# Properties

 Label 19.2.e Level 19 Weight 2 Character orbit e Rep. character $$\chi_{19}(4,\cdot)$$ Character field $$\Q(\zeta_{9})$$ Dimension 6 Newform subspaces 1 Sturm bound 3 Trace bound 0

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 19.e (of order $$9$$ and degree $$6$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$19$$ Character field: $$\Q(\zeta_{9})$$ Newform subspaces: $$1$$ Sturm bound: $$3$$ Trace bound: $$0$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(19, [\chi])$$.

Total New Old
Modular forms 18 18 0
Cusp forms 6 6 0
Eisenstein series 12 12 0

## Trace form

 $$6q - 6q^{2} - 3q^{3} - 6q^{5} + 3q^{6} + 6q^{8} + 3q^{9} + O(q^{10})$$ $$6q - 6q^{2} - 3q^{3} - 6q^{5} + 3q^{6} + 6q^{8} + 3q^{9} + 9q^{10} + 3q^{12} - 3q^{13} - 3q^{14} + 3q^{15} - 18q^{16} + 3q^{17} - 6q^{18} - 12q^{19} - 6q^{20} + 6q^{23} + 15q^{24} + 15q^{26} + 6q^{27} + 6q^{28} - 3q^{29} + 9q^{31} + 9q^{32} - 9q^{33} + 6q^{35} - 24q^{36} - 15q^{38} - 24q^{39} + 21q^{41} - 3q^{42} - 3q^{43} + 9q^{44} - 15q^{45} - 18q^{46} - 3q^{47} - 3q^{48} + 15q^{49} - 15q^{50} + 3q^{51} + 15q^{52} - 3q^{53} + 30q^{54} + 18q^{55} - 6q^{56} + 24q^{57} + 36q^{58} + 12q^{59} - 6q^{60} - 12q^{61} - 12q^{62} + 12q^{63} - 12q^{64} - 12q^{65} - 9q^{66} - 30q^{67} - 15q^{68} - 12q^{69} - 9q^{70} - 6q^{71} - 12q^{72} - 12q^{73} + 15q^{74} + 30q^{75} + 36q^{76} - 18q^{77} + 15q^{78} - 39q^{79} + 3q^{80} + 6q^{81} - 54q^{82} + 3q^{84} + 24q^{86} - 21q^{87} + 9q^{88} - 12q^{89} + 18q^{90} + 15q^{91} + 42q^{92} + 9q^{93} + 18q^{94} + 39q^{95} + 18q^{96} + 18q^{97} - 9q^{98} + 9q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(19, [\chi])$$ into newform subspaces

