# Properties

 Label 15.10.b Level 15 Weight 10 Character orbit b Rep. character $$\chi_{15}(4,\cdot)$$ Character field $$\Q$$ Dimension 8 Newform subspaces 1 Sturm bound 20 Trace bound 0

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$15 = 3 \cdot 5$$ Weight: $$k$$ $$=$$ $$10$$ Character orbit: $$[\chi]$$ $$=$$ 15.b (of order $$2$$ and degree $$1$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$5$$ Character field: $$\Q$$ Newform subspaces: $$1$$ Sturm bound: $$20$$ Trace bound: $$0$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{10}(15, [\chi])$$.

Total New Old
Modular forms 20 8 12
Cusp forms 16 8 8
Eisenstein series 4 0 4

## Trace form

 $$8q - 1194q^{4} - 690q^{5} + 486q^{6} - 52488q^{9} + O(q^{10})$$ $$8q - 1194q^{4} - 690q^{5} + 486q^{6} - 52488q^{9} + 67090q^{10} - 71988q^{11} + 416364q^{14} + 80190q^{15} - 1505630q^{16} + 851584q^{19} + 2078100q^{20} - 1593108q^{21} + 1242702q^{24} + 1695500q^{25} - 877524q^{26} - 73572q^{29} + 3086100q^{30} + 474088q^{31} - 8124388q^{34} - 36357180q^{35} + 7833834q^{36} + 12959676q^{39} - 15313390q^{40} + 93320088q^{41} - 74555892q^{44} + 4527090q^{45} - 9664072q^{46} + 51329600q^{49} + 67798200q^{50} - 108196236q^{51} - 3188646q^{54} + 64428480q^{55} - 67781220q^{56} + 236526036q^{59} + 63172710q^{60} - 357427760q^{61} - 12137026q^{64} + 19848300q^{65} + 23317308q^{66} + 167059584q^{69} + 200900520q^{70} - 156890664q^{71} - 1523381796q^{74} - 528573600q^{75} + 1098697344q^{76} + 863922280q^{79} + 630213180q^{80} + 344373768q^{81} + 529023636q^{84} - 2223350420q^{85} + 997642392q^{86} + 357382224q^{89} - 440177490q^{90} + 214754328q^{91} - 721679824q^{94} + 1698584640q^{95} - 475022718q^{96} + 472313268q^{99} + O(q^{100})$$

## Decomposition of $$S_{10}^{\mathrm{new}}(15, [\chi])$$ into newform subspaces

Label Dim. $$A$$ Field CM Traces $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$
15.10.b.a $$8$$ $$7.726$$ $$\mathbb{Q}[x]/(x^{8} + \cdots)$$ None $$0$$ $$0$$ $$-690$$ $$0$$ $$q-\beta _{3}q^{2}+\beta _{4}q^{3}+(-149-\beta _{1})q^{4}+\cdots$$

