Space of modular forms of level 1008, weight 4, and trivial character

• Label: 1008.4.a
• Level: $1008$
• Weight: $4$
• Character orbit: 1008.a
• Rep. character: $\chi_{1008}(1,\cdot)$
• Character field: $\Q$
• Dimension: $45$
• Newform subspaces: $34$
• Sturm bound: $768$
• Trace bound: $13$

Defining parameters

Level:	\( N \)	\(=\)	\( 1008 = 2^{4} \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1008.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 34 \)
Sturm bound:			\(768\)
Trace bound:			\(13\)
Distinguishing \(T_p\):			\(5\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(1008))\).

	Total	New	Old
Modular forms	600	45	555
Cusp forms	552	45	507
Eisenstein series	48	0	48

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

\(2\)	\(3\)	\(7\)	Fricke		Total				Cusp				Eisenstein
					All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)	\(+\)		\(78\)	\(4\)	\(74\)		\(72\)	\(4\)	\(68\)		\(6\)	\(0\)	\(6\)
\(+\)	\(+\)	\(-\)	\(-\)		\(72\)	\(4\)	\(68\)		\(66\)	\(4\)	\(62\)		\(6\)	\(0\)	\(6\)
\(+\)	\(-\)	\(+\)	\(-\)		\(72\)	\(7\)	\(65\)		\(66\)	\(7\)	\(59\)		\(6\)	\(0\)	\(6\)
\(+\)	\(-\)	\(-\)	\(+\)		\(78\)	\(7\)	\(71\)		\(72\)	\(7\)	\(65\)		\(6\)	\(0\)	\(6\)
\(-\)	\(+\)	\(+\)	\(-\)		\(76\)	\(4\)	\(72\)		\(70\)	\(4\)	\(66\)		\(6\)	\(0\)	\(6\)
\(-\)	\(+\)	\(-\)	\(+\)		\(74\)	\(6\)	\(68\)		\(68\)	\(6\)	\(62\)		\(6\)	\(0\)	\(6\)
\(-\)	\(-\)	\(+\)	\(+\)		\(74\)	\(7\)	\(67\)		\(68\)	\(7\)	\(61\)		\(6\)	\(0\)	\(6\)
\(-\)	\(-\)	\(-\)	\(-\)		\(76\)	\(6\)	\(70\)		\(70\)	\(6\)	\(64\)		\(6\)	\(0\)	\(6\)
Plus space			\(+\)		\(304\)	\(24\)	\(280\)		\(280\)	\(24\)	\(256\)		\(24\)	\(0\)	\(24\)
Minus space			\(-\)		\(296\)	\(21\)	\(275\)		\(272\)	\(21\)	\(251\)		\(24\)	\(0\)	\(24\)

Trace form

45 * q + 2 * q^5 + 7 * q^7 + 56 * q^11 - 46 * q^13 + 50 * q^17 - 24 * q^19 - 220 * q^23 + 1147 * q^25 - 242 * q^29 + 264 * q^31 + 210 * q^35 + 62 * q^37 + 138 * q^41 - 892 * q^43 - 1536 * q^47 + 2205 * q^49 - 98 * q^53 + 1080 * q^55 + 1080 * q^59 + 834 * q^61 - 580 * q^65 - 228 * q^67 - 2900 * q^71 + 458 * q^73 + 476 * q^77 + 3016 * q^79 + 1112 * q^83 - 1588 * q^85 + 1730 * q^89 - 546 * q^91 - 1848 * q^95 + 906 * q^97

