Space of modular forms of level 175, weight 10, and trivial character

• Label: 175.10.a
• Level: $175$
• Weight: $10$
• Character orbit: 175.a
• Rep. character: $\chi_{175}(1,\cdot)$
• Character field: $\Q$
• Dimension: $85$
• Newform subspaces: $13$
• Sturm bound: $200$
• Trace bound: $2$

Defining parameters

Level:	\( N \)	\(=\)	\( 175 = 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	175.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 13 \)
Sturm bound:			\(200\)
Trace bound:			\(2\)
Distinguishing \(T_p\):			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{10}(\Gamma_0(175))\).

	Total	New	Old
Modular forms	186	85	101
Cusp forms	174	85	89
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
\(+\)	\(+\)	$+$	\(19\)
\(+\)	\(-\)	$-$	\(22\)
\(-\)	\(+\)	$-$	\(23\)
\(-\)	\(-\)	$+$	\(21\)
Plus space		\(+\)	\(40\)
Minus space		\(-\)	\(45\)

Trace form

\( 85 q - 17 q^{2} + 294 q^{3} + 22273 q^{4} - 11386 q^{6} + 2401 q^{7} - 7905 q^{8} + 522233 q^{9} + O(q^{10}) \)

Decomposition of \(S_{10}^{\mathrm{new}}(\Gamma_0(175))\) 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
175.10.a.a	$175$	$10$	175.a	1.a	$1$	$1$	$1$	$90.131$	\(\Q\)	None	✓	$1$	$1$	\(-28\)	\(116\)	\(0\)	\(-2401\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-28q^{2}+116q^{3}+272q^{4}-3248q^{6}+\cdots\)
175.10.a.b	$175$	$10$	175.a	1.a	$1$	$2$	$2$	$90.131$	\(\Q(\sqrt{193}) \)	None	✓	$1$	$0$	\(6\)	\(86\)	\(0\)	\(4802\)		$+$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(3+\beta )q^{2}+(43-11\beta )q^{3}+(-310+\cdots)q^{4}+\cdots\)
175.10.a.c	$175$	$10$	175.a	1.a	$1$	$2$	$2$	$90.131$	\(\Q(\sqrt{2}) \)	None	✓	$1$	$1$	\(24\)	\(174\)	\(0\)	\(-4802\)		$+$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q+(12+\beta )q^{2}+(87+54\beta )q^{3}+(-360+\cdots)q^{4}+\cdots\)
175.10.a.d	$175$	$10$	175.a	1.a	$1$	$3$	$3$	$90.131$	\(\mathbb{Q}[x]/(x^{3} - \cdots)\)	None	✓	$1$	$1$	\(-21\)	\(-84\)	\(0\)	\(-7203\)		$+$	$+$	$+$	$2\cdot 3$	$\mathrm{SU}(2)$	\(q+(-7+\beta _{2})q^{2}+(-28+\beta _{1}+\beta _{2})q^{3}+\cdots\)
175.10.a.e	$175$	$10$	175.a	1.a	$1$	$4$	$4$	$90.131$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓	$1$	$0$	\(19\)	\(18\)	\(0\)	\(9604\)		$+$	$-$	$-$	$2\cdot 3^{2}\cdot 5$	$\mathrm{SU}(2)$	\(q+(5+\beta _{1})q^{2}+(4-\beta _{2})q^{3}+(435+9\beta _{1}+\cdots)q^{4}+\cdots\)
