Properties

Label 11.5.d.a
Level 11
Weight 5
Character orbit 11.d
Analytic conductor 1.137
Analytic rank 0
Dimension 12
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) \(=\) \( 11 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 11.d (of order \(10\), degree \(4\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.13706959392\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(3\) over \(\Q(\zeta_{10})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -1 - \beta_{2} - 2 \beta_{3} - \beta_{4} + \beta_{7} ) q^{2} + ( 1 + \beta_{2} + 3 \beta_{3} + 3 \beta_{4} + \beta_{9} ) q^{3} + ( -2 + \beta_{1} - 2 \beta_{3} - 7 \beta_{4} + \beta_{6} - \beta_{11} ) q^{4} + ( 4 \beta_{2} + 4 \beta_{4} - \beta_{6} - 2 \beta_{7} + 2 \beta_{8} - \beta_{9} + \beta_{10} + \beta_{11} ) q^{5} + ( 12 - 6 \beta_{1} - \beta_{2} + 6 \beta_{3} + 13 \beta_{4} - \beta_{5} - 6 \beta_{8} - 3 \beta_{9} - 2 \beta_{10} + 2 \beta_{11} ) q^{6} + ( -17 + 5 \beta_{1} - 24 \beta_{2} - 7 \beta_{3} - 12 \beta_{4} + 4 \beta_{5} - 5 \beta_{6} - 5 \beta_{7} + 5 \beta_{8} - 2 \beta_{9} - 2 \beta_{10} - \beta_{11} ) q^{7} + ( -19 + 5 \beta_{1} + 19 \beta_{2} + 7 \beta_{3} - 7 \beta_{4} - 5 \beta_{5} - 5 \beta_{7} - 2 \beta_{8} + 2 \beta_{9} + 6 \beta_{10} + \beta_{11} ) q^{8} + ( 7 - 12 \beta_{1} + 38 \beta_{2} + 7 \beta_{4} + \beta_{5} + 15 \beta_{6} + 6 \beta_{7} + 3 \beta_{8} + 4 \beta_{9} - \beta_{10} - 4 \beta_{11} ) q^{9} +O(q^{10})\) \( q + ( -1 - \beta_{2} - 2 \beta_{3} - \beta_{4} + \beta_{7} ) q^{2} + ( 1 + \beta_{2} + 3 \beta_{3} + 3 \beta_{4} + \beta_{9} ) q^{3} + ( -2 + \beta_{1} - 2 \beta_{3} - 7 \beta_{4} + \beta_{6} - \beta_{11} ) q^{4} + ( 4 \beta_{2} + 4 \beta_{4} - \beta_{6} - 2 \beta_{7} + 2 \beta_{8} - \beta_{9} + \beta_{10} + \beta_{11} ) q^{5} + ( 12 - 6 \beta_{1} - \beta_{2} + 6 \beta_{3} + 13 \beta_{4} - \beta_{5} - 6 \beta_{8} - 3 \beta_{9} - 2 \beta_{10} + 2 \beta_{11} ) q^{6} + ( -17 + 5 \beta_{1} - 24 \beta_{2} - 7 \beta_{3} - 12 \beta_{4} + 4 \beta_{5} - 5 \beta_{6} - 5 \beta_{7} + 5 \beta_{8} - 2 \beta_{9} - 2 \beta_{10} - \beta_{11} ) q^{7} + ( -19 + 5 \beta_{1} + 19 \beta_{2} + 7 \beta_{3} - 7 \beta_{4} - 5 \beta_{5} - 5 \beta_{7} - 2 \beta_{8} + 2 \beta_{9} + 6 \beta_{10} + \beta_{11} ) q^{8} + ( 7 - 12 \beta_{1} + 38 \beta_{2} + 7 \beta_{4} + \beta_{5} + 15 \beta_{6} + 6 \beta_{7} + 3 \beta_{8} + 4 \beta_{9} - \beta_{10} - 4 \beta_{11} ) q^{9} + ( 17 + 9 \beta_{1} - 7 \beta_{2} + 41 \beta_{3} + 34 \beta_{4} + 2 \beta_{5} - 5 \beta_{6} - 5 \beta_{7} + 5 \beta_{9} - 5 \beta_{10} - 4 \beta_{11} ) q^{10} + ( -10 + 7 \beta_{1} + 7 \beta_{2} - 9 \beta_{3} - 51 \beta_{4} - 3 \beta_{5} - \beta_{6} + 18 \beta_{7} + 2 \beta_{8} - \beta_{9} + 5 \beta_{10} + 7 \beta_{11} ) q^{11} + ( -2 - 9 \beta_{1} - 127 \beta_{2} - 127 \beta_{3} + 6 \beta_{5} - 6 \beta_{6} + 24 \beta_{7} - 18 \beta_{8} - \beta_{9} - \beta_{10} ) q^{12} + ( 16 - 6 \beta_{2} - 12 \beta_{3} - 28 \beta_{4} - 5 \beta_{5} - 28 \beta_{6} - 15 \beta_{7} + 28 \beta_{8} + \beta_{9} - 3 \beta_{10} + \beta_{11} ) q^{13} + ( 101 - 14 \beta_{1} + 101 \beta_{2} + 59 \beta_{3} + 59 \beta_{4} + 38 \beta_{6} + 24 \beta_{7} - 19 \beta_{8} - 5 \beta_{9} ) q^{14} + ( 94 + 18 \beta_{1} + 94 \beta_{3} - 51 \beta_{4} + 18 \beta_{6} - 9 \beta_{7} - 9 \beta_{8} + \beta_{10} - 3 \beta_{11} ) q^{15} + ( 9 \beta_{1} + 130 \beta_{2} - 15 \beta_{3} + 130 \beta_{4} + 3 \beta_{5} - 21 \beta_{6} - 42 \beta_{7} + 33 \beta_{8} - 3 \beta_{9} ) q^{16} + ( -176 - 29 \beta_{1} - 71 \beta_{2} - 88 \beta_{3} - 105 \beta_{4} + \beta_{5} + 5 \beta_{6} - 29 \beta_{8} + 9 \beta_{9} + 8 \beta_{10} - 8 \beta_{11} ) q^{17} + ( -313 + 30 \beta_{1} - 200 \beta_{2} + 113 \beta_{3} - 100 \beta_{4} - 20 \beta_{5} - 30 \beta_{6} - 36 \beta_{7} + 36 \beta_{8} + 6 \beta_{9} + 10 \beta_{10} + 7 \beta_{11} ) q^{18} + ( -68 + 15 \beta_{1} + 68 \beta_{2} - 48 \beta_{3} + 48 \beta_{4} + 27 \beta_{5} - 15 \beta_{7} - 28 \beta_{8} - 11 \beta_{9} - 32 \beta_{10} - 5 \beta_{11} ) q^{19} + ( 120 - 44 \beta_{1} + 96 \beta_{2} + 120 \beta_{4} + 2 \beta_{5} + 20 \beta_{6} + 22 \beta_{7} - 24 \beta_{8} - 14 \beta_{9} - 2 \beta_{10} + 14 \beta_{11} ) q^{20} + ( 172 - 21 \beta_{1} + 106 \beta_{2} + 238 \beta_{3} + 344 \beta_{4} - 14 \beta_{5} + 12 \beta_{6} + 12 \beta_{7} - 27 \beta_{9} + 27 \beta_{10} + 28 \beta_{11} ) q^{21} + ( -35 + 19 \beta_{1} + 162 \beta_{2} + 51 \beta_{3} - 349 \beta_{4} + 6 \beta_{5} - 20 \beta_{6} - 25 \beta_{7} + 40 \beta_{8} + 13 \beta_{9} - 21 \beta_{10} - 36 \beta_{11} ) q^{22} + ( -40 + 8 \beta_{1} - 362 \beta_{2} - 362 \beta_{3} - 28 \beta_{5} - 8 \beta_{6} - 8 \beta_{7} + 16 \beta_{8} + 8 \beta_{9} + 8 \beta_{10} ) q^{23} + ( 113 - 236 \beta_{2} - 472 \beta_{3} - 585 \beta_{4} + 25 \beta_{5} + 54 \beta_{6} + 51 \beta_{7} - 54 \beta_{8} - 3 \beta_{9} + 19 \beta_{10} - 3 \beta_{11} ) q^{24} + ( 297 + 43 \beta_{1} + 297 \beta_{2} + 383 \beta_{3} + 383 \beta_{4} + 10 \beta_{5} - 66 \beta_{6} - 23 \beta_{7} + 33 \beta_{8} + 9 \beta_{9} - 10 \beta_{11} ) q^{25} + ( 525 - 13 \beta_{1} + 525 \beta_{3} + 352 \beta_{4} - 13 \beta_{6} + 22 \beta_{7} + 22 \beta_{8} - 13 \beta_{10} + 35 \beta_{11} ) q^{26} + ( -30 \beta_{1} + 182 \beta_{2} - 220 \beta_{3} + 182 \beta_{4} - 23 \beta_{5} + 51 \beta_{6} + 102 \beta_{7} - 72 \beta_{8} + 46 \beta_{9} - 23 \beta_{10} - 23 \beta_{11} ) q^{27} + ( -472 + 62 \beta_{1} - 