# Properties

 Label 15.5.f Level 15 Weight 5 Character orbit f Rep. character $$\chi_{15}(7,\cdot)$$ Character field $$\Q(\zeta_{4})$$ Dimension 8 Newform subspaces 1 Sturm bound 10 Trace bound 0

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$15 = 3 \cdot 5$$ Weight: $$k$$ $$=$$ $$5$$ Character orbit: $$[\chi]$$ $$=$$ 15.f (of order $$4$$ and degree $$2$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$5$$ Character field: $$\Q(i)$$ Newform subspaces: $$1$$ Sturm bound: $$10$$ Trace bound: $$0$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{5}(15, [\chi])$$.

Total New Old
Modular forms 20 8 12
Cusp forms 12 8 4
Eisenstein series 8 0 8

## Trace form

 $$8q - 84q^{5} + 36q^{6} + 20q^{7} + 180q^{8} + O(q^{10})$$ $$8q - 84q^{5} + 36q^{6} + 20q^{7} + 180q^{8} + 104q^{10} - 288q^{11} - 360q^{12} - 340q^{13} + 144q^{15} + 620q^{16} + 900q^{17} + 564q^{20} + 792q^{21} - 1100q^{22} - 1560q^{23} - 1204q^{25} - 3024q^{26} + 3580q^{28} - 2664q^{30} - 512q^{31} + 4980q^{32} + 2700q^{33} + 6600q^{35} + 2484q^{36} - 3820q^{37} - 7680q^{38} - 2952q^{40} - 2712q^{41} - 7380q^{42} - 1240q^{43} - 1944q^{45} + 13528q^{46} + 4800q^{47} + 3600q^{48} + 3744q^{50} + 6264q^{51} - 1240q^{52} + 1020q^{53} - 3644q^{55} - 30720q^{56} - 5400q^{57} + 2340q^{58} - 1044q^{60} - 4760q^{61} + 28680q^{62} + 540q^{63} - 1212q^{65} + 10008q^{66} - 8920q^{67} - 1920q^{68} + 7380q^{70} + 7536q^{71} - 4860q^{72} + 11600q^{73} - 5976q^{75} + 4344q^{76} - 360q^{77} - 4680q^{78} + 10644q^{80} - 5832q^{81} - 27200q^{82} - 32400q^{83} - 15628q^{85} + 14592q^{86} + 10620q^{87} - 14340q^{88} + 8964q^{90} + 16528q^{91} - 31800q^{92} + 14040q^{93} + 18864q^{95} - 4068q^{96} + 58640q^{97} + 46440q^{98} + O(q^{100})$$

## Decomposition of $$S_{5}^{\mathrm{new}}(15, [\chi])$$ into newform subspaces

Label Dim. $$A$$ Field CM Traces $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$
15.5.f.a $$8$$ $$1.551$$ $$\mathbb{Q}[x]/(x^{8} - \cdots)$$ None $$0$$ $$0$$ $$-84$$ $$20$$ $$q+\beta _{1}q^{2}+\beta _{5}q^{3}+(-\beta _{1}-12\beta _{2}-2\beta _{3}+\cdots)q^{4}+\cdots$$

