# Properties

 Label 13.7.d Level 13 Weight 7 Character orbit d Rep. character $$\chi_{13}(5,\cdot)$$ Character field $$\Q(\zeta_{4})$$ Dimension 12 Newform subspaces 1 Sturm bound 8 Trace bound 0

# Related objects

## Defining parameters

 Level: $$N$$ = $$13$$ Weight: $$k$$ = $$7$$ 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: $$8$$ Trace bound: $$0$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{7}(13, [\chi])$$.

Total New Old
Modular forms 16 16 0
Cusp forms 12 12 0
Eisenstein series 4 4 0

## Trace form

 $$12q + 6q^{2} - 4q^{3} + 108q^{5} - 640q^{6} + 398q^{7} - 912q^{8} + 1940q^{9} + O(q^{10})$$ $$12q + 6q^{2} - 4q^{3} + 108q^{5} - 640q^{6} + 398q^{7} - 912q^{8} + 1940q^{9} + 1686q^{11} - 3926q^{13} + 5484q^{14} - 15268q^{15} + 2132q^{16} + 15254q^{18} + 1766q^{19} - 19044q^{20} + 3428q^{21} + 28832q^{22} + 31608q^{24} - 58266q^{26} - 20464q^{27} + 4092q^{28} - 90108q^{29} + 61014q^{31} - 64932q^{32} - 44452q^{33} + 259896q^{34} + 158772q^{35} - 40212q^{37} + 137852q^{39} - 104196q^{40} - 190416q^{41} - 959204q^{42} + 489372q^{44} - 151444q^{45} - 44412q^{46} + 562446q^{47} + 930308q^{48} + 82422q^{50} - 578500q^{52} + 509136q^{53} - 871432q^{54} - 1264036q^{55} + 939908q^{57} - 1019980q^{58} - 994458q^{59} + 2407804q^{60} + 1013696q^{61} + 865778q^{63} - 1130064q^{65} - 418352q^{66} - 1442386q^{67} - 2313132q^{68} + 2958968q^{70} - 655866q^{71} - 1706508q^{72} + 2588228q^{73} + 3373752q^{74} + 246984q^{76} + 77480q^{78} - 75316q^{79} - 2685408q^{80} - 4016140q^{81} + 894966q^{83} - 3220504q^{84} + 105396q^{85} + 3704832q^{86} + 2109064q^{87} - 977376q^{89} + 1088750q^{91} + 3682872q^{92} - 216268q^{93} - 6238300q^{94} + 896384q^{96} + 983388q^{97} + 1039302q^{98} + 2894714q^{99} + O(q^{100})$$

## Decomposition of $$S_{7}^{\mathrm{new}}(13, [\chi])$$ into newform subspaces

Label Dim. $$A$$ Field CM Traces $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$
13.7.d.a $$12$$ $$2.991$$ $$\mathbb{Q}[x]/(x^{12} - \cdots)$$ None $$6$$ $$-4$$ $$108$$ $$398$$ $$q-\beta _{2}q^{2}+(\beta _{1}+\beta _{2}-\beta _{3}-\beta _{7})q^{3}+\cdots$$