Properties

Label 43.7.b.b
Level 43
Weight 7
Character orbit 43.b
Analytic conductor 9.892
Analytic rank 0
Dimension 20
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 43 \)
Weight: \( k \) = \( 7 \)
Character orbit: \([\chi]\) = 43.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(9.89232559565\)
Analytic rank: \(0\)
Dimension: \(20\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} + \cdots)\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{15} \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q + \beta_{1} q^{2} -\beta_{3} q^{3} + ( -40 + \beta_{2} ) q^{4} + ( \beta_{1} + \beta_{13} ) q^{5} + ( -13 + \beta_{4} ) q^{6} + ( 2 \beta_{1} + \beta_{3} - \beta_{15} ) q^{7} + ( -32 \beta_{1} + \beta_{3} - \beta_{13} + \beta_{14} ) q^{8} + ( -136 - 2 \beta_{2} - \beta_{7} ) q^{9} +O(q^{10})\) \( q + \beta_{1} q^{2} -\beta_{3} q^{3} + ( -40 + \beta_{2} ) q^{4} + ( \beta_{1} + \beta_{13} ) q^{5} + ( -13 + \beta_{4} ) q^{6} + ( 2 \beta_{1} + \beta_{3} - \beta_{15} ) q^{7} + ( -32 \beta_{1} + \beta_{3} - \beta_{13} + \beta_{14} ) q^{8} + ( -136 - 2 \beta_{2} - \beta_{7} ) q^{9} + ( -99 + 4 \beta_{2} - \beta_{8} - \beta_{10} ) q^{10} + ( 130 + 2 \beta_{2} - \beta_{4} - \beta_{5} - \beta_{6} ) q^{11} + ( -18 \beta_{1} + 26 \beta_{3} - \beta_{12} - \beta_{13} ) q^{12} + ( -280 - 3 \beta_{2} + \beta_{4} - \beta_{5} + 2 \beta_{7} - \beta_{8} + 2 \beta_{9} + 2 \beta_{10} + \beta_{11} ) q^{13} + ( -191 + 6 \beta_{2} - 2 \beta_{4} + \beta_{5} - \beta_{7} - \beta_{8} - 2 \beta_{9} + 5 \beta_{10} - \beta_{11} ) q^{14} + ( 201 + 3 \beta_{2} + \beta_{4} + \beta_{5} - 2 \beta_{6} + \beta_{7} + 3 \beta_{9} + 3 \beta_{10} - 2 \beta_{11} ) q^{15} + ( 750 - 28 \beta_{2} - 3 \beta_{4} - 6 \beta_{6} - 3 \beta_{7} + 2 \beta_{8} - 5 \beta_{9} - 4 \beta_{10} + \beta_{11} ) q^{16} + ( 167 - 20 \beta_{2} - 4 \beta_{4} - \beta_{5} - \beta_{6} + \beta_{7} + 2 \beta_{8} - 2 \beta_{9} - 3 \beta_{10} - 3 \beta_{11} ) q^{17} + ( -16 \beta_{1} + 21 \beta_{3} + 3 \beta_{13} - 5 \beta_{14} + 2 \beta_{15} - \beta_{16} ) q^{18} + ( -2 \beta_{1} - 8 \beta_{3} + 8 \beta_{13} + 2 \beta_{14} + \beta_{15} - \beta_{19} ) q^{19} + ( -255 \beta_{1} + 30 \beta_{3} - \beta_{12} - 47 \beta_{13} + 5 \beta_{14} + 4 \beta_{15} + \beta_{18} ) q^{20} + ( 529 - 2 \beta_{2} + 4 \beta_{4} + \beta_{5} - 7 \beta_{6} - 2 \beta_{8} + 2 \beta_{9} + 3 \beta_{10} + 3 \beta_{11} ) q^{21} + ( 31 \beta_{1} - 68 \beta_{3} + \beta_{12} + 10 \beta_{13} - 2 \beta_{14} - 14 \beta_{15} - \beta_{16} + \beta_{17} ) q^{22} + ( -35 + 4 \beta_{2} - 6 \beta_{4} - \beta_{5} - 5 \beta_{6} - 8 \beta_{7} - 3 \beta_{8} - 8 \beta_{9} - \beta_{10} + 5 \beta_{11} ) q^{23} + ( 1356 - 24 \beta_{2} - 12 \beta_{4} + \beta_{5} - 6 \beta_{6} - 6 \beta_{7} + 4 \beta_{8} - 18 \beta_{9} - 4 \beta_{10} - 4 \beta_{11} ) q^{24} + ( -2900 + 25 \beta_{2} + 2 \beta_{4} + 15 \beta_{6} + 7 \beta_{7} - 5 \beta_{8} + 3 \beta_{9} + 3 \beta_{10} - 6 \beta_{11} ) q^{25} + ( -179 \beta_{1} + 60 \beta_{3} - 5 \beta_{12} - 44 \beta_{13} + 8 \beta_{14} - 12 \beta_{15} + \beta_{16} + \beta_{17} + \beta_{18} + \beta_{19} ) q^{26} + ( -63 \beta_{1} + 136 \beta_{3} - \beta_{12} + 20 \beta_{13} - 7 \beta_{14} + 6 \beta_{15} - \beta_{16} - \beta_{17} - \beta_{18} + \beta_{19} ) q^{27} + ( -345 \beta_{1} - 45 \beta_{3} + 4 \beta_{12} - 38 \beta_{13} + 9 \beta_{14} + 26 \beta_{15} - \beta_{17} + \beta_{18} - \beta_{19} ) q^{28} + ( -28 \beta_{1} + 120 \beta_{3} - 5 \beta_{12} + 20 \beta_{13} + 2 \beta_{14} - 12 \beta_{15} + \beta_{16} - \beta_{18} + 2 \beta_{19} ) q^{29} + ( -83 \beta_{1} + 32 \beta_{3} - 2 \beta_{12} - 9 \beta_{13} + 12 \beta_{14} + 24 \beta_{15} + 2 \beta_{16} - \beta_{17} - 2 \beta_{19} ) q^{30} + ( 825 + 97 \beta_{2} + 46 \beta_{4} - \beta_{5} - 8 \beta_{6} + 2 \beta_{8} + 43 \beta_{9} + 12 \beta_{10} + 7 \beta_{11} ) q^{31} + ( 534 \beta_{1} - 148 \beta_{3} + 9 \beta_{12} + 101 \beta_{13} - 25 \beta_{14} - 16 \beta_{15} - 3 \beta_{16} - 2 \beta_{18} + \beta_{19} ) q^{32} + ( 615 \beta_{1} + 100 \beta_{3} - 6 \beta_{12} + 68 \beta_{13} + \beta_{14} - 7 \beta_{15} + 4 \beta_{16} - \beta_{17} - 2 \beta_{18} ) q^{33} + ( 1319 \beta_{1} - 407 \beta_{3} + 11 \beta_{12} + 149 \beta_{13} - 20 \beta_{14} - 30 \beta_{15} + \beta_{17} - 2 \beta_{18} - 3 \beta_{19} ) q^{34} + ( -7557 + 68 \beta_{2} + 42 \beta_{4} - 11 \beta_{5} + 57 \beta_{6} + 7 \beta_{7} - 9 \beta_{8} + 29 \beta_{9} - 30 \beta_{10} - 2 \beta_{11} ) q^{35} + ( -6646 + 97 \beta_{2} + 19 \beta_{4} + \beta_{5} + 42 \beta_{6} + 62 \beta_{7} - 7 \beta_{8} + 72 \beta_{9} + 41 \beta_{10} - 12 \beta_{11} ) q^{36} + ( 556 \beta_{1} - 248 \beta_{3} + 10 \beta_{12} - 63 \beta_{13} - \beta_{14} - 13 \beta_{15} + 4 \beta_{16} - \beta_{17} + 2 \beta_{18} + 3 \beta_{19} ) q^{37} + ( 150 - 76 \beta_{2} - 9 \beta_{4} - 11 \beta_{5} - 12 \beta_{6} - 13 \beta_{7} - 18 \beta_{8} - 5 \beta_{9} - 24 \beta_{10} + 61 \beta_{11} ) q^{38} + ( -489 \beta_{1} - 517 \beta_{3} + 15 \beta_{12} + 37 \beta_{13} + 48 \beta_{14} + 21 \beta_{15} + 5 \beta_{16} - \beta_{18} + \beta_{19} ) q^{39} + ( 20074 - 458 \beta_{2} - 58 \beta_{4} - 11 \beta_{5} - 58 \beta_{6} - 18 \beta_{7} + 85 \beta_{8} - 61 \beta_{9} - 111 \beta_{10} - 22 \beta_{11} ) q^{40} + ( -7108 + 447 \beta_{2} - 58 \beta_{4} + 65 \beta_{6} - 35 \beta_{7} - 7 \beta_{8} - 55 \beta_{9} + 57 \beta_{10} - 6 \beta_{11} ) q^{41} + ( 681 \beta_{1} + 386 \beta_{3} - 11 \beta_{12} - 152 \beta_{13} - 39 \beta_{14} + 28 \beta_{15} + \beta_{16} - \beta_{17} + 2 \beta_{18} + 3 \beta_{19} ) q^{42} + ( 2875 - 1434 \beta_{1} + 82 \beta_{2} + 380 \beta_{3} + 15 \beta_{4} + 12 \beta_{5} + 72 \beta_{6} + 21 \beta_{7} + 21 \beta_{8} + 23 \beta_{9} - 30 \beta_{10} + 14 \beta_{11} - 14 \beta_{12} - 57 \beta_{13} + 23 \beta_{14} - 26 \beta_{15} - 4 \beta_{16} + \beta_{17} + 2 \beta_{18} - 2 \beta_{19} ) q^{43} + ( 4428 + 367 \beta_{2} + 68 \beta_{4} + 88 \beta_{5} + 20 \beta_{6} - 4 \beta_{7} - 29 \beta_{8} - \beta_{9} + 111 \beta_{10} - 46 \beta_{11} ) q^{44} + ( 262 \beta_{1} - 598 \beta_{3} + 21 \beta_{12} + 78 \beta_{13} + 2 \beta_{14} + 23 \beta_{15} - \beta_{16} + 2 \beta_{17} - 3 \beta_{18} + 5 \beta_{19} ) q^{45} + ( 203 \beta_{1} - 159 \beta_{3} + 6 \beta_{12} - 