Space of modular forms of level 1200, weight 2, and Character 1200.o

• Label: 1200.2.o
• Level: $1200$
• Weight: $2$
• Character orbit: 1200.o
• Rep. character: $\chi_{1200}(1199,\cdot)$
• Character field: $\Q$
• Dimension: $36$
• Newform subspaces: $9$
• Sturm bound: $480$
• Trace bound: $21$

Defining parameters

Level:	\( N \)	\(=\)	\( 1200 = 2^{4} \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1200.o (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 60 \)
Character field:			\(\Q\)
Newform subspaces:			\( 9 \)
Sturm bound:			\(480\)
Trace bound:			\(21\)
Distinguishing \(T_p\):			\(7\), \(11\), \(17\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(1200, [\chi])\).

	Total	New	Old
Modular forms	276	36	240
Cusp forms	204	36	168
Eisenstein series	72	0	72

Trace form

\( 36 q + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(1200, [\chi])\) 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
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
1200.2.o.a	$1200$	$2$	1200.o	60.h	$2$	$4$	$4$	$9.582$	\(\Q(\zeta_{12})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{4}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\zeta_{12}^{2}q^{3}+\zeta_{12}^{3}q^{7}-3q^{9}-\zeta_{12}^{3}q^{11}+\cdots\)
1200.2.o.b	$1200$	$2$	1200.o	60.h	$2$	$4$	$4$	$9.582$	\(\Q(\zeta_{12})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{4}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\zeta_{12}^{2}q^{3}+\zeta_{12}^{3}q^{7}-3q^{9}+\zeta_{12}^{3}q^{11}+\cdots\)
1200.2.o.c	$1200$	$2$	1200.o	60.h	$2$	$4$	$4$	$9.582$	\(\Q(\zeta_{12})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+(\zeta_{12}+\zeta_{12}^{3})q^{3}+(-3+3\zeta_{12}^{2}+\cdots)q^{9}+\cdots\)
1200.2.o.d	$1200$	$2$	1200.o	60.h	$2$	$4$	$4$	$9.582$	\(\Q(\zeta_{12})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+(\zeta_{12}+\zeta_{12}^{3})q^{3}+(-3+3\zeta_{12}^{2}+\cdots)q^{9}+\cdots\)
1200.2.o.e	$1200$	$2$	1200.o	60.h	$2$	$4$	$4$	$9.582$	\(\Q(i, \sqrt{6})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{3}q^{3}+(\beta _{1}-\beta _{3})q^{7}-3\beta _{2}q^{9}+(2\beta _{1}+\cdots)q^{11}+\cdots\)
1200.2.o.f	$1200$	$2$	1200.o	60.h	$2$	$4$	$4$	$9.582$	\(\Q(i, \sqrt{6})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{3}q^{3}+(\beta _{1}-\beta _{3})q^{7}-3\beta _{2}q^{9}+(-2\beta _{1}+\cdots)q^{11}+\cdots\)
1200.2.o.g	$1200$	$2$	1200.o	60.h	$2$	$4$	$4$	$9.582$	\(\Q(\zeta_{12})\)	\(\Q(\sqrt{-3}) \)		$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{U}(1)[D_{2}]$	\(q-\zeta_{12}^{3}q^{3}+3\zeta_{12}^{3}q^{7}+3q^{9}+7\zeta_{12}q^{13}+\cdots\)
1200.2.o.h	$1200$	$2$	1200.o	60.h	$2$	$4$	$4$	$9.582$	\(\Q(\zeta_{12})\)	\(\Q(\sqrt{-3}) \)		$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}\cdot 5^{2}$	$\mathrm{U}(1)[D_{2}]$	\(q+\zeta_{12}q^{3}+\zeta_{12}q^{7}+3q^{9}-\zeta_{12}^{3}q^{13}+\cdots\)
1200.2.o.i	$1200$	$2$	1200.o	60.h	$2$	$4$	$4$	$9.582$	\(\Q(\zeta_{12})\)	\(\Q(\sqrt{-3}) \)		$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{4}$	$\mathrm{U}(1)[D_{2}]$	\(q+\zeta_{12}q^{3}+2\zeta_{12}q^{7}+3q^{9}-\zeta_{12}^{3}q^{13}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(1200, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(1200, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(60, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(120, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(240, [\chi])\)\(^{\oplus 2}\)