Space of modular forms of level 399, weight 2, and Character 399.bq

• Label: 399.2.bq
• Level: $399$
• Weight: $2$
• Character orbit: 399.bq
• Rep. character: $\chi_{399}(4,\cdot)$
• Character field: $\Q(\zeta_{9})$
• Dimension: $162$
• Newform subspaces: $2$
• Sturm bound: $106$
• Trace bound: $1$

Defining parameters

Level:	\( N \)	\(=\)	\( 399 = 3 \cdot 7 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	399.bq (of order \(9\) and degree \(6\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 133 \)
Character field:			\(\Q(\zeta_{9})\)
Newform subspaces:			\( 2 \)
Sturm bound:			\(106\)
Trace bound:			\(1\)
Distinguishing \(T_p\):			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(399, [\chi])\).

	Total	New	Old
Modular forms	342	162	180
Cusp forms	294	162	132
Eisenstein series	48	0	48

Trace form

\( 162 q + 12 q^{7} + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(399, [\chi])\) into newform subspaces

Label	Level	Weight	Char	Prim	Char order	Dim	Rel. Dim	$A$	Field	CM	Self-dual	Twist minimal	Largest	Maximal	Minimal twist	Inner twists	Rank*	Traces				Coefficient ring index	Sato-Tate	$q$-expansion
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
399.2.bq.a	$399$	$2$	399.bq	133.w	$9$	$72$	$12$	$3.186$		None		✓			399.2.bp.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(6\)		$\mathrm{SU}(2)[C_{9}]$
399.2.bq.b	$399$	$2$	399.bq	133.w	$9$	$90$	$15$	$3.186$		None		✓	✓		399.2.bp.b	$2$	$0$	\(0\)	\(0\)	\(0\)	\(6\)		$\mathrm{SU}(2)[C_{9}]$

Decomposition of \(S_{2}^{\mathrm{old}}(399, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(399, [\chi]) \simeq \) \(S_{2}^{\mathrm{new}}(133, [\chi])\)\(^{\oplus 2}\)