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

• Label: 399.2.bd
• Level: $399$
• Weight: $2$
• Character orbit: 399.bd
• Rep. character: $\chi_{399}(68,\cdot)$
• Character field: $\Q(\zeta_{6})$
• Dimension: $100$
• Newform subspaces: $3$
• Sturm bound: $106$
• Trace bound: $7$

Defining parameters

Level:	\( N \)	\(=\)	\( 399 = 3 \cdot 7 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	399.bd (of order \(6\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 399 \)
Character field:			\(\Q(\zeta_{6})\)
Newform subspaces:			\( 3 \)
Sturm bound:			\(106\)
Trace bound:			\(7\)
Distinguishing \(T_p\):			\(2\), \(13\)

Dimensions

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

	Total	New	Old
Modular forms	116	116	0
Cusp forms	100	100	0
Eisenstein series	16	16	0

Trace form

\( 100 q + 50 q^{4} - 12 q^{6} - 11 q^{7} - 2 q^{9} + O(q^{10}) \)

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.bd.a	$399$	$2$	399.bd	399.ad	$6$	$2$	$1$	$3.186$	\(\Q(\sqrt{-3}) \)	\(\Q(\sqrt{-3}) \)		✓			399.2.p.a	$4$	$1$	\(0\)	\(0\)	\(0\)	\(-5\)	$1$	$\mathrm{U}(1)[D_{6}]$	\(q+(-1+2\zeta_{6})q^{3}-2\zeta_{6}q^{4}+(-3+\zeta_{6})q^{7}+\cdots\)
399.2.bd.b	$399$	$2$	399.bd	399.ad	$6$	$2$	$1$	$3.186$	\(\Q(\sqrt{-3}) \)	\(\Q(\sqrt{-3}) \)		✓			399.2.p.b	$4$	$0$	\(0\)	\(0\)	\(0\)	\(4\)	$1$	$\mathrm{U}(1)[D_{6}]$	\(q+(-1+2\zeta_{6})q^{3}-2\zeta_{6}q^{4}+(1+2\zeta_{6})q^{7}+\cdots\)
399.2.bd.c	$399$	$2$	399.bd	399.ad	$6$	$96$	$48$	$3.186$		None		✓	✓		399.2.p.c	$4$	$0$	\(0\)	\(0\)	\(0\)	\(-10\)		$\mathrm{SU}(2)[C_{6}]$