# Properties

 Label 13.5.d Level 13 Weight 5 Character orbit d Rep. character $$\chi_{13}(5,\cdot)$$ Character field $$\Q(\zeta_{4})$$ Dimension 6 Newform subspaces 1 Sturm bound 5 Trace bound 0

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$13$$ Weight: $$k$$ $$=$$ $$5$$ Character orbit: $$[\chi]$$ $$=$$ 13.d (of order $$4$$ and degree $$2$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$13$$ Character field: $$\Q(i)$$ Newform subspaces: $$1$$ Sturm bound: $$5$$ Trace bound: $$0$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{5}(13, [\chi])$$.

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

## Trace form

 $$6q - 2q^{2} - 4q^{3} - 14q^{5} + 32q^{6} + 48q^{7} - 96q^{8} - 58q^{9} + O(q^{10})$$ $$6q - 2q^{2} - 4q^{3} - 14q^{5} + 32q^{6} + 48q^{7} - 96q^{8} - 58q^{9} - 32q^{11} - 244q^{14} + 404q^{15} + 1044q^{16} - 802q^{18} + 732q^{19} + 428q^{20} - 2128q^{21} - 1632q^{22} - 24q^{24} + 910q^{26} + 236q^{27} + 1884q^{28} + 4184q^{29} - 3468q^{31} + 2092q^{32} + 2324q^{33} - 5304q^{34} - 4204q^{35} - 1758q^{37} + 1196q^{39} - 708q^{40} + 4750q^{41} + 9532q^{42} - 3956q^{44} + 830q^{45} + 516q^{46} - 6872q^{47} - 9436q^{48} - 322q^{50} + 3900q^{52} + 2108q^{53} - 184q^{54} + 6408q^{55} - 5800q^{57} + 6516q^{58} + 4372q^{59} + 1324q^{60} + 5988q^{61} - 652q^{63} - 5018q^{65} - 4592q^{66} + 72q^{67} - 10572q^{68} + 7368q^{70} - 14672q^{71} - 7980q^{72} + 5874q^{73} + 1544q^{74} + 3576q^{76} + 5720q^{78} + 2616q^{79} - 12080q^{80} - 19450q^{81} + 19264q^{83} + 6296q^{84} + 4164q^{85} + 29376q^{86} + 35584q^{87} - 986q^{89} - 30888q^{91} + 5304q^{92} - 9520q^{93} - 36156q^{94} + 20720q^{96} - 23154q^{97} - 41426q^{98} + 17492q^{99} + O(q^{100})$$

