# Properties

 Label 75.4.e Level $75$ Weight $4$ Character orbit 75.e Rep. character $\chi_{75}(32,\cdot)$ Character field $\Q(\zeta_{4})$ Dimension $32$ Newform subspaces $4$ Sturm bound $40$ Trace bound $6$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$75 = 3 \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 75.e (of order $$4$$ and degree $$2$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$15$$ Character field: $$\Q(i)$$ Newform subspaces: $$4$$ Sturm bound: $$40$$ Trace bound: $$6$$ Distinguishing $$T_p$$: $$2$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{4}(75, [\chi])$$.

Total New Old
Modular forms 72 40 32
Cusp forms 48 32 16
Eisenstein series 24 8 16

## Trace form

 $$32q + 6q^{3} + 12q^{6} + 16q^{7} + O(q^{10})$$ $$32q + 6q^{3} + 12q^{6} + 16q^{7} - 132q^{12} - 68q^{13} - 784q^{16} + 240q^{18} + 972q^{21} + 500q^{22} - 702q^{27} - 508q^{28} - 896q^{31} + 240q^{33} + 2364q^{36} + 1156q^{37} - 540q^{42} - 548q^{43} - 1496q^{46} + 1116q^{48} - 1128q^{51} - 224q^{52} - 684q^{57} - 60q^{58} - 2216q^{61} - 1428q^{63} - 1380q^{66} - 404q^{67} + 1800q^{72} + 2512q^{73} + 10248q^{76} + 360q^{78} + 1332q^{81} - 2800q^{82} + 1680q^{87} - 2460q^{88} - 3536q^{91} - 3408q^{93} - 10164q^{96} - 1904q^{97} + O(q^{100})$$

## Decomposition of $$S_{4}^{\mathrm{new}}(75, [\chi])$$ into newform subspaces

Label Dim. $$A$$ Field CM Traces $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$
75.4.e.a $$4$$ $$4.425$$ $$\Q(i, \sqrt{6})$$ $$\Q(\sqrt{-3})$$ $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{1}q^{3}-8\beta _{2}q^{4}-6\beta _{3}q^{7}+3^{3}\beta _{2}q^{9}+\cdots$$
75.4.e.b $$4$$ $$4.425$$ $$\Q(i, \sqrt{6})$$ $$\Q(\sqrt{-15})$$ $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{1}q^{2}-\beta _{3}q^{3}+19\beta _{2}q^{4}+3^{3}q^{6}+\cdots$$
75.4.e.c $$8$$ $$4.425$$ 8.0.$$\cdots$$.8 None $$0$$ $$6$$ $$0$$ $$16$$ $$q-\beta _{3}q^{2}+(1+\beta _{2}-\beta _{5}+\beta _{6}-\beta _{7})q^{3}+\cdots$$
75.4.e.d $$16$$ $$4.425$$ $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{3}q^{2}-\beta _{8}q^{3}+(6\beta _{1}+\beta _{5})q^{4}+(-6+\cdots)q^{6}+\cdots$$

## Decomposition of $$S_{4}^{\mathrm{old}}(75, [\chi])$$ into lower level spaces

$$S_{4}^{\mathrm{old}}(75, [\chi]) \cong$$ $$S_{4}^{\mathrm{new}}(15, [\chi])$$$$^{\oplus 2}$$