Properties

Label 74.7.d.a
Level $74$
Weight $7$
Character orbit 74.d
Analytic conductor $17.024$
Analytic rank $0$
Dimension $18$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 74 = 2 \cdot 37 \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 74.d (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(17.0240021879\)
Analytic rank: \(0\)
Dimension: \(18\)
Relative dimension: \(9\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{18} + \cdots)\)
Defining polynomial: \(x^{18} + 8470 x^{16} + 28007049 x^{14} + 45282701078 x^{12} + 36580026955844 x^{10} + 13599755691108102 x^{8} + 2140498199687229457 x^{6} + 117348928221103362406 x^{4} + 1596692663909213634249 x^{2} + 657611720884512028944\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{10}\cdot 5^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{17}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 4 - 4 \beta_{6} ) q^{2} + ( \beta_{1} - 4 \beta_{6} ) q^{3} -32 \beta_{6} q^{4} + ( 16 + 16 \beta_{6} + \beta_{7} ) q^{5} + ( -16 + 4 \beta_{1} + 4 \beta_{2} - 16 \beta_{6} ) q^{6} + ( -7 + 3 \beta_{2} - \beta_{4} ) q^{7} + ( -128 - 128 \beta_{6} ) q^{8} + ( -224 + \beta_{3} - \beta_{7} - \beta_{8} ) q^{9} +O(q^{10})\) \( q + ( 4 - 4 \beta_{6} ) q^{2} + ( \beta_{1} - 4 \beta_{6} ) q^{3} -32 \beta_{6} q^{4} + ( 16 + 16 \beta_{6} + \beta_{7} ) q^{5} + ( -16 + 4 \beta_{1} + 4 \beta_{2} - 16 \beta_{6} ) q^{6} + ( -7 + 3 \beta_{2} - \beta_{4} ) q^{7} + ( -128 - 128 \beta_{6} ) q^{8} + ( -224 + \beta_{3} - \beta_{7} - \beta_{8} ) q^{9} + ( 128 + 4 \beta_{7} + 4 \beta_{8} ) q^{10} + ( \beta_{1} + 124 \beta_{6} - \beta_{7} + \beta_{8} + \beta_{9} ) q^{11} + ( -128 + 32 \beta_{2} ) q^{12} + ( -375 - 2 \beta_{1} - 2 \beta_{2} - \beta_{4} - 375 \beta_{6} - 2 \beta_{7} - \beta_{10} - \beta_{11} - 2 \beta_{14} + \beta_{15} ) q^{13} + ( -28 - 12 \beta_{1} + 12 \beta_{2} - 4 \beta_{4} + 28 \beta_{6} + 4 \beta_{10} ) q^{14} + ( -104 + 45 \beta_{1} - 45 \beta_{2} + 5 \beta_{4} + 104 \beta_{6} + 16 \beta_{8} - 5 \beta_{10} + 5 \beta_{12} ) q^{15} -1024 q^{16} + ( -519 + 23 \beta_{1} + 23 \beta_{2} + \beta_{3} - 3 \beta_{4} - 519 \beta_{6} + \beta_{7} - 3 \beta_{10} - 8 \beta_{11} + \beta_{14} + \beta_{15} + \beta_{17} ) q^{17} + ( -896 + 4 \beta_{3} + 896 \beta_{6} - 8 \beta_{8} - 4 \beta_{17} ) q^{18} + ( 429 - 31 \beta_{1} - 31 \beta_{2} - 3 \beta_{4} - 2 \beta_{5} + 429 \beta_{6} + 17 \beta_{7} + 2 \beta_{9} - 3 \beta_{10} - 4 \beta_{11} - \beta_{15} ) q^{19} + ( 512 - 512 \beta_{6} + 32 \beta_{8} ) q^{20} + ( -28 \beta_{1} + 3176 \beta_{6} + 26 \beta_{7} - 26 \beta_{8} + 5 \beta_{9} - 3 \beta_{10} - 5 \beta_{11} + 5 \beta_{12} + 6 \beta_{13} + 6 \beta_{14} - 5 \beta_{17} ) q^{21} + ( 496 + 4 \beta_{1} + 4 \beta_{2} - 4 \beta_{5} + 496 \beta_{6} - 8 \beta_{7} + 4 \beta_{9} ) q^{22} + ( -2782 - 40 \beta_{1} - 40 \beta_{2} + 3 \beta_{3} - 3 \beta_{4} + 2 \beta_{5} - 2782 \beta_{6} - 19 \beta_{7} - 2 \beta_{9} - 3 \beta_{10} - 7 \beta_{11} - 11 \beta_{14} - 12 \beta_{15} + 3 \beta_{17} ) q^{23} + ( -512 - 128 \beta_{1} + 128 \beta_{2} + 512 \beta_{6} ) q^{24} + ( -144 \beta_{1} + 1362 \beta_{6} - \beta_{7} + \beta_{8} + 8 \beta_{9} - 7 \beta_{10} - 3 \beta_{11} + 3 \beta_{12} + 4 \beta_{13} + 4 \beta_{14} + 5 \beta_{15} + 5 \beta_{16} - 4 \beta_{17} ) q^{25} + ( -3000 - 16 \beta_{2} - 8 \beta_{4} - 8 \beta_{7} - 8 \beta_{8} - 4 \beta_{11} - 4 \beta_{12} + 8 \beta_{13} - 8 \beta_{14} + 4 \beta_{15} - 4 \beta_{16} ) q^{26} + ( -345 \beta_{1} - 1867 \beta_{6} + 23 \beta_{7} - 23 \beta_{8} - \beta_{9} + 19 \beta_{10} - 4 \beta_{11} + 4 \beta_{12} - 2 \beta_{13} - 2 \beta_{14} - 7 \beta_{15} - 7 \beta_{16} + 2 \beta_{17} ) q^{27} + ( -96 \beta_{1} + 224 \beta_{6} + 32 \beta_{10} ) q^{28} + ( -2277 + 272 \beta_{1} - 272 \beta_{2} + 5 \beta_{3} + 16 \beta_{4} - 2 \beta_{5} + 2277 \beta_{6} + 41 \beta_{8} - 2 \beta_{9} - 16 \beta_{10} + 23 \beta_{12} - 3 \beta_{13} + 4 \beta_{16} - 5 \beta_{17} ) q^{29} + ( 360 \beta_{1} + 832 \beta_{6} - 64 \beta_{7} + 64 \beta_{8} - 40 \beta_{10} - 20 \beta_{11} + 20 \beta_{12} ) q^{30} + ( -1019 - 82 \beta_{1} + 82 \beta_{2} + 4 \beta_{3} - \beta_{4} + 6 \beta_{5} + 1019 \beta_{6} + 66 \beta_{8} + 6 \beta_{9} + \beta_{10} - 8 \beta_{12} + 3 \beta_{13} + 9 \beta_{16} - 4 \beta_{17} ) q^{31} + ( -4096 + 4096 \beta_{6} ) q^{32} + ( -560 - 206 \beta_{2} + \beta_{3} + 43 \beta_{4} - 12 \beta_{5} + 50 \beta_{7} + 50 \beta_{8} + 15 \beta_{11} + 15 \beta_{12} - 13 \beta_{13} + 13 \beta_{14} + 3 \beta_{15} - 3 \beta_{16} ) q^{33} + ( -4152 + 184 \beta_{2} + 8 \beta_{3} - 24 \beta_{4} + 4 \beta_{7} + 4 \beta_{8} - 32 \beta_{11} - 32 \beta_{12} - 4 \beta_{13} + 4 \beta_{14} + 4 \beta_{15} - 4 \beta_{16} ) q^{34} + ( 3183 + 636 \beta_{1} + 636 \beta_{2} - 9 \beta_{3} - 57 \beta_{4} - 8 \beta_{5} + 3183 \beta_{6} - 81 \beta_{7} + 8 \beta_{9} - 57 \beta_{10} - 36 \beta_{11} - 2 \beta_{14} + 35 \beta_{15} - 9 \beta_{17} ) q^{35} + ( 7168 \beta_{6} + 32 \beta_{7} - 32 \beta_{8} - 32 \beta_{17} ) q^{36} + ( -13366 + 58 \beta_{1} + 170 \beta_{2} + 21 \beta_{3} - 34 \beta_{4} - 9 \beta_{5} - 3347 \beta_{6} + 104 \beta_{7} + 42 \beta_{8} - 20 \beta_{9} + 18 \beta_{10} + 21 \beta_{11} - 52 \beta_{12} + 8 \beta_{13} - 11 \beta_{14} + 15 \beta_{17} ) q^{37} + ( 3432 - 248 \beta_{2} - 24 \beta_{4} - 16 \beta_{5} + 68 \beta_{7} + 68 \beta_{8} - 16 \beta_{11} - 16 \beta_{12} - 4 \beta_{15} + 4 \beta_{16} ) q^{38} + ( 1519 - 587 \beta_{1} + 587 \beta_{2} + \beta_{3} - 112 \beta_{4} + 7 \beta_{5} - 1519 \beta_{6} - 97 \beta_{8} + 7 \beta_{9} + 112 \beta_{10} - 52 \beta_{12} + 19 \beta_{13} - 28 \beta_{16} - \beta_{17} ) q^{39} + ( -4096 \beta_{6} - 128 \beta_{7} + 128 \beta_{8} ) q^{40} + ( 1152 \beta_{1} - 2521 \beta_{6} - 60 \beta_{7} + 60 \beta_{8} + 5 \beta_{9} + 44 \beta_{10} - 7 \beta_{11} + 7 \beta_{12} + 