Space of modular forms of level 630, weight 4, and trivial character

• Label: 630.4.a
• Level: $630$
• Weight: $4$
• Character orbit: 630.a
• Rep. character: $\chi_{630}(1,\cdot)$
• Character field: $\Q$
• Dimension: $30$
• Newform subspaces: $27$
• Sturm bound: $576$
• Trace bound: $11$

Defining parameters

Level:	\( N \)	\(=\)	\( 630 = 2 \cdot 3^{2} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	630.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 27 \)
Sturm bound:			\(576\)
Trace bound:			\(11\)
Distinguishing \(T_p\):			\(11\), \(13\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(630))\).

	Total	New	Old
Modular forms	448	30	418
Cusp forms	416	30	386
Eisenstein series	32	0	32

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		Total				Cusp				Eisenstein
						All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)	\(+\)	\(+\)		\(32\)	\(2\)	\(30\)		\(30\)	\(2\)	\(28\)		\(2\)	\(0\)	\(2\)
\(+\)	\(+\)	\(+\)	\(-\)	\(-\)		\(26\)	\(1\)	\(25\)		\(24\)	\(1\)	\(23\)		\(2\)	\(0\)	\(2\)
\(+\)	\(+\)	\(-\)	\(+\)	\(-\)		\(26\)	\(1\)	\(25\)		\(24\)	\(1\)	\(23\)		\(2\)	\(0\)	\(2\)
\(+\)	\(+\)	\(-\)	\(-\)	\(+\)		\(28\)	\(2\)	\(26\)		\(26\)	\(2\)	\(24\)		\(2\)	\(0\)	\(2\)
\(+\)	\(-\)	\(+\)	\(+\)	\(-\)		\(27\)	\(2\)	\(25\)		\(25\)	\(2\)	\(23\)		\(2\)	\(0\)	\(2\)
\(+\)	\(-\)	\(+\)	\(-\)	\(+\)		\(28\)	\(2\)	\(26\)		\(26\)	\(2\)	\(24\)		\(2\)	\(0\)	\(2\)
\(+\)	\(-\)	\(-\)	\(+\)	\(+\)		\(29\)	\(2\)	\(27\)		\(27\)	\(2\)	\(25\)		\(2\)	\(0\)	\(2\)
\(+\)	\(-\)	\(-\)	\(-\)	\(-\)		\(28\)	\(2\)	\(26\)		\(26\)	\(2\)	\(24\)		\(2\)	\(0\)	\(2\)
\(-\)	\(+\)	\(+\)	\(+\)	\(-\)		\(28\)	\(1\)	\(27\)		\(26\)	\(1\)	\(25\)		\(2\)	\(0\)	\(2\)
\(-\)	\(+\)	\(+\)	\(-\)	\(+\)		\(28\)	\(2\)	\(26\)		\(26\)	\(2\)	\(24\)		\(2\)	\(0\)	\(2\)
\(-\)	\(+\)	\(-\)	\(+\)	\(+\)		\(26\)	\(2\)	\(24\)		\(24\)	\(2\)	\(22\)		\(2\)	\(0\)	\(2\)
\(-\)	\(+\)	\(-\)	\(-\)	\(-\)		\(30\)	\(1\)	\(29\)		\(28\)	\(1\)	\(27\)		\(2\)	\(0\)	\(2\)
\(-\)	\(-\)	\(+\)	\(+\)	\(+\)		\(29\)	\(3\)	\(26\)		\(27\)	\(3\)	\(24\)		\(2\)	\(0\)	\(2\)
\(-\)	\(-\)	\(+\)	\(-\)	\(-\)		\(28\)	\(2\)	\(26\)		\(26\)	\(2\)	\(24\)		\(2\)	\(0\)	\(2\)
\(-\)	\(-\)	\(-\)	\(+\)	\(-\)		\(27\)	\(2\)	\(25\)		\(25\)	\(2\)	\(23\)		\(2\)	\(0\)	\(2\)
\(-\)	\(-\)	\(-\)	\(-\)	\(+\)		\(28\)	\(3\)	\(25\)		\(26\)	\(3\)	\(23\)		\(2\)	\(0\)	\(2\)
Plus space				\(+\)		\(228\)	\(18\)	\(210\)		\(212\)	\(18\)	\(194\)		\(16\)	\(0\)	\(16\)
Minus space				\(-\)		\(220\)	\(12\)	\(208\)		\(204\)	\(12\)	\(192\)		\(16\)	\(0\)	\(16\)

