Space of modular forms of level 624, weight 2, and trivial character

• Label: 624.2.a
• Level: $624$
• Weight: $2$
• Character orbit: 624.a
• Rep. character: $\chi_{624}(1,\cdot)$
• Character field: $\Q$
• Dimension: $12$
• Newform subspaces: $11$
• Sturm bound: $224$
• Trace bound: $7$

Defining parameters

Level:	\( N \)	\(=\)	\( 624 = 2^{4} \cdot 3 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	624.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 11 \)
Sturm bound:			\(224\)
Trace bound:			\(7\)
Distinguishing \(T_p\):			\(5\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(624))\).

	Total	New	Old
Modular forms	124	12	112
Cusp forms	101	12	89
Eisenstein series	23	0	23

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		Total				Cusp				Eisenstein
					All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)	\(+\)		\(12\)	\(2\)	\(10\)		\(10\)	\(2\)	\(8\)		\(2\)	\(0\)	\(2\)
\(+\)	\(+\)	\(-\)	\(-\)		\(18\)	\(1\)	\(17\)		\(15\)	\(1\)	\(14\)		\(3\)	\(0\)	\(3\)
\(+\)	\(-\)	\(+\)	\(-\)		\(19\)	\(2\)	\(17\)		\(16\)	\(2\)	\(14\)		\(3\)	\(0\)	\(3\)
\(+\)	\(-\)	\(-\)	\(+\)		\(13\)	\(1\)	\(12\)		\(10\)	\(1\)	\(9\)		\(3\)	\(0\)	\(3\)
\(-\)	\(+\)	\(+\)	\(-\)		\(15\)	\(2\)	\(13\)		\(12\)	\(2\)	\(10\)		\(3\)	\(0\)	\(3\)
\(-\)	\(+\)	\(-\)	\(+\)		\(17\)	\(1\)	\(16\)		\(14\)	\(1\)	\(13\)		\(3\)	\(0\)	\(3\)
\(-\)	\(-\)	\(+\)	\(+\)		\(16\)	\(0\)	\(16\)		\(13\)	\(0\)	\(13\)		\(3\)	\(0\)	\(3\)
\(-\)	\(-\)	\(-\)	\(-\)		\(14\)	\(3\)	\(11\)		\(11\)	\(3\)	\(8\)		\(3\)	\(0\)	\(3\)
Plus space			\(+\)		\(58\)	\(4\)	\(54\)		\(47\)	\(4\)	\(43\)		\(11\)	\(0\)	\(11\)
Minus space			\(-\)		\(66\)	\(8\)	\(58\)		\(54\)	\(8\)	\(46\)		\(12\)	\(0\)	\(12\)

Trace form

12 * q + 4 * q^7 + 12 * q^9 + 8 * q^11 + 4 * q^15 - 4 * q^19 - 16 * q^23 + 20 * q^25 - 16 * q^29 + 20 * q^31 + 8 * q^33 - 16 * q^37 + 4 * q^39 + 16 * q^43 + 24 * q^47 + 20 * q^49 + 8 * q^51 - 8 * q^53 - 16 * q^55 - 8 * q^61 + 4 * q^63 + 20 * q^67 - 16 * q^69 - 16 * q^71 + 8 * q^73 + 8 * q^75 - 8 * q^77 + 16 * q^79 + 12 * q^81 - 48 * q^83 - 12 * q^91 + 16 * q^93 - 8 * q^97 + 8 * q^99

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(624))\) 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					A-L signs			Fricke sign	Coefficient ring index	Sato-Tate	$q$-expansion
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$		2	3	13
624.2.a.a	$624$	$2$	624.a	1.a	$1$	$1$	$1$	$4.983$	\(\Q\)	None	✓				312.2.a.d	$1$	$1$	\(0\)	\(-1\)	\(-4\)	\(4\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-4q^{5}+4q^{7}+q^{9}+2q^{11}+\cdots\)
624.2.a.b	$624$	$2$	624.a	1.a	$1$	$1$	$1$	$4.983$	\(\Q\)	None	✓		✓	✓	156.2.a.b	$1$	$1$	\(0\)	\(-1\)	\(0\)	\(-2\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-2q^{7}+q^{9}+q^{13}-6q^{17}+\cdots\)
624.2.a.c	$624$	$2$	624.a	1.a	$1$	$1$	$1$	$4.983$	\(\Q\)	None	✓				312.2.a.e	$1$	$1$	\(0\)	\(-1\)	\(0\)	\(0\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+q^{9}-6q^{11}-q^{13}+2q^{17}+\cdots\)
624.2.a.d	$624$	$2$	624.a	1.a	$1$	$1$	$1$	$4.983$	\(\Q\)	None	✓		✓	✓	312.2.a.f	$1$	$0$	\(0\)	\(-1\)	\(2\)	\(0\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+2q^{5}+q^{9}+q^{13}-2q^{15}+\cdots\)
624.2.a.e	$624$	$2$	624.a	1.a	$1$	$1$	$1$	$4.983$	\(\Q\)	None	✓				156.2.a.a	$1$	$0$	\(0\)	\(1\)	\(-4\)	\(2\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-4q^{5}+2q^{7}+q^{9}+4q^{11}+\cdots\)
624.2.a.f	$624$	$2$	624.a	1.a	$1$	$1$	$1$	$4.983$	\(\Q\)	None	✓		✓	✓	312.2.a.a	$1$	$1$	\(0\)	\(1\)	\(-2\)	\(-4\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-2q^{5}-4q^{7}+q^{9}+q^{13}-2q^{15}+\cdots\)
624.2.a.g	$624$	$2$	624.a	1.a	$1$	$1$	$1$	$4.983$	\(\Q\)	None	✓				312.2.a.b	$1$	$0$	\(0\)	\(1\)	\(0\)	\(4\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+4q^{7}+q^{9}+2q^{11}-q^{13}+\cdots\)
624.2.a.h	$624$	$2$	624.a	1.a	$1$	$1$	$1$	$4.983$	\(\Q\)	None	✓				78.2.a.a	$1$	$0$	\(0\)	\(1\)	\(2\)	\(-4\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+2q^{5}-4q^{7}+q^{9}+4q^{11}+\cdots\)
624.2.a.i	$624$	$2$	624.a	1.a	$1$	$1$	$1$	$4.983$	\(\Q\)	None	✓				39.2.a.a	$1$	$0$	\(0\)	\(1\)	\(2\)	\(4\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+2q^{5}+4q^{7}+q^{9}-4q^{11}+\cdots\)
624.2.a.j	$624$	$2$	624.a	1.a	$1$	$1$	$1$	$4.983$	\(\Q\)	None	✓				312.2.a.c	$1$	$0$	\(0\)	\(1\)	\(4\)	\(0\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+4q^{5}+q^{9}+2q^{11}-q^{13}+\cdots\)
624.2.a.k	$624$	$2$	624.a	1.a	$1$	$2$	$2$	$4.983$	\(\Q(\sqrt{2}) \)	None	✓		✓	✓	39.2.a.b	$1$	$0$	\(0\)	\(-2\)	\(0\)	\(0\)		$-$	$+$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q-q^{3}+\beta q^{5}+\beta q^{7}+q^{9}+2q^{11}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(624))\) into lower level spaces

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