}$$ $$+ ( 5054574 + 1946 \beta_{1} + 23156 \beta_{2} + 210 \beta_{3} - 4914 \beta_{4} + 210 \beta_{5} - 210 \beta_{6} - 9394 \beta_{7} + 9646 \beta_{8} + 6286 \beta_{9} - 1680 \beta_{10} - 1414 \beta_{11} - 714 \beta_{12} - 210 \beta_{13} - 210 \beta_{14} + 952 \beta_{15} - 322 \beta_{16} + 210 \beta_{17} - 742 \beta_{18} + 210 \beta_{19} ) q^{84}$$ $$+ ( 3428970 + 1345571 \beta_{1} + 30203 \beta_{2} - 1120 \beta_{3} + 752 \beta_{4} - 18407 \beta_{5} - 36750 \beta_{6} - 4145 \beta_{7} + 14428 \beta_{8} - 4655 \beta_{9} + 1353 \beta_{10} + 7202 \beta_{11} - 671 \beta_{13} + 176 \beta_{14} + 1023 \beta_{15} + 911 \beta_{17} + 847 \beta_{18} - 559 \beta_{19} ) q^{85}$$ $$+ ( 4715178 + 4114 \beta_{1} + 18875 \beta_{2} - 384 \beta_{3} - 3796 \beta_{4} - 384 \beta_{5} + 384 \beta_{6} + 27178 \beta_{7} - 14443 \beta_{8} + 614 \beta_{9} + 3072 \beta_{10} - 4396 \beta_{11} - 230 \beta_{12} + 384 \beta_{13} + 384 \beta_{14} - 1050 \beta_{15} - 102 \beta_{16} - 384 \beta_{17} + 666 \beta_{18} - 384 \beta_{19} ) q^{86}$$ $$+ ( -433490 \beta_{1} - 378 \beta_{2} - 3704 \beta_{3} - 11668 \beta_{5} + 131850 \beta_{6} + 1172 \beta_{7} + 378 \beta_{9} - 378 \beta_{10} - 11004 \beta_{13} - 756 \beta_{14} - 1172 \beta_{15} + 416 \beta_{17} + 1928 \beta_{18} + 416 \beta_{19} ) q^{87}$$ $$+ ( 205088 \beta_{1} - 560 \beta_{2} - 568 \beta_{3} - 35600 \beta_{5} + 167040 \beta_{6} + 17184 \beta_{7} + 560 \beta_{9} - 560 \beta_{10} + 4928 \beta_{11} - 3840 \beta_{13} + 576 \beta_{14} + 3360 \beta_{15} - 1088 \beta_{17} - 192 \beta_{18} - 736 \beta_{19} ) q^{88}$$ $$+ ( 5253568 + 3716 \beta_{1} - 9306 \beta_{2} + 5504 \beta_{4} + 24856 \beta_{7} + 772 \beta_{8} - 4626 \beta_{9} - 1488 \beta_{10} - 3716 \beta_{11} ) q^{89}$$ $$+ ( -11557684 - 990995 \beta_{1} + 39560 \beta_{2} + 4215 \beta_{3} + 501 \beta_{4} + 19209 \beta_{5} - 149916 \beta_{6} + 4038 \beta_{7} - 236 \beta_{8} + 5851 \beta_{9} - 132 \beta_{10} - 10378 \beta_{11} - 240 \beta_{12} - 135 \beta_{13} + 141 \beta_{14} - 858 \beta_{15} + 288 \beta_{16} - 141 \beta_{17} + 294 \beta_{18} + 1419 \beta_{19} ) q^{90}$$ $$+ ( -1612 + 2428 \beta_{1} + 1431 \beta_{2} + 14 \beta_{3} + 1600 \beta_{4} + 14 \beta_{5} - 14 \beta_{6} - 5656 \beta_{7} + 34938 \beta_{8} + 4615 \beta_{9} - 1799 \beta_{10} - 1600 \beta_{11} + 1188 \beta_{12} - 14 \beta_{13} - 14 \beta_{14} + 856 \beta_{15} - 814 \beta_{16} + 14 \beta_{17} - 842 \beta_{18} + 14 \beta_{19} ) q^{91}$$ $$+ ( -164489 \beta_{1} + 2002 \beta_{2} - 9 \beta_{3} + 16501 \beta_{5} - 571048 \beta_{6} + 37370 \beta_{7} - 2002 \beta_{9} + 2002 \beta_{10} + 4536 \beta_{11} + 3425 \beta_{13} - 1408 \beta_{14} - 5668 \beta_{15} - 1152 \beta_{17} - 2847 \beta_{18} - 251 \beta_{19} ) q^{92}$$ $$+ ( -3617332 \beta_{1} + 820 \beta_{2} - 3856 \beta_{3} + 49996 \beta_{5} + 98472 \beta_{6} + 6172 \beta_{7} - 820 \beta_{9} + 820 \beta_{10} + 2816 \beta_{11} + 508 \beta_{13} + 896 \beta_{14} + 1284 \beta_{15} - 1132 \beta_{17} + 388 \beta_{18} + 2924 \beta_{19} ) q^{93}$$ $$+ ( -9622174 + 2905 \beta_{1} - 38338 \beta_{2} + 80 \beta_{3} - 1026 \beta_{4} + 80 \beta_{5} - 80 \beta_{6} + 19013 \beta_{7} + 14755 \beta_{8} - 2965 \beta_{9} - 640 \beta_{10} - 2878 \beta_{11} - 1323 \beta_{12} - 80 \beta_{13} - 80 \beta_{14} + 187 \beta_{15} + 53 \beta_{16} + 80 \beta_{17} - 107 \beta_{18} + 80 \beta_{19} ) q^{94}$$ $$+ ( 2980 - 1946364 \beta_{1} - 6171 \beta_{2} + 3655 \beta_{3} - 1216 \beta_{4} - 36951 \beta_{5} + 93699 \beta_{6} + 3477 \beta_{7} - 20031 \beta_{8} - 2748 \beta_{9} + 1724 \beta_{10} + 1216 \beta_{11} - 2820 \beta_{12} + 12032 \beta_{13} - 392 \beta_{14} + 171 \beta_{15} + 155 \beta_{16} + 1763 \beta_{17} + 221 \beta_{18} + 1763 \beta_{19} ) q^{95}$$ $$+ ( 1446572 - 11620 \beta_{1} - 7280 \beta_{2} - 1020 \beta_{3} + 6076 \beta_{4} - 1020 \beta_{5} + 1020 \beta_{6} - 36188 \beta_{7} - 2964 \beta_{8} - 9756 \beta_{9} + 1152 \beta_{10} + 11604 \beta_{11} - 1212 \beta_{12} + 1020 \beta_{13} + 1020 \beta_{14} - 2056 \beta_{15} - 1004 \beta_{16} - 1020 \beta_{17} + 1036 \beta_{18} - 1020 \beta_{19} ) q^{96}$$ $$+ ( 1305395 \beta_{1} - 311 \beta_{2} + 8984 \beta_{3} - 16187 \beta_{5} - 34446 \beta_{6} - 23565 \beta_{7} + 311 \beta_{9} - 311 \beta_{10} - 4882 \beta_{11} - 1923 \beta_{13} - 1152 \beta_{14} - 381 \beta_{15} - 1301 \beta_{17} + 771 \beta_{18} - 1003 \beta_{19} ) q^{97}$$ $$+ ( 482377 \beta_{1} + 1365 \beta_{2} - 15925 \beta_{3} - 62020 \beta_{5} - 346395 \beta_{6} - 56371 \beta_{7} - 1365 \beta_{9} + 1365 \beta_{10} - 10878 \beta_{11} - 2611 \beta_{13} + 903 \beta_{14} - 21 \beta_{15} - 903 \beta_{17} - 1708 \beta_{18} + 1925 \beta_{19} ) q^{98}$$ $$+ ( 25209 - 564 \beta_{1} - 62943 \beta_{2} - 138 \beta_{3} - 768 \beta_{4} - 138 \beta_{5} + 138 \beta_{6} + 2376 \beta_{7} - 14883 \beta_{8} - 453 \beta_{9} + 2757 \beta_{10} + 768 \beta_{11} - 4503 \beta_{12} + 138 \beta_{13} + 138 \beta_{14} - 72 \beta_{15} - 342 \beta_{16} - 138 \beta_{17} - 66 \beta_{18} - 138 \beta_{19} ) q^{99}$$ $$+O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$20q$$ $$\mathstrut -\mathstrut 