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

• Label: 392.3.o
• Level: $392$
• Weight: $3$
• Character orbit: 392.o
• Rep. character: $\chi_{392}(129,\cdot)$
• Character field: $\Q(\zeta_{6})$
• Dimension: $40$
• Newform subspaces: $4$
• Sturm bound: $168$
• Trace bound: $9$

Defining parameters

Level:	\( N \)	\(=\)	\( 392 = 2^{3} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	392.o (of order \(6\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 7 \)
Character field:			\(\Q(\zeta_{6})\)
Newform subspaces:			\( 4 \)
Sturm bound:			\(168\)
Trace bound:			\(9\)
Distinguishing \(T_p\):			\(3\)

Dimensions

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

	Total	New	Old
Modular forms	256	40	216
Cusp forms	192	40	152
Eisenstein series	64	0	64

Trace form

\( 40 q + 52 q^{9} + O(q^{10}) \)

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.o.a	$392$	$3$	392.o	7.d	$6$	$8$	$4$	$10.681$	8.0.339738624.1	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{6}]$	\(q+(-\beta _{3}-\beta _{7})q^{3}+(\beta _{1}+\beta _{3})q^{5}+(-7+\cdots)q^{9}+\cdots\)
392.3.o.b	$392$	$3$	392.o	7.d	$6$	$8$	$4$	$10.681$	8.0.339738624.1	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{10}$	$\mathrm{SU}(2)[C_{6}]$	\(q+\beta _{5}q^{3}+(-\beta _{1}+\beta _{2})q^{5}+(\beta _{3}+\beta _{6}+\cdots)q^{9}+\cdots\)
392.3.o.c	$392$	$3$	392.o	7.d	$6$	$8$	$4$	$10.681$	8.0.\(\cdots\).2	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{6}$	$\mathrm{SU}(2)[C_{6}]$	\(q-\beta _{2}q^{3}+\beta _{7}q^{5}+(5-5\beta _{1}-2\beta _{4}+\cdots)q^{9}+\cdots\)
392.3.o.d	$392$	$3$	392.o	7.d	$6$	$16$	$8$	$10.681$	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}\cdot 7^{4}$	$\mathrm{SU}(2)[C_{6}]$	\(q+\beta _{2}q^{3}+(-\beta _{5}-\beta _{9}-\beta _{10})q^{5}+(8+\cdots)q^{9}+\cdots\)

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

\( S_{3}^{\mathrm{old}}(392, [\chi]) \cong \) \(S_{3}^{\mathrm{new}}(14, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(28, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(49, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(56, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(98, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(196, [\chi])\)\(^{\oplus 2}\)