# Properties

 Label 325.2.b Level $325$ Weight $2$ Character orbit 325.b Rep. character $\chi_{325}(274,\cdot)$ Character field $\Q$ Dimension $18$ Newform subspaces $6$ Sturm bound $70$ Trace bound $6$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$325 = 5^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 325.b (of order $$2$$ and degree $$1$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$5$$ Character field: $$\Q$$ Newform subspaces: $$6$$ Sturm bound: $$70$$ Trace bound: $$6$$ Distinguishing $$T_p$$: $$2$$, $$3$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(325, [\chi])$$.

Total New Old
Modular forms 42 18 24
Cusp forms 30 18 12
Eisenstein series 12 0 12

## Trace form

 $$18 q - 10 q^{4} - 12 q^{6} - 14 q^{9} + O(q^{10})$$ $$18 q - 10 q^{4} - 12 q^{6} - 14 q^{9} + 12 q^{11} - 4 q^{14} + 2 q^{16} + 8 q^{19} - 12 q^{21} + 8 q^{24} + 6 q^{26} + 28 q^{29} + 8 q^{31} + 18 q^{36} - 8 q^{39} - 76 q^{44} - 28 q^{46} - 22 q^{49} + 92 q^{54} - 44 q^{56} - 48 q^{59} - 44 q^{61} + 70 q^{64} - 8 q^{66} + 28 q^{69} + 52 q^{71} + 4 q^{74} - 40 q^{76} - 12 q^{79} - 30 q^{81} + 80 q^{84} + 84 q^{86} + 16 q^{89} + 4 q^{94} + 32 q^{96} - 76 q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(325, [\chi])$$ into newform subspaces

Label Dim $A$ Field CM Traces $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
325.2.b.a $2$ $2.595$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+2iq^{2}-iq^{3}-2q^{4}+2q^{6}+2iq^{7}+\cdots$$
325.2.b.b $2$ $2.595$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+iq^{2}-2iq^{3}+q^{4}+2q^{6}+4iq^{7}+\cdots$$
325.2.b.c $2$ $2.595$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+iq^{3}+2q^{4}+4iq^{7}+2q^{9}-6q^{11}+\cdots$$
325.2.b.d $4$ $2.595$ $$\Q(\zeta_{8})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+(\zeta_{8}+\zeta_{8}^{2})q^{2}+2\zeta_{8}^{2}q^{3}+(-1-2\zeta_{8}^{3})q^{4}+\cdots$$
325.2.b.e $4$ $2.595$ $$\Q(\zeta_{12})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q-\zeta_{12}^{2}q^{2}+(-\zeta_{12}-\zeta_{12}^{2})q^{3}-q^{4}+\cdots$$
325.2.b.f $4$ $2.595$ $$\Q(\zeta_{8})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+(\zeta_{8}+\zeta_{8}^{2})q^{2}-\zeta_{8}^{2}q^{3}+(-1-2\zeta_{8}^{3})q^{4}+\cdots$$

## Decomposition of $$S_{2}^{\mathrm{old}}(325, [\chi])$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(325, [\chi]) \cong$$ $$S_{2}^{\mathrm{new}}(65, [\chi])$$$$^{\oplus 2}$$