# Properties

 Label 19.4.c Level 19 Weight 4 Character orbit c Rep. character $$\chi_{19}(7,\cdot)$$ Character field $$\Q(\zeta_{3})$$ Dimension 8 Newform subspaces 2 Sturm bound 6 Trace bound 2

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$19$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 19.c (of order $$3$$ and degree $$2$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$19$$ Character field: $$\Q(\zeta_{3})$$ Newform subspaces: $$2$$ Sturm bound: $$6$$ Trace bound: $$2$$ Distinguishing $$T_p$$: $$2$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{4}(19, [\chi])$$.

Total New Old
Modular forms 12 12 0
Cusp forms 8 8 0
Eisenstein series 4 4 0

## Trace form

 $$8q - 3q^{2} - 2q^{3} - 7q^{4} - 5q^{5} - q^{6} + 12q^{7} + 18q^{8} - 4q^{9} + O(q^{10})$$ $$8q - 3q^{2} - 2q^{3} - 7q^{4} - 5q^{5} - q^{6} + 12q^{7} + 18q^{8} - 4q^{9} - 32q^{10} - 10q^{11} + 42q^{12} - 73q^{13} - 142q^{14} + 91q^{15} - 43q^{16} + 187q^{17} + 60q^{18} - 139q^{19} + 668q^{20} - 130q^{21} - 121q^{22} - 115q^{23} - 39q^{24} + 75q^{25} - 320q^{26} - 212q^{27} - 454q^{28} - 137q^{29} - 128q^{30} + 576q^{31} - 465q^{32} - 737q^{33} + 916q^{34} - 324q^{35} + 938q^{36} + 1260q^{37} - 498q^{38} + 1962q^{39} - 690q^{40} - 278q^{41} - 266q^{42} + 43q^{43} + 433q^{44} - 1772q^{45} - 1012q^{46} - 537q^{47} + 39q^{48} - 760q^{49} + 1270q^{50} - 805q^{51} - 1074q^{52} + 623q^{53} + 53q^{54} + 1170q^{55} + 2076q^{56} - 717q^{57} + 3352q^{58} - 316q^{59} - 1006q^{60} - 1077q^{61} - 496q^{62} + 912q^{63} - 2602q^{64} + 186q^{65} - 863q^{66} + 184q^{67} - 1540q^{68} + 1542q^{69} - 1908q^{70} - 1995q^{71} + 1854q^{72} - 116q^{73} + 908q^{74} + 3014q^{75} + 1583q^{76} + 1364q^{77} - 748q^{78} + 483q^{79} + 470q^{80} + 104q^{81} - 143q^{82} - 890q^{83} - 828q^{84} + 1575q^{85} + 654q^{86} - 1710q^{87} - 1470q^{88} - 713q^{89} + 1588q^{90} - 1356q^{91} + 1408q^{92} - 792q^{93} - 2184q^{94} - 2143q^{95} + 286q^{96} + 870q^{97} - 2491q^{98} - 1642q^{99} + O(q^{100})$$