752q^{4}$$ $$\mathstrut -\mathstrut 1420q^{5}$$ $$\mathstrut +\mathstrut 3408q^{6}$$ $$\mathstrut +\mathstrut 2556q^{9}$$ $$\mathstrut +\mathstrut O(q^{10})$$ $$20q$$ $$\mathstrut -\mathstrut 752q^{4}$$ $$\mathstrut -\mathstrut 1420q^{5}$$ $$\mathstrut +\mathstrut 3408q^{6}$$ $$\mathstrut +\mathstrut 2556q^{9}$$ $$\mathstrut -\mathstrut 4160q^{10}$$ $$\mathstrut +\mathstrut 8848q^{14}$$ $$\mathstrut -\mathstrut 59200q^{16}$$ $$\mathstrut -\mathstrut 4240q^{20}$$ $$\mathstrut +\mathstrut 410256q^{21}$$ $$\mathstrut -\mathstrut 156672q^{24}$$ $$\mathstrut -\mathstrut 657260q^{25}$$ $$\mathstrut -\mathstrut 440448q^{26}$$ $$\mathstrut +\mathstrut 660136q^{29}$$ $$\mathstrut +\mathstrut 667920q^{30}$$ $$\mathstrut +\mathstrut 4342528q^{34}$$ $$\mathstrut -\mathstrut 7191312q^{36}$$ $$\mathstrut -\mathstrut 945280q^{40}$$ $$\mathstrut +\mathstrut 7068520q^{41}$$ $$\mathstrut +\mathstrut 2666880q^{44}$$ $$\mathstrut -\mathstrut 18729060q^{45}$$ $$\mathstrut +\mathstrut 561168q^{46}$$ $$\mathstrut +\mathstrut 11719036q^{49}$$ $$\mathstrut +\mathstrut 3914880q^{50}$$ $$\mathstrut +\mathstrut 37110816q^{54}$$ $$\mathstrut -\mathstrut 35044352q^{56}$$ $$\mathstrut -\mathstrut 22734720q^{60}$$ $$\mathstrut -\mathstrut 17660440q^{61}$$ $$\mathstrut +\mathstrut 20201728q^{64}$$ $$\mathstrut -\mathstrut 44202240q^{65}$$ $$\mathstrut +\mathstrut 31902720q^{66}$$ $$\mathstrut +\mathstrut 111747216q^{69}$$ $$\mathstrut -\mathstrut 47166000q^{70}$$ $$\mathstrut +\mathstrut 19114368q^{74}$$ $$\mathstrut -\mathstrut 54998400q^{76}$$ $$\mathstrut +\mathstrut 7968320q^{80}$$ $$\mathstrut -\mathstrut 154212444q^{81}$$ $$\mathstrut +\mathstrut 101289216q^{84}$$ $$\mathstrut +\mathstrut 68800000q^{85}$$ $$\mathstrut +\mathstrut 94429648q^{86}$$ $$\mathstrut +\mathstrut 105006376q^{89}$$ $$\mathstrut -\mathstrut 230808000q^{90}$$ $$\mathstrut -\mathstrut 192757872q^{94}$$ $$\mathstrut +\mathstrut 28850688q^{96}$$ $$\mathstrut +\mathstrut O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{20}\mathstrut +\mathstrut$$ $$94$$ $$x^{18}\mathstrut +\mathstrut$$ $$5343$$ $$x^{16}\mathstrut +\mathstrut$$ $$172772$$ $$x^{14}\mathstrut +\mathstrut$$ $$36131456$$ $$x^{12}\mathstrut +\mathstrut$$ $$3044563968$$ $$x^{10}\mathstrut +\mathstrut$$ $$147994443776$$ $$x^{8}\mathstrut +\mathstrut$$ $$2898633162752$$ $$x^{6}\mathstrut +\mathstrut$$ $$367168164200448$$ $$x^{4}\mathstrut +\mathstrut$$ $$26458647810801664$$ $$x^{2}\mathstrut +\mathstrut$$ $$1152921504606846976$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$2 \nu$$ $$\beta_{2}$$ $$=$$ $$4 \nu^{2} + 38$$ $$\beta_{3}$$ $$=$$ $$8 \nu^{3} + 76 \nu$$ $$\beta_{4}$$ $$=$$ $$($$$$-$$$$3302597$$ $$\nu^{18}\mathstrut -\mathstrut$$ $$1391343702$$ $$\nu^{16}\mathstrut -\mathstrut$$ $$29876009883$$ $$\nu^{14}\mathstrut +\mathstrut$$ $$1448679820940$$ $$\nu^{12}\mathstrut -\mathstrut$$ $$186292651580544$$ $$\nu^{10}\mathstrut -\mathstrut$$ $$38687379151309824$$ $$\nu^{8}\mathstrut -\mathstrut$$ $$677837057835925504$$ $$\nu^{6}\mathstrut +\mathstrut$$ $$16327481853927751680$$ $$\nu^{4}\mathstrut -\mathstrut$$ $$1614929506130542460928$$ $$\nu^{2}\mathstrut -\mathstrut$$ $$379959887896387285680128$$$$)/$$$$12647057731702751232$$ $$\beta_{5}$$ $$=$$ $$($$$$10120793$$ $$\nu^{19}\mathstrut -\mathstrut$$ $$1538239314$$ $$\nu^{17}\mathstrut -\mathstrut$$ $$72882621561$$ $$\nu^{15}\mathstrut +\mathstrut$$ $$834264549700$$ $$\nu^{13}\mathstrut +\mathstrut$$ $$124572188012160$$ $$\nu^{11}\mathstrut -\mathstrut$$ $$46269733902222336$$ $$\nu^{9}\mathstrut -\mathstrut$$ $$2016433934423293952$$ $$\nu^{7}\mathstrut +\mathstrut$$ $$342252525333774336$$ $$\nu^{5}\mathstrut +\mathstrut$$ $$1227114972205899841536$$ $$\nu^{3}\mathstrut -\mathstrut$$ $$476580632743716635803648$$ $$\nu$$$$)/$$$$16\!\cdots\!96$$ $$\beta_{6}$$ $$=$$ $$($$$$-$$$$21623225$$ $$\nu^{19}\mathstrut +\mathstrut$$ $$457010706$$ $$\nu^{17}\mathstrut +\mathstrut$$ $$11425127385$$ $$\nu^{15}\mathstrut -\mathstrut$$ $$2821562731204$$ $$\nu^{13}\mathstrut -\mathstrut$$ $$540171803713152$$ $$\nu^{11}\mathstrut +\mathstrut$$ $$11249843890652160$$ $$\nu^{9}\mathstrut +\mathstrut$$ $$314137908512030720$$ $$\nu^{7}\mathstrut -\mathstrut$$ $$33683583372833587200$$ $$\nu^{5}\mathstrut -\mathstrut$$ $$5450441813486387331072$$ $$\nu^{3}\mathstrut +\mathstrut$$ $$201380656501864768995328$$ $$\nu$$$$)/$$$$32\!\cdots\!92$$ $$\beta_{7}$$ $$=$$ $$($$$$11547117$$ $$\nu^{19}\mathstrut -\mathstrut$$ $$488855936$$ $$\nu^{18}\mathstrut +\mathstrut$$ $$1694508294$$ $$\nu^{17}\mathstrut -\mathstrut$$ $$65873566976$$ $$\nu^{16}\mathstrut +\mathstrut$$ $$36909113715$$ $$\nu^{15}\mathstrut -\mathstrut$$ $$1089638299264$$ $$\nu^{14}\mathstrut +\mathstrut$$ $$3116676587796$$ $$\nu^{13}\mathstrut -\mathstrut$$ $$62406029653504$$ $$\nu^{12}\mathstrut +\mathstrut$$ $$413930864189568$$ $$\nu^{11}\mathstrut -\mathstrut$$ $$20884934598574080$$ $$\nu^{10}\mathstrut +\mathstrut$$ $$42049065119198208$$ $$\nu^{9}\mathstrut -\mathstrut$$ $$1859300525651656704$$ $$\nu^{8}\mathstrut +\mathstrut$$ $$814895842762162176$$ $$\nu^{7}\mathstrut -\mathstrut$$ $$26832143584627523584$$ $$\nu^{6}\mathstrut +\mathstrut$$ $$24633116361196830720$$ $$\nu^{5}\mathstrut -\mathstrut$$ $$730004290065343184896$$ $$\nu^{4}\mathstrut +\mathstrut$$ $$3878837848709397479424$$ $$\nu^{3}\mathstrut -\mathstrut$$ $$218323408975256549851136$$ $$\nu^{2}\mathstrut +\mathstrut$$ $$312734663541792914276352$$ $$\nu\mathstrut -\mathstrut$$ $$16442352117641685688647680$$$$)/$$$$17\!