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

• Label: 52.2.l
• Level: $52$
• Weight: $2$
• Character orbit: 52.l
• Rep. character: $\chi_{52}(7,\cdot)$
• Character field: $\Q(\zeta_{12})$
• Dimension: $20$
• Newform subspaces: $2$
• Sturm bound: $14$
• Trace bound: $1$

Defining parameters

Level:	\( N \)	\(=\)	\( 52 = 2^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	52.l (of order \(12\) and degree \(4\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 52 \)
Character field:			\(\Q(\zeta_{12})\)
Newform subspaces:			\( 2 \)
Sturm bound:			\(14\)
Trace bound:			\(1\)
Distinguishing \(T_p\):			\(3\)

Dimensions

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

	Total	New	Old
Modular forms	36	36	0
Cusp forms	20	20	0
Eisenstein series	16	16	0

Trace form

\( 20 q - 4 q^{2} - 6 q^{4} - 6 q^{5} - 14 q^{6} + 2 q^{8} - 2 q^{9} + O(q^{10}) \)

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.l.a	$52$	$2$	52.l	52.l	$12$	$4$	$1$	$0.415$	\(\Q(\zeta_{12})\)	\(\Q(\sqrt{-1}) \)		$4$	$0$	\(-2\)	\(0\)	\(6\)	\(0\)	$1$	$\mathrm{U}(1)[D_{12}]$	\(q+(-\zeta_{12}-\zeta_{12}^{2}+\zeta_{12}^{3})q^{2}+2\zeta_{12}q^{4}+\cdots\)
52.2.l.b	$52$	$2$	52.l	52.l	$12$	$16$	$4$	$0.415$	16.0.\(\cdots\).1	None		$4$	$0$	\(-2\)	\(0\)	\(-12\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{12}]$	\(q-\beta _{12}q^{2}+(\beta _{3}+\beta _{12}-\beta _{13}+\beta _{14}+\cdots)q^{3}+\cdots\)