Properties

Label 18.5.d.a
Level 18
Weight 5
Character orbit 18.d
Analytic conductor 1.861
Analytic rank 0
Dimension 8
CM No
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 18 = 2 \cdot 3^{2} \)
Weight: \( k \) = \( 5 \)
Character orbit: \([\chi]\) = 18.d (of order \(6\) and degree \(2\))

Newform invariants

Self dual: No
Analytic conductor: \(1.86065933551\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: 8.0.221456830464.4
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{4}\cdot 3^{4} \)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \( -\beta_{3} q^{2} \) \( + ( 2 + \beta_{1} - 3 \beta_{2} - \beta_{5} ) q^{3} \) \( + ( 8 - 8 \beta_{2} ) q^{4} \) \( + ( -\beta_{1} + 4 \beta_{2} - 3 \beta_{3} + 2 \beta_{4} - 3 \beta_{5} - \beta_{6} ) q^{5} \) \( + ( 4 - 3 \beta_{1} + 4 \beta_{2} - 2 \beta_{3} - \beta_{4} + \beta_{6} + \beta_{7} ) q^{6} \) \( + ( 2 + 6 \beta_{1} - 9 \beta_{2} + 5 \beta_{3} + 4 \beta_{5} - \beta_{6} - \beta_{7} ) q^{7} \) \( + ( -8 \beta_{1} - 8 \beta_{3} ) q^{8} \) \( + ( -23 + 6 \beta_{1} + 24 \beta_{2} + 23 \beta_{3} - 6 \beta_{4} + 2 \beta_{5} + \beta_{6} - \beta_{7} ) q^{9} \) \(+O(q^{10})\) \( q\) \( -\beta_{3} q^{2} \) \( + ( 2 + \beta_{1} - 3 \beta_{2} - \beta_{5} ) q^{3} \) \( + ( 8 - 8 \beta_{2} ) q^{4} \) \( + ( -\beta_{1} + 4 \beta_{2} - 3 \beta_{3} + 2 \beta_{4} - 3 \beta_{5} - \beta_{6} ) q^{5} \) \( + ( 4 - 3 \beta_{1} + 4 \beta_{2} - 2 \beta_{3} - \beta_{4} + \beta_{6} + \beta_{7} ) q^{6} \) \( + ( 2 + 6 \beta_{1} - 9 \beta_{2} + 5 \beta_{3} + 4 \beta_{5} - \beta_{6} - \beta_{7} ) q^{7} \) \( + ( -8 \beta_{1} - 8 \beta_{3} ) q^{8} \) \( + ( -23 + 6 \beta_{1} + 24 \beta_{2} + 23 \beta_{3} - 6 \beta_{4} + 2 \beta_{5} + \beta_{6} - \beta_{7} ) q^{9} \) \( + ( 4 + 2 \beta_{1} - 4 \beta_{2} + 4 \beta_{3} - \beta_{4} + 8 \beta_{5} + 3 \beta_{6} + \beta_{7} ) q^{10} \) \( + ( -122 - 6 \beta_{1} + 67 \beta_{2} - 17 \beta_{3} + 12 \beta_{4} - 4 \beta_{5} + \beta_{6} - 2 \beta_{7} ) q^{11} \) \( + ( -8 + 8 \beta_{1} - 24 \beta_{2} - 8 \beta_{4} ) q^{12} \) \( + ( 10 + 17 \beta_{1} - 12 \beta_{2} + 33 \beta_{3} - 10 \beta_{4} + 15 \beta_{5} - \beta_{6} + 6 \beta_{7} ) q^{13} \) \( + ( 24 - 12 \beta_{1} + 28 \beta_{2} - 6 \beta_{3} + 11 \beta_{4} - 4 \beta_{6} - 3 \beta_{7} ) q^{14} \) \( + ( 120 - 51 \beta_{1} + 36 \beta_{2} + 3 \beta_{3} - 6 \beta_{4} - 3 \beta_{5} - 3 \beta_{6} - 6 \beta_{7} ) q^{15} \) \( -64 \beta_{2} q^{16} \) \( + ( 103 - 13 \beta_{1} - 205 \beta_{2} - 26 \beta_{3} + 20 \beta_{4} - 25 \beta_{5} + 6 \beta_{6} + 7 \beta_{7} ) q^{17} \) \( + ( -128 + 27 \beta_{1} + 156 \beta_{2} + 11 \beta_{3} + 3 \beta_{4} - 16 \beta_{5} - 2 \beta_{6} + 5 \beta_{7} ) q^{18} \) \( + ( 7 + 76 \beta_{1} + \beta_{2} - 73 \beta_{3} - 