208 \beta_{13} - 75 \beta_{14} + 64 \beta_{15} - 9 \beta_{16} + \beta_{17} + 3 \beta_{18} + 5 \beta_{19} ) q^{46} + ( 2441 - 349 \beta_{2} + 78 \beta_{4} + 24 \beta_{5} + 15 \beta_{6} - 72 \beta_{7} + 8 \beta_{8} - 90 \beta_{9} - 244 \beta_{10} + 39 \beta_{11} ) q^{47} + ( 2176 \beta_{1} + 815 \beta_{3} - 26 \beta_{12} + 268 \beta_{13} - 105 \beta_{14} + 114 \beta_{15} - 5 \beta_{16} - \beta_{17} - 4 \beta_{18} - 4 \beta_{19} ) q^{48} + ( -980 - 190 \beta_{2} + 51 \beta_{4} - 11 \beta_{5} + 19 \beta_{6} + 106 \beta_{7} + 12 \beta_{8} - 9 \beta_{9} + 65 \beta_{10} + 77 \beta_{11} ) q^{49} + ( -4695 \beta_{1} + 119 \beta_{3} - 4 \beta_{12} - 473 \beta_{13} + 164 \beta_{14} + 82 \beta_{15} + 7 \beta_{16} + 5 \beta_{18} - 6 \beta_{19} ) q^{50} + ( 3335 \beta_{1} - 555 \beta_{3} + 17 \beta_{12} - 67 \beta_{13} - 36 \beta_{14} - 160 \beta_{15} + 7 \beta_{16} + 2 \beta_{17} + \beta_{18} - 4 \beta_{19} ) q^{51} + ( 920 - 865 \beta_{2} - 264 \beta_{4} + 88 \beta_{5} - 58 \beta_{6} - 151 \beta_{7} + 96 \beta_{8} - 142 \beta_{9} + 102 \beta_{10} - 61 \beta_{11} ) q^{52} + ( 20915 + 877 \beta_{2} - 116 \beta_{4} - 12 \beta_{5} - 71 \beta_{6} + 64 \beta_{7} - 19 \beta_{8} + 63 \beta_{9} - 227 \beta_{10} - 34 \beta_{11} ) q^{53} + ( 8528 + 268 \beta_{2} - 148 \beta_{4} - 120 \beta_{5} + 16 \beta_{6} + 250 \beta_{7} - 97 \beta_{8} + 139 \beta_{9} + 47 \beta_{10} - 20 \beta_{11} ) q^{54} + ( 2437 \beta_{1} - 1639 \beta_{3} + 73 \beta_{12} + 386 \beta_{13} - 89 \beta_{14} + 55 \beta_{15} - 11 \beta_{16} - \beta_{17} + \beta_{18} - 8 \beta_{19} ) q^{55} + ( 22514 - 724 \beta_{2} + 143 \beta_{4} - 120 \beta_{5} - 106 \beta_{6} + 61 \beta_{7} + 73 \beta_{8} - 28 \beta_{9} + 151 \beta_{10} + 43 \beta_{11} ) q^{56} + ( -4440 - 335 \beta_{2} + 125 \beta_{4} + 42 \beta_{5} + 45 \beta_{6} - 38 \beta_{7} + 31 \beta_{8} + 182 \beta_{9} + 389 \beta_{10} - 34 \beta_{11} ) q^{57} + ( 4477 - 30 \beta_{2} - 414 \beta_{4} + 42 \beta_{5} - 32 \beta_{6} - 140 \beta_{7} - 78 \beta_{8} - 147 \beta_{9} + 22 \beta_{10} - 110 \beta_{11} ) q^{58} + ( -45903 - 596 \beta_{2} + 239 \beta_{4} + 25 \beta_{5} + 107 \beta_{6} - 65 \beta_{7} + 61 \beta_{8} + 94 \beta_{9} - 114 \beta_{10} - 2 \beta_{11} ) q^{59} + ( 21524 - 624 \beta_{2} - 148 \beta_{4} - 120 \beta_{5} - 290 \beta_{6} - 83 \beta_{7} + 26 \beta_{8} + 108 \beta_{9} - 56 \beta_{10} + 59 \beta_{11} ) q^{60} + ( 5164 \beta_{1} + 1058 \beta_{3} - 97 \beta_{12} + 412 \beta_{13} - 196 \beta_{14} + 225 \beta_{15} + 5 \beta_{16} + 2 \beta_{17} + 3 \beta_{18} ) q^{61} + ( -5922 \beta_{1} + 3699 \beta_{3} - 94 \beta_{12} + 152 \beta_{13} + 124 \beta_{14} - 328 \beta_{15} - \beta_{16} + \beta_{17} - 2 \beta_{18} + 7 \beta_{19} ) q^{62} + ( -1422 \beta_{1} + 171 \beta_{3} - 8 \beta_{12} - 305 \beta_{13} - 5 \beta_{14} - 391 \beta_{15} - 6 \beta_{16} - \beta_{17} - 8 \beta_{18} + 3 \beta_{19} ) q^{63} + ( -7994 + 434 \beta_{2} + 482 \beta_{4} + 42 \beta_{5} - 100 \beta_{6} + 258 \beta_{7} - 211 \beta_{8} + 113 \beta_{9} + 69 \beta_{10} + 28 \beta_{11} ) q^{64} + ( -7476 \beta_{1} - 766 \beta_{3} + 49 \beta_{12} - 562 \beta_{13} + 254 \beta_{14} + 141 \beta_{15} - \beta_{16} - 12 \beta_{17} - 3 \beta_{18} - 4 \beta_{19} ) q^{65} + ( -62209 + 930 \beta_{2} - 573 \beta_{4} - 119 \beta_{5} - 100 \beta_{6} - 369 \beta_{7} - 233 \beta_{8} - 203 \beta_{9} - 103 \beta_{10} + 85 \beta_{11} ) q^{66} + ( 54762 + 754 \beta_{2} + 226 \beta_{4} + 41 \beta_{5} - 173 \beta_{6} - 12 \beta_{7} + 112 \beta_{8} - 397 \beta_{9} + 2 \beta_{10} - 118 \beta_{11} ) q^{67} + ( -129760 + 1917 \beta_{2} + 609 \beta_{4} + 77 \beta_{5} + 220 \beta_{6} + 27 \beta_{7} - 322 \beta_{8} + 127 \beta_{9} + 56 \beta_{10} + 5 \beta_{11} ) q^{68} + ( 2399 \beta_{1} + 4334 \beta_{3} - 94 \beta_{12} + 985 \beta_{13} - 88 \beta_{14} + 321 \beta_{15} + 2 \beta_{16} - 12 \beta_{17} - 6 \beta_{18} - 4 \beta_{19} ) q^{69} + ( -13018 \beta_{1} + 3574 \beta_{3} - 78 \beta_{12} - 1177 \beta_{13} + 439 \beta_{14} - 520 \beta_{15} - 4 \beta_{16} + 11 \beta_{17} + 9 \beta_{18} - 2 \beta_{19} ) q^{70} + ( -3708 \beta_{1} - 3604 \beta_{3} + 116 \beta_{12} + 1071 \beta_{13} + 275 \beta_{14} + 146 \beta_{15} - 2 \beta_{16} + 11 \beta_{17} - 4 \beta_{18} + 9 \beta_{19} ) q^{71} + ( -16459 \beta_{1} + 984 \beta_{3} - 86 \beta_{12} - 239 \beta_{13} + 419 \beta_{14} - 74 \beta_{15} - \beta_{16} - \beta_{17} + 7 \beta_{18} - 12 \beta_{19} ) q^{72} + ( -2784 \beta_{1} - 940 \beta_{3} - 61 \beta_{12} - 879 \beta_{13} + 21 \beta_{14} + 415 \beta_{15} - 9 \beta_{16} + 11 \beta_{17} - \beta_{18} + 11 \beta_{19} ) q^{73} + ( -61498 + 548 \beta_{2} + 727 \beta_{4} - 131 \beta_{5} - 80 \beta_{6} - 253 \beta_{7} + 241 \beta_{8} - 30 \beta_{9} - 25 \beta_{10} - 141 \beta_{11} ) q^{74} + ( -1505 \beta_{1} - 1419 \beta_{3} + 75 \beta_{12} - 341 \beta_{13} + 156 \beta_{14} - 417 \beta_{15} + 21 \beta_{16} + 2 \beta_{17} + 19 \beta_{18} - 6 \beta_{19} ) q^{75} + ( 5730 \beta_{1} - 537 \beta_{3} - 65 \beta_{12} - 663 \beta_{13} - 278 \beta_{14} - 366 \beta_{15} - 24 \beta_{16} + 11 \beta_{17} + 18 \beta_{18} - 3 \beta_{19} ) q^{76} + ( 16205 \beta_{1} - 1338 \beta_{3} + 44 \beta_{12} - 441 \beta_{13} - 530 \beta_{14} - 249 \beta_{15} + 8 \beta_{16} - 12 \beta_{17} + 16 \beta_{18} + 25 \beta_{19} ) q^{77} + ( 43363 - 2934 \beta_{2} + 1038 \beta_{4} - 33 \beta_{5} - 236 \beta_{6} - 569 \beta_{7} - 85 \beta_{8} - 120 \beta_{9} - 407 \beta_{10} + 143 \beta_{11} ) q^{78} + ( -20131 + 1224 \beta_{2} - 1175 \beta_{4} + 66 \beta_{5} - 128 \beta_{6} + 204 \beta_{7} + 93 \beta_{8} - 96 \beta_{9} + 203 \beta_{10} + 163 \beta_{11} ) q^{79} + ( 31863 \beta_{1} - 4591 \beta_{3} + 162 \beta_{12} + 3624 \beta_{13} - 786 \beta_{14} - 1150 \beta_{15} - 29 \beta_{16} + 11 \beta_{17} - 21 \beta_{18} - 22 \beta_{19} ) q^{80} + ( 23788 - 1735 \beta_{2} - 1204 \beta_{4} - 33 \beta_{5} + 334 \beta_{6} + 276 \beta_{7} - 7 \beta_{8} + 115 \beta_{9} - 124 \beta_{10} - 161 \beta_{11} ) q^{81} + ( -30821 \beta_{1} - 3239 \beta_{3} + 112 \beta_{12} + 157 \beta_{13} + 652 \beta_{14} + 1086 \beta_{15} - 35 \beta_{16} + 7 \beta_{18} - 6 \beta_{19} ) q^{82} + ( 67693 + 160 \beta_{2} + 60 \beta_{4} - 34 \beta_{5} + 150 \beta_{6} + 203 \beta_{7} + 153 \beta_{8} + 