Space of modular forms of level 1224, weight 1, and Character 1224.bu

• Label: 1224.1.bu
• Level: $1224$
• Weight: $1$
• Character orbit: 1224.bu
• Rep. character: $\chi_{1224}(53,\cdot)$
• Character field: $\Q(\zeta_{8})$
• Dimension: $8$
• Newform subspaces: $2$
• Sturm bound: $216$
• Trace bound: $14$

Defining parameters

Level:	\( N \)	\(=\)	\( 1224 = 2^{3} \cdot 3^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	1224.bu (of order \(8\) and degree \(4\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 408 \)
Character field:			\(\Q(\zeta_{8})\)
Newform subspaces:			\( 2 \)
Sturm bound:			\(216\)
Trace bound:			\(14\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{1}(1224, [\chi])\).

	Total	New	Old
Modular forms	40	8	32
Cusp forms	8	8	0
Eisenstein series	32	0	32

The following table gives the dimensions of subspaces with specified projective image type.

	\(D_n\)	\(A_4\)	\(S_4\)	\(A_5\)
Dimension	8	0	0	0

Trace form

\( 8 q + 8 q^{7} + O(q^{10}) \)

Decomposition of \(S_{1}^{\mathrm{new}}(1224, [\chi])\) into newform subspaces

Label	Level	Weight	Char	Prim	Char order	Dim	Rel. Dim	$A$	Field	Image	CM	RM	Self-dual	Inner twists	Rank*	Traces				Coefficient ring index	Sato-Tate	$q$-expansion
																$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
1224.1.bu.a	$1224$	$1$	1224.bu	408.ae	$8$	$4$	$1$	$0.611$	\(\Q(\zeta_{8})\)	$D_{8}$	None	\(\Q(\sqrt{2}) \)		$4$	$0$	\(0\)	\(0\)	\(0\)	\(4\)	$1$		\(q-\zeta_{8}q^{2}+\zeta_{8}^{2}q^{4}+(1-\zeta_{8}^{3})q^{7}-\zeta_{8}^{3}q^{8}+\cdots\)
1224.1.bu.b	$1224$	$1$	1224.bu	408.ae	$8$	$4$	$1$	$0.611$	\(\Q(\zeta_{8})\)	$D_{8}$	None	\(\Q(\sqrt{2}) \)		$4$	$0$	\(0\)	\(0\)	\(0\)	\(4\)	$1$		\(q+\zeta_{8}q^{2}+\zeta_{8}^{2}q^{4}+(1-\zeta_{8}^{3})q^{7}+\zeta_{8}^{3}q^{8}+\cdots\)