# Properties

 Label 325.2.f Level $325$ Weight $2$ Character orbit 325.f Rep. character $\chi_{325}(18,\cdot)$ Character field $\Q(\zeta_{4})$ Dimension $38$ Newform subspaces $4$ Sturm bound $70$ Trace bound $1$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$325 = 5^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 325.f (of order $$4$$ and degree $$2$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$65$$ Character field: $$\Q(i)$$ Newform subspaces: $$4$$ Sturm bound: $$70$$ Trace bound: $$1$$ Distinguishing $$T_p$$: $$2$$

## Dimensions

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

Total New Old
Modular forms 82 46 36
Cusp forms 58 38 20
Eisenstein series 24 8 16

## Trace form

 $$38 q + 4 q^{3} - 34 q^{4} - 4 q^{6} + 4 q^{7} + O(q^{10})$$ $$38 q + 4 q^{3} - 34 q^{4} - 4 q^{6} + 4 q^{7} - 20 q^{11} - 10 q^{13} + 18 q^{16} - 14 q^{17} - 18 q^{18} - 12 q^{19} - 8 q^{22} + 20 q^{23} + 52 q^{24} - 36 q^{26} - 20 q^{27} + 12 q^{28} - 16 q^{31} - 2 q^{34} + 44 q^{37} - 8 q^{38} + 8 q^{39} + 14 q^{41} - 20 q^{42} + 4 q^{43} + 48 q^{44} - 28 q^{46} - 28 q^{47} + 16 q^{48} - 50 q^{49} + 42 q^{52} + 14 q^{53} - 36 q^{54} - 24 q^{58} + 16 q^{59} - 8 q^{61} + 40 q^{62} + 6 q^{64} + 56 q^{66} - 2 q^{68} + 16 q^{69} + 4 q^{71} + 22 q^{72} - 12 q^{76} + 20 q^{77} - 4 q^{78} + 34 q^{81} - 34 q^{82} - 60 q^{83} + 24 q^{84} + 84 q^{86} - 16 q^{87} + 16 q^{88} - 6 q^{89} - 48 q^{91} - 44 q^{92} + 20 q^{93} - 212 q^{96} + 36 q^{99} + O(q^{100})$$

## 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.f.a $2$ $2.595$ $$\Q(\sqrt{-1})$$ None $$0$$ $$-2$$ $$0$$ $$4$$ $$q-iq^{2}+(-1+i)q^{3}+q^{4}+(1+i)q^{6}+\cdots$$
325.2.f.b $8$ $2.595$ 8.0.619810816.2 None $$0$$ $$6$$ $$0$$ $$0$$ $$q-\beta _{6}q^{2}+(1+\beta _{2}-\beta _{3})q^{3}+(-1-\beta _{4}+\cdots)q^{4}+\cdots$$
325.2.f.c $12$ $2.595$ $$\mathbb{Q}[x]/(x^{12} + \cdots)$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{7}q^{2}-\beta _{4}q^{3}+(-2+\beta _{5})q^{4}+(-1+\cdots)q^{6}+\cdots$$
325.2.f.d $16$ $2.595$ $$\mathbb{Q}[x]/(x^{16} + \cdots)$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q-\beta _{13}q^{2}+\beta _{3}q^{3}+\beta _{1}q^{4}+(1+\beta _{4}+\cdots)q^{6}+\cdots$$

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

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