278 \beta_{9} + 326 \beta_{10} + 140 \beta_{11} ) q^{83} + ( -32426 + 2082 \beta_{2} - 500 \beta_{4} - 78 \beta_{5} - 390 \beta_{6} + 103 \beta_{7} + 265 \beta_{8} + 121 \beta_{9} + 223 \beta_{10} - 57 \beta_{11} ) q^{84} + ( 29658 \beta_{1} + 528 \beta_{3} + 43 \beta_{12} + 91 \beta_{13} - 937 \beta_{14} + 37 \beta_{15} - 17 \beta_{16} + 11 \beta_{17} - 21 \beta_{18} - 28 \beta_{19} ) q^{85} + ( 152972 - 3081 \beta_{1} - 2972 \beta_{2} - 386 \beta_{3} - 1016 \beta_{4} + 152 \beta_{5} - 164 \beta_{6} + 128 \beta_{7} + 247 \beta_{8} - 337 \beta_{9} + 35 \beta_{10} - 58 \beta_{11} - 31 \beta_{12} + 1483 \beta_{13} + 411 \beta_{14} - 634 \beta_{15} + 33 \beta_{16} - 12 \beta_{17} - 21 \beta_{18} + 14 \beta_{19} ) q^{86} + ( 106171 - 2928 \beta_{2} + 822 \beta_{4} + 176 \beta_{5} + 364 \beta_{6} + 238 \beta_{7} - 17 \beta_{8} - 267 \beta_{9} - 523 \beta_{10} - 63 \beta_{11} ) q^{87} + ( -15343 \beta_{1} + 1715 \beta_{3} + 14 \beta_{12} - 2423 \beta_{13} + 590 \beta_{14} + 1864 \beta_{15} + 20 \beta_{16} - 24 \beta_{17} + 29 \beta_{18} - 46 \beta_{19} ) q^{88} + ( 18569 \beta_{1} - 6410 \beta_{3} - 2 \beta_{12} - 717 \beta_{13} - 298 \beta_{14} + 967 \beta_{15} + 78 \beta_{16} - 12 \beta_{17} - 30 \beta_{18} - 41 \beta_{19} ) q^{89} + ( -34325 + 350 \beta_{2} + 1673 \beta_{4} + 305 \beta_{5} + 300 \beta_{6} + 73 \beta_{7} - 320 \beta_{8} + 432 \beta_{9} + 82 \beta_{10} - 159 \beta_{11} ) q^{90} + ( 11312 \beta_{1} + 4089 \beta_{3} - 58 \beta_{12} - 935 \beta_{13} - 521 \beta_{14} + 1393 \beta_{15} + 28 \beta_{16} + 11 \beta_{17} + 38 \beta_{18} + 37 \beta_{19} ) q^{91} + ( -25366 + 3375 \beta_{2} + 691 \beta_{4} + 55 \beta_{5} + 262 \beta_{6} + 720 \beta_{7} + 314 \beta_{8} + 349 \beta_{9} + 370 \beta_{10} - 144 \beta_{11} ) q^{92} + ( -30805 \beta_{1} + 4614 \beta_{3} + 68 \beta_{12} - 2240 \beta_{13} + 285 \beta_{14} + 492 \beta_{15} - 54 \beta_{16} + \beta_{17} - 52 \beta_{18} + 57 \beta_{19} ) q^{93} + ( 25775 \beta_{1} + 8377 \beta_{3} - 19 \beta_{12} - 2212 \beta_{13} - 1110 \beta_{14} - 1082 \beta_{15} - 48 \beta_{16} - 24 \beta_{17} - 8 \beta_{18} + 39 \beta_{19} ) q^{94} + ( -135220 + 945 \beta_{2} - 1640 \beta_{4} + 165 \beta_{5} + 690 \beta_{6} - 355 \beta_{7} + 145 \beta_{8} + 655 \beta_{9} + 470 \beta_{10} + 195 \beta_{11} ) q^{95} + ( -125234 + 6756 \beta_{2} - 2583 \beta_{4} - 153 \beta_{5} + 184 \beta_{6} + 517 \beta_{7} - 343 \beta_{8} - 492 \beta_{9} - 389 \beta_{10} - 31 \beta_{11} ) q^{96} + ( 391391 - 3421 \beta_{2} - 1288 \beta_{4} - 9 \beta_{5} - 602 \beta_{6} + 269 \beta_{7} - 187 \beta_{8} - 129 \beta_{9} - 340 \beta_{10} + 343 \beta_{11} ) q^{97} + ( 9684 \beta_{1} + 1710 \beta_{3} - 107 \beta_{12} + 3120 \beta_{13} - 51 \beta_{14} - 380 \beta_{15} + 95 \beta_{16} + 11 \beta_{17} - 12 \beta_{18} + 77 \beta_{19} ) q^{98} + ( 187473 - 3396 \beta_{2} + 3012 \beta_{4} - 8 \beta_{5} + 196 \beta_{6} + 221 \beta_{7} + 385 \beta_{8} + 178 \beta_{9} + 388 \beta_{10} - 170 \beta_{11} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20q - 796q^{4} - 258q^{6} - 2736q^{9} + O(q^{10}) \) \( 20q - 796q^{4} - 258q^{6} - 2736q^{9} - 1962q^{10} + 2616q^{11} - 5612q^{13} - 3876q^{14} + 4048q^{15} + 14900q^{16} + 3328q^{17} + 10544q^{21} - 836q^{23} + 26966q^{24} - 57904q^{25} + 17116q^{31} - 150472q^{35} - 132110q^{36} + 2266q^{38} + 401286q^{40} - 141936q^{41} + 58160q^{43} + 88544q^{44} + 48452q^{47} - 20644q^{49} + 12292q^{52} + 425212q^{53} + 173740q^{54} + 447864q^{56} - 92904q^{57} + 86502q^{58} - 918856q^{59} + 429848q^{60} - 156892q^{64} - 1246176q^{66} + 1098496q^{67} - 2589854q^{68} - 1224106q^{74} + 855716q^{78} - 401492q^{79} + 470644q^{81} + 1354360q^{83} - 637268q^{84} + 3046356q^{86} + 2116740q^{87} - 683716q^{90} - 485998q^{92} - 2711660q^{95} - 2480062q^{96} + 7814560q^{97} + 3744560q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{20} + 1038 x^{18} + 455829 x^{16} + 110435384 x^{14} + 16133606976 x^{12} + 1458210485616 x^{10} + 80362197690736 x^{8} + 2545997652841536 x^{6} + 40210452531479040 x^{4} + 212033222410436608 x^{2} + 64236717122519040\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} + 104 \)
\(\beta_{3}\)\(=\)\((\)\(91588349793300851406844468650523 \nu^{19} + 81757820632325281471962537674807738 \nu^{17} + 29083626316883677223728194446984311095 \nu^{15} + 5132538731195956653903570402785391828776 \nu^{13} + 429623605065855600282133135491936283961024 \nu^{11} + 6327035425774001457021604057429277591585488 \nu^{9} - 1588811726380311830176629936566467794395186480 \nu^{7} - 111461246421953596484086749379797899021088850240 \nu^{5} - 2513947526196903731840426724315903917844715944448 \nu^{3} - 11924989281913840194301078664310430083810164637696 \nu\)\()/ \)\(19\!\cdots\!04\)\( \)
\(\beta_{4}\)\(=\)\((\)\(18907509166365060068667643159709 \nu^{18} + 17990056223077921267332830757833718 \nu^{16} + 7076783875218101560707725455839598689 \nu^{14} + 1488674480226805644946125281356754112056 \nu^{12} + 180721671311197878262129723522206469361920 \nu^{10} + 12711722725337833116515126664741821265928752 \nu^{8} + 489552514237680600890864898555010415288879792 \nu^{6} + 8802210963063815003156553464979561664674396992 \nu^{4} + 44523809977047820106416827812636485021873809920 \nu^{2} + 12042389162706312045650589345382859063719968768\)\()/ \)\(28\!\cdots\!76\)\( \)
\(\beta_{5}\)\(=\)\((\)\(-80171392817838448320457362171609 \nu^{18} - 86823983370373985539041129430834078 \nu^{16} - 38480118644236878984659210438353747661 \nu^{14} - 9000456466281556249815259907562237171256 \nu^{12} - 1193271256591433371621969805786675393306688 \nu^{10} - 89351850586232775523541776087124546130462704 \nu^{8} - 3533474545706883273481454518912783704449395696 \nu^{6} - 62137507814300435116110936635164799513494625856 \nu^{4} - 250243411759364711958487333940416555190249087488 \nu^{2} + 703466742335876829074388527174116415313151229952\)\()/ \)\(11\!\cdots\!04\)\( \)
\(\beta_{6}\)\(=\)\((\)\(-127046434404630632331195430848283 \nu^{18} - 126280759645812998493911807382559802 \nu^{16} - 52344461749732812482778194102338275639 \nu^{14} - 11723209312062567248526805999489431154088 \nu^{12} - 1534408262797790186014055598307753079366336 \nu^{10} - 118314341232656954852115790564153323888628432 \nu^{8} - 5132454951013185470386648610748294919983751888 \nu^{6} - 111175201558745993618364891426273941684984952512 \nu^{4} - 921177889434015627934535576048792625030257305088 \nu^{2} - 2018682533154411611641491851450389623018182443008\)\()/ \)\(11\!\cdots\!04\)\( \)