## Decomposition of $$S_{5}^{\mathrm{old}}(15, [\chi])$$ into lower level spaces

$$S_{5}^{\mathrm{old}}(15, [\chi]) \cong$$ $$S_{5}^{\mathrm{new}}(5, [\chi])$$$$^{\oplus 2}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - 60 T^{3} - 523 T^{4} - 420 T^{5} + 1800 T^{6} + 9480 T^{7} + 180804 T^{8} + 151680 T^{9} + 460800 T^{10} - 1720320 T^{11} - 34275328 T^{12} - 62914560 T^{13} + 4294967296 T^{16}$$
$3$ $$( 1 + 729 T^{4} )^{2}$$
$5$ $$1 + 84 T + 4130 T^{2} + 145500 T^{3} + 4037250 T^{4} + 90937500 T^{5} + 1613281250 T^{6} + 20507812500 T^{7} + 152587890625 T^{8}$$
$7$ $$1 - 20 T + 200 T^{2} + 199700 T^{3} - 6227200 T^{4} + 79376300 T^{5} + 19597959000 T^{6} - 40112935980 T^{7} + 30640225075198 T^{8} - 96311159287980 T^{9} + 112978333641159000 T^{10} + 1098670165252736300 T^{11} -$$$$20\!\cdots\!00$$$$T^{12} +$$$$15\!\cdots\!00$$$$T^{13} +$$$$38\!\cdots\!00$$$$T^{14} -$$$$91\!\cdots\!20$$$$T^{15} +$$$$11\!\cdots\!01$$$$T^{16}$$
$11$ $$( 1 + 144 T + 41822 T^{2} + 6150240 T^{3} + 823068186 T^{4} + 90045663840 T^{5} + 8964917121182 T^{6} + 451933686247824 T^{7} + 45949729863572161 T^{8} )^{2}$$
$13$ $$1 + 340 T + 57800 T^{2} + 10577660 T^{3} + 3418178672 T^{4} + 809516994020 T^{5} + 133608496263000 T^{6} + 23843087441136780 T^{7} + 4251050341381084894 T^{8} +$$$$68\!\cdots\!80$$$$T^{9} +$$$$10\!\cdots\!00$$$$T^{10} +$$$$18\!\cdots\!20$$$$T^{11} +$$$$22\!\cdots\!52$$$$T^{12} +$$$$20\!\cdots\!60$$$$T^{13} +$$$$31\!\cdots\!00$$$$T^{14} +$$$$52\!\cdots\!40$$$$T^{15} +$$$$44\!\cdots\!81$$$$T^{16}$$
$17$ $$1 - 900 T + 405000 T^{2} - 162344700 T^{3} + 70846529936 T^{4} - 26992676052900 T^{5} + 8778464632575000 T^{6} - 2859179891204281500 T^{7} +$$$$88\!\cdots\!86$$$$T^{8} -$$$$23\!\cdots\!00$$$$T^{9} +$$$$61\!\cdots\!00$$$$T^{10} -$$$$15\!\cdots\!00$$$$T^{11} +$$$$34\!\cdots\!16$$$$T^{12} -$$$$65\!\cdots\!00$$$$T^{13} +$$$$13\!\cdots\!00$$$$T^{14} -$$$$25\!\cdots\!00$$$$T^{15} +$$$$23\!\cdots\!61$$$$T^{16}$$
$19$ $$1 - 475280 T^{2} + 118243003996 T^{4} - 19617731389431920 T^{6} +$$$$27\!\cdots\!66$$$$T^{8} -$$$$33\!\cdots\!20$$$$T^{10} +$$$$34\!\cdots\!76$$$$T^{12} -$$$$23\!\cdots\!80$$$$T^{14} +$$$$83\!\cdots\!61$$$$T^{16}$$
$23$ $$1 + 1560 T + 1216800 T^{2} + 797095800 T^{3} + 382745999300 T^{4} + 92493403026600 T^{5} - 3754766037924000 T^{6} - 28706182737150138360 T^{7} -$$$$24\!\cdots\!22$$$$T^{8} -$$$$80\!\cdots\!60$$$$T^{9} -$$$$29\!\cdots\!00$$$$T^{10} +$$$$20\!\cdots\!00$$$$T^{11} +$$$$23\!\cdots\!00$$$$T^{12} +$$$$13\!\cdots\!00$$$$T^{13} +$$$$58\!\cdots\!00$$$$T^{14} +$$$$20\!\cdots\!60$$$$T^{15} +$$$$37\!\cdots\!21$$$$T^{16}$$
