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

Downloads

Learn more about

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:

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 \(S_{9}^{\mathrm{new}}(20, [\chi])\).