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

• Label: 1200.2.w
• Level: $1200$
• Weight: $2$
• Character orbit: 1200.w
• Rep. character: $\chi_{1200}(607,\cdot)$
• Character field: $\Q(\zeta_{4})$
• Dimension: $36$
• Newform subspaces: $5$
• Sturm bound: $480$
• Trace bound: $41$

Defining parameters

Level:	\( N \)	\(=\)	\( 1200 = 2^{4} \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1200.w (of order \(4\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 20 \)
Character field:			\(\Q(i)\)
Newform subspaces:			\( 5 \)
Sturm bound:			\(480\)
Trace bound:			\(41\)
Distinguishing \(T_p\):			\(7\), \(13\)

Dimensions

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

	Total	New	Old
Modular forms	552	36	516
Cusp forms	408	36	372
Eisenstein series	144	0	144

Trace form

\( 36 q - 12 q^{13} - 12 q^{17} + 24 q^{33} + 12 q^{37} + 96 q^{41} + 12 q^{53} - 12 q^{73} + 48 q^{77} - 36 q^{81} - 48 q^{93} - 12 q^{97}+O(q^{100}) \)

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	Twist minimal	Largest	Maximal	Minimal twist	Inner twists	Rank*	Traces				Coefficient ring index	Sato-Tate	$q$-expansion
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
1200.2.w.a	$1200$	$2$	1200.w	20.e	$4$	$4$	$2$	$9.582$	\(\Q(\zeta_{8})\)	None					240.2.w.a	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+\zeta_{8}^{3}q^{3}+4\zeta_{8}q^{7}-\zeta_{8}^{2}q^{9}+(-4\zeta_{8}+\cdots)q^{11}+\cdots\)
1200.2.w.b	$1200$	$2$	1200.w	20.e	$4$	$8$	$4$	$9.582$	\(\Q(\zeta_{24})\)	None					240.2.w.b	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{6}$	$\mathrm{SU}(2)[C_{4}]$	\(q-\beta_{3} q^{3}+(\beta_{4}-2\beta_1)q^{7}-\beta_{2} q^{9}+\cdots\)
1200.2.w.c	$1200$	$2$	1200.w	20.e	$4$	$8$	$4$	$9.582$	\(\Q(\zeta_{24})\)	None		✓			1200.2.w.c	$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{8}$	$\mathrm{SU}(2)[C_{4}]$	\(q-\beta_1 q^{3}+2\beta_{3} q^{7}+\beta_{2} q^{9}-\beta_{4} q^{11}+\cdots\)
1200.2.w.d	$1200$	$2$	1200.w	20.e	$4$	$8$	$4$	$9.582$	\(\Q(\zeta_{24})\)	None		✓			1200.2.w.d	$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{4}$	$\mathrm{SU}(2)[C_{4}]$	\(q+\beta_1 q^{3}+\beta_{5} q^{7}+\beta_{3} q^{9}-2\beta_{4} q^{11}+\cdots\)
1200.2.w.e	$1200$	$2$	1200.w	20.e	$4$	$8$	$4$	$9.582$	\(\Q(\zeta_{24})\)	None		✓			1200.2.w.e	$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{4}$	$\mathrm{SU}(2)[C_{4}]$	\(q+\beta_1 q^{3}+\beta_{5} q^{7}+\beta_{3} q^{9}+2\beta_{4} q^{11}+\cdots\)

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

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