Space of modular forms of level 1600, weight 2, and Character 1600.s

• Label: 1600.2.s
• Level: $1600$
• Weight: $2$
• Character orbit: 1600.s
• Rep. character: $\chi_{1600}(207,\cdot)$
• Character field: $\Q(\zeta_{4})$
• Dimension: $68$
• Newform subspaces: $5$
• Sturm bound: $480$
• Trace bound: $1$

Defining parameters

Level:	\( N \)	\(=\)	\( 1600 = 2^{6} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1600.s (of order \(4\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 80 \)
Character field:			\(\Q(i)\)
Newform subspaces:			\( 5 \)
Sturm bound:			\(480\)
Trace bound:			\(1\)
Distinguishing \(T_p\):			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(1600, [\chi])\).

	Total	New	Old
Modular forms	528	76	452
Cusp forms	432	68	364
Eisenstein series	96	8	88

Trace form

\( 68 q - 4 q^{3} - 4 q^{7} + 60 q^{9} + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(1600, [\chi])\) into newform subspaces

Label	Level	Weight	Char	Prim	Char order	Dim	Rel. Dim	$A$	Field	CM	Self-dual	Inner twists	Rank*	Traces				Coefficient ring index	Sato-Tate	$q$-expansion
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
1600.2.s.a	$1600$	$2$	1600.s	80.s	$4$	$2$	$1$	$12.776$	\(\Q(\sqrt{-1}) \)	None		$2$	$1$	\(0\)	\(-4\)	\(0\)	\(-6\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q-2q^{3}+(-3-3i)q^{7}+q^{9}+(1-i)q^{11}+\cdots\)
1600.2.s.b	$1600$	$2$	1600.s	80.s	$4$	$8$	$4$	$12.776$	\(\Q(\zeta_{24})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{4}]$	\(q+(\zeta_{24}+\zeta_{24}^{5}+\zeta_{24}^{7})q^{3}+(-\zeta_{24}+\cdots)q^{7}+\cdots\)
1600.2.s.c	$1600$	$2$	1600.s	80.s	$4$	$16$	$8$	$12.776$	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{14}$	$\mathrm{SU}(2)[C_{4}]$	\(q-\beta _{4}q^{3}-\beta _{7}q^{7}+(1-\beta _{12})q^{9}+(-1+\cdots)q^{11}+\cdots\)
1600.2.s.d	$1600$	$2$	1600.s	80.s	$4$	$18$	$9$	$12.776$	\(\mathbb{Q}[x]/(x^{18} + \cdots)\)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(2\)	$2^{15}$	$\mathrm{SU}(2)[C_{4}]$	\(q+\beta _{1}q^{3}+\beta _{11}q^{7}+(1+\beta _{3})q^{9}+(-\beta _{13}+\cdots)q^{11}+\cdots\)
1600.2.s.e	$1600$	$2$	1600.s	80.s	$4$	$24$	$12$	$12.776$		None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$\mathrm{SU}(2)[C_{4}]$

Decomposition of \(S_{2}^{\mathrm{old}}(1600, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(1600, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(80, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(160, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(320, [\chi])\)\(^{\oplus 2}\)