Trace form

30 * q + 4 * q^2 + 120 * q^4 + 16 * q^8 - 172 * q^11 - 32 * q^13 + 480 * q^16 - 140 * q^17 - 260 * q^19 - 160 * q^22 + 72 * q^23 + 750 * q^25 - 80 * q^26 + 432 * q^29 + 1136 * q^31 + 64 * q^32 + 504 * q^34 + 70 * q^35 + 396 * q^37 - 488 * q^38 + 652 * q^41 + 456 * q^43 - 688 * q^44 + 16 * q^46 - 224 * q^47 + 1470 * q^49 + 100 * q^50 - 128 * q^52 + 2324 * q^53 - 360 * q^55 + 1272 * q^58 - 244 * q^59 - 496 * q^61 + 560 * q^62 + 1920 * q^64 - 440 * q^65 + 3088 * q^67 - 560 * q^68 - 140 * q^70 + 64 * q^71 + 332 * q^73 + 296 * q^74 - 1040 * q^76 + 616 * q^77 + 172 * q^79 + 2856 * q^82 + 316 * q^83 - 1060 * q^85 + 2064 * q^86 - 640 * q^88 + 1532 * q^89 - 1344 * q^91 + 288 * q^92 + 560 * q^94 - 2060 * q^95 + 716 * q^97 + 196 * q^98