76 \beta_{2} - 236 \beta_{3} - 396 \beta_{4} + 16 \beta_{5} - 26 \beta_{6} + 62 \beta_{8} + 18 \beta_{9} + 2 \beta_{10} - 2 \beta_{11} ) q^{28} + ( -502 - 96 \beta_{1} - 640 \beta_{2} - 138 \beta_{3} - 320 \beta_{4} + 22 \beta_{5} + 96 \beta_{6} + 95 \beta_{7} - 95 \beta_{8} + 14 \beta_{9} - 11 \beta_{10} - 18 \beta_{11} ) q^{29} + ( -430 - 81 \beta_{1} + 430 \beta_{2} - 231 \beta_{3} + 231 \beta_{4} - 37 \beta_{5} + 81 \beta_{7} + 75 \beta_{8} + 14 \beta_{9} + 46 \beta_{10} + 9 \beta_{11} ) q^{30} + ( -35 + 208 \beta_{1} + 258 \beta_{2} - 35 \beta_{4} - 38 \beta_{5} - 111 \beta_{6} - 104 \beta_{7} + 97 \beta_{8} - 25 \beta_{9} + 38 \beta_{10} + 25 \beta_{11} ) q^{31} + ( 89 + 45 \beta_{1} - 166 \beta_{2} + 344 \beta_{3} + 178 \beta_{4} + 33 \beta_{5} - 41 \beta_{6} - 41 \beta_{7} + 31 \beta_{9} - 31 \beta_{10} - 66 \beta_{11} ) q^{32} + ( 549 - 117 \beta_{1} + 158 \beta_{2} + 372 \beta_{3} - 169 \beta_{4} + 47 \beta_{5} + 78 \beta_{6} - 18 \beta_{7} - 189 \beta_{8} - 54 \beta_{9} - 5 \beta_{10} + 37 \beta_{11} ) q^{33} + ( -36 + 37 \beta_{1} - 290 \beta_{2} - 290 \beta_{3} + 13 \beta_{5} + 84 \beta_{6} - 158 \beta_{7} + 74 \beta_{8} - 29 \beta_{9} - 29 \beta_{10} ) q^{34} + ( 174 - 49 \beta_{2} - 98 \beta_{3} - 272 \beta_{4} - 21 \beta_{5} - 3 \beta_{6} - 99 \beta_{7} + 3 \beta_{8} - 11 \beta_{9} - 43 \beta_{10} - 11 \beta_{11} ) q^{35} + ( 558 - 78 \beta_{1} + 558 \beta_{2} + 965 \beta_{3} + 965 \beta_{4} - 67 \beta_{5} - 66 \beta_{6} - 144 \beta_{7} + 33 \beta_{8} - 23 \beta_{9} + 67 \beta_{11} ) q^{36} + ( 34 - 166 \beta_{1} + 34 \beta_{3} + 6 \beta_{4} - 166 \beta_{6} - 11 \beta_{7} - 11 \beta_{8} + 61 \beta_{10} - 32 \beta_{11} ) q^{37} + ( 47 \beta_{1} + 33 \beta_{2} - 757 \beta_{3} + 33 \beta_{4} + 68 \beta_{5} + 29 \beta_{6} + 58 \beta_{7} - 105 \beta_{8} - 108 \beta_{9} + 40 \beta_{10} + 40 \beta_{11} ) q^{38} + ( -934 + 183 \beta_{1} - 318 \beta_{2} - 467 \beta_{3} - 616 \beta_{4} - 28 \beta_{5} + 84 \beta_{6} + 183 \beta_{8} - 59 \beta_{9} - 31 \beta_{10} + 31 \beta_{11} ) q^{39} + ( -334 - 42 \beta_{1} - 440 \beta_{2} - 106 \beta_{3} - 220 \beta_{4} + 12 \beta_{5} + 42 \beta_{6} + 144 \beta_{7} - 144 \beta_{8} - 48 \beta_{9} - 6 \beta_{10} + 18 \beta_{11} ) q^{40} + ( -330 + 79 \beta_{1} + 330 \beta_{2} + 381 \beta_{3} - 381 \beta_{4} - 19 \beta_{5} - 79 \beta_{7} + 128 \beta_{8} + 18 \beta_{9} + 2 \beta_{10} - 17 \beta_{11} ) q^{41} + ( -15 - 725 \beta_{2} - 15 \beta_{4} + 79 \beta_{5} - 189 \beta_{6} - 189 \beta_{8} + 147 \beta_{9} - 79 \beta_{10} - 147 \beta_{11} ) q^{42} + ( 516 - 189 \beta_{1} + 615 \beta_{2} + 417 \beta_{3} + 1032 \beta_{4} - 14 \beta_{5} + 188 \beta_{6} + 188 \beta_{7} + 61 \beta_{9} - 61 \beta_{10} + 28 \beta_{11} ) q^{43} + ( -251 + 3 \beta_{1} + 740 \beta_{2} + 224 \beta_{3} + 382 \beta_{4} - 149 \beta_{5} - 13 \beta_{6} - 107 \beta_{7} + 224 \beta_{8} + 75 \beta_{9} + 109 \beta_{10} + 58 \beta_{11} ) q^{44} + ( 350 + 9 \beta_{1} - 146 \beta_{2} - 146 \beta_{3} + 87 \beta_{5} - 57 \beta_{6} + 39 \beta_{7} + 18 \beta_{8} + 61 \beta_{9} + 61 \beta_{10} ) q^{45} + ( 224 - 18 \beta_{2} - 36 \beta_{3} - 260 \beta_{4} - 60 \beta_{5} + 258 \beta_{6} + 206 \beta_{7} - 258 \beta_{8} + 56 \beta_{9} + 52 \beta_{10} + 56 \beta_{11} ) q^{46} + ( -503 + 294 \beta_{1} - 503 \beta_{2} - 942 \beta_{3} - 942 \beta_{4} + 129 \beta_{5} - 198 \beta_{6} + 96 \beta_{7} + 99 \beta_{8} + 80 \beta_{9} - 129 \beta_{11} ) q^{47} + ( -103 + 189 \beta_{1} - 103 \beta_{3} - 443 \beta_{4} + 189 \beta_{6} + 99 \beta_{7} + 99 \beta_{8} - 121 \beta_{10} - 145 \beta_{11} ) q^{48} + ( -154 \beta_{1} - 370 \beta_{2} + 481 \beta_{3} - 370 \beta_{4} - 88 \beta_{5} + 21 \beta_{6} + 42 \beta_{7} + 112 \beta_{8} + 21 \beta_{9} + 67 \beta_{10} + 67 \beta_{11} ) q^{49} + ( 1398 - 92 \beta_{1} + 527 \beta_{2} + 699 \beta_{3} + 871 \beta_{4} - 39 \beta_{5} - 248 \beta_{6} - 92 \beta_{8} - 70 \beta_{9} - 31 \beta_{10} + 31 \beta_{11} ) q^{50} + ( 1030 - 45 \beta_{1} + 416 \beta_{2} - 614 \beta_{3} + 208 \beta_{4} + 50 \beta_{5} + 45 \beta_{6} - 147 \beta_{7} + 147 \beta_{8} - 54 \beta_{9} - 25 \beta_{10} + 2 \beta_{11} ) q^{51} + ( 1018 - 234 \beta_{1} - 1018 \beta_{2} - 664 \beta_{3} + 664 \beta_{4} + 42 \beta_{5} + 234 \beta_{7} - 396 \beta_{8} - 36 \beta_{9} - 12 \beta_{10} + 30 \beta_{11} ) q^{52} + ( 236 - 210 \beta_{1} + 756 \beta_{2} + 236 \beta_{4} + 29 \beta_{5} + 262 \beta_{6} + 105 \beta_{7} + 52 \beta_{8} - 99 \beta_{9} - 29 \beta_{10} + 99 \beta_{11} ) q^{53} + ( -985 + 195 \beta_{1} - 809 \beta_{2} - 1161 \beta_{3} - 1970 \beta_{4} - 64 \beta_{5} - 228 \beta_{6} - 228 \beta_{7} - 136 \beta_{9} + 136 \beta_{10} + 128 \beta_{11} ) q^{54} + ( -186 + 293 \beta_{1} - 1093 \beta_{2} + 497 \beta_{3} + 521 \beta_{4} - 25 \beta_{5} - 177 \beta_{6} - 48 \beta_{7} - 31 \beta_{8} + 32 \beta_{9} - 39 \beta_{10} - 15 \beta_{11} ) q^{55} + ( -1148 + 22 \beta_{1} + 1134 \beta_{2} + 1134 \beta_{3} - 28 \beta_{5} - 66 \beta_{6} + 22 \beta_{7} + 44 \beta_{8} - 44 \beta_{9} - 44 \beta_{10} ) q^{56} + ( -890 + 793 \beta_{2} + 1586 \beta_{3} + 2476 \beta_{4} + 68 \beta_{5} - 519 \beta_{6} - 261 \beta_{7} + 519 \beta_{8} - 73 \beta_{9} - 78 \beta_{10} - 73 \beta_{11} ) q^{57} + ( -1127 - 548 \beta_{1} - 1127 \beta_{2} - 1765 \beta_{3} - 1765 \beta_{4} + 60 \beta_{5} + 742 \beta_{6} + 194 \beta_{7} - 371 \beta_{8} - 89 \beta_{9} - 