# Properties

 Label 9.6.c Level 9 Weight 6 Character orbit c Rep. character $$\chi_{9}(4,\cdot)$$ Character field $$\Q(\zeta_{3})$$ Dimension 8 Newform subspaces 1 Sturm bound 6 Trace bound 0

# Related objects

## Defining parameters

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

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{6}(9, [\chi])$$.

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

## Trace form

 $$8q + 3q^{2} - 12q^{3} - 49q^{4} + 78q^{5} + 171q^{6} + 28q^{7} - 750q^{8} - 414q^{9} + O(q^{10})$$ $$8q + 3q^{2} - 12q^{3} - 49q^{4} + 78q^{5} + 171q^{6} + 28q^{7} - 750q^{8} - 414q^{9} + 60q^{10} + 444q^{11} + 2724q^{12} - 182q^{13} + 1392q^{14} - 2052q^{15} - 289q^{16} - 4356q^{17} - 8100q^{18} + 952q^{19} + 6684q^{20} + 8670q^{21} + 1011q^{22} + 8844q^{23} + 549q^{24} - 1654q^{25} - 24888q^{26} - 10152q^{27} - 1604q^{28} + 12018q^{29} + 22104q^{30} + 1132q^{31} + 8703q^{32} - 8820q^{33} + 10125q^{34} - 16224q^{35} - 14589q^{36} - 15176q^{37} - 11145q^{38} + 8220q^{39} - 8736q^{40} + 1248q^{41} + 10098q^{42} - 6092q^{43} + 49530q^{44} + 43038q^{45} + 45960q^{46} - 60q^{47} - 57405q^{48} + 9090q^{49} - 57057q^{50} - 9396q^{51} - 32510q^{52} + 20952q^{53} + 8181q^{54} - 36120q^{55} - 61170q^{56} - 104298q^{57} + 8328q^{58} + 2076q^{59} + 88308q^{60} + 48142q^{61} + 241764q^{62} + 133524q^{63} - 20926q^{64} - 13146q^{65} - 100998q^{66} - 7148q^{67} - 123129q^{68} - 125982q^{69} - 654q^{70} - 71856q^{71} + 35451q^{72} + 122452q^{73} - 160320q^{74} - 18732q^{75} - 49571q^{76} + 39534q^{77} + 181422q^{78} - 59516q^{79} + 124512q^{80} + 194562q^{81} - 233598q^{82} + 117696q^{83} + 108354q^{84} + 28836q^{85} - 15915q^{86} - 142956q^{87} + 104523q^{88} - 451728q^{89} - 539892q^{90} + 111392q^{91} + 134034q^{92} + 113898q^{93} + 169464q^{94} + 294888q^{95} + 42768q^{96} + 33976q^{97} + 57654q^{98} + 33696q^{99} + O(q^{100})$$