## Decomposition of $$S_{10}^{\mathrm{old}}(15, [\chi])$$ into lower level spaces

$$S_{10}^{\mathrm{old}}(15, [\chi]) \cong$$ $$S_{10}^{\mathrm{new}}(5, [\chi])$$$$^{\oplus 2}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - 1451 T^{2} + 1581940 T^{4} - 1146579392 T^{6} + 685942325248 T^{8} - 300568908136448 T^{10} + 108710089027747840 T^{12} - 26138892237258358784 T^{14} +$$$$47\!\cdots\!96$$$$T^{16}$$
$3$ $$( 1 + 6561 T^{2} )^{4}$$
$5$ $$1 + 690 T - 609700 T^{2} - 1699931250 T^{3} - 2538785156250 T^{4} - 3320178222656250 T^{5} - 2325820922851562500 T^{6} +$$$$51\!\cdots\!50$$$$T^{7} +$$$$14\!\cdots\!25$$$$T^{8}$$
$7$ $$1 - 187079228 T^{2} + 15253712953314292 T^{4} -$$$$75\!\cdots\!16$$$$T^{6} +$$$$30\!\cdots\!14$$$$T^{8} -$$$$12\!\cdots\!84$$$$T^{10} +$$$$40\!\cdots\!92$$$$T^{12} -$$$$80\!\cdots\!72$$$$T^{14} +$$$$70\!\cdots\!01$$$$T^{16}$$
$11$ $$( 1 + 35994 T + 5523704672 T^{2} + 186668393765490 T^{3} + 15626722625814770286 T^{4} +$$$$44\!\cdots\!90$$$$T^{5} +$$$$30\!\cdots\!32$$$$T^{6} +$$$$47\!\cdots\!74$$$$T^{7} +$$$$30\!\cdots\!61$$$$T^{8} )^{2}$$
$13$ $$1 - 57204710684 T^{2} +$$$$16\!\cdots\!00$$$$T^{4} -$$$$29\!\cdots\!48$$$$T^{6} +$$$$37\!\cdots\!18$$$$T^{8} -$$$$33\!\cdots\!92$$$$T^{10} +$$$$20\!\cdots\!00$$$$T^{12} -$$$$81\!\cdots\!76$$$$T^{14} +$$$$15\!\cdots\!81$$$$T^{16}$$
$17$ $$1 - 118674378380 T^{2} +$$$$17\!\cdots\!36$$$$T^{4} -$$$$19\!\cdots\!60$$$$T^{6} +$$$$18\!\cdots\!86$$$$T^{8} -$$$$27\!\cdots\!40$$$$T^{10} +$$$$34\!\cdots\!16$$$$T^{12} -$$$$33\!\cdots\!20$$$$T^{14} +$$$$39\!\cdots\!61$$$$T^{16}$$
$19$ $$( 1 - 425792 T + 640639588492 T^{2} - 117316391220409664 T^{3} +$$$$17\!\cdots\!54$$$$T^{4} -$$$$37\!\cdots\!56$$$$T^{5} +$$$$66\!\cdots\!72$$$$T^{6} -$$$$14\!\cdots\!88$$$$T^{7} +$$$$10\!\cdots\!81$$$$T^{8} )^{2}$$
$23$ $$1 - 10272938895992 T^{2} +$$$$48\!\cdots\!32$$$$T^{4} -$$$$14\!\cdots\!84$$$$T^{6} +$$$$30\!\cdots\!94$$$$T^{8} -$$$$46\!\cdots\!96$$$$T^{10} +$$$$51\!\cdots\!52$$$$T^{12} -$$$$35\!\cdots\!28$$$$T^{14} +$$$$11\!\cdots\!21$$$$T^{16}$$
$29$ $$( 1 + 36786 T + 11698955610980 T^{2} - 38232367405598309058 T^{3} +$$$$21\!\cdots\!18$$$$T^{4} -$$$$55\!\cdots\!02$$$$T^{5} +$$$$24\!\cdots\!80$$$$T^{6} +$$$$11\!\cdots\!74$$$$T^{7} +$$$$44\!\cdots\!21$$$$T^{8} )^{2}$$
$31$ $$( 1 - 237044 T + 50199397014268 T^{2} -$$$$19\!\cdots\!72$$$$T^{3} +$$$$11\!\cdots\!74$$$$T^{4} -$$$$51\!\cdots\!12$$$$T^{5} +$$$$35\!\cdots\!88$$$$T^{6} -$$$$43\!\cdots\!84$$$$T^{7} +$$$$48\!\cdots\!81$$$$T^{8} )^{2}$$
$37$ $$1 - 624655700715068 T^{2} +$$$$20\!\cdots\!12$$$$T^{4} -$$$$45\!\cdots\!76$$$$T^{6} +$$$$70\!\cdots\!14$$$$T^{8} -$$$$77\!\cdots\!04$$$$T^{10} +$$$$59\!\cdots\!92$$$$T^{12} -$$$$30\!\cdots\!52$$$$T^{14} +$$$$81\!\cdots\!81$$$$T^{16}$$
$41$ $$( 1 - 46660044 T + 1629598509234068 T^{2} -$$$$39\!\cdots\!32$$$$T^{3} +$$$$78\!\cdots\!54$$$$T^{4} -$$$$12\!\cdots\!52$$$$T^{5} +$$$$17\!\cdots\!28$$$$T^{6} -$$$$16\!\cdots\!64$$$$T^{7} +$$$$11\!\cdots\!41$$$$T^{8} )^{2}$$