## Decomposition of $$S_{5}^{\mathrm{new}}(13, [\chi])$$ into newform subspaces

Label Dim. $$A$$ Field CM Traces $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$
13.5.d.a $$6$$ $$1.344$$ 6.0.$$\cdots$$.1 None $$-2$$ $$-4$$ $$-14$$ $$48$$ $$q-\beta _{1}q^{2}+(-1+\beta _{4})q^{3}+(-\beta _{1}-\beta _{2}+\cdots)q^{4}+\cdots$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + 2 T + 2 T^{2} + 44 T^{3} - 175 T^{4} - 1162 T^{5} - 1006 T^{6} - 18592 T^{7} - 44800 T^{8} + 180224 T^{9} + 131072 T^{10} + 2097152 T^{11} + 16777216 T^{12}$$
$3$ $$( 1 + 2 T + 138 T^{2} + 180 T^{3} + 11178 T^{4} + 13122 T^{5} + 531441 T^{6} )^{2}$$
$5$ $$1 + 14 T + 98 T^{2} + 8726 T^{3} + 769304 T^{4} + 5342354 T^{5} + 37472702 T^{6} + 3338971250 T^{7} + 300509375000 T^{8} + 2130371093750 T^{9} + 14953613281250 T^{10} + 1335144042968750 T^{11} + 59604644775390625 T^{12}$$
$7$ $$1 - 48 T + 1152 T^{2} - 39716 T^{3} - 5764800 T^{4} + 372144048 T^{5} - 10433184376 T^{6} + 893517859248 T^{7} - 33232924804800 T^{8} - 549720562474916 T^{9} + 38284336016180352 T^{10} - 3830028782285376048 T^{11} +$$$$19\!\cdots\!01$$$$T^{12}$$
$11$ $$1 + 32 T + 512 T^{2} + 344936 T^{3} - 63068881 T^{4} - 5548427368 T^{5} - 85767986656 T^{6} - 81234525094888 T^{7} - 13519374757082161 T^{8} + 1082556930552634856 T^{9} + 23526261690148946432 T^{10} +$$$$21\!\cdots\!32$$$$T^{11} +$$$$98\!\cdots\!41$$$$T^{12}$$
$13$ $$1 + 13689 T^{2} - 5026736 T^{3} + 390971529 T^{4} + 23298085122481 T^{6}$$
$17$ $$1 - 182400 T^{2} + 31740762624 T^{4} - 2752934184177374 T^{6} +$$$$22\!\cdots\!84$$$$T^{8} -$$$$88\!\cdots\!00$$$$T^{10} +$$$$33\!\cdots\!21$$$$T^{12}$$
$19$ $$1 - 732 T + 267912 T^{2} - 58326044 T^{3} + 980477859 T^{4} + 1915050167496 T^{5} + 36465197577488 T^{6} + 249571252878246216 T^{7} + 16652007528631209219 T^{8} -$$$$12\!\cdots\!84$$$$T^{9} +$$$$77\!\cdots\!72$$$$T^{10} -$$$$27\!\cdots\!32$$$$T^{11} +$$$$48\!\cdots\!21$$$$T^{12}$$
$23$ $$1 - 1524906 T^{2} + 1007971632387 T^{4} - 369112562746452020 T^{6} +$$$$78\!\cdots\!47$$$$T^{8} -$$$$93\!\cdots\!66$$$$T^{10} +$$$$48\!\cdots\!41$$$$T^{12}$$
$29$ $$( 1 - 2092 T + 2568557 T^{2} - 2263548328 T^{3} + 1816691563517 T^{4} - 1046515495914412 T^{5} + 353814783205469041 T^{6} )^{2}$$
$31$ $$1 + 3468 T + 6013512 T^{2} + 8766749740 T^{3} + 12186993284643 T^{4} + 14076253382920248 T^{5} + 13957766872742357648 T^{6} +$$$$12\!\cdots\!08$$$$T^{7} +$$$$10\!\cdots\!63$$$$T^{8} +$$$$69\!\cdots\!40$$$$T^{9} +$$$$43\!\cdots\!72$$$$T^{10} +$$$$23\!\cdots\!68$$$$T^{11} +$$$$62\!\cdots\!21$$$$T^{12}$$
$37$ $$1 + 1758 T + 1545282 T^{2} + 3527449702 T^{3} + 9479025269208 T^{4} + 9214471485611970 T^{5} + 7772724445723511006 T^{6} +$$$$17\!\cdots\!70$$$$T^{7} +$$$$33\!\cdots\!68$$$$T^{8} +$$$$23\!\cdots\!62$$$$T^{9} +$$$$19\!\cdots\!62$$$$T^{10} +$$$$40\!\cdots\!58$$$$T^{11} +$$$$43\!\cdots\!61$$$$T^{12}$$
$41$ $$1 - 4750 T + 11281250 T^{2} - 23149979278 T^{3} + 37142272482563 T^{4} - 45662503590367708 T^{5} + 65846400896247469892 T^{6} -$$$$12\!\cdots\!88$$$$T^{7} +$$$$29\!\cdots\!23$$$$T^{8} -$$$$52\!\cdots\!18$$$$T^{9} +$$$$71\!\cdots\!50$$$$T^{10} -$$$$85\!\cdots\!50$$$$T^{11} +$$$$50\!\cdots\!61$$$$T^{12}$$
