Space of modular forms of level 1050, weight 2, and Character 1050.b

• Label: 1050.2.b
• Level: $1050$
• Weight: $2$
• Character orbit: 1050.b
• Rep. character: $\chi_{1050}(251,\cdot)$
• Character field: $\Q$
• Dimension: $52$
• Newform subspaces: $6$
• Sturm bound: $480$
• Trace bound: $7$

Defining parameters

Level:	\( N \)	\(=\)	\( 1050 = 2 \cdot 3 \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1050.b (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 21 \)
Character field:			\(\Q\)
Newform subspaces:			\( 6 \)
Sturm bound:			\(480\)
Trace bound:			\(7\)
Distinguishing \(T_p\):			\(11\), \(17\), \(37\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(1050, [\chi])\).

	Total	New	Old
Modular forms	264	52	212
Cusp forms	216	52	164
Eisenstein series	48	0	48

Trace form

\( 52 q - 52 q^{4} - 8 q^{7} + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(1050, [\chi])\) 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
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
1050.2.b.a	$1050$	$2$	1050.b	21.c	$2$	$4$	$4$	$8.384$	\(\Q(i, \sqrt{5})\)	None		$2$	$0$	\(0\)	\(-2\)	\(0\)	\(-6\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{2}q^{2}+(-1+\beta _{3})q^{3}-q^{4}+(-\beta _{1}+\cdots)q^{6}+\cdots\)
1050.2.b.b	$1050$	$2$	1050.b	21.c	$2$	$4$	$4$	$8.384$	\(\Q(i, \sqrt{6})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(4\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{2}q^{2}-\beta _{3}q^{3}-q^{4}-\beta _{1}q^{6}+(1+\cdots)q^{7}+\cdots\)
1050.2.b.c	$1050$	$2$	1050.b	21.c	$2$	$4$	$4$	$8.384$	\(\Q(i, \sqrt{5})\)	None		$2$	$0$	\(0\)	\(2\)	\(0\)	\(-6\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{2}q^{2}+(1-\beta _{3})q^{3}-q^{4}+(\beta _{1}+\beta _{2}+\cdots)q^{6}+\cdots\)
1050.2.b.d	$1050$	$2$	1050.b	21.c	$2$	$12$	$12$	$8.384$	\(\mathbb{Q}[x]/(x^{12} + \cdots)\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(-4\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{4}q^{2}-\beta _{9}q^{3}-q^{4}+\beta _{8}q^{6}+(\beta _{1}+\cdots)q^{7}+\cdots\)
1050.2.b.e	$1050$	$2$	1050.b	21.c	$2$	$12$	$12$	$8.384$	\(\mathbb{Q}[x]/(x^{12} + \cdots)\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(4\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{4}q^{2}-\beta _{9}q^{3}-q^{4}-\beta _{8}q^{6}+(\beta _{7}+\cdots)q^{7}+\cdots\)
1050.2.b.f	$1050$	$2$	1050.b	21.c	$2$	$16$	$16$	$8.384$	16.0.\(\cdots\).7	None		$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{16}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{10}q^{2}-\beta _{7}q^{3}-q^{4}-\beta _{2}q^{6}+\beta _{9}q^{7}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(1050, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(1050, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(21, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(42, [\chi])\)\(^{\oplus 3}\)