## Decomposition of $$S_{4}^{\mathrm{new}}(19, [\chi])$$ into newform subspaces

Label Dim. $$A$$ Field CM Traces $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$
19.4.c.a $$4$$ $$1.121$$ $$\Q(\sqrt{-3}, \sqrt{55})$$ None $$-2$$ $$0$$ $$14$$ $$-28$$ $$q+\beta _{2}q^{2}+(\beta _{1}+\beta _{3})q^{3}+(7+7\beta _{2})q^{4}+\cdots$$
19.4.c.b $$4$$ $$1.121$$ $$\Q(\sqrt{-3}, \sqrt{73})$$ None $$-1$$ $$-2$$ $$-19$$ $$40$$ $$q-\beta _{1}q^{2}-\beta _{2}q^{3}+(-11+\beta _{1}+10\beta _{2}+\cdots)q^{4}+\cdots$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ ($$( 1 + T - 7 T^{2} + 8 T^{3} + 64 T^{4} )^{2}$$)($$1 + T + 3 T^{2} - 18 T^{3} - 68 T^{4} - 144 T^{5} + 192 T^{6} + 512 T^{7} + 4096 T^{8}$$)
$3$ ($$1 + T^{2} - 728 T^{4} + 729 T^{6} + 531441 T^{8}$$)($$( 1 + T - 26 T^{2} + 27 T^{3} + 729 T^{4} )^{2}$$)
$5$ ($$1 - 14 T - 48 T^{2} + 84 T^{3} + 19411 T^{4} + 10500 T^{5} - 750000 T^{6} - 27343750 T^{7} + 244140625 T^{8}$$)($$1 + 19 T + 39 T^{2} + 1368 T^{3} + 42934 T^{4} + 171000 T^{5} + 609375 T^{6} + 37109375 T^{7} + 244140625 T^{8}$$)
$7$ ($$( 1 + 14 T + 680 T^{2} + 4802 T^{3} + 117649 T^{4} )^{2}$$)($$( 1 - 20 T + 494 T^{2} - 6860 T^{3} + 117649 T^{4} )^{2}$$)
$11$ ($$( 1 - 28 T + 163 T^{2} - 37268 T^{3} + 1771561 T^{4} )^{2}$$)($$( 1 + 33 T + 2770 T^{2} + 43923 T^{3} + 1771561 T^{4} )^{2}$$)
$13$ ($$1 - 28 T - 286 T^{2} + 93072 T^{3} - 5404357 T^{4} + 204479184 T^{5} - 1380467374 T^{6} - 296925982444 T^{7} + 23298085122481 T^{8}$$)($$1 + 101 T + 3713 T^{2} + 211494 T^{3} + 14855738 T^{4} + 464652318 T^{5} + 17921941817 T^{6} + 1071054436673 T^{7} + 23298085122481 T^{8}$$)
$17$ ($$1 - 112 T + 3102 T^{2} + 43008 T^{3} + 3385123 T^{4} + 211298304 T^{5} + 74874739038 T^{6} - 13281842167664 T^{7} + 582622237229761 T^{8}$$)($$1 - 75 T + 2441 T^{2} + 498150 T^{3} - 41635338 T^{4} + 2447410950 T^{5} + 58919805929 T^{6} - 8894090737275 T^{7} + 582622237229761 T^{8}$$)
$19$ ($$1 + 196 T + 18867 T^{2} + 1344364 T^{3} + 47045881 T^{4}$$)($$1 - 57 T + 7942 T^{2} - 390963 T^{3} + 47045881 T^{4}$$)
$23$ ($$1 + 114 T - 11892 T^{2} + 63156 T^{3} + 313254323 T^{4} + 768419052 T^{5} - 1760442791988 T^{6} + 205331403406782 T^{7} + 21914624432020321 T^{8}$$)($$1 + T - 19059 T^{2} - 5274 T^{3} + 215235544 T^{4} - 64168758 T^{5} - 2821416008451 T^{6} + 1801152661463 T^{7} + 21914624432020321 T^{8}$$)
$29$ ($$1 + 222 T - 9120 T^{2} + 2136972 T^{3} + 1614216419 T^{4} + 52118610108 T^{5} - 5424788687520 T^{6} + 3220586406642918 T^{7} + 353814783205469041 T^{8}$$)($$1 - 85 T - 12681 T^{2} + 2454120 T^{3} - 374785010 T^{4} + 59853532680 T^{5} - 7542954533601 T^{6} - 1233107407948865 T^{7} + 353814783205469041 T^{8}$$)
$31$ ($$( 1 - 266 T + 74576 T^{2} - 7924406 T^{3} + 887503681 T^{4} )^{2}$$)($$( 1 - 22 T + 59046 T^{2} - 655402 T^{3} + 887503681 T^{4} )^{2}$$)
$37$ ($$( 1 - 182 T + 106892 T^{2} - 9218846 T^{3} + 2565726409 T^{4} )^{2}$$)($$( 1 - 448 T + 127830 T^{2} - 22692544 T^{3} + 2565726409 T^{4} )^{2}$$)
$41$ ($$1 + 154 T - 56475 T^{2} - 8878254 T^{3} + 45961804 T^{4} - 611898143934 T^{5} - 268262137010475 T^{6} + 50416817896669994 T^{7} + 22563490300366186081 T^{8}$$)($$1 + 124 T - 126237 T^{2} + 467604 T^{3} + 14244408232 T^{4} + 32227735284 T^{5} - 599638909071117 T^{6} + 40595359864851164 T^{7} + 22563490300366186081 T^{8}$$)
$43$ ($$1 + 268 T - 8126 T^{2} - 21189152 T^{3} - 5639871317 T^{4} - 1684685908064 T^{5} - 51367396136174 T^{6} + 134694819999073924 T^{7} + 39959630797262576401 T^{8}$$)($$1 - 311 T - 78425 T^{2} - 5017052 T^{3} + 16664761720 T^{4} - 398890753364 T^{5} - 495752897117825 T^{6} - 156306302312358173 T^{7} + 39959630797262576401 T^{8}$$)
