# Properties

 Label 19.3.d Level 19 Weight 3 Character orbit d Rep. character $$\chi_{19}(8,\cdot)$$ Character field $$\Q(\zeta_{6})$$ Dimension 6 Newform subspaces 1 Sturm bound 5 Trace bound 0

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$19$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 19.d (of order $$6$$ and degree $$2$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$19$$ Character field: $$\Q(\zeta_{6})$$ Newform subspaces: $$1$$ Sturm bound: $$5$$ Trace bound: $$0$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{3}(19, [\chi])$$.

Total New Old
Modular forms 10 10 0
Cusp forms 6 6 0
Eisenstein series 4 4 0

## Trace form

 $$6q - 3q^{2} - 9q^{3} + 5q^{4} - 2q^{5} + q^{6} + 14q^{9} + O(q^{10})$$ $$6q - 3q^{2} - 9q^{3} + 5q^{4} - 2q^{5} + q^{6} + 14q^{9} - 60q^{10} + 26q^{11} + 30q^{13} + 54q^{14} - 18q^{15} + q^{16} - 42q^{17} + 25q^{19} + 108q^{20} - 102q^{21} - 39q^{22} + 8q^{23} - 83q^{24} - 17q^{25} - 148q^{26} + 32q^{28} - 12q^{29} + 304q^{30} + 51q^{32} + 123q^{33} - 6q^{34} - 38q^{35} - 54q^{36} - 14q^{38} - 44q^{39} - 96q^{40} + 63q^{41} - 92q^{42} - 34q^{43} - 69q^{44} - 28q^{45} + 58q^{47} - 147q^{48} + 18q^{49} + 132q^{51} + 162q^{52} - 12q^{53} + 29q^{54} - 28q^{55} - 16q^{57} + 172q^{58} - 147q^{59} - 222q^{60} + 58q^{61} - 116q^{62} + 86q^{63} + 166q^{64} + 11q^{66} + 201q^{67} - 84q^{68} - 198q^{70} - 102q^{71} + 210q^{72} + 7q^{73} + 174q^{74} - 173q^{76} - 376q^{77} + 450q^{78} + 134q^{80} + 253q^{81} - 145q^{82} + 146q^{83} - 90q^{85} - 270q^{86} - 568q^{87} - 72q^{89} - 438q^{90} - 216q^{91} + 72q^{92} - 160q^{93} + 558q^{95} + 126q^{96} + 21q^{97} + 411q^{98} - 56q^{99} + O(q^{100})$$

