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

• Label: 1200.3.j
• Level: $1200$
• Weight: $3$
• Character orbit: 1200.j
• Rep. character: $\chi_{1200}(799,\cdot)$
• Character field: $\Q$
• Dimension: $36$
• Newform subspaces: $6$
• Sturm bound: $720$
• Trace bound: $29$

Defining parameters

Level:	\( N \)	\(=\)	\( 1200 = 2^{4} \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1200.j (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 20 \)
Character field:			\(\Q\)
Newform subspaces:			\( 6 \)
Sturm bound:			\(720\)
Trace bound:			\(29\)
Distinguishing \(T_p\):			\(7\), \(13\)

Dimensions

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

	Total	New	Old
Modular forms	516	36	480
Cusp forms	444	36	408
Eisenstein series	72	0	72

Trace form

\( 36 q + 108 q^{9} - 120 q^{29} - 24 q^{41} + 780 q^{49} + 120 q^{61} + 324 q^{81} + 792 q^{89}+O(q^{100}) \)

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	Twist minimal	Largest	Maximal	Minimal twist	Inner twists	Rank*	Traces				Coefficient ring index	Sato-Tate	$q$-expansion
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
1200.3.j.a	$1200$	$3$	1200.j	20.d	$2$	$4$	$4$	$32.698$	\(\Q(\zeta_{12})\)	None					48.3.g.a	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{4}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta_1 q^{3}-4\beta_1 q^{7}+3 q^{9}-6\beta_{2} q^{11}+\cdots\)
1200.3.j.b	$1200$	$3$	1200.j	20.d	$2$	$4$	$4$	$32.698$	\(\Q(\zeta_{12})\)	None		✓			1200.3.e.b	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta_{3} q^{3}-3\beta_{3} q^{7}+3 q^{9}+2\beta_{2} q^{11}+\cdots\)
1200.3.j.c	$1200$	$3$	1200.j	20.d	$2$	$4$	$4$	$32.698$	\(\Q(\zeta_{12})\)	None		✓			1200.3.e.c	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta_{3} q^{3}-\beta_{3} q^{7}+3 q^{9}-2\beta_{2} q^{11}+\cdots\)
1200.3.j.d	$1200$	$3$	1200.j	20.d	$2$	$8$	$8$	$32.698$	\(\Q(i, \sqrt{3}, \sqrt{11})\)	None		✓			1200.3.e.k	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{12}\cdot 3^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{1}q^{3}+(-\beta _{1}+\beta _{5})q^{7}+3q^{9}-\beta _{4}q^{11}+\cdots\)
1200.3.j.e	$1200$	$3$	1200.j	20.d	$2$	$8$	$8$	$32.698$	\(\Q(i, \sqrt{3}, \sqrt{5})\)	None					240.3.e.b	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{14}\cdot 3^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{1}q^{3}+(-2\beta _{1}-\beta _{6})q^{7}+3q^{9}+\cdots\)
1200.3.j.f	$1200$	$3$	1200.j	20.d	$2$	$8$	$8$	$32.698$	\(\Q(i, \sqrt{3}, \sqrt{5})\)	None					240.3.e.a	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{14}\cdot 3^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{1}q^{3}+(-2\beta _{1}+\beta _{6})q^{7}+3q^{9}+\cdots\)

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

\( S_{3}^{\mathrm{old}}(1200, [\chi]) \simeq \) \(S_{3}^{\mathrm{new}}(20, [\chi])\)\(^{\oplus 12}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(60, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(80, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(100, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(240, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(300, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(400, [\chi])\)\(^{\oplus 2}\)