## Decomposition of $$S_{6}^{\mathrm{new}}(9, [\chi])$$ into newform subspaces

Label Dim. $$A$$ Field CM Traces $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$
9.6.c.a $$8$$ $$1.443$$ $$\mathbb{Q}[x]/(x^{8} + \cdots)$$ None $$3$$ $$-12$$ $$78$$ $$28$$ $$q+(1-\beta _{1}+\beta _{4}+\beta _{5})q^{2}+(-3+3\beta _{1}+\cdots)q^{3}+\cdots$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - 3 T - 35 T^{2} + 300 T^{3} - 272 T^{4} - 7056 T^{5} + 47392 T^{6} + 37824 T^{7} - 1500608 T^{8} + 1210368 T^{9} + 48529408 T^{10} - 231211008 T^{11} - 285212672 T^{12} + 10066329600 T^{13} - 37580963840 T^{14} - 103079215104 T^{15} + 1099511627776 T^{16}$$
$3$ $$1 + 12 T + 279 T^{2} + 6156 T^{3} + 29160 T^{4} + 1495908 T^{5} + 16474671 T^{6} + 172186884 T^{7} + 3486784401 T^{8}$$
$5$ $$1 - 78 T - 2381 T^{2} + 191094 T^{3} + 7253005 T^{4} - 66450636 T^{5} - 21330994286 T^{6} + 411629383632 T^{7} - 16622006693174 T^{8} + 1286341823850000 T^{9} - 208310491074218750 T^{10} - 2027912475585937500 T^{11} +$$$$69\!\cdots\!25$$$$T^{12} +$$$$56\!\cdots\!50$$$$T^{13} -$$$$22\!\cdots\!25$$$$T^{14} -$$$$22\!\cdots\!50$$$$T^{15} +$$$$90\!\cdots\!25$$$$T^{16}$$
$7$ $$1 - 28 T - 37767 T^{2} - 688508 T^{3} + 752155661 T^{4} + 29978584680 T^{5} - 6360743348222 T^{6} - 326958516789136 T^{7} + 37987499418745374 T^{8} - 5495191791675008752 T^{9} -$$$$17\!\cdots\!78$$$$T^{10} +$$$$14\!\cdots\!40$$$$T^{11} +$$$$60\!\cdots\!61$$$$T^{12} -$$$$92\!\cdots\!56$$$$T^{13} -$$$$85\!\cdots\!83$$$$T^{14} -$$$$10\!\cdots\!04$$$$T^{15} +$$$$63\!\cdots\!01$$$$T^{16}$$
$11$ $$1 - 444 T - 421874 T^{2} + 122372328 T^{3} + 147603288625 T^{4} - 24043789512072 T^{5} - 33881306185300610 T^{6} + 1343287267601416260 T^{7} +$$$$64\!\cdots\!56$$$$T^{8} +$$$$21\!\cdots\!60$$$$T^{9} -$$$$87\!\cdots\!10$$$$T^{10} -$$$$10\!\cdots\!72$$$$T^{11} +$$$$99\!\cdots\!25$$$$T^{12} +$$$$13\!\cdots\!28$$$$T^{13} -$$$$73\!\cdots\!74$$$$T^{14} -$$$$12\!\cdots\!44$$$$T^{15} +$$$$45\!\cdots\!01$$$$T^{16}$$
$13$ $$1 + 182 T - 1009641 T^{2} - 162494414 T^{3} + 553084541681 T^{4} + 66263508549516 T^{5} - 217064082764017418 T^{6} - 12452829446314121656 T^{7} +$$$$75\!\cdots\!98$$$$T^{8} -$$$$46\!\cdots\!08$$$$T^{9} -$$$$29\!\cdots\!82$$$$T^{10} +$$$$33\!\cdots\!12$$$$T^{11} +$$$$10\!\cdots\!81$$$$T^{12} -$$$$11\!\cdots\!02$$$$T^{13} -$$$$26\!\cdots\!09$$$$T^{14} +$$$$17\!\cdots\!74$$$$T^{15} +$$$$36\!\cdots\!01$$$$T^{16}$$
