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}$