2 \beta_{13} + 2 \beta_{14} + 20 \beta_{15} + 20 \beta_{16} - 9 \beta_{17} ) q^{41} + ( 12704 - 112 \beta_{1} - 112 \beta_{2} - 20 \beta_{3} - 12 \beta_{4} - 20 \beta_{5} + 12704 \beta_{6} + 208 \beta_{7} + 20 \beta_{9} - 12 \beta_{10} - 40 \beta_{11} + 48 \beta_{14} - 20 \beta_{17} ) q^{42} + ( -3635 + 137 \beta_{1} + 137 \beta_{2} + 9 \beta_{3} - 14 \beta_{4} - 11 \beta_{5} - 3635 \beta_{6} - 129 \beta_{7} + 11 \beta_{9} - 14 \beta_{10} + 52 \beta_{11} - 18 \beta_{14} - 41 \beta_{15} + 9 \beta_{17} ) q^{43} + ( 3968 + 32 \beta_{2} - 32 \beta_{5} - 32 \beta_{7} - 32 \beta_{8} ) q^{44} + ( -26787 + 135 \beta_{1} + 135 \beta_{2} + 30 \beta_{3} + 45 \beta_{5} - 26787 \beta_{6} - 207 \beta_{7} - 45 \beta_{9} - 15 \beta_{11} - 90 \beta_{14} - 15 \beta_{15} + 30 \beta_{17} ) q^{45} + ( -22256 - 320 \beta_{2} + 24 \beta_{3} - 24 \beta_{4} + 16 \beta_{5} - 76 \beta_{7} - 76 \beta_{8} - 28 \beta_{11} - 28 \beta_{12} + 44 \beta_{13} - 44 \beta_{14} - 48 \beta_{15} + 48 \beta_{16} ) q^{46} + ( 12623 + 275 \beta_{2} - 8 \beta_{3} + 143 \beta_{4} + 24 \beta_{5} + 232 \beta_{7} + 232 \beta_{8} + 17 \beta_{11} + 17 \beta_{12} - 32 \beta_{13} + 32 \beta_{14} + \beta_{15} - \beta_{16} ) q^{47} + ( -1024 \beta_{1} + 4096 \beta_{6} ) q^{48} + ( 43753 + 242 \beta_{2} - 31 \beta_{3} - 85 \beta_{4} - 35 \beta_{5} + 232 \beta_{7} + 232 \beta_{8} + 48 \beta_{11} + 48 \beta_{12} - 68 \beta_{13} + 68 \beta_{14} - 2 \beta_{15} + 2 \beta_{16} ) q^{49} + ( 5448 - 576 \beta_{1} - 576 \beta_{2} - 16 \beta_{3} - 28 \beta_{4} - 32 \beta_{5} + 5448 \beta_{6} - 8 \beta_{7} + 32 \beta_{9} - 28 \beta_{10} - 24 \beta_{11} + 32 \beta_{14} + 40 \beta_{15} - 16 \beta_{17} ) q^{50} + ( -21782 - 1234 \beta_{1} + 1234 \beta_{2} - 4 \beta_{3} + 135 \beta_{4} + 44 \beta_{5} + 21782 \beta_{6} - 781 \beta_{8} + 44 \beta_{9} - 135 \beta_{10} + 100 \beta_{12} + 90 \beta_{13} + 3 \beta_{16} + 4 \beta_{17} ) q^{51} + ( -12000 + 64 \beta_{1} - 64 \beta_{2} - 32 \beta_{4} + 12000 \beta_{6} - 64 \beta_{8} + 32 \beta_{10} - 32 \beta_{12} + 64 \beta_{13} - 32 \beta_{16} ) q^{52} + ( 3240 - 1236 \beta_{2} - 25 \beta_{3} + 51 \beta_{4} + 47 \beta_{5} + 114 \beta_{7} + 114 \beta_{8} + 63 \beta_{11} + 63 \beta_{12} - 48 \beta_{13} + 48 \beta_{14} + 47 \beta_{15} - 47 \beta_{16} ) q^{53} + ( -7468 - 1380 \beta_{1} - 1380 \beta_{2} + 8 \beta_{3} + 76 \beta_{4} + 4 \beta_{5} - 7468 \beta_{6} + 184 \beta_{7} - 4 \beta_{9} + 76 \beta_{10} - 32 \beta_{11} - 16 \beta_{14} - 56 \beta_{15} + 8 \beta_{17} ) q^{54} + ( 8712 - 859 \beta_{1} + 859 \beta_{2} + 39 \beta_{3} + 27 \beta_{4} - 12 \beta_{5} - 8712 \beta_{6} - 704 \beta_{8} - 12 \beta_{9} - 27 \beta_{10} + 11 \beta_{12} - 67 \beta_{13} - 85 \beta_{16} - 39 \beta_{17} ) q^{55} + ( 896 - 384 \beta_{1} - 384 \beta_{2} + 128 \beta_{4} + 896 \beta_{6} + 128 \beta_{10} ) q^{56} + ( 27699 + 1378 \beta_{1} - 1378 \beta_{2} - 53 \beta_{3} + 274 \beta_{4} + 9 \beta_{5} - 27699 \beta_{6} + 175 \beta_{8} + 9 \beta_{9} - 274 \beta_{10} + 210 \beta_{12} + \beta_{13} + 15 \beta_{16} + 53 \beta_{17} ) q^{57} + ( 2176 \beta_{1} + 18216 \beta_{6} - 164 \beta_{7} + 164 \beta_{8} - 16 \beta_{9} - 128 \beta_{10} - 92 \beta_{11} + 92 \beta_{12} - 12 \beta_{13} - 12 \beta_{14} + 16 \beta_{15} + 16 \beta_{16} - 40 \beta_{17} ) q^{58} + ( -9978 - 339 \beta_{1} - 339 \beta_{2} + 58 \beta_{3} + 57 \beta_{4} + 96 \beta_{5} - 9978 \beta_{6} + 132 \beta_{7} - 96 \beta_{9} + 57 \beta_{10} + 104 \beta_{11} - 118 \beta_{14} + 10 \beta_{15} + 58 \beta_{17} ) q^{59} + ( 3328 + 1440 \beta_{1} + 1440 \beta_{2} - 160 \beta_{4} + 3328 \beta_{6} - 512 \beta_{7} - 160 \beta_{10} - 160 \beta_{11} ) q^{60} + ( 28457 + 7 \beta_{1} - 7 \beta_{2} - 43 \beta_{3} - 216 \beta_{4} + 46 \beta_{5} - 28457 \beta_{6} - 374 \beta_{8} + 46 \beta_{9} + 216 \beta_{10} - 102 \beta_{12} - 13 \beta_{13} + 75 \beta_{16} + 43 \beta_{17} ) q^{61} + ( -656 \beta_{1} + 8152 \beta_{6} - 264 \beta_{7} + 264 \beta_{8} + 48 \beta_{9} + 8 \beta_{10} + 32 \beta_{11} - 32 \beta_{12} + 12 \beta_{13} + 12 \beta_{14} + 36 \beta_{15} + 36 \beta_{16} - 32 \beta_{17} ) q^{62} + ( 25466 - 6842 \beta_{2} - 50 \beta_{3} + 556 \beta_{4} + 241 \beta_{7} + 241 \beta_{8} + 371 \beta_{11} + 371 \beta_{12} + 11 \beta_{13} - 11 \beta_{14} - 83 \beta_{15} + 83 \beta_{16} ) q^{63} + 32768 \beta_{6} q^{64} + ( 366 \beta_{1} - 39882 \beta_{6} - 168 \beta_{7} + 168 \beta_{8} - 47 \beta_{9} - 382 \beta_{10} + 137 \beta_{11} - 137 \beta_{12} - 121 \beta_{13} - 121 \beta_{14} + 25 \beta_{15} + 25 \beta_{16} + 196 \beta_{17} ) q^{65} + ( -2240 + 824 \beta_{1} - 824 \beta_{2} + 4 \beta_{3} + 172 \beta_{4} - 48 \beta_{5} + 2240 \beta_{6} + 400 \beta_{8} - 48 \beta_{9} - 172 \beta_{10} + 120 \beta_{12} - 104 \beta_{13} - 24 \beta_{16} - 4 \beta_{17} ) q^{66} + ( 1214 \beta_{1} - 79749 \beta_{6} - 204 \beta_{7} + 204 \beta_{8} - 71 \beta_{9} + 193 \beta_{10} + 84 \beta_{11} - 84 \beta_{12} + 123 \beta_{13} + 123 \beta_{14} - 44 \beta_{15} - 44 \beta_{16} + 262 \beta_{17} ) q^{67} + ( -16608 - 736 \beta_{1} + 736 \beta_{2} + 32 \beta_{3} - 96 \beta_{4} + 16608 \beta_{6} + 32 \beta_{8} + 96 \beta_{10} - 256 \beta_{12} - 32 \beta_{13} - 32 \beta_{16} - 32 \beta_{17} ) q^{68} + ( 32815 - 2616 \beta_{1} + 2616 \beta_{2} - 182 \beta_{3} - 392 \beta_{4} + 88 \beta_{5} - 32815 \beta_{6} - 924 \beta_{8} + 88 \beta_{9} + 392 \beta_{10} - 320 \beta_{12} + \beta_{13} + 83 \beta_{16} + 182 \beta_{17} ) q^{69} + ( 25464 + 5088 \beta_{2} - 72 \beta_{3} - 456 \beta_{4} - 64 \beta_{5} - 324 \beta_{7} - 324 \beta_{8} - 144 \beta_{11} - 144 \beta_{12} + 8 \beta_{13} - 8 \beta_{14} + 140 \beta_{15} - 140 \beta_{16} ) q^{70} + ( -10431 - 173 \beta_{2} + 146 \beta_{3} - 15 \beta_{4} + 70 \beta_{5} - 1052 \beta_{7} - 1052 \beta_{8} - 20 \beta_{11} - 20 \beta_{12} - 156 \beta_{13} + 156 \beta_{14} - 19 \beta_{15} + 19 \beta_{16} ) q^{71} + ( 28672 - 128 \beta_{3} + 28672 \beta_{6} + 256 \beta_{7} - 128 \beta_{17} ) q^{72} + ( -1188 \beta_{1} + 1027 \beta_{6} + 1190 \beta_{7} - 1190 \beta_{8} - 88 \beta_{9} + 102 \beta_{10} + 436 \beta_{11} - 436 \beta_{12} - 14 \beta_{13} - 14 \beta_{14} + 67 \beta_{15} + 67 \beta_{16} - 343 \beta_{17} ) q^{73} + ( -66852 - 448 \beta_{1} + 912 \beta_{2} + 144 \beta_{3} - 64 \beta_{4} + 44 \beta_{5} + 40076 \beta_{6} + 248 \beta_{7} + 584 \beta_{8} - 116 \beta_{9} + 208 \beta_{10} + 292 \beta_{11} - 124 \beta_{12} + 76 \beta_{13} - 12 \beta_{14} - 24 \beta_{17} ) q^{74} + ( 138397 - 2279 \beta_{2} - 64 \beta_{3} + 713 \beta_{4} - 53 \beta_{5} + 224 \beta_{7} + 224 \beta_{8} + 252 \beta_{11} + 252 \beta_{12} + 11 \beta_{13} - 11 \beta_{14} + 30 \beta_{15} - 30 \beta_{16} ) q^{75} + ( 13728 + 992 \beta_{1} - 992 \beta_{2} - 96 \beta_{4} - 64 \beta_{5} - 13728 \beta_{6} + 544 \beta_{8} - 64 \beta_{9} + 96 \beta_{10} - 128 \beta_{12} + 32 \beta_{16} ) q^{76} + ( 5284 \beta_{1} - 56532 \beta_{6} + 319 \beta_{7} - 319 \beta_{8} - 273 \beta_{9} - 113 \beta_{10} + 55 \beta_{11} - 55 \beta_{12} + 47 \beta_{13} + 47 \beta_{14} - 30 \beta_{15} - 30 \beta_{16} + 165 \beta_{17} ) q^{77} + ( -4696 \beta_{1} - 12152 \beta_{6} + 388 \beta_{7} - 388 \beta_{8} + 56 \beta_{9} + 896 \beta_{10} + 208 \beta_{11} - 208 \beta_{12} + 76 \beta_{13} + 76 \beta_{14} - 112 \beta_{15} - 112 \beta_{16} - 8 \beta_{17} ) q^{78} + ( 95299 + 3977 \beta_{1} + 3977 \beta_{2} - 217 \beta_{3} + 47 \beta_{4} + 115 \beta_{5} + 95299 \beta_{6} + 1394 \beta_{7} - 115 \beta_{9} + 47 \beta_{10} + 91 \beta_{11} + 193 \beta_{14} - 167 \beta_{15} - 217 \beta_{17} ) q^{79} + ( -16384 - 16384 \beta_{6} - 1024 \beta_{7} ) q^{80} + ( 153731 - 844 \beta_{2} + 186 \beta_{3} + 322 \beta_{4} - 25 \beta_{5} + 93 \beta_{7} + 93 \beta_{8} - \beta_{11} - \beta_{12} - 136 \beta_{13} + 136 \beta_{14} - 90 \beta_{15} + 90 \beta_{16} ) q^{81} + ( -10084 + 4608 \beta_{1} + 4608 \beta_{2} - 36 \beta_{3} + 176 \beta_{4} - 20 \beta_{5} - 10084 \beta_{6} - 480 \beta_{7} + 20 \beta_{9} + 176 \beta_{10} - 56 \beta_{11} + 16 \beta_{14} + 160 \beta_{15} - 36 \beta_{17} ) q^{82} + ( 129491 + 7245 \beta_{2} - 618 \beta_{3} - 845 \beta_{4} - 162 \beta_{5} - 1484 \beta_{7} - 1484 \beta_{8} + 124 \beta_{11} + 124 \beta_{12} - 66 \beta_{13} + 66 \beta_{14} + 229 \beta_{15} - 229 \beta_{16} ) q^{83} + ( 101632 - 896 \beta_{2} - 160 \beta_{3} - 96 \beta_{4} - 160 \beta_{5} + 832 \beta_{7} + 832 \beta_{8} - 160 \beta_{11} - 160 \beta_{12} - 192 \beta_{13} + 192 \beta_{14} ) q^{84} + ( 15876 \beta_{1} + 36812 \beta_{6} - 361 \beta_{7} + 361 \beta_{8} - 132 \beta_{9} - 592 \beta_{10} - 93 \beta_{11} + 93 \beta_{12} + 44 \beta_{13} + 44 \beta_{14} - 100 \beta_{15} - 100 \beta_{16} + 86 \beta_{17} ) q^{85} + ( -29080 + 1096 \beta_{2} + 72 \beta_{3} - 112 \beta_{4} - 88 \beta_{5} - 516 \beta_{7} - 516 \beta_{8} + 208 \beta_{11} + 208 \beta_{12} + 72 \beta_{13} - 72 \beta_{14} - 164 \beta_{15} + 164 \beta_{16} ) q^{86} + ( -229010 - 5472 \beta_{1} - 5472 \beta_{2} + 228 \beta_{3} - 151 \beta_{4} + 193 \beta_{5} - 229010 \beta_{6} - 3714 \beta_{7} - 193 \beta_{9} - 151 \beta_{10} - 236 \beta_{11} - 421 \beta_{14} - 67 \beta_{15} + 228 \beta_{17} ) q^{87} + ( 15872 - 128 \beta_{1} + 128 \beta_{2} - 128 \beta_{5} - 15872 \beta_{6} - 256 \beta_{8} - 128 \beta_{9} ) q^{88} + ( 63521 + 854 \beta_{1} - 854 \beta_{2} - 192 \beta_{3} + 44 \beta_{4} + 77 \beta_{5} - 63521 \beta_{6} + 1847 \beta_{8} + 77 \beta_{9} - 44 \beta_{10} - 772 \beta_{12} - 219 \beta_{13} - 27 \beta_{16} + 192 \beta_{17} ) q^{89} + ( -214296 + 1080 \beta_{2} + 240 \beta_{3} + 360 \beta_{5} - 828 \beta_{7} - 828 \beta_{8} - 60 \beta_{11} - 60 \beta_{12} + 360 \beta_{13} - 360 \beta_{14} - 60 \beta_{15} + 60 \beta_{16} ) q^{90} + ( 29329 - 16425 \beta_{1} - 16425 \beta_{2} - 176 \beta_{3} + 1222 \beta_{4} + 70 \beta_{5} + 29329 \beta_{6} + 5319 \beta_{7} - 70 \beta_{9} + 1222 \beta_{10} + 180 \beta_{11} - 132 \beta_{14} + 371 \beta_{15} - 176 \beta_{17} ) q^{91} + ( -89024 + 1280 \beta_{1} - 1280 \beta_{2} + 96 \beta_{3} - 96 \beta_{4} + 64 \beta_{5} + 89024 \beta_{6} - 608 \beta_{8} + 64 \beta_{9} + 96 \beta_{10} - 224 \beta_{12} + 352 \beta_{13} + 384 \beta_{16} - 96 \beta_{17} ) q^{92} + ( 88150 - 181 \beta_{1} - 181 \beta_{2} + 39 \beta_{3} - 137 \beta_{4} - 123 \beta_{5} + 88150 \beta_{6} + 527 \beta_{7} + 123 \beta_{9} - 137 \beta_{10} - 145 \beta_{11} + 128 \beta_{14} + 115 \beta_{15} + 39 \beta_{17} ) q^{93} + ( 50492 - 1100 \beta_{1} + 1100 \beta_{2} - 32 \beta_{3} + 572 \beta_{4} + 96 \beta_{5} - 50492 \beta_{6} + 1856 \beta_{8} + 96 \beta_{9} - 572 \beta_{10} + 136 \beta_{12} - 256 \beta_{13} - 8 \beta_{16} + 32 \beta_{17} ) q^{94} + ( -840 \beta_{1} + 336840 \beta_{6} + 1315 \beta_{7} - 1315 \beta_{8} + 250 \beta_{9} + 80 \beta_{10} - 45 \beta_{11} + 45 \beta_{12} + 60 \beta_{13} + 60 \beta_{14} - 65 \beta_{15} - 65 \beta_{16} - 250 \beta_{17} ) q^{95} + ( 16384 - 4096 \beta_{1} - 4096 \beta_{2} + 16384 \beta_{6} ) q^{96} + ( -91889 + 397 \beta_{1} + 397 \beta_{2} + 298 \beta_{3} - 430 \beta_{4} - 267 \beta_{5} - 91889 \beta_{6} - 3616 \beta_{7} + 267 \beta_{9} - 430 \beta_{10} + 94 \beta_{11} + 126 \beta_{14} + 236 \beta_{15} + 298 \beta_{17} ) q^{97} + ( 175012 - 968 \beta_{1} + 968 \beta_{2} - 124 \beta_{3} - 340 \beta_{4} - 140 \beta_{5} - 175012 \beta_{6} + 1856 \beta_{8} - 140 \beta_{9} + 340 \beta_{10} + 384 \beta_{12} - 544 \beta_{13} + 16 \beta_{16} + 124 \beta_{17} ) q^{98} + ( 718 \beta_{1} - 71764 \beta_{6} - 4292 \beta_{7} + 4292 \beta_{8} + 237 \beta_{9} - 1724 \beta_{10} - 568 \beta_{11} + 568 \beta_{12} - 446 \beta_{13} - 446 \beta_{14} - 29 \beta_{15} - 29 \beta_{16} + 70 \beta_{17} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18q + 72q^{2} + 294q^{5} - 256q^{6} - 104q^{7} - 2304q^{8} - 4042q^{9} + O(q^{10}) \) \( 18q + 72q^{2} + 294q^{5} - 256q^{6} - 104q^{7} - 2304q^{8} - 4042q^{9} + 2352q^{10} - 2048q^{12} - 6766q^{13} - 416q^{14} - 2136q^{15} - 18432q^{16} - 9134q^{17} - 16168q^{18} + 7578q^{19} + 9408q^{20} + 8928q^{22} - 50578q^{23} - 8192q^{24} - 54128q^{26} - 42950q^{29} - 17358q^{31} - 73728q^{32} - 11056q^{33} - 73072q^{34} + 62152q^{35} - 238242q^{37} + 60624q^{38} + 31572q^{39} + 229024q^{42} - 65470q^{43} + 71424q^{44} - 482358q^{45} - 404624q^{46} + 232192q^{47} + 791686q^{49} + 93752q^{50} - 386848q^{51} - 216512q^{52} + 49972q^{53} - 144560q^{54} + 160168q^{55} + 13312q^{56} + 488476q^{57} - 181570q^{59} + 68352q^{60} + 508802q^{61} + 404788q^{63} - 44224q^{66} - 292288q^{68} + 604532q^{69} + 497216q^{70} - 202632q^{71} + 517376q^{72} - 1191224q^{74} + 2476628q^{75} + 242496q^{76} + 1752858q^{79} - 301056q^{80} + 2760658q^{81} - 145808q^{82} + 2371616q^{83} + 1832192q^{84} - 523760q^{86} - 4188080q^{87} + 285696q^{88} + 1148346q^{89} - 3858864q^{90} + 433120q^{91} - 1618496q^{92} + 1589664q^{93} + 928768q^{94} + 262144q^{96} - 1670270q^{97} + 3166744q^{98} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{18} + 8470 x^{16} + 28007049 x^{14} + 45282701078 x^{12} + 36580026955844 x^{10} + 13599755691108102 x^{8} + 2140498199687229457 x^{6} + 117348928221103362406 x^{4} + 1596692663909213634249 x^{2} + 657611720884512028944\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\(-65285699492967695899218651215 \nu^{16} - 547367128967772941778063219864587 \nu^{14} - 1782904871642358507069246048037609740 \nu^{12} - 2813208893823245560689659957694343457278 \nu^{10} - 2172571750214881426241678230870620738383902 \nu^{8} - 732621727526351353644660282046547033780474844 \nu^{6} - 94105867536987384846618673287945534887203487651 \nu^{4} - 3081517034507571101947622490876940133441468954663 \nu^{2} - 4921257925665401457031200541511071743997455234288\)\()/ \)\(55\!\cdots\!24\)\( \)
\(\beta_{3}\)\(=\)\((\)\(-\)\(91\!\cdots\!07\)\( \nu^{16} - \)\(77\!\cdots\!47\)\( \nu^{14} - \)\(25\!\cdots\!60\)\( \nu^{12} - \)\(40\!\cdots\!66\)\( \nu^{10} - \)\(31\!\cdots\!34\)\( \nu^{8} - \)\(11\!\cdots\!08\)\( \nu^{6} - \)\(14\!\cdots\!87\)\( \nu^{4} - \)\(52\!\cdots\!19\)\( \nu^{2} + \)\(16\!\cdots\!08\)\(\)\()/ \)\(21\!\cdots\!60\)\( \)
\(\beta_{4}\)\(=\)\((\)\(-\)\(13\!\cdots\!73\)\( \nu^{16} - \)\(11\!\cdots\!13\)\( \nu^{14} - \)\(37\!\cdots\!40\)\( \nu^{12} - \)\(60\!\cdots\!74\)\( \nu^{10} - \)\(48\!\cdots\!86\)\( \nu^{8} - \)\(18\!\cdots\!72\)\( \nu^{6} - \)\(28\!\cdots\!93\)\( \nu^{4} - \)\(15\!\cdots\!81\)\( \nu^{2} - \)\(12\!\cdots\!88\)\(\)\()/ \)\(21\!\cdots\!60\)\( \)
\(\beta_{5}\)\(=\)\((\)\(-\)\(19\!\cdots\!19\)\( \nu^{16} - \)\(16\!\cdots\!99\)\( \nu^{14} - \)\(53\!\cdots\!00\)\( \nu^{12} - \)\(86\!\cdots\!82\)\( \nu^{10} - \)\(68\!\cdots\!58\)\( \nu^{8} - \)\(24\!\cdots\!56\)\( \nu^{6} - \)\(35\!\cdots\!79\)\( \nu^{4} - \)\(13\!\cdots\!23\)\( \nu^{2} - \)\(14\!\cdots\!04\)\(\)\()/ \)\(53\!\cdots\!40\)\( \)
\(\beta_{6}\)\(=\)\((\)\(-47976809463306235107812730208725294281 \nu^{17} - 405945030494258320912910160725780394466925 \nu^{15} - 1340179690883323730779651179789192819560221908 \nu^{13} - 2161089344674311592099595006427508100309716283698 \nu^{11} - 1736957542550563876081046060025461947614608026342930 \nu^{9} - 638544561979467423724339429181817620066436921239572356 \nu^{7} - 97997447155114320476374052666277617737790411816651617685 \nu^{5} - 5026715836281384982318719951054910588453638210610039206833 \nu^{3} - 56848659288659020433788355071061867950266517166844759112080 \nu\)\()/ \)\(35\!\cdots\!72\)\( \)
\(\beta_{7}\)\(=\)\((\)\(\)\(39\!\cdots\!09\)\( \nu^{17} - \)\(80\!\cdots\!53\)\( \nu^{16} + \)\(33\!\cdots\!69\)\( \nu^{15} - \)\(67\!\cdots\!53\)\( \nu^{14} + \)\(11\!\cdots\!40\)\( \nu^{13} - \)\(22\!\cdots\!40\)\( \nu^{12} + \)\(17\!\cdots\!22\)\( \nu^{11} - \)\(35\!\cdots\!74\)\( \nu^{10} + \)\(14\!\cdots\!98\)\( \nu^{9} - \)\(28\!\cdots\!26\)\( \nu^{8} + \)\(53\!\cdots\!16\)\( \nu^{7} - \)\(98\!\cdots\!12\)\( \nu^{6} + \)\(81\!\cdots\!89\)\( \nu^{5} - \)\(13\!\cdots\!93\)\( \nu^{4} + \)\(43\!\cdots\!93\)\( \nu^{3} - \)\(50\!\cdots\!01\)\( \nu^{2} + \)\(59\!\cdots\!44\)\( \nu - \)\(39\!\cdots\!08\)\(\)\()/ \)\(48\!\cdots\!80\)\( \)
\(\beta_{8}\)\(=\)\((\)\(-\)\(39\!\cdots\!09\)\( \nu^{17} - \)\(80\!\cdots\!53\)\( \nu^{16} - \)\(33\!\cdots\!69\)\( \nu^{15} - \)\(67\!\cdots\!53\)\( \nu^{14} - \)\(11\!\cdots\!40\)\( \nu^{13} - \)\(22\!\cdots\!40\)\( \nu^{12} - \)\(17\!\cdots\!22\)\( \nu^{11} - \)\(35\!\cdots\!74\)\( \nu^{10} - \)\(14\!\cdots\!98\)\( \nu^{9} - \)\(28\!\cdots\!26\)\( \nu^{8} - \)\(53\!\cdots\!16\)\( \nu^{7} - \)\(98\!\cdots\!12\)\( \nu^{6} - \)\(81\!\cdots\!89\)\( \nu^{5} - \)\(13\!\cdots\!93\)\( \nu^{4} - \)\(43\!\cdots\!93\)\( \nu^{3} - \)\(50\!\cdots\!01\)\( \nu^{2} - \)\(59\!\cdots\!44\)\( \nu - \)\(39\!\cdots\!08\)\(\)\()/ \)\(48\!\cdots\!80\)\( \)
\(\beta_{9}\)\(=\)\((\)\(\)\(87\!\cdots\!81\)\( \nu^{17} + \)\(73\!\cdots\!21\)\( \nu^{15} + \)\(24\!\cdots\!60\)\( \nu^{13} + \)\(39\!\cdots\!98\)\( \nu^{11} + \)\(31\!\cdots\!82\)\( \nu^{9} + \)\(11\!\cdots\!64\)\( \nu^{7} + \)\(16\!\cdots\!81\)\( \nu^{5} + \)\(69\!\cdots\!57\)\( \nu^{3} - \)\(55\!\cdots\!64\)\( \nu\)\()/ \)\(22\!\cdots\!80\)\( \)
\(\beta_{10}\)\(=\)\((\)\(\)\(11\!\cdots\!97\)\( \nu^{17} + \)\(93\!\cdots\!37\)\( \nu^{15} + \)\(30\!\cdots\!80\)\( \nu^{13} + \)\(50\!\cdots\!46\)\( \nu^{11} + \)\(40\!\cdots\!34\)\( \nu^{9} + \)\(15\!\cdots\!88\)\( \nu^{7} + \)\(24\!\cdots\!77\)\( \nu^{5} + \)\(13\!\cdots\!09\)\( \nu^{3} + \)\(20\!\cdots\!12\)\( \nu\)\()/ \)\(22\!\cdots\!80\)\( \)