11 \beta_{4} - 11 \beta_{5} - 3 \beta_{6} + 2 \beta_{7} ) q^{19} \) \( + ( 16 - 8 \beta_{2} - 8 \beta_{3} - 16 \beta_{5} - 8 \beta_{6} - 8 \beta_{7} ) q^{20} \) \( + ( 180 - 12 \beta_{1} - 243 \beta_{2} - 93 \beta_{3} + 4 \beta_{4} + 14 \beta_{5} + 3 \beta_{6} - 3 \beta_{7} ) q^{21} \) \( + ( 72 + 49 \beta_{1} - 64 \beta_{2} + 102 \beta_{3} + 22 \beta_{4} - 24 \beta_{5} + 4 \beta_{6} - 6 \beta_{7} ) q^{22} \) \( + ( 114 + 49 \beta_{1} + 86 \beta_{2} + 27 \beta_{3} - 32 \beta_{4} + 15 \beta_{5} + 13 \beta_{6} + 6 \beta_{7} ) q^{23} \) \( + ( 64 - 16 \beta_{1} - 32 \beta_{2} + 8 \beta_{3} - 8 \beta_{4} + 8 \beta_{7} ) q^{24} \) \( + ( -6 - 330 \beta_{1} + 212 \beta_{2} - 177 \beta_{3} + 12 \beta_{4} - 12 \beta_{5} - 3 \beta_{6} + 3 \beta_{7} ) q^{25} \) \( + ( -68 + 16 \beta_{1} + 140 \beta_{2} + 42 \beta_{3} - 37 \beta_{4} + 56 \beta_{5} - 15 \beta_{6} - 11 \beta_{7} ) q^{26} \) \( + ( -93 + 189 \beta_{1} - 114 \beta_{2} + 189 \beta_{3} + 39 \beta_{4} + 18 \beta_{5} - 3 \beta_{6} - 9 \beta_{7} ) q^{27} \) \( + ( -48 + 8 \beta_{1} + 8 \beta_{2} - 32 \beta_{3} + 32 \beta_{4} + 8 \beta_{5} - 8 \beta_{7} ) q^{28} \) \( + ( -278 + 36 \beta_{1} + 103 \beta_{2} + 133 \beta_{3} - 72 \beta_{4} + 62 \beta_{5} + 13 \beta_{6} + 31 \beta_{7} ) q^{29} \) \( + ( -444 + 24 \beta_{1} + 36 \beta_{2} - 156 \beta_{3} + 33 \beta_{4} - 24 \beta_{5} + 3 \beta_{6} + 15 \beta_{7} ) q^{30} \) \( + ( -386 + 147 \beta_{1} + 368 \beta_{2} + 285 \beta_{3} - 6 \beta_{4} - 33 \beta_{5} - 9 \beta_{6} - 30 \beta_{7} ) q^{31} \) \( -64 \beta_{1} q^{32} \) \( + ( -441 - 393 \beta_{1} + 96 \beta_{2} - 108 \beta_{3} + 78 \beta_{4} + 33 \beta_{5} + 12 \beta_{6} + 9 \beta_{7} ) q^{33} \) \( + ( 4 - 204 \beta_{1} - 56 \beta_{2} - 71 \beta_{3} - 54 \beta_{4} + 8 \beta_{5} + 25 \beta_{6} - 2 \beta_{7} ) q^{34} \) \( + ( 172 + 215 \beta_{1} - 361 \beta_{2} + 244 \beta_{3} - 52 \beta_{4} + 41 \beta_{5} - 6 \beta_{6} - 23 \beta_{7} ) q^{35} \) \( + ( 64 + 168 \beta_{1} + 144 \beta_{2} + 176 \beta_{3} - 48 \beta_{4} + 56 \beta_{5} + 16 \beta_{6} + 8 \beta_{7} ) q^{36} \) \( + ( -42 + 146 \beta_{1} + 44 \beta_{2} - 218 \beta_{3} + 26 \beta_{4} - 76 \beta_{5} - 30 \beta_{6} - 14 \beta_{7} ) q^{37} \) \( + ( 1148 + 18 \beta_{1} - 592 \beta_{2} + 21 \beta_{3} - 36 \beta_{4} + 40 \beta_{5} + 11 \beta_{6} + 20 \beta_{7} ) q^{38} \) \( + ( 338 + 181 \beta_{1} + 636 \beta_{2} - 393 \beta_{3} + 32 \beta_{4} - 21 \beta_{5} - 39 \beta_{6} - 24 \beta_{7} ) q^{39} \) \( + ( -32 \beta_{1} + 32 \beta_{2} - 48 \beta_{3} + 40 \beta_{4} + 16 \beta_{6} + 24 \beta_{7} ) q^{40} \) \( + ( -15 + 32 \beta_{1} + \beta_{2} - 28 \beta_{4} - 24 \beta_{5} + 8 \beta_{6} + 12 \beta_{7} ) q^{41} \) \( + ( 704 - 256 \beta_{1} - 772 \beta_{2} - 216 \beta_{3} + 39 \beta_{4} - 48 \beta_{5} - 14 \beta_{6} - 15 \beta_{7} ) q^{42} \) \( + ( -60 - 288 \beta_{1} + 85 \beta_{2} - 231 \beta_{3} + 54 \beta_{4} - 120 \beta_{5} + 3 \beta_{6} + 30 \beta_{7} ) q^{43} \) \( + ( -520 - 104 \beta_{1} + 1024 \beta_{2} - 136 \beta_{3} + 40 \beta_{4} - 80 \beta_{5} + 24 \beta_{6} + 8 \beta_{7} ) q^{44} \) \( + ( 1242 + 351 \beta_{1} - 1674 \beta_{2} + 567 \beta_{3} - 81 \beta_{5} - 27 \beta_{6} ) q^{45} \) \( + ( 236 + 136 \beta_{1} + 4 \beta_{2} - 118 \beta_{3} - 59 \beta_{4} - 56 \beta_{5} - 15 \beta_{6} + 11 \beta_{7} ) q^{46} \) \( + ( 306 - 84 \beta_{1} - 69 \beta_{2} - 147 \beta_{3} + 168 \beta_{4} - 126 \beta_{5} - 21 \beta_{6} - 63 \beta_{7} ) q^{47} \) \( + ( -192 - 64 \beta_{4} + 64 \beta_{5} ) q^{48} \) \( + ( 639 + 217 \beta_{1} - 637 \beta_{2} + 435 \beta_{3} - 26 \beta_{4} + 57 \beta_{5} + \beta_{6} + 30 \beta_{7} ) q^{49} \) \( + ( -1272 + 209 \beta_{1} - 1332 \beta_{2} + 42 \beta_{3} - 21 \beta_{4} + 48 \beta_{5} + 12 \beta_{6} - 3 \beta_{7} ) q^{50} \) \( + ( -2220 - 288 \beta_{1} + 1320 \beta_{2} - 18 \beta_{3} - 222 \beta_{4} - 6 \beta_{5} - 24 \beta_{6} - 9 \beta_{7} ) q^{51} \) \( + ( 16 + 144 \beta_{1} + 8 \beta_{2} + 40 \beta_{3} + 96 \beta_{4} + 32 \beta_{5} - 56 \beta_{6} - 8 \beta_{7} ) q^{52} \) \( + ( -112 - 476 \beta_{1} + 280 \beta_{2} - 532 \beta_{3} + 112 \beta_{4} - 56 \beta_{5} + 56 \beta_{7} ) q^{53} \) \( + ( 72 - 180 \beta_{1} + 1632 \beta_{2} + 33 \beta_{3} + 120 \beta_{4} - 48 \beta_{5} - 18 \beta_{6} - 48 \beta_{7} ) q^{54} \) \( + ( -306 + 249 \beta_{1} - 213 \beta_{2} + 90 \beta_{3} - 102 \beta_{4} + 387 \beta_{5} + 150 \beta_{6} + 63 \beta_{7} ) q^{55} \) \( + ( 352 - 48 \beta_{1} - 128 \beta_{2} - 16 \beta_{3} + 96 \beta_{4} - 64 \beta_{5} - 8 \beta_{6} - 32 \beta_{7} ) q^{56} \) \( + ( 622 - 34 \beta_{1} + 1452 \beta_{2} + 39 \beta_{3} - 150 \beta_{4} + 34 \beta_{5} + 141 \beta_{6} + 57 \beta_{7} ) q^{57} \) \( + ( -528 + 268 \beta_{1} + 404 \beta_{2} + 474 \beta_{3} - 227 \beta_{4} + 144 \beta_{5} - 62 \beta_{6} - 21 \beta_{7} ) q^{58} \) \( + ( 777 - 515 \beta_{1} + 845 \beta_{2} - 138 \beta_{3} + 295 \beta_{4} + 36 \beta_{5} - 104 \beta_{6} - 87 \beta_{7} ) q^{59} \) \( + ( 1344 + 48 \beta_{1} - 1080 \beta_{2} + 552 \beta_{3} - 96 \beta_{4} + 96 \beta_{5} + 24 \beta_{6} - 24 \beta_{7} ) q^{60} \) \( + ( 270 + 246 \beta_{1} - 1577 \beta_{2} + 513 \beta_{3} - 240 \beta_{4} + 540 \beta_{5} - 15 \beta_{6} - 135 \beta_{7} ) q^{61} \) \( + ( -1236 + 284 \beta_{1} + 2508 \beta_{2} + 182 \beta_{3} + 171 \beta_{4} - 168 \beta_{5} + 33 \beta_{6} + 69 \beta_{7} ) q^{62} \) \( + ( -352 - 186 \beta_{1} - 1011 \beta_{2} - 425 \beta_{3} - 270 \beta_{4} + 46 \beta_{5} + 101 \beta_{6} + 121 \beta_{7} ) q^{63} \) \( -512 q^{64} \) \( + ( -3790 + 48 \beta_{1} + 1847 \beta_{2} - 697 \beta_{3} - 96 \beta_{4} - 62 \beta_{5} - 55 \beta_{6} - 31 \beta_{7} ) q^{65} \) \( + ( -2148 + 45 \beta_{1} - 720 \beta_{2} + 465 \beta_{3} + 48 \beta_{4} - 24 \beta_{5} - 33 \beta_{6} - 120 \beta_{7} ) q^{66} \) \( + ( 1831 - 39 \beta_{1} - 1639 \beta_{2} + 18 \beta_{3} + 285 \beta_{4} - 90 \beta_{5} + 96 \beta_{6} + 99 \beta_{7} ) q^{67} \) \( + ( -1032 - 8 \beta_{1} - 808 \beta_{2} - 120 \beta_{3} - 32 \beta_{4} - 216 \beta_{5} - 8 \beta_{6} + 48 \beta_{7} ) q^{68} \) \( + ( -1278 + 909 \beta_{1} + 1038 \beta_{2} + 357 \beta_{3} + 114 \beta_{4} - 213 \beta_{5} + 63 \beta_{6} + 60 \beta_{7} ) q^{69} \) \( + ( -68 - 378 \beta_{1} + 1672 \beta_{2} - 332 \beta_{3} + 150 \beta_{4} - 136 \beta_{5} - 41 \beta_{6} + 34 \beta_{7} ) q^{70} \) \( + ( 1690 - 538 \beta_{1} - 3430 \beta_{2} - 392 \beta_{3} - 244 \beta_{4} + 242 \beta_{5} - 48 \beta_{6} - 98 \beta_{7} ) q^{71} \) \( + ( 160 + 216 \beta_{1} + 960 \beta_{2} - 64 \beta_{3} - 48 \beta_{4} - 64 \beta_{5} - 56 \beta_{6} - 16 \beta_{7} ) q^{72} \) \( + ( 2405 - 681 \beta_{1} + 303 \beta_{2} + 108 \beta_{3} + 372 \beta_{4} - 369 \beta_{5} - 168 \beta_{6} - 135 \beta_{7} ) q^{73} \) \( + ( 2608 - 24 \beta_{1} - 1280 \beta_{2} + 50 \beta_{3} + 48 \beta_{4} + 128 \beta_{5} + 76 \beta_{6} + 64 \beta_{7} ) q^{74} \) \( + ( 2715 - 486 \beta_{1} - 1323 \beta_{2} + 459 \beta_{3} + 23 \beta_{4} - 239 \beta_{5} - 189 \beta_{6} + 135 \beta_{7} ) q^{75} \) \( + ( 136 - 496 \beta_{1} - 216 \beta_{2} - 1032 \beta_{3} - 136 \beta_{4} + 72 \beta_{5} - 40 \beta_{6} - 24 \beta_{7} ) q^{76} \) \( + ( 2712 + 589 \beta_{1} + 3044 \beta_{2} - 183 \beta_{3} - 32 \beta_{4} - 315 \beta_{5} - 17 \beta_{6} + 66 \beta_{7} ) q^{77} \) \( + ( 4508 + 532 \beta_{1} - 2836 \beta_{2} - 374 \beta_{3} + 179 \beta_{4} + 120 \beta_{5} + 21 \beta_{6} + 13 \beta_{7} ) q^{78} \) \( + ( -302 + 1374 \beta_{1} - 1129 \beta_{2} + 379 \beta_{3} + 12 \beta_{4} - 604 \beta_{5} + 145 \beta_{6} + 151 \beta_{7} ) q^{79} \) \( + ( 128 + 64 \beta_{1} - 320 \beta_{2} + 128 \beta_{3} - 128 \beta_{4} + 64 \beta_{5} - 64 \beta_{7} ) q^{80} \) \( + ( -2046 - 657 \beta_{1} - 1107 \beta_{2} - 1329 \beta_{3} + 234 \beta_{4} + 147 \beta_{5} - 3 \beta_{6} - 204 \beta_{7} ) q^{81} \) \( + ( 160 + 65 \beta_{1} - 64 \beta_{2} + 79 \beta_{3} - 136 \beta_{4} + 32 \beta_{5} + 24 \beta_{6} + 40 \beta_{7} ) q^{82} \) \( + ( -582 + 24 \beta_{1} + 267 \beta_{2} + 1413 \beta_{3} - 48 \beta_{4} + 114 \beta_{5} + 45 \beta_{6} + 57 \beta_{7} ) q^{83} \) \( + ( -512 - 856 \beta_{1} - 1320 \beta_{2} - 768 \beta_{3} + 96 \beta_{4} - 8 \beta_{5} + 48 \beta_{6} + 24 \beta_{7} ) q^{84} \) \( + ( -4098 - 1104 \beta_{1} + 3918 \beta_{2} - 2298 \beta_{3} - 150 \beta_{4} - 150 \beta_{5} - 90 \beta_{6} - 210 \beta_{7} ) q^{85} \) \( + ( -936 + 121 \beta_{1} - 1272 \beta_{2} + 288 \beta_{3} - 276 \beta_{4} + 216 \beta_{5} + 120 \beta_{6} + 36 \beta_{7} ) q^{86} \) \( + ( -1080 + 1308 \beta_{1} + 4857 \beta_{2} - 639 \beta_{3} + 258 \beta_{4} + 60 \beta_{5} - 141 \beta_{6} - 9 \beta_{7} ) q^{87} \) \( + ( -64 + 1008 \beta_{1} - 640 \beta_{2} + 488 \beta_{3} - 96 \beta_{4} - 128 \beta_{5} + 80 \beta_{6} + 32 \beta_{7} ) q^{88} \) \( + ( 3256 + 2324 \beta_{1} - 6544 \beta_{2} + 2380 \beta_{3} - 100 \beta_{4} + 80 \beta_{5} - 12 \beta_{6} - 44 \beta_{7} ) q^{89} \) \( + ( -2052 - 1674 \beta_{1} + 4860 \beta_{2} - 1134 \beta_{3} - 81 \beta_{4} + 216 \beta_{5} + 81 \beta_{6} + 81 \beta_{7} ) q^{90} \) \( + ( -5780 - 1227 \beta_{1} - 93 \beta_{2} + 1506 \beta_{3} - 372 \beta_{4} - 93 \beta_{5} + 93 \beta_{7} ) q^{91} \) \( + ( 1808 + 96 \beta_{1} - 1000 \beta_{2} - 88 \beta_{3} - 192 \beta_{4} + 208 \beta_{5} + 56 \beta_{6} + 104 \beta_{7} ) q^{92} \) \( + ( 230 + 787 \beta_{1} - 2928 \beta_{2} + 2169 \beta_{3} + 500 \beta_{4} + 315 \beta_{5} - 9 \beta_{6} - 288 \beta_{7} ) q^{93} \) \( + ( -426 \beta_{1} + 252 \beta_{2} - 726 \beta_{3} + 483 \beta_{4} - 336 \beta_{5} + 126 \beta_{6} + 21 \beta_{7} ) q^{94} \) \( + ( 2262 - 788 \beta_{1} + 950 \beta_{2} + 924 \beta_{3} - 476 \beta_{4} + 1044 \beta_{5} + 268 \beta_{6} - 60 \beta_{7} ) q^{95} \) \( + ( 256 + 64 \beta_{1} - 512 \beta_{2} + 192 \beta_{3} - 64 \beta_{6} ) q^{96} \) \( + ( -20 + 1188 \beta_{1} + 7997 \beta_{2} + 436 \beta_{3} + 276 \beta_{4} - 40 \beta_{5} - 128 \beta_{6} + 10 \beta_{7} ) q^{97} \) \( + ( -1516 - 521 \beta_{1} + 3028 \beta_{2} - 403 \beta_{3} - 179 \beta_{4} + 232 \beta_{5} - 57 \beta_{6} - 61 \beta_{7} ) q^{98} \) \( + ( -2181 - 2268 \beta_{1} - 1008 \beta_{2} - 1950 \beta_{3} + 369 \beta_{4} - 33 \beta_{5} - 372 \beta_{6} + 93 \beta_{7} ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(8q \) \(\mathstrut +\mathstrut 6q^{3} \) \(\mathstrut +\mathstrut 32q^{4} \) \(\mathstrut +\mathstrut 18q^{5} \) \(\mathstrut +\mathstrut 48q^{6} \) \(\mathstrut -\mathstrut 26q^{7} \) \(\mathstrut -\mathstrut 78q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(8q \) \(\mathstrut +\mathstrut 6q^{3} \) \(\mathstrut +\mathstrut 32q^{4} \) \(\mathstrut +\mathstrut 18q^{5} \) \(\mathstrut +\mathstrut 48q^{6} \) \(\mathstrut -\mathstrut 26q^{7} \) \(\mathstrut -\mathstrut 78q^{9} \) \(\mathstrut -\mathstrut 720q^{11} \) \(\mathstrut -\mathstrut 144q^{12} \) \(\mathstrut +\mathstrut 10q^{13} \) \(\mathstrut +\mathstrut 288q^{14} \) \(\mathstrut +\mathstrut 1134q^{15} \) \(\mathstrut -\mathstrut 256q^{16} \) \(\mathstrut -\mathstrut 