Properties

Label 19.8.a.b
Level 19
Weight 8
Character orbit 19.a
Self dual Yes
Analytic conductor 5.935
Analytic rank 0
Dimension 6
CM No
Inner twists 1

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) = \( 19 \)
Weight: \( k \) = \( 8 \)
Character orbit: \([\chi]\) = 19.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(5.9353154842\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2\cdot 3 \)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \( + ( 2 + \beta_{1} ) q^{2} \) \( + ( 6 + \beta_{1} + \beta_{4} ) q^{3} \) \( + ( 57 + 6 \beta_{1} + \beta_{2} ) q^{4} \) \( + ( 32 + 7 \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} ) q^{5} \) \( + ( 148 + 12 \beta_{1} - 6 \beta_{2} - 5 \beta_{3} + \beta_{4} + 4 \beta_{5} ) q^{6} \) \( + ( 352 - 2 \beta_{1} + 3 \beta_{2} + 7 \beta_{3} - 8 \beta_{4} - \beta_{5} ) q^{7} \) \( + ( 972 + 14 \beta_{1} + 14 \beta_{2} + \beta_{3} - 15 \beta_{4} - 8 \beta_{5} ) q^{8} \) \( + ( 1011 - 57 \beta_{1} - 2 \beta_{2} - 10 \beta_{3} + 15 \beta_{4} - 10 \beta_{5} ) q^{9} \) \(+O(q^{10})\) \( q\) \( + ( 2 + \beta_{1} ) q^{2} \) \( + ( 6 + \beta_{1} + \beta_{4} ) q^{3} \) \( + ( 57 + 6 \beta_{1} + \beta_{2} ) q^{4} \) \( + ( 32 + 7 \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} ) q^{5} \) \( + ( 148 + 12 \beta_{1} - 6 \beta_{2} - 5 \beta_{3} + \beta_{4} + 4 \beta_{5} ) q^{6} \) \( + ( 352 - 2 \beta_{1} + 3 \beta_{2} + 7 \beta_{3} - 8 \beta_{4} - \beta_{5} ) q^{7} \) \( + ( 972 + 14 \beta_{1} + 14 \beta_{2} + \beta_{3} - 15 \beta_{4} - 8 \beta_{5} ) q^{8} \) \( + ( 1011 - 57 \beta_{1} - 2 \beta_{2} - 10 \beta_{3} + 15 \beta_{4} - 10 \beta_{5} ) q^{9} \) \( + ( 1408 - 67 \beta_{1} + 4 \beta_{2} + 11 \beta_{3} - \beta_{4} + 22 \beta_{5} ) q^{10} \) \( + ( 1314 - 223 \beta_{1} - 15 \beta_{2} + 15 \beta_{3} + 49 \beta_{4} + 41 \beta_{5} ) q^{11} \) \( + ( 1154 - 36 \beta_{2} - 49 \beta_{3} + 35 \beta_{4} - 28 \beta_{5} ) q^{12} \) \( + ( 1136 - 55 \beta_{1} - 24 \beta_{2} - 46 \beta_{3} + 35 \beta_{4} - 92 \beta_{5} ) q^{13} \) \( + ( 1404 + 207 \beta_{1} + 82 \beta_{2} + 110 \beta_{3} - 140 \beta_{4} + 10 \beta_{5} ) q^{14} \) \( + ( 716 - 174 \beta_{1} - 10 \beta_{2} + 42 \beta_{3} + 62 \beta_{4} + 150 \beta_{5} ) q^{15} \) \( + ( -1922 + 852 \beta_{1} + 80 \beta_{2} + 75 \beta_{3} - 261 \beta_{4} - 84 \beta_{5} ) q^{16} \) \( + ( 786 + 326 \beta_{1} + 31 \beta_{2} - 151 \beta_{3} + 2 \beta_{4} - 57 \beta_{5} ) q^{17} \) \( + ( -10286 + 851 \beta_{1} - 218 \beta_{2} - 207 \beta_{3} + 135 \beta_{4} + 96 \beta_{5} ) q^{18} \) \( -6859 q^{19} \) \( + ( -11501 + 574 \beta_{1} + 177 \beta_{2} + 57 \beta_{3} + \beta_{4} - 40 \beta_{5} ) q^{20} \) \( + ( -13150 - 1563 \beta_{1} + 52 \beta_{2} + 310 \beta_{3} + 155 \beta_{4} + 160 \beta_{5} ) q^{21} \) \( + ( -37612 - 11 \beta_{1} - 548 \beta_{2} + 13 \beta_{3} + 217 \beta_{4} + 26 \beta_{5} ) q^{22} \) \( + ( 758 - 1601 \beta_{1} + 574 \beta_{2} + 176 \beta_{3} - 371 \beta_{4} - 638 \beta_{5} ) q^{23} \) \( + ( -24700 - 1220 \beta_{1} + 102 \beta_{2} - 145 \beta_{3} + 951 \beta_{4} - 196 \beta_{5} ) q^{24} \) \( + ( -7835 - 2215 \beta_{1} - 405 \beta_{2} + 113 \beta_{3} - 307 \beta_{4} + 523 \beta_{5} ) q^{25} \) \( + ( -17244 - 846 \beta_{1} - 814 \beta_{2} - 935 \beta_{3} + 671 \beta_{4} + 884 \beta_{5} ) q^{26} \) \( + ( 34410 - 1727 \beta_{1} + 642 \beta_{2} - 102 \beta_{3} - 287 \beta_{4} - 294 \beta_{5} ) q^{27} \) \( + ( 14375 + 3050 \beta_{1} + 1599 \beta_{2} + 1016 \beta_{3} - 1636 \beta_{4} - 308 \beta_{5} ) q^{28} \) \( + ( 63844 + 195 \beta_{1} - 330 \beta_{2} - 840 \beta_{3} - 1315 \beta_{4} + 10 \beta_{5} ) q^{29} \) \( + ( -24276 + 1454 \beta_{1} - 196 \beta_{2} + 550 \beta_{3} + 158 \beta_{4} - 836 \beta_{5} ) q^{30} \) \( + ( 42024 + 2776 \beta_{1} - 2148 \beta_{2} - 980 \beta_{3} + 748 \beta_{4} + 812 \beta_{5} ) q^{31} \) \( + ( 44140 - 964 \beta_{1} + 1350 \beta_{2} + 1755 \beta_{3} - 693 \beta_{4} + 780 \beta_{5} ) q^{32} \) \( + ( 81472 + 856 \beta_{1} + 394 \beta_{2} + 1446 \beta_{3} + 44 \beta_{4} - 1638 \beta_{5} ) q^{33} \) \( + ( 45432 + 8657 \beta_{1} + 238 \beta_{2} - 1660 \beta_{3} + 1178 \beta_{4} - 878 \beta_{5} ) q^{34} \) \( + ( 120966 + 3701 \beta_{1} - 937 \beta_{2} + 505 \beta_{3} - 219 \beta_{4} + 367 \beta_{5} ) q^{35} \) \( + ( -24211 - 2030 \beta_{1} - 1501 \beta_{2} - 1395 \beta_{3} + 4257 \beta_{4} + 948 \beta_{5} ) q^{36} \) \( + ( 173342 + 16264 \beta_{1} - 100 \beta_{2} - 1100 \beta_{3} - 1484 \beta_{4} + 1580 \beta_{5} ) q^{37} \) \( + ( -13718 - 6859 \beta_{1} ) q^{38} \) \( + ( 191362 - 3807 \beta_{1} + 2954 \beta_{2} - 640 \beta_{3} + 7123 \beta_{4} + 2102 \beta_{5} ) q^{39} \) \( + ( -90440 + 6930 \beta_{1} + 1408 \beta_{2} - 786 \beta_{3} - 3330 \beta_{4} - 3372 \beta_{5} ) q^{40} \) \( + ( 85598 - 10946 \beta_{1} - 1012 \beta_{2} + 1960 \beta_{3} - 5218 \beta_{4} - 1372 \beta_{5} ) q^{41} \) \( + ( -280612 - 24480 \beta_{1} - 1442 \beta_{2} + 2857 \beta_{3} - 3865 \beta_{4} + 1084 \beta_{5} ) q^{42} \) \( + ( 29502 + 8223 \beta_{1} + 4393 \beta_{2} + 3087 \beta_{3} + 3807 \beta_{4} - 111 \beta_{5} ) q^{43} \) \( + ( -268449 - 42002 \beta_{1} - 3903 \beta_{2} - 3345 \beta_{3} + 2087 \beta_{4} - 152 \beta_{5} ) q^{44} \) \( + ( -78448 - 2973 \beta_{1} - 769 \beta_{2} - 939 \beta_{3} - 4057 \beta_{4} - 2265 \beta_{5} ) q^{45} \) \( + ( -260184 + 6274 \beta_{1} + 