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

• Label: 3600.1.dx
• Level: $3600$
• Weight: $1$
• Character orbit: 3600.dx
• Rep. character: $\chi_{3600}(287,\cdot)$
• Character field: $\Q(\zeta_{20})$
• Dimension: $16$
• Newform subspaces: $1$
• Sturm bound: $720$
• Trace bound: $0$

Defining parameters

Level:	\( N \)	\(=\)	\( 3600 = 2^{4} \cdot 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	3600.dx (of order \(20\) and degree \(8\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 300 \)
Character field:			\(\Q(\zeta_{20})\)
Newform subspaces:			\( 1 \)
Sturm bound:			\(720\)
Trace bound:			\(0\)

Dimensions

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

	Total	New	Old
Modular forms	256	16	240
Cusp forms	64	16	48
Eisenstein series	192	0	192

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

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

Trace form

\( 16 q - 4 q^{13} + 4 q^{37} + 4 q^{73} - 12 q^{85} + 4 q^{97}+O(q^{100}) \)

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	Twist minimal	Largest	Maximal	Minimal twist	Inner twists	Rank*	Traces				Coefficient ring index	Sato-Tate	$q$-expansion
																				$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
3600.1.dx.a	$3600$	$1$	3600.dx	300.u	$20$	$16$	$2$	$1.797$	\(\Q(\zeta_{40})\)	$D_{20}$	\(\Q(\sqrt{-1}) \)	None		✓	✓	✓	3600.1.dx.a	$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$		\(q-\zeta_{40}^{7}q^{5}+(-\zeta_{40}^{4}+\zeta_{40}^{18})q^{13}+\cdots\)

Decomposition of \(S_{1}^{\mathrm{old}}(3600, [\chi])\) into lower level spaces

\( S_{1}^{\mathrm{old}}(3600, [\chi]) \simeq \) \(S_{1}^{\mathrm{new}}(900, [\chi])\)\(^{\oplus 3}\)