\(\beta_{11}\)\(=\)\((\)\(-\)\(63\!\cdots\!33\)\( \nu^{17} + \)\(79\!\cdots\!41\)\( \nu^{16} - \)\(53\!\cdots\!13\)\( \nu^{15} + \)\(67\!\cdots\!21\)\( \nu^{14} - \)\(17\!\cdots\!60\)\( \nu^{13} + \)\(22\!\cdots\!40\)\( \nu^{12} - \)\(28\!\cdots\!14\)\( \nu^{11} + \)\(35\!\cdots\!78\)\( \nu^{10} - \)\(23\!\cdots\!86\)\( \nu^{9} + \)\(28\!\cdots\!42\)\( \nu^{8} - \)\(86\!\cdots\!92\)\( \nu^{7} + \)\(10\!\cdots\!04\)\( \nu^{6} - \)\(13\!\cdots\!73\)\( \nu^{5} + \)\(16\!\cdots\!01\)\( \nu^{4} - \)\(77\!\cdots\!41\)\( \nu^{3} + \)\(83\!\cdots\!77\)\( \nu^{2} - \)\(11\!\cdots\!28\)\( \nu + \)\(74\!\cdots\!76\)\(\)\()/ \)\(91\!\cdots\!20\)\( \)
\(\beta_{12}\)\(=\)\((\)\(\)\(63\!\cdots\!33\)\( \nu^{17} + \)\(79\!\cdots\!41\)\( \nu^{16} + \)\(53\!\cdots\!13\)\( \nu^{15} + \)\(67\!\cdots\!21\)\( \nu^{14} + \)\(17\!\cdots\!60\)\( \nu^{13} + \)\(22\!\cdots\!40\)\( \nu^{12} + \)\(28\!\cdots\!14\)\( \nu^{11} + \)\(35\!\cdots\!78\)\( \nu^{10} + \)\(23\!\cdots\!86\)\( \nu^{9} + \)\(28\!\cdots\!42\)\( \nu^{8} + \)\(86\!\cdots\!92\)\( \nu^{7} + \)\(10\!\cdots\!04\)\( \nu^{6} + \)\(13\!\cdots\!73\)\( \nu^{5} + \)\(16\!\cdots\!01\)\( \nu^{4} + \)\(77\!\cdots\!41\)\( \nu^{3} + \)\(83\!\cdots\!77\)\( \nu^{2} + \)\(11\!\cdots\!28\)\( \nu + \)\(74\!\cdots\!76\)\(\)\()/ \)\(91\!\cdots\!20\)\( \)
\(\beta_{13}\)\(=\)\((\)\(-\)\(10\!\cdots\!77\)\( \nu^{17} + \)\(44\!\cdots\!67\)\( \nu^{16} - \)\(87\!\cdots\!69\)\( \nu^{15} + \)\(37\!\cdots\!11\)\( \nu^{14} - \)\(28\!\cdots\!88\)\( \nu^{13} + \)\(12\!\cdots\!16\)\( \nu^{12} - \)\(46\!\cdots\!30\)\( \nu^{11} + \)\(20\!\cdots\!74\)\( \nu^{10} - \)\(37\!\cdots\!58\)\( \nu^{9} + \)\(16\!\cdots\!26\)\( \nu^{8} - \)\(13\!\cdots\!52\)\( \nu^{7} + \)\(60\!\cdots\!56\)\( \nu^{6} - \)\(21\!\cdots\!61\)\( \nu^{5} + \)\(92\!\cdots\!95\)\( \nu^{4} - \)\(11\!\cdots\!29\)\( \nu^{3} + \)\(33\!\cdots\!91\)\( \nu^{2} - \)\(11\!\cdots\!32\)\( \nu - \)\(72\!\cdots\!08\)\(\)\()/ \)\(12\!\cdots\!92\)\( \)
\(\beta_{14}\)\(=\)\((\)\(-\)\(10\!\cdots\!77\)\( \nu^{17} - \)\(44\!\cdots\!67\)\( \nu^{16} - \)\(87\!\cdots\!69\)\( \nu^{15} - \)\(37\!\cdots\!11\)\( \nu^{14} - \)\(28\!\cdots\!88\)\( \nu^{13} - \)\(12\!\cdots\!16\)\( \nu^{12} - \)\(46\!\cdots\!30\)\( \nu^{11} - \)\(20\!\cdots\!74\)\( \nu^{10} - \)\(37\!\cdots\!58\)\( \nu^{9} - \)\(16\!\cdots\!26\)\( \nu^{8} - \)\(13\!\cdots\!52\)\( \nu^{7} - \)\(60\!\cdots\!56\)\( \nu^{6} - \)\(21\!\cdots\!61\)\( \nu^{5} - \)\(92\!\cdots\!95\)\( \nu^{4} - \)\(11\!\cdots\!29\)\( \nu^{3} - \)\(33\!\cdots\!91\)\( \nu^{2} - \)\(11\!\cdots\!32\)\( \nu + \)\(72\!\cdots\!08\)\(\)\()/ \)\(12\!\cdots\!92\)\( \)
\(\beta_{15}\)\(=\)\((\)\(\)\(23\!\cdots\!57\)\( \nu^{17} - \)\(16\!\cdots\!44\)\( \nu^{16} + \)\(19\!\cdots\!97\)\( \nu^{15} - \)\(14\!\cdots\!04\)\( \nu^{14} + \)\(65\!\cdots\!40\)\( \nu^{13} - \)\(47\!\cdots\!60\)\( \nu^{12} + \)\(10\!\cdots\!46\)\( \nu^{11} - \)\(78\!\cdots\!52\)\( \nu^{10} + \)\(86\!\cdots\!34\)\( \nu^{9} - \)\(65\!\cdots\!28\)\( \nu^{8} + \)\(32\!\cdots\!88\)\( \nu^{7} - \)\(26\!\cdots\!36\)\( \nu^{6} + \)\(51\!\cdots\!37\)\( \nu^{5} - \)\(46\!\cdots\!44\)\( \nu^{4} + \)\(29\!\cdots\!89\)\( \nu^{3} - \)\(29\!\cdots\!68\)\( \nu^{2} + \)\(43\!\cdots\!92\)\( \nu - \)\(28\!\cdots\!24\)\(\)\()/ \)\(22\!\cdots\!80\)\( \)
\(\beta_{16}\)\(=\)\((\)\(\)\(23\!\cdots\!57\)\( \nu^{17} + \)\(16\!\cdots\!44\)\( \nu^{16} + \)\(19\!\cdots\!97\)\( \nu^{15} + \)\(14\!\cdots\!04\)\( \nu^{14} + \)\(65\!\cdots\!40\)\( \nu^{13} + \)\(47\!\cdots\!60\)\( \nu^{12} + \)\(10\!\cdots\!46\)\( \nu^{11} + \)\(78\!\cdots\!52\)\( \nu^{10} + \)\(86\!\cdots\!34\)\( \nu^{9} + \)\(65\!\cdots\!28\)\( \nu^{8} + \)\(32\!\cdots\!88\)\( \nu^{7} + \)\(26\!\cdots\!36\)\( \nu^{6} + \)\(51\!\cdots\!37\)\( \nu^{5} + \)\(46\!\cdots\!44\)\( \nu^{4} + \)\(29\!\cdots\!89\)\( \nu^{3} + \)\(29\!\cdots\!68\)\( \nu^{2} + \)\(43\!\cdots\!92\)\( \nu + \)\(28\!\cdots\!24\)\(\)\()/ \)\(22\!\cdots\!80\)\( \)
\(\beta_{17}\)\(=\)\((\)\(-\)\(49\!\cdots\!89\)\( \nu^{17} - \)\(41\!\cdots\!09\)\( \nu^{15} - \)\(13\!\cdots\!20\)\( \nu^{13} - \)\(22\!\cdots\!42\)\( \nu^{11} - \)\(17\!\cdots\!58\)\( \nu^{9} - \)\(65\!\cdots\!96\)\( \nu^{7} - \)\(10\!\cdots\!69\)\( \nu^{5} - \)\(51\!\cdots\!93\)\( \nu^{3} - \)\(56\!\cdots\!64\)\( \nu\)\()/ \)\(45\!\cdots\!60\)\( \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(-\beta_{8} - \beta_{7} + \beta_{3} - 8 \beta_{2} - 937\)
\(\nu^{3}\)\(=\)\(14 \beta_{17} - 7 \beta_{16} - 7 \beta_{15} - 2 \beta_{14} - 2 \beta_{13} + 4 \beta_{12} - 4 \beta_{11} + 19 \beta_{10} - \beta_{9} - 11 \beta_{8} + 11 \beta_{7} - 7343 \beta_{6} - 1851 \beta_{1}\)
\(\nu^{4}\)\(=\)\(-22 \beta_{16} + 22 \beta_{15} + 168 \beta_{14} - 168 \beta_{13} + 63 \beta_{12} + 63 \beta_{11} + 2008 \beta_{8} + 2008 \beta_{7} - 41 \beta_{5} + 18 \beta_{4} - 2129 \beta_{3} + 28260 \beta_{2} + 1733781\)
\(\nu^{5}\)\(=\)\(-42316 \beta_{17} + 15693 \beta_{16} + 15693 \beta_{15} + 4569 \beta_{14} + 4569 \beta_{13} - 4530 \beta_{12} + 4530 \beta_{11} - 54810 \beta_{10} + 1119 \beta_{9} + 14426 \beta_{8} - 14426 \beta_{7} + 26272990 \beta_{6} + 3738743 \beta_{1}\)
\(\nu^{6}\)\(=\)\(143194 \beta_{16} - 143194 \beta_{15} - 513599 \beta_{14} + 513599 \beta_{13} - 116539 \beta_{12} - 116539 \beta_{11} - 4569103 \beta_{8} - 4569103 \beta_{7} + 208402 \beta_{5} + 309913 \beta_{4} + 4407785 \beta_{3} - 78938382 \beta_{2} - 3499517624\)
\(\nu^{7}\)\(=\)\(110488610 \beta_{17} - 32413330 \beta_{16} - 32413330 \beta_{15} - 11864139 \beta_{14} - 11864139 \beta_{13} - 1001274 \beta_{12} + 1001274 \beta_{11} + 138550746 \beta_{10} - 2785046 \beta_{9} - 9787910 \beta_{8} + 9787910 \beta_{7} - 73905766182 \beta_{6} - 7783735593 \beta_{1}\)
\(\nu^{8}\)\(=\)\(-506435436 \beta_{16} + 506435436 \beta_{15} + 1332026025 \beta_{14} - 1332026025 \beta_{13} + 76645848 \beta_{12} + 76645848 \beta_{11} + 10452257053 \beta_{8} + 10452257053 \beta_{7} - 659318844 \beta_{5} - 1752784998 \beta_{4} - 9256097563 \beta_{3} + 202515470504 \beta_{2} + 7281799905607\)
