Properties

Label 19.5.d.a
Level $19$
Weight $5$
Character orbit 19.d
Analytic conductor $1.964$
Analytic rank $0$
Dimension $10$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 19 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 19.d (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.96402929859\)
Analytic rank: \(0\)
Dimension: \(10\)
Relative dimension: \(5\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} + \cdots)\)
Defining polynomial: \(x^{10} + 109 x^{8} + 4107 x^{6} + 61507 x^{4} + 300520 x^{2} + 108300\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta_{1} q^{2} + ( \beta_{3} + \beta_{5} ) q^{3} + ( 6 \beta_{3} - \beta_{4} - \beta_{9} ) q^{4} + ( 2 + \beta_{1} + \beta_{2} - 3 \beta_{3} - \beta_{5} - \beta_{6} - \beta_{8} ) q^{5} + ( 1 - \beta_{1} - 8 \beta_{3} + \beta_{4} - \beta_{5} - \beta_{6} - \beta_{7} + \beta_{9} ) q^{6} + ( -4 + \beta_{1} - \beta_{2} + \beta_{4} + \beta_{5} - \beta_{6} ) q^{7} + ( -6 - \beta_{1} - \beta_{2} + 12 \beta_{3} - \beta_{4} - \beta_{5} - \beta_{6} + \beta_{7} + \beta_{8} - 2 \beta_{9} ) q^{8} + ( -2 + \beta_{1} - 10 \beta_{3} + 2 \beta_{4} + 3 \beta_{5} + 2 \beta_{6} + \beta_{7} + 2 \beta_{9} ) q^{9} +O(q^{10})\) \( q -\beta_{1} q^{2} + ( \beta_{3} + \beta_{5} ) q^{3} + ( 6 \beta_{3} - \beta_{4} - \beta_{9} ) q^{4} + ( 2 + \beta_{1} + \beta_{2} - 3 \beta_{3} - \beta_{5} - \beta_{6} - \beta_{8} ) q^{5} + ( 1 - \beta_{1} - 8 \beta_{3} + \beta_{4} - \beta_{5} - \beta_{6} - \beta_{7} + \beta_{9} ) q^{6} + ( -4 + \beta_{1} - \beta_{2} + \beta_{4} + \beta_{5} - \beta_{6} ) q^{7} + ( -6 - \beta_{1} - \beta_{2} + 12 \beta_{3} - \beta_{4} - \beta_{5} - \beta_{6} + \beta_{7} + \beta_{8} - 2 \beta_{9} ) q^{8} + ( -2 + \beta_{1} - 10 \beta_{3} + 2 \beta_{4} + 3 \beta_{5} + 2 \beta_{6} + \beta_{7} + 2 \beta_{9} ) q^{9} + ( 21 - 2 \beta_{1} - 4 \beta_{2} - 4 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} + 9 \beta_{6} - 2 \beta_{7} + 4 \beta_{8} + 2 \beta_{9} ) q^{10} + ( 8 - \beta_{1} + \beta_{2} - 4 \beta_{4} - 5 \beta_{5} + 5 \beta_{6} + 4 \beta_{7} - 4 \beta_{8} ) q^{11} + ( 4 + 10 \beta_{1} + 10 \beta_{2} - 8 \beta_{3} + 2 \beta_{4} - 4 \beta_{5} - 4 \beta_{6} + \beta_{7} + \beta_{8} + 4 \beta_{9} ) q^{12} + ( -80 - 2 \beta_{1} + 8 \beta_{2} + 45 \beta_{3} + 5 \beta_{4} + 2 \beta_{5} + 6 \beta_{6} - 2 \beta_{7} + 4 \beta_{8} - 5 \beta_{9} ) q^{13} + ( -33 + 13 \beta_{1} + 2 \beta_{2} - 36 \beta_{3} + 6 \beta_{4} + \beta_{5} - 2 \beta_{6} - 4 \beta_{7} + 2 \beta_{8} + 3 \beta_{9} ) q^{14} + ( 53 + 3 \beta_{1} - 30 \beta_{2} - 35 \beta_{3} - 3 \beta_{4} - 3 \beta_{5} - 11 \beta_{6} + 3 \beta_{7} - 6 \beta_{8} + 3 \beta_{9} ) q^{15} + ( 54 - 10 \beta_{1} - 20 \beta_{2} - 64 \beta_{3} - 10 \beta_{5} - 20 \beta_{6} + 5 \beta_{9} ) q^{16} + ( -61 + 4 \beta_{1} + 5 \beta_{2} + 63 \beta_{3} + 2 \beta_{5} + 7 \beta_{6} - 3 \beta_{8} - 17 \beta_{9} ) q^{17} + ( 38 + 43 \beta_{1} + 43 \beta_{2} - 76 \beta_{3} + 2 \beta_{4} - 11 \beta_{5} - 11 \beta_{6} - 5 \beta_{7} - 5 \beta_{8} + 4 \beta_{9} ) q^{18} + ( -30 - 44 \beta_{1} - 22 \beta_{2} + 140 \beta_{3} - 10 \beta_{4} + 27 \beta_{5} + 19 \beta_{6} - 7 \beta_{7} + 8 \beta_{8} - 4 \beta_{9} ) q^{19} + ( -58 - 32 \beta_{1} + 32 \beta_{2} + \beta_{4} + 8 \beta_{5} - 8 \beta_{6} + 9 \beta_{7} - 9 \beta_{8} ) q^{20} + ( 71 - 7 \beta_{1} - \beta_{2} + 84 \beta_{3} - 2 \beta_{4} + 11 \beta_{5} + \beta_{6} + 2 \beta_{7} - \beta_{8} - \beta_{9} ) q^{21} + ( 139 - 23 \beta_{1} - \beta_{2} + 102 \beta_{3} - 36 \beta_{4} - 39 \beta_{5} + \beta_{6} + 2 \beta_{7} - \beta_{8} - 18 \beta_{9} ) q^{22} + ( -37 - 21 \beta_{1} - 10 \beta_{2} + 69 \beta_{3} + 7 \beta_{4} + 75 \beta_{5} + 37 \beta_{6} - \beta_{7} + 7 \beta_{9} ) q^{23} + ( 121 + 33 \beta_{1} + 76 \beta_{2} - 112 \beta_{3} + 9 \beta_{5} + 8 \beta_{6} + 10 \beta_{8} - 10 \beta_{9} ) q^{24} + ( 18 - 159 \beta_{1} - 78 \beta_{2} - 114 \beta_{3} + 21 \beta_{4} - 33 \beta_{5} - 18 \beta_{6} - 3 \beta_{7} + 21 \beta_{9} ) q^{25} + ( 78 + 80 \beta_{1} - 80 \beta_{2} + 13 \beta_{4} + 29 \beta_{5} - 29 \beta_{6} + \beta_{7} - \beta_{8} ) q^{26} + ( -37 + 9 \beta_{1} + 9 \beta_{2} + 74 \beta_{3} - 6 \beta_{4} - 56 \beta_{5} - 56 \beta_{6} + 3 \beta_{7} + 3 \beta_{8} - 12 \beta_{9} ) q^{27} + ( -7 + 74 \beta_{1} + 37 \beta_{2} - 312 \beta_{3} + 19 \beta_{4} + 14 \beta_{5} + 7 \beta_{6} + 19 \beta_{9} ) q^{28} + ( 315 + 3 \beta_{1} - 113 \beta_{2} - 129 \beta_{3} - 6 \beta_{4} - 3 \beta_{5} + 63 \beta_{6} + 3 \beta_{7} - 6 \beta_{8} + 6 \beta_{9} ) q^{29} + ( -606 - 47 \beta_{1} + 47 \beta_{2} + 14 \beta_{4} - 30 \beta_{5} + 30 \beta_{6} - 17 \beta_{7} + 17 \beta_{8} ) q^{30} + ( -198 + 207 \beta_{1} + 207 \beta_{2} + 396 \beta_{3} + 11 \beta_{4} - 67 \beta_{5} - 67 \beta_{6} - 4 \beta_{7} - 4 \beta_{8} + 22 \beta_{9} ) q^{31} + ( -318 + 11 \beta_{1} + 104 \beta_{2} + 174 \beta_{3} - \beta_{4} - 11 \beta_{5} + 52 \beta_{6} + 11 \beta_{7} - 22 \beta_{8} + \beta_{9} ) q^{32} + ( -262 + 152 \beta_{1} - 10 \beta_{2} - 254 \beta_{3} - 26 \beta_{4} - 12 \beta_{5} + 10 \beta_{6} + 20 \beta_{7} - 10 \beta_{8} - 13 \beta_{9} ) q^{33} + ( -32 - 18 \beta_{1} - 249 \beta_{2} + 28 \beta_{3} + 5 \beta_{4} + 18 \beta_{5} - 12 \beta_{6} - 18 \beta_{7} + 36 \beta_{8} - 5 \beta_{9} ) q^{34} + ( 106 - 56 \beta_{1} - 119 \beta_{2} - 120 \beta_{3} - 14 \beta_{5} - 21 \beta_{6} - 7 \beta_{8} + 4 \beta_{9} ) q^{35} + ( 688 + 6 \beta_{1} + 21 \beta_{2} - 654 \beta_{3} + 34 \beta_{5} + 59 \beta_{6} + 9 \beta_{8} + 36 \beta_{9} ) q^{36} + ( 88 + 156 \beta_{1} + 156 \beta_{2} - 176 \beta_{3} - 11 \beta_{4} + 30 \beta_{5} + 30 \beta_{6} + 21 \beta_{7} + 21 \beta_{8} - 22 \beta_{9} ) q^{37} + ( -456 - 228 \beta_{1} - 171 \beta_{2} + 798 \beta_{3} + 19 \beta_{5} - 19 \beta_{6} + 38 \beta_{7} - 38 \beta_{8} - 19 \beta_{9} ) q^{38} + ( -445 - 140 \beta_{1} + 140 \beta_{2} - 10 \beta_{4} + 5 \beta_{5} - 5 \beta_{6} - 47 \beta_{7} + 47 \beta_{8} ) q^{39} + ( 539 + 41 \beta_{1} - 5 \beta_{2} + 606 \beta_{3} - 18 \beta_{4} + 57 \beta_{5} + 5 \beta_{6} + 10 \beta_{7} - 5 \beta_{8} - 9 \beta_{9} ) q^{40} + ( -81 - 170 \beta_{1} + 5 \beta_{2} - 63 \beta_{3} + 104 \beta_{4} + 28 \beta_{5} - 5 \beta_{6} - 10 \beta_{7} + 5 \beta_{8} + 52 \beta_{9} ) q^{41} + ( 23 - 163 \beta_{1} - 75 \beta_{2} + 82 \beta_{3} - 5 \beta_{4} - 33 \beta_{5} - 23 \beta_{6} - 13 \beta_{7} - 5 \beta_{9} ) q^{42} + ( 1209 + 217 \beta_{1} + 401 \beta_{2} - 1253 \beta_{3} - 44 \beta_{5} - 55 \beta_{6} - 33 \beta_{8} - 8 \beta_{9} ) q^{43} + ( -43 - 510 \beta_{1} - 267 \beta_{2} + 1044 \beta_{3} - 58 \beta_{4} + 62 \beta_{5} + 