Space of modular forms of level 1388, weight 3, and Character 1388.d

• Label: 1388.3.d
• Level: $1388$
• Weight: $3$
• Character orbit: 1388.d
• Rep. character: $\chi_{1388}(693,\cdot)$
• Character field: $\Q$
• Dimension: $58$
• Newform subspaces: $2$
• Sturm bound: $522$
• Trace bound: $1$

Defining parameters

Level:	\( N \)	\(=\)	\( 1388 = 2^{2} \cdot 347 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1388.d (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 347 \)
Character field:			\(\Q\)
Newform subspaces:			\( 2 \)
Sturm bound:			\(522\)
Trace bound:			\(1\)
Distinguishing \(T_p\):			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{3}(1388, [\chi])\).

	Total	New	Old
Modular forms	351	58	293
Cusp forms	345	58	287
Eisenstein series	6	0	6

Trace form

\( 58 q + 4 q^{3} + 186 q^{9} + O(q^{10}) \)

Decomposition of \(S_{3}^{\mathrm{new}}(1388, [\chi])\) into newform subspaces

Label	Level	Weight	Char	Prim	Char order	Dim	Rel. Dim	$A$	Field	CM	Self-dual	Twist minimal	Minimal twist	Inner twists	Rank*	Traces				Coefficient ring index	Sato-Tate	$q$-expansion
																$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
1388.3.d.a	$1388$	$3$	1388.d	347.b	$2$	$10$	$10$	$37.820$	\(\mathbb{Q}[x]/(x^{10} - \cdots)\)	\(\Q(\sqrt{-347}) \)	✓	✓	1388.3.d.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{6}\cdot 5^{2}$	$\mathrm{U}(1)[D_{2}]$	\(q+\beta _{1}q^{3}+(9+2\beta _{4}+\beta _{6})q^{9}+(-\beta _{4}+\cdots)q^{11}+\cdots\)
1388.3.d.b	$1388$	$3$	1388.d	347.b	$2$	$48$	$48$	$37.820$		None		✓	1388.3.d.b	$2$	$0$	\(0\)	\(4\)	\(0\)	\(0\)		$\mathrm{SU}(2)[C_{2}]$

Decomposition of \(S_{3}^{\mathrm{old}}(1388, [\chi])\) into lower level spaces

\( S_{3}^{\mathrm{old}}(1388, [\chi]) \simeq \) \(S_{3}^{\mathrm{new}}(347, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(694, [\chi])\)\(^{\oplus 2}\)