Space of modular forms of level 7, weight 10, and trivial character

• Label: 7.10.a
• Level: $7$
• Weight: $10$
• Character orbit: 7.a
• Rep. character: $\chi_{7}(1,\cdot)$
• Character field: $\Q$
• Dimension: $5$
• Newform subspaces: $2$
• Sturm bound: $6$
• Trace bound: $1$

Defining parameters

Level:	\( N \)	\(=\)	\( 7 \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	7.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 2 \)
Sturm bound:			\(6\)
Trace bound:			\(1\)
Distinguishing \(T_p\):			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{10}(\Gamma_0(7))\).

	Total	New	Old
Modular forms	7	5	2
Cusp forms	5	5	0
Eisenstein series	2	0	2

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

\(7\)	Dim
\(+\)	\(2\)
\(-\)	\(3\)

Trace form

\( 5 q + 15 q^{2} - 2 q^{3} + 937 q^{4} - 684 q^{5} + 926 q^{6} + 2401 q^{7} + 16671 q^{8} - 14963 q^{9} + O(q^{10}) \)

Decomposition of \(S_{10}^{\mathrm{new}}(\Gamma_0(7))\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$		7
7.10.a.a	$7$	$10$	7.a	1.a	$1$	$2$	$2$	$3.605$	\(\Q(\sqrt{193}) \)	None	✓	✓	✓	✓	7.10.a.a	$1$	$1$	\(-6\)	\(-86\)	\(-2238\)	\(-4802\)		$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q+(-3-\beta )q^{2}+(-43+11\beta )q^{3}+(-310+\cdots)q^{4}+\cdots\)
7.10.a.b	$7$	$10$	7.a	1.a	$1$	$3$	$3$	$3.605$	\(\mathbb{Q}[x]/(x^{3} - \cdots)\)	None	✓	✓	✓	✓	7.10.a.b	$1$	$0$	\(21\)	\(84\)	\(1554\)	\(7203\)		$-$	$-$	$2\cdot 3$	$\mathrm{SU}(2)$	\(q+(7-\beta _{2})q^{2}+(28-\beta _{1}-\beta _{2})q^{3}+(519+\cdots)q^{4}+\cdots\)