# Properties

 Label 1875.2.a Level $1875$ Weight $2$ Character orbit 1875.a Rep. character $\chi_{1875}(1,\cdot)$ Character field $\Q$ Dimension $80$ Newform subspaces $16$ Sturm bound $500$ Trace bound $3$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$1875 = 3 \cdot 5^{4}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1875.a (trivial) Character field: $$\Q$$ Newform subspaces: $$16$$ Sturm bound: $$500$$ Trace bound: $$3$$ Distinguishing $$T_p$$: $$2$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_0(1875))$$.

Total New Old
Modular forms 280 80 200
Cusp forms 221 80 141
Eisenstein series 59 0 59

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
$$+$$$$+$$$+$$$18$$
$$+$$$$-$$$-$$$22$$
$$-$$$$+$$$-$$$26$$
$$-$$$$-$$$+$$$14$$
Plus space$$+$$$$32$$
Minus space$$-$$$$48$$

## Trace form

 $$80 q + 80 q^{4} + 80 q^{9} + O(q^{10})$$ $$80 q + 80 q^{4} + 80 q^{9} + 80 q^{16} + 10 q^{19} + 10 q^{21} - 20 q^{26} - 20 q^{29} + 10 q^{31} - 20 q^{34} + 80 q^{36} + 10 q^{39} - 20 q^{41} + 60 q^{44} + 60 q^{46} + 90 q^{49} + 60 q^{56} - 10 q^{61} + 140 q^{64} + 40 q^{74} + 40 q^{76} + 40 q^{79} + 80 q^{81} + 40 q^{84} + 60 q^{86} - 20 q^{89} + 50 q^{91} - 40 q^{94} - 40 q^{96} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_0(1875))$$ into newform subspaces

Label Dim $A$ Field CM Traces A-L signs $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 3 5
1875.2.a.a $2$ $14.972$ $$\Q(\sqrt{5})$$ None $$-2$$ $$-2$$ $$0$$ $$0$$ $+$ $+$ $$q-q^{2}-q^{3}-q^{4}+q^{6}+(2-4\beta )q^{7}+\cdots$$
1875.2.a.b $2$ $14.972$ $$\Q(\sqrt{5})$$ None $$-1$$ $$-2$$ $$0$$ $$4$$ $+$ $+$ $$q-\beta q^{2}-q^{3}+(-1+\beta )q^{4}+\beta q^{6}+\cdots$$
1875.2.a.c $2$ $14.972$ $$\Q(\sqrt{5})$$ None $$1$$ $$2$$ $$0$$ $$-4$$ $-$ $-$ $$q+\beta q^{2}+q^{3}+(-1+\beta )q^{4}+\beta q^{6}+\cdots$$
1875.2.a.d $2$ $14.972$ $$\Q(\sqrt{5})$$ None $$2$$ $$2$$ $$0$$ $$0$$ $-$ $+$ $$q+q^{2}+q^{3}-q^{4}+q^{6}+(-2+4\beta )q^{7}+\cdots$$
1875.2.a.e $4$ $14.972$ 4.4.5125.1 None $$-2$$ $$-4$$ $$0$$ $$-2$$ $+$ $+$ $$q-\beta _{1}q^{2}-q^{3}+(2+\beta _{1}+\beta _{2})q^{4}+\beta _{1}q^{6}+\cdots$$
1875.2.a.f $4$ $14.972$ $$\Q(\zeta_{15})^+$$ None $$-1$$ $$4$$ $$0$$ $$-5$$ $-$ $-$ $$q+(\beta _{2}+\beta _{3})q^{2}+q^{3}+\beta _{1}q^{4}+(\beta _{2}+\beta _{3})q^{6}+\cdots$$
1875.2.a.g $4$ $14.972$ $$\Q(\zeta_{15})^+$$ None $$1$$ $$-4$$ $$0$$ $$5$$ $+$ $+$ $$q+(-\beta _{2}-\beta _{3})q^{2}-q^{3}+\beta _{1}q^{4}+(\beta _{2}+\cdots)q^{6}+\cdots$$
1875.2.a.h $4$ $14.972$ 4.4.5125.1 None $$2$$ $$4$$ $$0$$ $$2$$ $-$ $+$ $$q+\beta _{1}q^{2}+q^{3}+(2+\beta _{1}+\beta _{2})q^{4}+\beta _{1}q^{6}+\cdots$$
1875.2.a.i $6$ $14.972$ 6.6.46840000.1 None $$-1$$ $$6$$ $$0$$ $$2$$ $-$ $+$ $$q-\beta _{1}q^{2}+q^{3}+(2+\beta _{2})q^{4}-\beta _{1}q^{6}+\cdots$$
1875.2.a.j $6$ $14.972$ 6.6.44400625.1 None $$0$$ $$-6$$ $$0$$ $$-6$$ $+$ $+$ $$q-\beta _{1}q^{2}-q^{3}+(1-\beta _{3}+\beta _{4})q^{4}+\beta _{1}q^{6}+\cdots$$
1875.2.a.k $6$ $14.972$ 6.6.44400625.1 None $$0$$ $$6$$ $$0$$ $$6$$ $-$ $+$ $$q+\beta _{1}q^{2}+q^{3}+(1-\beta _{3}+\beta _{4})q^{4}+\beta _{1}q^{6}+\cdots$$
1875.2.a.l $6$ $14.972$ 6.6.46840000.1 None $$1$$ $$-6$$ $$0$$ $$-2$$ $+$ $-$ $$q+\beta _{1}q^{2}-q^{3}+(2+\beta _{2})q^{4}-\beta _{1}q^{6}+\cdots$$
1875.2.a.m $8$ $14.972$ 8.8.5444000000.1 None $$-4$$ $$8$$ $$0$$ $$-8$$ $-$ $-$ $$q+(-1+\beta _{1})q^{2}+q^{3}+(1-\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots$$
1875.2.a.n $8$ $14.972$ 8.8.$$\cdots$$.1 None $$-1$$ $$-8$$ $$0$$ $$-12$$ $+$ $-$ $$q-\beta _{1}q^{2}-q^{3}+(1+\beta _{4}+\beta _{5})q^{4}+\beta _{1}q^{6}+\cdots$$
1875.2.a.o $8$ $14.972$ 8.8.$$\cdots$$.1 None $$1$$ $$8$$ $$0$$ $$12$$ $-$ $+$ $$q+\beta _{1}q^{2}+q^{3}+(1+\beta _{4}+\beta _{5})q^{4}+\beta _{1}q^{6}+\cdots$$
1875.2.a.p $8$ $14.972$ 8.8.5444000000.1 None $$4$$ $$-8$$ $$0$$ $$8$$ $+$ $-$ $$q+\beta _{1}q^{2}-q^{3}+(\beta _{1}+\beta _{2})q^{4}-\beta _{1}q^{6}+\cdots$$

## Decomposition of $$S_{2}^{\mathrm{old}}(\Gamma_0(1875))$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(\Gamma_0(1875)) \simeq$$ $$S_{2}^{\mathrm{new}}(\Gamma_0(15))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_0(75))$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_0(125))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_0(375))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_0(625))$$$$^{\oplus 2}$$