Space of modular forms of level 507, weight 2, and Character 507.e

• Label: 507.2.e
• Level: $507$
• Weight: $2$
• Character orbit: 507.e
• Rep. character: $\chi_{507}(22,\cdot)$
• Character field: $\Q(\zeta_{3})$
• Dimension: $50$
• Newform subspaces: $12$
• Sturm bound: $121$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 507 = 3 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	507.e (of order \(3\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 13 \)
Character field:			\(\Q(\zeta_{3})\)
Newform subspaces:			\( 12 \)
Sturm bound:			\(121\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(2\), \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(507, [\chi])\).

	Total	New	Old
Modular forms	150	50	100
Cusp forms	94	50	44
Eisenstein series	56	0	56

Trace form

\( 50 q + 2 q^{2} + q^{3} - 24 q^{4} + 8 q^{5} - q^{7} - 12 q^{8} - 25 q^{9} + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(507, [\chi])\) 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
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
507.2.e.a	$507$	$2$	507.e	13.c	$3$	$2$	$1$	$4.048$	\(\Q(\sqrt{-3}) \)	None		$2$	$0$	\(-1\)	\(1\)	\(4\)	\(4\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-1+\zeta_{6})q^{2}+(1-\zeta_{6})q^{3}+\zeta_{6}q^{4}+\cdots\)
507.2.e.b	$507$	$2$	507.e	13.c	$3$	$2$	$1$	$4.048$	\(\Q(\sqrt{-3}) \)	None		$2$	$0$	\(1\)	\(1\)	\(-4\)	\(-4\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(1-\zeta_{6})q^{2}+(1-\zeta_{6})q^{3}+\zeta_{6}q^{4}+\cdots\)
507.2.e.c	$507$	$2$	507.e	13.c	$3$	$2$	$1$	$4.048$	\(\Q(\sqrt{-3}) \)	None		$2$	$0$	\(1\)	\(1\)	\(2\)	\(2\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(1-\zeta_{6})q^{2}+(1-\zeta_{6})q^{3}+\zeta_{6}q^{4}+\cdots\)
507.2.e.d	$507$	$2$	507.e	13.c	$3$	$4$	$2$	$4.048$	\(\Q(\sqrt{2}, \sqrt{-3})\)	None		$2$	$0$	\(-2\)	\(-2\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-1+\beta _{1}-\beta _{2})q^{2}+(-1-\beta _{2})q^{3}+\cdots\)
507.2.e.e	$507$	$2$	507.e	13.c	$3$	$4$	$2$	$4.048$	\(\Q(\zeta_{12})\)	None		$4$	$0$	\(0\)	\(2\)	\(0\)	\(0\)	$3$	$\mathrm{SU}(2)[C_{3}]$	\(q-\zeta_{12}^{2}q^{2}+\zeta_{12}q^{3}+(-1+\zeta_{12}+\cdots)q^{4}+\cdots\)
507.2.e.f	$507$	$2$	507.e	13.c	$3$	$4$	$2$	$4.048$	\(\Q(\zeta_{12})\)	None		$4$	$0$	\(0\)	\(2\)	\(0\)	\(0\)	$3$	$\mathrm{SU}(2)[C_{3}]$	\(q+\zeta_{12}q^{3}+(2-2\zeta_{12})q^{4}-2\zeta_{12}^{3}q^{5}+\cdots\)
507.2.e.g	$507$	$2$	507.e	13.c	$3$	$4$	$2$	$4.048$	\(\Q(\sqrt{-3}, \sqrt{17})\)	None		$2$	$0$	\(1\)	\(-2\)	\(6\)	\(-3\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+\beta _{1}q^{2}-\beta _{2}q^{3}+(-2+\beta _{1}+2\beta _{2}+\cdots)q^{4}+\cdots\)
507.2.e.h	$507$	$2$	507.e	13.c	$3$	$4$	$2$	$4.048$	\(\Q(\sqrt{2}, \sqrt{-3})\)	None		$2$	$0$	\(2\)	\(-2\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(1+\beta _{1}+\beta _{2})q^{2}+(-1-\beta _{2})q^{3}+\cdots\)
507.2.e.i	$507$	$2$	507.e	13.c	$3$	$6$	$3$	$4.048$	6.0.64827.1	None		$2$	$0$	\(-3\)	\(3\)	\(12\)	\(-2\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-2+2\beta _{1}+\beta _{4}+2\beta _{5})q^{2}+(1-\beta _{5})q^{3}+\cdots\)
507.2.e.j	$507$	$2$	507.e	13.c	$3$	$6$	$3$	$4.048$	6.0.64827.1	None		$2$	$0$	\(-1\)	\(-3\)	\(8\)	\(-10\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q-\beta _{1}q^{2}+(-1+\beta _{5})q^{3}+(\beta _{1}-\beta _{2}+\cdots)q^{4}+\cdots\)
507.2.e.k	$507$	$2$	507.e	13.c	$3$	$6$	$3$	$4.048$	6.0.64827.1	None		$2$	$0$	\(1\)	\(-3\)	\(-8\)	\(10\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+\beta _{1}q^{2}+(-1+\beta _{5})q^{3}+(\beta _{1}-\beta _{2}+\cdots)q^{4}+\cdots\)
507.2.e.l	$507$	$2$	507.e	13.c	$3$	$6$	$3$	$4.048$	6.0.64827.1	None		$2$	$0$	\(3\)	\(3\)	\(-12\)	\(2\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(2-2\beta _{1}-\beta _{4}-2\beta _{5})q^{2}+(1-\beta _{5})q^{3}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(507, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(507, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(39, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(169, [\chi])\)\(^{\oplus 2}\)