Space of modular forms of level 630, weight 2, and trivial character

• Label: 630.2.a
• Level: $630$
• Weight: $2$
• Character orbit: 630.a
• Rep. character: $\chi_{630}(1,\cdot)$
• Character field: $\Q$
• Dimension: $10$
• Newform subspaces: $10$
• Sturm bound: $288$
• Trace bound: $13$

Defining parameters

Level:	\( N \)	\(=\)	\( 630 = 2 \cdot 3^{2} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	630.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 10 \)
Sturm bound:			\(288\)
Trace bound:			\(13\)
Distinguishing \(T_p\):			\(11\), \(13\), \(17\), \(19\), \(29\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(630))\).

	Total	New	Old
Modular forms	160	10	150
Cusp forms	129	10	119
Eisenstein series	31	0	31

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

\(2\)	\(3\)	\(5\)	\(7\)	Fricke	Dim
\(+\)	\(+\)	\(+\)	\(-\)	\(-\)	\(1\)
\(+\)	\(+\)	\(-\)	\(+\)	\(-\)	\(1\)
\(+\)	\(-\)	\(+\)	\(+\)	\(-\)	\(1\)
\(+\)	\(-\)	\(+\)	\(-\)	\(+\)	\(1\)
\(+\)	\(-\)	\(-\)	\(+\)	\(+\)	\(1\)
\(+\)	\(-\)	\(-\)	\(-\)	\(-\)	\(1\)
\(-\)	\(+\)	\(+\)	\(+\)	\(-\)	\(1\)
\(-\)	\(+\)	\(-\)	\(-\)	\(-\)	\(1\)
\(-\)	\(-\)	\(+\)	\(-\)	\(-\)	\(1\)
\(-\)	\(-\)	\(-\)	\(+\)	\(-\)	\(1\)
Plus space				\(+\)	\(2\)
Minus space				\(-\)	\(8\)

Trace form

\( 10 q - 2 q^{2} + 10 q^{4} - 2 q^{8} + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(630))\) 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	5	7
630.2.a.a	$630$	$2$	630.a	1.a	$1$	$1$	$1$	$5.031$	\(\Q\)	None	✓		✓	✓	210.2.a.e	$1$	$0$	\(-1\)	\(0\)	\(-1\)	\(-1\)		$+$	$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}-q^{5}-q^{7}-q^{8}+q^{10}+\cdots\)
630.2.a.b	$630$	$2$	630.a	1.a	$1$	$1$	$1$	$5.031$	\(\Q\)	None	✓		✓	✓	210.2.a.c	$1$	$1$	\(-1\)	\(0\)	\(-1\)	\(1\)		$+$	$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}-q^{5}+q^{7}-q^{8}+q^{10}+\cdots\)
630.2.a.c	$630$	$2$	630.a	1.a	$1$	$1$	$1$	$5.031$	\(\Q\)	None	✓	✓	✓	✓	630.2.a.c	$1$	$0$	\(-1\)	\(0\)	\(-1\)	\(1\)		$+$	$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}-q^{5}+q^{7}-q^{8}+q^{10}+\cdots\)
630.2.a.d	$630$	$2$	630.a	1.a	$1$	$1$	$1$	$5.031$	\(\Q\)	None	✓		✓	✓	70.2.a.a	$1$	$1$	\(-1\)	\(0\)	\(1\)	\(-1\)		$+$	$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}+q^{5}-q^{7}-q^{8}-q^{10}+\cdots\)
630.2.a.e	$630$	$2$	630.a	1.a	$1$	$1$	$1$	$5.031$	\(\Q\)	None	✓	✓	✓	✓	630.2.a.e	$1$	$0$	\(-1\)	\(0\)	\(1\)	\(-1\)		$+$	$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}+q^{5}-q^{7}-q^{8}-q^{10}+\cdots\)
630.2.a.f	$630$	$2$	630.a	1.a	$1$	$1$	$1$	$5.031$	\(\Q\)	None	✓		✓	✓	210.2.a.d	$1$	$0$	\(-1\)	\(0\)	\(1\)	\(1\)		$+$	$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}+q^{5}+q^{7}-q^{8}-q^{10}+\cdots\)
630.2.a.g	$630$	$2$	630.a	1.a	$1$	$1$	$1$	$5.031$	\(\Q\)	None	✓	✓	✓	✓	630.2.a.e	$1$	$0$	\(1\)	\(0\)	\(-1\)	\(-1\)		$-$	$+$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}-q^{5}-q^{7}+q^{8}-q^{10}+\cdots\)
630.2.a.h	$630$	$2$	630.a	1.a	$1$	$1$	$1$	$5.031$	\(\Q\)	None	✓		✓	✓	210.2.a.b	$1$	$0$	\(1\)	\(0\)	\(-1\)	\(1\)		$-$	$-$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}-q^{5}+q^{7}+q^{8}-q^{10}+\cdots\)
630.2.a.i	$630$	$2$	630.a	1.a	$1$	$1$	$1$	$5.031$	\(\Q\)	None	✓		✓	✓	210.2.a.a	$1$	$0$	\(1\)	\(0\)	\(1\)	\(-1\)		$-$	$-$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}+q^{5}-q^{7}+q^{8}+q^{10}+\cdots\)
630.2.a.j	$630$	$2$	630.a	1.a	$1$	$1$	$1$	$5.031$	\(\Q\)	None	✓	✓	✓	✓	630.2.a.c	$1$	$0$	\(1\)	\(0\)	\(1\)	\(1\)		$-$	$+$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}+q^{5}+q^{7}+q^{8}+q^{10}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(630))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(630)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(21))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(30))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(35))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(42))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(45))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(63))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(70))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(90))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(105))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(126))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(210))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(315))\)\(^{\oplus 2}\)