Space of modular forms of level 408, weight 1, and Character 408.u

• Label: 408.1.u
• Level: $408$
• Weight: $1$
• Character orbit: 408.u
• Rep. character: $\chi_{408}(89,\cdot)$
• Character field: $\Q(\zeta_{4})$
• Dimension: $4$
• Newform subspaces: $1$
• Sturm bound: $72$
• Trace bound: $0$

Defining parameters

Level:	\( N \)	\(=\)	\( 408 = 2^{3} \cdot 3 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	408.u (of order \(4\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 51 \)
Character field:			\(\Q(i)\)
Newform subspaces:			\( 1 \)
Sturm bound:			\(72\)
Trace bound:			\(0\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{1}(408, [\chi])\).

	Total	New	Old
Modular forms	20	4	16
Cusp forms	4	4	0
Eisenstein series	16	0	16

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

	\(D_n\)	\(A_4\)	\(S_4\)	\(A_5\)
Dimension	0	0	4	0

Trace form

\( 4 q - 4 q^{7} + O(q^{10}) \)

Decomposition of \(S_{1}^{\mathrm{new}}(408, [\chi])\) 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
																$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
408.1.u.a	$408$	$1$	408.u	51.f	$4$	$4$	$2$	$0.204$	\(\Q(\zeta_{8})\)	$S_{4}$	None	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(-4\)	$1$		\(q-\zeta_{8}q^{3}-\zeta_{8}q^{5}+(-1-\zeta_{8}^{2})q^{7}+\cdots\)