3850 \beta_{2} + 2275 \beta_{3} - 13007 \beta_{4} + 1712 \beta_{5} ) q^{46} \) \( + ( -41010 - 4623 \beta_{1} - 3369 \beta_{2} - 4251 \beta_{3} + 4605 \beta_{4} - 3297 \beta_{5} ) q^{47} \) \( + ( -477824 - 20360 \beta_{1} - 3186 \beta_{2} - 419 \beta_{3} - 3907 \beta_{4} + 7372 \beta_{5} ) q^{48} \) \( + ( -112003 + 44308 \beta_{1} + 305 \beta_{2} + 655 \beta_{3} + 16 \beta_{4} + 89 \beta_{5} ) q^{49} \) \( + ( -386342 - 33982 \beta_{1} - 1624 \beta_{2} + 3829 \beta_{3} + 5981 \beta_{4} - 2314 \beta_{5} ) q^{50} \) \( + ( -27210 + 31327 \beta_{1} - 1144 \beta_{2} - 8360 \beta_{3} + 5287 \beta_{4} - 2552 \beta_{5} ) q^{51} \) \( + ( -443792 - 3284 \beta_{1} - 7266 \beta_{2} - 4979 \beta_{3} + 22273 \beta_{4} + 4652 \beta_{5} ) q^{52} \) \( + ( 546984 - 9731 \beta_{1} + 11216 \beta_{2} + 7174 \beta_{3} + 55 \beta_{4} + 3348 \beta_{5} ) q^{53} \) \( + ( -232064 + 63114 \beta_{1} + 4434 \beta_{2} + 175 \beta_{3} - 9575 \beta_{4} - 4160 \beta_{5} ) q^{54} \) \( + ( -285370 + 59925 \beta_{1} - 6325 \beta_{2} + 7017 \beta_{3} + 13977 \beta_{4} + 2547 \beta_{5} ) q^{55} \) \( + ( 603804 + 46314 \beta_{1} + 16890 \beta_{2} + 4935 \beta_{3} - 20817 \beta_{4} - 9408 \beta_{5} ) q^{56} \) \( + ( -41154 - 6859 \beta_{1} - 6859 \beta_{4} ) q^{57} \) \( + ( 135128 + 75704 \beta_{1} + 5950 \beta_{2} - 2125 \beta_{3} + 13745 \beta_{4} - 9440 \beta_{5} ) q^{58} \) \( + ( 384390 + 1473 \beta_{1} - 6212 \beta_{2} + 3200 \beta_{3} + 1465 \beta_{4} + 15940 \beta_{5} ) q^{59} \) \( + ( 133938 - 49892 \beta_{1} - 1898 \beta_{2} - 3370 \beta_{3} - 13946 \beta_{4} - 4240 \beta_{5} ) q^{60} \) \( + ( 119140 - 41363 \beta_{1} - 14377 \beta_{2} - 11495 \beta_{3} - 20987 \beta_{4} + 2087 \beta_{5} ) q^{61} \) \( + ( 428356 - 17716 \beta_{1} - 18188 \beta_{2} - 13252 \beta_{3} + 47164 \beta_{4} + 4216 \beta_{5} ) q^{62} \) \( + ( -487690 - 97435 \beta_{1} + 497 \beta_{2} + 2943 \beta_{3} - 6323 \beta_{4} - 5679 \beta_{5} ) q^{63} \) \( + ( 417752 - 39288 \beta_{1} + 8542 \beta_{2} + 15105 \beta_{3} - 6255 \beta_{4} + 3420 \beta_{5} ) q^{64} \) \( + ( 866772 - 117248 \beta_{1} - 3992 \beta_{2} - 10780 \beta_{3} - 30684 \beta_{4} - 400 \beta_{5} ) q^{65} \) \( + ( 407356 + 11866 \beta_{1} + 232 \beta_{2} + 9720 \beta_{3} - 28132 \beta_{4} + 24972 \beta_{5} ) q^{66} \) \( + ( -531006 - 34413 \beta_{1} + 9634 \beta_{2} + 5046 \beta_{3} - 16101 \beta_{4} - 30078 \beta_{5} ) q^{67} \) \( + ( 1327473 + 100534 \beta_{1} - 5947 \beta_{2} - 5558 \beta_{3} + 14638 \beta_{4} + 5604 \beta_{5} ) q^{68} \) \( + ( -194130 - 224661 \beta_{1} + 28318 \beta_{2} + 12440 \beta_{3} + 17097 \beta_{4} - 7342 \beta_{5} ) q^{69} \) \( + ( 954508 + 66783 \beta_{1} - 