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(630))\) 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
630.4.a.a	$630$	$4$	630.a	1.a	$1$	$1$	$1$	$37.171$	\(\Q\)	None	✓				210.4.a.i	$1$	$1$	\(-2\)	\(0\)	\(-5\)	\(-7\)		$+$	$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+4q^{4}-5q^{5}-7q^{7}-8q^{8}+\cdots\)
630.4.a.b	$630$	$4$	630.a	1.a	$1$	$1$	$1$	$37.171$	\(\Q\)	None	✓				70.4.a.e	$1$	$1$	\(-2\)	\(0\)	\(-5\)	\(-7\)		$+$	$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+4q^{4}-5q^{5}-7q^{7}-8q^{8}+\cdots\)
630.4.a.c	$630$	$4$	630.a	1.a	$1$	$1$	$1$	$37.171$	\(\Q\)	None	✓	✓	✓	✓	630.4.a.c	$1$	$1$	\(-2\)	\(0\)	\(-5\)	\(7\)		$+$	$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+4q^{4}-5q^{5}+7q^{7}-8q^{8}+\cdots\)
630.4.a.d	$630$	$4$	630.a	1.a	$1$	$1$	$1$	$37.171$	\(\Q\)	None	✓				210.4.a.j	$1$	$0$	\(-2\)	\(0\)	\(5\)	\(-7\)		$+$	$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+4q^{4}+5q^{5}-7q^{7}-8q^{8}+\cdots\)
630.4.a.e	$630$	$4$	630.a	1.a	$1$	$1$	$1$	$37.171$	\(\Q\)	None	✓				210.4.a.g	$1$	$0$	\(-2\)	\(0\)	\(5\)	\(-7\)		$+$	$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+4q^{4}+5q^{5}-7q^{7}-8q^{8}+\cdots\)
630.4.a.f	$630$	$4$	630.a	1.a	$1$	$1$	$1$	$37.171$	\(\Q\)	None	✓	✓	✓	✓	630.4.a.f	$1$	$1$	\(-2\)	\(0\)	\(5\)	\(-7\)		$+$	$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+4q^{4}+5q^{5}-7q^{7}-8q^{8}+\cdots\)
630.4.a.g	$630$	$4$	630.a	1.a	$1$	$1$	$1$	$37.171$	\(\Q\)	None	✓	✓			630.4.a.g	$1$	$0$	\(-2\)	\(0\)	\(5\)	\(7\)		$+$	$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+4q^{4}+5q^{5}+7q^{7}-8q^{8}+\cdots\)
630.4.a.h	$630$	$4$	630.a	1.a	$1$	$1$	$1$	$37.171$	\(\Q\)	None	✓				210.4.a.h	$1$	$1$	\(-2\)	\(0\)	\(5\)	\(7\)		$+$	$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+4q^{4}+5q^{5}+7q^{7}-8q^{8}+\cdots\)
630.4.a.i	$630$	$4$	630.a	1.a	$1$	$1$	$1$	$37.171$	\(\Q\)	None	✓	✓			630.4.a.i	$1$	$0$	\(-2\)	\(0\)	\(5\)	\(7\)		$+$	$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+4q^{4}+5q^{5}+7q^{7}-8q^{8}+\cdots\)
630.4.a.j	$630$	$4$	630.a	1.a	$1$	$1$	$1$	$37.171$	\(\Q\)	None	✓				70.4.a.f	$1$	$1$	\(-2\)	\(0\)	\(5\)	\(7\)		$+$	$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+4q^{4}+5q^{5}+7q^{7}-8q^{8}+\cdots\)
630.4.a.k	$630$	$4$	630.a	1.a	$1$	$1$	$1$	$37.171$	\(\Q\)	None	✓				210.4.a.f	$1$	$0$	\(2\)	\(0\)	\(-5\)	\(-7\)		$-$	$-$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+4q^{4}-5q^{5}-7q^{7}+8q^{8}+\cdots\)
630.4.a.l	$630$	$4$	630.a	1.a	$1$	$1$	$1$	$37.171$	\(\Q\)	None	✓	✓	✓	✓	630.4.a.f	$1$	$1$	\(2\)	\(0\)	\(-5\)	\(-7\)		$-$	$+$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+4q^{4}-5q^{5}-7q^{7}+8q^{8}+\cdots\)
630.4.a.m	$630$	$4$	630.a	1.a	$1$	$1$	$1$	$37.171$	\(\Q\)	None	✓				70.4.a.b	$1$	$0$	\(2\)	\(0\)	\(-5\)	\(-7\)		$-$	$-$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+4q^{4}-5q^{5}-7q^{7}+8q^{8}+\cdots\)
630.4.a.n	$630$	$4$	630.a	1.a	$1$	$1$	$1$	$37.171$	\(\Q\)	None	✓				210.4.a.b	$1$	$0$	\(2\)	\(0\)	\(-5\)	\(-7\)		$-$	$-$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+4q^{4}-5q^{5}-7q^{7}+8q^{8}+\cdots\)
630.4.a.o	$630$	$4$	630.a	1.a	$1$	$1$	$1$	$37.171$	\(\Q\)	None	✓				70.4.a.d	$1$	$1$	\(2\)	\(0\)	\(-5\)	\(7\)		$-$	$-$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+4q^{4}-5q^{5}+7q^{7}+8q^{8}+\cdots\)
630.4.a.p	$630$	$4$	630.a	1.a	$1$	$1$	$1$	$37.171$	\(\Q\)	None	✓	✓			630.4.a.i	$1$	$0$	\(2\)	\(0\)	\(-5\)	\(7\)		$-$	$+$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+4q^{4}-5q^{5}+7q^{7}+8q^{8}+\cdots\)