43 \beta_{6} + 24 \beta_{7} - 58 \beta_{9} ) q^{44} + ( 848 + 176 \beta_{1} - 176 \beta_{2} - 26 \beta_{4} + 14 \beta_{5} - 14 \beta_{6} - 4 \beta_{7} + 4 \beta_{8} ) q^{45} + ( 108 + 75 \beta_{1} + 75 \beta_{2} - 216 \beta_{3} + 36 \beta_{4} - 58 \beta_{5} - 58 \beta_{6} - 43 \beta_{7} - 43 \beta_{8} + 72 \beta_{9} ) q^{46} + ( 5 + 63 \beta_{1} + 40 \beta_{2} - 489 \beta_{3} - 87 \beta_{4} + 7 \beta_{5} - 5 \beta_{6} - 17 \beta_{7} - 87 \beta_{9} ) q^{47} + ( 1469 + 25 \beta_{1} - 95 \beta_{2} - 774 \beta_{3} + 20 \beta_{4} - 25 \beta_{5} - 29 \beta_{6} + 25 \beta_{7} - 50 \beta_{8} - 20 \beta_{9} ) q^{48} + ( -1751 - 53 \beta_{1} + 53 \beta_{2} - 30 \beta_{4} + \beta_{5} - \beta_{6} + 9 \beta_{7} - 9 \beta_{8} ) q^{49} + ( -1764 + 288 \beta_{1} + 288 \beta_{2} + 3528 \beta_{3} - 69 \beta_{4} + 84 \beta_{5} + 84 \beta_{6} - 138 \beta_{9} ) q^{50} + ( -404 - 14 \beta_{1} + 109 \beta_{2} + 145 \beta_{3} - 52 \beta_{4} + 14 \beta_{5} - 142 \beta_{6} - 14 \beta_{7} + 28 \beta_{8} + 52 \beta_{9} ) q^{51} + ( -1406 + 156 \beta_{1} + 12 \beta_{2} - 1346 \beta_{3} + 80 \beta_{4} + 84 \beta_{5} - 12 \beta_{6} - 24 \beta_{7} + 12 \beta_{8} + 40 \beta_{9} ) q^{52} + ( -36 + 32 \beta_{1} - 430 \beta_{2} + 3 \beta_{3} + 19 \beta_{4} - 32 \beta_{5} + 34 \beta_{6} + 32 \beta_{7} - 64 \beta_{8} - 19 \beta_{9} ) q^{53} + ( -287 - 138 \beta_{1} - 214 \beta_{2} + 310 \beta_{3} + 23 \beta_{5} - 16 \beta_{6} + 62 \beta_{8} - 83 \beta_{9} ) q^{54} + ( 2275 - 51 \beta_{1} - 84 \beta_{2} - 2302 \beta_{3} - 27 \beta_{5} - 72 \beta_{6} + 18 \beta_{8} + 41 \beta_{9} ) q^{55} + ( 408 + 295 \beta_{1} + 295 \beta_{2} - 816 \beta_{3} + 15 \beta_{4} + 21 \beta_{5} + 21 \beta_{6} + 6 \beta_{7} + 6 \beta_{8} + 30 \beta_{9} ) q^{56} + ( -1327 - 121 \beta_{1} + 82 \beta_{2} + 1335 \beta_{3} + 115 \beta_{4} - 73 \beta_{5} + 76 \beta_{6} - 43 \beta_{7} + 22 \beta_{8} + 141 \beta_{9} ) q^{57} + ( -1934 - 168 \beta_{1} + 168 \beta_{2} + 14 \beta_{4} - 39 \beta_{5} + 39 \beta_{6} + 66 \beta_{7} - 66 \beta_{8} ) q^{58} + ( 1290 - 156 \beta_{1} + 53 \beta_{2} + 885 \beta_{3} + 100 \beta_{4} - 299 \beta_{5} - 53 \beta_{6} - 106 \beta_{7} + 53 \beta_{8} + 50 \beta_{9} ) q^{59} + ( 523 + 211 \beta_{1} - 2 \beta_{2} + 562 \beta_{3} + 38 \beta_{4} + 35 \beta_{5} + 2 \beta_{6} + 4 \beta_{7} - 2 \beta_{8} + 19 \beta_{9} ) q^{60} + ( 11 + 95 \beta_{1} - 5 \beta_{2} + 127 \beta_{3} - 36 \beta_{4} - 127 \beta_{5} - 11 \beta_{6} + 105 \beta_{7} - 36 \beta_{9} ) q^{61} + ( 4263 - 169 \beta_{1} - 288 \beta_{2} - 4152 \beta_{3} + 111 \beta_{5} + 172 \beta_{6} + 50 \beta_{8} + 197 \beta_{9} ) q^{62} + ( -12 - 80 \beta_{1} - 48 \beta_{2} + 1112 \beta_{3} - 58 \beta_{4} + 8 \beta_{5} + 12 \beta_{6} + 16 \beta_{7} - 58 \beta_{9} ) q^{63} + ( 2156 + 60 \beta_{1} - 60 \beta_{2} - 145 \beta_{4} - 240 \beta_{5} + 240 \beta_{6} ) q^{64} + ( -1473 - 58 \beta_{1} - 58 \beta_{2} + 2946 \beta_{3} - 29 \beta_{4} + 488 \beta_{5} + 488 \beta_{6} + 88 \beta_{7} + 88 \beta_{8} - 58 \beta_{9} ) q^{65} + ( 108 + 373 \beta_{1} + 186 \beta_{2} - 3034 \beta_{3} + 41 \beta_{4} - 217 \beta_{5} - 108 \beta_{6} + \beta_{7} + 41 \beta_{9} ) q^{66} + ( 1777 - 43 \beta_{1} - 194 \beta_{2} - 1075 \beta_{3} + 98 \beta_{4} + 43 \beta_{5} - 459 \beta_{6} - 43 \beta_{7} + 86 \beta_{8} - 