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(1008))\) 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	7
1008.4.a.a	$1008$	$4$	1008.a	1.a	$1$	$1$	$1$	$59.474$	\(\Q\)	None	✓				126.4.a.e	$1$	$0$	\(0\)	\(0\)	\(-22\)	\(7\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-22q^{5}+7q^{7}+26q^{11}-54q^{13}+\cdots\)
1008.4.a.b	$1008$	$4$	1008.a	1.a	$1$	$1$	$1$	$59.474$	\(\Q\)	None	✓				42.4.a.a	$1$	$0$	\(0\)	\(0\)	\(-18\)	\(-7\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-18q^{5}-7q^{7}-72q^{11}-34q^{13}+\cdots\)
1008.4.a.c	$1008$	$4$	1008.a	1.a	$1$	$1$	$1$	$59.474$	\(\Q\)	None	✓				7.4.a.a	$1$	$1$	\(0\)	\(0\)	\(-16\)	\(7\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2^{4}q^{5}+7q^{7}-8q^{11}+28q^{13}+\cdots\)
1008.4.a.d	$1008$	$4$	1008.a	1.a	$1$	$1$	$1$	$59.474$	\(\Q\)	None	✓				84.4.a.b	$1$	$1$	\(0\)	\(0\)	\(-14\)	\(7\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-14q^{5}+7q^{7}+4q^{11}+54q^{13}+\cdots\)
1008.4.a.e	$1008$	$4$	1008.a	1.a	$1$	$1$	$1$	$59.474$	\(\Q\)	None	✓				56.4.a.b	$1$	$0$	\(0\)	\(0\)	\(-8\)	\(7\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-8q^{5}+7q^{7}+56q^{11}-28q^{13}+\cdots\)
1008.4.a.f	$1008$	$4$	1008.a	1.a	$1$	$1$	$1$	$59.474$	\(\Q\)	None	✓				28.4.a.b	$1$	$0$	\(0\)	\(0\)	\(-6\)	\(-7\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-6q^{5}-7q^{7}-12q^{11}-82q^{13}+\cdots\)
1008.4.a.g	$1008$	$4$	1008.a	1.a	$1$	$1$	$1$	$59.474$	\(\Q\)	None	✓				126.4.a.b	$1$	$1$	\(0\)	\(0\)	\(-6\)	\(-7\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-6q^{5}-7q^{7}+30q^{11}+2q^{13}-66q^{17}+\cdots\)
1008.4.a.h	$1008$	$4$	1008.a	1.a	$1$	$1$	$1$	$59.474$	\(\Q\)	None	✓				84.4.a.a	$1$	$0$	\(0\)	\(0\)	\(-6\)	\(-7\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-6q^{5}-7q^{7}+6^{2}q^{11}+62q^{13}+\cdots\)
1008.4.a.i	$1008$	$4$	1008.a	1.a	$1$	$1$	$1$	$59.474$	\(\Q\)	None	✓				168.4.a.c	$1$	$0$	\(0\)	\(0\)	\(-4\)	\(7\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-4q^{5}+7q^{7}-26q^{11}+2q^{13}+6^{2}q^{17}+\cdots\)
1008.4.a.j	$1008$	$4$	1008.a	1.a	$1$	$1$	$1$	$59.474$	\(\Q\)	None	✓				42.4.a.b	$1$	$1$	\(0\)	\(0\)	\(-2\)	\(7\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{5}+7q^{7}-8q^{11}-42q^{13}+2q^{17}+\cdots\)
1008.4.a.k	$1008$	$4$	1008.a	1.a	$1$	$1$	$1$	$59.474$	\(\Q\)	None	✓				168.4.a.b	$1$	$1$	\(0\)	\(0\)	\(2\)	\(-7\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{5}-7q^{7}+12q^{11}-66q^{13}+\cdots\)
1008.4.a.l	$1008$	$4$	1008.a	1.a	$1$	$1$	$1$	$59.474$	\(\Q\)	None	✓				168.4.a.f	$1$	$0$	\(0\)	\(0\)	\(2\)	\(7\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{5}+7q^{7}+52q^{11}+86q^{13}+\cdots\)
1008.4.a.m	$1008$	$4$	1008.a	1.a	$1$	$1$	$1$	$59.474$	\(\Q\)	None	✓				21.4.a.b	$1$	$1$	\(0\)	\(0\)	\(4\)	\(7\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+4q^{5}+7q^{7}+62q^{11}-62q^{13}+\cdots\)
1008.4.a.n	$1008$	$4$	1008.a	1.a	$1$	$1$	$1$	$59.474$	\(\Q\)	None	✓				126.4.a.b	$1$	$1$	\(0\)	\(0\)	\(6\)	\(-7\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+6q^{5}-7q^{7}-30q^{11}+2q^{13}+66q^{17}+\cdots\)
