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 \(6\), degree \(2\), minimal)

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: \( 2^{4}\cdot 3^{4} \)
Twist minimal: yes
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})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + 6q^{3} + 32q^{4} + 18q^{5} + 48q^{6} - 26q^{7} - 78q^{9} + O(q^{10}) \) \( 8q + 6q^{3} + 32q^{4} + 18q^{5} + 48q^{6} - 26q^{7} - 78q^{9} - 720q^{11} - 144q^{12} + 10q^{13} + 288q^{14} + 1134q^{15} - 256q^{16} - 384q^{18} + 100q^{19} + 144q^{20} + 438q^{21} + 336q^{22} + 1278q^{23} + 384q^{24} + 794q^{25} - 1296q^{27} - 416q^{28} - 1854q^{29} - 3456q^{30} - 1478q^{31} - 3384q^{33} - 96q^{34} + 1056q^{36} - 32q^{37} + 6768q^{38} + 5274q^{39} - 36q^{41} + 2592q^{42} - 68q^{43} + 3402q^{45} + 2112q^{46} + 2214q^{47} - 1536q^{48} + 2442q^{49} - 15552q^{50} - 12006q^{51} - 80q^{52} + 7056q^{54} - 3996q^{55} + 2304q^{56} + 10902q^{57} - 2400q^{58} + 9108q^{59} + 6480q^{60} - 4478q^{61} - 6654q^{63} - 4096q^{64} - 22554q^{65} - 19872q^{66} + 7504q^{67} - 11088q^{68} - 5994q^{69} + 6048q^{70} + 5376q^{72} + 20716q^{73} + 15264q^{74} + 16590q^{75} + 400q^{76} + 34434q^{77} + 24096q^{78} - 6050q^{79} - 21150q^{81} + 1152q^{82} - 3834q^{83} - 9600q^{84} - 16092q^{85} - 12528q^{86} + 10170q^{87} - 2688q^{88} + 2592q^{90} - 45868q^{91} + 10224q^{92} - 10926q^{93} + 672q^{94} + 20880q^{95} + 31336q^{97} - 22338q^{99} + O(q^{100}) \)

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

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

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 Parity Ord Mult Type
1.a even 1 1 trivial
9.d odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 18.5.d.a 8
3.b odd 2 1 54.5.d.a 8
4.b odd 2 1 144.5.q.b 8
9.c even 3 1 54.5.d.a 8
9.c even 3 1 162.5.b.c 8
9.d odd 6 1 inner 18.5.d.a 8
9.d odd 6 1 162.5.b.c 8
12.b even 2 1 432.5.q.b 8
36.f odd 6 1 432.5.q.b 8
36.f odd 6 1 1296.5.e.e 8
36.h even 6 1 144.5.q.b 8
36.h even 6 1 1296.5.e.e 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
18.5.d.a 8 1.a even 1 1 trivial
18.5.d.a 8 9.d odd 6 1 inner
54.5.d.a 8 3.b odd 2 1
54.5.d.a 8 9.c even 3 1
144.5.q.b 8 4.b odd 2 1
144.5.q.b 8 36.h even 6 1
162.5.b.c 8 9.c even 3 1
162.5.b.c 8 9.d odd 6 1
432.5.q.b 8 12.b even 2 1
432.5.q.b 8 36.f odd 6 1
1296.5.e.e 8 36.f odd 6 1
1296.5.e.e 8 36.h even 6 1

Hecke kernels

This newform subspace is the entire newspace \(S_{5}^{\mathrm{new}}(18, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - 8 T^{2} + 64 T^{4} )^{2} \)
$3$ \( 1 - 6 T + 57 T^{2} + 162 T^{3} + 4212 T^{4} + 13122 T^{5} + 373977 T^{6} - 3188646 T^{7} + 43046721 T^{8} \)
$5$ \( 1 - 18 T + 1015 T^{2} - 16326 T^{3} + 558925 T^{4} - 5405940 T^{5} - 314464142 T^{6} + 3511352520 T^{7} - 306662326214 T^{8} + 2194595325000 T^{9} - 122837555468750 T^{10} - 1319809570312500 T^{11} + 85285186767578125 T^{12} - 1556968688964843750 T^{13} + 60498714447021484375 T^{14} - \)\(67\!\cdots\!50\)\( T^{15} + \)\(23\!\cdots\!25\)\( T^{16} \)