98 \beta_{9} ) q^{67} + ( -5092 - \beta_{1} + \beta_{2} + 112 \beta_{4} + 221 \beta_{5} - 221 \beta_{6} + 15 \beta_{7} - 15 \beta_{8} ) q^{68} + ( -2603 - 70 \beta_{1} - 70 \beta_{2} + 5206 \beta_{3} + 76 \beta_{4} + 182 \beta_{5} + 182 \beta_{6} + 17 \beta_{7} + 17 \beta_{8} + 152 \beta_{9} ) q^{69} + ( -2668 - 17 \beta_{1} + 104 \beta_{2} + 1390 \beta_{3} + 52 \beta_{4} + 17 \beta_{5} + 78 \beta_{6} - 17 \beta_{7} + 34 \beta_{8} - 52 \beta_{9} ) q^{70} + ( -1683 + 31 \beta_{1} + 7 \beta_{2} - 1875 \beta_{3} - 256 \beta_{4} - 178 \beta_{5} - 7 \beta_{6} - 14 \beta_{7} + 7 \beta_{8} - 128 \beta_{9} ) q^{71} + ( 268 - \beta_{1} + 494 \beta_{2} - 102 \beta_{3} - 26 \beta_{4} + \beta_{5} + 62 \beta_{6} - \beta_{7} + 2 \beta_{8} + 26 \beta_{9} ) q^{72} + ( 325 + 178 \beta_{1} + 167 \beta_{2} - 171 \beta_{3} + 154 \beta_{5} + 497 \beta_{6} - 189 \beta_{8} + 62 \beta_{9} ) q^{73} + ( 3549 + 175 \beta_{1} + 269 \beta_{2} - 3810 \beta_{3} - 261 \beta_{5} - 441 \beta_{6} - 81 \beta_{8} + 27 \beta_{9} ) q^{74} + ( 1668 - 387 \beta_{1} - 387 \beta_{2} - 3336 \beta_{3} + 18 \beta_{4} - 129 \beta_{5} - 129 \beta_{6} - 126 \beta_{7} - 126 \beta_{8} + 36 \beta_{9} ) q^{75} + ( -1631 + 50 \beta_{1} - 317 \beta_{2} + 3482 \beta_{3} - 56 \beta_{4} - 358 \beta_{5} - 437 \beta_{6} - 81 \beta_{7} + 117 \beta_{8} - 144 \beta_{9} ) q^{76} + ( -2108 + 426 \beta_{1} - 426 \beta_{2} + 69 \beta_{4} - 6 \beta_{5} + 6 \beta_{6} + 7 \beta_{7} - 7 \beta_{8} ) q^{77} + ( 2316 + 105 \beta_{1} - 99 \beta_{2} + 2912 \beta_{3} - 102 \beta_{4} + 398 \beta_{5} + 99 \beta_{6} + 198 \beta_{7} - 99 \beta_{8} - 51 \beta_{9} ) q^{78} + ( 1547 - 455 \beta_{1} - 19 \beta_{2} + 1897 \beta_{3} - 160 \beta_{4} + 312 \beta_{5} + 19 \beta_{6} + 38 \beta_{7} - 19 \beta_{8} - 80 \beta_{9} ) q^{79} + ( 389 - 101 \beta_{1} + 49 \beta_{2} - 822 \beta_{3} + 25 \beta_{4} - 579 \beta_{5} - 389 \beta_{6} - 199 \beta_{7} + 25 \beta_{9} ) q^{80} + ( 2883 + 126 \beta_{1} + 169 \beta_{2} - 3067 \beta_{3} - 184 \beta_{5} - 285 \beta_{6} - 83 \beta_{8} - 292 \beta_{9} ) q^{81} + ( -32 + 1171 \beta_{1} + 665 \beta_{2} + 2590 \beta_{3} + 44 \beta_{4} + 223 \beta_{5} + 32 \beta_{6} - 159 \beta_{7} + 44 \beta_{9} ) q^{82} + ( 752 - 538 \beta_{1} + 538 \beta_{2} + 356 \beta_{4} + 124 \beta_{5} - 124 \beta_{6} + \beta_{7} - \beta_{8} ) q^{83} + ( -650 - 146 \beta_{1} - 146 \beta_{2} + 1300 \beta_{3} - 61 \beta_{4} - 34 \beta_{5} - 34 \beta_{6} + 25 \beta_{7} + 25 \beta_{8} - 122 \beta_{9} ) q^{84} + ( -288 - 1398 \beta_{1} - 774 \beta_{2} - 2247 \beta_{3} + 41 \beta_{4} + 426 \beta_{5} + 288 \beta_{6} + 150 \beta_{7} + 41 \beta_{9} ) q^{85} + ( 8550 - 85 \beta_{1} + 931 \beta_{2} - 4044 \beta_{3} - 231 \beta_{4} + 85 \beta_{5} + 292 \beta_{6} - 85 \beta_{7} + 170 \beta_{8} + 231 \beta_{9} ) q^{86} + ( -3399 + 221 \beta_{1} - 221 \beta_{2} - 239 \beta_{4} + 72 \beta_{5} - 72 \beta_{6} + 95 \beta_{7} - 95 \beta_{8} ) q^{87} + ( -4006 - 1044 \beta_{1} - 1044 \beta_{2} + 8012 \beta_{3} - 42 \beta_{4} + 248 \beta_{5} + 248 \beta_{6} - 25 \beta_{7} - 25 \beta_{8} - 84 \beta_{9} ) q^{88} + ( -3372 + 27 \beta_{1} - 738 \beta_{2} + 1503 \beta_{3} - 13 \beta_{4} - 27 \beta_{5} - 312 \beta_{6} + 27 \beta_{7} - 54 \beta_{8} + 13 \beta_{9} ) q^{89} + ( -3854 - 1142 \beta_{1} - 