384q^{18} \) \(\mathstrut +\mathstrut 100q^{19} \) \(\mathstrut +\mathstrut 144q^{20} \) \(\mathstrut +\mathstrut 438q^{21} \) \(\mathstrut +\mathstrut 336q^{22} \) \(\mathstrut +\mathstrut 1278q^{23} \) \(\mathstrut +\mathstrut 384q^{24} \) \(\mathstrut +\mathstrut 794q^{25} \) \(\mathstrut -\mathstrut 1296q^{27} \) \(\mathstrut -\mathstrut 416q^{28} \) \(\mathstrut -\mathstrut 1854q^{29} \) \(\mathstrut -\mathstrut 3456q^{30} \) \(\mathstrut -\mathstrut 1478q^{31} \) \(\mathstrut -\mathstrut 3384q^{33} \) \(\mathstrut -\mathstrut 96q^{34} \) \(\mathstrut +\mathstrut 1056q^{36} \) \(\mathstrut -\mathstrut 32q^{37} \) \(\mathstrut +\mathstrut 6768q^{38} \) \(\mathstrut +\mathstrut 5274q^{39} \) \(\mathstrut -\mathstrut 36q^{41} \) \(\mathstrut +\mathstrut 2592q^{42} \) \(\mathstrut -\mathstrut 68q^{43} \) \(\mathstrut +\mathstrut 3402q^{45} \) \(\mathstrut +\mathstrut 2112q^{46} \) \(\mathstrut +\mathstrut 2214q^{47} \) \(\mathstrut -\mathstrut 1536q^{48} \) \(\mathstrut +\mathstrut 2442q^{49} \) \(\mathstrut -\mathstrut 15552q^{50} \) \(\mathstrut -\mathstrut 12006q^{51} \) \(\mathstrut -\mathstrut 80q^{52} \) \(\mathstrut +\mathstrut 7056q^{54} \) \(\mathstrut -\mathstrut 3996q^{55} \) \(\mathstrut +\mathstrut 2304q^{56} \) \(\mathstrut +\mathstrut 10902q^{57} \) \(\mathstrut -\mathstrut 2400q^{58} \) \(\mathstrut +\mathstrut 9108q^{59} \) \(\mathstrut +\mathstrut 6480q^{60} \) \(\mathstrut -\mathstrut 4478q^{61} \) \(\mathstrut -\mathstrut 6654q^{63} \) \(\mathstrut -\mathstrut 4096q^{64} \) \(\mathstrut -\mathstrut 22554q^{65} \) \(\mathstrut -\mathstrut 19872q^{66} \) \(\mathstrut +\mathstrut 7504q^{67} \) \(\mathstrut -\mathstrut 11088q^{68} \) \(\mathstrut -\mathstrut 5994q^{69} \) \(\mathstrut +\mathstrut 6048q^{70} \) \(\mathstrut +\mathstrut 5376q^{72} \) \(\mathstrut +\mathstrut 20716q^{73} \) \(\mathstrut +\mathstrut 15264q^{74} \) \(\mathstrut +\mathstrut 16590q^{75} \) \(\mathstrut +\mathstrut 400q^{76} \) \(\mathstrut +\mathstrut 34434q^{77} \) \(\mathstrut +\mathstrut 24096q^{78} \) \(\mathstrut -\mathstrut 6050q^{79} \) \(\mathstrut -\mathstrut 21150q^{81} \) \(\mathstrut +\mathstrut 1152q^{82} \) \(\mathstrut -\mathstrut 3834q^{83} \) \(\mathstrut -\mathstrut 9600q^{84} \) \(\mathstrut -\mathstrut 16092q^{85} \) \(\mathstrut -\mathstrut 12528q^{86} \) \(\mathstrut +\mathstrut 10170q^{87} \) \(\mathstrut -\mathstrut 2688q^{88} \) \(\mathstrut +\mathstrut 2592q^{90} \) \(\mathstrut -\mathstrut 45868q^{91} \) \(\mathstrut +\mathstrut 10224q^{92} \) \(\mathstrut -\mathstrut 10926q^{93} \) \(\mathstrut +\mathstrut 672q^{94} \) \(\mathstrut +\mathstrut 20880q^{95} \) \(\mathstrut +\mathstrut 31336q^{97} \) \(\mathstrut -\mathstrut 22338q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8}\mathstrut -\mathstrut \) \(4\) \(x^{7}\mathstrut +\mathstrut \) \(38\) \(x^{6}\mathstrut -\mathstrut \) \(100\) \(x^{5}\mathstrut +\mathstrut \) \(449\) \(x^{4}\mathstrut -\mathstrut \) \(736\) \(x^{3}\mathstrut +\mathstrut \) \(1900\) \(x^{2}\mathstrut -\mathstrut \) \(1548\) \(x\mathstrut +\mathstrut \) \(2307\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( -19 \nu^{7} - 286 \nu^{6} + 545 \nu^{5} - 8755 \nu^{4} + 14939 \nu^{3} - 71959 \nu^{2} + 65748 \nu - 147099 \)\()/4230\)
\(\beta_{2}\)\(=\)\((\)\( -8 \nu^{7} + 28 \nu^{6} - 290 \nu^{5} + 655 \nu^{4} - 2912 \nu^{3} + 3727 \nu^{2} - 7344 \nu + 3777 \)\()/1410\)
\(\beta_{3}\)\(=\)\((\)\( -19 \nu^{7} + 419 \nu^{6} - 1570 \nu^{5} + 10985 \nu^{4} - 21016 \nu^{3} + 78911 \nu^{2} - 67497 \nu + 146886 \)\()/4230\)
\(\beta_{4}\)\(=\)\((\)\( 179 \nu^{7} - 979 \nu^{6} + 6665 \nu^{5} - 23380 \nu^{4} + 70091 \nu^{3} - 149926 \nu^{2} + 205212 \nu - 201981 \)\()/4230\)
\(\beta_{5}\)\(=\)\((\)\( -217 \nu^{7} + 407 \nu^{6} - 5575 \nu^{5} + 5870 \nu^{4} - 40213 \nu^{3} + 18698 \nu^{2} - 73716 \nu + 843 \)\()/4230\)
\(\beta_{6}\)\(=\)\((\)\( 467 \nu^{7} - 1987 \nu^{6} + 14990 \nu^{5} - 36385 \nu^{4} + 122048 \nu^{3} - 129703 \nu^{2} + 188301 \nu + 78702 \)\()/4230\)
\(\beta_{7}\)\(=\)\((\)\( 532 \nu^{7} - 1157 \nu^{6} + 14350 \nu^{5} - 13595 \nu^{4} + 101998 \nu^{3} + 17587 \nu^{2} + 197916 \nu + 235632 \)\()/4230\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{7}\mathstrut -\mathstrut \) \(\beta_{6}\mathstrut +\mathstrut \) \(2\) \(\beta_{5}\mathstrut +\mathstrut \) \(\beta_{4}\mathstrut -\mathstrut \) \(5\) \(\beta_{3}\mathstrut -\mathstrut \) \(5\) \(\beta_{1}\mathstrut +\mathstrut \) \(10\)\()/18\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{5}\mathstrut +\mathstrut \) \(\beta_{4}\mathstrut -\mathstrut \) \(2\) \(\beta_{1}\mathstrut -\mathstrut \) \(22\)\()/3\)
\(\nu^{3}\)\(=\)\((\)\(-\)\(8\) \(\beta_{7}\mathstrut +\mathstrut \) \(11\) \(\beta_{6}\mathstrut -\mathstrut \) \(16\) \(\beta_{5}\mathstrut +\mathstrut \) \(4\) \(\beta_{4}\mathstrut +\mathstrut \) \(103\) \(\beta_{3}\mathstrut +\mathstrut \) \(60\) \(\beta_{2}\mathstrut +\mathstrut \) \(88\) \(\beta_{1}\mathstrut -\mathstrut \) \(242\)\()/18\)
\(\nu^{4}\)\(=\)\((\)\(\beta_{7}\mathstrut +\mathstrut \) \(2\) \(\beta_{6}\mathstrut -\mathstrut \) \(15\) \(\beta_{5}\mathstrut -\mathstrut \) \(16\) \(\beta_{4}\mathstrut +\mathstrut \) \(22\) \(\beta_{3}\mathstrut +\mathstrut \) \(18\) \(\beta_{2}\mathstrut +\mathstrut \) \(53\) \(\beta_{1}\mathstrut +\mathstrut \) \(177\)\()/3\)