$43$ $$1 - 5149872 T^{2} + 18545289223032 T^{4} - 84377003006781422498 T^{6} +$$$$21\!\cdots\!32$$$$T^{8} -$$$$70\!\cdots\!72$$$$T^{10} +$$$$15\!\cdots\!01$$$$T^{12}$$
$47$ $$1 + 6872 T + 23612192 T^{2} + 59503259324 T^{3} + 95575957382048 T^{4} + 82680226895549288 T^{5} + 81739598027076446408 T^{6} +$$$$40\!\cdots\!28$$$$T^{7} +$$$$22\!\cdots\!28$$$$T^{8} +$$$$69\!\cdots\!84$$$$T^{9} +$$$$13\!\cdots\!32$$$$T^{10} +$$$$19\!\cdots\!72$$$$T^{11} +$$$$13\!\cdots\!81$$$$T^{12}$$
$53$ $$( 1 - 1054 T + 11036969 T^{2} - 24267726304 T^{3} + 87086994192089 T^{4} - 65621713693574494 T^{5} +$$$$49\!\cdots\!41$$$$T^{6} )^{2}$$
$59$ $$1 - 4372 T + 9557192 T^{2} + 8760544940 T^{3} - 22022350167805 T^{4} - 699590778075744808 T^{5} +$$$$33\!\cdots\!36$$$$T^{6} -$$$$84\!\cdots\!88$$$$T^{7} -$$$$32\!\cdots\!05$$$$T^{8} +$$$$15\!\cdots\!40$$$$T^{9} +$$$$20\!\cdots\!72$$$$T^{10} -$$$$11\!\cdots\!72$$$$T^{11} +$$$$31\!\cdots\!61$$$$T^{12}$$
$61$ $$( 1 - 2994 T + 39378039 T^{2} - 81711006860 T^{3} + 545222066885799 T^{4} - 573971695113859314 T^{5} +$$$$26\!\cdots\!21$$$$T^{6} )^{2}$$
$67$ $$1 - 72 T + 2592 T^{2} + 2528502352 T^{3} + 569584534669983 T^{4} - 216259992437699016 T^{5} + 17291018413684499168 T^{6} -$$$$43\!\cdots\!36$$$$T^{7} +$$$$23\!\cdots\!03$$$$T^{8} +$$$$20\!\cdots\!72$$$$T^{9} +$$$$42\!\cdots\!52$$$$T^{10} -$$$$23\!\cdots\!72$$$$T^{11} +$$$$66\!\cdots\!21$$$$T^{12}$$
$71$ $$1 + 14672 T + 107633792 T^{2} + 664874016500 T^{3} + 4498446529994480 T^{4} + 28160075410127913488 T^{5} +$$$$15\!\cdots\!76$$$$T^{6} +$$$$71\!\cdots\!28$$$$T^{7} +$$$$29\!\cdots\!80$$$$T^{8} +$$$$10\!\cdots\!00$$$$T^{9} +$$$$44\!\cdots\!32$$$$T^{10} +$$$$15\!\cdots\!72$$$$T^{11} +$$$$26\!\cdots\!81$$$$T^{12}$$
$73$ $$1 - 5874 T + 17251938 T^{2} - 193617003890 T^{3} + 3002766228450819 T^{4} - 10485445128171427236 T^{5} +$$$$28\!\cdots\!92$$$$T^{6} -$$$$29\!\cdots\!76$$$$T^{7} +$$$$24\!\cdots\!39$$$$T^{8} -$$$$44\!\cdots\!90$$$$T^{9} +$$$$11\!\cdots\!18$$$$T^{10} -$$$$10\!\cdots\!74$$$$T^{11} +$$$$52\!\cdots\!41$$$$T^{12}$$
$79$ $$( 1 - 1308 T + 107661369 T^{2} - 109363076192 T^{3} + 4193419043120889 T^{4} - 1984378323357781788 T^{5} +$$$$59\!\cdots\!41$$$$T^{6} )^{2}$$
$83$ $$1 - 19264 T + 185550848 T^{2} - 1498697581912 T^{3} + 8779394341391855 T^{4} - 36613623736591560520 T^{5} +$$$$19\!\cdots\!12$$$$T^{6} -$$$$17\!\cdots\!20$$$$T^{7} +$$$$19\!\cdots\!55$$$$T^{8} -$$$$16\!\cdots\!32$$$$T^{9} +$$$$94\!\cdots\!88$$$$T^{10} -$$$$46\!\cdots\!64$$$$T^{11} +$$$$11\!\cdots\!21$$$$T^{12}$$
$89$ $$1 + 986 T + 486098 T^{2} - 154474746070 T^{3} + 1975473144822275 T^{4} + 28274168611850630084 T^{5} +$$$$38\!\cdots\!24$$$$T^{6} +$$$$17\!\cdots\!44$$$$T^{7} +$$$$77\!\cdots\!75$$$$T^{8} -$$$$38\!\cdots\!70$$$$T^{9} +$$$$75\!\cdots\!78$$$$T^{10} +$$$$95\!\cdots\!86$$$$T^{11} +$$$$61\!\cdots\!41$$$$T^{12}$$
$97$ $$1 + 23154 T + 268053858 T^{2} + 2366383285666 T^{3} + 18858271611968655 T^{4} +$$$$17\!\cdots\!56$$$$T^{5} +$$$$18\!\cdots\!12$$$$T^{6} +$$$$15\!\cdots\!36$$$$T^{7} +$$$$14\!\cdots\!55$$$$T^{8} +$$$$16\!\cdots\!06$$$$T^{9} +$$$$16\!\cdots\!18$$$$T^{10} +$$$$12\!\cdots\!54$$$$T^{11} +$$$$48\!\cdots\!81$$$$T^{12}$$