$47$ ($$1 + 126 T - 183364 T^{2} - 1059156 T^{3} + 27269068323 T^{4} - 109964753388 T^{5} - 1976520039586756 T^{6} + 141010439610948642 T^{7} +$$$$11\!\cdots\!41$$$$T^{8}$$)($$1 + 411 T - 53197 T^{2} + 5947992 T^{3} + 21019305612 T^{4} + 617538373416 T^{5} - 573421917856813 T^{6} + 459962624445237237 T^{7} +$$$$11\!\cdots\!41$$$$T^{8}$$)
$53$ ($$1 - 884 T + 331458 T^{2} - 134583696 T^{3} + 63993013963 T^{4} - 20036416909392 T^{5} + 7346554811096082 T^{6} - 2916991015153085572 T^{7} +$$$$49\!\cdots\!41$$$$T^{8}$$)($$1 + 261 T - 67795 T^{2} - 42239718 T^{3} - 13832852190 T^{4} - 6288522496686 T^{5} - 1502632862740555 T^{6} + 861238297460356713 T^{7} +$$$$49\!\cdots\!41$$$$T^{8}$$)
$59$ ($$1 + 112 T - 392055 T^{2} - 689808 T^{3} + 118943542984 T^{4} - 141672077232 T^{5} - 16537089116622255 T^{6} + 970255531689353168 T^{7} +$$$$17\!\cdots\!81$$$$T^{8}$$)($$1 + 204 T - 189673 T^{2} - 36611676 T^{3} + 2767015416 T^{4} - 7519269405204 T^{5} - 8000508357289393 T^{6} + 1767251147005607556 T^{7} +$$$$17\!\cdots\!81$$$$T^{8}$$)
$61$ ($$1 + 546 T - 184120 T^{2} + 15437604 T^{3} + 113364517539 T^{4} + 3504042793524 T^{5} - 9485931327347320 T^{6} + 6385003766687440986 T^{7} +$$$$26\!\cdots\!21$$$$T^{8}$$)($$1 + 531 T - 241597 T^{2} + 36955476 T^{3} + 158592815262 T^{4} + 8388190897956 T^{5} - 12447167884494517 T^{6} + 6209591575294928871 T^{7} +$$$$26\!\cdots\!21$$$$T^{8}$$)
$67$ ($$1 - 740 T + 27469 T^{2} + 60232300 T^{3} + 15380056192 T^{4} + 18115647244900 T^{5} + 2484801299800261 T^{6} - 20132835453258260780 T^{7} +$$$$81\!\cdots\!61$$$$T^{8}$$)($$1 + 556 T + 109279 T^{2} - 223327964 T^{3} - 143492232488 T^{4} - 67168788436532 T^{5} + 9885201545046151 T^{6} + 15126833124339990532 T^{7} +$$$$81\!\cdots\!61$$$$T^{8}$$)
$71$ ($$1 + 432 T - 478834 T^{2} - 21757248 T^{3} + 247939283379 T^{4} - 7787158388928 T^{5} - 61338771351028114 T^{6} + 19806552310369981392 T^{7} +$$$$16\!\cdots\!41$$$$T^{8}$$)($$1 + 1563 T + 1116569 T^{2} + 954333414 T^{3} + 756871198320 T^{4} + 341566426538154 T^{5} + 143032805917387049 T^{6} + 71661206622935835453 T^{7} +$$$$16\!\cdots\!41$$$$T^{8}$$)
$73$ ($$1 + 350 T - 684179 T^{2} + 10025750 T^{3} + 451742200252 T^{4} + 3900187187750 T^{5} - 103539699608181731 T^{6} + 20605055347893769550 T^{7} +$$$$22\!\cdots\!21$$$$T^{8}$$)($$1 - 234 T - 595639 T^{2} + 29867526 T^{3} + 250378414884 T^{4} + 11618975361942 T^{5} - 90140567212553671 T^{6} - 13775951289734691642 T^{7} +$$$$22\!\cdots\!21$$$$T^{8}$$)
$79$ ($$1 - 152 T - 796270 T^{2} + 25339008 T^{3} + 416895123299 T^{4} + 12493119165312 T^{5} - 193563248207706670 T^{6} - 18217442589357984488 T^{7} +$$$$59\!\cdots\!41$$$$T^{8}$$)($$1 - 331 T - 886369 T^{2} - 3261012 T^{3} + 694771263500 T^{4} - 1607806095468 T^{5} - 215465184862693249 T^{6} - 39670878270246663589 T^{7} +$$$$59\!\cdots\!41$$$$T^{8}$$)
$83$ ($$( 1 + 1904 T + 1982503 T^{2} + 1088682448 T^{3} + 326940373369 T^{4} )^{2}$$)($$( 1 - 1459 T + 1202686 T^{2} - 834237233 T^{3} + 326940373369 T^{4} )^{2}$$)
$89$ ($$1 + 112 T - 805650 T^{2} - 66275328 T^{3} + 163616999539 T^{4} - 46722051704832 T^{5} - 400392977062729650 T^{6} + 39239917215238343408 T^{7} +$$$$24\!\cdots\!21$$$$T^{8}$$)($$1 + 601 T - 1025139 T^{2} - 14182398 T^{3} + 1170321796870 T^{4} - 9998150935662 T^{5} - 509474903634468579 T^{6} +$$$$21\!\cdots\!09$$$$T^{7} +$$$$24\!\cdots\!21$$$$T^{8}$$)
$97$ ($$1 - 546 T - 1390339 T^{2} + 74742486 T^{3} + 1745825858028 T^{4} + 68215448925078 T^{5} - 1158113464360980931 T^{6} -$$$$41\!\cdots\!82$$$$T^{7} +$$$$69\!\cdots\!41$$$$T^{8}$$)($$1 - 324 T - 376477 T^{2} + 435421332 T^{3} - 696983774568 T^{4} + 397397293340436 T^{5} - 313594801499655133 T^{6} -$$$$24\!\cdots\!08$$$$T^{7} +$$$$69\!\cdots\!41$$$$T^{8}$$)