$17$ $$( 1 + 2178 T + 4502417 T^{2} + 6473305458 T^{3} + 7835841277908 T^{4} + 9191168067679506 T^{5} + 9076845209277885233 T^{6} +$$$$62\!\cdots\!54$$$$T^{7} +$$$$40\!\cdots\!01$$$$T^{8} )^{2}$$
$19$ $$( 1 - 476 T + 5379895 T^{2} - 1785507140 T^{3} + 16120413258280 T^{4} - 4421092443846860 T^{5} + 32984492705012310895 T^{6} -$$$$72\!\cdots\!24$$$$T^{7} +$$$$37\!\cdots\!01$$$$T^{8} )^{2}$$
$23$ $$1 - 8844 T + 29643313 T^{2} - 58247545548 T^{3} + 172348926961837 T^{4} - 574153269020383320 T^{5} +$$$$14\!\cdots\!02$$$$T^{6} -$$$$52\!\cdots\!88$$$$T^{7} +$$$$17\!\cdots\!06$$$$T^{8} -$$$$33\!\cdots\!84$$$$T^{9} +$$$$60\!\cdots\!98$$$$T^{10} -$$$$15\!\cdots\!40$$$$T^{11} +$$$$29\!\cdots\!37$$$$T^{12} -$$$$64\!\cdots\!64$$$$T^{13} +$$$$21\!\cdots\!37$$$$T^{14} -$$$$40\!\cdots\!08$$$$T^{15} +$$$$29\!\cdots\!01$$$$T^{16}$$
$29$ $$1 - 12018 T + 15291415 T^{2} + 58796364810 T^{3} + 2512265879646529 T^{4} - 12711396421291774644 T^{5} -$$$$20\!\cdots\!74$$$$T^{6} -$$$$10\!\cdots\!36$$$$T^{7} +$$$$21\!\cdots\!82$$$$T^{8} -$$$$21\!\cdots\!64$$$$T^{9} -$$$$86\!\cdots\!74$$$$T^{10} -$$$$10\!\cdots\!56$$$$T^{11} +$$$$44\!\cdots\!29$$$$T^{12} +$$$$21\!\cdots\!90$$$$T^{13} +$$$$11\!\cdots\!15$$$$T^{14} -$$$$18\!\cdots\!82$$$$T^{15} +$$$$31\!\cdots\!01$$$$T^{16}$$
$31$ $$1 - 1132 T - 60658635 T^{2} + 62431167700 T^{3} + 1166392900733297 T^{4} - 126919079777751576 T^{5} -$$$$56\!\cdots\!38$$$$T^{6} -$$$$20\!\cdots\!72$$$$T^{7} +$$$$28\!\cdots\!06$$$$T^{8} -$$$$60\!\cdots\!72$$$$T^{9} -$$$$46\!\cdots\!38$$$$T^{10} -$$$$29\!\cdots\!76$$$$T^{11} +$$$$78\!\cdots\!97$$$$T^{12} +$$$$12\!\cdots\!00$$$$T^{13} -$$$$33\!\cdots\!35$$$$T^{14} -$$$$17\!\cdots\!32$$$$T^{15} +$$$$45\!\cdots\!01$$$$T^{16}$$
$37$ $$( 1 + 7588 T + 181007992 T^{2} + 1702118220604 T^{3} + 15462039197985406 T^{4} +$$$$11\!\cdots\!28$$$$T^{5} +$$$$87\!\cdots\!08$$$$T^{6} +$$$$25\!\cdots\!84$$$$T^{7} +$$$$23\!\cdots\!01$$$$T^{8} )^{2}$$
$41$ $$1 - 1248 T - 412882754 T^{2} + 256860139200 T^{3} + 101750473497083809 T^{4} - 31706402029277721984 T^{5} -$$$$17\!\cdots\!98$$$$T^{6} +$$$$13\!\cdots\!52$$$$T^{7} +$$$$23\!\cdots\!00$$$$T^{8} +$$$$15\!\cdots\!52$$$$T^{9} -$$$$23\!\cdots\!98$$$$T^{10} -$$$$49\!\cdots\!84$$$$T^{11} +$$$$18\!\cdots\!09$$$$T^{12} +$$$$53\!\cdots\!00$$$$T^{13} -$$$$99\!\cdots\!54$$$$T^{14} -$$$$34\!\cdots\!48$$$$T^{15} +$$$$32\!\cdots\!01$$$$T^{16}$$