1008.4.a.o	$1008$	$4$	1008.a	1.a	$1$	$1$	$1$	$59.474$	\(\Q\)	None	✓				28.4.a.a	$1$	$1$	\(0\)	\(0\)	\(8\)	\(7\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+8q^{5}+7q^{7}-40q^{11}-12q^{13}+\cdots\)
1008.4.a.p	$1008$	$4$	1008.a	1.a	$1$	$1$	$1$	$59.474$	\(\Q\)	None	✓				168.4.a.a	$1$	$1$	\(0\)	\(0\)	\(10\)	\(-7\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+10q^{5}-7q^{7}-12q^{11}+30q^{13}+\cdots\)
1008.4.a.q	$1008$	$4$	1008.a	1.a	$1$	$1$	$1$	$59.474$	\(\Q\)	None	✓				168.4.a.e	$1$	$0$	\(0\)	\(0\)	\(10\)	\(7\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+10q^{5}+7q^{7}-52q^{11}-10q^{13}+\cdots\)
1008.4.a.r	$1008$	$4$	1008.a	1.a	$1$	$1$	$1$	$59.474$	\(\Q\)	None	✓				14.4.a.b	$1$	$0$	\(0\)	\(0\)	\(12\)	\(-7\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+12q^{5}-7q^{7}+48q^{11}+56q^{13}+\cdots\)
1008.4.a.s	$1008$	$4$	1008.a	1.a	$1$	$1$	$1$	$59.474$	\(\Q\)	None	✓				14.4.a.a	$1$	$1$	\(0\)	\(0\)	\(14\)	\(7\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+14q^{5}+7q^{7}-28q^{11}+18q^{13}+\cdots\)
1008.4.a.t	$1008$	$4$	1008.a	1.a	$1$	$1$	$1$	$59.474$	\(\Q\)	None	✓				168.4.a.d	$1$	$1$	\(0\)	\(0\)	\(16\)	\(-7\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2^{4}q^{5}-7q^{7}-18q^{11}-54q^{13}+\cdots\)
1008.4.a.u	$1008$	$4$	1008.a	1.a	$1$	$1$	$1$	$59.474$	\(\Q\)	None	✓				56.4.a.a	$1$	$0$	\(0\)	\(0\)	\(16\)	\(7\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2^{4}q^{5}+7q^{7}+24q^{11}-68q^{13}+\cdots\)
1008.4.a.v	$1008$	$4$	1008.a	1.a	$1$	$1$	$1$	$59.474$	\(\Q\)	None	✓				21.4.a.a	$1$	$0$	\(0\)	\(0\)	\(18\)	\(-7\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+18q^{5}-7q^{7}-6^{2}q^{11}-34q^{13}+\cdots\)
1008.4.a.w	$1008$	$4$	1008.a	1.a	$1$	$1$	$1$	$59.474$	\(\Q\)	None	✓				126.4.a.e	$1$	$0$	\(0\)	\(0\)	\(22\)	\(7\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+22q^{5}+7q^{7}-26q^{11}-54q^{13}+\cdots\)
1008.4.a.x	$1008$	$4$	1008.a	1.a	$1$	$2$	$2$	$59.474$	\(\Q(\sqrt{57}) \)	None	✓				56.4.a.c	$1$	$1$	\(0\)	\(0\)	\(-22\)	\(-14\)		$+$	$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(-11-\beta )q^{5}-7q^{7}+(18+6\beta )q^{11}+\cdots\)
1008.4.a.y	$1008$	$4$	1008.a	1.a	$1$	$2$	$2$	$59.474$	\(\Q(\sqrt{177}) \)	None	✓				168.4.a.h	$1$	$1$	\(0\)	\(0\)	\(-14\)	\(-14\)		$+$	$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(-7-\beta )q^{5}-7q^{7}+(9-3\beta )q^{11}+\cdots\)
1008.4.a.z	$1008$	$4$	1008.a	1.a	$1$	$2$	$2$	$59.474$	\(\Q(\sqrt{22}) \)	None	✓				504.4.a.k	$1$	$1$	\(0\)	\(0\)	\(-12\)	\(14\)		$+$	$+$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(-6+\beta )q^{5}+7q^{7}+(2+\beta )q^{11}+\cdots\)
1008.4.a.ba	$1008$	$4$	1008.a	1.a	$1$	$2$	$2$	$59.474$	\(\Q(\sqrt{57}) \)	None	✓		✓		21.4.a.c	$1$	$0$	\(0\)	\(0\)	\(-6\)	\(-14\)		$-$	$-$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q+(-3-\beta )q^{5}-7q^{7}+(-3+5\beta )q^{11}+\cdots\)
1008.4.a.bb	$1008$	$4$	1008.a	1.a	$1$	$2$	$2$	$59.474$	\(\Q(\sqrt{30}) \)	None	✓				504.4.a.l	$1$	$0$	\(0\)	\(0\)	\(-4\)	\(-14\)		$+$	$+$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q+(-2+\beta )q^{5}-7q^{7}+(18+3\beta )q^{11}+\cdots\)
