Space of modular forms of level 175, weight 2, and trivial character

• Label: 175.2.a
• Level: $175$
• Weight: $2$
• Character orbit: 175.a
• Rep. character: $\chi_{175}(1,\cdot)$
• Character field: $\Q$
• Dimension: $9$
• Newform subspaces: $6$
• Sturm bound: $40$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 175 = 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	175.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 6 \)
Sturm bound:			\(40\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(175))\).

	Total	New	Old
Modular forms	26	9	17
Cusp forms	15	9	6
Eisenstein series	11	0	11

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		Total				Cusp				Eisenstein
				All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)		\(5\)	\(1\)	\(4\)		\(3\)	\(1\)	\(2\)		\(2\)	\(0\)	\(2\)
\(+\)	\(-\)	\(-\)		\(8\)	\(4\)	\(4\)		\(5\)	\(4\)	\(1\)		\(3\)	\(0\)	\(3\)
\(-\)	\(+\)	\(-\)		\(8\)	\(3\)	\(5\)		\(5\)	\(3\)	\(2\)		\(3\)	\(0\)	\(3\)
\(-\)	\(-\)	\(+\)		\(5\)	\(1\)	\(4\)		\(2\)	\(1\)	\(1\)		\(3\)	\(0\)	\(3\)
Plus space		\(+\)		\(10\)	\(2\)	\(8\)		\(5\)	\(2\)	\(3\)		\(5\)	\(0\)	\(5\)
Minus space		\(-\)		\(16\)	\(7\)	\(9\)		\(10\)	\(7\)	\(3\)		\(6\)	\(0\)	\(6\)

Trace form

9 * q + q^2 + 5 * q^4 + 4 * q^6 + q^7 + 9 * q^8 + 9 * q^9 - 4 * q^12 - 10 * q^13 - q^14 - 7 * q^16 + 2 * q^17 - 7 * q^18 - 4 * q^19 - 4 * q^21 - 8 * q^22 + 8 * q^23 - 24 * q^24 - 2 * q^26 + 12 * q^27 + 7 * q^28 + 14 * q^29 - 12 * q^31 + 9 * q^32 + 12 * q^33 - 2 * q^34 - 31 * q^36 - 14 * q^37 - 20 * q^38 - 20 * q^39 + 22 * q^41 - 8 * q^42 - 6 * q^44 + 6 * q^46 - 4 * q^47 - 28 * q^48 + 9 * q^49 - 4 * q^51 + 6 * q^52 - 10 * q^53 - 8 * q^54 + 9 * q^56 + 12 * q^57 + 26 * q^58 + 32 * q^59 - 14 * q^61 + 5 * q^63 - 9 * q^64 + 10 * q^68 - 12 * q^69 + 8 * q^71 + 5 * q^72 + 6 * q^73 + 8 * q^74 - 16 * q^76 + 4 * q^77 + 20 * q^78 - 20 * q^79 + 33 * q^81 + 18 * q^82 - 20 * q^83 + 14 * q^86 - 28 * q^87 - 4 * q^88 + 54 * q^89 - 6 * q^91 - 24 * q^92 + 4 * q^93 + 12 * q^94 + 8 * q^96 + 10 * q^97 + q^98 + 12 * q^99

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(175))\) into newform subspaces

Label	Level	Weight	Char	Prim	Char order	Dim	Rel. Dim	$A$	Field	CM	Self-dual	Twist minimal	Largest	Maximal	Minimal twist	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
175.2.a.a	$175$	$2$	175.a	1.a	$1$	$1$	$1$	$1.397$	\(\Q\)	None	✓		✓	✓	35.2.b.a	$1$	$1$	\(-2\)	\(-1\)	\(0\)	\(1\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}-q^{3}+2q^{4}+2q^{6}+q^{7}-2q^{9}+\cdots\)
175.2.a.b	$175$	$2$	175.a	1.a	$1$	$1$	$1$	$1.397$	\(\Q\)	None	✓		✓	✓	35.2.a.a	$1$	$1$	\(0\)	\(-1\)	\(0\)	\(-1\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-2q^{4}-q^{7}-2q^{9}-3q^{11}+\cdots\)
175.2.a.c	$175$	$2$	175.a	1.a	$1$	$1$	$1$	$1.397$	\(\Q\)	None	✓				35.2.b.a	$1$	$0$	\(2\)	\(1\)	\(0\)	\(-1\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+q^{3}+2q^{4}+2q^{6}-q^{7}-2q^{9}+\cdots\)
175.2.a.d	$175$	$2$	175.a	1.a	$1$	$2$	$2$	$1.397$	\(\Q(\sqrt{5}) \)	None	✓	✓	✓		175.2.a.d	$1$	$0$	\(-1\)	\(2\)	\(0\)	\(-2\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta q^{2}+(2-2\beta )q^{3}+(-1+\beta )q^{4}+\cdots\)
175.2.a.e	$175$	$2$	175.a	1.a	$1$	$2$	$2$	$1.397$	\(\Q(\sqrt{5}) \)	None	✓	✓			175.2.a.d	$1$	$0$	\(1\)	\(-2\)	\(0\)	\(2\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{2}+(-2+2\beta )q^{3}+(-1+\beta )q^{4}+\cdots\)
175.2.a.f	$175$	$2$	175.a	1.a	$1$	$2$	$2$	$1.397$	\(\Q(\sqrt{17}) \)	None	✓				35.2.a.b	$1$	$0$	\(1\)	\(1\)	\(0\)	\(2\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{2}+(1-\beta )q^{3}+(2+\beta )q^{4}-4q^{6}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(175))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(175)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(35))\)\(^{\oplus 2}\)