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

• Label: 507.2.f
• Level: $507$
• Weight: $2$
• Character orbit: 507.f
• Rep. character: $\chi_{507}(239,\cdot)$
• Character field: $\Q(\zeta_{4})$
• Dimension: $84$
• Newform subspaces: $7$
• Sturm bound: $121$
• Trace bound: $7$

Defining parameters

Level:	\( N \)	\(=\)	\( 507 = 3 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	507.f (of order \(4\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 39 \)
Character field:			\(\Q(i)\)
Newform subspaces:			\( 7 \)
Sturm bound:			\(121\)
Trace bound:			\(7\)
Distinguishing \(T_p\):			\(2\), \(5\), \(7\)

Dimensions

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

	Total	New	Old
Modular forms	148	124	24
Cusp forms	92	84	8
Eisenstein series	56	40	16

Trace form

\( 84 q + 4 q^{3} + 4 q^{6} - 4 q^{7} + 4 q^{9} + O(q^{10}) \)

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	Twist minimal	Largest	Maximal	Minimal twist	Inner twists	Rank*	Traces				Coefficient ring index	Sato-Tate	$q$-expansion
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
507.2.f.a	$507$	$2$	507.f	39.f	$4$	$4$	$2$	$4.048$	\(\Q(\zeta_{8})\)	None					39.2.f.a	$4$	$0$	\(0\)	\(-4\)	\(0\)	\(-4\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+\zeta_{8}q^{2}+(-1-\zeta_{8}-\zeta_{8}^{3})q^{3}-\zeta_{8}^{2}q^{4}+\cdots\)
507.2.f.b	$507$	$2$	507.f	39.f	$4$	$4$	$2$	$4.048$	\(\Q(\zeta_{12})\)	\(\Q(\sqrt{-3}) \)					39.2.k.a	$4$	$0$	\(0\)	\(0\)	\(0\)	\(-2\)	$2$	$\mathrm{U}(1)[D_{4}]$	\(q+(\beta_{3}-\beta_{2}+\beta_1)q^{3}+2\beta_{2} q^{4}+\cdots\)
507.2.f.c	$507$	$2$	507.f	39.f	$4$	$4$	$2$	$4.048$	\(\Q(\zeta_{12})\)	\(\Q(\sqrt{-3}) \)					39.2.k.a	$4$	$0$	\(0\)	\(0\)	\(0\)	\(2\)	$2$	$\mathrm{U}(1)[D_{4}]$	\(q+(\beta_{3}-\beta_{2}+\beta_1)q^{3}+2\beta_{2} q^{4}+\cdots\)
507.2.f.d	$507$	$2$	507.f	39.f	$4$	$8$	$4$	$4.048$	8.0.56070144.2	\(\Q(\sqrt{-39}) \)		✓			507.2.f.d	$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{6}$	$\mathrm{U}(1)[D_{4}]$	\(q-\beta _{1}q^{2}+\beta _{4}q^{3}+(-2\beta _{2}-\beta _{5})q^{4}+\cdots\)
507.2.f.e	$507$	$2$	507.f	39.f	$4$	$8$	$4$	$4.048$	8.0.56070144.2	None					39.2.k.b	$4$	$0$	\(0\)	\(4\)	\(0\)	\(-8\)	$2^{2}$	$\mathrm{SU}(2)[C_{4}]$	\(q+(\beta _{1}+\beta _{2}+\beta _{4})q^{2}+(\beta _{4}+\beta _{5})q^{3}+\cdots\)
507.2.f.f	$507$	$2$	507.f	39.f	$4$	$8$	$4$	$4.048$	8.0.56070144.2	None					39.2.k.b	$4$	$0$	\(0\)	\(4\)	\(0\)	\(8\)	$2^{2}$	$\mathrm{SU}(2)[C_{4}]$	\(q+(\beta _{1}+\beta _{2}+\beta _{4})q^{2}+(1-\beta _{2}-\beta _{4}+\cdots)q^{3}+\cdots\)
507.2.f.g	$507$	$2$	507.f	39.f	$4$	$48$	$24$	$4.048$		None		✓	✓		507.2.f.g	$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$\mathrm{SU}(2)[C_{4}]$

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

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