60 \beta_{11} ) q^{58} + ( -1636 + 115 \beta_{1} - 1636 \beta_{3} - 766 \beta_{4} + 115 \beta_{6} - 308 \beta_{7} - 308 \beta_{8} + 43 \beta_{10} + 171 \beta_{11} ) q^{59} + ( 330 \beta_{1} + 268 \beta_{2} + 2162 \beta_{3} + 268 \beta_{4} - 276 \beta_{6} - 552 \beta_{7} + 222 \beta_{8} + 142 \beta_{9} - 142 \beta_{10} - 142 \beta_{11} ) q^{60} + ( 836 - 426 \beta_{1} - 116 \beta_{2} + 418 \beta_{3} + 952 \beta_{4} + 20 \beta_{5} + 469 \beta_{6} - 426 \beta_{8} + 169 \beta_{9} + 149 \beta_{10} - 149 \beta_{11} ) q^{61} + ( 1595 + 435 \beta_{1} + 4112 \beta_{2} + 2517 \beta_{3} + 2056 \beta_{4} - 170 \beta_{5} - 435 \beta_{6} - 484 \beta_{7} + 484 \beta_{8} + 188 \beta_{9} + 85 \beta_{10} - 9 \beta_{11} ) q^{62} + ( 456 + 429 \beta_{1} - 456 \beta_{2} + 1265 \beta_{3} - 1265 \beta_{4} - 22 \beta_{5} - 429 \beta_{7} + 72 \beta_{8} - 11 \beta_{9} + 66 \beta_{10} + 44 \beta_{11} ) q^{63} + ( -862 - 176 \beta_{1} - 1157 \beta_{2} - 862 \beta_{4} - 121 \beta_{5} + 473 \beta_{6} + 88 \beta_{7} + 297 \beta_{8} - 196 \beta_{9} + 121 \beta_{10} + 196 \beta_{11} ) q^{64} + ( -378 - 22 \beta_{1} + 368 \beta_{2} - 1124 \beta_{3} - 756 \beta_{4} + 93 \beta_{5} - 149 \beta_{6} - 149 \beta_{7} + 32 \beta_{9} - 32 \beta_{10} - 186 \beta_{11} ) q^{65} + ( -1967 - 129 \beta_{1} - 2846 \beta_{2} - 5012 \beta_{3} - 157 \beta_{4} + 445 \beta_{5} - 24 \beta_{6} + 663 \beta_{7} + 81 \beta_{8} - 90 \beta_{9} - 210 \beta_{10} - 206 \beta_{11} ) q^{66} + ( 2285 - 416 \beta_{1} + 1521 \beta_{2} + 1521 \beta_{3} - 139 \beta_{5} - 79 \beta_{6} + 911 \beta_{7} - 832 \beta_{8} - 152 \beta_{9} - 152 \beta_{10} ) q^{67} + ( -419 + 167 \beta_{2} + 334 \beta_{3} + 753 \beta_{4} - 12 \beta_{5} - 151 \beta_{6} + 25 \beta_{7} + 151 \beta_{8} + 33 \beta_{9} + 54 \beta_{10} + 33 \beta_{11} ) q^{68} + ( -1508 + 252 \beta_{1} - 1508 \beta_{2} - 1090 \beta_{3} - 1090 \beta_{4} - 430 \beta_{5} + 252 \beta_{7} - 100 \beta_{9} + 430 \beta_{11} ) q^{69} + ( -343 - 46 \beta_{1} - 343 \beta_{3} + 1821 \beta_{4} - 46 \beta_{6} + 299 \beta_{7} + 299 \beta_{8} + 212 \beta_{10} + 259 \beta_{11} ) q^{70} + ( -195 \beta_{1} - 2769 \beta_{2} - 659 \beta_{3} - 2769 \beta_{4} + 193 \beta_{5} + 139 \beta_{6} + 278 \beta_{7} - 83 \beta_{8} - 64 \beta_{9} - 129 \beta_{10} - 129 \beta_{11} ) q^{71} + ( 5842 - 420 \beta_{1} + 2667 \beta_{2} + 2921 \beta_{3} + 3175 \beta_{4} + 220 \beta_{5} - 369 \beta_{6} - 420 \beta_{8} + 215 \beta_{9} - 5 \beta_{10} + 5 \beta_{11} ) q^{72} + ( 2059 + 505 \beta_{1} + 174 \beta_{2} - 1885 \beta_{3} + 87 \beta_{4} - 52 \beta_{5} - 505 \beta_{6} - 112 \beta_{7} + 112 \beta_{8} + 134 \beta_{9} + 26 \beta_{10} - 41 \beta_{11} ) q^{73} + ( 2837 + 306 \beta_{1} - 2837 \beta_{2} - 37 \beta_{3} + 37 \beta_{4} + 186 \beta_{5} - 306 \beta_{7} + 401 \beta_{8} - 23 \beta_{9} - 326 \beta_{10} - 140 \beta_{11} ) q^{74} + ( 261 - 408 \beta_{1} + 1869 \beta_{2} + 261 \beta_{4} - 168 \beta_{5} + 393 \beta_{6} + 204 \beta_{7} - 15 \beta_{8} + 192 \beta_{9} + 168 \beta_{10} - 192 \beta_{11} ) q^{75} + ( -3205 + 574 \beta_{1} - 2261 \beta_{2} - 4149 \beta_{3} - 6410 \beta_{4} + 29 \beta_{5} + 371 \beta_{6} + 371 \beta_{7} + 72 \beta_{9} - 72 \beta_{10} - 58 \beta_{11} ) q^{76} + ( 1758 - 234 \beta_{1} + 1262 \beta_{2} + 1800 \beta_{3} + 2500 \beta_{4} - 214 \beta_{5} + 640 \beta_{6} + 547 \beta_{7} - 389 \beta_{8} - 108 \beta_{9} + 23 \beta_{10} - 102 \beta_{11} ) q^{77} + ( -1710 + 168 \beta_{1} + 4567 \beta_{2} + 4567 \beta_{3} - 295 \beta_{5} + 147 \beta_{6} - 483 \beta_{7} + 336 \beta_{8} + 407 \beta_{9} + 407 \beta_{10} ) q^{78} + ( -1901 + 948 \beta_{2} + 1896 \beta_{3} + 3797 \beta_{4} + 234 \beta_{5} - 69 \beta_{6} - 40 \beta_{7} + 69 \beta_{8} + 17 \beta_{9} + 268 \beta_{10} + 17 \beta_{11} ) q^{79} + ( -2204 + 100 \beta_{1} - 2204 \beta_{2} - 2860 \beta_{3} - 2860 \beta_{4} + 232 \beta_{5} - 264 \beta_{6} - 164 \beta_{7} + 132 \beta_{8} + 212 \beta_{9} - 232 \beta_{11} ) q^{80} + ( -3378 + 228 \beta_{1} - 3378 \beta_{3} - 2734 \beta_{4} + 228 \beta_{6} - 240 \beta_{7} - 240 \beta_{8} - 354 \beta_{10} - 158 \beta_{11} ) q^{81} + ( -64 \beta_{1} + 2941 \beta_{2} + 4310 \beta_{3} + 2941 \beta_{4} - 297 \beta_{5} - 388 \beta_{6} - 776 \beta_{7} + 840 \beta_{8} + 133 \beta_{9} + 164 \beta_{10} + 164 \beta_{11} ) q^{82} + ( -2156 + 361 \beta_{1} - 4070 \beta_{2} - 1078 \beta_{3} + 1914 \beta_{4} - 17 \beta_{5} - 280 \beta_{6} + 361 \beta_{8} - 219 \beta_{9} - 202 \beta_{10} + 202 \beta_{11} ) q^{83} + ( 2936 - 852 \beta_{1} + 2308 \beta_{2} - 628 \beta_{3} + 1154 \beta_{4} + 364 \beta_{5} + 852 \beta_{6} + 678 \beta_{7} - 678 \beta_{8} - 540 \beta_{9} - 182 \beta_{10} + 88 \beta_{11} ) q^{84} + ( -194 - 255 \beta_{1} + 194 \beta_{2} + 1602 \beta_{3} - 1602 \beta_{4} - 78 \beta_{5} + 255 \beta_{7} - 693 \beta_{8} + 87 \beta_{9} - 18 \beta_{10} - 96 \beta_{11} ) q^{85} + ( -2297 - 374 \beta_{1} - 4946 \beta_{2} - 2297 \beta_{4} + 71 \beta_{5} - 610 \beta_{6} + 187 \beta_{7} - 984 \beta_{8} - 117 \beta_{9} - 71 \beta_{10} + 117 \beta_{11} ) q^{86} + ( 1664 - 447 \beta_{1} + 2291 \beta_{2} + 1037 \beta_{3} + 3328 \beta_{4} - 169 \beta_{5} - 369 \beta_{6} - 369 \beta_{7} - 234 \beta_{9} + 234 \beta_{10} + 338 \beta_{11} ) q^{87} + ( -1496 + 814 \beta_{1} - 847 \beta_{2} - 440 \beta_{3} + 1496 \beta_{4} - 341 \beta_{5} - 957 \beta_{6} - 1716 \beta_{7} + 671 \beta_{8} - 