Space of modular forms of level 1050, weight 3, and Character 1050.f

• Label: 1050.3.f
• Level: $1050$
• Weight: $3$
• Character orbit: 1050.f
• Rep. character: $\chi_{1050}(601,\cdot)$
• Character field: $\Q$
• Dimension: $52$
• Newform subspaces: $5$
• Sturm bound: $720$
• Trace bound: $7$

Defining parameters

Level:	\( N \)	\(=\)	\( 1050 = 2 \cdot 3 \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1050.f (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 7 \)
Character field:			\(\Q\)
Newform subspaces:			\( 5 \)
Sturm bound:			\(720\)
Trace bound:			\(7\)
Distinguishing \(T_p\):			\(11\), \(23\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{3}(1050, [\chi])\).

	Total	New	Old
Modular forms	504	52	452
Cusp forms	456	52	404
Eisenstein series	48	0	48

Trace form

\( 52 q + 104 q^{4} - 8 q^{7} - 156 q^{9} + O(q^{10}) \)

Decomposition of \(S_{3}^{\mathrm{new}}(1050, [\chi])\) 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
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
1050.3.f.a	$1050$	$3$	1050.f	7.b	$2$	$4$	$4$	$28.610$	\(\Q(\sqrt{2}, \sqrt{-3})\)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(-8\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{1}q^{2}+\beta _{2}q^{3}+2q^{4}+\beta _{3}q^{6}+(-2+\cdots)q^{7}+\cdots\)
1050.3.f.b	$1050$	$3$	1050.f	7.b	$2$	$8$	$8$	$28.610$	8.0.3317760000.3	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{6}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{1}q^{2}+\beta _{5}q^{3}+2q^{4}-\beta _{3}q^{6}+(2\beta _{1}+\cdots)q^{7}+\cdots\)
1050.3.f.c	$1050$	$3$	1050.f	7.b	$2$	$12$	$12$	$28.610$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(-10\)	$2^{6}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{5}q^{2}-\beta _{2}q^{3}+2q^{4}-\beta _{4}q^{6}+(-1+\cdots)q^{7}+\cdots\)
1050.3.f.d	$1050$	$3$	1050.f	7.b	$2$	$12$	$12$	$28.610$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(10\)	$2^{6}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{5}q^{2}+\beta _{2}q^{3}+2q^{4}-\beta _{4}q^{6}+(1+\cdots)q^{7}+\cdots\)
1050.3.f.e	$1050$	$3$	1050.f	7.b	$2$	$16$	$16$	$28.610$	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{20}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{5}q^{2}-\beta _{1}q^{3}+2q^{4}-\beta _{8}q^{6}+(\beta _{1}+\cdots)q^{7}+\cdots\)

Decomposition of \(S_{3}^{\mathrm{old}}(1050, [\chi])\) into lower level spaces

\( S_{3}^{\mathrm{old}}(1050, [\chi]) \cong \) \(S_{3}^{\mathrm{new}}(7, [\chi])\)\(^{\oplus 12}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(14, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(21, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(35, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(42, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(105, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(175, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(525, [\chi])\)\(^{\oplus 2}\)