Space of modular forms of level 624, weight 4, and trivial character

• Label: 624.4.a
• Level: $624$
• Weight: $4$
• Character orbit: 624.a
• Rep. character: $\chi_{624}(1,\cdot)$
• Character field: $\Q$
• Dimension: $36$
• Newform subspaces: $21$
• Sturm bound: $448$
• Trace bound: $7$

Defining parameters

Level:	\( N \)	\(=\)	\( 624 = 2^{4} \cdot 3 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	624.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 21 \)
Sturm bound:			\(448\)
Trace bound:			\(7\)
Distinguishing \(T_p\):			\(5\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(624))\).

	Total	New	Old
Modular forms	348	36	312
Cusp forms	324	36	288
Eisenstein series	24	0	24

The following table gives the dimensions of the cuspidal new subspaces with specified eigenvalues for the Atkin-Lehner operators and the Fricke involution.

\(2\)	\(3\)	\(13\)	Fricke	Dim
\(+\)	\(+\)	\(+\)	$+$	\(4\)
\(+\)	\(+\)	\(-\)	$-$	\(5\)
\(+\)	\(-\)	\(+\)	$-$	\(4\)
\(+\)	\(-\)	\(-\)	$+$	\(5\)
\(-\)	\(+\)	\(+\)	$-$	\(4\)
\(-\)	\(+\)	\(-\)	$+$	\(5\)
\(-\)	\(-\)	\(+\)	$+$	\(6\)
\(-\)	\(-\)	\(-\)	$-$	\(3\)
Plus space			\(+\)	\(20\)
Minus space			\(-\)	\(16\)

Trace form

\( 36 q - 28 q^{7} + 324 q^{9} + O(q^{10}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(624))\) into newform subspaces