\(\beta_{7}\)\(=\)\((\)\(-315973643637116540231992019646133 \nu^{18} - 309796850399216996079260676219158950 \nu^{16} - 126356219385873473510838742048198633945 \nu^{14} - 27772976418584591366529512189000872466008 \nu^{12} - 3558567351299433313420425597336857813503808 \nu^{10} - 268047422978885943834822996657692346506961968 \nu^{8} - 11325530227529034298485697967702720627394975792 \nu^{6} - 234813793445692594213507957816059656552225719616 \nu^{4} - 1609712483118880255843708671765365705852748831232 \nu^{2} - 655264188823563513861725645733791307042406170624\)\()/ \)\(22\!\cdots\!08\)\( \)
\(\beta_{8}\)\(=\)\((\)\(-359719798692979524022579290862255 \nu^{18} - 365048353697340697620993350834698706 \nu^{16} - 154231137816721978031246551910917530971 \nu^{14} - 35073295267344947920334996248135254065032 \nu^{12} - 4627885853038352963441405240834608640014784 \nu^{10} - 355113639599728672774814918199463609572322192 \nu^{8} - 14952867779654746219135501310049287641088747408 \nu^{6} - 296123493188721239124423680318773487017973343168 \nu^{4} - 1742644099046754657420401694280589172920414219776 \nu^{2} - 283002973287713511659813140477450530316802162688\)\()/ \)\(22\!\cdots\!08\)\( \)
\(\beta_{9}\)\(=\)\((\)\(219432965221798280856528488291789 \nu^{18} + 219908818355486516230484434205591670 \nu^{16} + 92055251589185048798403311777140989137 \nu^{14} + 20864286575190037024303373081239705626904 \nu^{12} + 2770691442389911515943350715086443424425536 \nu^{10} + 217310445836292882200399195555985378664146096 \nu^{8} + 9588652351002632170655917801267978066794682032 \nu^{6} + 208036838014454015785458499867453169504989190976 \nu^{4} + 1534900265958043150734943647510053495275622943232 \nu^{2} + 1182191913316948879822397434203066186879140855808\)\()/ \)\(11\!\cdots\!04\)\( \)
\(\beta_{10}\)\(=\)\((\)\(-887379944524499461895342023349561 \nu^{18} - 882778046141926937769915989553560542 \nu^{16} - 365957705492440212265985790763902602413 \nu^{14} - 81844553288011854537048538312368494802488 \nu^{12} - 10662866170547256350128019170718183695382080 \nu^{10} - 812511510226284920334162144108078760346961392 \nu^{8} - 34201892665776249585554880920522584696818146800 \nu^{6} - 678895548137427889535037388340367713411434165824 \nu^{4} - 3900249563859214410292655082035667154537594724864 \nu^{2} - 971469953766943213860417491090773635588133257216\)\()/ \)\(22\!\cdots\!08\)\( \)
\(\beta_{11}\)\(=\)\((\)\(-2654448452268211757993222422040623 \nu^{18} - 2614925611211698053281568693549113298 \nu^{16} - 1072175416593646137886286085894870297307 \nu^{14} - 236858010340518397667693631749120107368584 \nu^{12} - 30440059648216237408334589657509738385464768 \nu^{10} - 2285547321663186808716522221262647251403877264 \nu^{8} - 94832101346810178316105462716079433792205107088 \nu^{6} - 1867989636210462512339521505790865337072062550976 \nu^{4} - 11082464416810395189056914879112888066277436714496 \nu^{2} - 3206241042746945965995708655871789467530367303680\)\()/ \)\(22\!\cdots\!08\)\( \)
\(\beta_{12}\)\(=\)\((\)\(-2364501764615352601846563595876363 \nu^{19} - 2610376832562020554331587238040733146 \nu^{17} - 1242034947456774351205309144878031725543 \nu^{15} - 333320282137854552377064700238487536736424 \nu^{13} - 55234573970670773779120253138080457412201152 \nu^{11} - 5783031138625495377605179018455663340012907472 \nu^{9} - 373135204748884815564382683500680477824900428752 \nu^{7} - 13670953015258383989701906722769597707738014198464 \nu^{5} - 234695442571800703111626888909701496335245158770176 \nu^{3} - 1006680484928922381137508146075379229883354233405440 \nu\)\()/ \)\(19\!\cdots\!04\)\( \)
\(\beta_{13}\)\(=\)\((\)\(-901144402369524353293151670003901 \nu^{19} - 898806297191853561798032408247169302 \nu^{17} - 374164844059079399132464025268152584321 \nu^{15} - 84259355378060174271228720528660697853720 \nu^{13} - 11109119392161842830675696675759543820699456 \nu^{11} - 865529490564226335096508596117981282652746672 \nu^{9} - 38167606882890130726227633124342985344679242672 \nu^{7} - 852438560238121818696038120252608205363585935168 \nu^{5} - 7634865669920766036309654352941980575392812804608 \nu^{3} - 25348093247334961509985659301067270300587978686464 \nu\)\()/ \)\(66\!\cdots\!68\)\( \)
\(\beta_{14}\)\(=\)\((\)\(-127046434404630632331195430848283 \nu^{19} - 126280759645812998493911807382559802 \nu^{17} - 52344461749732812482778194102338275639 \nu^{15} - 11723209312062567248526805999489431154088 \nu^{13} - 1534408262797790186014055598307753079366336 \nu^{11} - 118314341232656954852115790564153323888628432 \nu^{9} - 5132454951013185470386648610748294919983751888 \nu^{7} - 111175201558745993618364891426273941684984952512 \nu^{5} - 917776000815368794411024112555067168912037438976 \nu^{3} - 1463040725442095469467952814141898457042270978048 \nu\)\()/ \)\(90\!\cdots\!32\)\( \)
\(\beta_{15}\)\(=\)\((\)\(-4354312975552347961089852236821765 \nu^{19} - 4317720724410074944680881435606150150 \nu^{17} - 1780545696062365940131059546794013120617 \nu^{15} - 394733953899769354280161598264089255705048 \nu^{13} - 50636811523048762867354030823342757725006144 \nu^{11} - 3744449669103776113096732008480978667900095280 \nu^{9} - 147215178026197961083737891379690519596924158768 \nu^{7} - 2353219668082975872709182550228395615344260213056 \nu^{5} + 2881501633869827592132969482972224740909722502656 \nu^{3} + 137460735046434595931540747703008549989747654754304 \nu\)\()/ \)\(19\!\cdots\!04\)\( \)
\(\beta_{16}\)\(=\)\((\)\(-14363071541163991603129067998783115 \nu^{19} - 14189511582179833427638740974735135450 \nu^{17} - 5839637674744072773302105593299499216999 \nu^{15} - 1297243846677763415093027557757935456874152 \nu^{13} - 168301309937882177362157936317490698365582016 \nu^{11} - 12859696347947016617598137220318917457557570512 \nu^{9} - 551690239831749131830721891760945352495402789840 \nu^{7} - 11576707102464481316308546075314568574825589123776 \nu^{5} - 78021934652599275420055805219432515660817443153408 \nu^{3} - 17451752072715529085189608000572229483346540593152 \nu\)\()/ \)\(99\!\cdots\!52\)\( \)