20 \beta_{2} - 3844 \beta_{3} + 344 \beta_{4} - 30 \beta_{5} + 20 \beta_{6} + 40 \beta_{7} - 20 \beta_{8} + 172 \beta_{9} ) q^{90} + ( 1554 - 38 \beta_{1} + 776 \beta_{2} - 634 \beta_{3} - 58 \beta_{4} + 38 \beta_{5} + 210 \beta_{6} - 38 \beta_{7} + 76 \beta_{8} + 58 \beta_{9} ) q^{91} + ( 2189 + 255 \beta_{1} + 648 \beta_{2} - 2250 \beta_{3} - 61 \beta_{5} - 260 \beta_{6} + 138 \beta_{8} + 271 \beta_{9} ) q^{92} + ( 2261 - 195 \beta_{1} - 319 \beta_{2} - 1880 \beta_{3} + 381 \beta_{5} + 691 \beta_{6} + 71 \beta_{8} - 347 \beta_{9} ) q^{93} + ( 148 - 597 \beta_{1} - 597 \beta_{2} - 296 \beta_{3} - 18 \beta_{4} + 76 \beta_{5} + 76 \beta_{6} + 109 \beta_{7} + 109 \beta_{8} - 36 \beta_{9} ) q^{94} + ( -3985 + 1388 \beta_{1} + 1378 \beta_{2} + 6519 \beta_{3} - 277 \beta_{4} + 157 \beta_{5} + 228 \beta_{6} + 112 \beta_{7} - 71 \beta_{8} - 202 \beta_{9} ) q^{95} + ( -4154 + 219 \beta_{1} - 219 \beta_{2} - 126 \beta_{4} - 131 \beta_{5} + 131 \beta_{6} - 54 \beta_{7} + 54 \beta_{8} ) q^{96} + ( 3269 + 104 \beta_{1} - 32 \beta_{2} + 2919 \beta_{3} + 60 \beta_{4} - 414 \beta_{5} + 32 \beta_{6} + 64 \beta_{7} - 32 \beta_{8} + 30 \beta_{9} ) q^{97} + ( 1494 + 1495 \beta_{1} - 11 \beta_{2} + 1374 \beta_{3} - 200 \beta_{4} - 142 \beta_{5} + 11 \beta_{6} + 22 \beta_{7} - 11 \beta_{8} - 100 \beta_{9} ) q^{98} + ( 10 + 1469 \beta_{1} + 762 \beta_{2} - 878 \beta_{3} + 106 \beta_{4} + 35 \beta_{5} - 10 \beta_{6} - 55 \beta_{7} + 106 \beta_{9} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10q - 3q^{2} + 9q^{3} + 29q^{4} + 8q^{5} - 35q^{6} - 24q^{7} - 58q^{9} + O(q^{10}) \) \( 10q - 3q^{2} + 9q^{3} + 29q^{4} + 8q^{5} - 35q^{6} - 24q^{7} - 58q^{9} + 144q^{10} + 50q^{11} - 624q^{13} - 474q^{14} + 504q^{15} + 285q^{16} - 292q^{17} + 305q^{19} - 652q^{20} + 1158q^{21} + 1629q^{22} + 98q^{23} + 505q^{24} - 681q^{25} + 1524q^{26} - 1472q^{28} + 2598q^{29} - 6656q^{30} - 2745q^{32} - 3441q^{33} + 486q^{34} + 694q^{35} + 3402q^{36} - 342q^{38} - 5552q^{39} + 8784q^{40} - 1407q^{41} + 292q^{42} + 5424q^{43} + 4151q^{44} + 9572q^{45} - 2416q^{47} + 11481q^{48} - 17826q^{49} - 3342q^{51} - 19962q^{52} + 1122q^{53} - 1039q^{54} + 11424q^{55} - 7906q^{57} - 20236q^{58} + 15387q^{59} + 8886q^{60} + 860q^{61} + 21636q^{62} + 5318q^{63} + 19710q^{64} - 13921q^{66} + 14763q^{67} - 48844q^{68} - 20334q^{70} - 27264q^{71} + 354q^{72} + 1561q^{73} + 17094q^{74} + 1955q^{76} - 18392q^{77} + 40266q^{78} + 24750q^{79} - 2002q^{80} + 14311q^{81} + 14479q^{82} + 6002q^{83} - 14944q^{85} + 59946q^{86} - 31996q^{87} - 22566q^{89} - 60630q^{90} + 8724q^{91} + 9572q^{92} + 12476q^{93} - 7312q^{95} - 41850q^{96} + 46287q^{97} + 25515q^{98} - 2048q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{10} + 109 x^{8} + 4107 x^{6} + 61507 x^{4} + 300520 x^{2} + 108300\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{6} + 56 \nu^{4} + 649 \nu^{2} + 300 \nu + 570 \)\()/600\)
\(\beta_{2}\)\(=\)\((\)\( -\nu^{6} - 56 \nu^{4} - 649 \nu^{2} + 300 \nu - 570 \)\()/600\)
\(\beta_{3}\)\(=\)\((\)\( \nu^{9} + 109 \nu^{7} + 3917 \nu^{5} + 50867 \nu^{3} + 177210 \nu + 57000 \)\()/114000\)
\(\beta_{4}\)\(=\)\( \nu^{2} + 22 \)
\(\beta_{5}\)\(=\)\((\)\( -\nu^{9} - 204 \nu^{7} - 285 \nu^{6} - 12087 \nu^{5} - 24510 \nu^{4} - 263572 \nu^{3} - 552615 \nu^{2} - 1713360 \nu - 1786950 \)\()/171000\)
