# Properties

 Label 5.9.c Level $5$ Weight $9$ Character orbit 5.c Rep. character $\chi_{5}(2,\cdot)$ Character field $\Q(\zeta_{4})$ Dimension $6$ Newform subspaces $1$ Sturm bound $4$ Trace bound $0$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$5$$ Weight: $$k$$ $$=$$ $$9$$ Character orbit: $$[\chi]$$ $$=$$ 5.c (of order $$4$$ and degree $$2$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$5$$ Character field: $$\Q(i)$$ Newform subspaces: $$1$$ Sturm bound: $$4$$ Trace bound: $$0$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{9}(5, [\chi])$$.

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

## Trace form

 $$6q - 2q^{2} - 72q^{3} + 220q^{5} + 1752q^{6} - 2352q^{7} - 8220q^{8} + O(q^{10})$$ $$6q - 2q^{2} - 72q^{3} + 220q^{5} + 1752q^{6} - 2352q^{7} - 8220q^{8} + 30870q^{10} + 23192q^{11} - 45912q^{12} - 119142q^{13} + 241440q^{15} + 218616q^{16} - 265502q^{17} - 454062q^{18} + 412260q^{20} + 231672q^{21} - 35664q^{22} + 28888q^{23} - 340350q^{25} - 801388q^{26} + 392040q^{27} + 1305192q^{28} - 2549760q^{30} - 747648q^{31} + 3033928q^{32} + 4269096q^{33} - 4971680q^{35} - 3972804q^{36} - 454002q^{37} + 1443720q^{38} + 2683500q^{40} + 2489432q^{41} + 4223856q^{42} + 792648q^{43} + 210690q^{45} - 3149928q^{46} - 15313352q^{47} - 21677712q^{48} + 29537650q^{50} + 35567712q^{51} - 735732q^{52} - 13509122q^{53} + 4448040q^{55} - 18454800q^{56} - 34625520q^{57} - 23903520q^{58} + 13688520q^{60} + 24111192q^{61} + 53913416q^{62} + 44837688q^{63} - 30943610q^{65} - 55047936q^{66} - 32827752q^{67} + 8118692q^{68} - 44156280q^{70} - 13992928q^{71} + 82596420q^{72} + 111859638q^{73} - 126793200q^{75} - 56470800q^{76} + 26260136q^{77} + 31125576q^{78} + 23045920q^{80} + 65834226q^{81} + 38023056q^{82} - 14768432q^{83} - 19713030q^{85} - 135560008q^{86} - 133207680q^{87} - 44555040q^{88} + 135147990q^{90} + 167542032q^{91} + 69931048q^{92} - 96798024q^{93} + 239661000q^{95} + 184867872q^{96} - 186656202q^{97} - 345959698q^{98} + O(q^{100})$$

## Decomposition of $$S_{9}^{\mathrm{new}}(5, [\chi])$$ into newform subspaces

Label Dim. $$A$$ Field CM Traces $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$
