Space of modular forms of level 1875, weight 4, and trivial character

• 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: $3$

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.

\(3\)	\(5\)	Fricke	Dim
\(+\)	\(+\)	$+$	\(62\)
\(+\)	\(-\)	$-$	\(58\)
\(-\)	\(+\)	$-$	\(54\)
\(-\)	\(-\)	$+$	\(66\)
Plus space		\(+\)	\(128\)
Minus space		\(-\)	\(112\)

Trace form

\( 240 q + 960 q^{4} + 2160 q^{9} + O(q^{10}) \)

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

Label	Level	Weight	Char	Prim	Char order	Dim	Rel. Dim	$A$	Field	CM	Self-dual	Inner twists	Rank*	Traces					A-L signs		Fricke sign	Coefficient ring index	Sato-Tate	$q$-expansion
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$		3	5
1875.4.a.a	$1875$	$4$	1875.a	1.a	$1$	$10$	$10$	$110.629$	\(\mathbb{Q}[x]/(x^{10} - \cdots)\)	None	✓	$1$	$1$	\(-1\)	\(-30\)	\(0\)	\(51\)		$+$	$-$	$-$	$2^{2}\cdot 5$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}-3q^{3}+(4+\beta _{2}-\beta _{3}+\beta _{4}+\cdots)q^{4}+\cdots\)
1875.4.a.b	$1875$	$4$	1875.a	1.a	$1$	$10$	$10$	$110.629$	\(\mathbb{Q}[x]/(x^{10} - \cdots)\)	None	✓	$1$	$0$	\(-1\)	\(30\)	\(0\)	\(21\)		$-$	$-$	$+$	$5$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+3q^{3}+(2+\beta _{2}-\beta _{3})q^{4}+\cdots\)
1875.4.a.c	$1875$	$4$	1875.a	1.a	$1$	$10$	$10$	$110.629$	\(\mathbb{Q}[x]/(x^{10} - \cdots)\)	None	✓	$1$	$0$	\(1\)	\(-30\)	\(0\)	\(-21\)		$+$	$+$	$+$	$5$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}-3q^{3}+(2+\beta _{2}-\beta _{3})q^{4}+\cdots\)
1875.4.a.d	$1875$	$4$	1875.a	1.a	$1$	$10$	$10$	$110.629$	\(\mathbb{Q}[x]/(x^{10} - \cdots)\)	None	✓	$1$	$1$	\(1\)	\(30\)	\(0\)	\(-51\)		$-$	$+$	$-$	$2^{2}\cdot 5$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+3q^{3}+(4+\beta _{2}-\beta _{3}+\beta _{4}+\cdots)q^{4}+\cdots\)
1875.4.a.e	$1875$	$4$	1875.a	1.a	$1$	$14$	$14$	$110.629$	\(\mathbb{Q}[x]/(x^{14} - \cdots)\)	None	✓	$1$	$1$	\(-8\)	\(42\)	\(0\)	\(-29\)		$-$	$+$	$-$	$2^{2}\cdot 5^{3}$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{2}+3q^{3}+(4-\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)
1875.4.a.f	$1875$	$4$	1875.a	1.a	$1$	$14$	$14$	$110.629$	\(\mathbb{Q}[x]/(x^{14} - \cdots)\)	None	✓	$1$	$0$	\(0\)	\(-42\)	\(0\)	\(27\)		$+$	$+$	$+$	$2^{2}\cdot 5^{3}$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}-3q^{3}+(4+\beta _{2})q^{4}+3\beta _{1}q^{6}+\cdots\)
1875.4.a.g	$1875$	$4$	1875.a	1.a	$1$	$14$	$14$	$110.629$	\(\mathbb{Q}[x]/(x^{14} - \cdots)\)	None	✓	$1$	$1$	\(0\)	\(42\)	\(0\)	\(-27\)		$-$	$+$	$-$	$2^{2}\cdot 5^{3}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+3q^{3}+(4+\beta _{2})q^{4}+3\beta _{1}q^{6}+\cdots\)
1875.4.a.h	$1875$	$4$	1875.a	1.a	$1$	$14$	$14$	$110.629$	\(\mathbb{Q}[x]/(x^{14} - \cdots)\)	None	✓	$1$	$0$	\(8\)	\(-42\)	\(0\)	\(29\)		$+$	$+$	$+$	$2^{2}\cdot 5^{3}$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{2}-3q^{3}+(4-\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)
1875.4.a.i	$1875$	$4$	1875.a	1.a	$1$	$16$	$16$	$110.629$	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)	None	✓	$1$	$1$	\(-1\)	\(48\)	\(0\)	\(-52\)		$-$	$+$	$-$	$2^{4}\cdot 5^{6}$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+3q^{3}+(2+\beta _{2})q^{4}-3\beta _{1}q^{6}+\cdots\)
1875.4.a.j	$1875$	$4$	1875.a	1.a	$1$	$16$	$16$	$110.629$	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)	None	✓	$1$	$1$	\(1\)	\(-48\)	\(0\)	\(52\)		$+$	$-$	$-$	$2^{4}\cdot 5^{6}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}-3q^{3}+(2+\beta _{2})q^{4}-3\beta _{1}q^{6}+\cdots\)
1875.4.a.k	$1875$	$4$	1875.a	1.a	$1$	$24$	$24$	$110.629$		None	✓	$1$	$0$	\(-1\)	\(-72\)	\(0\)	\(-62\)		$+$	$+$	$+$		$\mathrm{SU}(2)$
1875.4.a.l	$1875$	$4$	1875.a	1.a	$1$	$24$	$24$	$110.629$		None	✓	$1$	$0$	\(1\)	\(72\)	\(0\)	\(62\)		$-$	$-$	$+$		$\mathrm{SU}(2)$
1875.4.a.m	$1875$	$4$	1875.a	1.a	$1$	$32$	$32$	$110.629$		None	✓	$1$	$1$	\(-8\)	\(-96\)	\(0\)	\(-56\)		$+$	$-$	$-$		$\mathrm{SU}(2)$
1875.4.a.n	$1875$	$4$	1875.a	1.a	$1$	$32$	$32$	$110.629$		None	✓	$1$	$0$	\(8\)	\(96\)	\(0\)	\(56\)		$-$	$-$	$+$		$\mathrm{SU}(2)$

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

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