Label Dim. $$A$$ Field CM Traces $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$
19.2.e.a $$6$$ $$0.152$$ $$\Q(\zeta_{18})$$ None $$-6$$ $$-3$$ $$-6$$ $$0$$ $$q+(-1+\zeta_{18}-\zeta_{18}^{2})q^{2}+(-1+\zeta_{18}^{2}+\cdots)q^{3}+\cdots$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + 6 T + 18 T^{2} + 36 T^{3} + 54 T^{4} + 69 T^{5} + 91 T^{6} + 138 T^{7} + 216 T^{8} + 288 T^{9} + 288 T^{10} + 192 T^{11} + 64 T^{12}$$
$3$ $$1 + 3 T + 3 T^{2} + T^{3} - 12 T^{4} - 30 T^{5} - 35 T^{6} - 90 T^{7} - 108 T^{8} + 27 T^{9} + 243 T^{10} + 729 T^{11} + 729 T^{12}$$
$5$ $$1 + 6 T + 18 T^{2} + 45 T^{3} + 81 T^{4} + 87 T^{5} + 109 T^{6} + 435 T^{7} + 2025 T^{8} + 5625 T^{9} + 11250 T^{10} + 18750 T^{11} + 15625 T^{12}$$
$7$ $$1 - 18 T^{2} + 2 T^{3} + 198 T^{4} - 18 T^{5} - 1581 T^{6} - 126 T^{7} + 9702 T^{8} + 686 T^{9} - 43218 T^{10} + 117649 T^{12}$$
$11$ $$1 - 24 T^{2} - 18 T^{3} + 312 T^{4} + 216 T^{5} - 3593 T^{6} + 2376 T^{7} + 37752 T^{8} - 23958 T^{9} - 351384 T^{10} + 1771561 T^{12}$$
$13$ $$1 + 3 T + 24 T^{2} + 26 T^{3} + 315 T^{4} + 261 T^{5} + 4905 T^{6} + 3393 T^{7} + 53235 T^{8} + 57122 T^{9} + 685464 T^{10} + 1113879 T^{11} + 4826809 T^{12}$$
$17$ $$1 - 3 T - 72 T^{3} - 117 T^{4} + 1275 T^{5} + 1369 T^{6} + 21675 T^{7} - 33813 T^{8} - 353736 T^{9} - 4259571 T^{11} + 24137569 T^{12}$$
$19$ $$1 + 12 T + 78 T^{2} + 385 T^{3} + 1482 T^{4} + 4332 T^{5} + 6859 T^{6}$$
$23$ $$1 - 6 T + 36 T^{2} - 54 T^{3} + 576 T^{4} + 516 T^{5} + 4969 T^{6} + 11868 T^{7} + 304704 T^{8} - 657018 T^{9} + 10074276 T^{10} - 38618058 T^{11} + 148035889 T^{12}$$
$29$ $$1 + 3 T + 36 T^{2} + 378 T^{3} + 1872 T^{4} + 9921 T^{5} + 94159 T^{6} + 287709 T^{7} + 1574352 T^{8} + 9219042 T^{9} + 25462116 T^{10} + 61533447 T^{11} + 594823321 T^{12}$$
$31$ $$1 - 9 T - 18 T^{2} + 119 T^{3} + 2187 T^{4} - 3402 T^{5} - 67065 T^{6} - 105462 T^{7} + 2101707 T^{8} + 3545129 T^{9} - 16623378 T^{10} - 257662359 T^{11} + 887503681 T^{12}$$
$37$ $$( 1 + 90 T^{2} - 17 T^{3} + 3330 T^{4} + 50653 T^{6} )^{2}$$
$41$ $$1 - 21 T + 162 T^{2} - 180 T^{3} - 4707 T^{4} + 28401 T^{5} - 103463 T^{6} + 1164441 T^{7} - 7912467 T^{8} - 12405780 T^{9} + 457773282 T^{10} - 2432980221 T^{11} + 4750104241 T^{12}$$
$43$ $$1 + 3 T + 60 T^{2} + 8 T^{3} + 2691 T^{4} + 2799 T^{5} + 147141 T^{6} + 120357 T^{7} + 4975659 T^{8} + 636056 T^{9} + 205128060 T^{10} + 441025329 T^{11} + 6321363049 T^{12}$$
$47$ $$1 + 3 T + 54 T^{2} + 306 T^{3} + 4635 T^{4} + 25239 T^{5} + 190171 T^{6} + 1186233 T^{7} + 10238715 T^{8} + 31769838 T^{9} + 263502774 T^{10} + 688035021 T^{11} + 10779215329 T^{12}$$
$53$ $$1 + 3 T + 234 T^{3} - 1521 T^{4} - 27771 T^{5} - 29411 T^{6} - 1471863 T^{7} - 4272489 T^{8} + 34837218 T^{9} + 1254586479 T^{11} + 22164361129 T^{12}$$
$59$ $$1 - 12 T + 18 T^{2} + 1080 T^{3} - 7614 T^{4} - 29982 T^{5} + 754273 T^{6} - 1768938 T^{7} - 26504334 T^{8} + 221809320 T^{9} + 218112498 T^{10} - 8579091588 T^{11} + 42180533641 T^{12}$$
$61$ $$1 + 12 T + 24 T^{2} - 586 T^{3} - 4140 T^{4} + 44676 T^{5} + 736335 T^{6} + 2725236 T^{7} - 15404940 T^{8} - 133010866 T^{9} + 332300184 T^{10} + 10135155612 T^{11} + 51520374361 T^{12}$$
$67$ $$1 + 30 T + 348 T^{2} + 1322 T^{3} - 6408 T^{4} - 55008 T^{5} - 101691 T^{6} - 3685536 T^{7} - 28765512 T^{8} + 397608686 T^{9} + 7012590108 T^{10} + 40503753210 T^{11} + 90458382169 T^{12}$$
$71$ $$1 + 6 T - 36 T^{2} + 594 T^{3} + 3240 T^{4} + 14892 T^{5} + 665785 T^{6} + 1057332 T^{7} + 16332840 T^{8} + 212599134 T^{9} - 914820516 T^{10} + 10825376106 T^{11} + 128100283921 T^{12}$$
$73$ $$1 + 12 T + 96 T^{2} + 512 T^{3} + 7776 T^{4} + 102924 T^{5} + 981639 T^{6} + 7513452 T^{7} + 41438304 T^{8} + 199176704 T^{9} + 2726231136 T^{10} + 24876859116 T^{11} + 151334226289 T^{12}$$
$79$ $$1 + 39 T + 708 T^{2} + 8198 T^{3} + 67068 T^{4} + 403623 T^{5} + 2596617 T^{6} + 31886217 T^{7} + 418571388 T^{8} + 4041933722 T^{9} + 27576657348 T^{10} + 120005199561 T^{11} + 243087455521 T^{12}$$
$83$ $$1 - 60 T^{2} + 918 T^{3} - 1380 T^{4} - 27540 T^{5} + 1055455 T^{6} - 2285820 T^{7} - 9506820 T^{8} + 524900466 T^{9} - 2847499260 T^{10} + 326940373369 T^{12}$$
$89$ $$1 + 12 T + 54 T^{2} + 1035 T^{3} - 279 T^{4} - 69891 T^{5} + 3961 T^{6} - 6220299 T^{7} - 2209959 T^{8} + 729642915 T^{9} + 3388081014 T^{10} + 67008713388 T^{11} + 496981290961 T^{12}$$
$97$ $$1 - 18 T + 234 T^{2} - 3310 T^{3} + 39204 T^{4} - 371520 T^{5} + 3748107 T^{6} - 36037440 T^{7} + 368870436 T^{8} - 3020947630 T^{9} + 20715851754 T^{10} - 154572124626 T^{11} + 832972004929 T^{12}$$