\cdots\!40$$ $$\beta_{8}$$ $$=$$ $$($$$$607811$$ $$\nu^{18}\mathstrut +\mathstrut$$ $$30995610$$ $$\nu^{16}\mathstrut +\mathstrut$$ $$223223901$$ $$\nu^{14}\mathstrut +\mathstrut$$ $$58863964588$$ $$\nu^{12}\mathstrut +\mathstrut$$ $$18819124901760$$ $$\nu^{10}\mathstrut +\mathstrut$$ $$857972427985920$$ $$\nu^{8}\mathstrut +\mathstrut$$ $$7078665453568000$$ $$\nu^{6}\mathstrut +\mathstrut$$ $$607645999476768768$$ $$\nu^{4}\mathstrut +\mathstrut$$ $$188843249831686176768$$ $$\nu^{2}\mathstrut +\mathstrut$$ $$6004190878028206702592$$$$)/$$$$395220554115710976$$ $$\beta_{9}$$ $$=$$ $$($$$$103924053$$ $$\nu^{19}\mathstrut +\mathstrut$$ $$6141831296$$ $$\nu^{18}\mathstrut +\mathstrut$$ $$15250574646$$ $$\nu^{17}\mathstrut -\mathstrut$$ $$1454773281024$$ $$\nu^{16}\mathstrut +\mathstrut$$ $$332182023435$$ $$\nu^{15}\mathstrut -\mathstrut$$ $$22767559211136$$ $$\nu^{14}\mathstrut +\mathstrut$$ $$28050089290164$$ $$\nu^{13}\mathstrut +\mathstrut$$ $$1172064818696704$$ $$\nu^{12}\mathstrut +\mathstrut$$ $$3725377777706112$$ $$\nu^{11}\mathstrut +\mathstrut$$ $$43868445489315840$$ $$\nu^{10}\mathstrut +\mathstrut$$ $$378441586072783872$$ $$\nu^{9}\mathstrut -\mathstrut$$ $$38978511953512759296$$ $$\nu^{8}\mathstrut +\mathstrut$$ $$7334062584859459584$$ $$\nu^{7}\mathstrut -\mathstrut$$ $$691245163541468020736$$ $$\nu^{6}\mathstrut +\mathstrut$$ $$221698047250771476480$$ $$\nu^{5}\mathstrut +\mathstrut$$ $$16366106950298203324416$$ $$\nu^{4}\mathstrut +\mathstrut$$ $$34909540638384577314816$$ $$\nu^{3}\mathstrut +\mathstrut$$ $$475522968361368022941696$$ $$\nu^{2}\mathstrut +\mathstrut$$ $$2814611971876136228487168$$ $$\nu\mathstrut -\mathstrut$$ $$410578305182332705036042240$$$$)/$$$$80\!\cdots\!80$$ $$\beta_{10}$$ $$=$$ $$($$$$-$$$$103924053$$ $$\nu^{19}\mathstrut -\mathstrut$$ $$9822609536$$ $$\nu^{18}\mathstrut -\mathstrut$$ $$15250574646$$ $$\nu^{17}\mathstrut +\mathstrut$$ $$1108780126464$$ $$\nu^{16}\mathstrut -\mathstrut$$ $$332182023435$$ $$\nu^{15}\mathstrut +\mathstrut$$ $$3101161074816$$ $$\nu^{14}\mathstrut -\mathstrut$$ $$28050089290164$$ $$\nu^{13}\mathstrut -\mathstrut$$ $$1808000236777984$$ $$\nu^{12}\mathstrut -\mathstrut$$ $$3725377777706112$$ $$\nu^{11}\mathstrut -\mathstrut$$ $$176860322513633280$$ $$\nu^{10}\mathstrut -\mathstrut$$ $$378441586072783872$$ $$\nu^{9}\mathstrut +\mathstrut$$ $$27772147149810302976$$ $$\nu^{8}\mathstrut -\mathstrut$$ $$7334062584859459584$$ $$\nu^{7}\mathstrut +\mathstrut$$ $$146510435249863786496$$ $$\nu^{6}\mathstrut -\mathstrut$$ $$221698047250771476480$$ $$\nu^{5}\mathstrut -\mathstrut$$ $$27035332821498143440896$$ $$\nu^{4}\mathstrut -\mathstrut$$ $$34909540638384577314816$$ $$\nu^{3}\mathstrut -\mathstrut$$ $$1729858154191646890131456$$ $$\nu^{2}\mathstrut -\mathstrut$$ $$2814611971876136228487168$$ $$\nu\mathstrut +\mathstrut$$ $$323841747964459628426690560$$$$)/$$$$80\!\cdots\!80$$ $$\beta_{11}$$ $$=$$ $$($$$$11547117$$ $$\nu^{19}\mathstrut +\mathstrut$$ $$488855936$$ $$\nu^{18}\mathstrut +\mathstrut$$ $$1694508294$$ $$\nu^{17}\mathstrut +\mathstrut$$ $$65873566976$$ $$\nu^{16}\mathstrut +\mathstrut$$ $$36909113715$$ $$\nu^{15}\mathstrut +\mathstrut$$ $$1089638299264$$ $$\nu^{14}\mathstrut +\mathstrut$$ $$3116676587796$$ $$\nu^{13}\mathstrut +\mathstrut$$ $$62406029653504$$ $$\nu^{12}\mathstrut +\mathstrut$$ $$413930864189568$$ $$\nu^{11}\mathstrut +\mathstrut$$ $$20884934598574080$$ $$\nu^{10}\mathstrut +\mathstrut$$ $$42049065119198208$$ $$\nu^{9}\mathstrut +\mathstrut$$ $$1859300525651656704$$ $$\nu^{8}\mathstrut +\mathstrut$$ $$814895842762162176$$ $$\nu^{7}\mathstrut +\mathstrut$$ $$26832143584627523584$$ $$\nu^{6}\mathstrut +\mathstrut$$ $$24633116361196830720$$ $$\nu^{5}\mathstrut +\mathstrut$$ $$730004290065343184896$$ $$\nu^{4}\mathstrut +\mathstrut$$ $$3878837848709397479424$$ $$\nu^{3}\mathstrut +\mathstrut$$ $$218323408975256549851136$$ $$\nu^{2}\mathstrut +\mathstrut$$ $$313454140603863115235328$$ $$\nu\mathstrut +\mathstrut$$ $$16442352117641685688647680$$$$)/$$$$35\!\cdots\!88$$ $$\beta_{12}$$ $$=$$ $$($$$$11250533$$ $$\nu^{18}\mathstrut +\mathstrut$$ $$356503830$$ $$\nu^{16}\mathstrut +\mathstrut$$ $$6095187195$$ $$\nu^{14}\mathstrut +\mathstrut$$ $$1380868386292$$ $$\nu^{12}\mathstrut +\mathstrut$$ $$320541156988032$$ $$\nu^{10}\mathstrut +\mathstrut$$ $$7748665416698880$$ $$\nu^{8}\mathstrut +\mathstrut$$ $$102852276112064512$$ $$\nu^{6}\mathstrut +\mathstrut$$ $$17957218619410612224$$ $$\nu^{4}\mathstrut +\mathstrut$$ $$3135517970758599966720$$ $$\nu^{2}\mathstrut +\mathstrut$$ $$16797154910545217847296$$$$)/$$$$1580882216462843904$$ $$\beta_{13}$$ $$=$$ $$($$$$32459431$$ $$\nu^{19}\mathstrut -\mathstrut$$ $$1527461550$$ $$\nu^{17}\mathstrut -\mathstrut$$ $$42771655815$$ $$\nu^{15}\mathstrut +\mathstrut$$ $$3554605976252$$ $$\nu^{13}\mathstrut +\mathstrut$$ $$745081528209792$$ $$\nu^{11}\mathstrut -\mathstrut$$ $$47291577343933440$$ $$\nu^{9}\mathstrut -\mathstrut$$ $$930922400653508608$$ $$\nu^{7}\mathstrut +\mathstrut$$ $$33103730770905661440$$ $$\nu^{5}\mathstrut +\mathstrut$$ $$7655738336341515042816$$ $$\nu^{3}\mathstrut -\mathstrut$$ $$616847663822970445365248$$ $$\nu$$$$)/$$$$26\!