Space of modular forms of level 392, weight 3, and Character 392.j

• Label: 392.3.j
• Level: $392$
• Weight: $3$
• Character orbit: 392.j
• Rep. character: $\chi_{392}(117,\cdot)$
• Character field: $\Q(\zeta_{6})$
• Dimension: $152$
• Newform subspaces: $6$
• Sturm bound: $168$
• Trace bound: $2$

Defining parameters

Level:	\( N \)	\(=\)	\( 392 = 2^{3} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	392.j (of order \(6\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 56 \)
Character field:			\(\Q(\zeta_{6})\)
Newform subspaces:			\( 6 \)
Sturm bound:			\(168\)
Trace bound:			\(2\)
Distinguishing \(T_p\):			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{3}(392, [\chi])\).

	Total	New	Old
Modular forms	240	168	72
Cusp forms	208	152	56
Eisenstein series	32	16	16

Trace form

\( 152 q + 3 q^{2} + 7 q^{4} + 18 q^{8} - 202 q^{9} + O(q^{10}) \)

Decomposition of \(S_{3}^{\mathrm{new}}(392, [\chi])\) 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
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
392.3.j.a	$392$	$3$	392.j	56.j	$6$	$4$	$2$	$10.681$	\(\Q(\sqrt{2}, \sqrt{-3})\)	\(\Q(\sqrt{-14}) \)		$8$	$0$	\(-4\)	\(0\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{U}(1)[D_{6}]$	\(q+2\beta _{2}q^{2}+\beta _{1}q^{3}+(-4-4\beta _{2})q^{4}+\cdots\)
392.3.j.b	$392$	$3$	392.j	56.j	$6$	$4$	$2$	$10.681$	\(\Q(\sqrt{-3}, \sqrt{-7})\)	\(\Q(\sqrt{-7}) \)		$8$	$0$	\(-3\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{U}(1)[D_{6}]$	\(q+(-\beta _{2}-\beta _{3})q^{2}+(-2+3\beta _{1}+2\beta _{2}+\cdots)q^{4}+\cdots\)
392.3.j.c	$392$	$3$	392.j	56.j	$6$	$4$	$2$	$10.681$	\(\Q(\sqrt{-3}, \sqrt{7})\)	\(\Q(\sqrt{-14}) \)		$8$	$0$	\(4\)	\(0\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{U}(1)[D_{6}]$	\(q-2\beta _{2}q^{2}+\beta _{1}q^{3}+(-4-4\beta _{2})q^{4}+\cdots\)
392.3.j.d	$392$	$3$	392.j	56.j	$6$	$16$	$8$	$10.681$	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)	None		$8$	$0$	\(4\)	\(0\)	\(0\)	\(0\)	$2^{10}\cdot 3^{4}$	$\mathrm{SU}(2)[C_{6}]$	\(q+(\beta _{2}+\beta _{5})q^{2}+(-\beta _{3}+\beta _{11})q^{3}+(2+\cdots)q^{4}+\cdots\)
392.3.j.e	$392$	$3$	392.j	56.j	$6$	$28$	$14$	$10.681$		None		$4$	$0$	\(-2\)	\(0\)	\(0\)	\(0\)		$\mathrm{SU}(2)[C_{6}]$
392.3.j.f	$392$	$3$	392.j	56.j	$6$	$96$	$48$	$10.681$		None		$8$	$0$	\(4\)	\(0\)	\(0\)	\(0\)		$\mathrm{SU}(2)[C_{6}]$

Decomposition of \(S_{3}^{\mathrm{old}}(392, [\chi])\) into lower level spaces

\( S_{3}^{\mathrm{old}}(392, [\chi]) \cong \) \(S_{3}^{\mathrm{new}}(56, [\chi])\)\(^{\oplus 2}\)