$29$ $$1 - 3993236 T^{2} + 7931637401512 T^{4} - 9869586507702691580 T^{6} +$$$$83\!\cdots\!06$$$$T^{8} -$$$$49\!\cdots\!80$$$$T^{10} +$$$$19\!\cdots\!52$$$$T^{12} -$$$$49\!\cdots\!16$$$$T^{14} +$$$$62\!\cdots\!41$$$$T^{16}$$
$31$ $$( 1 + 256 T + 1349068 T^{2} - 180990272 T^{3} + 1174676381974 T^{4} - 167148316987712 T^{5} + 1150608006098454988 T^{6} +$$$$20\!\cdots\!16$$$$T^{7} +$$$$72\!\cdots\!81$$$$T^{8} )^{2}$$
$37$ $$1 + 3820 T + 7296200 T^{2} + 10974405380 T^{3} + 13446367756400 T^{4} + 10277980840235420 T^{5} + 1373285107740096600 T^{6} -$$$$20\!\cdots\!20$$$$T^{7} -$$$$46\!\cdots\!82$$$$T^{8} -$$$$38\!\cdots\!20$$$$T^{9} +$$$$48\!\cdots\!00$$$$T^{10} +$$$$67\!\cdots\!20$$$$T^{11} +$$$$16\!\cdots\!00$$$$T^{12} +$$$$25\!\cdots\!80$$$$T^{13} +$$$$31\!\cdots\!00$$$$T^{14} +$$$$31\!\cdots\!20$$$$T^{15} +$$$$15\!\cdots\!81$$$$T^{16}$$
$41$ $$( 1 + 1356 T + 8325968 T^{2} + 6593365668 T^{3} + 29888933841054 T^{4} + 18631275563373348 T^{5} + 66482231940054114128 T^{6} +$$$$30\!\cdots\!36$$$$T^{7} +$$$$63\!\cdots\!41$$$$T^{8} )^{2}$$
$43$ $$1 + 1240 T + 768800 T^{2} + 4155585560 T^{3} + 45689035751396 T^{4} + 43012988909739080 T^{5} + 26844821235643471200 T^{6} +$$$$14\!\cdots\!20$$$$T^{7} +$$$$79\!\cdots\!06$$$$T^{8} +$$$$49\!\cdots\!20$$$$T^{9} +$$$$31\!\cdots\!00$$$$T^{10} +$$$$17\!\cdots\!80$$$$T^{11} +$$$$62\!\cdots\!96$$$$T^{12} +$$$$19\!\cdots\!60$$$$T^{13} +$$$$12\!\cdots\!00$$$$T^{14} +$$$$67\!\cdots\!40$$$$T^{15} +$$$$18\!\cdots\!01$$$$T^{16}$$
$47$ $$1 - 4800 T + 11520000 T^{2} - 34026427200 T^{3} + 139196198193956 T^{4} - 374777003457297600 T^{5} +$$$$77\!\cdots\!00$$$$T^{6} -$$$$20\!\cdots\!00$$$$T^{7} +$$$$52\!\cdots\!26$$$$T^{8} -$$$$98\!\cdots\!00$$$$T^{9} +$$$$18\!\cdots\!00$$$$T^{10} -$$$$43\!\cdots\!00$$$$T^{11} +$$$$78\!\cdots\!76$$$$T^{12} -$$$$94\!\cdots\!00$$$$T^{13} +$$$$15\!\cdots\!00$$$$T^{14} -$$$$31\!\cdots\!00$$$$T^{15} +$$$$32\!\cdots\!41$$$$T^{16}$$
$53$ $$1 - 1020 T + 520200 T^{2} - 3711679620 T^{3} + 27373065594800 T^{4} - 31066466654514780 T^{5} + 24336610065951787800 T^{6} -$$$$12\!\cdots\!80$$$$T^{7} -$$$$51\!\cdots\!42$$$$T^{8} -$$$$95\!\cdots\!80$$$$T^{9} +$$$$15\!\cdots\!00$$$$T^{10} -$$$$15\!\cdots\!80$$$$T^{11} +$$$$10\!\cdots\!00$$$$T^{12} -$$$$11\!\cdots\!20$$$$T^{13} +$$$$12\!\cdots\!00$$$$T^{14} -$$$$19\!\cdots\!20$$$$T^{15} +$$$$15\!\cdots\!41$$$$T^{16}$$
$59$ $$1 - 68266460 T^{2} + 2308748573454136 T^{4} -$$$$49\!\cdots\!40$$$$T^{6} +$$$$71\!\cdots\!06$$$$T^{8} -$$$$72\!\cdots\!40$$$$T^{10} +$$$$49\!\cdots\!76$$$$T^{12} -$$$$21\!\cdots\!60$$$$T^{14} +$$$$46\!\cdots\!81$$$$T^{16}$$
