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 \(10\) and degree \(4\))

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: \( 1 \)
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.

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

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

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

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. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
11.d Odd 1 yes

Hecke kernels

There are no other newforms in \(S_{5}^{\mathrm{new}}(11, [\chi])\).