\(\nu^{9}\)\(=\)\(-273091103384 \beta_{17} + 67670207269 \beta_{16} + 67670207269 \beta_{15} + 32867115281 \beta_{14} + 32867115281 \beta_{13} + 23483558474 \beta_{12} - 23483558474 \beta_{11} - 338639389483 \beta_{10} + 10106563273 \beta_{9} - 34379083441 \beta_{8} + 34379083441 \beta_{7} + 190330441591403 \beta_{6} + 16510404730719 \beta_{1}\)
\(\nu^{10}\)\(=\)\(1470333351580 \beta_{16} - 1470333351580 \beta_{15} - 3291295829811 \beta_{14} + 3291295829811 \beta_{13} + 425322984441 \beta_{12} + 425322984441 \beta_{11} - 23709740994160 \beta_{8} - 23709740994160 \beta_{7} + 1802623183331 \beta_{5} + 6496060812180 \beta_{4} + 19739852092115 \beta_{3} - 497760635062680 \beta_{2} - 15447311291061267\)
\(\nu^{11}\)\(=\)\(655046564680222 \beta_{17} - 144203763819243 \beta_{16} - 144203763819243 \beta_{15} - 90905309850390 \beta_{14} - 90905309850390 \beta_{13} - 93562756280718 \beta_{12} + 93562756280718 \beta_{11} + 818095662960240 \beta_{10} - 35320421155575 \beta_{9} + 211842445328554 \beta_{8} - 211842445328554 \beta_{7} - 468828908740652836 \beta_{6} - 35525749891243823 \beta_{1}\)
\(\nu^{12}\)\(=\)\(-3901054034928430 \beta_{16} + 3901054034928430 \beta_{15} + 7966641685471148 \beta_{14} - 7966641685471148 \beta_{13} - 2546545941050909 \beta_{12} - 2546545941050909 \beta_{11} + 53622088750003429 \beta_{8} + 53622088750003429 \beta_{7} - 4634648505205594 \beta_{5} - 20427071298264109 \beta_{4} - 42645663883247123 \beta_{3} + 1193218895720415054 \beta_{2} + 33260163828231002300\)
\(\nu^{13}\)\(=\)\(-1542831710135714096 \beta_{17} + 313076878589822260 \beta_{16} + 313076878589822260 \beta_{15} + 245956470677508678 \beta_{14} + 245956470677508678 \beta_{13} + 292383138701046384 \beta_{12} - 292383138701046384 \beta_{11} - 1968019151048706564 \beta_{10} + 112471983721616468 \beta_{9} - 781185027720988324 \beta_{8} + 781185027720988324 \beta_{7} + 1125397883012634363828 \beta_{6} + 77327588909664398601 \beta_{1}\)
\(\nu^{14}\)\(=\)\(9863361536820312984 \beta_{16} - 9863361536820312984 \beta_{15} - 19099174815260248362 \beta_{14} + 19099174815260248362 \beta_{13} + 9612294749812460472 \beta_{12} + 9612294749812460472 \beta_{11} - 121523828574980213317 \beta_{8} - 121523828574980213317 \beta_{7} + 11569909190855397204 \beta_{5} + 58872884996333788488 \beta_{4} + 93097929084357805981 \beta_{3} - 2815850690274827089520 \beta_{2} - 72471676293911637015985\)
\(\nu^{15}\)\(=\)\(3592073221495162402130 \beta_{17} - 690109081884434375743 \beta_{16} - 690109081884434375743 \beta_{15} - 649435871629036871492 \beta_{14} - 649435871629036871492 \beta_{13} - 829774913749450441040 \beta_{12} + 829774913749450441040 \beta_{11} + 4727887567738709001955 \beta_{10} - 332347992390402002629 \beta_{9} + 2425373738928848212093 \beta_{8} - 2425373738928848212093 \beta_{7} - 2658385147869480204247127 \beta_{6} - 169899541413250083652011 \beta_{1}\)
\(\nu^{16}\)\(=\)\(-24241080386846798968906 \beta_{16} + 24241080386846798968906 \beta_{15} + 45586412105438354158698 \beta_{14} - 45586412105438354158698 \beta_{13} - 30708037200637173122493 \beta_{12} - 30708037200637173122493 \beta_{11} + 276764788190644023455752 \beta_{8} + 276764788190644023455752 \beta_{7} - 28450490103523357416221 \beta_{5} - 160847244784565351490702 \beta_{4} - 204950623793612995413173 \beta_{3} + 6577630523084595075781476 \beta_{2} + 159433570148683075245625005\)
\(\nu^{17}\)\(=\)\(-8300978493809457511836880 \beta_{17} + 1539466420434824459622645 \beta_{16} + 1539466420434824459622645 \beta_{15} + 1678169015598633563987871 \beta_{14} + 1678169015598633563987871 \beta_{13} + 2239994086789314087415014 \beta_{12} - 2239994086789314087415014 \beta_{11} - 11354107177345041942558666 \beta_{10} + 930282902945963082641235 \beta_{9} - 6895913979940064129478922 \beta_{8} + 6895913979940064129478922 \beta_{7} + 6214719860918864868480787438 \beta_{6} + 376152153752231132474778047 \beta_{1}\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/74\mathbb{Z}\right)^\times\).

\(n\) \(39\)
\(\chi(n)\) \(-\beta_{6}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
31.1
48.2475i
46.4181i
14.7444i
4.39870i
0.651946i
8.36225i
19.7872i
38.5266i
42.4806i
42.4806i
38.5266i
19.7872i
8.36225i
0.651946i
4.39870i
14.7444i
46.4181i
48.2475i
4.00000 + 4.00000i 44.2475i 32.0000i 111.595 111.595i 176.990 176.990i 600.642 −128.000 + 128.000i −1228.84 892.760
31.2 4.00000 + 4.00000i 42.4181i 32.0000i −29.2519 + 29.2519i 169.672 169.672i −271.084 −128.000 + 128.000i −1070.29 −234.015
31.3 4.00000 + 4.00000i 10.7444i 32.0000i 60.1111 60.1111i 42.9776 42.9776i −174.087 −128.000 + 128.000i 613.558 480.889
31.4 4.00000 + 4.00000i 0.398699i 32.0000i 44.3245 44.3245i 1.59480 1.59480i −329.540 −128.000 + 128.000i 728.841 354.596
31.5 4.00000 + 4.00000i 4.65195i 32.0000i −62.1985 + 62.1985i −18.6078 + 18.6078i 564.672 −128.000 + 128.000i 707.359 −497.588
31.6 4.00000 + 4.00000i 12.3623i 32.0000i −142.025 + 142.025i −49.4490 + 49.4490i −327.411 −128.000 + 128.000i 576.175 −1136.20
31.7 4.00000 + 4.00000i 23.7872i 32.0000i 114.190 114.190i −95.1488 + 95.1488i 375.130 −128.000 + 128.000i 163.169 913.518
31.8 4.00000 + 4.00000i 42.5266i 32.0000i −72.1936 + 72.1936i −170.107 + 170.107i 66.7198 −128.000 + 128.000i −1079.52 −577.548
31.9 4.00000 + 4.00000i 46.4806i 32.0000i 122.449 122.449i −185.923 + 185.923i −557.043 −128.000 + 128.000i −1431.45 979.593
43.1 4.00000 4.00000i 46.4806i 32.0000i 122.449 + 122.449i −185.923 185.923i −557.043 −128.000 128.000i −1431.45 979.593
43.2 4.00000 4.00000i 42.5266i 32.0000i −72.1936 72.1936i −170.107 170.107i 66.7198 −128.000 128.000i −1079.52 −577.548
43.3 4.00000 4.00000i 23.7872i 32.0000i 114.190 + 114.190i −95.1488 95.1488i 375.130 −128.000 128.000i 163.169 913.518
