# Properties

 Label 48.9.g Level $48$ Weight $9$ Character orbit 48.g Rep. character $\chi_{48}(31,\cdot)$ Character field $\Q$ Dimension $8$ Newform subspaces $3$ Sturm bound $72$ Trace bound $5$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$48 = 2^{4} \cdot 3$$ Weight: $$k$$ $$=$$ $$9$$ Character orbit: $$[\chi]$$ $$=$$ 48.g (of order $$2$$ and degree $$1$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$4$$ Character field: $$\Q$$ Newform subspaces: $$3$$ Sturm bound: $$72$$ Trace bound: $$5$$ Distinguishing $$T_p$$: $$5$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{9}(48, [\chi])$$.

Total New Old
Modular forms 70 8 62
Cusp forms 58 8 50
Eisenstein series 12 0 12

## Trace form

 $$8 q + 1008 q^{5} - 17496 q^{9} + O(q^{10})$$ $$8 q + 1008 q^{5} - 17496 q^{9} + 59984 q^{13} + 115920 q^{17} - 121824 q^{21} + 2112280 q^{25} + 206640 q^{29} + 2760480 q^{33} - 3960560 q^{37} + 13472208 q^{41} - 2204496 q^{45} + 6149768 q^{49} - 42196176 q^{53} + 775008 q^{57} + 16762000 q^{61} - 65829024 q^{65} - 89954800 q^{73} + 39265920 q^{77} + 38263752 q^{81} + 147417696 q^{85} - 18967536 q^{89} + 129711456 q^{93} - 36206320 q^{97} + O(q^{100})$$

## Decomposition of $$S_{9}^{\mathrm{new}}(48, [\chi])$$ into newform subspaces

Label Dim $A$ Field CM Traces $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
48.9.g.a $2$ $19.554$ $$\Q(\sqrt{-3})$$ None $$0$$ $$0$$ $$-180$$ $$0$$ $$q+3^{3}\zeta_{6}q^{3}-90q^{5}+532\zeta_{6}q^{7}-3^{7}q^{9}+\cdots$$
48.9.g.b $2$ $19.554$ $$\Q(\sqrt{-3})$$ None $$0$$ $$0$$ $$1452$$ $$0$$ $$q-3\zeta_{6}q^{3}+726q^{5}-14^{2}\zeta_{6}q^{7}-3^{7}q^{9}+\cdots$$
48.9.g.c $4$ $19.554$ $$\Q(\sqrt{-3}, \sqrt{1801})$$ None $$0$$ $$0$$ $$-264$$ $$0$$ $$q-3^{3}\beta _{1}q^{3}+(-66+\beta _{2})q^{5}+(772\beta _{1}+\cdots)q^{7}+\cdots$$

## Decomposition of $$S_{9}^{\mathrm{old}}(48, [\chi])$$ into lower level spaces

$$S_{9}^{\mathrm{old}}(48, [\chi]) \cong$$ $$S_{9}^{\mathrm{new}}(4, [\chi])$$$$^{\oplus 6}$$$$\oplus$$$$S_{9}^{\mathrm{new}}(12, [\chi])$$$$^{\oplus 3}$$$$\oplus$$$$S_{9}^{\mathrm{new}}(16, [\chi])$$$$^{\oplus 2}$$