Space of modular forms of level 35, weight 10, and trivial character

• Label: 35.10.a
• Level: $35$
• Weight: $10$
• Character orbit: 35.a
• Rep. character: $\chi_{35}(1,\cdot)$
• Character field: $\Q$
• Dimension: $18$
• Newform subspaces: $5$
• Sturm bound: $40$
• Trace bound: $1$

Defining parameters

Level:	\( N \)	\(=\)	\( 35 = 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	35.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 5 \)
Sturm bound:			\(40\)
Trace bound:			\(1\)
Distinguishing \(T_p\):			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{10}(\Gamma_0(35))\).

	Total	New	Old
Modular forms	38	18	20
Cusp forms	34	18	16
Eisenstein series	4	0	4

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
\(+\)	\(+\)	$+$	\(4\)
\(+\)	\(-\)	$-$	\(5\)
\(-\)	\(+\)	$-$	\(6\)
\(-\)	\(-\)	$+$	\(3\)
Plus space		\(+\)	\(7\)
Minus space		\(-\)	\(11\)

Trace form

\( 18 q + 2 q^{2} - 292 q^{3} + 5122 q^{4} + 4304 q^{6} - 4802 q^{7} - 8766 q^{8} + 200470 q^{9} + O(q^{10}) \)

Decomposition of \(S_{10}^{\mathrm{new}}(\Gamma_0(35))\) 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
35.10.a.a	$35$	$10$	35.a	1.a	$1$	$1$	$1$	$18.026$	\(\Q\)	None	✓	$1$	$1$	\(28\)	\(-116\)	\(625\)	\(2401\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+28q^{2}-116q^{3}+272q^{4}+5^{4}q^{5}+\cdots\)
35.10.a.b	$35$	$10$	35.a	1.a	$1$	$2$	$2$	$18.026$	\(\Q(\sqrt{2}) \)	None	✓	$1$	$1$	\(-24\)	\(-174\)	\(1250\)	\(4802\)		$-$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q+(-12+\beta )q^{2}+(-87+54\beta )q^{3}+\cdots\)
35.10.a.c	$35$	$10$	35.a	1.a	$1$	$4$	$4$	$18.026$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓	$1$	$1$	\(-19\)	\(-18\)	\(-2500\)	\(-9604\)		$+$	$+$	$+$	$2\cdot 3^{2}\cdot 5$	$\mathrm{SU}(2)$	\(q+(-5-\beta _{1})q^{2}+(-4+\beta _{2})q^{3}+(435+\cdots)q^{4}+\cdots\)
35.10.a.d	$35$	$10$	35.a	1.a	$1$	$5$	$5$	$18.026$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓	$1$	$0$	\(2\)	\(140\)	\(-3125\)	\(12005\)		$+$	$-$	$-$	$2^{3}\cdot 5$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(28+\beta _{2})q^{3}+(168-4\beta _{1}+\cdots)q^{4}+\cdots\)
35.10.a.e	$35$	$10$	35.a	1.a	$1$	$6$	$6$	$18.026$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓	$1$	$0$	\(15\)	\(-124\)	\(3750\)	\(-14406\)		$-$	$+$	$-$	$2^{3}\cdot 3^{2}\cdot 5$	$\mathrm{SU}(2)$	\(q+(3-\beta _{1})q^{2}+(-20-\beta _{1}+\beta _{2})q^{3}+\cdots\)

Decomposition of \(S_{10}^{\mathrm{old}}(\Gamma_0(35))\) into lower level spaces

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