Space of modular forms of level 1175, weight 4, and trivial character

• Label: 1175.4.a
• Level: $1175$
• Weight: $4$
• Character orbit: 1175.a
• Rep. character: $\chi_{1175}(1,\cdot)$
• Character field: $\Q$
• Dimension: $219$
• Newform subspaces: $12$
• Sturm bound: $480$
• Trace bound: $2$

Defining parameters

Level:	\( N \)	\(=\)	\( 1175 = 5^{2} \cdot 47 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1175.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 12 \)
Sturm bound:			\(480\)
Trace bound:			\(2\)
Distinguishing \(T_p\):			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(1175))\).

	Total	New	Old
Modular forms	366	219	147
Cusp forms	354	219	135
Eisenstein series	12	0	12

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

\(5\)	\(47\)	Fricke	Dim
\(+\)	\(+\)	$+$	\(59\)
\(+\)	\(-\)	$-$	\(44\)
\(-\)	\(+\)	$-$	\(53\)
\(-\)	\(-\)	$+$	\(63\)
Plus space		\(+\)	\(122\)
Minus space		\(-\)	\(97\)

Trace form

\( 219 q - 6 q^{2} + 2 q^{3} + 890 q^{4} + 32 q^{6} + 18 q^{7} - 30 q^{8} + 1959 q^{9} + O(q^{10}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(1175))\) 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
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$		5	47
1175.4.a.a	$1175$	$4$	1175.a	1.a	$1$	$3$	$3$	$69.327$	3.3.1101.1	None	✓	$1$	$0$	\(5\)	\(5\)	\(0\)	\(45\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(2+\beta _{2})q^{2}+(1+\beta _{1}-\beta _{2})q^{3}+(2+\cdots)q^{4}+\cdots\)
1175.4.a.b	$1175$	$4$	1175.a	1.a	$1$	$8$	$8$	$69.327$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓	$1$	$1$	\(-3\)	\(-7\)	\(0\)	\(-39\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+(-1-\beta _{5})q^{3}+(5+\beta _{2}-\beta _{3}+\cdots)q^{4}+\cdots\)
1175.4.a.c	$1175$	$4$	1175.a	1.a	$1$	$8$	$8$	$69.327$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓	$1$	$1$	\(3\)	\(10\)	\(0\)	\(15\)		$+$	$-$	$-$	$2^{3}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(2-\beta _{1}-\beta _{3})q^{3}+(1+\beta _{1}+\cdots)q^{4}+\cdots\)
1175.4.a.d	$1175$	$4$	1175.a	1.a	$1$	$10$	$10$	$69.327$	\(\mathbb{Q}[x]/(x^{10} - \cdots)\)	None	✓	$1$	$1$	\(-7\)	\(-8\)	\(0\)	\(-9\)		$+$	$-$	$-$	$2^{3}$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{2}+(-1+\beta _{6})q^{3}+(4+\cdots)q^{4}+\cdots\)
1175.4.a.e	$1175$	$4$	1175.a	1.a	$1$	$13$	$13$	$69.327$	\(\mathbb{Q}[x]/(x^{13} - \cdots)\)	None	✓	$1$	$0$	\(3\)	\(10\)	\(0\)	\(19\)		$+$	$+$	$+$	$2^{3}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(1+\beta _{4})q^{3}+(4+\beta _{2})q^{4}+\cdots\)
1175.4.a.f	$1175$	$4$	1175.a	1.a	$1$	$15$	$15$	$69.327$	\(\mathbb{Q}[x]/(x^{15} - \cdots)\)	None	✓	$1$	$0$	\(-7\)	\(-8\)	\(0\)	\(-13\)		$+$	$+$	$+$	$2^{3}$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+(-1-\beta _{4})q^{3}+(5+\beta _{2})q^{4}+\cdots\)
1175.4.a.g	$1175$	$4$	1175.a	1.a	$1$	$18$	$18$	$69.327$	\(\mathbb{Q}[x]/(x^{18} - \cdots)\)	None	✓	$1$	$1$	\(-1\)	\(-7\)	\(0\)	\(-6\)		$+$	$-$	$-$	$2^{3}\cdot 3$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}-\beta _{4}q^{3}+(3+\beta _{2})q^{4}+(-1+\cdots)q^{6}+\cdots\)
1175.4.a.h	$1175$	$4$	1175.a	1.a	$1$	$18$	$18$	$69.327$	\(\mathbb{Q}[x]/(x^{18} - \cdots)\)	None	✓	$1$	$1$	\(1\)	\(7\)	\(0\)	\(6\)		$-$	$+$	$-$	$2^{3}\cdot 3$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+\beta _{4}q^{3}+(3+\beta _{2})q^{4}+(-1+\cdots)q^{6}+\cdots\)
1175.4.a.i	$1175$	$4$	1175.a	1.a	$1$	$28$	$28$	$69.327$		None	✓	$1$	$0$	\(-1\)	\(-7\)	\(0\)	\(-6\)		$+$	$+$	$+$		$\mathrm{SU}(2)$
1175.4.a.j	$1175$	$4$	1175.a	1.a	$1$	$28$	$28$	$69.327$		None	✓	$1$	$0$	\(1\)	\(7\)	\(0\)	\(6\)		$-$	$-$	$+$		$\mathrm{SU}(2)$
1175.4.a.k	$1175$	$4$	1175.a	1.a	$1$	$35$	$35$	$69.327$		None	✓	$1$	$1$	\(-8\)	\(-12\)	\(0\)	\(-84\)		$-$	$+$	$-$		$\mathrm{SU}(2)$
1175.4.a.l	$1175$	$4$	1175.a	1.a	$1$	$35$	$35$	$69.327$		None	✓	$1$	$0$	\(8\)	\(12\)	\(0\)	\(84\)		$-$	$-$	$+$		$\mathrm{SU}(2)$

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(1175))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(1175)) \cong \) \(S_{4}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(25))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(47))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(235))\)\(^{\oplus 2}\)