$7$ \( 1 + 26 T - 5685 T^{2} + 19510 T^{3} + 19835513 T^{4} - 334899084 T^{5} - 45037679042 T^{6} + 451520935736 T^{7} + 90278690161986 T^{8} + 1084101766702136 T^{9} - 259633257179000642 T^{10} - 4635434404995823884 T^{11} + \)\(65\!\cdots\!13\)\( T^{12} + \)\(15\!\cdots\!10\)\( T^{13} - \)\(10\!\cdots\!85\)\( T^{14} + \)\(11\!\cdots\!26\)\( T^{15} + \)\(11\!\cdots\!01\)\( T^{16} \)
$11$ \( 1 + 720 T + 274582 T^{2} + 73283040 T^{3} + 15179596969 T^{4} + 2592718024176 T^{5} + 383108185780198 T^{6} + 51080491982360160 T^{7} + 6354846385614434932 T^{8} + \)\(74\!\cdots\!60\)\( T^{9} + \)\(82\!\cdots\!38\)\( T^{10} + \)\(81\!\cdots\!96\)\( T^{11} + \)\(69\!\cdots\!09\)\( T^{12} + \)\(49\!\cdots\!40\)\( T^{13} + \)\(27\!\cdots\!62\)\( T^{14} + \)\(10\!\cdots\!20\)\( T^{15} + \)\(21\!\cdots\!21\)\( T^{16} \)
$13$ \( 1 - 10 T - 34545 T^{2} - 8013734 T^{3} + 152683301 T^{4} + 264027759132 T^{5} + 34691703939994 T^{6} - 3939272844487792 T^{7} - 880372532161082934 T^{8} - \)\(11\!\cdots\!12\)\( T^{9} + \)\(28\!\cdots\!74\)\( T^{10} + \)\(61\!\cdots\!92\)\( T^{11} + \)\(10\!\cdots\!41\)\( T^{12} - \)\(15\!\cdots\!34\)\( T^{13} - \)\(18\!\cdots\!45\)\( T^{14} - \)\(15\!\cdots\!10\)\( T^{15} + \)\(44\!\cdots\!81\)\( T^{16} \)
$17$ \( 1 - 249218 T^{2} + 30482460289 T^{4} - 3595961342423810 T^{6} + \)\(36\!\cdots\!56\)\( T^{8} - \)\(25\!\cdots\!10\)\( T^{10} + \)\(14\!\cdots\!09\)\( T^{12} - \)\(84\!\cdots\!78\)\( T^{14} + \)\(23\!\cdots\!61\)\( T^{16} \)
$19$ \( ( 1 - 50 T + 226153 T^{2} - 12428354 T^{3} + 33132078964 T^{4} - 1619675521634 T^{5} + 3840883732411273 T^{6} - 110665745953308050 T^{7} + \)\(28\!\cdots\!81\)\( T^{8} )^{2} \)
$23$ \( 1 - 1278 T + 1655467 T^{2} - 1419907842 T^{3} + 1186049018761 T^{4} - 817434220578732 T^{5} + 550327618738318606 T^{6} - \)\(32\!\cdots\!56\)\( T^{7} + \)\(18\!\cdots\!30\)\( T^{8} - \)\(89\!\cdots\!96\)\( T^{9} + \)\(43\!\cdots\!86\)\( T^{10} - \)\(17\!\cdots\!72\)\( T^{11} + \)\(72\!\cdots\!21\)\( T^{12} - \)\(24\!\cdots\!42\)\( T^{13} + \)\(79\!\cdots\!47\)\( T^{14} - \)\(17\!\cdots\!18\)\( T^{15} + \)\(37\!\cdots\!21\)\( T^{16} \)
$29$ \( 1 + 1854 T + 3116071 T^{2} + 3652934346 T^{3} + 3878113467181 T^{4} + 4083633101435148 T^{5} + 4044828167734324498 T^{6} + \)\(39\!\cdots\!28\)\( T^{7} + \)\(34\!\cdots\!30\)\( T^{8} + \)\(28\!\cdots\!68\)\( T^{9} + \)\(20\!\cdots\!78\)\( T^{10} + \)\(14\!\cdots\!68\)\( T^{11} + \)\(97\!\cdots\!01\)\( T^{12} + \)\(64\!\cdots\!46\)\( T^{13} + \)\(39\!\cdots\!51\)\( T^{14} + \)\(16\!\cdots\!94\)\( T^{15} + \)\(62\!\cdots\!41\)\( T^{16} \)
