Space of modular forms of level 394, weight 3, and Character 394.f

• Label: 394.3.f
• Level: $394$
• Weight: $3$
• Character orbit: 394.f
• Rep. character: $\chi_{394}(69,\cdot)$
• Character field: $\Q(\zeta_{28})$
• Dimension: $396$
• Newform subspaces: $2$
• Sturm bound: $148$
• Trace bound: $1$

Defining parameters

Level:	\( N \)	\(=\)	\( 394 = 2 \cdot 197 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	394.f (of order \(28\) and degree \(12\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 197 \)
Character field:			\(\Q(\zeta_{28})\)
Newform subspaces:			\( 2 \)
Sturm bound:			\(148\)
Trace bound:			\(1\)
Distinguishing \(T_p\):			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{3}(394, [\chi])\).

	Total	New	Old
Modular forms	1212	396	816
Cusp forms	1164	396	768
Eisenstein series	48	0	48

Trace form

\( 396 q + 2 q^{2} - 2 q^{5} - 4 q^{8} + O(q^{10}) \)

Decomposition of \(S_{3}^{\mathrm{new}}(394, [\chi])\) 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
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
394.3.f.a	$394$	$3$	394.f	197.f	$28$	$192$	$16$	$10.736$		None		$2$	$0$	\(-32\)	\(0\)	\(2\)	\(28\)		$\mathrm{SU}(2)[C_{28}]$
394.3.f.b	$394$	$3$	394.f	197.f	$28$	$204$	$17$	$10.736$		None		$2$	$0$	\(34\)	\(0\)	\(-4\)	\(-28\)		$\mathrm{SU}(2)[C_{28}]$

Decomposition of \(S_{3}^{\mathrm{old}}(394, [\chi])\) into lower level spaces

\( S_{3}^{\mathrm{old}}(394, [\chi]) \simeq \) \(S_{3}^{\mathrm{new}}(197, [\chi])\)\(^{\oplus 2}\)