# Properties

 Label 75.9.d Level $75$ Weight $9$ Character orbit 75.d Rep. character $\chi_{75}(74,\cdot)$ Character field $\Q$ Dimension $46$ Newform subspaces $4$ Sturm bound $90$ Trace bound $4$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$75 = 3 \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$9$$ Character orbit: $$[\chi]$$ $$=$$ 75.d (of order $$2$$ and degree $$1$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$15$$ Character field: $$\Q$$ Newform subspaces: $$4$$ Sturm bound: $$90$$ Trace bound: $$4$$ Distinguishing $$T_p$$: $$2$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{9}(75, [\chi])$$.

Total New Old
Modular forms 86 50 36
Cusp forms 74 46 28
Eisenstein series 12 4 8

## Trace form

 $$46 q + 5160 q^{4} - 2 q^{6} + 11492 q^{9} + O(q^{10})$$ $$46 q + 5160 q^{4} - 2 q^{6} + 11492 q^{9} + 511032 q^{16} + 359214 q^{19} + 918774 q^{21} - 287334 q^{24} - 3218858 q^{31} - 1006364 q^{34} + 13513874 q^{36} - 2402834 q^{39} - 36479184 q^{46} - 50793772 q^{49} - 22870682 q^{51} + 24424268 q^{54} + 50274102 q^{61} - 41461504 q^{64} - 62032210 q^{66} + 36826668 q^{69} + 400879388 q^{76} - 212023636 q^{79} - 177577364 q^{81} + 767549976 q^{84} + 11622338 q^{91} - 894613664 q^{94} - 66741466 q^{96} + 664033870 q^{99} + O(q^{100})$$

## Decomposition of $$S_{9}^{\mathrm{new}}(75, [\chi])$$ into newform subspaces

Label Dim $A$ Field CM Traces $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
75.9.d.a $2$ $30.553$ $$\Q(\sqrt{-1})$$ $$\Q(\sqrt{-3})$$ $$0$$ $$0$$ $$0$$ $$0$$ $$q+3^{4}iq^{3}-2^{8}q^{4}+4273iq^{7}-3^{8}q^{9}+\cdots$$
75.9.d.b $4$ $30.553$ $$\Q(i, \sqrt{14})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{2}q^{2}+(-9\beta _{1}+3\beta _{2})q^{3}+248q^{4}+\cdots$$
75.9.d.c $20$ $30.553$ $$\mathbb{Q}[x]/(x^{20} + \cdots)$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{1}q^{2}+(-2\beta _{1}-\beta _{4})q^{3}+(79-\beta _{2}+\cdots)q^{4}+\cdots$$
75.9.d.d $20$ $30.553$ $$\mathbb{Q}[x]/(x^{20} + \cdots)$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q-\beta _{1}q^{2}+(-\beta _{1}-\beta _{3})q^{3}+(155-\beta _{2}+\cdots)q^{4}+\cdots$$

## Decomposition of $$S_{9}^{\mathrm{old}}(75, [\chi])$$ into lower level spaces

$$S_{9}^{\mathrm{old}}(75, [\chi]) \simeq$$ $$S_{9}^{\mathrm{new}}(15, [\chi])$$$$^{\oplus 2}$$