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

• Label: 1600.2.o
• Level: $1600$
• Weight: $2$
• Character orbit: 1600.o
• Rep. character: $\chi_{1600}(543,\cdot)$
• Character field: $\Q(\zeta_{4})$
• Dimension: $72$
• Newform subspaces: $12$
• Sturm bound: $480$
• Trace bound: $39$

Defining parameters

Level:	\( N \)	\(=\)	\( 1600 = 2^{6} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1600.o (of order \(4\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 40 \)
Character field:			\(\Q(i)\)
Newform subspaces:			\( 12 \)
Sturm bound:			\(480\)
Trace bound:			\(39\)
Distinguishing \(T_p\):			\(3\), \(7\), \(79\)

Dimensions

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

	Total	New	Old
Modular forms	552	72	480
Cusp forms	408	72	336
Eisenstein series	144	0	144

Trace form

\( 72 q + O(q^{10}) \)

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.o.a	$1600$	$2$	1600.o	40.k	$4$	$2$	$1$	$12.776$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(-2\)	\(0\)	\(-2\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+(-1+i)q^{3}+(-1+i)q^{7}+iq^{9}+\cdots\)
1600.2.o.b	$1600$	$2$	1600.o	40.k	$4$	$2$	$1$	$12.776$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(-2\)	\(0\)	\(2\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+(-1+i)q^{3}+(1-i)q^{7}+iq^{9}+4q^{11}+\cdots\)
1600.2.o.c	$1600$	$2$	1600.o	40.k	$4$	$2$	$1$	$12.776$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(2\)	\(0\)	\(-2\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+(1-i)q^{3}+(-1+i)q^{7}+iq^{9}-4q^{11}+\cdots\)
1600.2.o.d	$1600$	$2$	1600.o	40.k	$4$	$2$	$1$	$12.776$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(2\)	\(0\)	\(2\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+(1-i)q^{3}+(1-i)q^{7}+iq^{9}-4q^{11}+\cdots\)
1600.2.o.e	$1600$	$2$	1600.o	40.k	$4$	$8$	$4$	$12.776$	8.0.49787136.1	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(-4\)	$2^{7}$	$\mathrm{SU}(2)[C_{4}]$	\(q-\beta _{2}q^{3}+(-\beta _{1}+\beta _{4})q^{7}+(2\beta _{1}+\beta _{4}+\cdots)q^{9}+\cdots\)
1600.2.o.f	$1600$	$2$	1600.o	40.k	$4$	$8$	$4$	$12.776$	8.0.\(\cdots\).1	None		$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{4}$	$\mathrm{SU}(2)[C_{4}]$	\(q+\beta _{7}q^{3}+2\beta _{6}q^{7}-4\beta _{3}q^{9}-\beta _{4}q^{11}+\cdots\)
1600.2.o.g	$1600$	$2$	1600.o	40.k	$4$	$8$	$4$	$12.776$	\(\Q(\zeta_{24})\)	None		$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{10}$	$\mathrm{SU}(2)[C_{4}]$	\(q+\zeta_{24}q^{3}+\zeta_{24}^{2}q^{7}+\zeta_{24}^{3}q^{9}-\zeta_{24}^{7}q^{11}+\cdots\)
1600.2.o.h	$1600$	$2$	1600.o	40.k	$4$	$8$	$4$	$12.776$	\(\Q(\zeta_{24})\)	None		$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{4}$	$\mathrm{SU}(2)[C_{4}]$	\(q-\zeta_{24}^{5}q^{3}+2\zeta_{24}^{3}q^{9}-\zeta_{24}^{6}q^{11}+\cdots\)
1600.2.o.i	$1600$	$2$	1600.o	40.k	$4$	$8$	$4$	$12.776$	\(\Q(\zeta_{24})\)	None		$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{4}$	$\mathrm{SU}(2)[C_{4}]$	\(q-\zeta_{24}^{5}q^{3}+2\zeta_{24}^{3}q^{9}-\zeta_{24}^{6}q^{11}+\cdots\)
1600.2.o.j	$1600$	$2$	1600.o	40.k	$4$	$8$	$4$	$12.776$	\(\Q(\zeta_{24})\)	None		$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{10}$	$\mathrm{SU}(2)[C_{4}]$	\(q+\zeta_{24}q^{3}+\zeta_{24}^{2}q^{7}+\zeta_{24}^{3}q^{9}+\zeta_{24}^{7}q^{11}+\cdots\)
1600.2.o.k	$1600$	$2$	1600.o	40.k	$4$	$8$	$4$	$12.776$	8.0.\(\cdots\).1	None		$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{4}$	$\mathrm{SU}(2)[C_{4}]$	\(q+\beta _{7}q^{3}-2\beta _{6}q^{7}-4\beta _{3}q^{9}-\beta _{4}q^{11}+\cdots\)
1600.2.o.l	$1600$	$2$	1600.o	40.k	$4$	$8$	$4$	$12.776$	8.0.49787136.1	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(4\)	$2^{7}$	$\mathrm{SU}(2)[C_{4}]$	\(q-\beta _{2}q^{3}+(\beta _{1}-\beta _{4})q^{7}+(2\beta _{1}+\beta _{4}+\cdots)q^{9}+\cdots\)

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

\( S_{2}^{\mathrm{old}}(1600, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(40, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(80, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(160, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(200, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(320, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(400, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(800, [\chi])\)\(^{\oplus 2}\)