Space of modular forms of level 471, weight 1, and Character 471.d

• Label: 471.1.d
• Level: $471$
• Weight: $1$
• Character orbit: 471.d
• Rep. character: $\chi_{471}(470,\cdot)$
• Character field: $\Q$
• Dimension: $7$
• Newform subspaces: $3$
• Sturm bound: $52$
• Trace bound: $1$

Defining parameters

Level:	\( N \)	\(=\)	\( 471 = 3 \cdot 157 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	471.d (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 471 \)
Character field:			\(\Q\)
Newform subspaces:			\( 3 \)
Sturm bound:			\(52\)
Trace bound:			\(1\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{1}(471, [\chi])\).

	Total	New	Old
Modular forms	9	9	0
Cusp forms	7	7	0
Eisenstein series	2	2	0

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

	\(D_n\)	\(A_4\)	\(S_4\)	\(A_5\)
Dimension	7	0	0	0

Trace form

\( 7 q - q^{3} + 5 q^{4} + 7 q^{9} + O(q^{10}) \)

Decomposition of \(S_{1}^{\mathrm{new}}(471, [\chi])\) 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
																$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
471.1.d.a	$471$	$1$	471.d	471.d	$2$	$1$	$1$	$0.235$	\(\Q\)	$D_{2}$	\(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-471}) \)	\(\Q(\sqrt{157}) \)	✓	$4$	$0$	\(0\)	\(1\)	\(0\)	\(0\)	$1$		\(q+q^{3}-q^{4}+q^{9}-q^{12}+2q^{13}+q^{16}+\cdots\)
471.1.d.b	$471$	$1$	471.d	471.d	$2$	$2$	$2$	$0.235$	\(\Q(\sqrt{2}) \)	$D_{4}$	\(\Q(\sqrt{-471}) \)	None	✓	$4$	$0$	\(0\)	\(2\)	\(0\)	\(0\)	$1$		\(q-\beta q^{2}+q^{3}+q^{4}+\beta q^{5}-\beta q^{6}+q^{9}+\cdots\)
471.1.d.c	$471$	$1$	471.d	471.d	$2$	$4$	$4$	$0.235$	\(\Q(\zeta_{16})^+\)	$D_{8}$	\(\Q(\sqrt{-471}) \)	None	✓	$4$	$0$	\(0\)	\(-4\)	\(0\)	\(0\)	$1$		\(q-\beta _{1}q^{2}-q^{3}+(1+\beta _{2})q^{4}+\beta _{3}q^{5}+\cdots\)