\(\nu^{5}\)\(=\)\((\)\(97\) \(\beta_{7}\mathstrut -\mathstrut \) \(118\) \(\beta_{6}\mathstrut +\mathstrut \) \(122\) \(\beta_{5}\mathstrut -\mathstrut \) \(239\) \(\beta_{4}\mathstrut -\mathstrut \) \(1250\) \(\beta_{3}\mathstrut -\mathstrut \) \(1392\) \(\beta_{2}\mathstrut -\mathstrut \) \(821\) \(\beta_{1}\mathstrut +\mathstrut \) \(3940\)\()/18\)
\(\nu^{6}\)\(=\)\((\)\(-\)\(16\) \(\beta_{7}\mathstrut -\mathstrut \) \(53\) \(\beta_{6}\mathstrut +\mathstrut \) \(194\) \(\beta_{5}\mathstrut +\mathstrut \) \(180\) \(\beta_{4}\mathstrut -\mathstrut \) \(505\) \(\beta_{3}\mathstrut -\mathstrut \) \(690\) \(\beta_{2}\mathstrut -\mathstrut \) \(894\) \(\beta_{1}\mathstrut -\mathstrut \) \(1271\)\()/3\)
\(\nu^{7}\)\(=\)\((\)\(-\)\(1367\) \(\beta_{7}\mathstrut +\mathstrut \) \(1061\) \(\beta_{6}\mathstrut -\mathstrut \) \(934\) \(\beta_{5}\mathstrut +\mathstrut \) \(5005\) \(\beta_{4}\mathstrut +\mathstrut \) \(12613\) \(\beta_{3}\mathstrut +\mathstrut \) \(19800\) \(\beta_{2}\mathstrut +\mathstrut \) \(3991\) \(\beta_{1}\mathstrut -\mathstrut \) \(56654\)\()/18\)

Character Values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/18\mathbb{Z}\right)^\times\).

\(n\) \(11\)
\(\chi(n)\) \(1 - \beta_{2}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
5.1
0.500000 1.74753i
0.500000 + 3.16175i
0.500000 + 2.20403i
0.500000 3.61825i
0.500000 + 1.74753i
0.500000 3.16175i
0.500000 2.20403i
0.500000 + 3.61825i
−2.44949 + 1.41421i −7.80760 4.47676i 4.00000 6.92820i −32.5033 18.7658i 25.4557 0.0758456i −1.35458 2.34620i 22.6274i 40.9173 + 69.9055i 106.155
5.2 −2.44949 + 1.41421i 6.85811 5.82806i 4.00000 6.92820i 37.0033 + 21.3638i −8.55675 + 23.9746i −19.8424 34.3680i 22.6274i 13.0674 79.9390i −120.852
5.3 2.44949 1.41421i −1.67960 8.84189i 4.00000 6.92820i 6.41371 + 3.70296i −16.6185 19.2828i 30.1882 + 52.2875i 22.6274i −75.3579 + 29.7016i 20.9471
5.4 2.44949 1.41421i 5.62909 + 7.02235i 4.00000 6.92820i −1.91371 1.10488i 23.7195 + 9.24044i −21.9913 38.0900i 22.6274i −17.6268 + 79.0588i −6.25016
11.1 −2.44949 1.41421i −7.80760 + 4.47676i 4.00000 + 6.92820i −32.5033 + 18.7658i 25.4557 + 0.0758456i −1.35458 + 2.34620i 22.6274i 40.9173 69.9055i 106.155
11.2 −2.44949 1.41421i 6.85811 + 5.82806i 4.00000 + 6.92820i 37.0033 21.3638i −8.55675 23.9746i −19.8424 + 34.3680i 22.6274i 13.0674 + 79.9390i −120.852
11.3 2.44949 + 1.41421i −1.67960 + 8.84189i 4.00000 + 6.92820i 6.41371 3.70296i −16.6185 + 19.2828i 30.1882 52.2875i 22.6274i −75.3579 29.7016i 20.9471
11.4 2.44949 + 1.41421i 5.62909 7.02235i 4.00000 + 6.92820i −1.91371 + 1.10488i 23.7195 9.24044i −21.9913 + 38.0900i 22.6274i −17.6268 79.0588i −6.25016
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 11.4
Significant digits:
Format:

Inner twists

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

Hecke kernels

There are no other newforms in \(S_{5}^{\mathrm{new}}(18, [\chi])\).