$61$ $$( 1 + 2380 T + 35645776 T^{2} + 28234432420 T^{3} + 552631006355806 T^{4} + 390929462012565220 T^{5} +$$$$68\!\cdots\!56$$$$T^{6} +$$$$63\!\cdots\!80$$$$T^{7} +$$$$36\!\cdots\!61$$$$T^{8} )^{2}$$
$67$ $$1 + 8920 T + 39783200 T^{2} + 240429760280 T^{3} + 743487179691236 T^{4} - 1082655630131333560 T^{5} -$$$$10\!\cdots\!00$$$$T^{6} -$$$$87\!\cdots\!40$$$$T^{7} -$$$$61\!\cdots\!14$$$$T^{8} -$$$$17\!\cdots\!40$$$$T^{9} -$$$$41\!\cdots\!00$$$$T^{10} -$$$$88\!\cdots\!60$$$$T^{11} +$$$$12\!\cdots\!16$$$$T^{12} +$$$$79\!\cdots\!80$$$$T^{13} +$$$$26\!\cdots\!00$$$$T^{14} +$$$$12\!\cdots\!20$$$$T^{15} +$$$$27\!\cdots\!61$$$$T^{16}$$
$71$ $$( 1 - 3768 T + 59990708 T^{2} - 219866356776 T^{3} + 1753029620362470 T^{4} - 5587173721023900456 T^{5} +$$$$38\!\cdots\!88$$$$T^{6} -$$$$61\!\cdots\!88$$$$T^{7} +$$$$41\!\cdots\!21$$$$T^{8} )^{2}$$
$73$ $$1 - 11600 T + 67280000 T^{2} - 460834857520 T^{3} + 3747670062415100 T^{4} - 21684328079009172880 T^{5} +$$$$10\!\cdots\!00$$$$T^{6} -$$$$63\!\cdots\!00$$$$T^{7} +$$$$38\!\cdots\!78$$$$T^{8} -$$$$18\!\cdots\!00$$$$T^{9} +$$$$85\!\cdots\!00$$$$T^{10} -$$$$49\!\cdots\!80$$$$T^{11} +$$$$24\!\cdots\!00$$$$T^{12} -$$$$85\!\cdots\!20$$$$T^{13} +$$$$35\!\cdots\!00$$$$T^{14} -$$$$17\!\cdots\!00$$$$T^{15} +$$$$42\!\cdots\!21$$$$T^{16}$$
$79$ $$1 - 192979448 T^{2} + 19833375036960508 T^{4} -$$$$13\!\cdots\!96$$$$T^{6} +$$$$60\!\cdots\!70$$$$T^{8} -$$$$19\!\cdots\!56$$$$T^{10} +$$$$45\!\cdots\!68$$$$T^{12} -$$$$67\!\cdots\!88$$$$T^{14} +$$$$52\!\cdots\!41$$$$T^{16}$$
$83$ $$1 + 32400 T + 524880000 T^{2} + 6014129561040 T^{3} + 60088627758116132 T^{4} +$$$$56\!\cdots\!80$$$$T^{5} +$$$$48\!\cdots\!00$$$$T^{6} +$$$$37\!\cdots\!80$$$$T^{7} +$$$$27\!\cdots\!14$$$$T^{8} +$$$$18\!\cdots\!80$$$$T^{9} +$$$$11\!\cdots\!00$$$$T^{10} +$$$$60\!\cdots\!80$$$$T^{11} +$$$$30\!\cdots\!92$$$$T^{12} +$$$$14\!\cdots\!40$$$$T^{13} +$$$$59\!\cdots\!00$$$$T^{14} +$$$$17\!\cdots\!00$$$$T^{15} +$$$$25\!\cdots\!61$$$$T^{16}$$
$89$ $$1 - 372973760 T^{2} + 67192056713379196 T^{4} -$$$$75\!\cdots\!40$$$$T^{6} +$$$$56\!\cdots\!26$$$$T^{8} -$$$$29\!\cdots\!40$$$$T^{10} +$$$$10\!\cdots\!56$$$$T^{12} -$$$$22\!\cdots\!60$$$$T^{14} +$$$$24\!\cdots\!21$$$$T^{16}$$
$97$ $$1 - 58640 T + 1719324800 T^{2} - 34474607566960 T^{3} + 547427197512783356 T^{4} -$$$$74\!\cdots\!80$$$$T^{5} +$$$$91\!\cdots\!00$$$$T^{6} -$$$$10\!\cdots\!20$$$$T^{7} +$$$$10\!\cdots\!26$$$$T^{8} -$$$$89\!\cdots\!20$$$$T^{9} +$$$$71\!\cdots\!00$$$$T^{10} -$$$$51\!\cdots\!80$$$$T^{11} +$$$$33\!\cdots\!76$$$$T^{12} -$$$$18\!\cdots\!60$$$$T^{13} +$$$$82\!\cdots\!00$$$$T^{14} -$$$$24\!\cdots\!40$$$$T^{15} +$$$$37\!\cdots\!41$$$$T^{16}$$