Space of modular forms of level 392, weight 2, and Character 392.e

• Label: 392.2.e
• Level: $392$
• Weight: $2$
• Character orbit: 392.e
• Rep. character: $\chi_{392}(195,\cdot)$
• Character field: $\Q$
• Dimension: $36$
• Newform subspaces: $5$
• Sturm bound: $112$
• Trace bound: $9$

Defining parameters

Level:	\( N \)	\(=\)	\( 392 = 2^{3} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	392.e (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 56 \)
Character field:			\(\Q\)
Newform subspaces:			\( 5 \)
Sturm bound:			\(112\)
Trace bound:			\(9\)
Distinguishing \(T_p\):			\(3\), \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(392, [\chi])\).

	Total	New	Old
Modular forms	64	44	20
Cusp forms	48	36	12
Eisenstein series	16	8	8

Trace form

\( 36 q + 4 q^{2} + 4 q^{4} - 8 q^{8} - 24 q^{9} + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(392, [\chi])\) 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
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
392.2.e.a	$392$	$2$	392.e	56.e	$2$	$4$	$4$	$3.130$	4.0.2048.2	\(\Q(\sqrt{-2}) \)		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$7$	$\mathrm{U}(1)[D_{2}]$	\(q-\beta _{2}q^{2}-\beta _{3}q^{3}+2q^{4}-\beta _{1}q^{6}-2\beta _{2}q^{8}+\cdots\)
392.2.e.b	$392$	$2$	392.e	56.e	$2$	$4$	$4$	$3.130$	4.0.2048.2	\(\Q(\sqrt{-2}) \)		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{U}(1)[D_{2}]$	\(q+\beta _{2}q^{2}+\beta _{1}q^{3}+2q^{4}+(-\beta _{1}+\beta _{3})q^{6}+\cdots\)
392.2.e.c	$392$	$2$	392.e	56.e	$2$	$8$	$8$	$3.130$	8.0.339738624.1	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{5}q^{2}+(\beta _{4}+\beta _{7})q^{3}+(-1+\beta _{6}+\cdots)q^{4}+\cdots\)
392.2.e.d	$392$	$2$	392.e	56.e	$2$	$8$	$8$	$3.130$	8.0.\(\cdots\).10	None		$4$	$0$	\(4\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+(1-\beta _{5})q^{2}-\beta _{2}q^{3}+(\beta _{3}-\beta _{5}+\beta _{6}+\cdots)q^{4}+\cdots\)
392.2.e.e	$392$	$2$	392.e	56.e	$2$	$12$	$12$	$3.130$	12.0.\(\cdots\).2	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{8}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{5}q^{2}-\beta _{10}q^{3}+\beta _{1}q^{4}-\beta _{9}q^{5}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(392, [\chi])\) into lower level spaces

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