\(\beta_{6}\)\(=\)\((\)\( -\nu^{9} - 204 \nu^{7} + 285 \nu^{6} - 12087 \nu^{5} + 24510 \nu^{4} - 263572 \nu^{3} + 552615 \nu^{2} - 1713360 \nu + 1786950 \)\()/171000\)
\(\beta_{7}\)\(=\)\((\)\( 23 \nu^{9} + 95 \nu^{8} + 2127 \nu^{7} + 7980 \nu^{6} + 65961 \nu^{5} + 202065 \nu^{4} + 857771 \nu^{3} + 1498340 \nu^{2} + 5198730 \nu + 1259700 \)\()/171000\)
\(\beta_{8}\)\(=\)\((\)\( 23 \nu^{9} - 95 \nu^{8} + 2127 \nu^{7} - 7980 \nu^{6} + 65961 \nu^{5} - 202065 \nu^{4} + 857771 \nu^{3} - 1498340 \nu^{2} + 5198730 \nu - 1259700 \)\()/171000\)
\(\beta_{9}\)\(=\)\((\)\( 11 \nu^{9} + 1104 \nu^{7} + 37767 \nu^{5} + 497882 \nu^{3} - 28500 \nu^{2} + 1895160 \nu - 627000 \)\()/57000\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{2} + \beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{4} - 22\)
\(\nu^{3}\)\(=\)\(-2 \beta_{9} + \beta_{8} + \beta_{7} - \beta_{6} - \beta_{5} - \beta_{4} + 12 \beta_{3} - 33 \beta_{2} - 33 \beta_{1} - 6\)
\(\nu^{4}\)\(=\)\(10 \beta_{6} - 10 \beta_{5} - 43 \beta_{4} + 10 \beta_{2} - 10 \beta_{1} + 756\)
\(\nu^{5}\)\(=\)\(126 \beta_{9} - 53 \beta_{8} - 53 \beta_{7} + 23 \beta_{6} + 23 \beta_{5} + 63 \beta_{4} - 1116 \beta_{3} + 1229 \beta_{2} + 1229 \beta_{1} + 558\)
\(\nu^{6}\)\(=\)\(-560 \beta_{6} + 560 \beta_{5} + 1759 \beta_{4} - 860 \beta_{2} + 860 \beta_{1} - 28628\)
\(\nu^{7}\)\(=\)\(-6358 \beta_{9} + 2319 \beta_{8} + 2319 \beta_{7} - 639 \beta_{6} - 639 \beta_{5} - 3179 \beta_{4} + 67908 \beta_{3} - 47977 \beta_{2} - 47977 \beta_{1} - 33954\)
\(\nu^{8}\)\(=\)\(-900 \beta_{8} + 900 \beta_{7} + 25770 \beta_{6} - 25770 \beta_{5} - 72067 \beta_{4} + 50970 \beta_{2} - 50970 \beta_{1} + 1130464\)
\(\nu^{9}\)\(=\)\(301214 \beta_{9} - 96037 \beta_{8} - 96037 \beta_{7} + 30427 \beta_{6} + 30427 \beta_{5} + 150607 \beta_{4} - 3527004 \beta_{3} + 1916901 \beta_{2} + 1916901 \beta_{1} + 1763502\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/19\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(\beta_{3}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
8.1
6.00126i
4.58432i
0.625165i
2.91518i
6.56352i
6.00126i
4.58432i
0.625165i
2.91518i
6.56352i
−5.19724 3.00063i −1.61920 0.934844i 10.0076 + 17.3336i −20.3151 + 35.1867i 5.61024 + 9.71722i −11.8590 24.0959i −38.7521 67.1207i 211.165 121.916i
8.2 −3.97014 2.29216i 10.1306 + 5.84893i 2.50801 + 4.34400i 20.8352 36.0877i −26.8134 46.4422i 24.1856 50.3541i 27.9200 + 48.3589i −165.438 + 95.5155i
8.3 −0.541408 0.312582i −8.70876 5.02800i −7.80458 13.5179i 4.61985 8.00181i 3.14333 + 5.44441i 0.274467 19.7609i 10.0617 + 17.4273i −5.00245 + 2.88816i
8.4 2.52462 + 1.45759i 8.18489 + 4.72555i −3.75086 6.49668i −7.00036 + 12.1250i 13.7758 + 23.8604i 19.8223 68.5118i 4.16163 + 7.20815i −35.3465 + 20.4073i
8.5 5.68417 + 3.28176i −3.48758 2.01356i 13.5399 + 23.4517i 5.86031 10.1504i −13.2160 22.8908i −44.4232 72.7220i −32.3912 56.1032i 66.6221 38.4643i
12.1 −5.19724 + 3.00063i −1.61920 + 0.934844i 10.0076 17.3336i −20.3151 35.1867i 5.61024 9.71722i −11.8590 24.0959i −38.7521 + 67.1207i 211.165 + 121.916i
12.2 −3.97014 + 2.29216i 10.1306 5.84893i 2.50801 4.34400i 20.8352 + 36.0877i −26.8134 + 46.4422i 24.1856 50.3541i 27.9200 48.3589i −165.438 95.5155i