$31$ \( 1 + 1478 T - 273609 T^{2} - 14717462 T^{3} + 1239820588133 T^{4} - 59691395608740 T^{5} - 148444619433025670 T^{6} - \)\(17\!\cdots\!40\)\( T^{7} - \)\(10\!\cdots\!22\)\( T^{8} - \)\(16\!\cdots\!40\)\( T^{9} - \)\(12\!\cdots\!70\)\( T^{10} - \)\(47\!\cdots\!40\)\( T^{11} + \)\(90\!\cdots\!73\)\( T^{12} - \)\(98\!\cdots\!62\)\( T^{13} - \)\(16\!\cdots\!89\)\( T^{14} + \)\(84\!\cdots\!98\)\( T^{15} + \)\(52\!\cdots\!61\)\( T^{16} \)
$37$ \( ( 1 + 16 T + 5152060 T^{2} + 1816976752 T^{3} + 11981316770374 T^{4} + 3405306966505072 T^{5} + 18096504895368227260 T^{6} + \)\(10\!\cdots\!96\)\( T^{7} + \)\(12\!\cdots\!41\)\( T^{8} )^{2} \)
$41$ \( 1 + 36 T + 10981366 T^{2} + 395313624 T^{3} + 74491912981249 T^{4} + 2021455678408920 T^{5} + \)\(33\!\cdots\!50\)\( T^{6} + \)\(74\!\cdots\!60\)\( T^{7} + \)\(10\!\cdots\!68\)\( T^{8} + \)\(21\!\cdots\!60\)\( T^{9} + \)\(26\!\cdots\!50\)\( T^{10} + \)\(45\!\cdots\!20\)\( T^{11} + \)\(47\!\cdots\!09\)\( T^{12} + \)\(71\!\cdots\!24\)\( T^{13} + \)\(55\!\cdots\!26\)\( T^{14} + \)\(51\!\cdots\!56\)\( T^{15} + \)\(40\!\cdots\!81\)\( T^{16} \)
$43$ \( 1 + 68 T - 10748538 T^{2} + 2055805384 T^{3} + 65375958867329 T^{4} - 16226764390392264 T^{5} - \)\(28\!\cdots\!82\)\( T^{6} + \)\(26\!\cdots\!32\)\( T^{7} + \)\(10\!\cdots\!76\)\( T^{8} + \)\(91\!\cdots\!32\)\( T^{9} - \)\(33\!\cdots\!82\)\( T^{10} - \)\(64\!\cdots\!64\)\( T^{11} + \)\(89\!\cdots\!29\)\( T^{12} + \)\(96\!\cdots\!84\)\( T^{13} - \)\(17\!\cdots\!38\)\( T^{14} + \)\(37\!\cdots\!68\)\( T^{15} + \)\(18\!\cdots\!01\)\( T^{16} \)
$47$ \( 1 - 2214 T + 16322779 T^{2} - 32521107258 T^{3} + 136095840713065 T^{4} - 267869759907619980 T^{5} + \)\(91\!\cdots\!74\)\( T^{6} - \)\(16\!\cdots\!08\)\( T^{7} + \)\(50\!\cdots\!62\)\( T^{8} - \)\(82\!\cdots\!48\)\( T^{9} + \)\(21\!\cdots\!14\)\( T^{10} - \)\(31\!\cdots\!80\)\( T^{11} + \)\(77\!\cdots\!65\)\( T^{12} - \)\(89\!\cdots\!58\)\( T^{13} + \)\(22\!\cdots\!99\)\( T^{14} - \)\(14\!\cdots\!54\)\( T^{15} + \)\(32\!\cdots\!41\)\( T^{16} \)
$53$ \( 1 - 47149064 T^{2} + 1030375014657436 T^{4} - \)\(13\!\cdots\!20\)\( T^{6} + \)\(12\!\cdots\!10\)\( T^{8} - \)\(86\!\cdots\!20\)\( T^{10} + \)\(39\!\cdots\!56\)\( T^{12} - \)\(11\!\cdots\!84\)\( T^{14} + \)\(15\!\cdots\!41\)\( T^{16} \)
$59$ \( 1 - 9108 T + 65431078 T^{2} - 344092862520 T^{3} + 1463362600081057 T^{4} - 4890955357569963960 T^{5} + \)\(14\!\cdots\!50\)\( T^{6} - \)\(36\!\cdots\!52\)\( T^{7} + \)\(11\!\cdots\!92\)\( T^{8} - \)\(44\!\cdots\!72\)\( T^{9} + \)\(21\!\cdots\!50\)\( T^{10} - \)\(87\!\cdots\!60\)\( T^{11} + \)\(31\!\cdots\!37\)\( T^{12} - \)\(89\!\cdots\!20\)\( T^{13} + \)\(20\!\cdots\!58\)\( T^{14} - \)\(34\!\cdots\!68\)\( T^{15} + \)\(46\!\cdots\!81\)\( T^{16} \)
$61$ \( 1 + 4478 T - 125937 T^{2} - 125289669758 T^{3} - 568176190027387 T^{4} - 742254823045377828 T^{5} + \)\(27\!\cdots\!38\)\( T^{6} + \)\(22\!\cdots\!92\)\( T^{7} + \)\(73\!\cdots\!38\)\( T^{8} + \)\(30\!\cdots\!72\)\( T^{9} + \)\(52\!\cdots\!78\)\( T^{10} - \)\(19\!\cdots\!88\)\( T^{11} - \)\(20\!\cdots\!07\)\( T^{12} - \)\(63\!\cdots\!58\)\( T^{13} - \)\(88\!\cdots\!17\)\( T^{14} + \)\(43\!\cdots\!18\)\( T^{15} + \)\(13\!\cdots\!21\)\( T^{16} \)
