Space of modular forms of level 52, weight 2, and trivial character

• Label: 52.2.a
• Level: $52$
• Weight: $2$
• Character orbit: 52.a
• Rep. character: $\chi_{52}(1,\cdot)$
• Character field: $\Q$
• Dimension: $1$
• Newform subspaces: $1$
• Sturm bound: $14$
• Trace bound: $0$

Defining parameters

Level:	\( N \)	\(=\)	\( 52 = 2^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	52.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 1 \)
Sturm bound:			\(14\)
Trace bound:			\(0\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(52))\).

	Total	New	Old
Modular forms	10	1	9
Cusp forms	5	1	4
Eisenstein series	5	0	5

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

\(2\)	\(13\)	Fricke	Dim
\(-\)	\(+\)	$-$	\(1\)
Plus space		\(+\)	\(0\)
Minus space		\(-\)	\(1\)

Trace form

\( q + 2 q^{5} - 2 q^{7} - 3 q^{9} + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(52))\) 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
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$		2	13
52.2.a.a	$52$	$2$	52.a	1.a	$1$	$1$	$1$	$0.415$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(0\)	\(2\)	\(-2\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{5}-2q^{7}-3q^{9}-2q^{11}-q^{13}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(52))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(52)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(26))\)\(^{\oplus 2}\)