# Properties

 Label 1875.4.a Level $1875$ Weight $4$ Character orbit 1875.a Rep. character $\chi_{1875}(1,\cdot)$ Character field $\Q$ Dimension $240$ Newform subspaces $14$ Sturm bound $1000$ Trace bound $3$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$1875 = 3 \cdot 5^{4}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1875.a (trivial) Character field: $$\Q$$ Newform subspaces: $$14$$ Sturm bound: $$1000$$ Trace bound: $$3$$ Distinguishing $$T_p$$: $$2$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{4}(\Gamma_0(1875))$$.

Total New Old
Modular forms 780 240 540
Cusp forms 720 240 480
Eisenstein series 60 0 60

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.
$$+$$$$+$$$$+$$$$62$$
$$+$$$$-$$$$-$$$$58$$
$$-$$$$+$$$$-$$$$54$$
$$-$$$$-$$$$+$$$$66$$
Plus space$$+$$$$128$$
Minus space$$-$$$$112$$

## Trace form

 $$240 q + 960 q^{4} + 2160 q^{9} + O(q^{10})$$ $$240 q + 960 q^{4} + 2160 q^{9} + 3840 q^{16} - 180 q^{19} - 120 q^{21} + 440 q^{26} + 260 q^{29} + 180 q^{31} - 40 q^{34} + 8640 q^{36} + 420 q^{39} - 460 q^{41} - 2580 q^{44} - 2220 q^{46} + 12480 q^{49} - 420 q^{56} - 580 q^{61} + 16380 q^{64} + 380 q^{74} - 2880 q^{76} - 2520 q^{79} + 19440 q^{81} - 1920 q^{84} + 4980 q^{86} - 3340 q^{89} + 900 q^{91} + 3720 q^{94} + 3480 q^{96} + O(q^{100})$$

## Decomposition of $$S_{4}^{\mathrm{new}}(\Gamma_0(1875))$$ into newform subspaces

Label Dim $A$ Field CM Traces A-L signs $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 3 5
1875.4.a.a $10$ $110.629$ $$\mathbb{Q}[x]/(x^{10} - \cdots)$$ None $$-1$$ $$-30$$ $$0$$ $$51$$ $+$ $-$ $$q-\beta _{1}q^{2}-3q^{3}+(4+\beta _{2}-\beta _{3}+\beta _{4}+\cdots)q^{4}+\cdots$$
1875.4.a.b $10$ $110.629$ $$\mathbb{Q}[x]/(x^{10} - \cdots)$$ None $$-1$$ $$30$$ $$0$$ $$21$$ $-$ $-$ $$q-\beta _{1}q^{2}+3q^{3}+(2+\beta _{2}-\beta _{3})q^{4}+\cdots$$
1875.4.a.c $10$ $110.629$ $$\mathbb{Q}[x]/(x^{10} - \cdots)$$ None $$1$$ $$-30$$ $$0$$ $$-21$$ $+$ $+$ $$q+\beta _{1}q^{2}-3q^{3}+(2+\beta _{2}-\beta _{3})q^{4}+\cdots$$
1875.4.a.d $10$ $110.629$ $$\mathbb{Q}[x]/(x^{10} - \cdots)$$ None $$1$$ $$30$$ $$0$$ $$-51$$ $-$ $+$ $$q+\beta _{1}q^{2}+3q^{3}+(4+\beta _{2}-\beta _{3}+\beta _{4}+\cdots)q^{4}+\cdots$$
1875.4.a.e $14$ $110.629$ $$\mathbb{Q}[x]/(x^{14} - \cdots)$$ None $$-8$$ $$42$$ $$0$$ $$-29$$ $-$ $+$ $$q+(-1+\beta _{1})q^{2}+3q^{3}+(4-\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots$$
1875.4.a.f $14$ $110.629$ $$\mathbb{Q}[x]/(x^{14} - \cdots)$$ None $$0$$ $$-42$$ $$0$$ $$27$$ $+$ $+$ $$q-\beta _{1}q^{2}-3q^{3}+(4+\beta _{2})q^{4}+3\beta _{1}q^{6}+\cdots$$
1875.4.a.g $14$ $110.629$ $$\mathbb{Q}[x]/(x^{14} - \cdots)$$ None $$0$$ $$42$$ $$0$$ $$-27$$ $-$ $+$ $$q+\beta _{1}q^{2}+3q^{3}+(4+\beta _{2})q^{4}+3\beta _{1}q^{6}+\cdots$$
1875.4.a.h $14$ $110.629$ $$\mathbb{Q}[x]/(x^{14} - \cdots)$$ None $$8$$ $$-42$$ $$0$$ $$29$$ $+$ $+$ $$q+(1-\beta _{1})q^{2}-3q^{3}+(4-\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots$$
1875.4.a.i $16$ $110.629$ $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ None $$-1$$ $$48$$ $$0$$ $$-52$$ $-$ $+$ $$q-\beta _{1}q^{2}+3q^{3}+(2+\beta _{2})q^{4}-3\beta _{1}q^{6}+\cdots$$
1875.4.a.j $16$ $110.629$ $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ None $$1$$ $$-48$$ $$0$$ $$52$$ $+$ $-$ $$q+\beta _{1}q^{2}-3q^{3}+(2+\beta _{2})q^{4}-3\beta _{1}q^{6}+\cdots$$
1875.4.a.k $24$ $110.629$ None $$-1$$ $$-72$$ $$0$$ $$-62$$ $+$ $+$
1875.4.a.l $24$ $110.629$ None $$1$$ $$72$$ $$0$$ $$62$$ $-$ $-$
1875.4.a.m $32$ $110.629$ None $$-8$$ $$-96$$ $$0$$ $$-56$$ $+$ $-$
1875.4.a.n $32$ $110.629$ None $$8$$ $$96$$ $$0$$ $$56$$ $-$ $-$

## Decomposition of $$S_{4}^{\mathrm{old}}(\Gamma_0(1875))$$ into lower level spaces

$$S_{4}^{\mathrm{old}}(\Gamma_0(1875)) \cong$$ $$S_{4}^{\mathrm{new}}(\Gamma_0(5))$$$$^{\oplus 8}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_0(15))$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_0(25))$$$$^{\oplus 6}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_0(75))$$$$^{\oplus 3}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_0(125))$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_0(375))$$$$^{\oplus 2}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_0(625))$$$$^{\oplus 2}$$