# Properties

 Label 75.2.e Level $75$ Weight $2$ Character orbit 75.e Rep. character $\chi_{75}(32,\cdot)$ Character field $\Q(\zeta_{4})$ Dimension $8$ Newform subspaces $2$ Sturm bound $20$ Trace bound $6$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$75 = 3 \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 75.e (of order $$4$$ and degree $$2$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$15$$ Character field: $$\Q(i)$$ Newform subspaces: $$2$$ Sturm bound: $$20$$ Trace bound: $$6$$ Distinguishing $$T_p$$: $$2$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(75, [\chi])$$.

Total New Old
Modular forms 32 16 16
Cusp forms 8 8 0
Eisenstein series 24 8 16

## Trace form

 $$8 q - 12 q^{6} + O(q^{10})$$ $$8 q - 12 q^{6} + 4 q^{16} - 12 q^{21} - 4 q^{31} + 36 q^{36} - 24 q^{46} + 48 q^{51} - 44 q^{61} - 8 q^{76} - 72 q^{81} + 36 q^{91} - 36 q^{96} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(75, [\chi])$$ into newform subspaces

Label Dim $A$ Field CM Traces $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
75.2.e.a $4$ $0.599$ $$\Q(i, \sqrt{6})$$ $$\Q(\sqrt{-15})$$ $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{1}q^{2}+\beta _{3}q^{3}+\beta _{2}q^{4}-3q^{6}-\beta _{3}q^{8}+\cdots$$
75.2.e.b $4$ $0.599$ $$\Q(i, \sqrt{6})$$ $$\Q(\sqrt{-3})$$ $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{1}q^{3}-2\beta _{2}q^{4}+\beta _{3}q^{7}+3\beta _{2}q^{9}+\cdots$$