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

• Label: 1050.2.j
• Level: $1050$
• Weight: $2$
• Character orbit: 1050.j
• Rep. character: $\chi_{1050}(407,\cdot)$
• Character field: $\Q(\zeta_{4})$
• Dimension: $72$
• Newform subspaces: $6$
• Sturm bound: $480$
• Trace bound: $14$

Defining parameters

Level:	\( N \)	\(=\)	\( 1050 = 2 \cdot 3 \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1050.j (of order \(4\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 15 \)
Character field:			\(\Q(i)\)
Newform subspaces:			\( 6 \)
Sturm bound:			\(480\)
Trace bound:			\(14\)
Distinguishing \(T_p\):			\(11\), \(17\), \(29\)

Dimensions

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

	Total	New	Old
Modular forms	528	72	456
Cusp forms	432	72	360
Eisenstein series	96	0	96

Trace form

\( 72 q - 8 q^{3} + O(q^{10}) \)

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.j.a	$1050$	$2$	1050.j	15.e	$4$	$8$	$4$	$8.384$	8.0.40960000.1	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{4}]$	\(q+\beta _{7}q^{2}+(-\beta _{4}+\beta _{7})q^{3}-\beta _{2}q^{4}+\cdots\)
1050.2.j.b	$1050$	$2$	1050.j	15.e	$4$	$8$	$4$	$8.384$	8.0.40960000.1	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{4}]$	\(q+\beta _{7}q^{2}+(\beta _{1}-\beta _{6}-\beta _{7})q^{3}-\beta _{2}q^{4}+\cdots\)
1050.2.j.c	$1050$	$2$	1050.j	15.e	$4$	$12$	$6$	$8.384$	\(\mathbb{Q}[x]/(x^{12} + \cdots)\)	None		$2$	$0$	\(0\)	\(-4\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{4}]$	\(q-\beta _{1}q^{2}-\beta _{9}q^{3}+\beta _{7}q^{4}-\beta _{5}q^{6}+\cdots\)
1050.2.j.d	$1050$	$2$	1050.j	15.e	$4$	$12$	$6$	$8.384$	\(\mathbb{Q}[x]/(x^{12} + \cdots)\)	None		$2$	$0$	\(0\)	\(-4\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{4}]$	\(q+\beta _{2}q^{2}+\beta _{4}q^{3}-\beta _{7}q^{4}-\beta _{5}q^{6}+\cdots\)
1050.2.j.e	$1050$	$2$	1050.j	15.e	$4$	$16$	$8$	$8.384$	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{4}$	$\mathrm{SU}(2)[C_{4}]$	\(q-\beta _{11}q^{2}+(\beta _{11}-\beta _{13})q^{3}+\beta _{5}q^{4}+\cdots\)
1050.2.j.f	$1050$	$2$	1050.j	15.e	$4$	$16$	$8$	$8.384$	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{4}$	$\mathrm{SU}(2)[C_{4}]$	\(q+\beta _{11}q^{2}+(-\beta _{1}+\beta _{11})q^{3}+\beta _{5}q^{4}+\cdots\)

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

\( S_{2}^{\mathrm{old}}(1050, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(30, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(75, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(105, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(150, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(210, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(525, [\chi])\)\(^{\oplus 2}\)