5.9.c.a $$6$$ $$2.037$$ $$\mathbb{Q}[x]/(x^{6} - \cdots)$$ None $$-2$$ $$-72$$ $$220$$ $$-2352$$ $$q-\beta _{3}q^{2}+(-12-12\beta _{1}-\beta _{2}-\beta _{5})q^{3}+\cdots$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + 2 T + 2 T^{2} + 2912 T^{3} - 51136 T^{4} - 1135744 T^{5} + 2070656 T^{6} - 290750464 T^{7} - 3351248896 T^{8} + 48855252992 T^{9} + 8589934592 T^{10} + 2199023255552 T^{11} + 281474976710656 T^{12}$$
$3$ $$1 + 72 T + 2592 T^{2} + 88992 T^{3} - 43043121 T^{4} - 3659977224 T^{5} - 147990802464 T^{6} - 24013110566664 T^{7} - 1852865220656241 T^{8} + 25133969310517152 T^{9} + 4803028329503971872 T^{10} +$$$$87\!\cdots\!72$$$$T^{11} +$$$$79\!\cdots\!61$$$$T^{12}$$
$5$ $$1 - 220 T + 194375 T^{2} - 199375000 T^{3} + 75927734375 T^{4} - 33569335937500 T^{5} + 59604644775390625 T^{6}$$
$7$ $$1 + 2352 T + 2765952 T^{2} - 2361533048 T^{3} + 6289666320399 T^{4} + 128493496173721896 T^{5} +$$$$28\!\cdots\!96$$$$T^{6} +$$$$74\!\cdots\!96$$$$T^{7} +$$$$20\!\cdots\!99$$$$T^{8} -$$$$45\!\cdots\!48$$$$T^{9} +$$$$30\!\cdots\!52$$$$T^{10} +$$$$14\!\cdots\!52$$$$T^{11} +$$$$36\!\cdots\!01$$$$T^{12}$$
$11$ $$( 1 - 11596 T + 493098215 T^{2} - 5106545516920 T^{3} + 105699981590497415 T^{4} -$$$$53\!\cdots\!56$$$$T^{5} +$$$$98\!\cdots\!41$$$$T^{6} )^{2}$$
$13$ $$1 + 119142 T + 7097408082 T^{2} + 332686420223782 T^{3} + 13680551086291514559 T^{4} +$$$$46\!\cdots\!56$$$$T^{5} +$$$$13\!\cdots\!76$$$$T^{6} +$$$$38\!\cdots\!76$$$$T^{7} +$$$$91\!\cdots\!19$$$$T^{8} +$$$$18\!\cdots\!02$$$$T^{9} +$$$$31\!\cdots\!42$$$$T^{10} +$$$$43\!\cdots\!42$$$$T^{11} +$$$$29\!\cdots\!21$$$$T^{12}$$
$17$ $$1 + 265502 T + 35245656002 T^{2} + 3505767378301982 T^{3} +$$$$37\!\cdots\!19$$$$T^{4} +$$$$40\!\cdots\!76$$$$T^{5} +$$$$37\!\cdots\!76$$$$T^{6} +$$$$28\!\cdots\!16$$$$T^{7} +$$$$18\!\cdots\!39$$$$T^{8} +$$$$11\!\cdots\!22$$$$T^{9} +$$$$83\!\cdots\!22$$$$T^{10} +$$$$43\!\cdots\!02$$$$T^{11} +$$$$11\!\cdots\!41$$$$T^{12}$$
$19$ $$1 - 66391003446 T^{2} +$$$$21\!\cdots\!15$$$$T^{4} -$$$$42\!\cdots\!20$$$$T^{6} +$$$$60\!\cdots\!15$$$$T^{8} -$$$$55\!\cdots\!06$$$$T^{10} +$$$$23\!\cdots\!41$$$$T^{12}$$
