# Properties

 Label 13.3.d Level $13$ Weight $3$ Character orbit 13.d Rep. character $\chi_{13}(5,\cdot)$ Character field $\Q(\zeta_{4})$ Dimension $4$ Newform subspaces $1$ Sturm bound $3$ Trace bound $0$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$13$$ Weight: $$k$$ $$=$$ $$3$$ 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: $$3$$ Trace bound: $$0$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{3}(13, [\chi])$$.

Total New Old
Modular forms 8 8 0
Cusp forms 4 4 0
Eisenstein series 4 4 0

## Trace form

 $$4 q - 4 q^{2} - 4 q^{3} + 8 q^{5} - 16 q^{6} - 12 q^{7} + 36 q^{8} + 8 q^{9} + O(q^{10})$$ $$4 q - 4 q^{2} - 4 q^{3} + 8 q^{5} - 16 q^{6} - 12 q^{7} + 36 q^{8} + 8 q^{9} - 4 q^{11} + 8 q^{13} + 4 q^{14} - 28 q^{15} - 84 q^{16} + 32 q^{18} + 16 q^{20} + 32 q^{21} + 88 q^{22} + 24 q^{24} - 88 q^{26} - 52 q^{27} + 4 q^{28} + 40 q^{29} + 40 q^{31} + 20 q^{32} - 76 q^{33} - 108 q^{34} - 68 q^{35} + 40 q^{37} + 92 q^{39} + 84 q^{40} + 32 q^{41} + 76 q^{42} - 172 q^{44} + 56 q^{45} + 132 q^{46} - 4 q^{47} - 76 q^{48} - 128 q^{50} + 80 q^{52} - 80 q^{53} + 152 q^{54} + 64 q^{55} - 40 q^{57} - 140 q^{58} + 56 q^{59} - 116 q^{60} - 296 q^{61} - 64 q^{63} + 56 q^{65} + 112 q^{66} - 84 q^{67} + 444 q^{68} - 32 q^{70} + 284 q^{71} - 48 q^{72} + 100 q^{74} + 80 q^{76} - 232 q^{78} + 64 q^{79} - 88 q^{80} - 220 q^{81} - 52 q^{83} - 184 q^{84} - 144 q^{85} - 24 q^{86} + 160 q^{87} + 200 q^{89} + 156 q^{91} - 456 q^{92} + 80 q^{93} - 452 q^{94} + 320 q^{96} - 68 q^{97} + 224 q^{98} + 152 q^{99} + O(q^{100})$$

## Decomposition of $$S_{3}^{\mathrm{new}}(13, [\chi])$$ into newform subspaces

Label Dim $A$ Field CM Traces $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
13.3.d.a $4$ $0.354$ $$\Q(i, \sqrt{10})$$ None $$-4$$ $$-4$$ $$8$$ $$-12$$ $$q+(-1+\beta _{1}-\beta _{2})q^{2}+(-1-\beta _{1}+\beta _{3})q^{3}+\cdots$$