Space of modular forms of level 504, weight 2, and Character 504.r

• Label: 504.2.r
• Level: $504$
• Weight: $2$
• Character orbit: 504.r
• Rep. character: $\chi_{504}(169,\cdot)$
• Character field: $\Q(\zeta_{3})$
• Dimension: $36$
• Newform subspaces: $6$
• Sturm bound: $192$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 504 = 2^{3} \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	504.r (of order \(3\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 9 \)
Character field:			\(\Q(\zeta_{3})\)
Newform subspaces:			\( 6 \)
Sturm bound:			\(192\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(504, [\chi])\).

	Total	New	Old
Modular forms	208	36	172
Cusp forms	176	36	140
Eisenstein series	32	0	32

Trace form

36 * q - 2 * q^3 + 6 * q^9 + 14 * q^11 + 20 * q^15 + 12 * q^17 - 12 * q^19 + 4 * q^23 - 18 * q^25 - 20 * q^27 - 12 * q^29 - 18 * q^33 - 24 * q^35 - 28 * q^39 - 6 * q^41 + 6 * q^43 + 36 * q^45 - 12 * q^47 - 18 * q^49 + 38 * q^51 + 48 * q^53 + 30 * q^57 + 14 * q^59 - 4 * q^63 - 16 * q^65 + 6 * q^67 - 16 * q^69 + 24 * q^71 + 36 * q^73 - 22 * q^75 - 8 * q^77 + 6 * q^81 + 16 * q^83 - 24 * q^85 + 40 * q^87 - 48 * q^89 - 44 * q^93 + 12 * q^95 - 42 * q^97 + 36 * q^99

Decomposition of \(S_{2}^{\mathrm{new}}(504, [\chi])\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
504.2.r.a	$504$	$2$	504.r	9.c	$3$	$2$	$1$	$4.024$	\(\Q(\sqrt{-3}) \)	None		✓			504.2.r.a	$2$	$0$	\(0\)	\(0\)	\(1\)	\(-1\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-1+2\zeta_{6})q^{3}+(1-\zeta_{6})q^{5}-\zeta_{6}q^{7}+\cdots\)
504.2.r.b	$504$	$2$	504.r	9.c	$3$	$2$	$1$	$4.024$	\(\Q(\sqrt{-3}) \)	None		✓			504.2.r.b	$2$	$0$	\(0\)	\(3\)	\(-2\)	\(-1\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(2-\zeta_{6})q^{3}+(-2+2\zeta_{6})q^{5}-\zeta_{6}q^{7}+\cdots\)
504.2.r.c	$504$	$2$	504.r	9.c	$3$	$6$	$3$	$4.024$	\(\Q(\zeta_{18})\)	None		✓			504.2.r.c	$2$	$0$	\(0\)	\(0\)	\(3\)	\(-3\)	$3$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-\beta_{5}-\beta_{2})q^{3}+(-\beta_{3}+\beta_{2}-\beta_1+1)q^{5}+\cdots\)
504.2.r.d	$504$	$2$	504.r	9.c	$3$	$8$	$4$	$4.024$	8.0.508277025.1	None		✓			504.2.r.d	$2$	$0$	\(0\)	\(-4\)	\(4\)	\(-4\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-\beta _{4}-\beta _{6})q^{3}+(1+\beta _{1}+2\beta _{3}-\beta _{4}+\cdots)q^{5}+\cdots\)
504.2.r.e	$504$	$2$	504.r	9.c	$3$	$8$	$4$	$4.024$	8.0.2091141441.1	None		✓			504.2.r.e	$2$	$0$	\(0\)	\(-1\)	\(-3\)	\(4\)	$3$	$\mathrm{SU}(2)[C_{3}]$	\(q+\beta _{4}q^{3}+(\beta _{2}+\beta _{4}+\beta _{6})q^{5}+(1+\beta _{2}+\cdots)q^{7}+\cdots\)
504.2.r.f	$504$	$2$	504.r	9.c	$3$	$10$	$5$	$4.024$	10.0.\(\cdots\).1	None		✓	✓		504.2.r.f	$2$	$0$	\(0\)	\(0\)	\(-3\)	\(5\)	$3^{2}$	$\mathrm{SU}(2)[C_{3}]$	\(q+\beta _{6}q^{3}+(-\beta _{2}-\beta _{7})q^{5}+(1-\beta _{2}+\cdots)q^{7}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(504, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(504, [\chi]) \simeq \) \(S_{2}^{\mathrm{new}}(18, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(36, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(63, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(72, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(126, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(252, [\chi])\)\(^{\oplus 2}\)