630.4.a.q	$630$	$4$	630.a	1.a	$1$	$1$	$1$	$37.171$	\(\Q\)	None	✓				210.4.a.c	$1$	$1$	\(2\)	\(0\)	\(-5\)	\(7\)		$-$	$-$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+4q^{4}-5q^{5}+7q^{7}+8q^{8}+\cdots\)
630.4.a.r	$630$	$4$	630.a	1.a	$1$	$1$	$1$	$37.171$	\(\Q\)	None	✓	✓			630.4.a.g	$1$	$0$	\(2\)	\(0\)	\(-5\)	\(7\)		$-$	$+$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+4q^{4}-5q^{5}+7q^{7}+8q^{8}+\cdots\)
630.4.a.s	$630$	$4$	630.a	1.a	$1$	$1$	$1$	$37.171$	\(\Q\)	None	✓				70.4.a.a	$1$	$1$	\(2\)	\(0\)	\(5\)	\(-7\)		$-$	$-$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+4q^{4}+5q^{5}-7q^{7}+8q^{8}+\cdots\)
630.4.a.t	$630$	$4$	630.a	1.a	$1$	$1$	$1$	$37.171$	\(\Q\)	None	✓				210.4.a.d	$1$	$1$	\(2\)	\(0\)	\(5\)	\(-7\)		$-$	$-$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+4q^{4}+5q^{5}-7q^{7}+8q^{8}+\cdots\)
630.4.a.u	$630$	$4$	630.a	1.a	$1$	$1$	$1$	$37.171$	\(\Q\)	None	✓	✓	✓	✓	630.4.a.c	$1$	$1$	\(2\)	\(0\)	\(5\)	\(7\)		$-$	$+$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+4q^{4}+5q^{5}+7q^{7}+8q^{8}+\cdots\)
630.4.a.v	$630$	$4$	630.a	1.a	$1$	$1$	$1$	$37.171$	\(\Q\)	None	✓				210.4.a.a	$1$	$0$	\(2\)	\(0\)	\(5\)	\(7\)		$-$	$-$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+4q^{4}+5q^{5}+7q^{7}+8q^{8}+\cdots\)
630.4.a.w	$630$	$4$	630.a	1.a	$1$	$1$	$1$	$37.171$	\(\Q\)	None	✓				210.4.a.e	$1$	$0$	\(2\)	\(0\)	\(5\)	\(7\)		$-$	$-$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+4q^{4}+5q^{5}+7q^{7}+8q^{8}+\cdots\)
630.4.a.x	$630$	$4$	630.a	1.a	$1$	$1$	$1$	$37.171$	\(\Q\)	None	✓				70.4.a.c	$1$	$0$	\(2\)	\(0\)	\(5\)	\(7\)		$-$	$-$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+4q^{4}+5q^{5}+7q^{7}+8q^{8}+\cdots\)
630.4.a.y	$630$	$4$	630.a	1.a	$1$	$2$	$2$	$37.171$	\(\Q(\sqrt{3649}) \)	None	✓	✓	✓	✓	630.4.a.y	$1$	$0$	\(-4\)	\(0\)	\(-10\)	\(-14\)		$+$	$+$	$+$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q-2q^{2}+4q^{4}-5q^{5}-7q^{7}-8q^{8}+\cdots\)
630.4.a.z	$630$	$4$	630.a	1.a	$1$	$2$	$2$	$37.171$	\(\Q(\sqrt{106}) \)	None	✓		✓	✓	210.4.a.k	$1$	$0$	\(-4\)	\(0\)	\(-10\)	\(14\)		$+$	$-$	$+$	$-$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q-2q^{2}+4q^{4}-5q^{5}+7q^{7}-8q^{8}+\cdots\)
630.4.a.ba	$630$	$4$	630.a	1.a	$1$	$2$	$2$	$37.171$	\(\Q(\sqrt{3649}) \)	None	✓	✓	✓	✓	630.4.a.y	$1$	$0$	\(4\)	\(0\)	\(10\)	\(-14\)		$-$	$+$	$-$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q+2q^{2}+4q^{4}+5q^{5}-7q^{7}+8q^{8}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(630))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(630)) \simeq \) \(S_{4}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(6))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(7))\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(9))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(10))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(18))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(21))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(30))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(35))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(42))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(45))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(63))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(70))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(90))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(105))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(126))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(210))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(315))\)\(^{\oplus 2}\)