\cdots\!16$$ $$\beta_{14}$$ $$=$$ $$($$$$-$$$$4397900707$$ $$\nu^{19}\mathstrut -\mathstrut$$ $$16920576256$$ $$\nu^{18}\mathstrut -\mathstrut$$ $$1027539355098$$ $$\nu^{17}\mathstrut -\mathstrut$$ $$1855269723648$$ $$\nu^{16}\mathstrut -\mathstrut$$ $$17641443616509$$ $$\nu^{15}\mathstrut -\mathstrut$$ $$95809272635136$$ $$\nu^{14}\mathstrut -\mathstrut$$ $$1331282867958572$$ $$\nu^{13}\mathstrut -\mathstrut$$ $$3579926299599872$$ $$\nu^{12}\mathstrut -\mathstrut$$ $$273899347089016704$$ $$\nu^{11}\mathstrut -\mathstrut$$ $$366240911949201408$$ $$\nu^{10}\mathstrut -\mathstrut$$ $$29240798137031758848$$ $$\nu^{9}\mathstrut -\mathstrut$$ $$37218976597244903424$$ $$\nu^{8}\mathstrut -\mathstrut$$ $$506867138478902607872$$ $$\nu^{7}\mathstrut -\mathstrut$$ $$1195540294318363246592$$ $$\nu^{6}\mathstrut -\mathstrut$$ $$24410800717512044445696$$ $$\nu^{5}\mathstrut -\mathstrut$$ $$14694104124069148360704$$ $$\nu^{4}\mathstrut -\mathstrut$$ $$3103108019049450939875328$$ $$\nu^{3}\mathstrut -\mathstrut$$ $$1561971397786855494647808$$ $$\nu^{2}\mathstrut -\mathstrut$$ $$270781286964223331969007616$$ $$\nu\mathstrut -\mathstrut$$ $$152824712869997759306924032$$$$)/$$$$16\!\cdots\!60$$ $$\beta_{15}$$ $$=$$ $$($$$$-$$$$4073259013$$ $$\nu^{19}\mathstrut -\mathstrut$$ $$4973631424$$ $$\nu^{18}\mathstrut -\mathstrut$$ $$608485077462$$ $$\nu^{17}\mathstrut +\mathstrut$$ $$44798068608$$ $$\nu^{16}\mathstrut -\mathstrut$$ $$10172775434331$$ $$\nu^{15}\mathstrut +\mathstrut$$ $$8838463344576$$ $$\nu^{14}\mathstrut -\mathstrut$$ $$526301963730548$$ $$\nu^{13}\mathstrut -\mathstrut$$ $$555286437340928$$ $$\nu^{12}\mathstrut -\mathstrut$$ $$180374093711039616$$ $$\nu^{11}\mathstrut -\mathstrut$$ $$195039789293985792$$ $$\nu^{10}\mathstrut -\mathstrut$$ $$17119797791987715072$$ $$\nu^{9}\mathstrut -\mathstrut$$ $$2194382902581067776$$ $$\nu^{8}\mathstrut -\mathstrut$$ $$250653736953037979648$$ $$\nu^{7}\mathstrut +\mathstrut$$ $$57145141213026844672$$ $$\nu^{6}\mathstrut -\mathstrut$$ $$6091424925188886626304$$ $$\nu^{5}\mathstrut -\mathstrut$$ $$12449600986973969842176$$ $$\nu^{4}\mathstrut -\mathstrut$$ $$1886982719360489459023872$$ $$\nu^{3}\mathstrut -\mathstrut$$ $$2394234670402360502648832$$ $$\nu^{2}\mathstrut -\mathstrut$$ $$152961201235185611334221824$$ $$\nu\mathstrut -\mathstrut$$ $$24719306825100182450864128$$$$)/$$$$80\!\cdots\!80$$ $$\beta_{16}$$ $$=$$ $$($$$$875056263$$ $$\nu^{19}\mathstrut +\mathstrut$$ $$41392821248$$ $$\nu^{18}\mathstrut -\mathstrut$$ $$13654586990$$ $$\nu^{17}\mathstrut +\mathstrut$$ $$570874583040$$ $$\nu^{16}\mathstrut -\mathstrut$$ $$635936680551$$ $$\nu^{15}\mathstrut -\mathstrut$$ $$7313859204096$$ $$\nu^{14}\mathstrut +\mathstrut$$ $$44409908241980$$ $$\nu^{13}\mathstrut +\mathstrut$$ $$4015046027038720$$ $$\nu^{12}\mathstrut +\mathstrut$$ $$17462369935049088$$ $$\nu^{11}\mathstrut +\mathstrut$$ $$1185724563708837888$$ $$\nu^{10}\mathstrut -\mathstrut$$ $$537466764231932928$$ $$\nu^{9}\mathstrut +\mathstrut$$ $$13558381674104881152$$ $$\nu^{8}\mathstrut -\mathstrut$$ $$18470451322984857600$$ $$\nu^{7}\mathstrut -\mathstrut$$ $$93652815973575557120$$ $$\nu^{6}\mathstrut +\mathstrut$$ $$156706787651093528576$$ $$\nu^{5}\mathstrut +\mathstrut$$ $$33750100738878405083136$$ $$\nu^{4}\mathstrut +\mathstrut$$ $$148380910325528883363840$$ $$\nu^{3}\mathstrut +\mathstrut$$ $$11882536641470384874455040$$ $$\nu^{2}\mathstrut -\mathstrut$$ $$9520514294419540177584128$$ $$\nu\mathstrut -\mathstrut$$ $$42077881924465238713827328$$$$)/$$$$17\!\cdots\!40$$ $$\beta_{17}$$ $$=$$ $$($$$$14557212835$$ $$\nu^{19}\mathstrut +\mathstrut$$ $$3251847424$$ $$\nu^{18}\mathstrut +\mathstrut$$ $$1888412158938$$ $$\nu^{17}\mathstrut +\mathstrut$$ $$1868182445568$$ $$\nu^{16}\mathstrut +\mathstrut$$ $$18352861149693$$ $$\nu^{15}\mathstrut +\mathstrut$$ $$47097160769280$$ $$\nu^{14}\mathstrut +\mathstrut$$ $$1220994732918572$$ $$\nu^{13}\mathstrut +\mathstrut$$ $$574389925962752$$ $$\nu^{12}\mathstrut +\mathstrut$$ $$606312880330065792$$ $$\nu^{11}\mathstrut +\mathstrut$$ $$173813655573528576$$ $$\nu^{10}\mathstrut +\mathstrut$$ $$54748719137284592640$$ $$\nu^{9}\mathstrut +\mathstrut$$ $$45812348387432595456$$ $$\nu^{8}\mathstrut +\mathstrut$$ $$623357947096921014272$$ $$\nu^{7}\mathstrut +\mathstrut$$ $$838758159210996826112$$ $$\nu^{6}\mathstrut +\mathstrut$$ $$7878748810219955945472$$ $$\nu^{5}\mathstrut -\mathstrut$$ $$5189086142183673692160$$ $$\nu^{4}\mathstrut +\mathstrut$$ $$6342184416022792280997888$$ $$\nu^{3}\mathstrut +\mathstrut$$ $$1106262701527205840683008$$ $$\nu^{2}\mathstrut +\mathstrut$$ $$499895096886226898007359488$$ $$\nu\mathstrut +\mathstrut$$ $$394504893526125148691759104$$$$)/$$$$16\!