# Properties

 Label 75.2.b Level $75$ Weight $2$ Character orbit 75.b Rep. character $\chi_{75}(49,\cdot)$ Character field $\Q$ Dimension $4$ Newform subspaces $2$ Sturm bound $20$ Trace bound $4$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$75 = 3 \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 75.b (of order $$2$$ and degree $$1$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$5$$ Character field: $$\Q$$ Newform subspaces: $$2$$ Sturm bound: $$20$$ Trace bound: $$4$$ Distinguishing $$T_p$$: $$2$$

## Dimensions

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

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

## Trace form

 $$4 q - 2 q^{4} - 2 q^{6} - 4 q^{9} + O(q^{10})$$ $$4 q - 2 q^{4} - 2 q^{6} - 4 q^{9} - 4 q^{11} + 12 q^{14} - 10 q^{16} + 2 q^{19} + 6 q^{21} + 6 q^{24} + 8 q^{26} - 16 q^{29} - 6 q^{31} - 4 q^{34} + 2 q^{36} - 2 q^{39} + 4 q^{41} - 16 q^{44} + 24 q^{46} + 10 q^{49} - 8 q^{51} + 2 q^{54} + 28 q^{59} + 10 q^{61} + 2 q^{64} - 16 q^{66} + 12 q^{69} - 32 q^{71} - 28 q^{74} - 28 q^{76} + 4 q^{81} - 12 q^{84} - 4 q^{86} + 12 q^{89} - 6 q^{91} + 8 q^{94} + 26 q^{96} + 4 q^{99} + O(q^{100})$$

## 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.b.a $2$ $0.599$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+2iq^{2}+iq^{3}-2q^{4}-2q^{6}-3iq^{7}+\cdots$$
75.2.b.b $2$ $0.599$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+iq^{2}-iq^{3}+q^{4}+q^{6}+3iq^{8}+\cdots$$