# Properties

 Label 625.2.a Level $625$ Weight $2$ Character orbit 625.a Rep. character $\chi_{625}(1,\cdot)$ Character field $\Q$ Dimension $32$ Newform subspaces $7$ Sturm bound $125$ Trace bound $3$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$625 = 5^{4}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 625.a (trivial) Character field: $$\Q$$ Newform subspaces: $$7$$ Sturm bound: $$125$$ Trace bound: $$3$$ Distinguishing $$T_p$$: $$2$$, $$3$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_0(625))$$.

Total New Old
Modular forms 77 48 29
Cusp forms 48 32 16
Eisenstein series 29 16 13

The following table gives the dimensions of the cuspidal new subspaces with specified eigenvalues for the Atkin-Lehner operators and the Fricke involution.

$$5$$Dim
$$+$$$$14$$
$$-$$$$18$$

## Trace form

 $$32 q + 24 q^{4} + 4 q^{6} + 16 q^{9} + O(q^{10})$$ $$32 q + 24 q^{4} + 4 q^{6} + 16 q^{9} + 4 q^{11} + 8 q^{14} + 12 q^{16} - 10 q^{19} - 6 q^{21} + 20 q^{24} - 6 q^{26} + 10 q^{29} - 6 q^{31} + 18 q^{34} + 12 q^{36} + 2 q^{39} + 14 q^{41} - 22 q^{44} - 26 q^{46} - 26 q^{49} - 46 q^{51} - 50 q^{54} - 10 q^{56} - 30 q^{59} + 4 q^{61} - 46 q^{64} - 2 q^{66} + 32 q^{69} + 24 q^{71} - 12 q^{74} - 40 q^{76} - 40 q^{79} - 8 q^{81} - 52 q^{84} - 56 q^{86} + 30 q^{89} - 46 q^{91} + 48 q^{94} + 84 q^{96} + 2 q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_0(625))$$ into newform subspaces

Label Dim $A$ Field CM Traces A-L signs $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 5
625.2.a.a $2$ $4.991$ $$\Q(\sqrt{5})$$ None $$-1$$ $$-3$$ $$0$$ $$-1$$ $+$ $$q-\beta q^{2}+(-1-\beta )q^{3}+(-1+\beta )q^{4}+\cdots$$
625.2.a.b $2$ $4.991$ $$\Q(\sqrt{5})$$ None $$-1$$ $$2$$ $$0$$ $$-1$$ $+$ $$q-\beta q^{2}+q^{3}+(-1+\beta )q^{4}-\beta q^{6}+\cdots$$
625.2.a.c $2$ $4.991$ $$\Q(\sqrt{5})$$ None $$1$$ $$-2$$ $$0$$ $$1$$ $+$ $$q+\beta q^{2}-q^{3}+(-1+\beta )q^{4}-\beta q^{6}+\cdots$$
625.2.a.d $2$ $4.991$ $$\Q(\sqrt{5})$$ None $$1$$ $$3$$ $$0$$ $$1$$ $-$ $$q+\beta q^{2}+(1+\beta )q^{3}+(-1+\beta )q^{4}+(1+\cdots)q^{6}+\cdots$$
625.2.a.e $8$ $4.991$ 8.8.6152203125.1 None $$-5$$ $$-5$$ $$0$$ $$-10$$ $+$ $$q+(-1+\beta _{1})q^{2}+(\beta _{3}-\beta _{5}+\beta _{7})q^{3}+\cdots$$
625.2.a.f $8$ $4.991$ 8.8.$$\cdots$$.2 None $$0$$ $$0$$ $$0$$ $$0$$ $-$ $$q+\beta _{1}q^{2}+(-\beta _{5}+\beta _{6})q^{3}+(1+\beta _{2}+\cdots)q^{4}+\cdots$$
625.2.a.g $8$ $4.991$ 8.8.6152203125.1 None $$5$$ $$5$$ $$0$$ $$10$$ $-$ $$q+(1-\beta _{1})q^{2}+(-\beta _{3}+\beta _{5}-\beta _{7})q^{3}+\cdots$$

## Decomposition of $$S_{2}^{\mathrm{old}}(\Gamma_0(625))$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(\Gamma_0(625)) \cong$$ $$S_{2}^{\mathrm{new}}(\Gamma_0(125))$$$$^{\oplus 2}$$