175.10.a.f	$175$	$10$	175.a	1.a	$1$	$5$	$5$	$90.131$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓	$1$	$1$	\(-2\)	\(-140\)	\(0\)	\(-12005\)		$+$	$+$	$+$	$2^{3}\cdot 5$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+(-28-\beta _{2})q^{3}+(168-4\beta _{1}+\cdots)q^{4}+\cdots\)
175.10.a.g	$175$	$10$	175.a	1.a	$1$	$6$	$6$	$90.131$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓	$1$	$0$	\(-15\)	\(124\)	\(0\)	\(14406\)		$+$	$-$	$-$	$2^{3}\cdot 3^{2}\cdot 5$	$\mathrm{SU}(2)$	\(q+(-3+\beta _{1})q^{2}+(20+\beta _{1}-\beta _{2})q^{3}+\cdots\)
175.10.a.h	$175$	$10$	175.a	1.a	$1$	$8$	$8$	$90.131$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓	$1$	$1$	\(-27\)	\(69\)	\(0\)	\(19208\)		$-$	$-$	$+$	$2^{7}\cdot 5^{3}$	$\mathrm{SU}(2)$	\(q+(-3-\beta _{1})q^{2}+(8+\beta _{1}-\beta _{2})q^{3}+\cdots\)
175.10.a.i	$175$	$10$	175.a	1.a	$1$	$8$	$8$	$90.131$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓	$1$	$1$	\(27\)	\(-69\)	\(0\)	\(-19208\)		$+$	$+$	$+$	$2^{7}\cdot 5^{3}$	$\mathrm{SU}(2)$	\(q+(3+\beta _{1})q^{2}+(-8-\beta _{1}+\beta _{2})q^{3}+\cdots\)
175.10.a.j	$175$	$10$	175.a	1.a	$1$	$10$	$10$	$90.131$	\(\mathbb{Q}[x]/(x^{10} - \cdots)\)	None	✓	$1$	$0$	\(-22\)	\(77\)	\(0\)	\(-24010\)		$-$	$+$	$-$	$2^{7}\cdot 3^{4}\cdot 5^{6}$	$\mathrm{SU}(2)$	\(q+(-2-\beta _{1})q^{2}+(8+\beta _{3})q^{3}+(237+\cdots)q^{4}+\cdots\)
175.10.a.k	$175$	$10$	175.a	1.a	$1$	$10$	$10$	$90.131$	\(\mathbb{Q}[x]/(x^{10} - \cdots)\)	None	✓	$1$	$0$	\(22\)	\(-77\)	\(0\)	\(24010\)		$+$	$-$	$-$	$2^{7}\cdot 3^{4}\cdot 5^{6}$	$\mathrm{SU}(2)$	\(q+(2+\beta _{1})q^{2}+(-8-\beta _{3})q^{3}+(237+\cdots)q^{4}+\cdots\)
175.10.a.l	$175$	$10$	175.a	1.a	$1$	$13$	$13$	$90.131$	\(\mathbb{Q}[x]/(x^{13} - \cdots)\)	None	✓	$1$	$1$	\(-32\)	\(-158\)	\(0\)	\(31213\)		$-$	$-$	$+$	$2^{9}\cdot 3^{2}\cdot 5^{10}$	$\mathrm{SU}(2)$	\(q+(-2-\beta _{1})q^{2}+(-12+\beta _{3})q^{3}+(208+\cdots)q^{4}+\cdots\)
175.10.a.m	$175$	$10$	175.a	1.a	$1$	$13$	$13$	$90.131$	\(\mathbb{Q}[x]/(x^{13} - \cdots)\)	None	✓	$1$	$0$	\(32\)	\(158\)	\(0\)	\(-31213\)		$-$	$+$	$-$	$2^{9}\cdot 3^{2}\cdot 5^{10}$	$\mathrm{SU}(2)$	\(q+(2+\beta _{1})q^{2}+(12-\beta _{3})q^{3}+(208+5\beta _{1}+\cdots)q^{4}+\cdots\)

Decomposition of \(S_{10}^{\mathrm{old}}(\Gamma_0(175))\) into lower level spaces

\( S_{10}^{\mathrm{old}}(\Gamma_0(175)) \cong \) \(S_{10}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 4}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(7))\)\(^{\oplus 3}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(25))\)\(^{\oplus 2}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(35))\)\(^{\oplus 2}\)