# Properties

 Label 13.10.c Level $13$ Weight $10$ Character orbit 13.c Rep. character $\chi_{13}(3,\cdot)$ Character field $\Q(\zeta_{3})$ Dimension $18$ Newform subspaces $1$ Sturm bound $11$ Trace bound $0$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$13$$ Weight: $$k$$ $$=$$ $$10$$ Character orbit: $$[\chi]$$ $$=$$ 13.c (of order $$3$$ and degree $$2$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$13$$ Character field: $$\Q(\zeta_{3})$$ Newform subspaces: $$1$$ Sturm bound: $$11$$ Trace bound: $$0$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{10}(13, [\chi])$$.

Total New Old
Modular forms 22 22 0
Cusp forms 18 18 0
Eisenstein series 4 4 0

## Trace form

 $$18 q + 15 q^{2} + 161 q^{3} - 1793 q^{4} - 2280 q^{5} + 2118 q^{6} - 1939 q^{7} - 14478 q^{8} - 33654 q^{9} + O(q^{10})$$ $$18 q + 15 q^{2} + 161 q^{3} - 1793 q^{4} - 2280 q^{5} + 2118 q^{6} - 1939 q^{7} - 14478 q^{8} - 33654 q^{9} + 46923 q^{10} - 5433 q^{11} + 8712 q^{12} - 212524 q^{13} - 9900 q^{14} - 347428 q^{15} + 400127 q^{16} + 248589 q^{17} - 38738 q^{18} - 311001 q^{19} + 927069 q^{20} + 1553030 q^{21} + 1857242 q^{22} + 591609 q^{23} - 4492800 q^{24} + 7008998 q^{25} - 3801525 q^{26} - 11603482 q^{27} + 2697168 q^{28} + 11014155 q^{29} - 6597836 q^{30} - 23148076 q^{31} - 11868417 q^{32} + 14131427 q^{33} + 21859862 q^{34} + 21112794 q^{35} + 10792871 q^{36} - 29215749 q^{37} + 14572188 q^{38} - 71569875 q^{39} + 27322222 q^{40} + 3328377 q^{41} + 39828306 q^{42} + 6074381 q^{43} + 31824624 q^{44} + 32857342 q^{45} + 36693338 q^{46} - 45575052 q^{47} - 30270064 q^{48} + 10293266 q^{49} - 49601730 q^{50} - 136587494 q^{51} - 35278230 q^{52} - 29480016 q^{53} + 152965386 q^{54} - 18710998 q^{55} - 7665444 q^{56} + 523363230 q^{57} - 163479359 q^{58} - 32715855 q^{59} - 188638416 q^{60} - 220502845 q^{61} - 59980476 q^{62} + 166572574 q^{63} - 924604030 q^{64} + 128091756 q^{65} - 48128076 q^{66} + 112659045 q^{67} - 238942419 q^{68} + 86003951 q^{69} + 2040150992 q^{70} - 236450709 q^{71} + 995206683 q^{72} - 211881220 q^{73} - 455580507 q^{74} + 968954813 q^{75} - 365789708 q^{76} - 2399230890 q^{77} - 441111970 q^{78} - 817519096 q^{79} + 580424625 q^{80} + 176914851 q^{81} + 941792217 q^{82} + 2225691456 q^{83} - 1819004068 q^{84} + 1812284636 q^{85} + 291320076 q^{86} + 69564799 q^{87} + 3178375740 q^{88} - 1154379039 q^{89} - 10225809510 q^{90} - 1658338903 q^{91} - 4545506592 q^{92} + 3136878060 q^{93} + 2755131560 q^{94} + 1779441012 q^{95} + 5906965568 q^{96} - 3616470111 q^{97} + 8263323501 q^{98} + 2262149268 q^{99} + O(q^{100})$$

## Decomposition of $$S_{10}^{\mathrm{new}}(13, [\chi])$$ into newform subspaces

Label Dim $A$ Field CM Traces $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
13.10.c.a $18$ $6.695$ $$\mathbb{Q}[x]/(x^{18} - \cdots)$$ None $$15$$ $$161$$ $$-2280$$ $$-1939$$ $$q+(-2\beta _{2}-\beta _{3})q^{2}+(-18\beta _{2}-\beta _{6}+\cdots)q^{3}+\cdots$$