Space of modular forms of level 507, weight 4, and Character 507.b

• Label: 507.4.b
• Level: $507$
• Weight: $4$
• Character orbit: 507.b
• Rep. character: $\chi_{507}(337,\cdot)$
• Character field: $\Q$
• Dimension: $76$
• Newform subspaces: $11$
• Sturm bound: $242$
• Trace bound: $17$

Defining parameters

Level:	\( N \)	\(=\)	\( 507 = 3 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	507.b (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 13 \)
Character field:			\(\Q\)
Newform subspaces:			\( 11 \)
Sturm bound:			\(242\)
Trace bound:			\(17\)
Distinguishing \(T_p\):			\(2\), \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(507, [\chi])\).

	Total	New	Old
Modular forms	196	76	120
Cusp forms	168	76	92
Eisenstein series	28	0	28

Trace form

\( 76 q - 296 q^{4} + 684 q^{9} + O(q^{10}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(507, [\chi])\) 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
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
507.4.b.a	$507$	$4$	507.b	13.b	$2$	$2$	$2$	$29.914$	\(\Q(\sqrt{-3}) \)	None		$2$	$0$	\(0\)	\(-6\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q-3q^{3}+8q^{4}+3\zeta_{6}q^{5}-6\zeta_{6}q^{7}+\cdots\)
507.4.b.b	$507$	$4$	507.b	13.b	$2$	$2$	$2$	$29.914$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(-6\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q-3q^{3}+8q^{4}-6iq^{5}-iq^{7}+9q^{9}+\cdots\)
507.4.b.c	$507$	$4$	507.b	13.b	$2$	$2$	$2$	$29.914$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(6\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+3iq^{2}+3q^{3}-q^{4}-9iq^{5}+9iq^{6}+\cdots\)
507.4.b.d	$507$	$4$	507.b	13.b	$2$	$2$	$2$	$29.914$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(6\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+iq^{2}+3q^{3}+7q^{4}-7iq^{5}+3iq^{6}+\cdots\)
507.4.b.e	$507$	$4$	507.b	13.b	$2$	$4$	$4$	$29.914$	\(\Q(\sqrt{-3}, \sqrt{-17})\)	None		$2$	$0$	\(0\)	\(-12\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{1}q^{2}-3q^{3}-9q^{4}+(-2\beta _{1}+3\beta _{2}+\cdots)q^{5}+\cdots\)
507.4.b.f	$507$	$4$	507.b	13.b	$2$	$4$	$4$	$29.914$	\(\Q(i, \sqrt{14})\)	None		$2$	$0$	\(0\)	\(-12\)	\(0\)	\(0\)	$2^{3}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{2}q^{2}-3q^{3}+(-7-\beta _{3})q^{4}+(7\beta _{1}+\cdots)q^{5}+\cdots\)
507.4.b.g	$507$	$4$	507.b	13.b	$2$	$6$	$6$	$29.914$	6.0.158155776.1	None		$2$	$0$	\(0\)	\(18\)	\(0\)	\(0\)	$2^{6}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{4}q^{2}+3q^{3}+(-3+\beta _{1})q^{4}+(\beta _{2}+\cdots)q^{5}+\cdots\)
507.4.b.h	$507$	$4$	507.b	13.b	$2$	$8$	$8$	$29.914$	\(\mathbb{Q}[x]/(x^{8} + \cdots)\)	None		$2$	$0$	\(0\)	\(-24\)	\(0\)	\(0\)	$2^{6}\cdot 13^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{2}q^{2}-3q^{3}+(-6+\beta _{1})q^{4}+(\beta _{2}+\cdots)q^{5}+\cdots\)
507.4.b.i	$507$	$4$	507.b	13.b	$2$	$10$	$10$	$29.914$	\(\mathbb{Q}[x]/(x^{10} + \cdots)\)	None		$2$	$0$	\(0\)	\(30\)	\(0\)	\(0\)	$2^{5}\cdot 3^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{1}q^{2}+3q^{3}+(-6+\beta _{4})q^{4}+(\beta _{1}+\cdots)q^{5}+\cdots\)
507.4.b.j	$507$	$4$	507.b	13.b	$2$	$18$	$18$	$29.914$	\(\mathbb{Q}[x]/(x^{18} + \cdots)\)	None		$2$	$0$	\(0\)	\(-54\)	\(0\)	\(0\)	$13^{10}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{10}q^{2}-3q^{3}+(-4+\beta _{2})q^{4}+(2\beta _{10}+\cdots)q^{5}+\cdots\)
507.4.b.k	$507$	$4$	507.b	13.b	$2$	$18$	$18$	$29.914$	\(\mathbb{Q}[x]/(x^{18} + \cdots)\)	None		$2$	$0$	\(0\)	\(54\)	\(0\)	\(0\)	$13^{8}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{9}q^{2}+3q^{3}+(-5+\beta _{2})q^{4}-\beta _{15}q^{5}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(507, [\chi])\) into lower level spaces

\( S_{4}^{\mathrm{old}}(507, [\chi]) \cong \) \(S_{4}^{\mathrm{new}}(13, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(169, [\chi])\)\(^{\oplus 2}\)