$43$ $$1 + 6092 T - 493633434 T^{2} - 1717030281896 T^{3} + 152938627936979465 T^{4} +$$$$29\!\cdots\!88$$$$T^{5} -$$$$33\!\cdots\!22$$$$T^{6} -$$$$15\!\cdots\!68$$$$T^{7} +$$$$57\!\cdots\!20$$$$T^{8} -$$$$23\!\cdots\!24$$$$T^{9} -$$$$72\!\cdots\!78$$$$T^{10} +$$$$92\!\cdots\!16$$$$T^{11} +$$$$71\!\cdots\!65$$$$T^{12} -$$$$11\!\cdots\!28$$$$T^{13} -$$$$49\!\cdots\!66$$$$T^{14} +$$$$90\!\cdots\!44$$$$T^{15} +$$$$21\!\cdots\!01$$$$T^{16}$$
$47$ $$1 + 60 T - 695011823 T^{2} + 3293074754652 T^{3} + 272693815319492413 T^{4} -$$$$15\!\cdots\!00$$$$T^{5} -$$$$70\!\cdots\!78$$$$T^{6} +$$$$17\!\cdots\!56$$$$T^{7} +$$$$15\!\cdots\!82$$$$T^{8} +$$$$41\!\cdots\!92$$$$T^{9} -$$$$37\!\cdots\!22$$$$T^{10} -$$$$18\!\cdots\!00$$$$T^{11} +$$$$75\!\cdots\!13$$$$T^{12} +$$$$20\!\cdots\!64$$$$T^{13} -$$$$10\!\cdots\!27$$$$T^{14} +$$$$20\!\cdots\!80$$$$T^{15} +$$$$76\!\cdots\!01$$$$T^{16}$$
$53$ $$( 1 - 10476 T + 1064413976 T^{2} - 16431260960628 T^{3} + 549720542476264830 T^{4} -$$$$68\!\cdots\!04$$$$T^{5} +$$$$18\!\cdots\!24$$$$T^{6} -$$$$76\!\cdots\!32$$$$T^{7} +$$$$30\!\cdots\!01$$$$T^{8} )^{2}$$
$59$ $$1 - 2076 T - 2248193858 T^{2} + 15695201332392 T^{3} + 2863253985413117089 T^{4} -$$$$21\!\cdots\!96$$$$T^{5} -$$$$25\!\cdots\!86$$$$T^{6} +$$$$73\!\cdots\!56$$$$T^{7} +$$$$18\!\cdots\!76$$$$T^{8} +$$$$52\!\cdots\!44$$$$T^{9} -$$$$12\!\cdots\!86$$$$T^{10} -$$$$77\!\cdots\!04$$$$T^{11} +$$$$74\!\cdots\!89$$$$T^{12} +$$$$29\!\cdots\!08$$$$T^{13} -$$$$30\!\cdots\!58$$$$T^{14} -$$$$19\!\cdots\!24$$$$T^{15} +$$$$68\!\cdots\!01$$$$T^{16}$$
$61$ $$1 - 48142 T - 755431725 T^{2} + 43590251951830 T^{3} + 1116615498545065565 T^{4} -$$$$17\!\cdots\!08$$$$T^{5} -$$$$17\!\cdots\!86$$$$T^{6} +$$$$66\!\cdots\!20$$$$T^{7} +$$$$16\!\cdots\!34$$$$T^{8} +$$$$56\!\cdots\!20$$$$T^{9} -$$$$12\!\cdots\!86$$$$T^{10} -$$$$10\!\cdots\!08$$$$T^{11} +$$$$56\!\cdots\!65$$$$T^{12} +$$$$18\!\cdots\!30$$$$T^{13} -$$$$27\!\cdots\!25$$$$T^{14} -$$$$14\!\cdots\!42$$$$T^{15} +$$$$25\!\cdots\!01$$$$T^{16}$$
