Space of modular forms of level 624, weight 3, and Character 624.l

• Label: 624.3.l
• Level: $624$
• Weight: $3$
• Character orbit: 624.l
• Rep. character: $\chi_{624}(545,\cdot)$
• Character field: $\Q$
• Dimension: $54$
• Newform subspaces: $9$
• Sturm bound: $336$
• Trace bound: $13$

Defining parameters

Level:	\( N \)	\(=\)	\( 624 = 2^{4} \cdot 3 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	624.l (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 39 \)
Character field:			\(\Q\)
Newform subspaces:			\( 9 \)
Sturm bound:			\(336\)
Trace bound:			\(13\)
Distinguishing \(T_p\):			\(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{3}(624, [\chi])\).

	Total	New	Old
Modular forms	236	58	178
Cusp forms	212	54	158
Eisenstein series	24	4	20

Trace form

\( 54 q + 2 q^{3} - 2 q^{9} + O(q^{10}) \)

Decomposition of \(S_{3}^{\mathrm{new}}(624, [\chi])\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
624.3.l.a	$624$	$3$	624.l	39.d	$2$	$2$	$2$	$17.003$	\(\Q(\sqrt{3}) \)	\(\Q(\sqrt{-39}) \)	✓				39.3.d.b	$4$	$0$	\(0\)	\(-6\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{U}(1)[D_{2}]$	\(q-3q^{3}+\beta q^{5}+9q^{9}+3\beta q^{11}-13q^{13}+\cdots\)
624.3.l.b	$624$	$3$	624.l	39.d	$2$	$2$	$2$	$17.003$	\(\Q(\sqrt{-3}) \)	\(\Q(\sqrt{-3}) \)					156.3.g.a	$4$	$0$	\(0\)	\(-6\)	\(0\)	\(0\)	$2^{3}$	$\mathrm{U}(1)[D_{2}]$	\(q-3q^{3}-2\zeta_{6}q^{7}+9q^{9}+(11-\zeta_{6})q^{13}+\cdots\)
624.3.l.c	$624$	$3$	624.l	39.d	$2$	$2$	$2$	$17.003$	\(\Q(\sqrt{13}) \)	\(\Q(\sqrt{-39}) \)	✓				39.3.d.a	$4$	$0$	\(0\)	\(6\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{U}(1)[D_{2}]$	\(q+3q^{3}-\beta q^{5}+9q^{9}+\beta q^{11}+13q^{13}+\cdots\)
624.3.l.d	$624$	$3$	624.l	39.d	$2$	$4$	$4$	$17.003$	\(\Q(\sqrt{2}, \sqrt{-5})\)	None					78.3.d.b	$4$	$0$	\(0\)	\(-8\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-2-\beta _{3})q^{3}-\beta _{1}q^{5}-\beta _{2}q^{7}+(-1+\cdots)q^{9}+\cdots\)
624.3.l.e	$624$	$3$	624.l	39.d	$2$	$4$	$4$	$17.003$	\(\Q(\sqrt{-2}, \sqrt{15})\)	None					156.3.g.c	$4$	$0$	\(0\)	\(-4\)	\(0\)	\(0\)	$2^{4}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-1+\beta _{1})q^{3}-\beta _{3}q^{5}+(-7-2\beta _{1}+\cdots)q^{9}+\cdots\)
624.3.l.f	$624$	$3$	624.l	39.d	$2$	$4$	$4$	$17.003$	\(\Q(\sqrt{3}, \sqrt{-35})\)	None					39.3.d.c	$4$	$0$	\(0\)	\(2\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+(1-\beta _{1})q^{3}-3\beta _{2}q^{5}+(-\beta _{2}-2\beta _{3})q^{7}+\cdots\)
624.3.l.g	$624$	$3$	624.l	39.d	$2$	$4$	$4$	$17.003$	\(\Q(\sqrt{2}, \sqrt{-5})\)	None					78.3.d.a	$4$	$0$	\(0\)	\(8\)	\(0\)	\(0\)	$2\cdot 3^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(2+\beta _{3})q^{3}-\beta _{1}q^{5}+\beta _{2}q^{7}+(-1+\cdots)q^{9}+\cdots\)
624.3.l.h	$624$	$3$	624.l	39.d	$2$	$4$	$4$	$17.003$	\(\Q(i, \sqrt{11})\)	None					156.3.g.b	$4$	$0$	\(0\)	\(10\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+(3+\beta _{3})q^{3}+(2\beta _{1}-\beta _{2})q^{5}+7\beta _{2}q^{7}+\cdots\)
624.3.l.i	$624$	$3$	624.l	39.d	$2$	$28$	$28$	$17.003$		None			✓		312.3.l.a	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$\mathrm{SU}(2)[C_{2}]$

Decomposition of \(S_{3}^{\mathrm{old}}(624, [\chi])\) into lower level spaces

\( S_{3}^{\mathrm{old}}(624, [\chi]) \simeq \) \(S_{3}^{\mathrm{new}}(39, [\chi])\)\(^{\oplus 5}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(78, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(156, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(312, [\chi])\)\(^{\oplus 2}\)