12.3 −0.541408 + 0.312582i −8.70876 + 5.02800i −7.80458 + 13.5179i 4.61985 + 8.00181i 3.14333 5.44441i 0.274467 19.7609i 10.0617 17.4273i −5.00245 2.88816i
12.4 2.52462 1.45759i 8.18489 4.72555i −3.75086 + 6.49668i −7.00036 12.1250i 13.7758 23.8604i 19.8223 68.5118i 4.16163 7.20815i −35.3465 20.4073i
12.5 5.68417 3.28176i −3.48758 + 2.01356i 13.5399 23.4517i 5.86031 + 10.1504i −13.2160 + 22.8908i −44.4232 72.7220i −32.3912 + 56.1032i 66.6221 + 38.4643i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 12.5
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 19.5.d.a 10
3.b odd 2 1 171.5.p.a 10
4.b odd 2 1 304.5.r.a 10
19.d odd 6 1 inner 19.5.d.a 10
57.f even 6 1 171.5.p.a 10
76.f even 6 1 304.5.r.a 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
19.5.d.a 10 1.a even 1 1 trivial
19.5.d.a 10 19.d odd 6 1 inner
171.5.p.a 10 3.b odd 2 1
171.5.p.a 10 57.f even 6 1
304.5.r.a 10 4.b odd 2 1
304.5.r.a 10 76.f even 6 1

Hecke kernels

This newform subspace is the entire newspace \(S_{5}^{\mathrm{new}}(19, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 108300 + 279300 T + 209890 T^{2} - 77910 T^{3} - 18113 T^{4} + 7857 T^{5} + 2160 T^{6} - 159 T^{7} - 50 T^{8} + 3 T^{9} + T^{10} \)
$3$ \( 70073667 + 83905713 T + 37056123 T^{2} + 4270806 T^{3} - 700875 T^{4} - 132579 T^{5} + 17599 T^{6} + 1440 T^{7} - 133 T^{8} - 9 T^{9} + T^{10} \)
$5$ \( 6589766238916 - 800715483680 T + 131928778968 T^{2} - 5397491428 T^{3} + 745099952 T^{4} - 22819866 T^{5} + 3296657 T^{6} - 12016 T^{7} + 1935 T^{8} - 8 T^{9} + T^{10} \)
$7$ \( ( -69320 + 251520 T + 4202 T^{2} - 1474 T^{3} + 12 T^{4} + T^{5} )^{2} \)
$11$ \( ( -8646509620 + 231684376 T + 864245 T^{2} - 40067 T^{3} - 25 T^{4} + T^{5} )^{2} \)
$13$ \( \)\(45\!\cdots\!00\)\( + 7048028395184128800 T - 724730922892673984 T^{2} - 1121016177074808 T^{3} + 90590786414176 T^{4} + 154467622146 T^{5} - 3725379891 T^{6} - 9093552 T^{7} + 115219 T^{8} + 624 T^{9} + T^{10} \)
$17$ \( \)\(58\!\cdots\!00\)\( + \)\(14\!\cdots\!00\)\( T + 7854542082224928360 T^{2} - 74518301589690960 T^{3} + 3202257265135704 T^{4} + 9069968961882 T^{5} + 49544015037 T^{6} + 51610248 T^{7} + 274663 T^{8} + 292 T^{9} + T^{10} \)
$19$ \( \)\(37\!\cdots\!01\)\( - \)\(87\!\cdots\!05\)\( T + 19007948524940190668 T^{2} + 461344342700751847 T^{3} + 2401322961930031 T^{4} - 12805136074104 T^{5} + 18426216511 T^{6} + 27164167 T^{7} + 8588 T^{8} - 305 T^{9} + T^{10} \)
$23$ \( \)\(88\!\cdots\!76\)\( + \)\(41\!\cdots\!36\)\( T + \)\(20\!\cdots\!84\)\( T^{2} + 36790152635045685344 T^{3} + 107004829184934662 T^{4} + 93675713755596 T^{5} + 397308878771 T^{6} + 172317986 T^{7} + 744723 T^{8} - 98 T^{9} + T^{10} \)
$29$ \( \)\(61\!\cdots\!92\)\( - \)\(14\!\cdots\!80\)\( T + \)\(14\!\cdots\!12\)\( T^{2} - \)\(62\!\cdots\!80\)\( T^{3} + 1247589146783029756 T^{4} - 509154513177852 T^{5} - 1183857886575 T^{6} + 659572446 T^{7} + 1995991 T^{8} - 2598 T^{9} + T^{10} \)
$31$ \( \)\(15\!\cdots\!00\)\( + \)\(26\!\cdots\!80\)\( T^{2} + 12776336366159849068 T^{4} + 15854752268172 T^{6} + 6984856 T^{8} + T^{10} \)
$37$ \( \)\(15\!\cdots\!28\)\( + \)\(33\!\cdots\!52\)\( T^{2} + 6147900568104251724 T^{4} + 15349623791328 T^{6} + 7481496 T^{8} + T^{10} \)
