# Properties

 Label 40.2.d Level $40$ Weight $2$ Character orbit 40.d Rep. character $\chi_{40}(21,\cdot)$ Character field $\Q$ Dimension $4$ Newform subspaces $1$ Sturm bound $12$ Trace bound $0$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$40 = 2^{3} \cdot 5$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 40.d (of order $$2$$ and degree $$1$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$8$$ Character field: $$\Q$$ Newform subspaces: $$1$$ Sturm bound: $$12$$ Trace bound: $$0$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(40, [\chi])$$.

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

## Trace form

 $$4 q - 2 q^{2} + 4 q^{6} - 4 q^{7} - 8 q^{8} - 4 q^{9} + O(q^{10})$$ $$4 q - 2 q^{2} + 4 q^{6} - 4 q^{7} - 8 q^{8} - 4 q^{9} + 2 q^{10} - 4 q^{12} - 4 q^{14} + 4 q^{15} + 8 q^{16} + 14 q^{18} + 4 q^{20} + 4 q^{22} - 4 q^{23} + 8 q^{24} - 4 q^{25} - 12 q^{26} + 12 q^{28} - 8 q^{30} - 8 q^{31} + 8 q^{32} + 8 q^{33} - 12 q^{34} - 24 q^{36} - 20 q^{38} + 24 q^{39} - 8 q^{40} - 8 q^{41} - 4 q^{42} + 8 q^{44} + 20 q^{46} + 20 q^{47} - 24 q^{48} - 12 q^{49} + 2 q^{50} + 8 q^{54} - 8 q^{55} + 8 q^{56} + 8 q^{57} + 24 q^{58} + 12 q^{60} + 16 q^{62} - 20 q^{63} - 16 q^{66} + 24 q^{68} - 8 q^{70} + 8 q^{71} + 8 q^{72} + 16 q^{73} + 4 q^{74} - 16 q^{76} - 24 q^{78} - 32 q^{79} + 4 q^{81} - 8 q^{82} - 8 q^{84} + 20 q^{86} - 48 q^{87} - 16 q^{88} + 8 q^{89} + 10 q^{90} - 36 q^{92} - 4 q^{94} + 16 q^{95} + 32 q^{96} - 16 q^{97} + 18 q^{98} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(40, [\chi])$$ into newform subspaces

Label Dim $A$ Field CM Traces $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
40.2.d.a $4$ $0.319$ $$\Q(\zeta_{12})$$ None $$-2$$ $$0$$ $$0$$ $$-4$$ $$q+(-\zeta_{12}+\zeta_{12}^{2})q^{2}+(-1+\zeta_{12}+\cdots)q^{3}+\cdots$$