132 \beta_{9} + 77 \beta_{10} + 506 \beta_{11} ) q^{88} + ( -80 + 497 \beta_{1} - 437 \beta_{2} - 437 \beta_{3} + 466 \beta_{5} - 90 \beta_{6} - 904 \beta_{7} + 994 \beta_{8} - 185 \beta_{9} - 185 \beta_{10} ) q^{89} + ( 929 - 468 \beta_{2} - 936 \beta_{3} - 1865 \beta_{4} - 5 \beta_{5} + 555 \beta_{6} + 513 \beta_{7} - 555 \beta_{8} - 274 \beta_{9} - 553 \beta_{10} - 274 \beta_{11} ) q^{90} + ( 1279 - 483 \beta_{1} + 1279 \beta_{2} + 1121 \beta_{3} + 1121 \beta_{4} + 301 \beta_{5} - 490 \beta_{6} - 973 \beta_{7} + 245 \beta_{8} - 37 \beta_{9} - 301 \beta_{11} ) q^{91} + ( 870 - 38 \beta_{1} + 870 \beta_{3} + 1422 \beta_{4} - 38 \beta_{6} + 22 \beta_{7} + 22 \beta_{8} + 318 \beta_{10} - 314 \beta_{11} ) q^{92} + ( -21 \beta_{1} - 5090 \beta_{2} - 2802 \beta_{3} - 5090 \beta_{4} + 61 \beta_{5} + 471 \beta_{6} + 942 \beta_{7} - 921 \beta_{8} - 429 \beta_{9} + 368 \beta_{10} + 368 \beta_{11} ) q^{93} + ( 2102 + 506 \beta_{1} + 6506 \beta_{2} + 1051 \beta_{3} - 4404 \beta_{4} - 533 \beta_{5} + 1099 \beta_{6} + 506 \beta_{8} - 468 \beta_{9} + 65 \beta_{10} - 65 \beta_{11} ) q^{94} + ( 2046 - 798 \beta_{1} + 2638 \beta_{2} + 592 \beta_{3} + 1319 \beta_{4} - 362 \beta_{5} + 798 \beta_{6} + 261 \beta_{7} - 261 \beta_{8} + 208 \beta_{9} + 181 \beta_{10} + 77 \beta_{11} ) q^{95} + ( -1222 - 810 \beta_{1} + 1222 \beta_{2} - 3315 \beta_{3} + 3315 \beta_{4} - 391 \beta_{5} + 810 \beta_{7} + 279 \beta_{8} + 143 \beta_{9} + 496 \beta_{10} + 105 \beta_{11} ) q^{96} + ( 2929 + 2496 \beta_{1} + 1500 \beta_{2} + 2929 \beta_{4} + 728 \beta_{5} - 1460 \beta_{6} - 1248 \beta_{7} + 1036 \beta_{8} + 348 \beta_{9} - 728 \beta_{10} - 348 \beta_{11} ) q^{97} + ( 630 - 868 \beta_{1} - 1858 \beta_{2} + 3118 \beta_{3} + 1260 \beta_{4} + 46 \beta_{5} + 39 \beta_{6} + 39 \beta_{7} + 313 \beta_{9} - 313 \beta_{10} - 92 \beta_{11} ) q^{98} + ( 493 - 630 \beta_{1} - 36 \beta_{2} + 5485 \beta_{3} - 52 \beta_{4} + 105 \beta_{5} + 387 \beta_{6} - 267 \beta_{7} - 114 \beta_{8} + 849 \beta_{9} + 364 \beta_{10} - 113 \beta_{11} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12q - 5q^{2} - 6q^{3} + 7q^{4} - 18q^{5} + 75q^{6} - 80q^{7} - 245q^{8} + q^{9} + O(q^{10}) \) \( 12q - 5q^{2} - 6q^{3} + 7q^{4} - 18q^{5} + 75q^{6} - 80q^{7} - 245q^{8} + q^{9} - 43q^{11} + 594q^{12} + 250q^{13} + 610q^{14} + 1134q^{15} - 633q^{16} - 1250q^{17} - 3150q^{18} - 1025q^{19} + 752q^{20} - 35q^{22} + 1684q^{23} + 5345q^{24} + 197q^{25} + 3490q^{26} - 687q^{27} - 3580q^{28} - 2690q^{29} - 6740q^{30} - 1136q^{31} + 5939q^{33} + 2370q^{34} + 3610q^{35} - 514q^{36} - 336q^{37} + 1900q^{38} - 6880q^{39} - 2340q^{40} - 4550q^{41} + 1310q^{42} - 6268q^{44} + 5136q^{45} + 4150q^{46} + 24q^{47} + 344q^{48} + 827q^{49} + 8895q^{50} + 13155q^{51} + 14070q^{52} + 414q^{53} - 2738q^{55} - 21340q^{56} - 26925q^{57} + 2980q^{58} - 10011q^{59} - 6856q^{60} + 9460q^{61} - 6200q^{62} + 9150q^{63} - 2633q^{64} - 3210q^{66} + 12154q^{67} - 9400q^{68} - 9022q^{69} - 9380q^{70} + 17574q^{71} + 43045q^{72} + 27950q^{73} + 43270q^{74} - 1761q^{75} + 4090q^{77} - 42920q^{78} - 41540q^{79} - 2308q^{80} - 21080q^{81} - 28175q^{82} - 18665q^{83} + 26250q^{84} - 4230q^{85} - 10125q^{86} - 15125q^{88} + 5554q^{89} + 18400q^{90} + 7390q^{91} + 3904q^{92} + 36898q^{93} + 18920q^{94} + 14110q^{95} - 21140q^{96} + 20769q^{97} - 3269q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{12} + 115 x^{10} + 5030 x^{8} + 102975 x^{6} + 953170 x^{4} + 2910655 x^{2} + 73205\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\(-13052 \nu^{11} - 117304 \nu^{10} - 1219776 \nu^{9} - 10477907 \nu^{8} - 40135806 \nu^{7} - 321264966 \nu^{6} - 549385674 \nu^{5} - 3857093625 \nu^{4} - 2823880454 \nu^{3} - 13654448170 \nu^{2} - 5016493020 \nu - 1427754383\)\()/ 1339827192 \)
\(\beta_{3}\)\(=\)\((\)\(13052 \nu^{11} - 117304 \nu^{10} + 1219776 \nu^{9} - 10477907 \nu^{8} + 40135806 \nu^{7} - 321264966 \nu^{6} + 549385674 \nu^{5} - 3857093625 \nu^{4} + 2823880454 \nu^{3} - 13654448170 \nu^{2} + 5016493020 \nu - 1427754383\)\()/ 1339827192 \)
\(\beta_{4}\)\(=\)\((\)\(-31452 \nu^{11} + 117304 \nu^{10} - 2737673 \nu^{9} + 10477907 \nu^{8} - 79586472 \nu^{7} + 321264966 \nu^{6} - 827426403 \nu^{5} + 3857093625 \nu^{4} - 1134660378 \nu^{3} + 13654448170 \nu^{2} + 7973305285 \nu + 757840787\)\()/ 1339827192 \)
\(\beta_{5}\)\(=\)\((\)\( 45643 \nu^{10} + 4016197 \nu^{8} + 120327945 \nu^{6} + 1386775659 \nu^{4} + 4387794733 \nu^{2} - 676101899 \)\()/60901236\)
\(\beta_{6}\)\(=\)\((\)\(10664 \nu^{11} + 25564 \nu^{10} + 952537 \nu^{9} + 2319614 \nu^{8} + 29205906 \nu^{7} + 72240366 \nu^{6} + 350644875 \nu^{5} + 874263126 \nu^{4} + 1241313470 \nu^{3} + 2997579640 \nu^{2} + 129795853 \nu + 86861060\)\()/ 121802472 \)
\(\beta_{7}\)\(=\)\((\)\(10664 \nu^{11} - 25564 \nu^{10} + 952537 \nu^{9} - 2319614 \nu^{8} + 29205906 \nu^{7} - 72240366 \nu^{6} + 350644875 \nu^{5} - 874263126 \nu^{4} + 1241313470 \nu^{3} - 2997579640 \nu^{2} + 129795853 \nu - 86861060\)\()/ 121802472 \)
\(\beta_{8}\)\(=\)\((\)\(10664 \nu^{11} - 79937 \nu^{10} + 952537 \nu^{9} - 7147008 \nu^{8} + 29205906 \nu^{7} - 219213027 \nu^{6} + 350644875 \nu^{5} - 2622222042 \nu^{4} + 1241313470 \nu^{3} - 9047202395 \nu^{2} + 68894617 \nu - 209313060\)\()/ 121802472 \)
