# Properties

 Label 325.2.n Level $325$ Weight $2$ Character orbit 325.n Rep. character $\chi_{325}(101,\cdot)$ Character field $\Q(\zeta_{6})$ Dimension $38$ Newform subspaces $6$ Sturm bound $70$ Trace bound $3$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$325 = 5^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 325.n (of order $$6$$ and degree $$2$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$13$$ Character field: $$\Q(\zeta_{6})$$ Newform subspaces: $$6$$ 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 84 50 34
Cusp forms 60 38 22
Eisenstein series 24 12 12

## Trace form

 $$38q + 3q^{2} + 17q^{4} + 6q^{7} - 13q^{9} + O(q^{10})$$ $$38q + 3q^{2} + 17q^{4} + 6q^{7} - 13q^{9} - 18q^{11} - 16q^{12} + 5q^{13} - 24q^{14} - 23q^{16} - q^{17} + 18q^{19} + 12q^{22} + 4q^{23} + 54q^{24} - 39q^{26} + 12q^{27} + 18q^{28} + 3q^{29} + 3q^{32} - 42q^{33} + 11q^{36} - 21q^{37} + 4q^{38} + 10q^{39} + 27q^{41} - 4q^{42} + 10q^{43} + 60q^{46} - 38q^{48} - 13q^{49} + 12q^{51} + 4q^{52} + 30q^{53} - 18q^{54} - 42q^{56} - 45q^{58} - 36q^{59} + 21q^{61} - 10q^{62} + 24q^{63} - 14q^{64} - 100q^{66} - 12q^{67} + 17q^{68} + 36q^{69} - 36q^{71} + 45q^{72} - 15q^{74} + 30q^{76} + 36q^{77} + 80q^{78} - 48q^{79} - 19q^{81} - 13q^{82} - 204q^{84} - 16q^{87} + 18q^{88} + 12q^{89} + 42q^{91} - 56q^{92} + 12q^{93} - 4q^{94} + 18q^{97} - 93q^{98} + O(q^{100})$$

## 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.n.a $$2$$ $$2.595$$ $$\Q(\sqrt{-3})$$ None $$3$$ $$2$$ $$0$$ $$0$$ $$q+(1+\zeta_{6})q^{2}+(2-2\zeta_{6})q^{3}+\zeta_{6}q^{4}+\cdots$$
325.2.n.b $$4$$ $$2.595$$ $$\Q(\sqrt{-3}, \sqrt{-7})$$ None $$-3$$ $$2$$ $$0$$ $$6$$ $$q+(-1+\beta _{3})q^{2}+\beta _{2}q^{3}+(\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots$$
325.2.n.c $$4$$ $$2.595$$ $$\Q(\sqrt{-3}, \sqrt{-7})$$ None $$3$$ $$-2$$ $$0$$ $$-6$$ $$q+(1-\beta _{3})q^{2}-\beta _{2}q^{3}+(\beta _{1}+\beta _{2}-2\beta _{3})q^{4}+\cdots$$
325.2.n.d $$8$$ $$2.595$$ 8.0.22581504.2 None $$0$$ $$-2$$ $$0$$ $$6$$ $$q+(\beta _{3}+\beta _{5}+\beta _{7})q^{2}+(-1+\beta _{1}+2\beta _{2}+\cdots)q^{3}+\cdots$$
325.2.n.e $$10$$ $$2.595$$ $$\mathbb{Q}[x]/(x^{10} + \cdots)$$ None $$0$$ $$-3$$ $$0$$ $$-6$$ $$q-\beta _{3}q^{2}+(-1-\beta _{2}-\beta _{8}+\beta _{9})q^{3}+\cdots$$
325.2.n.f $$10$$ $$2.595$$ $$\mathbb{Q}[x]/(x^{10} + \cdots)$$ None $$0$$ $$3$$ $$0$$ $$6$$ $$q+(\beta _{1}+\beta _{3})q^{2}+(-\beta _{2}+\beta _{9})q^{3}+(2+\cdots)q^{4}+\cdots$$

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

$$S_{2}^{\mathrm{old}}(325, [\chi]) \cong$$ $$S_{2}^{\mathrm{new}}(13, [\chi])$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(65, [\chi])$$$$^{\oplus 2}$$