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

• Label: 507.2.j
• Level: $507$
• Weight: $2$
• Character orbit: 507.j
• Rep. character: $\chi_{507}(316,\cdot)$
• Character field: $\Q(\zeta_{6})$
• Dimension: $54$
• Newform subspaces: $9$
• Sturm bound: $121$
• Trace bound: $10$

Defining parameters

Level:	\( N \)	\(=\)	\( 507 = 3 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	507.j (of order \(6\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 13 \)
Character field:			\(\Q(\zeta_{6})\)
Newform subspaces:			\( 9 \)
Sturm bound:			\(121\)
Trace bound:			\(10\)
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	54	96
Cusp forms	94	54	40
Eisenstein series	56	0	56

Trace form

\( 54 q - q^{3} + 30 q^{4} + 3 q^{7} - 27 q^{9} + O(q^{10}) \)

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.j.a	$507$	$2$	507.j	13.e	$6$	$2$	$1$	$4.048$	\(\Q(\sqrt{-3}) \)	None		$2$	$0$	\(-3\)	\(1\)	\(0\)	\(-6\)	$1$	$\mathrm{SU}(2)[C_{6}]$	\(q+(-1-\zeta_{6})q^{2}+(1-\zeta_{6})q^{3}+\zeta_{6}q^{4}+\cdots\)
507.2.j.b	$507$	$2$	507.j	13.e	$6$	$2$	$1$	$4.048$	\(\Q(\sqrt{-3}) \)	None		$2$	$0$	\(0\)	\(1\)	\(0\)	\(3\)	$1$	$\mathrm{SU}(2)[C_{6}]$	\(q+(1-\zeta_{6})q^{3}-2\zeta_{6}q^{4}+(2-4\zeta_{6})q^{5}+\cdots\)
507.2.j.c	$507$	$2$	507.j	13.e	$6$	$2$	$1$	$4.048$	\(\Q(\sqrt{-3}) \)	None		$2$	$0$	\(3\)	\(1\)	\(0\)	\(6\)	$1$	$\mathrm{SU}(2)[C_{6}]$	\(q+(1+\zeta_{6})q^{2}+(1-\zeta_{6})q^{3}+\zeta_{6}q^{4}+\cdots\)
507.2.j.d	$507$	$2$	507.j	13.e	$6$	$4$	$2$	$4.048$	\(\Q(\zeta_{12})\)	None		$4$	$0$	\(0\)	\(2\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{6}]$	\(q+\zeta_{12}q^{2}+(1-\zeta_{12}^{2})q^{3}-\zeta_{12}^{2}q^{4}+\cdots\)
507.2.j.e	$507$	$2$	507.j	13.e	$6$	$4$	$2$	$4.048$	\(\Q(\zeta_{12})\)	None		$4$	$0$	\(0\)	\(2\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{6}]$	\(q+\zeta_{12}q^{2}+(1-\zeta_{12}^{2})q^{3}-\zeta_{12}^{2}q^{4}+\cdots\)
507.2.j.f	$507$	$2$	507.j	13.e	$6$	$8$	$4$	$4.048$	\(\Q(\zeta_{24})\)	None		$4$	$0$	\(0\)	\(-4\)	\(0\)	\(0\)	$2^{6}$	$\mathrm{SU}(2)[C_{6}]$	\(q+\zeta_{24}^{3}q^{2}-\zeta_{24}q^{3}+(1-\zeta_{24}-\zeta_{24}^{4}+\cdots)q^{4}+\cdots\)
507.2.j.g	$507$	$2$	507.j	13.e	$6$	$8$	$4$	$4.048$	8.0.1731891456.1	None		$4$	$0$	\(0\)	\(-4\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{6}]$	\(q+\beta _{1}q^{2}+\beta _{2}q^{3}+(2+3\beta _{2}-\beta _{4}+\beta _{5}+\cdots)q^{4}+\cdots\)
507.2.j.h	$507$	$2$	507.j	13.e	$6$	$12$	$6$	$4.048$	12.0.\(\cdots\).1	None		$4$	$0$	\(0\)	\(-6\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{6}]$	\(q+\beta _{1}q^{2}+(-1+\beta _{7})q^{3}+(1-\beta _{3}+\beta _{4}+\cdots)q^{4}+\cdots\)
507.2.j.i	$507$	$2$	507.j	13.e	$6$	$12$	$6$	$4.048$	12.0.\(\cdots\).1	None		$4$	$0$	\(0\)	\(6\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{6}]$	\(q+(-\beta _{1}+\beta _{8}+2\beta _{11})q^{2}+\beta _{7}q^{3}+\cdots\)

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

\( S_{2}^{\mathrm{old}}(507, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(13, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(39, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(169, [\chi])\)\(^{\oplus 2}\)