## Decomposition of $$S_{3}^{\mathrm{new}}(19, [\chi])$$ into newform subspaces

Label Dim. $$A$$ Field CM Traces $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$
19.3.d.a $$6$$ $$0.518$$ 6.0.6967728.1 None $$-3$$ $$-9$$ $$-2$$ $$0$$ $$q+(-1-\beta _{5})q^{2}+(\beta _{1}+2\beta _{2}+\beta _{3}+\beta _{5})q^{3}+\cdots$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + 3 T + 8 T^{2} + 15 T^{3} + 20 T^{4} + 27 T^{5} - T^{6} + 108 T^{7} + 320 T^{8} + 960 T^{9} + 2048 T^{10} + 3072 T^{11} + 4096 T^{12}$$
$3$ $$1 + 9 T + 47 T^{2} + 180 T^{3} + 463 T^{4} + 891 T^{5} + 2142 T^{6} + 8019 T^{7} + 37503 T^{8} + 131220 T^{9} + 308367 T^{10} + 531441 T^{11} + 531441 T^{12}$$
$5$ $$1 + 2 T - 27 T^{2} + 114 T^{3} + 238 T^{4} - 2656 T^{5} + 5401 T^{6} - 66400 T^{7} + 148750 T^{8} + 1781250 T^{9} - 10546875 T^{10} + 19531250 T^{11} + 244140625 T^{12}$$
$7$ $$( 1 + 69 T^{2} - 94 T^{3} + 3381 T^{4} + 117649 T^{6} )^{2}$$
$11$ $$( 1 - 13 T + 280 T^{2} - 3149 T^{3} + 33880 T^{4} - 190333 T^{5} + 1771561 T^{6} )^{2}$$
$13$ $$1 - 30 T + 779 T^{2} - 14370 T^{3} + 239570 T^{4} - 3505902 T^{5} + 48205499 T^{6} - 592497438 T^{7} + 6842358770 T^{8} - 69361245330 T^{9} + 635454231659 T^{10} - 4135754755470 T^{11} + 23298085122481 T^{12}$$
$17$ $$1 + 42 T + 333 T^{2} + 6726 T^{3} + 513306 T^{4} + 6959454 T^{5} + 21390509 T^{6} + 2011282206 T^{7} + 42871830426 T^{8} + 162349289094 T^{9} + 2322927227853 T^{10} + 84671743818858 T^{11} + 582622237229761 T^{12}$$
$19$ $$1 - 25 T + 1026 T^{2} - 16967 T^{3} + 370386 T^{4} - 3258025 T^{5} + 47045881 T^{6}$$
$23$ $$1 - 8 T - 1413 T^{2} + 3876 T^{3} + 1337428 T^{4} - 1325942 T^{5} - 814307171 T^{6} - 701423318 T^{7} + 374267188948 T^{8} + 573787105764 T^{9} - 110653422202053 T^{10} - 331412089709192 T^{11} + 21914624432020321 T^{12}$$
$29$ $$1 + 12 T + 1091 T^{2} + 12516 T^{3} + 342470 T^{4} + 25875306 T^{5} + 53097851 T^{6} + 21761132346 T^{7} + 242222524070 T^{8} + 7444808685636 T^{9} + 545768836540451 T^{10} + 5048486799602412 T^{11} + 353814783205469041 T^{12}$$
$31$ $$1 - 1222 T^{2} + 2797835 T^{4} - 2253009856 T^{6} + 2583859377035 T^{8} - 1042232847752902 T^{10} + 787662783788549761 T^{12}$$
$37$ $$1 - 5190 T^{2} + 13520751 T^{4} - 22697438888 T^{6} + 25340064214911 T^{8} - 18229768365849990 T^{10} + 6582952005840035281 T^{12}$$
$41$ $$1 - 63 T + 4739 T^{2} - 215208 T^{3} + 9562553 T^{4} - 413399385 T^{5} + 16368671414 T^{6} - 694924366185 T^{7} + 27021489327833 T^{8} - 1022260433497128 T^{9} + 37840560660804419 T^{10} - 845627536539601263 T^{11} + 22563490300366186081 T^{12}$$
$43$ $$1 + 34 T - 3475 T^{2} - 31338 T^{3} + 9673978 T^{4} - 28877782 T^{5} - 22052754731 T^{6} - 53395018918 T^{7} + 33073405660378 T^{8} - 198098875229562 T^{9} - 40616495964663475 T^{10} + 734790398651664466 T^{11} + 39959630797262576401 T^{12}$$
$47$ $$1 - 58 T - 3429 T^{2} + 99066 T^{3} + 16840372 T^{4} - 217034296 T^{5} - 36013746767 T^{6} - 479428759864 T^{7} + 82175643281332 T^{8} + 1067853745782714 T^{9} - 81648901963178469 T^{10} - 3050749669678142842 T^{11} +$$$$11\!\cdots\!41$$$$T^{12}$$
$53$ $$1 + 12 T + 7659 T^{2} + 91332 T^{3} + 36866070 T^{4} + 510480060 T^{5} + 120117765391 T^{6} + 1433938488540 T^{7} + 290891024879670 T^{8} + 2024315430633828 T^{9} + 476846968860613899 T^{10} + 2098649644386156588 T^{11} +$$$$49\!\cdots\!41$$$$T^{12}$$
$59$ $$1 + 147 T + 16901 T^{2} + 1425606 T^{3} + 106851335 T^{4} + 7433603271 T^{5} + 456697800146 T^{6} + 25876372986351 T^{7} + 1294756199526935 T^{8} + 60132821841811446 T^{9} + 2481581225950629221 T^{10} + 75134162735194285947 T^{11} +$$$$17\!\cdots\!81$$$$T^{12}$$
$61$ $$1 - 58 T - 5011 T^{2} + 428958 T^{3} + 12450718 T^{4} - 1018447028 T^{5} + 4642755289 T^{6} - 3789641391188 T^{7} + 172390661763838 T^{8} + 22100076745145838 T^{9} - 960645345429375091 T^{10} - 41373888876447190858 T^{11} +$$$$26\!\cdots\!21$$$$T^{12}$$
$67$ $$1 - 201 T + 28649 T^{2} - 3051582 T^{3} + 282765251 T^{4} - 23237371929 T^{5} + 1659244019666 T^{6} - 104312562589281 T^{7} + 5698036787496371 T^{8} - 276041170776041358 T^{9} + 11633432894320208009 T^{10} -$$$$36\!\cdots\!49$$$$T^{11} +$$$$81\!\cdots\!61$$$$T^{12}$$
$71$ $$1 + 102 T + 19395 T^{2} + 1624554 T^{3} + 208411794 T^{4} + 13817724054 T^{5} + 1288263668227 T^{6} + 69655146956214 T^{7} + 5296094025765714 T^{8} + 208105828644996234 T^{9} + 12524389738511534595 T^{10} +$$$$33\!\cdots\!02$$$$T^{11} +$$$$16\!\cdots\!41$$$$T^{12}$$
$73$ $$1 - 7 T - 12713 T^{2} + 22960 T^{3} + 94466917 T^{4} + 36786071 T^{5} - 563497579498 T^{6} + 196032972359 T^{7} + 2682694275492997 T^{8} + 3474633835595440 T^{9} - 10252527148249451753 T^{10} - 30083380807924903543 T^{11} +$$$$22\!\cdots\!21$$$$T^{12}$$
$79$ $$1 + 12035 T^{2} + 69730790 T^{4} + 4384013520 T^{5} + 462010537511 T^{6} + 27360628378320 T^{7} + 2716019918693990 T^{8} + 18258404527225461635 T^{10} +$$$$59\!\cdots\!41$$$$T^{12}$$
$83$ $$( 1 - 73 T + 15082 T^{2} - 608183 T^{3} + 103899898 T^{4} - 3464457433 T^{5} + 326940373369 T^{6} )^{2}$$
$89$ $$1 + 72 T + 9723 T^{2} + 575640 T^{3} + 40073838 T^{4} + 8635069188 T^{5} + 306038573143 T^{6} + 68398383038148 T^{7} + 2514322401590958 T^{8} + 286082310328790040 T^{9} + 38275452957841333563 T^{10} +$$$$22\!\cdots\!72$$$$T^{11} +$$$$24\!\cdots\!21$$$$T^{12}$$
$97$ $$1 - 21 T + 12695 T^{2} - 263508 T^{3} + 43921757 T^{4} - 6267660039 T^{5} - 57670086154 T^{6} - 58972413306951 T^{7} + 3888361567466717 T^{8} - 219494787074830932 T^{9} + 99496219480615519895 T^{10} -$$$$15\!\cdots\!29$$$$T^{11} +$$$$69\!\cdots\!41$$$$T^{12}$$