\(\beta_{9}\)\(=\)\((\)\(-169940 \nu^{11} - 259930 \nu^{10} - 15878131 \nu^{9} - 23961014 \nu^{8} - 530879748 \nu^{7} - 766470936 \nu^{6} - 7708194267 \nu^{5} - 9732816588 \nu^{4} - 46854588548 \nu^{3} - 36929122516 \nu^{2} - 104516796175 \nu + 4416548158\)\()/ 1339827192 \)
\(\beta_{10}\)\(=\)\((\)\(169940 \nu^{11} - 259930 \nu^{10} + 15878131 \nu^{9} - 23961014 \nu^{8} + 530879748 \nu^{7} - 766470936 \nu^{6} + 7708194267 \nu^{5} - 9732816588 \nu^{4} + 46854588548 \nu^{3} - 36929122516 \nu^{2} + 104516796175 \nu + 4416548158\)\()/ 1339827192 \)
\(\beta_{11}\)\(=\)\((\)\(-380623 \nu^{11} + 502073 \nu^{10} - 35561311 \nu^{9} + 44178167 \nu^{8} - 1181014629 \nu^{7} + 1323607395 \nu^{6} - 16702393701 \nu^{5} + 15254532249 \nu^{4} - 93304347409 \nu^{3} + 48265742063 \nu^{2} - 169552623955 \nu - 7437120889\)\()/ 1339827192 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{10} + \beta_{9} - 2 \beta_{8} + \beta_{7} + \beta_{6} - \beta_{5} + \beta_{3} + \beta_{2} - \beta_{1} - 19\)
\(\nu^{3}\)\(=\)\(\beta_{10} - \beta_{9} - 5 \beta_{7} - 5 \beta_{6} - 24 \beta_{4} - 9 \beta_{3} - 15 \beta_{2} - 25 \beta_{1} - 12\)
\(\nu^{4}\)\(=\)\(-31 \beta_{10} - 31 \beta_{9} + 90 \beta_{8} - 51 \beta_{7} - 39 \beta_{6} + 33 \beta_{5} - 113 \beta_{3} - 113 \beta_{2} + 45 \beta_{1} + 476\)
\(\nu^{5}\)\(=\)\(32 \beta_{11} - 33 \beta_{10} + 33 \beta_{9} + 271 \beta_{7} + 271 \beta_{6} - 16 \beta_{5} + 1300 \beta_{4} + 677 \beta_{3} + 623 \beta_{2} + 682 \beta_{1} + 650\)
\(\nu^{6}\)\(=\)\(954 \beta_{10} + 954 \beta_{9} - 3628 \beta_{8} + 2292 \beta_{7} + 1336 \beta_{6} - 1142 \beta_{5} + 5450 \beta_{3} + 5450 \beta_{2} - 1814 \beta_{1} - 12905\)
\(\nu^{7}\)\(=\)\(-2096 \beta_{11} + 1056 \beta_{10} - 1056 \beta_{9} - 11834 \beta_{7} - 11834 \beta_{6} + 1048 \beta_{5} - 56748 \beta_{4} - 34702 \beta_{3} - 22046 \beta_{2} - 19683 \beta_{1} - 28374\)
\(\nu^{8}\)\(=\)\(-30477 \beta_{10} - 30477 \beta_{9} + 140790 \beta_{8} - 95925 \beta_{7} - 44865 \beta_{6} + 40191 \beta_{5} - 226923 \beta_{3} - 226923 \beta_{2} + 70395 \beta_{1} + 369011\)
\(\nu^{9}\)\(=\)\(99264 \beta_{11} - 36579 \beta_{10} + 36579 \beta_{9} + 478665 \beta_{7} + 478665 \beta_{6} - 49632 \beta_{5} + 2301516 \beta_{4} + 1545459 \beta_{3} + 756057 \beta_{2} + 596465 \beta_{1} + 1150758\)
\(\nu^{10}\)\(=\)\(1012445 \beta_{10} + 1012445 \beta_{9} - 5366098 \beta_{8} + 3851627 \beta_{7} + 1514471 \beta_{6} - 1431005 \beta_{5} + 8936705 \beta_{3} + 8936705 \beta_{2} - 2683049 \beta_{1} - 11069600\)
\(\nu^{11}\)\(=\)\(-4178328 \beta_{11} + 1343903 \beta_{10} - 1343903 \beta_{9} - 18668485 \beta_{7} - 18668485 \beta_{6} + 2089164 \beta_{5} - 90111516 \beta_{4} - 64217769 \beta_{3} - 25893747 \beta_{2} - 18898340 \beta_{1} - 45055758\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/11\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(-\beta_{4}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2.1
4.56289i
2.38108i
5.04186i
4.56289i
2.38108i
5.04186i
6.10049i
0.159251i
5.08417i
6.10049i
0.159251i
5.08417i
−5.45760 1.77328i −10.4040 + 7.55891i 13.6966 + 9.95117i −9.33639 28.7345i 70.1847 22.8044i −33.6132 + 46.2646i −3.13669 4.31729i 26.0747 80.2495i 173.377i
2.2 −3.38257 1.09906i 12.7857 9.28936i −2.71041 1.96923i 5.48545 + 16.8825i −53.4582 + 17.3696i −19.6356 + 27.0262i 40.4526 + 55.6782i 52.1517 160.506i 63.1351i
2.3 3.67706 + 1.19475i −1.64569 + 1.19566i −0.850950 0.618251i −1.76709 5.43854i −7.47979 + 2.43033i −2.52826 + 3.47985i −38.7511 53.3363i −23.7517 + 73.1002i 22.1090i
6.1 −5.45760 + 1.77328i −10.4040 7.55891i 13.6966 9.95117i −9.33639 + 28.7345i 70.1847 + 22.8044i −33.6132 46.2646i −3.13669 + 4.31729i 26.0747 + 80.2495i 173.377i
6.2 −3.38257 + 1.09906i 12.7857 + 9.28936i −2.71041 + 1.96923i 5.48545 16.8825i −53.4582 17.3696i −19.6356 27.0262i 40.4526 55.6782i 52.1517 + 160.506i 63.1351i
6.3 3.67706 1.19475i −1.64569 1.19566i −0.850950 + 0.618251i −1.76709 + 5.43854i −7.47979 2.43033i −2.52826 3.47985i −38.7511 + 53.3363i −23.7517 73.1002i 22.1090i
7.1 −2.46775 + 3.39656i −2.26281 + 6.96422i −0.502581 1.54679i 14.7329 10.7041i −18.0704 24.8717i 47.9990 15.5958i −57.3923 18.6479i 22.1503 + 16.0931i 76.4560i
7.2 1.02443 1.41001i 2.68168 8.25338i 4.00561 + 12.3280i −8.06057 + 5.85635i −8.89011 12.2362i −56.0338 + 18.2065i 48.0070 + 15.5984i 4.60358 + 3.34469i 17.3649i
7.3 4.10644 5.65202i −4.15494 + 12.7876i −10.1383 31.2024i −10.0543 + 7.30486i 55.2138 + 75.9952i 23.8119 7.73696i −111.679 36.2869i −80.7285 58.6527i 86.8239i
8.1 −2.46775 3.39656i −2.26281 6.96422i −0.502581 + 1.54679i 14.7329 + 10.7041i −18.0704 + 24.8717i 47.9990 + 15.5958i −57.3923 + 18.6479i 22.1503 16.0931i 76.4560i
8.2 1.02443 + 1.41001i 2.68168 + 8.25338i 4.00561 12.3280i −8.06057 5.85635i −8.89011 + 12.2362i −56.0338 18.2065i 48.0070 15.5984i 4.60358 3.34469i 17.3649i
8.3 4.10644 + 5.65202i −4.15494 12.7876i −10.1383 + 31.2024i −10.0543 7.30486i 55.2138 75.9952i 23.8119 + 7.73696i −111.679 + 36.2869i −80.7285 + 58.6527i 86.8239i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 8.3
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.d odd 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 11.5.d.a 12
3.b odd 2 1 99.5.k.a 12
4.b odd 2 1 176.5.n.a 12
11.b odd 2 1 121.5.d.d 12
11.c even 5 1 121.5.b.b 12
11.c even 5 1 121.5.d.c 12
11.c even 5 1 121.5.d.d 12
11.c even 5 1 121.5.d.e 12
11.d odd 10 1 inner 11.5.d.a 12
11.d odd 10 1 121.5.b.b 12
11.d odd 10 1 121.5.d.c 12
11.d odd 10 1 121.5.d.e 12
