Space of modular forms of level 390, weight 2, and Character 390.x

• Label: 390.2.x
• Level: $390$
• Weight: $2$
• Character orbit: 390.x
• Rep. character: $\chi_{390}(49,\cdot)$
• Character field: $\Q(\zeta_{6})$
• Dimension: $24$
• Newform subspaces: $2$
• Sturm bound: $168$
• Trace bound: $2$

Defining parameters

Level:	\( N \)	\(=\)	\( 390 = 2 \cdot 3 \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	390.x (of order \(6\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 65 \)
Character field:			\(\Q(\zeta_{6})\)
Newform subspaces:			\( 2 \)
Sturm bound:			\(168\)
Trace bound:			\(2\)
Distinguishing \(T_p\):			\(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(390, [\chi])\).

	Total	New	Old
Modular forms	184	24	160
Cusp forms	152	24	128
Eisenstein series	32	0	32

Trace form

\( 24 q - 12 q^{4} + 12 q^{9} + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(390, [\chi])\) 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
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
390.2.x.a	$390$	$2$	390.x	65.l	$6$	$12$	$6$	$3.114$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None		$2$	$0$	\(-6\)	\(0\)	\(-2\)	\(-2\)	$1$	$\mathrm{SU}(2)[C_{6}]$	\(q+(-1-\beta _{6})q^{2}+\beta _{4}q^{3}+\beta _{6}q^{4}+(\beta _{3}+\cdots)q^{5}+\cdots\)
390.2.x.b	$390$	$2$	390.x	65.l	$6$	$12$	$6$	$3.114$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None		$2$	$0$	\(6\)	\(0\)	\(2\)	\(2\)	$1$	$\mathrm{SU}(2)[C_{6}]$	\(q+(1+\beta _{6})q^{2}-\beta _{4}q^{3}+\beta _{6}q^{4}+(\beta _{3}+\cdots)q^{5}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(390, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(390, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(65, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(130, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(195, [\chi])\)\(^{\oplus 2}\)