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}$