Space of modular forms of level 3600, weight 1, and Character 3600.bf

• Label: 3600.1.bf
• Level: $3600$
• Weight: $1$
• Character orbit: 3600.bf
• Rep. character: $\chi_{3600}(2557,\cdot)$
• Character field: $\Q(\zeta_{4})$
• Dimension: $4$
• Newform subspaces: $2$
• Sturm bound: $720$
• Trace bound: $2$

Defining parameters

Level:	\( N \)	\(=\)	\( 3600 = 2^{4} \cdot 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	3600.bf (of order \(4\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 80 \)
Character field:			\(\Q(i)\)
Newform subspaces:			\( 2 \)
Sturm bound:			\(720\)
Trace bound:			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{1}(3600, [\chi])\).

	Total	New	Old
Modular forms	52	8	44
Cusp forms	4	4	0
Eisenstein series	48	4	44

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

\( 4 q + 4 q^{4} + O(q^{10}) \)

Decomposition of \(S_{1}^{\mathrm{new}}(3600, [\chi])\) 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
																$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
3600.1.bf.a	$3600$	$1$	3600.bf	80.t	$4$	$2$	$1$	$1.797$	\(\Q(\sqrt{-1}) \)	$D_{4}$	\(\Q(\sqrt{-15}) \)	None		$4$	$0$	\(-2\)	\(0\)	\(0\)	\(0\)	$1$		\(q-q^{2}+q^{4}-q^{8}+q^{16}+(1-i)q^{17}+\cdots\)
3600.1.bf.b	$3600$	$1$	3600.bf	80.t	$4$	$2$	$1$	$1.797$	\(\Q(\sqrt{-1}) \)	$D_{4}$	\(\Q(\sqrt{-15}) \)	None		$4$	$0$	\(2\)	\(0\)	\(0\)	\(0\)	$1$		\(q+q^{2}+q^{4}+q^{8}+q^{16}+(-1+i+\cdots)q^{17}+\cdots\)