$23$ $$1 - 28888 T + 417258272 T^{2} - 2207314762520128 T^{3} +$$$$86\!\cdots\!39$$$$T^{4} -$$$$83\!\cdots\!64$$$$T^{5} +$$$$12\!\cdots\!16$$$$T^{6} -$$$$65\!\cdots\!84$$$$T^{7} +$$$$52\!\cdots\!79$$$$T^{8} -$$$$10\!\cdots\!48$$$$T^{9} +$$$$15\!\cdots\!12$$$$T^{10} -$$$$85\!\cdots\!88$$$$T^{11} +$$$$23\!\cdots\!81$$$$T^{12}$$
$29$ $$1 - 1726260912966 T^{2} +$$$$13\!\cdots\!15$$$$T^{4} -$$$$77\!\cdots\!20$$$$T^{6} +$$$$34\!\cdots\!15$$$$T^{8} -$$$$10\!\cdots\!06$$$$T^{10} +$$$$15\!\cdots\!61$$$$T^{12}$$
$31$ $$( 1 + 373824 T + 1438821265815 T^{2} + 103488660558742480 T^{3} +$$$$12\!\cdots\!15$$$$T^{4} +$$$$27\!\cdots\!44$$$$T^{5} +$$$$62\!\cdots\!21$$$$T^{6} )^{2}$$
$37$ $$1 + 454002 T + 103058908002 T^{2} + 1591257258997412242 T^{3} +$$$$36\!\cdots\!59$$$$T^{4} +$$$$11\!\cdots\!36$$$$T^{5} +$$$$25\!\cdots\!36$$$$T^{6} +$$$$39\!\cdots\!56$$$$T^{7} +$$$$45\!\cdots\!19$$$$T^{8} +$$$$68\!\cdots\!62$$$$T^{9} +$$$$15\!\cdots\!62$$$$T^{10} +$$$$24\!\cdots\!02$$$$T^{11} +$$$$18\!\cdots\!21$$$$T^{12}$$
$41$ $$( 1 - 1244716 T + 19779856453415 T^{2} - 17348876621060070520 T^{3} +$$$$15\!\cdots\!15$$$$T^{4} -$$$$79\!\cdots\!56$$$$T^{5} +$$$$50\!\cdots\!61$$$$T^{6} )^{2}$$
$43$ $$1 - 792648 T + 314145425952 T^{2} - 6701894073526462448 T^{3} +$$$$27\!\cdots\!99$$$$T^{4} -$$$$18\!\cdots\!04$$$$T^{5} +$$$$86\!\cdots\!96$$$$T^{6} -$$$$22\!\cdots\!04$$$$T^{7} +$$$$37\!\cdots\!99$$$$T^{8} -$$$$10\!\cdots\!48$$$$T^{9} +$$$$58\!\cdots\!52$$$$T^{10} -$$$$17\!\cdots\!48$$$$T^{11} +$$$$25\!\cdots\!01$$$$T^{12}$$
$47$ $$1 + 15313352 T + 117249374737952 T^{2} +$$$$85\!\cdots\!72$$$$T^{3} +$$$$63\!\cdots\!79$$$$T^{4} +$$$$35\!\cdots\!16$$$$T^{5} +$$$$17\!\cdots\!16$$$$T^{6} +$$$$85\!\cdots\!76$$$$T^{7} +$$$$36\!\cdots\!59$$$$T^{8} +$$$$11\!\cdots\!32$$$$T^{9} +$$$$37\!\cdots\!32$$$$T^{10} +$$$$11\!\cdots\!52$$$$T^{11} +$$$$18\!\cdots\!61$$$$T^{12}$$
$53$ $$1 + 13509122 T + 91248188605442 T^{2} +$$$$98\!\cdots\!42$$$$T^{3} +$$$$11\!\cdots\!79$$$$T^{4} +$$$$81\!\cdots\!76$$$$T^{5} +$$$$50\!\cdots\!36$$$$T^{6} +$$$$51\!\cdots\!36$$$$T^{7} +$$$$46\!\cdots\!59$$$$T^{8} +$$$$23\!\cdots\!02$$$$T^{9} +$$$$13\!\cdots\!22$$$$T^{10} +$$$$12\!\cdots\!22$$$$T^{11} +$$$$58\!\cdots\!61$$$$T^{12}$$
