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

• Label: 399.2.j
• Level: $399$
• Weight: $2$
• Character orbit: 399.j
• Rep. character: $\chi_{399}(58,\cdot)$
• Character field: $\Q(\zeta_{3})$
• Dimension: $48$
• Newform subspaces: $7$
• Sturm bound: $106$
• Trace bound: $4$

Defining parameters

Level:	\( N \)	\(=\)	\( 399 = 3 \cdot 7 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	399.j (of order \(3\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 7 \)
Character field:			\(\Q(\zeta_{3})\)
Newform subspaces:			\( 7 \)
Sturm bound:			\(106\)
Trace bound:			\(4\)
Distinguishing \(T_p\):			\(2\)

Dimensions

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

	Total	New	Old
Modular forms	116	48	68
Cusp forms	100	48	52
Eisenstein series	16	0	16

Trace form

\( 48 q + 4 q^{2} - 20 q^{4} - 2 q^{5} - 8 q^{6} + 6 q^{7} - 24 q^{8} - 24 q^{9} + O(q^{10}) \)

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	Inner twists	Rank*	Traces				Coefficient ring index	Sato-Tate	$q$-expansion
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
399.2.j.a	$399$	$2$	399.j	7.c	$3$	$2$	$1$	$3.186$	\(\Q(\sqrt{-3}) \)	None		$2$	$0$	\(-2\)	\(-1\)	\(-1\)	\(4\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q-2\zeta_{6}q^{2}+(-1+\zeta_{6})q^{3}+(-2+2\zeta_{6})q^{4}+\cdots\)
399.2.j.b	$399$	$2$	399.j	7.c	$3$	$2$	$1$	$3.186$	\(\Q(\sqrt{-3}) \)	None		$2$	$0$	\(0\)	\(-1\)	\(-2\)	\(1\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-1+\zeta_{6})q^{3}+(2-2\zeta_{6})q^{4}-2\zeta_{6}q^{5}+\cdots\)
399.2.j.c	$399$	$2$	399.j	7.c	$3$	$4$	$2$	$3.186$	\(\Q(\sqrt{2}, \sqrt{-3})\)	None		$2$	$0$	\(2\)	\(-2\)	\(2\)	\(-10\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(1+\beta _{1}+\beta _{2})q^{2}+\beta _{2}q^{3}+(2\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)
399.2.j.d	$399$	$2$	399.j	7.c	$3$	$8$	$4$	$3.186$	8.0.310217769.2	None		$2$	$0$	\(0\)	\(4\)	\(-4\)	\(2\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q-\beta _{1}q^{2}+(1-\beta _{5})q^{3}+(-\beta _{1}-\beta _{2}+\cdots)q^{4}+\cdots\)
399.2.j.e	$399$	$2$	399.j	7.c	$3$	$8$	$4$	$3.186$	8.0.542936601.2	None		$2$	$0$	\(2\)	\(-4\)	\(-4\)	\(6\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(\beta _{3}+\beta _{5})q^{2}-\beta _{4}q^{3}+(2-\beta _{1}+2\beta _{2}+\cdots)q^{4}+\cdots\)
399.2.j.f	$399$	$2$	399.j	7.c	$3$	$8$	$4$	$3.186$	8.0.42575625.1	None		$2$	$0$	\(2\)	\(-4\)	\(2\)	\(2\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q-\beta _{5}q^{2}+(-1-\beta _{6})q^{3}+(-\beta _{2}-\beta _{3}+\cdots)q^{4}+\cdots\)
399.2.j.g	$399$	$2$	399.j	7.c	$3$	$16$	$8$	$3.186$	\(\mathbb{Q}[x]/(x^{16} + \cdots)\)	None		$2$	$0$	\(0\)	\(8\)	\(5\)	\(1\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q-\beta _{1}q^{2}+(1-\beta _{5})q^{3}+(-1+\beta _{5}+\beta _{9}+\cdots)q^{4}+\cdots\)

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

\( S_{2}^{\mathrm{old}}(399, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(21, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(133, [\chi])\)\(^{\oplus 2}\)