# Properties

 Label 13.4.b Level $13$ Weight $4$ Character orbit 13.b Rep. character $\chi_{13}(12,\cdot)$ Character field $\Q$ Dimension $2$ Newform subspaces $1$ Sturm bound $4$ Trace bound $0$

# Related objects

## Defining parameters

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

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{4}(13, [\chi])$$.

Total New Old
Modular forms 4 4 0
Cusp forms 2 2 0
Eisenstein series 2 2 0

## Trace form

 $$2 q - 2 q^{3} - 2 q^{4} - 52 q^{9} + O(q^{10})$$ $$2 q - 2 q^{3} - 2 q^{4} - 52 q^{9} + 54 q^{10} + 2 q^{12} + 52 q^{13} + 90 q^{14} - 142 q^{16} - 90 q^{17} - 288 q^{22} + 324 q^{23} + 88 q^{25} + 234 q^{26} + 106 q^{27} - 288 q^{29} - 54 q^{30} - 270 q^{35} + 52 q^{36} - 36 q^{38} - 52 q^{39} + 378 q^{40} - 90 q^{42} - 194 q^{43} + 142 q^{48} + 236 q^{49} + 90 q^{51} - 52 q^{52} - 828 q^{53} + 864 q^{55} + 630 q^{56} + 752 q^{61} - 1584 q^{62} - 866 q^{64} - 702 q^{65} + 288 q^{66} + 90 q^{68} - 324 q^{69} + 1818 q^{74} - 88 q^{75} + 1440 q^{77} - 234 q^{78} - 1660 q^{79} + 1298 q^{81} + 1152 q^{82} + 288 q^{87} - 2016 q^{88} - 1404 q^{90} - 1170 q^{91} - 324 q^{92} + 666 q^{94} + 108 q^{95} + O(q^{100})$$

## Decomposition of $$S_{4}^{\mathrm{new}}(13, [\chi])$$ into newform subspaces

Label Dim $A$ Field CM Traces $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
13.4.b.a $2$ $0.767$ $$\Q(\sqrt{-1})$$ None $$0$$ $$-2$$ $$0$$ $$0$$ $$q+iq^{2}-q^{3}-q^{4}-3iq^{5}-iq^{6}+\cdots$$