33.f even 10 1 99.5.k.a 12
44.g even 10 1 176.5.n.a 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
11.5.d.a 12 1.a even 1 1 trivial
11.5.d.a 12 11.d odd 10 1 inner
99.5.k.a 12 3.b odd 2 1
99.5.k.a 12 33.f even 10 1
121.5.b.b 12 11.c even 5 1
121.5.b.b 12 11.d odd 10 1
121.5.d.c 12 11.c even 5 1
121.5.d.c 12 11.d odd 10 1
121.5.d.d 12 11.b odd 2 1
121.5.d.d 12 11.c even 5 1
121.5.d.e 12 11.c even 5 1
121.5.d.e 12 11.d odd 10 1
176.5.n.a 12 4.b odd 2 1
176.5.n.a 12 44.g even 10 1

Hecke kernels

This newform subspace is the entire newspace \(S_{5}^{\mathrm{new}}(11, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + 5 T + 33 T^{2} + 125 T^{3} + 551 T^{4} + 1810 T^{5} + 8610 T^{6} + 50940 T^{7} + 233740 T^{8} + 1132840 T^{9} + 4354088 T^{10} + 17654480 T^{11} + 63726544 T^{12} + 282471680 T^{13} + 1114646528 T^{14} + 4640112640 T^{15} + 15318384640 T^{16} + 53414461440 T^{17} + 144451829760 T^{18} + 485868175360 T^{19} + 2366526980096 T^{20} + 8589934592000 T^{21} + 36283883716608 T^{22} + 87960930222080 T^{23} + 281474976710656 T^{24} \)
$3$ \( 1 + 6 T - 104 T^{2} + 46 T^{3} + 10081 T^{4} - 91194 T^{5} - 1472814 T^{6} + 5513616 T^{7} + 56745846 T^{8} - 574747974 T^{9} + 106590006 T^{10} + 40654029276 T^{11} + 197200145376 T^{12} + 3292976371356 T^{13} + 699337029366 T^{14} - 305444638050534 T^{15} + 2442722600670966 T^{16} + 19224790261904016 T^{17} - 415966175342727534 T^{18} - 2086226211137713434 T^{19} + 18680296523815409121 T^{20} + 6904353223661959566 T^{21} - \)\(12\!\cdots\!04\)\( T^{22} + \)\(59\!\cdots\!86\)\( T^{23} + \)\(79\!\cdots\!61\)\( T^{24} \)
$5$ \( 1 + 18 T - 874 T^{2} - 8990 T^{3} + 924345 T^{4} - 2498082 T^{5} - 756868316 T^{6} + 6088380428 T^{7} + 472540127940 T^{8} - 4427489398370 T^{9} - 231003059512494 T^{10} + 256199910578148 T^{11} + 91821549616068396 T^{12} + 160124944111342500 T^{13} - 90235570122067968750 T^{14} - \)\(10\!\cdots\!50\)\( T^{15} + \)\(72\!\cdots\!00\)\( T^{16} + \)\(58\!\cdots\!00\)\( T^{17} - \)\(45\!\cdots\!00\)\( T^{18} - \)\(93\!\cdots\!50\)\( T^{19} + \)\(21\!\cdots\!25\)\( T^{20} - \)\(13\!\cdots\!50\)\( T^{21} - \)\(79\!\cdots\!50\)\( T^{22} + \)\(10\!\cdots\!50\)\( T^{23} + \)\(35\!\cdots\!25\)\( T^{24} \)
$7$ \( 1 + 80 T + 6388 T^{2} + 701930 T^{3} + 49215681 T^{4} + 3279603800 T^{5} + 239232786210 T^{6} + 14513581978500 T^{7} + 843606939238690 T^{8} + 50663633663881700 T^{9} + 2671940584129639018 T^{10} + \)\(13\!\cdots\!90\)\( T^{11} + \)\(72\!\cdots\!24\)\( T^{12} + \)\(33\!\cdots\!90\)\( T^{13} + \)\(15\!\cdots\!18\)\( T^{14} + \)\(70\!\cdots\!00\)\( T^{15} + \)\(28\!\cdots\!90\)\( T^{16} + \)\(11\!\cdots\!00\)\( T^{17} + \)\(45\!\cdots\!10\)\( T^{18} + \)\(15\!\cdots\!00\)\( T^{19} + \)\(54\!\cdots\!81\)\( T^{20} + \)\(18\!\cdots\!30\)\( T^{21} + \)\(40\!\cdots\!88\)\( T^{22} + \)\(12\!\cdots\!80\)\( T^{23} + \)\(36\!\cdots\!01\)\( T^{24} \)
$11$ \( 1 + 43 T + 18931 T^{2} - 249865 T^{3} + 113135 T^{4} - 53668078882 T^{5} - 2617669162966 T^{6} - 785754342911362 T^{7} + 24251492001935 T^{8} - 784183406349392665 T^{9} + \)\(86\!\cdots\!91\)\( T^{10} + \)\(28\!\cdots\!43\)\( T^{11} + \)\(98\!\cdots\!41\)\( T^{12} \)
$13$ \( 1 - 250 T + 98548 T^{2} - 25239340 T^{3} + 4920135411 T^{4} - 1056434505070 T^{5} + 134058098615820 T^{6} - 14343324685617900 T^{7} + 158785723907559280 T^{8} + \)\(58\!\cdots\!50\)\( T^{9} - \)\(16\!\cdots\!92\)\( T^{10} + \)\(38\!\cdots\!10\)\( T^{11} - \)\(73\!\cdots\!16\)\( T^{12} + \)\(11\!\cdots\!10\)\( T^{13} - \)\(13\!\cdots\!32\)\( T^{14} + \)\(13\!\cdots\!50\)\( T^{15} + \)\(10\!\cdots\!80\)\( T^{16} - \)\(27\!\cdots\!00\)\( T^{17} + \)\(72\!\cdots\!20\)\( T^{18} - \)\(16\!\cdots\!70\)\( T^{19} + \)\(21\!\cdots\!91\)\( T^{20} - \)\(31\!\cdots\!40\)\( T^{21} + \)\(35\!\cdots\!48\)\( T^{22} - \)\(25\!\cdots\!50\)\( T^{23} + \)\(29\!\cdots\!21\)\( T^{24} \)
$17$ \( 1 + 1250 T + 992428 T^{2} + 582264010 T^{3} + 283914863531 T^{4} + 119803876725030 T^{5} + 46223383217047820 T^{6} + 16729167329461506640 T^{7} + \)\(58\!\cdots\!20\)\( T^{8} + \)\(19\!\cdots\!90\)\( T^{9} + \)\(62\!\cdots\!28\)\( T^{10} + \)\(19\!\cdots\!80\)\( T^{11} + \)\(56\!\cdots\!04\)\( T^{12} + \)\(16\!\cdots\!80\)\( T^{13} + \)\(43\!\cdots\!48\)\( T^{14} + \)\(11\!\cdots\!90\)\( T^{15} + \)\(28\!\cdots\!20\)\( T^{16} + \)\(67\!\cdots\!40\)\( T^{17} + \)\(15\!\cdots\!20\)\( T^{18} + \)\(33\!\cdots\!30\)\( T^{19} + \)\(67\!\cdots\!91\)\( T^{20} + \)\(11\!\cdots\!10\)\( T^{21} + \)\(16\!\cdots\!28\)\( T^{22} + \)\(17\!\cdots\!50\)\( T^{23} + \)\(11\!\cdots\!41\)\( T^{24} \)
$19$ \( 1 + 1025 T + 619378 T^{2} + 353530160 T^{3} + 166899801006 T^{4} + 66114359518655 T^{5} + 21592456954609665 T^{6} + 5902640524678934610 T^{7} + \)\(15\!\cdots\!25\)\( T^{8} + \)\(24\!\cdots\!45\)\( T^{9} - \)\(10\!\cdots\!67\)\( T^{10} - \)\(11\!\cdots\!95\)\( T^{11} - \)\(61\!\cdots\!56\)\( T^{12} - \)\(15\!\cdots\!95\)\( T^{13} - \)\(17\!\cdots\!47\)\( T^{14} + \)\(53\!\cdots\!45\)\( T^{15} + \)\(46\!\cdots\!25\)\( T^{16} + \)\(22\!\cdots\!10\)\( T^{17} + \)\(10\!\cdots\!65\)\( T^{18} + \)\(42\!\cdots\!55\)\( T^{19} + \)\(13\!\cdots\!66\)\( T^{20} + \)\(38\!\cdots\!60\)\( T^{21} + \)\(87\!\cdots\!78\)\( T^{22} + \)\(18\!\cdots\!25\)\( T^{23} + \)\(23\!\cdots\!41\)\( T^{24} \)
