# Properties

 Label 325.2.e Level $325$ Weight $2$ Character orbit 325.e Rep. character $\chi_{325}(126,\cdot)$ Character field $\Q(\zeta_{3})$ Dimension $40$ Newform subspaces $5$ Sturm bound $70$ Trace bound $3$

# Related objects

## Defining parameters

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

## Dimensions

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

Total New Old
Modular forms 80 52 28
Cusp forms 56 40 16
Eisenstein series 24 12 12

## Trace form

 $$40 q + 2 q^{3} - 18 q^{4} - 10 q^{6} + 2 q^{7} + 12 q^{8} - 14 q^{9} + O(q^{10})$$ $$40 q + 2 q^{3} - 18 q^{4} - 10 q^{6} + 2 q^{7} + 12 q^{8} - 14 q^{9} - 6 q^{11} + 4 q^{12} + 4 q^{13} + 8 q^{14} - 18 q^{16} + 10 q^{17} - 8 q^{18} - 4 q^{19} - 24 q^{21} - 4 q^{22} + 6 q^{23} - 8 q^{24} + 20 q^{27} - 10 q^{28} - 12 q^{29} - 32 q^{31} - 2 q^{32} - 14 q^{33} + 80 q^{34} + 32 q^{36} + 6 q^{37} - 24 q^{38} + 12 q^{39} - 10 q^{41} - 4 q^{42} - 14 q^{43} - 8 q^{44} + 10 q^{46} - 24 q^{47} + 12 q^{48} - 12 q^{49} - 18 q^{52} - 40 q^{53} - 26 q^{54} + 2 q^{56} - 12 q^{57} + 32 q^{58} + 16 q^{59} - 4 q^{61} + 4 q^{62} + 12 q^{63} - 8 q^{64} + 124 q^{66} - 6 q^{67} + 34 q^{68} - 30 q^{71} + 12 q^{72} + 56 q^{73} - 2 q^{74} - 32 q^{76} - 4 q^{77} - 12 q^{78} + 40 q^{79} - 24 q^{81} - 20 q^{82} + 8 q^{83} - 4 q^{84} - 112 q^{86} - 18 q^{87} + 2 q^{88} + 16 q^{89} - 30 q^{91} + 20 q^{92} - 8 q^{93} + 10 q^{94} + 84 q^{96} - 22 q^{97} - 12 q^{98} + 40 q^{99} + O(q^{100})$$

## 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.e.a $$4$$ $$2.595$$ $$\Q(\sqrt{-3}, \sqrt{13})$$ None $$-1$$ $$2$$ $$0$$ $$-2$$ $$q-\beta _{1}q^{2}+(1+\beta _{2})q^{3}+(-1+\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots$$
325.2.e.b $$4$$ $$2.595$$ $$\Q(\sqrt{-3}, \sqrt{5})$$ None $$1$$ $$0$$ $$0$$ $$4$$ $$q+\beta _{1}q^{2}+(1-2\beta _{1}+\beta _{3})q^{3}+(\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots$$
325.2.e.c $$10$$ $$2.595$$ $$\mathbb{Q}[x]/(x^{10} + \cdots)$$ None $$0$$ $$-3$$ $$0$$ $$2$$ $$q+\beta _{1}q^{2}+(-1-\beta _{4}+\beta _{6}+\beta _{8})q^{3}+\cdots$$
325.2.e.d $$10$$ $$2.595$$ $$\mathbb{Q}[x]/(x^{10} + \cdots)$$ None $$0$$ $$3$$ $$0$$ $$-2$$ $$q-\beta _{1}q^{2}+(1+\beta _{4}-\beta _{6}-\beta _{8})q^{3}+(-\beta _{6}+\cdots)q^{4}+\cdots$$
325.2.e.e $$12$$ $$2.595$$ $$\mathbb{Q}[x]/(x^{12} + \cdots)$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{6}q^{2}+(\beta _{6}-\beta _{11})q^{3}+(-1+\beta _{2}+\cdots)q^{4}+\cdots$$

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

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