\cdots\!60$$ $$\beta_{18}$$ $$=$$ $$($$$$-$$$$8062974223$$ $$\nu^{19}\mathstrut -\mathstrut$$ $$9608144128$$ $$\nu^{18}\mathstrut -\mathstrut$$ $$554664723330$$ $$\nu^{17}\mathstrut +\mathstrut$$ $$347685480960$$ $$\nu^{16}\mathstrut -\mathstrut$$ $$7477151383569$$ $$\nu^{15}\mathstrut -\mathstrut$$ $$10614220241664$$ $$\nu^{14}\mathstrut -\mathstrut$$ $$740171595464540$$ $$\nu^{13}\mathstrut -\mathstrut$$ $$1777227490718720$$ $$\nu^{12}\mathstrut -\mathstrut$$ $$260817447307613568$$ $$\nu^{11}\mathstrut -\mathstrut$$ $$197271211788238848$$ $$\nu^{10}\mathstrut -\mathstrut$$ $$14890418145980408832$$ $$\nu^{9}\mathstrut +\mathstrut$$ $$10469155357945233408$$ $$\nu^{8}\mathstrut -\mathstrut$$ $$171203737292035850240$$ $$\nu^{7}\mathstrut -\mathstrut$$ $$501280209832509440$$ $$\nu^{6}\mathstrut -\mathstrut$$ $$6907454493244193243136$$ $$\nu^{5}\mathstrut -\mathstrut$$ $$19106176814806336536576$$ $$\nu^{4}\mathstrut -\mathstrut$$ $$2572151586144561722818560$$ $$\nu^{3}\mathstrut -\mathstrut$$ $$1639633678143530855301120$$ $$\nu^{2}\mathstrut -\mathstrut$$ $$111510004662339169127759872$$ $$\nu\mathstrut +\mathstrut$$ $$170111368032351097840467968$$$$)/$$$$80\!\cdots\!80$$ $$\beta_{19}$$ $$=$$ $$($$$$-$$$$12462645249$$ $$\nu^{19}\mathstrut +\mathstrut$$ $$5208440704$$ $$\nu^{18}\mathstrut -\mathstrut$$ $$771214807134$$ $$\nu^{17}\mathstrut -\mathstrut$$ $$940547583744$$ $$\nu^{16}\mathstrut -\mathstrut$$ $$3157211583711$$ $$\nu^{15}\mathstrut +\mathstrut$$ $$807475548288$$ $$\nu^{14}\mathstrut -\mathstrut$$ $$1068138105819876$$ $$\nu^{13}\mathstrut +\mathstrut$$ $$1215573223837184$$ $$\nu^{12}\mathstrut -\mathstrut$$ $$398376882905846400$$ $$\nu^{11}\mathstrut +\mathstrut$$ $$9306800401072128$$ $$\nu^{10}\mathstrut -\mathstrut$$ $$21386254801199717376$$ $$\nu^{9}\mathstrut -\mathstrut$$ $$27202860088810143744$$ $$\nu^{8}\mathstrut -\mathstrut$$ $$168951932796648554496$$ $$\nu^{7}\mathstrut -\mathstrut$$ $$240988012051815202816$$ $$\nu^{6}\mathstrut -\mathstrut$$ $$10667368249093924061184$$ $$\nu^{5}\mathstrut +\mathstrut$$ $$12536138204218247872512$$ $$\nu^{4}\mathstrut -\mathstrut$$ $$4043388926315402492903424$$ $$\nu^{3}\mathstrut -\mathstrut$$ $$325277002633778093359104$$ $$\nu^{2}\mathstrut -\mathstrut$$ $$164317707790091020727746560$$ $$\nu\mathstrut -\mathstrut$$ $$318092537091126269038297088$$$$)/$$$$80\!\cdots\!80$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$$$/2$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{2}\mathstrut -\mathstrut$$ $$38$$$$)/4$$ $$\nu^{3}$$ $$=$$ $$($$$$\beta_{3}\mathstrut -\mathstrut$$ $$38$$ $$\beta_{1}$$$$)/8$$ $$\nu^{4}$$ $$=$$ $$($$$$-$$$$\beta_{19}\mathstrut +\mathstrut$$ $$\beta_{18}\mathstrut -\mathstrut$$ $$\beta_{17}\mathstrut -\mathstrut$$ $$\beta_{16}\mathstrut -\mathstrut$$ $$2$$ $$\beta_{15}\mathstrut +\mathstrut$$ $$\beta_{14}\mathstrut +\mathstrut$$ $$\beta_{13}\mathstrut +\mathstrut$$ $$3$$ $$\beta_{12}\mathstrut -\mathstrut$$ $$\beta_{11}\mathstrut +\mathstrut$$ $$3$$ $$\beta_{9}\mathstrut +\mathstrut$$ $$\beta_{8}\mathstrut -\mathstrut$$ $$\beta_{7}\mathstrut +\mathstrut$$ $$\beta_{6}\mathstrut -\mathstrut$$ $$\beta_{5}\mathstrut -\mathstrut$$ $$3$$ $$\beta_{4}\mathstrut -\mathstrut$$ $$\beta_{3}\mathstrut -\mathstrut$$ $$40$$ $$\beta_{2}\mathstrut +\mathstrut$$ $$\beta_{1}\mathstrut -\mathstrut$$ $$2943$$$$)/16$$ $$\nu^{5}$$ $$=$$ $$($$$$-$$$$7$$ $$\beta_{19}\mathstrut -\mathstrut$$ $$\beta_{18}\mathstrut -\mathstrut$$ $$10$$ $$\beta_{17}\mathstrut -\mathstrut$$ $$2$$ $$\beta_{15}\mathstrut -\mathstrut$$ $$6$$ $$\beta_{14}\mathstrut -\mathstrut$$ $$27$$ $$\beta_{13}\mathstrut -\mathstrut$$ $$28$$ $$\beta_{11}\mathstrut -\mathstrut$$ $$132$$ $$\beta_{7}\mathstrut +\mathstrut$$ $$70$$ $$\beta_{6}\mathstrut +\mathstrut$$ $$7$$ $$\beta_{5}\mathstrut -\mathstrut$$ $$6$$ $$\beta_{3}\mathstrut -\mathstrut$$ $$729$$ $$\beta_{1}$$$$)/8$$ $$\nu^{6}$$ $$=$$ $$($$$$-$$$$9$$ $$\beta_{19}\mathstrut +\mathstrut$$ $$31$$ $$\beta_{18}\mathstrut -\mathstrut$$ $$9$$ $$\beta_{17}\mathstrut +\mathstrut$$ $$13$$ $$\beta_{16}\mathstrut -\mathstrut$$ $$40$$ $$\beta_{15}\mathstrut +\mathstrut$$ $$9$$ $$\beta_{14}\mathstrut +\mathstrut$$ $$9$$ $$\beta_{13}\mathstrut +\mathstrut$$ $$9$$ $$\beta_{12}\mathstrut -\mathstrut$$ $$51$$ $$\beta_{11}\mathstrut -\mathstrut$$ $$208$$ $$\beta_{10}\mathstrut -\mathstrut$$ $$379$$ $$\beta_{9}\mathstrut -\mathstrut$$ $$21$$ $$\beta_{8}\mathstrut +\mathstrut$$ $$619$$ $$\beta_{7}\mathstrut +\mathstrut$$ $$9$$ $$\beta_{6}\mathstrut -\mathstrut$$ $$9$$ $$\beta_{5}\mathstrut +\mathstrut$$ $$111$$ $$\beta_{4}\mathstrut -\mathstrut$$ $$9$$ $$\beta_{3}\mathstrut -\mathstrut$$ $$14$$ $$\beta_{2}\mathstrut +\mathstrut$$ $$29$$ $$\beta_{1}\mathstrut +\mathstrut$$ $$126275$$$$)/8$$ $$\nu^{7}$$ $$=$$ $$($$$$-$$$$274$$ $$\beta_{19}\mathstrut +\mathstrut$$ $$406$$ $$\beta_{18}\mathstrut +\mathstrut$$ $$108$$ $$\beta_{17}\mathstrut +\mathstrut$$ $$344$$ $$\beta_{15}\mathstrut -\mathstrut$$ $$76$$ $$\beta_{14}\mathstrut +\mathstrut$$ $$2082$$ $$\beta_{13}\mathstrut +\mathstrut$$ $$512$$ $$\beta_{11}\mathstrut -\mathstrut$$ $$302$$ $$\beta_{10}\mathstrut +\mathstrut$$ $$302$$ $$\beta_{9}\mathstrut +\mathstrut$$ $$1924$$ $$\beta_{7}\mathstrut +\mathstrut$$ $$24356$$ $$\beta_{6}\mathstrut -\mathstrut$$ $$13828$$ $$\beta_{5}\mathstrut -\mathstrut$$ $$551$$ $$\beta_{3}\mathstrut -\mathstrut$$ $$302$$ $$\beta_{2}\mathstrut +\mathstrut$$ $$75386$$ $$\beta_{1}$$$$)/8$$ $$\nu^{8}$$ $$=$$ $$($$$$139$$ $$\beta_{19}\mathstrut -\mathstrut$$ $$2451$$ $$\beta_{18}\mathstrut +\mathstrut$$ $$139$$ $$\beta_{17}\mathstrut -\mathstrut$$ $$2173$$ $$\beta_{16}\mathstrut +\mathstrut$$ $$2590$$ $$\beta_{15}\mathstrut -\mathstrut$$ $$139$$ $$\beta_{14}\mathstrut -\mathstrut$$ $$139$$ $$\beta_{13}\mathstrut -\mathstrut$$ $$8073$$ $$\beta_{12}\mathstrut -\mathstrut$$ $$1869$$ $$\beta_{11}\mathstrut -\mathstrut$$ $$5952$$ $$\beta_{10}\mathstrut +\mathstrut$$ $$9703$$ $$\beta_{9}\mathstrut +\mathstrut$$ $$56461$$ $$\beta_{8}\mathstrut -\mathstrut$$ $$24277$$ $$\beta_{7}\mathstrut -\mathstrut$$ $$139$$ $$\beta_{6}\mathstrut +\mathstrut$$ $$139$$ $$\beta_{5}\mathstrut -\mathstrut$$ $$1543$$ $$\beta_{4}\mathstrut +\mathstrut$$ $$139$$ $$\beta_{3}\mathstrut -\mathstrut$$ $$52512$$ $$\beta_{2}\mathstrut +\mathstrut$$ $$4181$$ $$\beta_{1}\mathstrut -\mathstrut$$ $$213160723$$$$)/16$$ $$\nu^{9}$$ $$=$$ $$($$$$3349$$ $$\beta_{19}\mathstrut +\mathstrut$$ $$9643$$ $$\beta_{18}\mathstrut +\mathstrut$$ $$16430$$ $$\beta_{17}\mathstrut +\mathstrut$$ $$11374$$ $$\beta_{15}\mathstrut +\mathstrut$$ $$7170$$ $$\beta_{14}\mathstrut -\mathstrut$$ $$183799$$ $$\beta_{13}\mathstrut -\mathstrut$$ $$7196$$ $$\beta_{11}\mathstrut -\mathstrut$$ $$6732$$ $$\beta_{10}\mathstrut +\mathstrut$$ $$6732$$ $$\beta_{9}\mathstrut -\mathstrut$$ $$62460$$ $$\beta_{7}\mathstrut -\mathstrut$$ $$4571346$$ $$\beta_{6}\mathstrut -\mathstrut$$ $$47921$$ $$\beta_{5}\mathstrut +\mathstrut$$ $$4584$$ $$\beta_{3}\mathstrut -\mathstrut$$ $$6732$$ $$\beta_{2}\mathstrut -\mathstrut$$ $$54768093$$ $$\beta_{1}$$$$)/8$$ $$\nu^{10}$$ $$=$$ $$($$$$-$$$$10014$$ $$\beta_{19}\mathstrut +\mathstrut$$ $$63685$$ $$\beta_{18}\mathstrut -\mathstrut$$ $$10014$$ $$\beta_{17}\mathstrut +\mathstrut$$ $$43657$$ $$\beta_{16}\mathstrut -\mathstrut$$ $$73699$$ $$\beta_{15}\mathstrut +\mathstrut$$ $$10014$$ $$\beta_{14}\mathstrut +\mathstrut$$ $$10014$$ $$\beta_{13}\mathstrut +\mathstrut$$ $$19965$$ $$\beta_{12}\mathstrut +\mathstrut$$ $$180161$$ $$\beta_{11}\mathstrut +\mathstrut$$ $$90488$$ $$\beta_{10}\mathstrut +\mathstrut$$ $$148659$$ $$\beta_{9}\mathstrut -\mathstrut$$ $$1059909$$ $$\beta_{8}\mathstrut -\mathstrut$$ $$963476$$ $$\beta_{7}\mathstrut +\mathstrut$$ $$10014$$ $$\beta_{6}\mathstrut -\mathstrut$$ $$10014$$ $$\beta_{5}\mathstrut -\mathstrut$$ $$87497$$ $$\beta_{4}\mathstrut -\mathstrut$$ $$10014$$ $$\beta_{3}\mathstrut -\mathstrut$$ $$14016463$$ $$\beta_{2}\mathstrut -\mathstrut$$ $$233832$$ $$\beta_{1}\mathstrut +\mathstrut$$ $$75246103$$$$)/4$$ $$\nu^{11}$$ $$=$$ $$($$$$1140020$$ $$\beta_{19}\mathstrut +\mathstrut$$ $$97392$$ $$\beta_{18}\mathstrut +\mathstrut$$ $$497136$$ $$\beta_{17}\mathstrut -\mathstrut$$ $$1893956$$ $$\beta_{15}\mathstrut -\mathstrut$$ $$118064$$ $$\beta_{14}\mathstrut +\mathstrut$$ $$10633504$$ $$\beta_{13}\mathstrut -\mathstrut$$ $$4176584$$ $$\beta_{11}\mathstrut +\mathstrut$$ $$580346$$ $$\beta_{10}\mathstrut -\mathstrut$$ $$580346$$ $$\beta_{9}\mathstrut -\mathstrut$$ $$17717220$$ $$\beta_{7}\mathstrut +\mathstrut$$ $$96666216$$ $$\beta_{6}\mathstrut +\mathstrut$$ $$7855422$$ $$\beta_{5}\mathstrut -\mathstrut$$ $$14399583$$ $$\beta_{3}\mathstrut +\mathstrut$$ $$580346$$ $$\beta_{2}\mathstrut +\mathstrut$$ $$97932580$$ $$\beta_{1}$$$$)/8$$ $$\nu^{12}$$ $$=$$ $$($$$$14017923$$ $$\beta_{19}\mathstrut -\mathstrut$$ $$21976067$$ $$\beta_{18}\mathstrut +\mathstrut$$ $$14017923$$ $$\beta_{17}\mathstrut +\mathstrut$$ $$6059779$$ $$\beta_{16}\mathstrut +\mathstrut$$ $$35993990$$ $$\beta_{15}\mathstrut -\mathstrut$$ $$14017923$$ $$\beta_{14}\mathstrut -\mathstrut$$ $$14017923$$ $$\beta_{13}\mathstrut -\mathstrut$$ $$36358857$$ $$\beta_{12}\mathstrut +\mathstrut$$ $$24442355$$ $$\beta_{11}\mathstrut -\mathstrut$$ $$10794240$$ $$\beta_{10}\mathstrut -\mathstrut$$ $$58728329$$ $$\beta_{9}\mathstrut +\mathstrut$$ $$74008941$$ $$\beta_{8}\mathstrut -\mathstrut$$ $$34301741$$ $$\beta_{7}\mathstrut -\mathstrut$$ $$14017923$$ $$\beta_{6}\mathstrut +\mathstrut$$ $$14017923$$ $$\beta_{5}\mathstrut +\mathstrut$$ $$138051865$$ $$\beta_{4}\mathstrut +\mathstrut$$ $$14017923$$ $$\beta_{3}\mathstrut +\mathstrut$$ $$235318232$$ $$\beta_{2}\mathstrut -\mathstrut$$ $$16484211$$ $$\beta_{1}\mathstrut +\mathstrut$$ $$213177952749$$$$)/16$$ $$\nu^{13}$$ $$=$$ $$($$$$66676517$$ $$\beta_{19}\mathstrut +\mathstrut$$ $$105062355$$ $$\beta_{18}\mathstrut +\mathstrut$$ $$125825694$$ $$\beta_{17}\mathstrut +\mathstrut$$ $$44106694$$ $$\beta_{15}\mathstrut +\mathstrut$$ $$43848850$$ $$\beta_{14}\mathstrut +\mathstrut$$ $$109690273$$ $$\beta_{13}\mathstrut +\mathstrut$$ $$798523348$$ $$\beta_{11}\mathstrut -\mathstrut$$ $$41117344$$ $$\beta_{10}\mathstrut +\mathstrut$$ $$41117344$$ $$\beta_{9}\mathstrut +\mathstrut$$ $$3814641324$$ $$\beta_{7}\mathstrut -\mathstrut$$ $$8922961490$$ $$\beta_{6}\mathstrut +\mathstrut$$ $$690161915$$ $$\beta_{5}\mathstrut +\mathstrut$$ $$29894022$$ $$\beta_{3}\mathstrut -\mathstrut$$ $$41117344$$ $$\beta_{2}\mathstrut +\mathstrut$$ $$50771922467$$ $$\beta_{1}$$$$)/8$$ $$\nu^{14}$$ $$=$$ $$($$$$344100561$$ $$\beta_{19}\mathstrut -\mathstrut$$ $$769664947$$ $$\beta_{18}\mathstrut +\mathstrut$$ $$344100561$$ $$\beta_{17}\mathstrut -\mathstrut$$ $$81463825$$ $$\beta_{16}\mathstrut +\mathstrut$$ $$1113765508$$ $$\beta_{15}\mathstrut -\mathstrut$$ $$344100561$$ $$\beta_{14}\mathstrut -\mathstrut$$ $$344100561$$ $$\beta_{13}\mathstrut -\mathstrut$$ $$182532189$$ $$\beta_{12}\mathstrut +\mathstrut$$ $$1266145519$$ $$\beta_{11}\mathstrut +\mathstrut$$ $$1949519472$$ $$\beta_{10}\mathstrut +\mathstrut$$ $$5786223599$$ $$\beta_{9}\mathstrut -\mathstrut$$ $$4093038615$$ $$\beta_{8}\mathstrut -\mathstrut$$ $$14429700235$$ $$\beta_{7}\mathstrut -\mathstrut$$ $$344100561$$ $$\beta_{6}\mathstrut +\mathstrut$$ $$344100561$$ $$\beta_{5}\mathstrut -\mathstrut$$ $$3530108107$$ $$\beta_{4}\mathstrut +\mathstrut$$ $$344100561$$ $$\beta_{3}\mathstrut +\mathstrut$$ $$23160611130$$ $$\beta_{2}\mathstrut -\mathstrut$$ $$840581133$$ $$\beta_{1}\mathstrut +\mathstrut$$ $$2477743922529$$$$)/8$$ $$\nu^{15}$$ $$=$$ $$($$$$9684699242$$ $$\beta_{19}\mathstrut -\mathstrut$$ $$12949656150$$ $$\beta_{18}\mathstrut -\mathstrut$$ $$2590174476$$ $$\beta_{17}\mathstrut -\mathstrut$$ $$9848016752$$ $$\beta_{15}\mathstrut +\mathstrut$$ $$3398721388$$ $$\beta_{14}\mathstrut -\mathstrut$$ $$28392840962$$ $$\beta_{13}\mathstrut -\mathstrut$$ $$14818724528$$ $$\beta_{11}\mathstrut +\mathstrut$$ $$9617817002$$ $$\beta_{10}\mathstrut -\mathstrut$$ $$9617817002$$ $$\beta_{9}\mathstrut -\mathstrut$$ $$57121528092$$ $$\beta_{7}\mathstrut -\mathstrut$$ $$622089709940$$ $$\beta_{6}\mathstrut +\mathstrut$$ $$105405457016$$ $$\beta_{5}\mathstrut +\mathstrut$$ $$17203410541$$ $$\beta_{3}\mathstrut +\mathstrut$$ $$9617817002$$ $$\beta_{2}\mathstrut +\mathstrut$$ $$985218925214$$ $$\beta_{1}$$$$)/8$$ $$\nu^{16}$$ $$=$$ $$($$$$-$$$$3613266745$$ $$\beta_{19}\mathstrut +\mathstrut$$ $$38623462337$$ $$\beta_{18}\mathstrut -\mathstrut$$ $$3613266745$$ $$\beta_{17}\mathstrut +\mathstrut$$ $$31396928847$$ $$\beta_{16}\mathstrut -\mathstrut$$ $$42236729082$$ $$\beta_{15}\mathstrut +\mathstrut$$ $$3613266745$$ $$\beta_{14}\mathstrut +\mathstrut$$ $$3613266745$$ $$\beta_{13}\mathstrut +\mathstrut$$ $$213846044307$$ $$\beta_{12}\mathstrut +\mathstrut$$ $$15751825087$$ $$\beta_{11}\mathstrut +\mathstrut$$ $$251228370240$$ $$\beta_{10}\mathstrut -\mathstrut$$ $$222541846109$$ $$\beta_{9}\mathstrut -\mathstrut$$ $$970768884671$$ $$\beta_{8}\mathstrut +\mathstrut$$ $$903791502855$$ $$\beta_{7}\mathstrut +\mathstrut$$ $$3613266745$$ $$\beta_{6}\mathstrut -\mathstrut$$ $$3613266745$$ $$\beta_{5}\mathstrut +\mathstrut$$ $$102416079325$$ $$\beta_{4}\mathstrut -\mathstrut$$ $$3613266745$$ $$\beta_{3}\mathstrut +\mathstrut$$ $$2015024656816$$ $$\beta_{2}\mathstrut -\mathstrut$$ $$50762020679$$ $$\beta_{1}\mathstrut +\mathstrut$$ $$1451820175978721$$$$)/16$$ $$\nu^{17}$$ $$=$$ $$($$$$-$$$$264166320887$$ $$\beta_{19}\mathstrut -\mathstrut$$ $$247665438841$$ $$\beta_{18}\mathstrut -\mathstrut$$ $$506965570042$$ $$\beta_{17}\mathstrut -\mathstrut$$ $$145026223306$$ $$\beta_{15}\mathstrut -\mathstrut$$ $$222830891798$$ $$\beta_{14}\mathstrut +\mathstrut$$ $$3444161822845$$ $$\beta_{13}\mathstrut +\mathstrut$$ $$693076259604$$ $$\beta_{11}\mathstrut +\mathstrut$$ $$103165004876$$ $$\beta_{10}\mathstrut -\mathstrut$$ $$103165004876$$ $$\beta_{9}\mathstrut +\mathstrut$$ $$3983130826116$$ $$\beta_{7}\mathstrut +\mathstrut$$ $$122827976301158$$ $$\beta_{6}\mathstrut +\mathstrut$$ $$4753135721619$$ $$\beta_{5}\mathstrut +\mathstrut$$ $$212262055644$$ $$\beta_{3}\mathstrut +\mathstrut$$ $$103165004876$$ $$\beta_{2}\mathstrut +\mathstrut$$ $$402665339208175$$ $$\beta_{1}$$$$)/8$$ $$\nu^{18}$$ $$=$$ $$($$$$206826061833$$ $$\beta_{19}\mathstrut -\mathstrut$$ $$1356326570262$$ $$\beta_{18}\mathstrut +\mathstrut$$ $$206826061833$$ $$\beta_{17}\mathstrut -\mathstrut$$ $$942674446596$$ $$\beta_{16}\mathstrut +\mathstrut$$ $$1563152632095$$ $$\beta_{15}\mathstrut -\mathstrut$$ $$206826061833$$ $$\beta_{14}\mathstrut -\mathstrut$$ $$206826061833$$ $$\beta_{13}\mathstrut -\mathstrut$$ $$383922347580$$ $$\beta_{12}\mathstrut -\mathstrut$$ $$5396804406460$$ $$\beta_{11}\mathstrut -\mathstrut$$ $$2789595237416$$ $$\beta_{10}\mathstrut -\mathstrut$$ $$3373248208270$$ $$\beta_{9}\mathstrut +\mathstrut$$ $$26701510478280$$ $$\beta_{8}\mathstrut +\mathstrut$$ $$27001827465579$$ $$\beta_{7}\mathstrut -\mathstrut$$ $$206826061833$$ $$\beta_{6}\mathstrut +\mathstrut$$ $$206826061833$$ $$\beta_{5}\mathstrut -\mathstrut$$ $$642850462448$$ $$\beta_{4}\mathstrut +\mathstrut$$ $$206826061833$$ $$\beta_{3}\mathstrut +\mathstrut$$ $$116255907068413$$ $$\beta_{2}\mathstrut +\mathstrut$$ $$6546304914889$$ $$\beta_{1}\mathstrut +\mathstrut$$ $$21061123527009492$$$$)/4$$ $$\nu^{19}$$ $$=$$ $$($$$$-$$$$28979466545816$$ $$\beta_{19}\mathstrut -\mathstrut$$ $$15746025254036$$ $$\beta_{18}\mathstrut -\mathstrut$$ $$15226592799096$$ $$\beta_{17}\mathstrut +\mathstrut$$ $$47319711735396$$ $$\beta_{15}\mathstrut +\mathstrut$$ $$6286559214840$$ $$\beta_{14}\mathstrut -\mathstrut$$ $$245477175628940$$ $$\beta_{13}\mathstrut +\mathstrut$$ $$54268117553448$$ $$\beta_{11}\mathstrut -\mathstrut$$ $$9760000253310$$ $$\beta_{10}\mathstrut +\mathstrut$$ $$9760000253310$$ $$\beta_{9}\mathstrut +\mathstrut$$ $$199913504082364$$ $$\beta_{7}\mathstrut -\mathstrut$$ $$2314559170464864$$ $$\beta_{6}\mathstrut -\mathstrut$$ $$366869223938318$$ $$\beta_{5}\mathstrut +\mathstrut$$ $$121054885930093$$ $$\beta_{3}\mathstrut -\mathstrut$$ $$9760000253310$$ $$\beta_{2}\mathstrut +\mathstrut$$ $$20527275242323824$$ $$\beta_{1}$$$$)/8$$

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
19.1
 −7.46306 − 2.88145i −7.46306 + 2.88145i −7.15078 − 3.58697i −7.15078 + 3.58697i −3.61621 − 7.13604i −3.61621 + 7.13604i −3.53165 − 7.17826i −3.53165 + 7.17826i −2.02968 − 7.73824i −2.02968 + 7.73824i 2.02968 − 7.73824i 2.02968 + 7.73824i 3.53165 − 7.17826i 3.53165 + 7.17826i 3.61621 − 7.13604i 3.61621 + 7.13604i 7.15078 − 3.58697i 7.15078 + 3.58697i 7.46306 − 2.88145i 7.46306 + 2.88145i
−14.9261 5.76290i −76.0331 189.578 + 172.035i 291.455 + 552.882i 1134.88 + 438.171i −269.408 −1838.24 3660.34i −779.974 −1164.08 9932.01i
19.2 −14.9261 + 5.76290i −76.0331 189.578 172.035i 291.455 552.882i 1134.88 438.171i −269.408 −1838.24 + 3660.34i −779.974 −1164.08 + 9932.01i
19.3 −14.3016 7.17394i 42.6663 153.069 + 205.197i −560.298 276.932i −610.194 306.085i 869.685 −717.054 4032.75i −4740.59 6026.43 + 7980.11i
19.4 −14.3016 + 7.17394i 42.6663 153.069 205.197i −560.298 + 276.932i −610.194 + 306.085i 869.685 −717.054 + 4032.75i −4740.59 6026.43 7980.11i
19.5 −7.23243 14.2721i 44.3270 −151.384 + 206.443i 530.961 329.705i −320.592 632.638i 2826.75 4041.25 + 667.476i −4596.11 −8545.71 5193.35i
19.6 −7.23243 + 14.2721i 44.3270 −151.384 206.443i 530.961 + 329.705i −320.592 + 632.638i 2826.75 4041.25 667.476i −4596.11 −8545.71 + 5193.35i
19.7 −7.06329 14.3565i −134.970 −156.220 + 202.809i −416.235 466.234i 953.329 + 1937.69i −1863.96 4015.06 + 810.276i 11655.8 −3753.51 + 9268.83i
19.8 −7.06329 + 14.3565i −134.970 −156.220 202.809i −416.235 + 466.234i 953.329 1937.69i −1863.96 4015.06 810.276i 11655.8 −3753.51 9268.83i
19.9 −4.05935 15.4765i 75.2390 −223.043 + 125.649i −200.884 + 591.837i −305.422 1164.44i −4411.35 2850.02 + 2941.87i −900.091 9975.01 + 706.503i
19.10 −4.05935 + 15.4765i 75.2390 −223.043 125.649i −200.884 591.837i −305.422 + 1164.44i −4411.35 2850.02 2941.87i −900.091 9975.01 706.503i
19.11 4.05935 15.4765i −75.2390 −223.043 125.649i −200.884 + 591.837i −305.422 + 1164.44i 4411.35 −2850.02 + 2941.87i −900.091 8344.09 + 5511.45i
19.12 4.05935 + 15.4765i −75.2390 −223.043 + 125.649i −200.884 591.837i −305.422 1164.44i 4411.35 −2850.02 2941.87i −900.091 8344.09 5511.45i
19.13 7.06329 14.3565i 134.970 −156.220 202.809i −416.235 466.234i 953.329 1937.69i 1863.96 −4015.06 + 810.276i 11655.8 −9633.48 + 2682.54i
19.14 7.06329 + 14.3565i 134.970 −156.220 + 202.809i −416.235 + 466.234i 953.329 + 1937.69i 1863.96 −4015.06 810.276i 11655.8 −9633.48 2682.54i
19.15 7.23243 14.2721i −44.3270 −151.384 206.443i 530.961 329.705i −320.592 + 632.638i −2826.75 −4041.25 + 667.476i −4596.11 −865.429 9962.48i
19.16 7.23243 + 14.2721i −44.3270 −151.384 + 206.443i 530.961 + 329.705i −320.592 632.638i −2826.75 −4041.25 667.476i −4596.11 −865.429 + 9962.48i
19.17 14.3016 7.17394i −42.6663 153.069 205.197i −560.298 276.932i −610.194 + 306.085i −869.685 717.054 4032.75i −4740.59 −9999.83 + 58.9826i
19.18 14.3016 + 7.17394i −42.6663 153.069 + 205.197i −560.298 + 276.932i −610.194 306.085i −869.685 717.054 + 4032.75i −4740.59 −9999.83 58.9826i
19.19 14.9261 5.76290i 76.0331 189.578 172.035i 291.455 + 552.882i 1134.88 438.171i 269.408 1838.24 3660.34i −779.974 7536.50 + 6572.76i
19.20 14.9261 + 5.76290i 76.0331 189.578 + 172.035i 291.455 552.882i 1134.88 + 438.171i 269.408 1838.24 + 3660.34i −779.974 7536.50 6572.76i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 19.20 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
5.b Even 1 yes
20.d Odd 1 yes

Hecke kernels

This newform can be constructed as the kernel of the linear operator $$T_{3}^{10}$$ $$\mathstrut -\mathstrut 33444 T_{3}^{8}$$ $$\mathstrut +\mathstrut 357004800 T_{3}^{6}$$ $$\mathstrut -\mathstrut$$$$16\!\cdots\!40$$$$T_{3}^{4}$$ $$\mathstrut +\mathstrut$$$$31\!\cdots\!00$$$$T_{3}^{2}$$ $$\mathstrut -\mathstrut$$$$21\!\cdots\!00$$ acting on $$S_{9}^{\mathrm{new}}(20, [\chi])$$.