\(\beta_{17}\)\(=\)\((\)\(-36490275094185505481433257919100579 \nu^{19} - 34407933989398688848578525499803880490 \nu^{17} - 13462334251387143881757793778649805217375 \nu^{15} - 2827132337573171354397996450137001617320936 \nu^{13} - 342465845072125523685913398102016644271173824 \nu^{11} - 23568167851592129270376628807647326868846806608 \nu^{9} - 799538154633717390737968697928410222143761256016 \nu^{7} - 5019173029307008885043019604593535424396978393280 \nu^{5} + 311932532138584875197779000709404457250508200634880 \nu^{3} + 3065667424925123291805001872624152302083595131715584 \nu\)\()/ \)\(19\!\cdots\!04\)\( \)
\(\beta_{18}\)\(=\)\((\)\(-5126753161175283613409481930482909 \nu^{19} - 5106097461621366673430669025184979694 \nu^{17} - 2126710781897199334351293428086404284273 \nu^{15} - 480887571005325401675155926688798041461776 \nu^{13} - 64079428332407953527422489153340532596209536 \nu^{11} - 5107836661330922479023235724013181248089957808 \nu^{9} - 236132320907678722362582783338078164951547394096 \nu^{7} - 5835791553168726737814704874552689107458319333824 \nu^{5} - 66090155597408219458497349765990956987613897928704 \nu^{3} - 288794669765668649873229640412562648808329157718016 \nu\)\()/ \)\(62\!\cdots\!72\)\( \)
\(\beta_{19}\)\(=\)\((\)\(-122976416262336968781959682282481041 \nu^{19} - 121266699718415430040151150036612682286 \nu^{17} - 49785431108662642309412108862005467202789 \nu^{15} - 11016846528714101246293430242316242674931960 \nu^{13} - 1419123083349072463782403960455407064597254208 \nu^{11} - 106902344832439145596479696707194129830398411888 \nu^{9} - 4456522727903482804490908362309536782560936625264 \nu^{7} - 88359001839784396989411060809904687733289722139712 \nu^{5} - 525961731000308349105119676431668251539105156477440 \nu^{3} + 64330859936660189223432290043888694991323006009344 \nu\)\()/ \)\(99\!\cdots\!52\)\( \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} - 104\)
\(\nu^{3}\)\(=\)\(\beta_{14} - \beta_{13} + \beta_{3} - 160 \beta_{1}\)
\(\nu^{4}\)\(=\)\(\beta_{11} - 4 \beta_{10} - 5 \beta_{9} + 2 \beta_{8} - 3 \beta_{7} - 6 \beta_{6} - 3 \beta_{4} - 220 \beta_{2} + 16622\)
\(\nu^{5}\)\(=\)\(\beta_{19} - 2 \beta_{18} - 3 \beta_{16} - 16 \beta_{15} - 281 \beta_{14} + 357 \beta_{13} + 9 \beta_{12} - 404 \beta_{3} + 29206 \beta_{1}\)
\(\nu^{6}\)\(=\)\(-292 \beta_{11} + 1349 \beta_{10} + 1713 \beta_{9} - 851 \beta_{8} + 1218 \beta_{7} + 1820 \beta_{6} + 42 \beta_{5} + 1442 \beta_{4} + 46258 \beta_{2} - 3033274\)
\(\nu^{7}\)\(=\)\(-292 \beta_{19} + 851 \beta_{18} - 42 \beta_{17} + 1260 \beta_{16} + 7204 \beta_{15} + 66615 \beta_{14} - 110162 \beta_{13} - 3714 \beta_{12} + 146496 \beta_{3} - 5714137 \beta_{1}\)
\(\nu^{8}\)\(=\)\(65312 \beta_{11} - 352073 \beta_{10} - 459101 \beta_{9} + 269135 \beta_{8} - 350790 \beta_{7} - 458084 \beta_{6} - 22710 \beta_{5} - 473594 \beta_{4} - 9833110 \beta_{2} + 593749010\)
\(\nu^{9}\)\(=\)\(65312 \beta_{19} - 269135 \beta_{18} + 22710 \beta_{17} - 373500 \beta_{16} - 2381628 \beta_{15} - 15222983 \beta_{14} + 31110930 \beta_{13} + 1136518 \beta_{12} - 44933520 \beta_{3} + 1168637677 \beta_{1}\)
\(\nu^{10}\)\(=\)\(-13509240 \beta_{11} + 85492437 \beta_{10} + 113671817 \beta_{9} - 75848979 \beta_{8} + 89088750 \beta_{7} + 109786316 \beta_{6} + 8260370 \beta_{5} + 133569506 \beta_{4} + 2129406214 \beta_{2} - 121523533746\)
\(\nu^{11}\)\(=\)\(-13509240 \beta_{19} + 75848979 \beta_{18} - 8260370 \beta_{17} + 97349120 \beta_{16} + 703275828 \beta_{15} + 3455914663 \beta_{14} - 8310178066 \beta_{13} - 309581062 \beta_{12} + 12441716052 \beta_{3} - 246764099129 \beta_{1}\)
\(\nu^{12}\)\(=\)\(2739131060 \beta_{11} - 20265491085 \beta_{10} - 27199582389 \beta_{9} + 20137988483 \beta_{8} - 21400030442 \beta_{7} - 25875901308 \beta_{6} - 2551036678 \beta_{5} - 34887414478 \beta_{4} - 469247624622 \beta_{2} + 25679915687010\)
\(\nu^{13}\)\(=\)\(2739131060 \beta_{19} - 20137988483 \beta_{18} + 2551036678 \beta_{17} - 23951067120 \beta_{16} - 195532370332 \beta_{15} - 786173342879 \beta_{14} + 2141324511038 \beta_{13} + 79485854290 \beta_{12} - 3241732484664 \beta_{3} + 53389945912937 \beta_{1}\)
\(\nu^{14}\)\(=\)\(-558241729520 \beta_{11} + 4765882154005 \beta_{10} + 6403567930657 \beta_{9} - 5159698077251 \beta_{8} + 5009455065830 \beta_{7} + 6062159715692 \beta_{6} + 723335924074 \beta_{5} + 8727749404554 \beta_{4} + 104923290065414 \beta_{2} - 5559769938055506\)
\(\nu^{15}\)\(=\)\(-558241729520 \beta_{19} + 5159698077251 \beta_{18} - 723335924074 \beta_{17} + 5732790989904 \beta_{16} + 52272133896292 \beta_{15} + 179664499005191 \beta_{14} - 538597830209946 \beta_{13} - 19732773682942 \beta_{12} + 813629720365308 \beta_{3} - 11772211200391753 \beta_{1}\)
\(\nu^{16}\)\(=\)\(115750227407084 \beta_{11} - 1118648450016477 \beta_{10} - 1496128810252909 \beta_{9} + 1291857923694915 \beta_{8} - 1159327567457666 \beta_{7} - 1417750625607164 \beta_{6} - 194971420770334 \beta_{5} - 2128293910697766 \beta_{4} - 23736430131100606 \beta_{2} + 1226537155683628322\)
\(\nu^{17}\)\(=\)\(115750227407084 \beta_{19} - 1291857923694915 \beta_{18} + 194971420770334 \beta_{17} - 1354298988228000 \beta_{16} - 13587898322759372 \beta_{15} - 41265817368279551 \beta_{14} + 133217490557733814 \beta_{13} + 4800530417238506 \beta_{12} - 199443099361421584 \beta_{3} + 2634374216721434217 \beta_{1}\)
\(\nu^{18}\)\(=\)\(-24547919685243416 \beta_{11} + 262650709457641013 \beta_{10} + 348460007019927961 \beta_{9} - 318394585244196179 \beta_{8} + 267207358039806830 \beta_{7} + 331599005614730508 \beta_{6} + 50858743758600450 \beta_{5} + 511002323944690834 \beta_{4} + 5419507585881166902 \beta_{2} - 274578681934689706514\)
\(\nu^{19}\)\(=\)\(-24547919685243416 \beta_{19} + 318394585244196179 \beta_{18} - 50858743758600450 \beta_{17} + 318066101798407280 \beta_{16} + 3459155857084311700 \beta_{15} + 9522817517042502439 \beta_{14} - 32558343082155530658 \beta_{13} - 1153308996523571558 \beta_{12} + 48155980031114918260 \beta_{3} - 596364668963675761209 \beta_{1}\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/43\mathbb{Z}\right)^\times\).

