Space of modular forms of level 624, weight 4, and Character 624.c

• Label: 624.4.c
• Level: $624$
• Weight: $4$
• Character orbit: 624.c
• Rep. character: $\chi_{624}(337,\cdot)$
• Character field: $\Q$
• Dimension: $42$
• Newform subspaces: $8$
• Sturm bound: $448$
• Trace bound: $13$

Defining parameters

Level:	\( N \)	\(=\)	\( 624 = 2^{4} \cdot 3 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	624.c (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 13 \)
Character field:			\(\Q\)
Newform subspaces:			\( 8 \)
Sturm bound:			\(448\)
Trace bound:			\(13\)
Distinguishing \(T_p\):			\(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(624, [\chi])\).

	Total	New	Old
Modular forms	348	42	306
Cusp forms	324	42	282
Eisenstein series	24	0	24

Trace form

\( 42 q + 6 q^{3} + 378 q^{9} + O(q^{10}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(624, [\chi])\) 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
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
624.4.c.a	$624$	$4$	624.c	13.b	$2$	$2$	$2$	$36.817$	\(\Q(\sqrt{-3}) \)	None		$2$	$0$	\(0\)	\(-6\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q-3q^{3}-2\zeta_{6}q^{5}+3\zeta_{6}q^{7}+9q^{9}+\cdots\)
624.4.c.b	$624$	$4$	624.c	13.b	$2$	$2$	$2$	$36.817$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(6\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+3q^{3}+4iq^{5}-7iq^{7}+9q^{9}-15iq^{11}+\cdots\)
624.4.c.c	$624$	$4$	624.c	13.b	$2$	$4$	$4$	$36.817$	4.0.1362828.1	None		$2$	$0$	\(0\)	\(-12\)	\(0\)	\(0\)	$2^{5}$	$\mathrm{SU}(2)[C_{2}]$	\(q-3q^{3}+(-\beta _{1}+\beta _{2})q^{5}+(-\beta _{1}+2\beta _{2}+\cdots)q^{7}+\cdots\)
624.4.c.d	$624$	$4$	624.c	13.b	$2$	$4$	$4$	$36.817$	\(\Q(i, \sqrt{17})\)	None		$2$	$0$	\(0\)	\(-12\)	\(0\)	\(0\)	$2^{3}\cdot 3^{4}$	$\mathrm{SU}(2)[C_{2}]$	\(q-3q^{3}+(\beta _{1}-\beta _{2})q^{5}+(-2\beta _{1}-\beta _{2}+\cdots)q^{7}+\cdots\)
624.4.c.e	$624$	$4$	624.c	13.b	$2$	$4$	$4$	$36.817$	4.0.5054412.1	None		$2$	$0$	\(0\)	\(12\)	\(0\)	\(0\)	$2^{4}$	$\mathrm{SU}(2)[C_{2}]$	\(q+3q^{3}-\beta _{2}q^{5}+3\beta _{1}q^{7}+9q^{9}+(\beta _{1}+\cdots)q^{11}+\cdots\)
624.4.c.f	$624$	$4$	624.c	13.b	$2$	$4$	$4$	$36.817$	4.0.47664588.1	None		$2$	$0$	\(0\)	\(12\)	\(0\)	\(0\)	$2^{4}$	$\mathrm{SU}(2)[C_{2}]$	\(q+3q^{3}+\beta _{1}q^{5}+\beta _{2}q^{7}+9q^{9}+(3\beta _{1}+\cdots)q^{11}+\cdots\)
624.4.c.g	$624$	$4$	624.c	13.b	$2$	$10$	$10$	$36.817$	\(\mathbb{Q}[x]/(x^{10} + \cdots)\)	None		$2$	$0$	\(0\)	\(-30\)	\(0\)	\(0\)	$2^{25}$	$\mathrm{SU}(2)[C_{2}]$	\(q-3q^{3}+\beta _{1}q^{5}-\beta _{3}q^{7}+9q^{9}+(\beta _{6}+\cdots)q^{11}+\cdots\)
624.4.c.h	$624$	$4$	624.c	13.b	$2$	$12$	$12$	$36.817$	\(\mathbb{Q}[x]/(x^{12} + \cdots)\)	None		$2$	$0$	\(0\)	\(36\)	\(0\)	\(0\)	$2^{35}$	$\mathrm{SU}(2)[C_{2}]$	\(q+3q^{3}+\beta _{5}q^{5}-\beta _{7}q^{7}+9q^{9}+\beta _{8}q^{11}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(624, [\chi])\) into lower level spaces

\( S_{4}^{\mathrm{old}}(624, [\chi]) \cong \) \(S_{4}^{\mathrm{new}}(13, [\chi])\)\(^{\oplus 10}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(26, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(52, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(104, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(208, [\chi])\)\(^{\oplus 2}\)