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

• Label: 390.2.b
• Level: $390$
• Weight: $2$
• Character orbit: 390.b
• Rep. character: $\chi_{390}(181,\cdot)$
• Character field: $\Q$
• Dimension: $12$
• Newform subspaces: $4$
• Sturm bound: $168$
• Trace bound: $13$

Defining parameters

Level:	\( N \)	\(=\)	\( 390 = 2 \cdot 3 \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	390.b (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 13 \)
Character field:			\(\Q\)
Newform subspaces:			\( 4 \)
Sturm bound:			\(168\)
Trace bound:			\(13\)
Distinguishing \(T_p\):			\(7\), \(11\)

Dimensions

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

	Total	New	Old
Modular forms	92	12	80
Cusp forms	76	12	64
Eisenstein series	16	0	16

Trace form

\( 12 q - 4 q^{3} - 12 q^{4} + 12 q^{9} + O(q^{10}) \)

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.b.a	$390$	$2$	390.b	13.b	$2$	$2$	$2$	$3.114$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(-2\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-iq^{2}-q^{3}-q^{4}+iq^{5}+iq^{6}+iq^{8}+\cdots\)
390.2.b.b	$390$	$2$	390.b	13.b	$2$	$2$	$2$	$3.114$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(-2\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-iq^{2}-q^{3}-q^{4}+iq^{5}+iq^{6}+iq^{8}+\cdots\)
390.2.b.c	$390$	$2$	390.b	13.b	$2$	$4$	$4$	$3.114$	\(\Q(i, \sqrt{13})\)	None		$2$	$0$	\(0\)	\(-4\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{1}q^{2}-q^{3}-q^{4}+\beta _{1}q^{5}-\beta _{1}q^{6}+\cdots\)
390.2.b.d	$390$	$2$	390.b	13.b	$2$	$4$	$4$	$3.114$	\(\Q(i, \sqrt{17})\)	None		$2$	$0$	\(0\)	\(4\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{1}q^{2}+q^{3}-q^{4}+\beta _{1}q^{5}-\beta _{1}q^{6}+\cdots\)

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

\( S_{2}^{\mathrm{old}}(390, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(26, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(39, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(65, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(78, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(130, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(195, [\chi])\)\(^{\oplus 2}\)