Space of modular forms of level 1600, weight 1, and Character 1600.bw

• Label: 1600.1.bw
• Level: $1600$
• Weight: $1$
• Character orbit: 1600.bw
• Rep. character: $\chi_{1600}(513,\cdot)$
• Character field: $\Q(\zeta_{20})$
• Dimension: $8$
• Newform subspaces: $1$
• Sturm bound: $240$
• Trace bound: $0$

Defining parameters

Level:	\( N \)	\(=\)	\( 1600 = 2^{6} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	1600.bw (of order \(20\) and degree \(8\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 25 \)
Character field:			\(\Q(\zeta_{20})\)
Newform subspaces:			\( 1 \)
Sturm bound:			\(240\)
Trace bound:			\(0\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{1}(1600, [\chi])\).

	Total	New	Old
Modular forms	120	24	96
Cusp forms	24	8	16
Eisenstein series	96	16	80

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

\( 8 q - 2 q^{13} + 2 q^{17} + 2 q^{25} + 2 q^{37} + 2 q^{45} + 2 q^{53} - 2 q^{65} - 2 q^{73} + 2 q^{81} + 8 q^{85} - 10 q^{89} - 2 q^{97}+O(q^{100}) \)

Decomposition of \(S_{1}^{\mathrm{new}}(1600, [\chi])\) 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
																				$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
1600.1.bw.a	$1600$	$1$	1600.bw	25.f	$20$	$8$	$1$	$0.799$	\(\Q(\zeta_{20})\)	$D_{20}$	\(\Q(\sqrt{-1}) \)	None			✓	✓	800.1.bo.a	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$		\(q+\zeta_{20}^{3}q^{5}-\zeta_{20}^{9}q^{9}+(\zeta_{20}-\zeta_{20}^{2}+\cdots)q^{13}+\cdots\)

Decomposition of \(S_{1}^{\mathrm{old}}(1600, [\chi])\) into lower level spaces

\( S_{1}^{\mathrm{old}}(1600, [\chi]) \simeq \) \(S_{1}^{\mathrm{new}}(800, [\chi])\)\(^{\oplus 2}\)