Space of modular forms of level 175, weight 4, and trivial character

• Label: 175.4.a
• Level: $175$
• Weight: $4$
• Character orbit: 175.a
• Rep. character: $\chi_{175}(1,\cdot)$
• Character field: $\Q$
• Dimension: $29$
• Newform subspaces: $10$
• Sturm bound: $80$
• Trace bound: $2$

Defining parameters

Level:	\( N \)	\(=\)	\( 175 = 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	175.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 10 \)
Sturm bound:			\(80\)
Trace bound:			\(2\)
Distinguishing \(T_p\):			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(175))\).

	Total	New	Old
Modular forms	66	29	37
Cusp forms	54	29	25
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.

\(5\)	\(7\)	Fricke	Dim
\(+\)	\(+\)	$+$	\(8\)
\(+\)	\(-\)	$-$	\(5\)
\(-\)	\(+\)	$-$	\(7\)
\(-\)	\(-\)	$+$	\(9\)
Plus space		\(+\)	\(17\)
Minus space		\(-\)	\(12\)

Trace form

\( 29 q - 5 q^{2} + 6 q^{3} + 137 q^{4} - 22 q^{6} - 7 q^{7} - 33 q^{8} + 233 q^{9} + O(q^{10}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(175))\) 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	7
175.4.a.a	$175$	$4$	175.a	1.a	$1$	$1$	$1$	$10.325$	\(\Q\)	None	✓	$1$	$0$	\(-1\)	\(8\)	\(0\)	\(-7\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+8q^{3}-7q^{4}-8q^{6}-7q^{7}+\cdots\)
175.4.a.b	$175$	$4$	175.a	1.a	$1$	$1$	$1$	$10.325$	\(\Q\)	None	✓	$1$	$1$	\(1\)	\(2\)	\(0\)	\(7\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+2q^{3}-7q^{4}+2q^{6}+7q^{7}+\cdots\)
175.4.a.c	$175$	$4$	175.a	1.a	$1$	$2$	$2$	$10.325$	\(\Q(\sqrt{2}) \)	None	✓	$1$	$1$	\(-8\)	\(-2\)	\(0\)	\(14\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-4+\beta )q^{2}+(-1-4\beta )q^{3}+(10+\cdots)q^{4}+\cdots\)
175.4.a.d	$175$	$4$	175.a	1.a	$1$	$2$	$2$	$10.325$	\(\Q(\sqrt{41}) \)	None	✓	$1$	$1$	\(-1\)	\(5\)	\(0\)	\(-14\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta q^{2}+(2+\beta )q^{3}+(2+\beta )q^{4}+(-10+\cdots)q^{6}+\cdots\)
175.4.a.e	$175$	$4$	175.a	1.a	$1$	$2$	$2$	$10.325$	\(\Q(\sqrt{41}) \)	None	✓	$1$	$1$	\(1\)	\(-5\)	\(0\)	\(14\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{2}+(-2-\beta )q^{3}+(2+\beta )q^{4}+(-10+\cdots)q^{6}+\cdots\)
175.4.a.f	$175$	$4$	175.a	1.a	$1$	$3$	$3$	$10.325$	3.3.14360.1	None	✓	$1$	$0$	\(3\)	\(-2\)	\(0\)	\(-21\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{2}+(-1-\beta _{1}+\beta _{2})q^{3}+\cdots\)
175.4.a.g	$175$	$4$	175.a	1.a	$1$	$4$	$4$	$10.325$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓	$1$	$0$	\(-4\)	\(3\)	\(0\)	\(-28\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{2}+(1-\beta _{3})q^{3}+(9+\beta _{2}+\cdots)q^{4}+\cdots\)
175.4.a.h	$175$	$4$	175.a	1.a	$1$	$4$	$4$	$10.325$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓	$1$	$0$	\(4\)	\(-3\)	\(0\)	\(28\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{2}+(-1+\beta _{3})q^{3}+(9+\beta _{2}+\cdots)q^{4}+\cdots\)
175.4.a.i	$175$	$4$	175.a	1.a	$1$	$5$	$5$	$10.325$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓	$1$	$1$	\(-4\)	\(-10\)	\(0\)	\(-35\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{2}+(-2+\beta _{3})q^{3}+(4+\cdots)q^{4}+\cdots\)
175.4.a.j	$175$	$4$	175.a	1.a	$1$	$5$	$5$	$10.325$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓	$1$	$0$	\(4\)	\(10\)	\(0\)	\(35\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{2}+(2-\beta _{3})q^{3}+(4-\beta _{1}+\cdots)q^{4}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(175))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(175)) \cong \) \(S_{4}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(7))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(25))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(35))\)\(^{\oplus 2}\)