656 \beta_{2} + 6309 \beta_{3} + 8877 \beta_{4} + 6990 \beta_{5} ) q^{70} \) \( + ( 1173072 - 62272 \beta_{1} + 10718 \beta_{2} - 3050 \beta_{3} + 18060 \beta_{4} - 1130 \beta_{5} ) q^{71} \) \( + ( 568712 - 145754 \beta_{1} - 14484 \beta_{2} - 7396 \beta_{3} + 29076 \beta_{4} - 3892 \beta_{5} ) q^{72} \) \( + ( -765114 + 150924 \beta_{1} - 11377 \beta_{2} - 18663 \beta_{3} + 3600 \beta_{4} + 11559 \beta_{5} ) q^{73} \) \( + ( 3304288 + 311250 \beta_{1} + 29492 \beta_{2} + 1060 \beta_{3} + 17956 \beta_{4} - 29736 \beta_{5} ) q^{74} \) \( + ( -1928858 + 107357 \beta_{1} - 2950 \beta_{2} + 20570 \beta_{3} - 29235 \beta_{4} + 3746 \beta_{5} ) q^{75} \) \( + ( -390963 - 41154 \beta_{1} - 6859 \beta_{2} ) q^{76} \) \( + ( 281490 - 47153 \beta_{1} - 33625 \beta_{2} + 1165 \beta_{3} + 60379 \beta_{4} + 25935 \beta_{5} ) q^{77} \) \( + ( -534320 + 446622 \beta_{1} - 24370 \beta_{2} - 32755 \beta_{3} - 23201 \beta_{4} - 21280 \beta_{5} ) q^{78} \) \( + ( -1362900 - 149726 \beta_{1} - 2812 \beta_{2} - 8060 \beta_{3} + 16702 \beta_{4} + 31508 \beta_{5} ) q^{79} \) \( + ( 2550478 - 106452 \beta_{1} + 7946 \beta_{2} - 7214 \beta_{3} - 25262 \beta_{4} + 7968 \beta_{5} ) q^{80} \) \( + ( -2785839 + 60324 \beta_{1} + 21796 \beta_{2} + 17900 \beta_{3} + 6036 \beta_{4} - 8236 \beta_{5} ) q^{81} \) \( + ( -1466552 - 141682 \beta_{1} + 15328 \beta_{2} + 40562 \beta_{3} - 17674 \beta_{4} + 16624 \beta_{5} ) q^{82} \) \( + ( -775216 + 281862 \beta_{1} - 2684 \beta_{2} + 7304 \beta_{3} - 36814 \beta_{4} - 17412 \beta_{5} ) q^{83} \) \( + ( -2873882 - 362424 \beta_{1} - 9508 \beta_{2} + 10025 \beta_{3} - 33107 \beta_{4} - 12388 \beta_{5} ) q^{84} \) \( + ( -2643474 - 55149 \beta_{1} + 29709 \beta_{2} - 17757 \beta_{3} - 31509 \beta_{4} - 37107 \beta_{5} ) q^{85} \) \( + ( 1782484 + 211837 \beta_{1} + 19472 \beta_{2} + 15895 \beta_{3} - 99465 \beta_{4} + 5890 \beta_{5} ) q^{86} \) \( + ( -4512206 + 434969 \beta_{1} - 11770 \beta_{2} - 25180 \beta_{3} + 69579 \beta_{4} + 14210 \beta_{5} ) q^{87} \) \( + ( -3844224 - 538806 \beta_{1} - 21492 \beta_{2} - 49908 \beta_{3} + 72540 \beta_{4} + 11004 \beta_{5} ) q^{88} \) \( + ( 2060330 - 248 \beta_{1} - 26542 \beta_{2} + 8590 \beta_{3} - 11452 \beta_{4} - 13438 \beta_{5} ) q^{89} \) \( + ( -696048 - 162163 \beta_{1} + 11540 \beta_{2} + 3331 \beta_{3} + 11951 \beta_{4} + 5062 \beta_{5} ) q^{90} \) \( + ( -2173758 - 432151 \beta_{1} - 10022 \beta_{2} - 4390 \beta_{3} - 38503 \beta_{4} - 33182 \beta_{5} ) q^{91} \) \( + ( 1479388 + 128764 \beta_{1} + 62062 \beta_{2} + 74243 \beta_{3} - 45433 \beta_{4} - 84 \beta_{5} ) q^{92} \) \( + ( 609584 + 691424 \beta_{1} - 43904 \beta_{2} - 27236 \beta_{3} + 58780 \beta_{4} + 62488 \beta_{5} ) q^{93} \) \( + ( -1724484 - 164889 \beta_{1} - 77952 \beta_{2} - 78795 \beta_{3} + 96261 \beta_{4} + 44334 \beta_{5} ) q^{94} \) \( + ( -219488 - 48013 \beta_{1} + 6859 \beta_{2} - 6859 \beta_{3} + 6859 \beta_{4} + 6859 \beta_{5} ) q^{95} \) \( + ( -1188292 - 411576 \beta_{1} - 9858 \beta_{2} + 52835 \beta_{3} - 50701 \beta_{4} - 42124 \beta_{5} ) q^{96} \) \( + ( 50334 + 198490 \beta_{1} + 71074 \beta_{2} + 81526 \beta_{3} + 10 \beta_{4} - 17758 \beta_{5} ) q^{97} \) \( + ( 7867414 + 59802 \beta_{1} + 47558 \beta_{2} + 7042 \beta_{3} - 12152 \beta_{4} + 1974 \beta_{5} ) q^{98} \) \( + ( 1899718 - 181701 \beta_{1} + 85661 \beta_{2} + 33575 \beta_{3} + 21631 \beta_{4} - 20611 \beta_{5} ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(6q \) \(\mathstrut +\mathstrut 15q^{2} \) \(\mathstrut +\mathstrut 40q^{3} \) \(\mathstrut +\mathstrut 357q^{4} \) \(\mathstrut +\mathstrut 219q^{5} \) \(\mathstrut +\mathstrut 925q^{6} \) \(\mathstrut +\mathstrut 2105q^{7} \) \(\mathstrut +\mathstrut 5835q^{8} \) \(\mathstrut +\mathstrut 5916q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(6q \) \(\mathstrut +\mathstrut 15q^{2} \) \(\mathstrut +\mathstrut 40q^{3} \) \(\mathstrut +\mathstrut 357q^{4} \) \(\mathstrut +\mathstrut 219q^{5} \) \(\mathstrut +\mathstrut 925q^{6} \) \(\mathstrut +\mathstrut 2105q^{7} \) \(\mathstrut +\mathstrut 5835q^{8} \) \(\mathstrut +\mathstrut 5916q^{9} \) \(\mathstrut +\mathstrut 8212q^{10} \) \(\mathstrut +\mathstrut 7257q^{11} \) \(\mathstrut +\mathstrut 7025q^{12} \) \(\mathstrut +\mathstrut 6850q^{13} \) \(\mathstrut +\mathstrut 8859q^{14} \) \(\mathstrut +\mathstrut 3650q^{15} \) \(\mathstrut -\mathstrut 9159q^{16} \) \(\mathstrut +\mathstrut 5415q^{17} \) \(\mathstrut -\mathstrut 58980q^{18} \) \(\mathstrut -\mathstrut 41154q^{19} \) \(\mathstrut -\mathstrut 67620q^{20} \) \(\mathstrut -\mathstrut 83290q^{21} \) \(\mathstrut -\mathstrut 223870q^{22} \) \(\mathstrut -\mathstrut 720q^{23} \) \(\mathstrut -\mathstrut 151113q^{24} \) \(\mathstrut -\mathstrut 53567q^{25} \) \(\mathstrut -\mathstrut 106527q^{26} \) \(\mathstrut +\mathstrut 199450q^{27} \) \(\mathstrut +\mathstrut 91615q^{28} \) \(\mathstrut +\mathstrut 381624q^{29} \) \(\mathstrut -\mathstrut 137776q^{30} \) \(\mathstrut +\mathstrut 264080q^{31} \) \(\mathstrut +\mathstrut 259155q^{32} \) \(\mathstrut +\mathstrut 496430q^{33} \) \(\mathstrut +\mathstrut 297463q^{34} \) \(\mathstrut +\mathstrut 739767q^{35} \) \(\mathstrut -\mathstrut 147282q^{36} \) \(\mathstrut +\mathstrut 1082300q^{37} \) \(\mathstrut -\mathstrut 102885q^{38} \) \(\mathstrut +\mathstrut 1129528q^{39} \) \(\mathstrut -\mathstrut 524232q^{40} \) \(\mathstrut +\mathstrut 485232q^{41} \) \(\mathstrut -\mathstrut 1753105q^{42} \) \(\mathstrut +\mathstrut 198705q^{43} \) \(\mathstrut -\mathstrut 1729290q^{44} \) \(\mathstrut -\mathstrut 478705q^{45} \) \(\mathstrut -\mathstrut 1565713q^{46} \) \(\mathstrut -\mathstrut 247125q^{47} \) \(\mathstrut -\mathstrut 2937955q^{48} \) \(\mathstrut -\mathstrut 538861q^{49} \) \(\mathstrut -\mathstrut 2396859q^{50} \) \(\mathstrut -\mathstrut 72176q^{51} \) \(\mathstrut -\mathstrut 2647795q^{52} \) \(\mathstrut +\mathstrut 3226770q^{53} \) \(\mathstrut -\mathstrut 1217249q^{54} \) \(\mathstrut -\mathstrut 1490553q^{55} \) \(\mathstrut +\mathstrut 3718965q^{56} \) \(\mathstrut -\mathstrut 274360q^{57} \) \(\mathstrut +\mathstrut 1048405q^{58} \) \(\mathstrut +\mathstrut 2305380q^{59} \) \(\mathstrut +\mathstrut 647440q^{60} \) \(\mathstrut +\mathstrut 585731q^{61} \) \(\mathstrut +\mathstrut 2583780q^{62} \) \(\mathstrut -\mathstrut 3209015q^{63} \) \(\mathstrut +\mathstrut 2380137q^{64} \) \(\mathstrut +\mathstrut 4809420q^{65} \) \(\mathstrut +\mathstrut 2420402q^{66} \) \(\mathstrut -\mathstrut 3264030q^{67} \) \(\mathstrut +\mathstrut 8276595q^{68} \) \(\mathstrut -\mathstrut 1867056q^{69} \) \(\mathstrut +\mathstrut 5936880q^{70} \) \(\mathstrut +\mathstrut 6833682q^{71} \) \(\mathstrut +\mathstrut 3040530q^{72} \) \(\mathstrut -\mathstrut 4160625q^{73} \) \(\mathstrut +\mathstrut 20750550q^{74} \) \(\mathstrut -\mathstrut 11237814q^{75} \) \(\mathstrut -\mathstrut 2448663q^{76} \) \(\mathstrut +\mathstrut 1659195q^{77} \) \(\mathstrut -\mathstrut 1839095q^{78} \) \(\mathstrut -\mathstrut 8680576q^{79} \) \(\mathstrut +\mathstrut 14904048q^{80} \) \(\mathstrut -\mathstrut 16541142q^{81} \) \(\mathstrut -\mathstrut 9240140q^{82} \) \(\mathstrut -\mathstrut 3785040q^{83} \) \(\mathstrut -\mathstrut 18290321q^{84} \) \(\mathstrut -\mathstrut 16108227q^{85} \) \(\mathstrut +\mathstrut 11192544q^{86} \) \(\mathstrut -\mathstrut 25742220q^{87} \) \(\mathstrut -\mathstrut 24666570q^{88} \) \(\mathstrut +\mathstrut 12473466q^{89} \) \(\mathstrut -\mathstrut 4688908q^{90} \) \(\mathstrut -\mathstrut 14289854q^{91} \) \(\mathstrut +\mathstrut 9179655q^{92} \) \(\mathstrut +\mathstrut 5742820q^{93} \) \(\mathstrut -\mathstrut 10757712q^{94} \) \(\mathstrut -\mathstrut 1502121q^{95} \) \(\mathstrut -\mathstrut 8195689q^{96} \) \(\mathstrut +\mathstrut 882830q^{97} \) \(\mathstrut +\mathstrut 47239200q^{98} \) \(\mathstrut +\mathstrut 10726225q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6}\mathstrut -\mathstrut \) \(3\) \(x^{5}\mathstrut -\mathstrut \) \(540\) \(x^{4}\mathstrut +\mathstrut \) \(610\) \(x^{3}\mathstrut +\mathstrut \) \(80412\) \(x^{2}\mathstrut +\mathstrut \) \(7680\) \(x\mathstrut -\mathstrut \) \(2267712\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 2 \nu - 181 \)