1008.4.a.bc	$1008$	$4$	1008.a	1.a	$1$	$2$	$2$	$59.474$	\(\Q(\sqrt{7}) \)	None	✓		✓		252.4.a.f	$2$	$1$	\(0\)	\(0\)	\(0\)	\(-14\)		$-$	$+$	$+$	$-$	$2\cdot 3$	$\mathrm{SU}(2)$	\(q+\beta q^{5}-7q^{7}-\beta q^{11}+26q^{13}-5\beta q^{17}+\cdots\)
1008.4.a.bd	$1008$	$4$	1008.a	1.a	$1$	$2$	$2$	$59.474$	\(\Q(\sqrt{7}) \)	None	✓				252.4.a.e	$2$	$0$	\(0\)	\(0\)	\(0\)	\(14\)		$-$	$+$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q+\beta q^{5}+7q^{7}+13\beta q^{11}-30q^{13}+\cdots\)
1008.4.a.be	$1008$	$4$	1008.a	1.a	$1$	$2$	$2$	$59.474$	\(\Q(\sqrt{19}) \)	None	✓				63.4.a.d	$2$	$0$	\(0\)	\(0\)	\(0\)	\(14\)		$-$	$+$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q+\beta q^{5}+7q^{7}+5\beta q^{11}+82q^{13}+\cdots\)
1008.4.a.bf	$1008$	$4$	1008.a	1.a	$1$	$2$	$2$	$59.474$	\(\Q(\sqrt{30}) \)	None	✓				504.4.a.l	$1$	$0$	\(0\)	\(0\)	\(4\)	\(-14\)		$+$	$+$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q+(2+\beta )q^{5}-7q^{7}+(-18+3\beta )q^{11}+\cdots\)
1008.4.a.bg	$1008$	$4$	1008.a	1.a	$1$	$2$	$2$	$59.474$	\(\Q(\sqrt{337}) \)	None	✓		✓		168.4.a.g	$1$	$0$	\(0\)	\(0\)	\(6\)	\(14\)		$+$	$-$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q+(3+\beta )q^{5}+7q^{7}+(13+\beta )q^{11}+(48+\cdots)q^{13}+\cdots\)
1008.4.a.bh	$1008$	$4$	1008.a	1.a	$1$	$2$	$2$	$59.474$	\(\Q(\sqrt{22}) \)	None	✓				504.4.a.k	$1$	$1$	\(0\)	\(0\)	\(12\)	\(14\)		$+$	$+$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(6+\beta )q^{5}+7q^{7}+(-2+\beta )q^{11}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(1008))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(1008)) \simeq \) \(S_{4}^{\mathrm{new}}(\Gamma_0(6))\)\(^{\oplus 16}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(7))\)\(^{\oplus 15}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(8))\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(9))\)\(^{\oplus 10}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(12))\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(16))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(18))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(21))\)\(^{\oplus 10}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(24))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(28))\)\(^{\oplus 9}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(36))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(42))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(48))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(56))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(63))\)\(^{\oplus 5}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(72))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(84))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(112))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(126))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(144))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(168))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(252))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(336))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(504))\)\(^{\oplus 2}\)