$41$ \( \)\(27\!\cdots\!75\)\( + \)\(60\!\cdots\!75\)\( T + \)\(22\!\cdots\!35\)\( T^{2} - \)\(50\!\cdots\!70\)\( T^{3} - 24652595202118929857 T^{4} + 34943904980180865 T^{5} + 23953610079357 T^{6} - 8228065650 T^{7} - 5188067 T^{8} + 1407 T^{9} + T^{10} \)
$43$ \( \)\(48\!\cdots\!96\)\( - \)\(96\!\cdots\!04\)\( T + \)\(14\!\cdots\!68\)\( T^{2} - \)\(11\!\cdots\!28\)\( T^{3} + \)\(82\!\cdots\!40\)\( T^{4} - 380613202146876782 T^{5} + 183252799307245 T^{6} - 64870697948 T^{7} + 26225353 T^{8} - 5424 T^{9} + T^{10} \)
$47$ \( \)\(16\!\cdots\!00\)\( + \)\(76\!\cdots\!00\)\( T + \)\(34\!\cdots\!00\)\( T^{2} + \)\(14\!\cdots\!00\)\( T^{3} + 54745330614722224850 T^{4} + 22327238184453150 T^{5} + 29563014788915 T^{6} + 6419955440 T^{7} + 9461271 T^{8} + 2416 T^{9} + T^{10} \)
$53$ \( \)\(42\!\cdots\!28\)\( - \)\(58\!\cdots\!44\)\( T + \)\(26\!\cdots\!00\)\( T^{2} + \)\(29\!\cdots\!92\)\( T^{3} - \)\(79\!\cdots\!64\)\( T^{4} - 76516199204402412 T^{5} + 187151731256385 T^{6} + 17423570538 T^{7} - 15109401 T^{8} - 1122 T^{9} + T^{10} \)
$59$ \( \)\(35\!\cdots\!63\)\( - \)\(72\!\cdots\!37\)\( T + \)\(62\!\cdots\!73\)\( T^{2} - \)\(28\!\cdots\!48\)\( T^{3} + \)\(62\!\cdots\!85\)\( T^{4} - 2709550708068190059 T^{5} - 1345303689902325 T^{6} + 64194471678 T^{7} + 74747929 T^{8} - 15387 T^{9} + T^{10} \)
$61$ \( \)\(14\!\cdots\!00\)\( - \)\(89\!\cdots\!80\)\( T + \)\(89\!\cdots\!56\)\( T^{2} - \)\(26\!\cdots\!40\)\( T^{3} + \)\(22\!\cdots\!28\)\( T^{4} - 592315141935115830 T^{5} + 349284302751273 T^{6} - 42689216700 T^{7} + 20649783 T^{8} - 860 T^{9} + T^{10} \)
$67$ \( \)\(13\!\cdots\!75\)\( + \)\(14\!\cdots\!75\)\( T - \)\(68\!\cdots\!35\)\( T^{2} - \)\(12\!\cdots\!20\)\( T^{3} + \)\(11\!\cdots\!77\)\( T^{4} - 19616713894959056487 T^{5} - 1231810567938345 T^{6} + 532810517694 T^{7} + 36557785 T^{8} - 14763 T^{9} + T^{10} \)
$71$ \( \)\(37\!\cdots\!00\)\( - \)\(56\!\cdots\!00\)\( T + \)\(21\!\cdots\!00\)\( T^{2} + \)\(89\!\cdots\!00\)\( T^{3} - \)\(31\!\cdots\!00\)\( T^{4} - 11214232765923310650 T^{5} + 2528262900871605 T^{6} + 1624177687680 T^{7} + 307347477 T^{8} + 27264 T^{9} + T^{10} \)
$73$ \( \)\(91\!\cdots\!25\)\( - \)\(55\!\cdots\!25\)\( T + \)\(39\!\cdots\!15\)\( T^{2} - \)\(21\!\cdots\!90\)\( T^{3} + \)\(20\!\cdots\!49\)\( T^{4} - 15051221378357383215 T^{5} + 7273880578183977 T^{6} - 232017879234 T^{7} + 96341611 T^{8} - 1561 T^{9} + T^{10} \)
$79$ \( \)\(48\!\cdots\!00\)\( - \)\(72\!\cdots\!00\)\( T + \)\(36\!\cdots\!00\)\( T^{2} - \)\(13\!\cdots\!00\)\( T^{3} - \)\(76\!\cdots\!72\)\( T^{4} + 9834036276435447408 T^{5} + 610597367294121 T^{6} - 1078378809750 T^{7} + 247758361 T^{8} - 24750 T^{9} + T^{10} \)
$83$ \( ( 705913210543561400 + 1262650869822640 T + 332146262435 T^{2} - 118400561 T^{3} - 3001 T^{4} + T^{5} )^{2} \)
$89$ \( \)\(56\!\cdots\!88\)\( + \)\(16\!\cdots\!80\)\( T + \)\(18\!\cdots\!52\)\( T^{2} + \)\(82\!\cdots\!20\)\( T^{3} + \)\(13\!\cdots\!64\)\( T^{4} - 5872085639186909976 T^{5} - 3488250668630175 T^{6} + 249448783842 T^{7} + 180795639 T^{8} + 22566 T^{9} + T^{10} \)
$97$ \( \)\(11\!\cdots\!47\)\( - \)\(34\!\cdots\!75\)\( T + \)\(46\!\cdots\!87\)\( T^{2} - \)\(34\!\cdots\!50\)\( T^{3} + \)\(15\!\cdots\!11\)\( T^{4} - \)\(45\!\cdots\!93\)\( T^{5} + 89842090652541225 T^{6} - 11673741460446 T^{7} + 966365581 T^{8} - 46287 T^{9} + T^{10} \)
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