43.4 4.00000 4.00000i 12.3623i 32.0000i −142.025 142.025i −49.4490 49.4490i −327.411 −128.000 128.000i 576.175 −1136.20
43.5 4.00000 4.00000i 4.65195i 32.0000i −62.1985 62.1985i −18.6078 18.6078i 564.672 −128.000 128.000i 707.359 −497.588
43.6 4.00000 4.00000i 0.398699i 32.0000i 44.3245 + 44.3245i 1.59480 + 1.59480i −329.540 −128.000 128.000i 728.841 354.596
43.7 4.00000 4.00000i 10.7444i 32.0000i 60.1111 + 60.1111i 42.9776 + 42.9776i −174.087 −128.000 128.000i 613.558 480.889
43.8 4.00000 4.00000i 42.4181i 32.0000i −29.2519 29.2519i 169.672 + 169.672i −271.084 −128.000 128.000i −1070.29 −234.015
43.9 4.00000 4.00000i 44.2475i 32.0000i 111.595 + 111.595i 176.990 + 176.990i 600.642 −128.000 128.000i −1228.84 892.760
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 43.9
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
37.d odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 74.7.d.a 18
37.d odd 4 1 inner 74.7.d.a 18
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
74.7.d.a 18 1.a even 1 1 trivial
74.7.d.a 18 37.d odd 4 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(42\!\cdots\!00\)\( T_{3}^{10} + \)\(17\!\cdots\!22\)\( T_{3}^{8} + \)\(29\!\cdots\!17\)\( T_{3}^{6} + \)\(19\!\cdots\!02\)\( T_{3}^{4} + \)\(30\!\cdots\!49\)\( T_{3}^{2} + \)\(47\!\cdots\!00\)\( \)">\(T_{3}^{18} + \cdots\) acting on \(S_{7}^{\mathrm{new}}(74, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 32 - 8 T + T^{2} )^{9} \)
$3$ \( \)\(47\!\cdots\!00\)\( + \)\(30\!\cdots\!49\)\( T^{2} + \)\(19\!\cdots\!02\)\( T^{4} + 2979392161707128817 T^{6} + 17465487582207222 T^{8} + 42593147514900 T^{10} + 49247171526 T^{12} + 29142265 T^{14} + 8582 T^{16} + T^{18} \)
$5$ \( \)\(30\!\cdots\!00\)\( + \)\(18\!\cdots\!00\)\( T + \)\(56\!\cdots\!00\)\( T^{2} - \)\(14\!\cdots\!00\)\( T^{3} + \)\(61\!\cdots\!00\)\( T^{4} - \)\(23\!\cdots\!00\)\( T^{5} + \)\(10\!\cdots\!00\)\( T^{6} - \)\(43\!\cdots\!00\)\( T^{7} + \)\(14\!\cdots\!50\)\( T^{8} - 73164454300761581250 T^{9} + 251955378125078125 T^{10} - 3533852834818500 T^{11} + 80139826763550 T^{12} - 521706571830 T^{13} + 1751679419 T^{14} - 2219088 T^{15} + 43218 T^{16} - 294 T^{17} + T^{18} \)
$7$ \( ( \)\(24\!\cdots\!96\)\( - 90830776559467038996 T - 3512742235078976196 T^{2} - 11149900127685710 T^{3} + 36584739571098 T^{4} + 166010950199 T^{5} - 98097986 T^{6} - 725990 T^{7} + 52 T^{8} + T^{9} )^{2} \)
$11$ \( \)\(83\!\cdots\!00\)\( + \)\(79\!\cdots\!61\)\( T^{2} + \)\(25\!\cdots\!34\)\( T^{4} + \)\(33\!\cdots\!73\)\( T^{6} + \)\(15\!\cdots\!90\)\( T^{8} + \)\(26\!\cdots\!36\)\( T^{10} + \)\(20\!\cdots\!26\)\( T^{12} + 81795322794945 T^{14} + 15001934 T^{16} + T^{18} \)
$13$ \( \)\(30\!\cdots\!92\)\( - \)\(16\!\cdots\!72\)\( T + \)\(42\!\cdots\!76\)\( T^{2} - \)\(61\!\cdots\!12\)\( T^{3} + \)\(50\!\cdots\!28\)\( T^{4} - \)\(16\!\cdots\!16\)\( T^{5} - \)\(10\!\cdots\!12\)\( T^{6} - \)\(11\!\cdots\!20\)\( T^{7} + \)\(66\!\cdots\!90\)\( T^{8} - \)\(11\!\cdots\!50\)\( T^{9} - \)\(48\!\cdots\!83\)\( T^{10} + \)\(38\!\cdots\!80\)\( T^{11} + \)\(60\!\cdots\!86\)\( T^{12} + 1673616904357065110 T^{13} + 234901520904939 T^{14} + 15128597344 T^{15} + 22889378 T^{16} + 6766 T^{17} + T^{18} \)
$17$ \( \)\(64\!\cdots\!72\)\( - \)\(11\!\cdots\!76\)\( T + \)\(10\!\cdots\!04\)\( T^{2} - \)\(48\!\cdots\!16\)\( T^{3} + \)\(10\!\cdots\!08\)\( T^{4} - \)\(12\!\cdots\!96\)\( T^{5} + \)\(93\!\cdots\!36\)\( T^{6} - \)\(51\!\cdots\!16\)\( T^{7} + \)\(11\!\cdots\!56\)\( T^{8} + \)\(21\!\cdots\!08\)\( T^{9} + \)\(31\!\cdots\!52\)\( T^{10} - \)\(60\!\cdots\!60\)\( T^{11} + \)\(14\!\cdots\!12\)\( T^{12} + 29722524206524186288 T^{13} + 3450734571401616 T^{14} - 171224126856 T^{15} + 41714978 T^{16} + 9134 T^{17} + T^{18} \)
$19$ \( \)\(28\!\cdots\!00\)\( + \)\(12\!\cdots\!00\)\( T + \)\(25\!\cdots\!00\)\( T^{2} + \)\(19\!\cdots\!80\)\( T^{3} + \)\(41\!\cdots\!88\)\( T^{4} + \)\(16\!\cdots\!08\)\( T^{5} + \)\(35\!\cdots\!24\)\( T^{6} + \)\(27\!\cdots\!36\)\( T^{7} + \)\(96\!\cdots\!32\)\( T^{8} - \)\(27\!\cdots\!72\)\( T^{9} + \)\(46\!\cdots\!68\)\( T^{10} + \)\(35\!\cdots\!96\)\( T^{11} + \)\(10\!\cdots\!96\)\( T^{12} - 29308181147562117104 T^{13} + 4351393078496432 T^{14} + 120642767016 T^{15} + 28713042 T^{16} - 7578 T^{17} + T^{18} \)
$23$ \( \)\(27\!\cdots\!00\)\( + \)\(19\!\cdots\!00\)\( T + \)\(69\!\cdots\!00\)\( T^{2} + \)\(93\!\cdots\!00\)\( T^{3} + \)\(86\!\cdots\!88\)\( T^{4} + \)\(14\!\cdots\!68\)\( T^{5} + \)\(39\!\cdots\!24\)\( T^{6} + \)\(53\!\cdots\!36\)\( T^{7} + \)\(40\!\cdots\!14\)\( T^{8} + \)\(17\!\cdots\!78\)\( T^{9} + \)\(76\!\cdots\!57\)\( T^{10} + \)\(59\!\cdots\!68\)\( T^{11} + \)\(45\!\cdots\!90\)\( T^{12} + \)\(18\!\cdots\!06\)\( T^{13} + 481167605160147655 T^{14} + 16480533444392 T^{15} + 1279067042 T^{16} + 50578 T^{17} + T^{18} \)
$29$ \( \)\(54\!\cdots\!88\)\( + \)\(44\!\cdots\!20\)\( T + \)\(18\!\cdots\!00\)\( T^{2} + \)\(39\!\cdots\!00\)\( T^{3} + \)\(50\!\cdots\!92\)\( T^{4} + \)\(44\!\cdots\!76\)\( T^{5} + \)\(64\!\cdots\!40\)\( T^{6} + \)\(11\!\cdots\!08\)\( T^{7} + \)\(14\!\cdots\!46\)\( T^{8} + \)\(92\!\cdots\!02\)\( T^{9} + \)\(29\!\cdots\!21\)\( T^{10} + \)\(45\!\cdots\!56\)\( T^{11} + \)\(10\!\cdots\!62\)\( T^{12} + \)\(59\!\cdots\!02\)\( T^{13} + 1563770698476546279 T^{14} - 2278019601168 T^{15} + 922351250 T^{16} + 42950 T^{17} + T^{18} \)
$31$ \( \)\(35\!\cdots\!32\)\( + \)\(43\!\cdots\!80\)\( T + \)\(27\!\cdots\!00\)\( T^{2} + \)\(22\!\cdots\!20\)\( T^{3} + \)\(50\!\cdots\!00\)\( T^{4} - \)\(76\!\cdots\!44\)\( T^{5} + \)\(21\!\cdots\!40\)\( T^{6} + \)\(10\!\cdots\!52\)\( T^{7} + \)\(18\!\cdots\!98\)\( T^{8} - \)\(15\!\cdots\!86\)\( T^{9} + \)\(39\!\cdots\!29\)\( T^{10} - \)\(17\!\cdots\!16\)\( T^{11} + \)\(10\!\cdots\!70\)\( T^{12} - \)\(56\!\cdots\!82\)\( T^{13} + 1056618606868053023 T^{14} - 23125783474232 T^{15} + 150650082 T^{16} + 17358 T^{17} + T^{18} \)