\(n\) \(3\)
\(\chi(n)\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
42.1
15.3591i
13.7504i
13.1339i
11.9161i
11.0119i
8.33048i
7.14115i
6.90119i
2.98785i
0.567672i
0.567672i
2.98785i
6.90119i
7.14115i
8.33048i
11.0119i
11.9161i
13.1339i
13.7504i
15.3591i
15.3591i 12.8491i −171.901 213.900i −197.350 274.602i 1657.26i 563.902 −3285.30
42.2 13.7504i 14.6903i −125.075 125.525i −201.998 485.131i 839.807i 513.196 1726.02
42.3 13.1339i 22.6278i −108.500 138.961i 297.193 295.441i 584.463i 216.980 1825.10
42.4 11.9161i 41.5647i −77.9927 111.502i 495.288 49.3353i 166.738i −998.624 −1328.67
42.5 11.0119i 50.0401i −57.2623 30.3065i −551.037 75.1486i 74.1946i −1775.01 333.733
42.6 8.33048i 16.8005i −5.39690 47.9107i 139.956 493.652i 488.192i 446.745 −399.119
42.7 7.14115i 19.7617i 13.0039 151.678i −141.122 62.8896i 549.897i 338.474 −1083.16
42.8 6.90119i 11.0801i 16.3736 107.518i −76.4660 594.412i 554.673i 606.231 742.004
42.9 2.98785i 41.6692i 55.0727 126.319i 124.502 69.5744i 355.772i −1007.32 377.422
42.10 0.567672i 31.6476i 63.6777 195.481i −17.9654 418.179i 72.4791i −272.568 110.969
42.11 0.567672i 31.6476i 63.6777 195.481i −17.9654 418.179i 72.4791i −272.568 110.969
42.12 2.98785i 41.6692i 55.0727 126.319i 124.502 69.5744i 355.772i −1007.32 377.422
42.13 6.90119i 11.0801i 16.3736 107.518i −76.4660 594.412i 554.673i 606.231 742.004
42.14 7.14115i 19.7617i 13.0039 151.678i −141.122 62.8896i 549.897i 338.474 −1083.16
42.15 8.33048i 16.8005i −5.39690 47.9107i 139.956 493.652i 488.192i 446.745 −399.119
42.16 11.0119i 50.0401i −57.2623 30.3065i −551.037 75.1486i 74.1946i −1775.01 333.733
42.17 11.9161i 41.5647i −77.9927 111.502i 495.288 49.3353i 166.738i −998.624 −1328.67
42.18 13.1339i 22.6278i −108.500 138.961i 297.193 295.441i 584.463i 216.980 1825.10
42.19 13.7504i 14.6903i −125.075 125.525i −201.998 485.131i 839.807i 513.196 1726.02
42.20 15.3591i 12.8491i −171.901 213.900i −197.350 274.602i 1657.26i 563.902 −3285.30
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 42.20
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
43.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 43.7.b.b 20
43.b odd 2 1 inner 43.7.b.b 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
43.7.b.b 20 1.a even 1 1 trivial
43.7.b.b 20 43.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{2}^{20} + \cdots\) acting on \(S_{7}^{\mathrm{new}}(43, [\chi])\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 - 242 T^{2} + 38293 T^{4} - 4675528 T^{6} + 480119360 T^{8} - 43302914704 T^{10} + 3491489747312 T^{12} - 260048564882368 T^{14} + 18169314061001216 T^{16} - 1212360600461180928 T^{18} + 78576420845084737536 T^{20} - \)\(49\!\cdots\!88\)\( T^{22} + \)\(30\!\cdots\!56\)\( T^{24} - \)\(17\!\cdots\!48\)\( T^{26} + \)\(98\!\cdots\!72\)\( T^{28} - \)\(49\!\cdots\!04\)\( T^{30} + \)\(22\!\cdots\!60\)\( T^{32} - \)\(90\!\cdots\!48\)\( T^{34} + \)\(30\!\cdots\!48\)\( T^{36} - \)\(78\!\cdots\!52\)\( T^{38} + \)\(13\!\cdots\!76\)\( T^{40} \)
$3$ \( 1 - 5922 T^{2} + 17916017 T^{4} - 36741838976 T^{6} + 57666038760582 T^{8} - 74196957195265770 T^{10} + 81787389614624858778 T^{12} - \)\(79\!\cdots\!30\)\( T^{14} + \)\(69\!\cdots\!81\)\( T^{16} - \)\(56\!\cdots\!70\)\( T^{18} + \)\(42\!\cdots\!26\)\( T^{20} - \)\(30\!\cdots\!70\)\( T^{22} + \)\(19\!\cdots\!61\)\( T^{24} - \)\(11\!\cdots\!30\)\( T^{26} + \)\(65\!\cdots\!58\)\( T^{28} - \)\(31\!\cdots\!70\)\( T^{30} + \)\(12\!\cdots\!62\)\( T^{32} - \)\(43\!\cdots\!56\)\( T^{34} + \)\(11\!\cdots\!57\)\( T^{36} - \)\(20\!\cdots\!42\)\( T^{38} + \)\(17\!\cdots\!01\)\( T^{40} \)
$5$ \( 1 - 127298 T^{2} + 8822084473 T^{4} - 428558542330656 T^{6} + 16198472909671906230 T^{8} - \)\(50\!\cdots\!50\)\( T^{10} + \)\(13\!\cdots\!50\)\( T^{12} - \)\(30\!\cdots\!50\)\( T^{14} + \)\(63\!\cdots\!25\)\( T^{16} - \)\(11\!\cdots\!50\)\( T^{18} + \)\(19\!\cdots\!50\)\( T^{20} - \)\(28\!\cdots\!50\)\( T^{22} + \)\(37\!\cdots\!25\)\( T^{24} - \)\(45\!\cdots\!50\)\( T^{26} + \)\(47\!\cdots\!50\)\( T^{28} - \)\(43\!\cdots\!50\)\( T^{30} + \)\(34\!\cdots\!50\)\( T^{32} - \)\(22\!\cdots\!00\)\( T^{34} + \)\(11\!\cdots\!25\)\( T^{36} - \)\(39\!\cdots\!50\)\( T^{38} + \)\(75\!\cdots\!25\)\( T^{40} \)
$7$ \( 1 - 1166168 T^{2} + 679026522922 T^{4} - 265497232908808176 T^{6} + \)\(78\!\cdots\!53\)\( T^{8} - \)\(18\!\cdots\!88\)\( T^{10} + \)\(38\!\cdots\!68\)\( T^{12} - \)\(67\!\cdots\!80\)\( T^{14} + \)\(10\!\cdots\!50\)\( T^{16} - \)\(14\!\cdots\!56\)\( T^{18} + \)\(17\!\cdots\!36\)\( T^{20} - \)\(20\!\cdots\!56\)\( T^{22} + \)\(20\!\cdots\!50\)\( T^{24} - \)\(17\!\cdots\!80\)\( T^{26} + \)\(14\!\cdots\!68\)\( T^{28} - \)\(96\!\cdots\!88\)\( T^{30} + \)\(55\!\cdots\!53\)\( T^{32} - \)\(25\!\cdots\!76\)\( T^{34} + \)\(91\!\cdots\!22\)\( T^{36} - \)\(21\!\cdots\!68\)\( T^{38} + \)\(25\!\cdots\!01\)\( T^{40} \)
$11$ \( ( 1 - 1308 T + 7313339 T^{2} - 3438068668 T^{3} + 19532935485360 T^{4} + 9588658946722524 T^{5} + 33611868196830806529 T^{6} + \)\(44\!\cdots\!20\)\( T^{7} + \)\(87\!\cdots\!39\)\( T^{8} + \)\(65\!\cdots\!76\)\( T^{9} + \)\(20\!\cdots\!36\)\( T^{10} + \)\(11\!\cdots\!36\)\( T^{11} + \)\(27\!\cdots\!19\)\( T^{12} + \)\(24\!\cdots\!20\)\( T^{13} + \)\(33\!\cdots\!89\)\( T^{14} + \)\(16\!\cdots\!24\)\( T^{15} + \)\(60\!\cdots\!60\)\( T^{16} - \)\(18\!\cdots\!28\)\( T^{17} + \)\(70\!\cdots\!59\)\( T^{18} - \)\(22\!\cdots\!28\)\( T^{19} + \)\(30\!\cdots\!01\)\( T^{20} )^{2} \)
$13$ \( ( 1 + 2806 T + 27952549 T^{2} + 54052614486 T^{3} + 334946569836432 T^{4} + 435014829194762046 T^{5} + \)\(24\!\cdots\!31\)\( T^{6} + \)\(20\!\cdots\!66\)\( T^{7} + \)\(13\!\cdots\!99\)\( T^{8} + \)\(81\!\cdots\!16\)\( T^{9} + \)\(70\!\cdots\!36\)\( T^{10} + \)\(39\!\cdots\!44\)\( T^{11} + \)\(32\!\cdots\!19\)\( T^{12} + \)\(23\!\cdots\!14\)\( T^{13} + \)\(13\!\cdots\!91\)\( T^{14} + \)\(11\!\cdots\!54\)\( T^{15} + \)\(42\!\cdots\!12\)\( T^{16} + \)\(32\!\cdots\!34\)\( T^{17} + \)\(82\!\cdots\!29\)\( T^{18} + \)\(39\!\cdots\!34\)\( T^{19} + \)\(68\!\cdots\!01\)\( T^{20} )^{2} \)