$67$ \( 1 - 7504 T - 22347594 T^{2} + 226068030400 T^{3} + 738296226968777 T^{4} - 4521766862440091376 T^{5} - \)\(18\!\cdots\!90\)\( T^{6} + \)\(18\!\cdots\!56\)\( T^{7} + \)\(57\!\cdots\!20\)\( T^{8} + \)\(38\!\cdots\!76\)\( T^{9} - \)\(76\!\cdots\!90\)\( T^{10} - \)\(37\!\cdots\!36\)\( T^{11} + \)\(12\!\cdots\!37\)\( T^{12} + \)\(75\!\cdots\!00\)\( T^{13} - \)\(14\!\cdots\!74\)\( T^{14} - \)\(10\!\cdots\!64\)\( T^{15} + \)\(27\!\cdots\!61\)\( T^{16} \)
$71$ \( 1 - 125290160 T^{2} + 8106610376011420 T^{4} - \)\(34\!\cdots\!88\)\( T^{6} + \)\(10\!\cdots\!18\)\( T^{8} - \)\(21\!\cdots\!68\)\( T^{10} + \)\(33\!\cdots\!20\)\( T^{12} - \)\(33\!\cdots\!60\)\( T^{14} + \)\(17\!\cdots\!41\)\( T^{16} \)
$73$ \( ( 1 - 10358 T + 106803217 T^{2} - 769838088062 T^{3} + 4439859073965124 T^{4} - 21862047555763898942 T^{5} + \)\(86\!\cdots\!77\)\( T^{6} - \)\(23\!\cdots\!18\)\( T^{7} + \)\(65\!\cdots\!61\)\( T^{8} )^{2} \)
$79$ \( 1 + 6050 T - 42146805 T^{2} + 267987207070 T^{3} + 3321952223354537 T^{4} - 14945323617952326060 T^{5} + \)\(48\!\cdots\!70\)\( T^{6} + \)\(67\!\cdots\!20\)\( T^{7} - \)\(27\!\cdots\!02\)\( T^{8} + \)\(26\!\cdots\!20\)\( T^{9} + \)\(73\!\cdots\!70\)\( T^{10} - \)\(88\!\cdots\!60\)\( T^{11} + \)\(76\!\cdots\!77\)\( T^{12} + \)\(24\!\cdots\!70\)\( T^{13} - \)\(14\!\cdots\!05\)\( T^{14} + \)\(82\!\cdots\!50\)\( T^{15} + \)\(52\!\cdots\!41\)\( T^{16} \)
$83$ \( 1 + 3834 T + 163867483 T^{2} + 609481897254 T^{3} + 15130998622630921 T^{4} + 49888358310056197140 T^{5} + \)\(10\!\cdots\!30\)\( T^{6} + \)\(30\!\cdots\!80\)\( T^{7} + \)\(54\!\cdots\!10\)\( T^{8} + \)\(14\!\cdots\!80\)\( T^{9} + \)\(23\!\cdots\!30\)\( T^{10} + \)\(53\!\cdots\!40\)\( T^{11} + \)\(76\!\cdots\!01\)\( T^{12} + \)\(14\!\cdots\!54\)\( T^{13} + \)\(18\!\cdots\!43\)\( T^{14} + \)\(20\!\cdots\!94\)\( T^{15} + \)\(25\!\cdots\!61\)\( T^{16} \)
$89$ \( 1 - 206196488 T^{2} + 19314405141094684 T^{4} - \)\(14\!\cdots\!60\)\( T^{6} + \)\(10\!\cdots\!86\)\( T^{8} - \)\(57\!\cdots\!60\)\( T^{10} + \)\(29\!\cdots\!24\)\( T^{12} - \)\(12\!\cdots\!08\)\( T^{14} + \)\(24\!\cdots\!21\)\( T^{16} \)
$97$ \( 1 - 31336 T + 307771782 T^{2} - 2149219106864 T^{3} + 52690258168920329 T^{4} - \)\(68\!\cdots\!16\)\( T^{5} + \)\(38\!\cdots\!30\)\( T^{6} - \)\(46\!\cdots\!00\)\( T^{7} + \)\(68\!\cdots\!04\)\( T^{8} - \)\(40\!\cdots\!00\)\( T^{9} + \)\(30\!\cdots\!30\)\( T^{10} - \)\(47\!\cdots\!56\)\( T^{11} + \)\(32\!\cdots\!09\)\( T^{12} - \)\(11\!\cdots\!64\)\( T^{13} + \)\(14\!\cdots\!42\)\( T^{14} - \)\(13\!\cdots\!96\)\( T^{15} + \)\(37\!\cdots\!41\)\( T^{16} \)
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