Space of modular forms of level 52, weight 2, and Character 52.e

• Label: 52.2.e
• Level: $52$
• Weight: $2$
• Character orbit: 52.e
• Rep. character: $\chi_{52}(9,\cdot)$
• Character field: $\Q(\zeta_{3})$
• Dimension: $4$
• Newform subspaces: $2$
• Sturm bound: $14$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 52 = 2^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	52.e (of order \(3\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 13 \)
Character field:			\(\Q(\zeta_{3})\)
Newform subspaces:			\( 2 \)
Sturm bound:			\(14\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(52, [\chi])\).

	Total	New	Old
Modular forms	20	4	16
Cusp forms	8	4	4
Eisenstein series	12	0	12

Trace form

\( 4 q - q^{3} - 2 q^{5} + 3 q^{7} - 7 q^{9} + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(52, [\chi])\) into newform subspaces

Label	Level	Weight	Char	Prim	Char order	Dim	Rel. Dim	$A$	Field	CM	Self-dual	Inner twists	Rank*	Traces				Coefficient ring index	Sato-Tate	$q$-expansion
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
52.2.e.a	$52$	$2$	52.e	13.c	$3$	$2$	$1$	$0.415$	\(\Q(\sqrt{-3}) \)	None		$2$	$0$	\(0\)	\(-3\)	\(4\)	\(-1\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-3+3\zeta_{6})q^{3}+2q^{5}-\zeta_{6}q^{7}-6\zeta_{6}q^{9}+\cdots\)
52.2.e.b	$52$	$2$	52.e	13.c	$3$	$2$	$1$	$0.415$	\(\Q(\sqrt{-3}) \)	None		$2$	$0$	\(0\)	\(2\)	\(-6\)	\(4\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(2-2\zeta_{6})q^{3}-3q^{5}+4\zeta_{6}q^{7}-\zeta_{6}q^{9}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(52, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(52, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(26, [\chi])\)\(^{\oplus 2}\)