$37$ \( \)\(48\!\cdots\!89\)\( + \)\(44\!\cdots\!82\)\( T + \)\(23\!\cdots\!81\)\( T^{2} + \)\(87\!\cdots\!00\)\( T^{3} + \)\(25\!\cdots\!56\)\( T^{4} + \)\(59\!\cdots\!04\)\( T^{5} + \)\(12\!\cdots\!60\)\( T^{6} + \)\(25\!\cdots\!24\)\( T^{7} + \)\(50\!\cdots\!78\)\( T^{8} + \)\(97\!\cdots\!24\)\( T^{9} + \)\(19\!\cdots\!42\)\( T^{10} + \)\(38\!\cdots\!04\)\( T^{11} + \)\(76\!\cdots\!40\)\( T^{12} + \)\(13\!\cdots\!64\)\( T^{13} + \)\(22\!\cdots\!44\)\( T^{14} + 3058065173196800 T^{15} + 32713323449 T^{16} + 238242 T^{17} + T^{18} \)
$41$ \( \)\(31\!\cdots\!00\)\( + \)\(12\!\cdots\!25\)\( T^{2} + \)\(15\!\cdots\!50\)\( T^{4} + \)\(51\!\cdots\!21\)\( T^{6} + \)\(68\!\cdots\!42\)\( T^{8} + \)\(43\!\cdots\!16\)\( T^{10} + \)\(14\!\cdots\!58\)\( T^{12} + \)\(26\!\cdots\!89\)\( T^{14} + 25770108898 T^{16} + T^{18} \)
$43$ \( \)\(60\!\cdots\!92\)\( - \)\(50\!\cdots\!64\)\( T + \)\(21\!\cdots\!44\)\( T^{2} + \)\(38\!\cdots\!44\)\( T^{3} + \)\(22\!\cdots\!52\)\( T^{4} + \)\(46\!\cdots\!68\)\( T^{5} + \)\(68\!\cdots\!84\)\( T^{6} + \)\(20\!\cdots\!36\)\( T^{7} + \)\(11\!\cdots\!60\)\( T^{8} + \)\(25\!\cdots\!20\)\( T^{9} + \)\(31\!\cdots\!24\)\( T^{10} + \)\(28\!\cdots\!48\)\( T^{11} + \)\(14\!\cdots\!12\)\( T^{12} + \)\(32\!\cdots\!72\)\( T^{13} + \)\(37\!\cdots\!88\)\( T^{14} + 467231794394088 T^{15} + 2143160450 T^{16} + 65470 T^{17} + T^{18} \)
$47$ \( ( \)\(69\!\cdots\!56\)\( + \)\(25\!\cdots\!56\)\( T - \)\(89\!\cdots\!52\)\( T^{2} - \)\(42\!\cdots\!48\)\( T^{3} + \)\(14\!\cdots\!06\)\( T^{4} + \)\(19\!\cdots\!63\)\( T^{5} + 1054290030584322 T^{6} - 22532077736 T^{7} - 116096 T^{8} + T^{9} )^{2} \)
$53$ \( ( \)\(21\!\cdots\!44\)\( + \)\(71\!\cdots\!32\)\( T - \)\(11\!\cdots\!12\)\( T^{2} - \)\(42\!\cdots\!06\)\( T^{3} + \)\(12\!\cdots\!34\)\( T^{4} + \)\(61\!\cdots\!45\)\( T^{5} + 2648858061459260 T^{6} - 66470441832 T^{7} - 24986 T^{8} + T^{9} )^{2} \)
$59$ \( \)\(26\!\cdots\!72\)\( - \)\(13\!\cdots\!84\)\( T + \)\(37\!\cdots\!24\)\( T^{2} + \)\(55\!\cdots\!04\)\( T^{3} + \)\(23\!\cdots\!28\)\( T^{4} - \)\(66\!\cdots\!32\)\( T^{5} + \)\(86\!\cdots\!92\)\( T^{6} - \)\(36\!\cdots\!80\)\( T^{7} + \)\(47\!\cdots\!16\)\( T^{8} + \)\(12\!\cdots\!20\)\( T^{9} + \)\(69\!\cdots\!16\)\( T^{10} - \)\(16\!\cdots\!12\)\( T^{11} + \)\(17\!\cdots\!88\)\( T^{12} + \)\(15\!\cdots\!12\)\( T^{13} + \)\(14\!\cdots\!76\)\( T^{14} - 16589594333213736 T^{15} + 16483832450 T^{16} + 181570 T^{17} + T^{18} \)
$61$ \( \)\(19\!\cdots\!88\)\( + \)\(43\!\cdots\!40\)\( T + \)\(49\!\cdots\!00\)\( T^{2} + \)\(95\!\cdots\!24\)\( T^{3} + \)\(54\!\cdots\!12\)\( T^{4} + \)\(13\!\cdots\!88\)\( T^{5} + \)\(19\!\cdots\!16\)\( T^{6} + \)\(15\!\cdots\!04\)\( T^{7} + \)\(59\!\cdots\!02\)\( T^{8} + \)\(39\!\cdots\!78\)\( T^{9} + \)\(89\!\cdots\!17\)\( T^{10} + \)\(10\!\cdots\!00\)\( T^{11} + \)\(49\!\cdots\!30\)\( T^{12} - \)\(84\!\cdots\!02\)\( T^{13} + \)\(10\!\cdots\!87\)\( T^{14} + 20202210670603040 T^{15} + 129439737602 T^{16} - 508802 T^{17} + T^{18} \)
$67$ \( \)\(51\!\cdots\!00\)\( + \)\(30\!\cdots\!64\)\( T^{2} + \)\(59\!\cdots\!84\)\( T^{4} + \)\(31\!\cdots\!72\)\( T^{6} + \)\(78\!\cdots\!20\)\( T^{8} + \)\(10\!\cdots\!41\)\( T^{10} + \)\(86\!\cdots\!10\)\( T^{12} + \)\(40\!\cdots\!19\)\( T^{14} + 989705505258 T^{16} + T^{18} \)
$71$ \( ( \)\(54\!\cdots\!04\)\( - \)\(10\!\cdots\!32\)\( T - \)\(13\!\cdots\!04\)\( T^{2} - \)\(54\!\cdots\!32\)\( T^{3} + \)\(20\!\cdots\!70\)\( T^{4} + \)\(99\!\cdots\!35\)\( T^{5} - 87113225998843130 T^{6} - 561329590332 T^{7} + 101316 T^{8} + T^{9} )^{2} \)
$73$ \( \)\(87\!\cdots\!00\)\( + \)\(50\!\cdots\!61\)\( T^{2} + \)\(12\!\cdots\!90\)\( T^{4} + \)\(88\!\cdots\!61\)\( T^{6} + \)\(17\!\cdots\!34\)\( T^{8} + \)\(15\!\cdots\!32\)\( T^{10} + \)\(69\!\cdots\!34\)\( T^{12} + \)\(16\!\cdots\!17\)\( T^{14} + 2027665527714 T^{16} + T^{18} \)
$79$ \( \)\(45\!\cdots\!08\)\( - \)\(59\!\cdots\!20\)\( T + \)\(39\!\cdots\!00\)\( T^{2} - \)\(14\!\cdots\!08\)\( T^{3} + \)\(40\!\cdots\!32\)\( T^{4} - \)\(15\!\cdots\!12\)\( T^{5} + \)\(76\!\cdots\!84\)\( T^{6} - \)\(27\!\cdots\!28\)\( T^{7} + \)\(60\!\cdots\!94\)\( T^{8} - \)\(87\!\cdots\!62\)\( T^{9} + \)\(13\!\cdots\!81\)\( T^{10} - \)\(34\!\cdots\!12\)\( T^{11} + \)\(77\!\cdots\!18\)\( T^{12} - \)\(95\!\cdots\!42\)\( T^{13} + \)\(71\!\cdots\!23\)\( T^{14} - 622707891676704056 T^{15} + 1536255584082 T^{16} - 1752858 T^{17} + T^{18} \)
$83$ \( ( \)\(29\!\cdots\!00\)\( - \)\(37\!\cdots\!20\)\( T + \)\(89\!\cdots\!56\)\( T^{2} + \)\(30\!\cdots\!92\)\( T^{3} - \)\(12\!\cdots\!18\)\( T^{4} + \)\(31\!\cdots\!67\)\( T^{5} + 2290738778767378474 T^{6} - 1579683164844 T^{7} - 1185808 T^{8} + T^{9} )^{2} \)
$89$ \( \)\(13\!\cdots\!00\)\( + \)\(26\!\cdots\!00\)\( T + \)\(25\!\cdots\!00\)\( T^{2} + \)\(92\!\cdots\!20\)\( T^{3} + \)\(15\!\cdots\!48\)\( T^{4} - \)\(11\!\cdots\!36\)\( T^{5} + \)\(25\!\cdots\!76\)\( T^{6} + \)\(10\!\cdots\!40\)\( T^{7} + \)\(23\!\cdots\!36\)\( T^{8} - \)\(30\!\cdots\!00\)\( T^{9} + \)\(24\!\cdots\!08\)\( T^{10} + \)\(37\!\cdots\!96\)\( T^{11} + \)\(10\!\cdots\!36\)\( T^{12} - \)\(18\!\cdots\!04\)\( T^{13} + \)\(15\!\cdots\!00\)\( T^{14} + 219344923395628352 T^{15} + 659349267858 T^{16} - 1148346 T^{17} + T^{18} \)
$97$ \( \)\(56\!\cdots\!08\)\( + \)\(11\!\cdots\!68\)\( T + \)\(11\!\cdots\!64\)\( T^{2} + \)\(26\!\cdots\!72\)\( T^{3} + \)\(47\!\cdots\!24\)\( T^{4} + \)\(24\!\cdots\!24\)\( T^{5} + \)\(12\!\cdots\!36\)\( T^{6} + \)\(34\!\cdots\!48\)\( T^{7} + \)\(57\!\cdots\!72\)\( T^{8} + \)\(59\!\cdots\!68\)\( T^{9} + \)\(57\!\cdots\!12\)\( T^{10} + \)\(83\!\cdots\!96\)\( T^{11} + \)\(12\!\cdots\!68\)\( T^{12} + \)\(12\!\cdots\!40\)\( T^{13} + \)\(70\!\cdots\!48\)\( T^{14} + 1821813191754924256 T^{15} + 1394900936450 T^{16} + 1670270 T^{17} + T^{18} \)
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