\(\beta_{3}\)\(=\)\((\)\( 23 \nu^{5} - 249 \nu^{4} - 7416 \nu^{3} + 52718 \nu^{2} + 454620 \nu - 510288 \)\()/11712\)
\(\beta_{4}\)\(=\)\((\)\( 43 \nu^{5} - 805 \nu^{4} - 13016 \nu^{3} + 219414 \nu^{2} + 775596 \nu - 7796880 \)\()/35136\)
\(\beta_{5}\)\(=\)\((\)\( -3 \nu^{5} + 59 \nu^{4} + 718 \nu^{3} - 14854 \nu^{2} - 11400 \nu + 407544 \)\()/1464\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2}\mathstrut +\mathstrut \) \(2\) \(\beta_{1}\mathstrut +\mathstrut \) \(181\)
\(\nu^{3}\)\(=\)\(-\)\(8\) \(\beta_{5}\mathstrut -\mathstrut \) \(15\) \(\beta_{4}\mathstrut +\mathstrut \) \(\beta_{3}\mathstrut +\mathstrut \) \(8\) \(\beta_{2}\mathstrut +\mathstrut \) \(246\) \(\beta_{1}\mathstrut +\mathstrut \) \(390\)
\(\nu^{4}\)\(=\)\(-\)\(20\) \(\beta_{5}\mathstrut -\mathstrut \) \(141\) \(\beta_{4}\mathstrut +\mathstrut \) \(67\) \(\beta_{3}\mathstrut +\mathstrut \) \(376\) \(\beta_{2}\mathstrut +\mathstrut \) \(1108\) \(\beta_{1}\mathstrut +\mathstrut \) \(45254\)
\(\nu^{5}\)\(=\)\(-\)\(2796\) \(\beta_{5}\mathstrut -\mathstrut \) \(6363\) \(\beta_{4}\mathstrut +\mathstrut \) \(1557\) \(\beta_{3}\mathstrut +\mathstrut \) \(4358\) \(\beta_{2}\mathstrut +\mathstrut \) \(66964\) \(\beta_{1}\mathstrut +\mathstrut \) \(222992\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−15.2573
−14.6171
−6.37480
5.78641
14.6629
18.7999
−13.2573 −51.9225 47.7563 −400.717 688.353 −4.24487 1063.82 508.948 5312.43
1.2 −12.6171 75.4805 31.1902 −68.7314 −952.342 325.908 1221.45 3510.30 867.188
1.3 −4.37480 −63.9788 −108.861 301.974 279.894 615.044 1036.22 1906.28 −1321.08
1.4 7.78641 37.1813 −67.3719 383.672 289.509 1025.33 −1521.24 −804.550 2987.43
1.5 16.6629 67.6316 149.652 −74.3884 1126.94 −1123.58 360.790 2387.04 −1239.53
1.6 20.7999 −24.3922 304.634 77.1906 −507.353 1266.54 3673.97 −1592.02 1605.55
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.6
Significant digits:
Format:

Inner twists

This newform does not admit any (nontrivial) inner twists.

Atkin-Lehner signs

\( p \) Sign
\(19\) \(1\)

Hecke kernels

This newform can be constructed as the kernel of the linear operator \(T_{2}^{6} \) \(\mathstrut -\mathstrut 15 T_{2}^{5} \) \(\mathstrut -\mathstrut 450 T_{2}^{4} \) \(\mathstrut +\mathstrut 4650 T_{2}^{3} \) \(\mathstrut +\mathstrut 64272 T_{2}^{2} \) \(\mathstrut -\mathstrut 289800 T_{2} \) \(\mathstrut -\mathstrut 1974784 \) acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(19))\).