$67$ $$1 + 7148 T - 3268771122 T^{2} - 16753922588168 T^{3} + 4825996342110423185 T^{4} +$$$$80\!\cdots\!92$$$$T^{5} -$$$$77\!\cdots\!58$$$$T^{6} +$$$$18\!\cdots\!60$$$$T^{7} +$$$$12\!\cdots\!20$$$$T^{8} +$$$$25\!\cdots\!20$$$$T^{9} -$$$$14\!\cdots\!42$$$$T^{10} +$$$$19\!\cdots\!56$$$$T^{11} +$$$$16\!\cdots\!85$$$$T^{12} -$$$$75\!\cdots\!76$$$$T^{13} -$$$$19\!\cdots\!78$$$$T^{14} +$$$$58\!\cdots\!64$$$$T^{15} +$$$$11\!\cdots\!01$$$$T^{16}$$
$71$ $$( 1 + 35928 T + 5285725772 T^{2} + 127659705990840 T^{3} + 12554595869440882758 T^{4} +$$$$23\!\cdots\!40$$$$T^{5} +$$$$17\!\cdots\!72$$$$T^{6} +$$$$21\!\cdots\!28$$$$T^{7} +$$$$10\!\cdots\!01$$$$T^{8} )^{2}$$
$73$ $$( 1 - 61226 T + 6621012721 T^{2} - 212085627051026 T^{3} + 16124371439127547300 T^{4} -$$$$43\!\cdots\!18$$$$T^{5} +$$$$28\!\cdots\!29$$$$T^{6} -$$$$54\!\cdots\!82$$$$T^{7} +$$$$18\!\cdots\!01$$$$T^{8} )^{2}$$
$79$ $$1 + 59516 T - 3717013179 T^{2} - 374196053042180 T^{3} - 3765094998545504239 T^{4} +$$$$30\!\cdots\!72$$$$T^{5} -$$$$95\!\cdots\!50$$$$T^{6} +$$$$90\!\cdots\!24$$$$T^{7} +$$$$17\!\cdots\!98$$$$T^{8} +$$$$27\!\cdots\!76$$$$T^{9} -$$$$90\!\cdots\!50$$$$T^{10} +$$$$87\!\cdots\!28$$$$T^{11} -$$$$33\!\cdots\!39$$$$T^{12} -$$$$10\!\cdots\!20$$$$T^{13} -$$$$31\!\cdots\!79$$$$T^{14} +$$$$15\!\cdots\!84$$$$T^{15} +$$$$80\!\cdots\!01$$$$T^{16}$$
$83$ $$1 - 117696 T - 5262215903 T^{2} + 507938387214576 T^{3} + 83906673955765181161 T^{4} -$$$$37\!\cdots\!00$$$$T^{5} -$$$$42\!\cdots\!38$$$$T^{6} +$$$$12\!\cdots\!52$$$$T^{7} +$$$$24\!\cdots\!34$$$$T^{8} +$$$$47\!\cdots\!36$$$$T^{9} -$$$$65\!\cdots\!62$$$$T^{10} -$$$$22\!\cdots\!00$$$$T^{11} +$$$$20\!\cdots\!61$$$$T^{12} +$$$$48\!\cdots\!68$$$$T^{13} -$$$$19\!\cdots\!47$$$$T^{14} -$$$$17\!\cdots\!72$$$$T^{15} +$$$$57\!\cdots\!01$$$$T^{16}$$
$89$ $$( 1 + 225864 T + 38022042668 T^{2} + 4189883418729720 T^{3} +$$$$36\!\cdots\!22$$$$T^{4} +$$$$23\!\cdots\!80$$$$T^{5} +$$$$11\!\cdots\!68$$$$T^{6} +$$$$39\!\cdots\!36$$$$T^{7} +$$$$97\!\cdots\!01$$$$T^{8} )^{2}$$
$97$ $$1 - 33976 T - 11979920730 T^{2} + 1870850391619312 T^{3} + 3904959771682438313 T^{4} -$$$$16\!\cdots\!64$$$$T^{5} +$$$$12\!\cdots\!22$$$$T^{6} +$$$$57\!\cdots\!68$$$$T^{7} -$$$$11\!\cdots\!00$$$$T^{8} +$$$$49\!\cdots\!76$$$$T^{9} +$$$$91\!\cdots\!78$$$$T^{10} -$$$$10\!\cdots\!52$$$$T^{11} +$$$$21\!\cdots\!13$$$$T^{12} +$$$$87\!\cdots\!84$$$$T^{13} -$$$$48\!\cdots\!70$$$$T^{14} -$$$$11\!\cdots\!68$$$$T^{15} +$$$$29\!\cdots\!01$$$$T^{16}$$