$59$ $$1 - 413223229068726 T^{2} +$$$$98\!\cdots\!15$$$$T^{4} -$$$$17\!\cdots\!20$$$$T^{6} +$$$$21\!\cdots\!15$$$$T^{8} -$$$$19\!\cdots\!06$$$$T^{10} +$$$$10\!\cdots\!21$$$$T^{12}$$
$61$ $$( 1 - 12055596 T + 200152007609415 T^{2} +$$$$24\!\cdots\!80$$$$T^{3} +$$$$38\!\cdots\!15$$$$T^{4} -$$$$44\!\cdots\!56$$$$T^{5} +$$$$70\!\cdots\!41$$$$T^{6} )^{2}$$
$67$ $$1 + 32827752 T + 538830650686752 T^{2} +$$$$16\!\cdots\!32$$$$T^{3} +$$$$54\!\cdots\!19$$$$T^{4} +$$$$88\!\cdots\!76$$$$T^{5} +$$$$12\!\cdots\!76$$$$T^{6} +$$$$36\!\cdots\!16$$$$T^{7} +$$$$90\!\cdots\!39$$$$T^{8} +$$$$10\!\cdots\!72$$$$T^{9} +$$$$14\!\cdots\!72$$$$T^{10} +$$$$36\!\cdots\!52$$$$T^{11} +$$$$44\!\cdots\!41$$$$T^{12}$$
$71$ $$( 1 + 6996464 T + 1071469029384215 T^{2} +$$$$14\!\cdots\!80$$$$T^{3} +$$$$69\!\cdots\!15$$$$T^{4} +$$$$29\!\cdots\!44$$$$T^{5} +$$$$26\!\cdots\!81$$$$T^{6} )^{2}$$
$73$ $$1 - 111859638 T + 6256289306745522 T^{2} -$$$$29\!\cdots\!78$$$$T^{3} +$$$$12\!\cdots\!39$$$$T^{4} -$$$$42\!\cdots\!64$$$$T^{5} +$$$$12\!\cdots\!16$$$$T^{6} -$$$$33\!\cdots\!84$$$$T^{7} +$$$$79\!\cdots\!79$$$$T^{8} -$$$$15\!\cdots\!98$$$$T^{9} +$$$$26\!\cdots\!62$$$$T^{10} -$$$$38\!\cdots\!38$$$$T^{11} +$$$$27\!\cdots\!81$$$$T^{12}$$
$79$ $$1 - 4185433961698566 T^{2} +$$$$55\!\cdots\!15$$$$T^{4} -$$$$46\!\cdots\!20$$$$T^{6} +$$$$12\!\cdots\!15$$$$T^{8} -$$$$22\!\cdots\!06$$$$T^{10} +$$$$12\!\cdots\!61$$$$T^{12}$$
$83$ $$1 + 14768432 T + 109053291869312 T^{2} +$$$$28\!\cdots\!12$$$$T^{3} +$$$$10\!\cdots\!19$$$$T^{4} +$$$$10\!\cdots\!16$$$$T^{5} +$$$$83\!\cdots\!56$$$$T^{6} +$$$$23\!\cdots\!56$$$$T^{7} +$$$$51\!\cdots\!39$$$$T^{8} +$$$$32\!\cdots\!52$$$$T^{9} +$$$$28\!\cdots\!32$$$$T^{10} +$$$$85\!\cdots\!32$$$$T^{11} +$$$$13\!\cdots\!41$$$$T^{12}$$
$89$ $$1 - 8165894455313286 T^{2} +$$$$62\!\cdots\!15$$$$T^{4} -$$$$26\!\cdots\!20$$$$T^{6} +$$$$97\!\cdots\!15$$$$T^{8} -$$$$19\!\cdots\!06$$$$T^{10} +$$$$37\!\cdots\!81$$$$T^{12}$$
$97$ $$1 + 186656202 T + 17420268872532402 T^{2} +$$$$93\!\cdots\!22$$$$T^{3} +$$$$66\!\cdots\!79$$$$T^{4} +$$$$11\!\cdots\!16$$$$T^{5} +$$$$13\!\cdots\!16$$$$T^{6} +$$$$87\!\cdots\!76$$$$T^{7} +$$$$41\!\cdots\!59$$$$T^{8} +$$$$45\!\cdots\!82$$$$T^{9} +$$$$65\!\cdots\!82$$$$T^{10} +$$$$55\!\cdots\!02$$$$T^{11} +$$$$23\!\cdots\!61$$$$T^{12}$$