Space of modular forms of level 4000, weight 1, and Character 4000.bf

• Label: 4000.1.bf
• Level: $4000$
• Weight: $1$
• Character orbit: 4000.bf
• Rep. character: $\chi_{4000}(799,\cdot)$
• Character field: $\Q(\zeta_{10})$
• Dimension: $16$
• Newform subspaces: $2$
• Sturm bound: $600$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 4000 = 2^{5} \cdot 5^{3} \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	4000.bf (of order \(10\) and degree \(4\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 100 \)
Character field:			\(\Q(\zeta_{10})\)
Newform subspaces:			\( 2 \)
Sturm bound:			\(600\)
Trace bound:			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{1}(4000, [\chi])\).

	Total	New	Old
Modular forms	232	16	216
Cusp forms	72	16	56
Eisenstein series	160	0	160

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

	\(D_n\)	\(A_4\)	\(S_4\)	\(A_5\)
Dimension	0	0	0	16

Trace form

\( 16 q + O(q^{10}) \)

Decomposition of \(S_{1}^{\mathrm{new}}(4000, [\chi])\) 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
																				$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
4000.1.bf.a	$4000$	$1$	4000.bf	100.h	$10$	$8$	$2$	$1.996$	\(\Q(\zeta_{20})\)	$A_{5}$	None	None					800.1.bh.a	$4$	$0$	\(0\)	\(-2\)	\(0\)	\(4\)	$1$		\(q-\zeta_{20}^{2}q^{3}+(-\zeta_{20}^{4}+\zeta_{20}^{6})q^{7}+\cdots\)
4000.1.bf.b	$4000$	$1$	4000.bf	100.h	$10$	$8$	$2$	$1.996$	\(\Q(\zeta_{20})\)	$A_{5}$	None	None					800.1.bh.a	$4$	$0$	\(0\)	\(2\)	\(0\)	\(-4\)	$1$		\(q+\zeta_{20}^{2}q^{3}+(\zeta_{20}^{4}-\zeta_{20}^{6})q^{7}+(-\zeta_{20}^{3}+\cdots)q^{13}+\cdots\)

Decomposition of \(S_{1}^{\mathrm{old}}(4000, [\chi])\) into lower level spaces

\( S_{1}^{\mathrm{old}}(4000, [\chi]) \simeq \) \(S_{1}^{\mathrm{new}}(400, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{1}^{\mathrm{new}}(500, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{1}^{\mathrm{new}}(2000, [\chi])\)\(^{\oplus 2}\)