Space of modular forms of level 1200, weight 3, and Character 1200.e

• Label: 1200.3.e
• Level: $1200$
• Weight: $3$
• Character orbit: 1200.e
• Rep. character: $\chi_{1200}(751,\cdot)$
• Character field: $\Q$
• Dimension: $38$
• Newform subspaces: $14$
• Sturm bound: $720$
• Trace bound: $13$

Defining parameters

Level:	\( N \)	\(=\)	\( 1200 = 2^{4} \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1200.e (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 4 \)
Character field:			\(\Q\)
Newform subspaces:			\( 14 \)
Sturm bound:			\(720\)
Trace bound:			\(13\)
Distinguishing \(T_p\):			\(7\), \(11\), \(13\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{3}(1200, [\chi])\).

	Total	New	Old
Modular forms	516	38	478
Cusp forms	444	38	406
Eisenstein series	72	0	72

Trace form

\( 38 q - 114 q^{9} + O(q^{10}) \)

Decomposition of \(S_{3}^{\mathrm{new}}(1200, [\chi])\) 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
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
1200.3.e.a	$1200$	$3$	1200.e	4.b	$2$	$2$	$2$	$32.698$	\(\Q(\sqrt{-3}) \)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q-\zeta_{6}q^{3}-4\zeta_{6}q^{7}-3q^{9}-8\zeta_{6}q^{11}+\cdots\)
1200.3.e.b	$1200$	$3$	1200.e	4.b	$2$	$2$	$2$	$32.698$	\(\Q(\sqrt{-3}) \)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q-\zeta_{6}q^{3}-3\zeta_{6}q^{7}-3q^{9}-2\zeta_{6}q^{11}+\cdots\)
1200.3.e.c	$1200$	$3$	1200.e	4.b	$2$	$2$	$2$	$32.698$	\(\Q(\sqrt{-3}) \)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+\zeta_{6}q^{3}-\zeta_{6}q^{7}-3q^{9}+2\zeta_{6}q^{11}+\cdots\)
1200.3.e.d	$1200$	$3$	1200.e	4.b	$2$	$2$	$2$	$32.698$	\(\Q(\sqrt{-3}) \)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+\zeta_{6}q^{3}-2\zeta_{6}q^{7}-3q^{9}-2\zeta_{6}q^{11}+\cdots\)
1200.3.e.e	$1200$	$3$	1200.e	4.b	$2$	$2$	$2$	$32.698$	\(\Q(\sqrt{-3}) \)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+\zeta_{6}q^{3}-2\zeta_{6}q^{7}-3q^{9}+2\zeta_{6}q^{11}+\cdots\)
1200.3.e.f	$1200$	$3$	1200.e	4.b	$2$	$2$	$2$	$32.698$	\(\Q(\sqrt{-3}) \)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+\zeta_{6}q^{3}-\zeta_{6}q^{7}-3q^{9}-2\zeta_{6}q^{11}+\cdots\)
1200.3.e.g	$1200$	$3$	1200.e	4.b	$2$	$2$	$2$	$32.698$	\(\Q(\sqrt{-3}) \)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q-\zeta_{6}q^{3}-3\zeta_{6}q^{7}-3q^{9}+2\zeta_{6}q^{11}+\cdots\)
1200.3.e.h	$1200$	$3$	1200.e	4.b	$2$	$2$	$2$	$32.698$	\(\Q(\sqrt{-3}) \)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q-\zeta_{6}q^{3}-4\zeta_{6}q^{7}-3q^{9}-12\zeta_{6}q^{11}+\cdots\)
1200.3.e.i	$1200$	$3$	1200.e	4.b	$2$	$2$	$2$	$32.698$	\(\Q(\sqrt{-3}) \)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q-\zeta_{6}q^{3}-4\zeta_{6}q^{7}-3q^{9}+8\zeta_{6}q^{11}+\cdots\)
1200.3.e.j	$1200$	$3$	1200.e	4.b	$2$	$4$	$4$	$32.698$	\(\Q(\sqrt{-3}, \sqrt{5})\)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{6}\cdot 3$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{1}q^{3}+(2\beta _{1}+\beta _{2})q^{7}-3q^{9}+(-2\beta _{1}+\cdots)q^{11}+\cdots\)
1200.3.e.k	$1200$	$3$	1200.e	4.b	$2$	$4$	$4$	$32.698$	\(\Q(\sqrt{-3}, \sqrt{11})\)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{4}\cdot 3$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{2}q^{3}+(\beta _{1}+\beta _{2})q^{7}-3q^{9}-\beta _{1}q^{11}+\cdots\)
1200.3.e.l	$1200$	$3$	1200.e	4.b	$2$	$4$	$4$	$32.698$	\(\Q(\sqrt{-3}, \sqrt{-7})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{6}\cdot 3$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{1}q^{3}+2\beta _{1}q^{7}-3q^{9}+\beta _{3}q^{11}+\cdots\)
1200.3.e.m	$1200$	$3$	1200.e	4.b	$2$	$4$	$4$	$32.698$	\(\Q(\sqrt{-3}, \sqrt{11})\)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{4}\cdot 3$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{2}q^{3}+(\beta _{1}+\beta _{2})q^{7}-3q^{9}+\beta _{1}q^{11}+\cdots\)
1200.3.e.n	$1200$	$3$	1200.e	4.b	$2$	$4$	$4$	$32.698$	\(\Q(\sqrt{-3}, \sqrt{5})\)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{6}\cdot 3$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{1}q^{3}+(2\beta _{1}+\beta _{2})q^{7}-3q^{9}+(2\beta _{1}+\cdots)q^{11}+\cdots\)

Decomposition of \(S_{3}^{\mathrm{old}}(1200, [\chi])\) into lower level spaces

\( S_{3}^{\mathrm{old}}(1200, [\chi]) \cong \) \(S_{3}^{\mathrm{new}}(12, [\chi])\)\(^{\oplus 9}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(16, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(24, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(48, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(60, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(120, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(240, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(300, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(600, [\chi])\)\(^{\oplus 2}\)