$43$ $$1 - 1288760154444440 T^{2} +$$$$10\!\cdots\!96$$$$T^{4} -$$$$57\!\cdots\!80$$$$T^{6} +$$$$31\!\cdots\!06$$$$T^{8} -$$$$14\!\cdots\!20$$$$T^{10} +$$$$65\!\cdots\!96$$$$T^{12} -$$$$20\!\cdots\!60$$$$T^{14} +$$$$40\!\cdots\!01$$$$T^{16}$$
$47$ $$1 - 4836473602067240 T^{2} +$$$$12\!\cdots\!56$$$$T^{4} -$$$$21\!\cdots\!80$$$$T^{6} +$$$$28\!\cdots\!26$$$$T^{8} -$$$$27\!\cdots\!20$$$$T^{10} +$$$$19\!\cdots\!76$$$$T^{12} -$$$$95\!\cdots\!60$$$$T^{14} +$$$$24\!\cdots\!41$$$$T^{16}$$
$53$ $$1 - 4341506689340012 T^{2} +$$$$22\!\cdots\!72$$$$T^{4} -$$$$59\!\cdots\!84$$$$T^{6} +$$$$28\!\cdots\!74$$$$T^{8} -$$$$64\!\cdots\!76$$$$T^{10} +$$$$26\!\cdots\!12$$$$T^{12} -$$$$56\!\cdots\!28$$$$T^{14} +$$$$14\!\cdots\!41$$$$T^{16}$$
$59$ $$( 1 - 118263018 T + 24386862278408432 T^{2} -$$$$17\!\cdots\!86$$$$T^{3} +$$$$26\!\cdots\!54$$$$T^{4} -$$$$15\!\cdots\!54$$$$T^{5} +$$$$18\!\cdots\!72$$$$T^{6} -$$$$76\!\cdots\!42$$$$T^{7} +$$$$56\!\cdots\!41$$$$T^{8} )^{2}$$
$61$ $$( 1 + 178713880 T + 47272829372304076 T^{2} +$$$$59\!\cdots\!20$$$$T^{3} +$$$$82\!\cdots\!06$$$$T^{4} +$$$$69\!\cdots\!20$$$$T^{5} +$$$$64\!\cdots\!56$$$$T^{6} +$$$$28\!\cdots\!80$$$$T^{7} +$$$$18\!\cdots\!61$$$$T^{8} )^{2}$$
$67$ $$1 - 128915336297443400 T^{2} +$$$$88\!\cdots\!36$$$$T^{4} -$$$$39\!\cdots\!00$$$$T^{6} +$$$$12\!\cdots\!86$$$$T^{8} -$$$$29\!\cdots\!00$$$$T^{10} +$$$$48\!\cdots\!16$$$$T^{12} -$$$$52\!\cdots\!00$$$$T^{14} +$$$$30\!\cdots\!61$$$$T^{16}$$
$71$ $$( 1 + 78445332 T + 132742682733898508 T^{2} +$$$$14\!\cdots\!24$$$$T^{3} +$$$$78\!\cdots\!70$$$$T^{4} +$$$$65\!\cdots\!44$$$$T^{5} +$$$$27\!\cdots\!88$$$$T^{6} +$$$$75\!\cdots\!12$$$$T^{7} +$$$$44\!\cdots\!21$$$$T^{8} )^{2}$$
$73$ $$1 - 112790784848235992 T^{2} +$$$$16\!\cdots\!32$$$$T^{4} -$$$$11\!\cdots\!84$$$$T^{6} +$$$$89\!\cdots\!94$$$$T^{8} -$$$$39\!\cdots\!96$$$$T^{10} +$$$$19\!\cdots\!52$$$$T^{12} -$$$$46\!\cdots\!28$$$$T^{14} +$$$$14\!\cdots\!21$$$$T^{16}$$
$79$ $$( 1 - 431961140 T + 452072920533003676 T^{2} -$$$$14\!\cdots\!80$$$$T^{3} +$$$$79\!\cdots\!66$$$$T^{4} -$$$$17\!\cdots\!20$$$$T^{5} +$$$$64\!\cdots\!36$$$$T^{6} -$$$$74\!\cdots\!60$$$$T^{7} +$$$$20\!\cdots\!21$$$$T^{8} )^{2}$$
$83$ $$1 - 981210392397464024 T^{2} +$$$$42\!\cdots\!20$$$$T^{4} -$$$$11\!\cdots\!28$$$$T^{6} +$$$$22\!\cdots\!98$$$$T^{8} -$$$$38\!\cdots\!52$$$$T^{10} +$$$$51\!\cdots\!20$$$$T^{12} -$$$$41\!\cdots\!96$$$$T^{14} +$$$$14\!\cdots\!61$$$$T^{16}$$
$89$ $$( 1 - 178691112 T + 708605882924008892 T^{2} -$$$$39\!\cdots\!44$$$$T^{3} +$$$$23\!\cdots\!94$$$$T^{4} -$$$$13\!\cdots\!96$$$$T^{5} +$$$$86\!\cdots\!52$$$$T^{6} -$$$$76\!\cdots\!48$$$$T^{7} +$$$$15\!\cdots\!61$$$$T^{8} )^{2}$$
$97$ $$1 - 1313769052010852360 T^{2} +$$$$18\!\cdots\!56$$$$T^{4} -$$$$20\!\cdots\!20$$$$T^{6} +$$$$15\!\cdots\!26$$$$T^{8} -$$$$11\!\cdots\!80$$$$T^{10} +$$$$62\!\cdots\!76$$$$T^{12} -$$$$25\!\cdots\!40$$$$T^{14} +$$$$11\!\cdots\!41$$$$T^{16}$$