Label	Level	Weight	Char	Prim	Char order	Dim	Rel. Dim	$A$	Field	CM	Self-dual	Inner twists	Rank*	Traces					A-L signs			Fricke sign	Coefficient ring index	Sato-Tate	$q$-expansion
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$		2	3	13
624.4.a.a	$624$	$4$	624.a	1.a	$1$	$1$	$1$	$36.817$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(-3\)	\(-16\)	\(8\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-3q^{3}-2^{4}q^{5}+8q^{7}+9q^{9}+38q^{11}+\cdots\)
624.4.a.b	$624$	$4$	624.a	1.a	$1$	$1$	$1$	$36.817$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(-3\)	\(-2\)	\(32\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-3q^{3}-2q^{5}+2^{5}q^{7}+9q^{9}+68q^{11}+\cdots\)
624.4.a.c	$624$	$4$	624.a	1.a	$1$	$1$	$1$	$36.817$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(-3\)	\(4\)	\(-4\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-3q^{3}+4q^{5}-4q^{7}+9q^{9}-2q^{11}+\cdots\)
624.4.a.d	$624$	$4$	624.a	1.a	$1$	$1$	$1$	$36.817$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(-3\)	\(10\)	\(8\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-3q^{3}+10q^{5}+8q^{7}+9q^{9}-40q^{11}+\cdots\)
624.4.a.e	$624$	$4$	624.a	1.a	$1$	$1$	$1$	$36.817$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(3\)	\(-20\)	\(32\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+3q^{3}-20q^{5}+2^{5}q^{7}+9q^{9}-50q^{11}+\cdots\)
624.4.a.f	$624$	$4$	624.a	1.a	$1$	$1$	$1$	$36.817$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(3\)	\(-16\)	\(-28\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+3q^{3}-2^{4}q^{5}-28q^{7}+9q^{9}-34q^{11}+\cdots\)
624.4.a.g	$624$	$4$	624.a	1.a	$1$	$1$	$1$	$36.817$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(3\)	\(-12\)	\(-2\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3q^{3}-12q^{5}-2q^{7}+9q^{9}+6^{2}q^{11}+\cdots\)
624.4.a.h	$624$	$4$	624.a	1.a	$1$	$1$	$1$	$36.817$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(3\)	\(-6\)	\(4\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3q^{3}-6q^{5}+4q^{7}+9q^{9}-6^{2}q^{11}+\cdots\)
624.4.a.i	$624$	$4$	624.a	1.a	$1$	$1$	$1$	$36.817$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(3\)	\(6\)	\(-20\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3q^{3}+6q^{5}-20q^{7}+9q^{9}-24q^{11}+\cdots\)
624.4.a.j	$624$	$4$	624.a	1.a	$1$	$2$	$2$	$36.817$	\(\Q(\sqrt{17}) \)	None	✓	$1$	$1$	\(0\)	\(-6\)	\(-18\)	\(10\)		$+$	$+$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q-3q^{3}+(-9-\beta )q^{5}+(5-5\beta )q^{7}+\cdots\)
624.4.a.k	$624$	$4$	624.a	1.a	$1$	$2$	$2$	$36.817$	\(\Q(\sqrt{7}) \)	None	✓	$1$	$0$	\(0\)	\(-6\)	\(-4\)	\(20\)		$+$	$+$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q-3q^{3}+(-2+\beta )q^{5}+(10+3\beta )q^{7}+\cdots\)
624.4.a.l	$624$	$4$	624.a	1.a	$1$	$2$	$2$	$36.817$	\(\Q(\sqrt{43}) \)	None	✓	$1$	$0$	\(0\)	\(-6\)	\(12\)	\(-44\)		$+$	$+$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q-3q^{3}+(6+\beta )q^{5}+(-22-\beta )q^{7}+\cdots\)
624.4.a.m	$624$	$4$	624.a	1.a	$1$	$2$	$2$	$36.817$	\(\Q(\sqrt{10}) \)	None	✓	$1$	$1$	\(0\)	\(-6\)	\(24\)	\(-8\)		$-$	$+$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q-3q^{3}+(12+\beta )q^{5}+(-4+3\beta )q^{7}+\cdots\)
624.4.a.n	$624$	$4$	624.a	1.a	$1$	$2$	$2$	$36.817$	\(\Q(\sqrt{55}) \)	None	✓	$1$	$1$	\(0\)	\(6\)	\(-4\)	\(-20\)		$+$	$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q+3q^{3}+(-2+\beta )q^{5}+(-10-\beta )q^{7}+\cdots\)
624.4.a.o	$624$	$4$	624.a	1.a	$1$	$2$	$2$	$36.817$	\(\Q(\sqrt{3}) \)	None	✓	$1$	$1$	\(0\)	\(6\)	\(-4\)	\(-4\)		$+$	$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q+3q^{3}+(-2+\beta )q^{5}+(-2+3\beta )q^{7}+\cdots\)
624.4.a.p	$624$	$4$	624.a	1.a	$1$	$2$	$2$	$36.817$	\(\Q(\sqrt{22}) \)	None	✓	$1$	$0$	\(0\)	\(6\)	\(0\)	\(-8\)		$-$	$-$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q+3q^{3}+\beta q^{5}+(-4+3\beta )q^{7}+9q^{9}+\cdots\)
624.4.a.q	$624$	$4$	624.a	1.a	$1$	$2$	$2$	$36.817$	\(\Q(\sqrt{113}) \)	None	✓	$1$	$0$	\(0\)	\(6\)	\(6\)	\(10\)		$+$	$-$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q+3q^{3}+(3+\beta )q^{5}+(5+\beta )q^{7}+9q^{9}+\cdots\)
624.4.a.r	$624$	$4$	624.a	1.a	$1$	$2$	$2$	$36.817$	\(\Q(\sqrt{14}) \)	None	✓	$1$	$0$	\(0\)	\(6\)	\(24\)	\(0\)		$-$	$-$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q+3q^{3}+(12+\beta )q^{5}-\beta q^{7}+9q^{9}+\cdots\)
624.4.a.s	$624$	$4$	624.a	1.a	$1$	$3$	$3$	$36.817$	3.3.13916.1	None	✓	$1$	$1$	\(0\)	\(-9\)	\(-4\)	\(-6\)		$+$	$+$	$-$	$-$	$2^{3}$	$\mathrm{SU}(2)$	\(q-3q^{3}+(-1+\beta _{2})q^{5}+(-2-\beta _{1}+\cdots)q^{7}+\cdots\)
624.4.a.t	$624$	$4$	624.a	1.a	$1$	$3$	$3$	$36.817$	3.3.3144.1	None	✓	$1$	$0$	\(0\)	\(-9\)	\(4\)	\(-30\)		$-$	$+$	$-$	$+$	$2^{3}$	$\mathrm{SU}(2)$	\(q-3q^{3}+(1-\beta _{2})q^{5}+(-11-3\beta _{1}+\cdots)q^{7}+\cdots\)
624.4.a.u	$624$	$4$	624.a	1.a	$1$	$3$	$3$	$36.817$	3.3.36248.1	None	✓	$1$	$0$	\(0\)	\(9\)	\(16\)	\(22\)		$+$	$-$	$-$	$+$	$2^{3}$	$\mathrm{SU}(2)$	\(q+3q^{3}+(5+\beta _{2})q^{5}+(8-\beta _{1}-\beta _{2})q^{7}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(624))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(624)) \cong \) \(S_{4}^{\mathrm{new}}(\Gamma_0(6))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(8))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(12))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(13))\)\(^{\oplus 10}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(16))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(24))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(26))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(39))\)\(^{\oplus 5}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(48))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(52))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(78))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(104))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(156))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(208))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(312))\)\(^{\oplus 2}\)