$17$ \( ( 1 - 1664 T + 134979316 T^{2} - 279496193526 T^{3} + 8578465814967234 T^{4} - 20710129877559965050 T^{5} + \)\(34\!\cdots\!43\)\( T^{6} - \)\(94\!\cdots\!36\)\( T^{7} + \)\(10\!\cdots\!31\)\( T^{8} - \)\(30\!\cdots\!44\)\( T^{9} + \)\(26\!\cdots\!95\)\( T^{10} - \)\(72\!\cdots\!36\)\( T^{11} + \)\(59\!\cdots\!91\)\( T^{12} - \)\(13\!\cdots\!24\)\( T^{13} + \)\(11\!\cdots\!03\)\( T^{14} - \)\(16\!\cdots\!50\)\( T^{15} + \)\(16\!\cdots\!54\)\( T^{16} - \)\(13\!\cdots\!14\)\( T^{17} + \)\(15\!\cdots\!56\)\( T^{18} - \)\(46\!\cdots\!56\)\( T^{19} + \)\(67\!\cdots\!01\)\( T^{20} )^{2} \)
$19$ \( 1 - 426664310 T^{2} + 92809437595177093 T^{4} - \)\(13\!\cdots\!40\)\( T^{6} + \)\(15\!\cdots\!10\)\( T^{8} - \)\(14\!\cdots\!42\)\( T^{10} + \)\(11\!\cdots\!58\)\( T^{12} - \)\(83\!\cdots\!90\)\( T^{14} + \)\(52\!\cdots\!57\)\( T^{16} - \)\(28\!\cdots\!18\)\( T^{18} + \)\(14\!\cdots\!42\)\( T^{20} - \)\(63\!\cdots\!98\)\( T^{22} + \)\(25\!\cdots\!97\)\( T^{24} - \)\(91\!\cdots\!90\)\( T^{26} + \)\(28\!\cdots\!78\)\( T^{28} - \)\(79\!\cdots\!42\)\( T^{30} + \)\(18\!\cdots\!10\)\( T^{32} - \)\(35\!\cdots\!40\)\( T^{34} + \)\(53\!\cdots\!33\)\( T^{36} - \)\(54\!\cdots\!10\)\( T^{38} + \)\(28\!\cdots\!01\)\( T^{40} \)
$23$ \( ( 1 + 418 T + 1035923986 T^{2} + 2202994994962 T^{3} + 514837202496385552 T^{4} + \)\(16\!\cdots\!08\)\( T^{5} + \)\(16\!\cdots\!55\)\( T^{6} + \)\(61\!\cdots\!48\)\( T^{7} + \)\(37\!\cdots\!91\)\( T^{8} + \)\(13\!\cdots\!30\)\( T^{9} + \)\(63\!\cdots\!85\)\( T^{10} + \)\(20\!\cdots\!70\)\( T^{11} + \)\(81\!\cdots\!11\)\( T^{12} + \)\(20\!\cdots\!12\)\( T^{13} + \)\(78\!\cdots\!55\)\( T^{14} + \)\(11\!\cdots\!92\)\( T^{15} + \)\(54\!\cdots\!72\)\( T^{16} + \)\(34\!\cdots\!98\)\( T^{17} + \)\(23\!\cdots\!66\)\( T^{18} + \)\(14\!\cdots\!62\)\( T^{19} + \)\(50\!\cdots\!01\)\( T^{20} )^{2} \)
$29$ \( 1 - 7119799082 T^{2} + 25311485826303973921 T^{4} - \)\(59\!\cdots\!28\)\( T^{6} + \)\(10\!\cdots\!78\)\( T^{8} - \)\(14\!\cdots\!26\)\( T^{10} + \)\(16\!\cdots\!90\)\( T^{12} - \)\(16\!\cdots\!14\)\( T^{14} + \)\(13\!\cdots\!41\)\( T^{16} - \)\(97\!\cdots\!06\)\( T^{18} + \)\(61\!\cdots\!10\)\( T^{20} - \)\(34\!\cdots\!46\)\( T^{22} + \)\(16\!\cdots\!21\)\( T^{24} - \)\(71\!\cdots\!94\)\( T^{26} + \)\(26\!\cdots\!90\)\( T^{28} - \)\(80\!\cdots\!26\)\( T^{30} + \)\(20\!\cdots\!98\)\( T^{32} - \)\(41\!\cdots\!68\)\( T^{34} + \)\(62\!\cdots\!41\)\( T^{36} - \)\(61\!\cdots\!02\)\( T^{38} + \)\(30\!\cdots\!01\)\( T^{40} \)
$31$ \( ( 1 - 8558 T + 4031881282 T^{2} - 32342675026006 T^{3} + 9021132388831175984 T^{4} - \)\(81\!\cdots\!68\)\( T^{5} + \)\(13\!\cdots\!27\)\( T^{6} - \)\(14\!\cdots\!00\)\( T^{7} + \)\(16\!\cdots\!79\)\( T^{8} - \)\(17\!\cdots\!38\)\( T^{9} + \)\(16\!\cdots\!25\)\( T^{10} - \)\(15\!\cdots\!78\)\( T^{11} + \)\(13\!\cdots\!19\)\( T^{12} - \)\(10\!\cdots\!00\)\( T^{13} + \)\(86\!\cdots\!67\)\( T^{14} - \)\(44\!\cdots\!68\)\( T^{15} + \)\(44\!\cdots\!04\)\( T^{16} - \)\(14\!\cdots\!66\)\( T^{17} + \)\(15\!\cdots\!62\)\( T^{18} - \)\(29\!\cdots\!18\)\( T^{19} + \)\(30\!\cdots\!01\)\( T^{20} )^{2} \)
$37$ \( 1 - 23384133878 T^{2} + \)\(27\!\cdots\!13\)\( T^{4} - \)\(22\!\cdots\!16\)\( T^{6} + \)\(14\!\cdots\!46\)\( T^{8} - \)\(72\!\cdots\!38\)\( T^{10} + \)\(31\!\cdots\!26\)\( T^{12} - \)\(12\!\cdots\!90\)\( T^{14} + \)\(40\!\cdots\!33\)\( T^{16} - \)\(12\!\cdots\!82\)\( T^{18} + \)\(33\!\cdots\!74\)\( T^{20} - \)\(80\!\cdots\!42\)\( T^{22} + \)\(17\!\cdots\!13\)\( T^{24} - \)\(34\!\cdots\!90\)\( T^{26} + \)\(59\!\cdots\!46\)\( T^{28} - \)\(89\!\cdots\!38\)\( T^{30} + \)\(11\!\cdots\!26\)\( T^{32} - \)\(12\!\cdots\!76\)\( T^{34} + \)\(98\!\cdots\!33\)\( T^{36} - \)\(54\!\cdots\!38\)\( T^{38} + \)\(15\!\cdots\!01\)\( T^{40} \)
$41$ \( ( 1 + 70968 T + 21026049068 T^{2} + 741643261844470 T^{3} + \)\(20\!\cdots\!02\)\( T^{4} + \)\(31\!\cdots\!54\)\( T^{5} + \)\(14\!\cdots\!99\)\( T^{6} + \)\(37\!\cdots\!20\)\( T^{7} + \)\(86\!\cdots\!23\)\( T^{8} - \)\(61\!\cdots\!80\)\( T^{9} + \)\(42\!\cdots\!55\)\( T^{10} - \)\(29\!\cdots\!80\)\( T^{11} + \)\(19\!\cdots\!63\)\( T^{12} + \)\(39\!\cdots\!20\)\( T^{13} + \)\(75\!\cdots\!39\)\( T^{14} + \)\(77\!\cdots\!54\)\( T^{15} + \)\(23\!\cdots\!82\)\( T^{16} + \)\(40\!\cdots\!70\)\( T^{17} + \)\(54\!\cdots\!28\)\( T^{18} + \)\(87\!\cdots\!48\)\( T^{19} + \)\(58\!\cdots\!01\)\( T^{20} )^{2} \)
$43$ \( 1 - 58160 T - 3625589906 T^{2} + 407094761387696 T^{3} + 20207651002758555917 T^{4} - \)\(66\!\cdots\!76\)\( T^{5} + \)\(24\!\cdots\!40\)\( T^{6} + \)\(29\!\cdots\!00\)\( T^{7} - \)\(22\!\cdots\!62\)\( T^{8} - \)\(11\!\cdots\!52\)\( T^{9} + \)\(21\!\cdots\!20\)\( T^{10} - \)\(70\!\cdots\!48\)\( T^{11} - \)\(88\!\cdots\!62\)\( T^{12} + \)\(75\!\cdots\!00\)\( T^{13} + \)\(38\!\cdots\!40\)\( T^{14} - \)\(67\!\cdots\!24\)\( T^{15} + \)\(12\!\cdots\!17\)\( T^{16} + \)\(16\!\cdots\!04\)\( T^{17} - \)\(92\!\cdots\!06\)\( T^{18} - \)\(93\!\cdots\!40\)\( T^{19} + \)\(10\!\cdots\!01\)\( T^{20} \)
$47$ \( ( 1 - 24226 T + 37335944065 T^{2} + 1379124453964116 T^{3} + \)\(76\!\cdots\!82\)\( T^{4} + \)\(62\!\cdots\!26\)\( T^{5} + \)\(12\!\cdots\!02\)\( T^{6} + \)\(13\!\cdots\!22\)\( T^{7} + \)\(18\!\cdots\!89\)\( T^{8} + \)\(19\!\cdots\!06\)\( T^{9} + \)\(21\!\cdots\!62\)\( T^{10} + \)\(20\!\cdots\!74\)\( T^{11} + \)\(21\!\cdots\!49\)\( T^{12} + \)\(16\!\cdots\!58\)\( T^{13} + \)\(17\!\cdots\!62\)\( T^{14} + \)\(91\!\cdots\!74\)\( T^{15} + \)\(12\!\cdots\!22\)\( T^{16} + \)\(23\!\cdots\!44\)\( T^{17} + \)\(68\!\cdots\!65\)\( T^{18} - \)\(47\!\cdots\!94\)\( T^{19} + \)\(21\!\cdots\!01\)\( T^{20} )^{2} \)
$53$ \( ( 1 - 212606 T + 117978826141 T^{2} - 27035287925250270 T^{3} + \)\(81\!\cdots\!56\)\( T^{4} - \)\(16\!\cdots\!34\)\( T^{5} + \)\(38\!\cdots\!83\)\( T^{6} - \)\(68\!\cdots\!38\)\( T^{7} + \)\(13\!\cdots\!03\)\( T^{8} - \)\(20\!\cdots\!72\)\( T^{9} + \)\(32\!\cdots\!92\)\( T^{10} - \)\(45\!\cdots\!88\)\( T^{11} + \)\(63\!\cdots\!23\)\( T^{12} - \)\(74\!\cdots\!82\)\( T^{13} + \)\(92\!\cdots\!23\)\( T^{14} - \)\(89\!\cdots\!66\)\( T^{15} + \)\(96\!\cdots\!76\)\( T^{16} - \)\(71\!\cdots\!30\)\( T^{17} + \)\(68\!\cdots\!01\)\( T^{18} - \)\(27\!\cdots\!14\)\( T^{19} + \)\(28\!\cdots\!01\)\( T^{20} )^{2} \)