$23$ \( ( 1 - 842 T + 1237966 T^{2} - 916714810 T^{3} + 731144219855 T^{4} - 446744415652252 T^{5} + 258141193233856964 T^{6} - \)\(12\!\cdots\!32\)\( T^{7} + \)\(57\!\cdots\!55\)\( T^{8} - \)\(20\!\cdots\!10\)\( T^{9} + \)\(75\!\cdots\!26\)\( T^{10} - \)\(14\!\cdots\!42\)\( T^{11} + \)\(48\!\cdots\!41\)\( T^{12} )^{2} \)
$29$ \( 1 + 2690 T + 5931748 T^{2} + 10598261380 T^{3} + 16033149600451 T^{4} + 21947778968733670 T^{5} + 27405046849774036980 T^{6} + \)\(31\!\cdots\!20\)\( T^{7} + \)\(34\!\cdots\!60\)\( T^{8} + \)\(34\!\cdots\!50\)\( T^{9} + \)\(33\!\cdots\!68\)\( T^{10} + \)\(30\!\cdots\!90\)\( T^{11} + \)\(26\!\cdots\!84\)\( T^{12} + \)\(21\!\cdots\!90\)\( T^{13} + \)\(16\!\cdots\!48\)\( T^{14} + \)\(12\!\cdots\!50\)\( T^{15} + \)\(85\!\cdots\!60\)\( T^{16} + \)\(56\!\cdots\!20\)\( T^{17} + \)\(34\!\cdots\!80\)\( T^{18} + \)\(19\!\cdots\!70\)\( T^{19} + \)\(10\!\cdots\!91\)\( T^{20} + \)\(46\!\cdots\!80\)\( T^{21} + \)\(18\!\cdots\!48\)\( T^{22} + \)\(59\!\cdots\!90\)\( T^{23} + \)\(15\!\cdots\!61\)\( T^{24} \)
$31$ \( 1 + 1136 T - 230384 T^{2} - 239729514 T^{3} + 1267435488321 T^{4} + 2677537350048516 T^{5} + 1095876078080394266 T^{6} - \)\(12\!\cdots\!44\)\( T^{7} + \)\(51\!\cdots\!26\)\( T^{8} + \)\(24\!\cdots\!96\)\( T^{9} + \)\(25\!\cdots\!46\)\( T^{10} + \)\(85\!\cdots\!26\)\( T^{11} - \)\(77\!\cdots\!44\)\( T^{12} + \)\(78\!\cdots\!46\)\( T^{13} + \)\(21\!\cdots\!86\)\( T^{14} + \)\(19\!\cdots\!56\)\( T^{15} + \)\(37\!\cdots\!06\)\( T^{16} - \)\(81\!\cdots\!44\)\( T^{17} + \)\(67\!\cdots\!86\)\( T^{18} + \)\(15\!\cdots\!56\)\( T^{19} + \)\(67\!\cdots\!81\)\( T^{20} - \)\(11\!\cdots\!34\)\( T^{21} - \)\(10\!\cdots\!84\)\( T^{22} + \)\(47\!\cdots\!56\)\( T^{23} + \)\(38\!\cdots\!41\)\( T^{24} \)
$37$ \( 1 + 336 T + 72946 T^{2} - 4475893024 T^{3} + 1626992896581 T^{4} - 4706417114238944 T^{5} + 10320605504003112056 T^{6} - \)\(11\!\cdots\!84\)\( T^{7} + \)\(22\!\cdots\!56\)\( T^{8} - \)\(44\!\cdots\!44\)\( T^{9} + \)\(30\!\cdots\!46\)\( T^{10} - \)\(47\!\cdots\!04\)\( T^{11} + \)\(15\!\cdots\!76\)\( T^{12} - \)\(89\!\cdots\!44\)\( T^{13} + \)\(10\!\cdots\!66\)\( T^{14} - \)\(29\!\cdots\!64\)\( T^{15} + \)\(27\!\cdots\!96\)\( T^{16} - \)\(26\!\cdots\!84\)\( T^{17} + \)\(44\!\cdots\!16\)\( T^{18} - \)\(38\!\cdots\!24\)\( T^{19} + \)\(24\!\cdots\!61\)\( T^{20} - \)\(12\!\cdots\!84\)\( T^{21} + \)\(39\!\cdots\!46\)\( T^{22} + \)\(33\!\cdots\!96\)\( T^{23} + \)\(18\!\cdots\!21\)\( T^{24} \)
$41$ \( 1 + 4550 T + 14401928 T^{2} + 23922427770 T^{3} + 36931794819271 T^{4} + 40163345738311410 T^{5} + \)\(12\!\cdots\!20\)\( T^{6} + \)\(30\!\cdots\!20\)\( T^{7} + \)\(86\!\cdots\!60\)\( T^{8} + \)\(14\!\cdots\!70\)\( T^{9} + \)\(24\!\cdots\!68\)\( T^{10} + \)\(25\!\cdots\!80\)\( T^{11} + \)\(43\!\cdots\!24\)\( T^{12} + \)\(71\!\cdots\!80\)\( T^{13} + \)\(19\!\cdots\!28\)\( T^{14} + \)\(32\!\cdots\!70\)\( T^{15} + \)\(54\!\cdots\!60\)\( T^{16} + \)\(54\!\cdots\!20\)\( T^{17} + \)\(65\!\cdots\!20\)\( T^{18} + \)\(57\!\cdots\!10\)\( T^{19} + \)\(15\!\cdots\!51\)\( T^{20} + \)\(27\!\cdots\!70\)\( T^{21} + \)\(46\!\cdots\!28\)\( T^{22} + \)\(41\!\cdots\!50\)\( T^{23} + \)\(25\!\cdots\!21\)\( T^{24} \)
$43$ \( 1 - 23356307 T^{2} + 286833712371781 T^{4} - \)\(23\!\cdots\!15\)\( T^{6} + \)\(14\!\cdots\!15\)\( T^{8} - \)\(69\!\cdots\!82\)\( T^{10} + \)\(26\!\cdots\!14\)\( T^{12} - \)\(81\!\cdots\!82\)\( T^{14} + \)\(19\!\cdots\!15\)\( T^{16} - \)\(37\!\cdots\!15\)\( T^{18} + \)\(53\!\cdots\!81\)\( T^{20} - \)\(50\!\cdots\!07\)\( T^{22} + \)\(25\!\cdots\!01\)\( T^{24} \)
$47$ \( 1 - 24 T - 10068774 T^{2} - 8140867214 T^{3} + 53779050778611 T^{4} + 233185473449483776 T^{5} - 81521520124710716194 T^{6} - \)\(19\!\cdots\!44\)\( T^{7} - \)\(12\!\cdots\!74\)\( T^{8} + \)\(86\!\cdots\!36\)\( T^{9} + \)\(22\!\cdots\!96\)\( T^{10} - \)\(14\!\cdots\!34\)\( T^{11} - \)\(15\!\cdots\!64\)\( T^{12} - \)\(69\!\cdots\!54\)\( T^{13} + \)\(54\!\cdots\!56\)\( T^{14} + \)\(10\!\cdots\!76\)\( T^{15} - \)\(72\!\cdots\!54\)\( T^{16} - \)\(54\!\cdots\!44\)\( T^{17} - \)\(11\!\cdots\!14\)\( T^{18} + \)\(15\!\cdots\!36\)\( T^{19} + \)\(17\!\cdots\!51\)\( T^{20} - \)\(12\!\cdots\!94\)\( T^{21} - \)\(77\!\cdots\!74\)\( T^{22} - \)\(89\!\cdots\!44\)\( T^{23} + \)\(18\!\cdots\!61\)\( T^{24} \)
$53$ \( 1 - 414 T - 13725134 T^{2} - 7014732824 T^{3} + 165958055930181 T^{4} + 208417442032136246 T^{5} - \)\(14\!\cdots\!44\)\( T^{6} - \)\(22\!\cdots\!24\)\( T^{7} + \)\(12\!\cdots\!16\)\( T^{8} + \)\(14\!\cdots\!66\)\( T^{9} - \)\(76\!\cdots\!74\)\( T^{10} - \)\(33\!\cdots\!74\)\( T^{11} + \)\(54\!\cdots\!16\)\( T^{12} - \)\(26\!\cdots\!94\)\( T^{13} - \)\(47\!\cdots\!14\)\( T^{14} + \)\(71\!\cdots\!06\)\( T^{15} + \)\(47\!\cdots\!36\)\( T^{16} - \)\(69\!\cdots\!24\)\( T^{17} - \)\(34\!\cdots\!64\)\( T^{18} + \)\(39\!\cdots\!06\)\( T^{19} + \)\(24\!\cdots\!21\)\( T^{20} - \)\(83\!\cdots\!04\)\( T^{21} - \)\(12\!\cdots\!34\)\( T^{22} - \)\(30\!\cdots\!34\)\( T^{23} + \)\(58\!\cdots\!61\)\( T^{24} \)
$59$ \( 1 + 10011 T + 24606746 T^{2} - 48335046044 T^{3} - 85055058906234 T^{4} + 1848432652355091201 T^{5} + \)\(68\!\cdots\!21\)\( T^{6} - \)\(10\!\cdots\!34\)\( T^{7} - \)\(82\!\cdots\!39\)\( T^{8} + \)\(15\!\cdots\!31\)\( T^{9} + \)\(13\!\cdots\!41\)\( T^{10} - \)\(16\!\cdots\!29\)\( T^{11} - \)\(23\!\cdots\!24\)\( T^{12} - \)\(19\!\cdots\!69\)\( T^{13} + \)\(19\!\cdots\!61\)\( T^{14} + \)\(27\!\cdots\!11\)\( T^{15} - \)\(17\!\cdots\!99\)\( T^{16} - \)\(26\!\cdots\!34\)\( T^{17} + \)\(21\!\cdots\!81\)\( T^{18} + \)\(70\!\cdots\!21\)\( T^{19} - \)\(39\!\cdots\!54\)\( T^{20} - \)\(27\!\cdots\!04\)\( T^{21} + \)\(16\!\cdots\!46\)\( T^{22} + \)\(82\!\cdots\!71\)\( T^{23} + \)\(10\!\cdots\!21\)\( T^{24} \)
