Space of modular forms of level 404, weight 1, and Character 404.d

• Label: 404.1.d
• Level: $404$
• Weight: $1$
• Character orbit: 404.d
• Rep. character: $\chi_{404}(403,\cdot)$
• Character field: $\Q$
• Dimension: $6$
• Newform subspaces: $2$
• Sturm bound: $51$
• Trace bound: $2$

Defining parameters

Level:	\( N \)	\(=\)	\( 404 = 2^{2} \cdot 101 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	404.d (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 404 \)
Character field:			\(\Q\)
Newform subspaces:			\( 2 \)
Sturm bound:			\(51\)
Trace bound:			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{1}(404, [\chi])\).

	Total	New	Old
Modular forms	8	8	0
Cusp forms	6	6	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	6	0	0	0

Trace form

6 * q + 6 * q^4 - 2 * q^5 - 2 * q^6 + 4 * q^9 - 2 * q^13 - 2 * q^14 + 6 * q^16 - 2 * q^17 - 2 * q^20 - 4 * q^21 - 2 * q^22 - 2 * q^24 + 4 * q^25 - 4 * q^30 - 4 * q^33 + 4 * q^36 - 2 * q^37 - 6 * q^45 + 4 * q^49 - 2 * q^52 - 4 * q^54 - 2 * q^56 + 6 * q^64 - 4 * q^65 - 2 * q^68 - 4 * q^70 - 4 * q^77 - 4 * q^78 - 2 * q^80 + 2 * q^81 - 4 * q^84 - 4 * q^85 - 2 * q^88 - 2 * q^96 - 2 * q^97

Decomposition of \(S_{1}^{\mathrm{new}}(404, [\chi])\) into newform subspaces

Label	Level	Weight	Char	Prim	Char order	Dim	Rel. Dim	$A$	Field	Image	CM	RM	Self-dual	Twist minimal	Largest	Maximal	Minimal twist	Inner twists	Rank*	Traces				Coefficient ring index	Sato-Tate	$q$-expansion
																				$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
404.1.d.a	$404$	$1$	404.d	404.d	$2$	$3$	$3$	$0.202$	\(\Q(\zeta_{14})^+\)	$D_{7}$	\(\Q(\sqrt{-101}) \)	None	✓	✓			404.1.d.a	$2$	$0$	\(-3\)	\(1\)	\(-1\)	\(1\)	$1$		\(q-q^{2}+\beta _{1}q^{3}+q^{4}+(-1+\beta _{1}-\beta _{2})q^{5}+\cdots\)
404.1.d.b	$404$	$1$	404.d	404.d	$2$	$3$	$3$	$0.202$	\(\Q(\zeta_{14})^+\)	$D_{7}$	\(\Q(\sqrt{-101}) \)	None	✓	✓			404.1.d.a	$2$	$0$	\(3\)	\(-1\)	\(-1\)	\(-1\)	$1$		\(q+q^{2}-\beta _{1}q^{3}+q^{4}+(-1+\beta _{1}-\beta _{2})q^{5}+\cdots\)