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

• Label: 507.2.k
• Level: $507$
• Weight: $2$
• Character orbit: 507.k
• Rep. character: $\chi_{507}(80,\cdot)$
• Character field: $\Q(\zeta_{12})$
• Dimension: $164$
• Newform subspaces: $11$
• Sturm bound: $121$
• Trace bound: $7$

Defining parameters

Level:	\( N \)	\(=\)	\( 507 = 3 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	507.k (of order \(12\) and degree \(4\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 39 \)
Character field:			\(\Q(\zeta_{12})\)
Newform subspaces:			\( 11 \)
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	300	244	56
Cusp forms	188	164	24
Eisenstein series	112	80	32

Trace form

\( 164 q + 2 q^{3} + 12 q^{4} + 2 q^{6} + 14 q^{7} + 2 q^{9} + O(q^{10}) \)

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.k.a	$507$	$2$	507.k	39.k	$12$	$4$	$1$	$4.048$	\(\Q(\zeta_{12})\)	\(\Q(\sqrt{-3}) \)					39.2.k.a	$4$	$0$	\(0\)	\(0\)	\(0\)	\(-8\)	$1$	$\mathrm{U}(1)[D_{12}]$	\(q+(\zeta_{12}+\zeta_{12}^{3})q^{3}+2\zeta_{12}q^{4}+(-1+\cdots)q^{7}+\cdots\)
507.2.k.b	$507$	$2$	507.k	39.k	$12$	$4$	$1$	$4.048$	\(\Q(\zeta_{12})\)	\(\Q(\sqrt{-3}) \)					39.2.k.a	$4$	$0$	\(0\)	\(0\)	\(0\)	\(8\)	$1$	$\mathrm{U}(1)[D_{12}]$	\(q+(\zeta_{12}+\zeta_{12}^{3})q^{3}+2\zeta_{12}q^{4}+(1+\zeta_{12}+\cdots)q^{7}+\cdots\)
507.2.k.c	$507$	$2$	507.k	39.k	$12$	$4$	$1$	$4.048$	\(\Q(\zeta_{12})\)	\(\Q(\sqrt{-3}) \)					39.2.k.a	$4$	$0$	\(0\)	\(0\)	\(0\)	\(10\)	$1$	$\mathrm{U}(1)[D_{12}]$	\(q+(-\zeta_{12}-\zeta_{12}^{3})q^{3}+2\zeta_{12}q^{4}+(2+\cdots)q^{7}+\cdots\)
507.2.k.d	$507$	$2$	507.k	39.k	$12$	$8$	$2$	$4.048$	8.0.56070144.2	None					39.2.k.b	$4$	$0$	\(0\)	\(-2\)	\(0\)	\(4\)	$1$	$\mathrm{SU}(2)[C_{12}]$	\(q+(-\beta _{1}-\beta _{2}+\beta _{3}-\beta _{4}+2\beta _{5}+\beta _{7})q^{2}+\cdots\)
507.2.k.e	$507$	$2$	507.k	39.k	$12$	$8$	$2$	$4.048$	8.0.56070144.2	None					39.2.k.b	$4$	$0$	\(0\)	\(-2\)	\(0\)	\(-4\)	$1$	$\mathrm{SU}(2)[C_{12}]$	\(q+(\beta _{2}-\beta _{5}-\beta _{7})q^{2}+(-\beta _{2}+\beta _{4})q^{3}+\cdots\)
507.2.k.f	$507$	$2$	507.k	39.k	$12$	$8$	$2$	$4.048$	8.0.56070144.2	None					39.2.k.b	$4$	$0$	\(0\)	\(-2\)	\(0\)	\(4\)	$1$	$\mathrm{SU}(2)[C_{12}]$	\(q+(\beta _{2}-\beta _{5}-\beta _{7})q^{2}+(\beta _{2}+\beta _{4}-\beta _{5}+\cdots)q^{3}+\cdots\)
507.2.k.g	$507$	$2$	507.k	39.k	$12$	$8$	$2$	$4.048$	8.0.56070144.2	\(\Q(\sqrt{-39}) \)		✓			507.2.f.d	$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{U}(1)[D_{12}]$	\(q-\beta _{6}q^{2}+(-\beta _{2}-2\beta _{3})q^{3}+(-2+2\beta _{3}+\cdots)q^{4}+\cdots\)
507.2.k.h	$507$	$2$	507.k	39.k	$12$	$8$	$2$	$4.048$	8.0.56070144.2	\(\Q(\sqrt{-39}) \)		✓			507.2.f.d	$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{U}(1)[D_{12}]$	\(q+\beta _{5}q^{2}+(2\beta _{2}+\beta _{3})q^{3}+(1+2\beta _{2}+\cdots)q^{4}+\cdots\)
507.2.k.i	$507$	$2$	507.k	39.k	$12$	$8$	$2$	$4.048$	\(\Q(\zeta_{24})\)	None					39.2.f.a	$8$	$0$	\(0\)	\(4\)	\(0\)	\(4\)	$1$	$\mathrm{SU}(2)[C_{12}]$	\(q+\zeta_{24}^{7}q^{2}+(\zeta_{24}+\zeta_{24}^{4}-\zeta_{24}^{7})q^{3}+\cdots\)
507.2.k.j	$507$	$2$	507.k	39.k	$12$	$8$	$2$	$4.048$	\(\Q(\zeta_{24})\)	None					39.2.f.a	$8$	$0$	\(0\)	\(4\)	\(0\)	\(-4\)	$1$	$\mathrm{SU}(2)[C_{12}]$	\(q+\zeta_{24}^{7}q^{2}+(-\zeta_{24}+\zeta_{24}^{4}+\zeta_{24}^{7})q^{3}+\cdots\)
507.2.k.k	$507$	$2$	507.k	39.k	$12$	$96$	$24$	$4.048$		None		✓	✓		507.2.f.g	$16$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$\mathrm{SU}(2)[C_{12}]$

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

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