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

• Label: 1050.2.d
• Level: $1050$
• Weight: $2$
• Character orbit: 1050.d
• Rep. character: $\chi_{1050}(1049,\cdot)$
• Character field: $\Q$
• Dimension: $48$
• Newform subspaces: $8$
• Sturm bound: $480$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 1050 = 2 \cdot 3 \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1050.d (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 105 \)
Character field:			\(\Q\)
Newform subspaces:			\( 8 \)
Sturm bound:			\(480\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(11\), \(13\), \(23\)

Dimensions

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

	Total	New	Old
Modular forms	264	48	216
Cusp forms	216	48	168
Eisenstein series	48	0	48

Trace form

\( 48 q + 48 q^{4} + 48 q^{16} + 20 q^{21} + 48 q^{39} + 56 q^{46} - 16 q^{49} + 40 q^{51} + 48 q^{64} - 32 q^{79} - 88 q^{81} + 20 q^{84} - 56 q^{91} - 96 q^{99}+O(q^{100}) \)

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	Twist minimal	Largest	Maximal	Minimal twist	Inner twists	Rank*	Traces				Coefficient ring index	Sato-Tate	$q$-expansion
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
1050.2.d.a	$1050$	$2$	1050.d	105.g	$2$	$4$	$4$	$8.384$	\(\Q(i, \sqrt{5})\)	None					210.2.b.a	$2$	$0$	\(-4\)	\(-2\)	\(0\)	\(6\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q-q^{2}+(-\beta _{2}-\beta _{3})q^{3}+q^{4}+(\beta _{2}+\beta _{3})q^{6}+\cdots\)
1050.2.d.b	$1050$	$2$	1050.d	105.g	$2$	$4$	$4$	$8.384$	\(\Q(i, \sqrt{6})\)	None					42.2.d.a	$4$	$0$	\(-4\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-q^{2}-\beta _{3}q^{3}+q^{4}+\beta _{3}q^{6}+(\beta _{1}+\beta _{2}+\cdots)q^{7}+\cdots\)
1050.2.d.c	$1050$	$2$	1050.d	105.g	$2$	$4$	$4$	$8.384$	\(\Q(i, \sqrt{5})\)	None					210.2.b.a	$2$	$0$	\(-4\)	\(2\)	\(0\)	\(-6\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q-q^{2}+(\beta _{2}+\beta _{3})q^{3}+q^{4}+(-\beta _{2}-\beta _{3})q^{6}+\cdots\)
1050.2.d.d	$1050$	$2$	1050.d	105.g	$2$	$4$	$4$	$8.384$	\(\Q(i, \sqrt{5})\)	None					210.2.b.a	$2$	$0$	\(4\)	\(-2\)	\(0\)	\(6\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+q^{2}+(-\beta _{2}-\beta _{3})q^{3}+q^{4}+(-\beta _{2}+\cdots)q^{6}+\cdots\)
1050.2.d.e	$1050$	$2$	1050.d	105.g	$2$	$4$	$4$	$8.384$	\(\Q(i, \sqrt{6})\)	None					42.2.d.a	$4$	$0$	\(4\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+q^{2}+\beta _{3}q^{3}+q^{4}+\beta _{3}q^{6}+(-\beta _{1}+\cdots)q^{7}+\cdots\)
1050.2.d.f	$1050$	$2$	1050.d	105.g	$2$	$4$	$4$	$8.384$	\(\Q(i, \sqrt{5})\)	None					210.2.b.a	$2$	$0$	\(4\)	\(2\)	\(0\)	\(-6\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+q^{2}+(\beta _{2}+\beta _{3})q^{3}+q^{4}+(\beta _{2}+\beta _{3})q^{6}+\cdots\)
1050.2.d.g	$1050$	$2$	1050.d	105.g	$2$	$12$	$12$	$8.384$	\(\mathbb{Q}[x]/(x^{12} + \cdots)\)	None		✓			1050.2.b.d	$4$	$0$	\(-12\)	\(0\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q-q^{2}+\beta _{10}q^{3}+q^{4}-\beta _{10}q^{6}+(-\beta _{1}+\cdots)q^{7}+\cdots\)
1050.2.d.h	$1050$	$2$	1050.d	105.g	$2$	$12$	$12$	$8.384$	\(\mathbb{Q}[x]/(x^{12} + \cdots)\)	None		✓			1050.2.b.d	$4$	$0$	\(12\)	\(0\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+q^{2}-\beta _{10}q^{3}+q^{4}-\beta _{10}q^{6}+(\beta _{1}+\cdots)q^{7}+\cdots\)

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

\( S_{2}^{\mathrm{old}}(1050, [\chi]) \simeq \) \(S_{2}^{\mathrm{new}}(105, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(210, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(525, [\chi])\)\(^{\oplus 2}\)