$59$ \( ( 1 + 459428 T + 395999778694 T^{2} + 137746690276768556 T^{3} + \)\(68\!\cdots\!29\)\( T^{4} + \)\(19\!\cdots\!76\)\( T^{5} + \)\(71\!\cdots\!96\)\( T^{6} + \)\(16\!\cdots\!84\)\( T^{7} + \)\(50\!\cdots\!02\)\( T^{8} + \)\(10\!\cdots\!08\)\( T^{9} + \)\(24\!\cdots\!24\)\( T^{10} + \)\(42\!\cdots\!28\)\( T^{11} + \)\(88\!\cdots\!62\)\( T^{12} + \)\(12\!\cdots\!64\)\( T^{13} + \)\(22\!\cdots\!56\)\( T^{14} + \)\(25\!\cdots\!76\)\( T^{15} + \)\(38\!\cdots\!89\)\( T^{16} + \)\(32\!\cdots\!36\)\( T^{17} + \)\(39\!\cdots\!74\)\( T^{18} + \)\(19\!\cdots\!08\)\( T^{19} + \)\(17\!\cdots\!01\)\( T^{20} )^{2} \)
$61$ \( 1 - 498959450864 T^{2} + \)\(12\!\cdots\!06\)\( T^{4} - \)\(20\!\cdots\!88\)\( T^{6} + \)\(26\!\cdots\!05\)\( T^{8} - \)\(28\!\cdots\!52\)\( T^{10} + \)\(25\!\cdots\!76\)\( T^{12} - \)\(19\!\cdots\!44\)\( T^{14} + \)\(13\!\cdots\!66\)\( T^{16} - \)\(81\!\cdots\!92\)\( T^{18} + \)\(44\!\cdots\!12\)\( T^{20} - \)\(21\!\cdots\!32\)\( T^{22} + \)\(94\!\cdots\!06\)\( T^{24} - \)\(36\!\cdots\!84\)\( T^{26} + \)\(12\!\cdots\!56\)\( T^{28} - \)\(37\!\cdots\!52\)\( T^{30} + \)\(93\!\cdots\!05\)\( T^{32} - \)\(19\!\cdots\!08\)\( T^{34} + \)\(30\!\cdots\!66\)\( T^{36} - \)\(32\!\cdots\!84\)\( T^{38} + \)\(17\!\cdots\!01\)\( T^{40} \)
$67$ \( ( 1 - 549248 T + 626059155535 T^{2} - 293286454957890652 T^{3} + \)\(18\!\cdots\!28\)\( T^{4} - \)\(79\!\cdots\!44\)\( T^{5} + \)\(37\!\cdots\!25\)\( T^{6} - \)\(13\!\cdots\!08\)\( T^{7} + \)\(51\!\cdots\!63\)\( T^{8} - \)\(17\!\cdots\!40\)\( T^{9} + \)\(54\!\cdots\!76\)\( T^{10} - \)\(15\!\cdots\!60\)\( T^{11} + \)\(42\!\cdots\!43\)\( T^{12} - \)\(10\!\cdots\!72\)\( T^{13} + \)\(24\!\cdots\!25\)\( T^{14} - \)\(48\!\cdots\!56\)\( T^{15} + \)\(10\!\cdots\!68\)\( T^{16} - \)\(14\!\cdots\!28\)\( T^{17} + \)\(28\!\cdots\!35\)\( T^{18} - \)\(22\!\cdots\!92\)\( T^{19} + \)\(36\!\cdots\!01\)\( T^{20} )^{2} \)
$71$ \( 1 - 1099385768348 T^{2} + \)\(68\!\cdots\!30\)\( T^{4} - \)\(30\!\cdots\!96\)\( T^{6} + \)\(10\!\cdots\!65\)\( T^{8} - \)\(28\!\cdots\!20\)\( T^{10} + \)\(68\!\cdots\!88\)\( T^{12} - \)\(13\!\cdots\!72\)\( T^{14} + \)\(24\!\cdots\!18\)\( T^{16} - \)\(38\!\cdots\!40\)\( T^{18} + \)\(52\!\cdots\!28\)\( T^{20} - \)\(62\!\cdots\!40\)\( T^{22} + \)\(66\!\cdots\!58\)\( T^{24} - \)\(61\!\cdots\!12\)\( T^{26} + \)\(49\!\cdots\!68\)\( T^{28} - \)\(34\!\cdots\!20\)\( T^{30} + \)\(20\!\cdots\!65\)\( T^{32} - \)\(96\!\cdots\!76\)\( T^{34} + \)\(35\!\cdots\!30\)\( T^{36} - \)\(94\!\cdots\!28\)\( T^{38} + \)\(14\!\cdots\!01\)\( T^{40} \)
$73$ \( 1 - 2077081529264 T^{2} + \)\(20\!\cdots\!78\)\( T^{4} - \)\(13\!\cdots\!76\)\( T^{6} + \)\(65\!\cdots\!45\)\( T^{8} - \)\(24\!\cdots\!96\)\( T^{10} + \)\(71\!\cdots\!00\)\( T^{12} - \)\(17\!\cdots\!16\)\( T^{14} + \)\(37\!\cdots\!50\)\( T^{16} - \)\(69\!\cdots\!28\)\( T^{18} + \)\(11\!\cdots\!72\)\( T^{20} - \)\(16\!\cdots\!88\)\( T^{22} + \)\(19\!\cdots\!50\)\( T^{24} - \)\(21\!\cdots\!76\)\( T^{26} + \)\(19\!\cdots\!00\)\( T^{28} - \)\(15\!\cdots\!96\)\( T^{30} + \)\(94\!\cdots\!45\)\( T^{32} - \)\(45\!\cdots\!16\)\( T^{34} + \)\(15\!\cdots\!58\)\( T^{36} - \)\(36\!\cdots\!84\)\( T^{38} + \)\(39\!\cdots\!01\)\( T^{40} \)
$79$ \( ( 1 + 200746 T + 1301669964385 T^{2} + 297228242564778316 T^{3} + \)\(87\!\cdots\!74\)\( T^{4} + \)\(19\!\cdots\!54\)\( T^{5} + \)\(40\!\cdots\!46\)\( T^{6} + \)\(79\!\cdots\!50\)\( T^{7} + \)\(13\!\cdots\!77\)\( T^{8} + \)\(24\!\cdots\!98\)\( T^{9} + \)\(37\!\cdots\!90\)\( T^{10} + \)\(58\!\cdots\!58\)\( T^{11} + \)\(80\!\cdots\!57\)\( T^{12} + \)\(11\!\cdots\!50\)\( T^{13} + \)\(14\!\cdots\!26\)\( T^{14} + \)\(16\!\cdots\!54\)\( T^{15} + \)\(18\!\cdots\!54\)\( T^{16} + \)\(14\!\cdots\!56\)\( T^{17} + \)\(15\!\cdots\!85\)\( T^{18} + \)\(59\!\cdots\!26\)\( T^{19} + \)\(72\!\cdots\!01\)\( T^{20} )^{2} \)
$83$ \( ( 1 - 677180 T + 2832058692499 T^{2} - 1643580370789010948 T^{3} + \)\(36\!\cdots\!36\)\( T^{4} - \)\(18\!\cdots\!20\)\( T^{5} + \)\(28\!\cdots\!05\)\( T^{6} - \)\(12\!\cdots\!84\)\( T^{7} + \)\(15\!\cdots\!87\)\( T^{8} - \)\(57\!\cdots\!92\)\( T^{9} + \)\(59\!\cdots\!84\)\( T^{10} - \)\(18\!\cdots\!48\)\( T^{11} + \)\(16\!\cdots\!07\)\( T^{12} - \)\(43\!\cdots\!56\)\( T^{13} + \)\(33\!\cdots\!05\)\( T^{14} - \)\(68\!\cdots\!80\)\( T^{15} + \)\(44\!\cdots\!16\)\( T^{16} - \)\(65\!\cdots\!72\)\( T^{17} + \)\(36\!\cdots\!59\)\( T^{18} - \)\(28\!\cdots\!20\)\( T^{19} + \)\(13\!\cdots\!01\)\( T^{20} )^{2} \)
$89$ \( 1 - 1519004027192 T^{2} + \)\(80\!\cdots\!22\)\( T^{4} - \)\(27\!\cdots\!40\)\( T^{6} + \)\(29\!\cdots\!49\)\( T^{8} - \)\(30\!\cdots\!96\)\( T^{10} + \)\(17\!\cdots\!00\)\( T^{12} - \)\(67\!\cdots\!60\)\( T^{14} + \)\(34\!\cdots\!98\)\( T^{16} - \)\(25\!\cdots\!32\)\( T^{18} + \)\(15\!\cdots\!80\)\( T^{20} - \)\(63\!\cdots\!72\)\( T^{22} + \)\(21\!\cdots\!18\)\( T^{24} - \)\(10\!\cdots\!60\)\( T^{26} + \)\(64\!\cdots\!00\)\( T^{28} - \)\(28\!\cdots\!96\)\( T^{30} + \)\(68\!\cdots\!29\)\( T^{32} - \)\(15\!\cdots\!40\)\( T^{34} + \)\(11\!\cdots\!42\)\( T^{36} - \)\(51\!\cdots\!52\)\( T^{38} + \)\(84\!\cdots\!01\)\( T^{40} \)
$97$ \( ( 1 - 3907280 T + 12196450176388 T^{2} - 26367002972894948630 T^{3} + \)\(49\!\cdots\!98\)\( T^{4} - \)\(77\!\cdots\!22\)\( T^{5} + \)\(10\!\cdots\!59\)\( T^{6} - \)\(13\!\cdots\!76\)\( T^{7} + \)\(15\!\cdots\!99\)\( T^{8} - \)\(15\!\cdots\!72\)\( T^{9} + \)\(15\!\cdots\!15\)\( T^{10} - \)\(13\!\cdots\!88\)\( T^{11} + \)\(10\!\cdots\!59\)\( T^{12} - \)\(78\!\cdots\!64\)\( T^{13} + \)\(52\!\cdots\!79\)\( T^{14} - \)\(31\!\cdots\!78\)\( T^{15} + \)\(16\!\cdots\!58\)\( T^{16} - \)\(73\!\cdots\!70\)\( T^{17} + \)\(28\!\cdots\!68\)\( T^{18} - \)\(75\!\cdots\!20\)\( T^{19} + \)\(16\!\cdots\!01\)\( T^{20} )^{2} \)
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