$61$ \( 1 - 9460 T + 57744648 T^{2} - 309062215210 T^{3} + 1619001804085751 T^{4} - 7693654169306140700 T^{5} + \)\(35\!\cdots\!40\)\( T^{6} - \)\(16\!\cdots\!00\)\( T^{7} + \)\(77\!\cdots\!60\)\( T^{8} - \)\(32\!\cdots\!20\)\( T^{9} + \)\(12\!\cdots\!28\)\( T^{10} - \)\(48\!\cdots\!10\)\( T^{11} + \)\(18\!\cdots\!84\)\( T^{12} - \)\(67\!\cdots\!10\)\( T^{13} + \)\(24\!\cdots\!68\)\( T^{14} - \)\(86\!\cdots\!20\)\( T^{15} + \)\(28\!\cdots\!60\)\( T^{16} - \)\(85\!\cdots\!00\)\( T^{17} + \)\(25\!\cdots\!40\)\( T^{18} - \)\(75\!\cdots\!00\)\( T^{19} + \)\(21\!\cdots\!71\)\( T^{20} - \)\(57\!\cdots\!10\)\( T^{21} + \)\(14\!\cdots\!48\)\( T^{22} - \)\(33\!\cdots\!60\)\( T^{23} + \)\(49\!\cdots\!81\)\( T^{24} \)
$67$ \( ( 1 - 6077 T + 35718991 T^{2} - 18219600105 T^{3} + 122534918608155 T^{4} + 2701855134548306278 T^{5} - \)\(61\!\cdots\!46\)\( T^{6} + \)\(54\!\cdots\!38\)\( T^{7} + \)\(49\!\cdots\!55\)\( T^{8} - \)\(14\!\cdots\!05\)\( T^{9} + \)\(58\!\cdots\!71\)\( T^{10} - \)\(20\!\cdots\!77\)\( T^{11} + \)\(66\!\cdots\!21\)\( T^{12} )^{2} \)
$71$ \( 1 - 17574 T + 104286776 T^{2} - 424728504824 T^{3} + 5029513991381961 T^{4} - 44618257451712179514 T^{5} + \)\(20\!\cdots\!06\)\( T^{6} - \)\(10\!\cdots\!44\)\( T^{7} + \)\(80\!\cdots\!46\)\( T^{8} - \)\(47\!\cdots\!54\)\( T^{9} + \)\(20\!\cdots\!26\)\( T^{10} - \)\(11\!\cdots\!94\)\( T^{11} + \)\(68\!\cdots\!36\)\( T^{12} - \)\(29\!\cdots\!14\)\( T^{13} + \)\(13\!\cdots\!86\)\( T^{14} - \)\(77\!\cdots\!14\)\( T^{15} + \)\(33\!\cdots\!66\)\( T^{16} - \)\(11\!\cdots\!44\)\( T^{17} + \)\(56\!\cdots\!86\)\( T^{18} - \)\(30\!\cdots\!54\)\( T^{19} + \)\(87\!\cdots\!01\)\( T^{20} - \)\(18\!\cdots\!04\)\( T^{21} + \)\(11\!\cdots\!76\)\( T^{22} - \)\(50\!\cdots\!94\)\( T^{23} + \)\(72\!\cdots\!61\)\( T^{24} \)
$73$ \( 1 - 27950 T + 454822888 T^{2} - 5337429236470 T^{3} + 50136942858006191 T^{4} - \)\(39\!\cdots\!10\)\( T^{5} + \)\(27\!\cdots\!20\)\( T^{6} - \)\(17\!\cdots\!80\)\( T^{7} + \)\(99\!\cdots\!60\)\( T^{8} - \)\(54\!\cdots\!10\)\( T^{9} + \)\(29\!\cdots\!08\)\( T^{10} - \)\(15\!\cdots\!80\)\( T^{11} + \)\(84\!\cdots\!04\)\( T^{12} - \)\(44\!\cdots\!80\)\( T^{13} + \)\(23\!\cdots\!48\)\( T^{14} - \)\(12\!\cdots\!10\)\( T^{15} + \)\(64\!\cdots\!60\)\( T^{16} - \)\(31\!\cdots\!80\)\( T^{17} + \)\(14\!\cdots\!20\)\( T^{18} - \)\(59\!\cdots\!10\)\( T^{19} + \)\(21\!\cdots\!11\)\( T^{20} - \)\(64\!\cdots\!70\)\( T^{21} + \)\(15\!\cdots\!88\)\( T^{22} - \)\(27\!\cdots\!50\)\( T^{23} + \)\(27\!\cdots\!81\)\( T^{24} \)
$79$ \( 1 + 41540 T + 884578158 T^{2} + 13528269571760 T^{3} + 171031570391119991 T^{4} + \)\(18\!\cdots\!60\)\( T^{5} + \)\(18\!\cdots\!10\)\( T^{6} + \)\(17\!\cdots\!80\)\( T^{7} + \)\(14\!\cdots\!70\)\( T^{8} + \)\(11\!\cdots\!40\)\( T^{9} + \)\(82\!\cdots\!08\)\( T^{10} + \)\(56\!\cdots\!20\)\( T^{11} + \)\(36\!\cdots\!24\)\( T^{12} + \)\(21\!\cdots\!20\)\( T^{13} + \)\(12\!\cdots\!88\)\( T^{14} + \)\(66\!\cdots\!40\)\( T^{15} + \)\(33\!\cdots\!70\)\( T^{16} + \)\(15\!\cdots\!80\)\( T^{17} + \)\(65\!\cdots\!10\)\( T^{18} + \)\(25\!\cdots\!60\)\( T^{19} + \)\(90\!\cdots\!31\)\( T^{20} + \)\(27\!\cdots\!60\)\( T^{21} + \)\(71\!\cdots\!58\)\( T^{22} + \)\(13\!\cdots\!40\)\( T^{23} + \)\(12\!\cdots\!61\)\( T^{24} \)
$83$ \( 1 + 18665 T + 224643628 T^{2} + 792700664890 T^{3} - 7907731388500564 T^{4} - \)\(17\!\cdots\!75\)\( T^{5} - \)\(12\!\cdots\!65\)\( T^{6} - \)\(28\!\cdots\!70\)\( T^{7} + \)\(49\!\cdots\!05\)\( T^{8} + \)\(50\!\cdots\!25\)\( T^{9} + \)\(19\!\cdots\!53\)\( T^{10} - \)\(10\!\cdots\!35\)\( T^{11} - \)\(13\!\cdots\!56\)\( T^{12} - \)\(49\!\cdots\!35\)\( T^{13} + \)\(44\!\cdots\!73\)\( T^{14} + \)\(53\!\cdots\!25\)\( T^{15} + \)\(25\!\cdots\!05\)\( T^{16} - \)\(67\!\cdots\!70\)\( T^{17} - \)\(14\!\cdots\!65\)\( T^{18} - \)\(97\!\cdots\!75\)\( T^{19} - \)\(20\!\cdots\!04\)\( T^{20} + \)\(96\!\cdots\!90\)\( T^{21} + \)\(13\!\cdots\!28\)\( T^{22} + \)\(51\!\cdots\!65\)\( T^{23} + \)\(13\!\cdots\!41\)\( T^{24} \)
$89$ \( ( 1 - 2777 T + 289401541 T^{2} - 984806913445 T^{3} + 37686355645869335 T^{4} - \)\(12\!\cdots\!62\)\( T^{5} + \)\(29\!\cdots\!14\)\( T^{6} - \)\(81\!\cdots\!42\)\( T^{7} + \)\(14\!\cdots\!35\)\( T^{8} - \)\(24\!\cdots\!45\)\( T^{9} + \)\(44\!\cdots\!01\)\( T^{10} - \)\(27\!\cdots\!77\)\( T^{11} + \)\(61\!\cdots\!41\)\( T^{12} )^{2} \)
$97$ \( 1 - 20769 T + 424679856 T^{2} - 8132949547004 T^{3} + 119819968442578026 T^{4} - \)\(16\!\cdots\!09\)\( T^{5} + \)\(21\!\cdots\!91\)\( T^{6} - \)\(25\!\cdots\!44\)\( T^{7} + \)\(28\!\cdots\!01\)\( T^{8} - \)\(31\!\cdots\!79\)\( T^{9} + \)\(31\!\cdots\!81\)\( T^{10} - \)\(31\!\cdots\!59\)\( T^{11} + \)\(30\!\cdots\!76\)\( T^{12} - \)\(27\!\cdots\!79\)\( T^{13} + \)\(24\!\cdots\!41\)\( T^{14} - \)\(21\!\cdots\!39\)\( T^{15} + \)\(17\!\cdots\!21\)\( T^{16} - \)\(13\!\cdots\!44\)\( T^{17} + \)\(10\!\cdots\!71\)\( T^{18} - \)\(70\!\cdots\!49\)\( T^{19} + \)\(45\!\cdots\!66\)\( T^{20} - \)\(27\!\cdots\!84\)\( T^{21} + \)\(12\!\cdots\!56\)\( T^{22} - \)\(54\!\cdots\!89\)\( T^{23} + \)\(23\!\cdots\!61\)\( T^{24} \)
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