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

• Label: 1600.1.bf
• Level: $1600$
• Weight: $1$
• Character orbit: 1600.bf
• Rep. character: $\chi_{1600}(319,\cdot)$
• Character field: $\Q(\zeta_{10})$
• Dimension: $4$
• 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.bf (of order \(10\) and degree \(4\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 100 \)
Character field:			\(\Q(\zeta_{10})\)
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	64	12	52
Cusp forms	16	4	12
Eisenstein series	48	8	40

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

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

Trace form

\( 4 q - q^{5} + q^{9} - q^{25} + 2 q^{29} + 5 q^{37} + 2 q^{41} + q^{45} - 4 q^{49} + 5 q^{53} - 2 q^{61} - 5 q^{65} - q^{81} - 5 q^{85} + 3 q^{89}+O(q^{100}) \)

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.bf.a	$1600$	$1$	1600.bf	100.h	$10$	$4$	$1$	$0.799$	\(\Q(\zeta_{10})\)	$D_{10}$	\(\Q(\sqrt{-1}) \)	None			✓	✓	400.1.x.a	$4$	$0$	\(0\)	\(0\)	\(-1\)	\(0\)	$1$		\(q-\zeta_{10}q^{5}+\zeta_{10}^{3}q^{9}+(\zeta_{10}^{2}-\zeta_{10}^{4}+\cdots)q^{13}+\cdots\)

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

\( S_{1}^{\mathrm{old}}(1600, [\chi]) \simeq \) \(S_{1}^{\mathrm{new}}(400, [\chi])\)\(^{\oplus 3}\)