# Properties

 Label 375.2.a Level $375$ Weight $2$ Character orbit 375.a Rep. character $\chi_{375}(1,\cdot)$ Character field $\Q$ Dimension $16$ Newform subspaces $6$ Sturm bound $100$ Trace bound $2$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$375 = 3 \cdot 5^{3}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 375.a (trivial) Character field: $$\Q$$ Newform subspaces: $$6$$ Sturm bound: $$100$$ Trace bound: $$2$$ Distinguishing $$T_p$$: $$2$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_0(375))$$.

Total New Old
Modular forms 60 16 44
Cusp forms 41 16 25
Eisenstein series 19 0 19

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

$$3$$$$5$$FrickeDim
$$+$$$$+$$$+$$$2$$
$$+$$$$-$$$-$$$6$$
$$-$$$$+$$$-$$$6$$
$$-$$$$-$$$+$$$2$$
Plus space$$+$$$$4$$
Minus space$$-$$$$12$$

## Trace form

 $$16 q + 18 q^{4} + 2 q^{6} + 16 q^{9} + O(q^{10})$$ $$16 q + 18 q^{4} + 2 q^{6} + 16 q^{9} + 4 q^{11} + 12 q^{14} + 30 q^{16} - 8 q^{19} - 16 q^{21} + 6 q^{24} + 12 q^{26} + 4 q^{29} - 4 q^{31} + 16 q^{34} + 18 q^{36} - 12 q^{39} + 16 q^{41} - 76 q^{44} - 84 q^{46} + 20 q^{49} + 8 q^{51} + 2 q^{54} - 60 q^{56} + 28 q^{59} - 38 q^{64} + 16 q^{66} + 12 q^{69} + 32 q^{71} - 68 q^{74} - 12 q^{76} - 40 q^{79} + 16 q^{81} - 52 q^{84} - 56 q^{86} + 32 q^{89} - 44 q^{91} - 2 q^{94} - 36 q^{96} + 4 q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_0(375))$$ into newform subspaces

Label Dim $A$ Field CM Traces A-L signs $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 3 5
375.2.a.a $2$ $2.994$ $$\Q(\sqrt{5})$$ None $$-3$$ $$2$$ $$0$$ $$-5$$ $-$ $-$ $$q+(-1-\beta )q^{2}+q^{3}+3\beta q^{4}+(-1+\cdots)q^{6}+\cdots$$
375.2.a.b $2$ $2.994$ $$\Q(\sqrt{5})$$ None $$-1$$ $$-2$$ $$0$$ $$-1$$ $+$ $+$ $$q-\beta q^{2}-q^{3}+(-1+\beta )q^{4}+\beta q^{6}+\cdots$$
375.2.a.c $2$ $2.994$ $$\Q(\sqrt{5})$$ None $$1$$ $$2$$ $$0$$ $$1$$ $-$ $+$ $$q+\beta q^{2}+q^{3}+(-1+\beta )q^{4}+\beta q^{6}+\cdots$$
375.2.a.d $2$ $2.994$ $$\Q(\sqrt{5})$$ None $$3$$ $$-2$$ $$0$$ $$5$$ $+$ $-$ $$q+(1+\beta )q^{2}-q^{3}+3\beta q^{4}+(-1-\beta )q^{6}+\cdots$$
375.2.a.e $4$ $2.994$ 4.4.2525.1 None $$-3$$ $$-4$$ $$0$$ $$4$$ $+$ $-$ $$q+(-1-\beta _{3})q^{2}-q^{3}+(1+\beta _{1}-\beta _{2}+\cdots)q^{4}+\cdots$$
375.2.a.f $4$ $2.994$ 4.4.2525.1 None $$3$$ $$4$$ $$0$$ $$-4$$ $-$ $+$ $$q+(1+\beta _{2}-\beta _{3})q^{2}+q^{3}+(2-\beta _{1}-\beta _{3})q^{4}+\cdots$$

## Decomposition of $$S_{2}^{\mathrm{old}}(\Gamma_0(375))$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(\Gamma_0(375)) \cong$$ $$S_{2}^{\mathrm{new}}(\Gamma_0(15))$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_0(75))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_0(125))$$$$^{\oplus 2}$$