Space of modular forms of level 882, weight 4, and trivial character

• Label: 882.4.a
• Level: $882$
• Weight: $4$
• Character orbit: 882.a
• Rep. character: $\chi_{882}(1,\cdot)$
• Character field: $\Q$
• Dimension: $51$
• Newform subspaces: $35$
• Sturm bound: $672$
• Trace bound: $13$

Defining parameters

Level:	\( N \)	\(=\)	\( 882 = 2 \cdot 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	882.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 35 \)
Sturm bound:			\(672\)
Trace bound:			\(13\)
Distinguishing \(T_p\):			\(5\), \(11\), \(13\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(882))\).

	Total	New	Old
Modular forms	536	51	485
Cusp forms	472	51	421
Eisenstein series	64	0	64

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
\(+\)	\(+\)	\(+\)	\(+\)		\(70\)	\(4\)	\(66\)		\(62\)	\(4\)	\(58\)		\(8\)	\(0\)	\(8\)
\(+\)	\(+\)	\(-\)	\(-\)		\(64\)	\(6\)	\(58\)		\(56\)	\(6\)	\(50\)		\(8\)	\(0\)	\(8\)
\(+\)	\(-\)	\(+\)	\(-\)		\(66\)	\(7\)	\(59\)		\(58\)	\(7\)	\(51\)		\(8\)	\(0\)	\(8\)
\(+\)	\(-\)	\(-\)	\(+\)		\(68\)	\(8\)	\(60\)		\(60\)	\(8\)	\(52\)		\(8\)	\(0\)	\(8\)
\(-\)	\(+\)	\(+\)	\(-\)		\(66\)	\(4\)	\(62\)		\(58\)	\(4\)	\(54\)		\(8\)	\(0\)	\(8\)
\(-\)	\(+\)	\(-\)	\(+\)		\(68\)	\(6\)	\(62\)		\(60\)	\(6\)	\(54\)		\(8\)	\(0\)	\(8\)
\(-\)	\(-\)	\(+\)	\(+\)		\(66\)	\(9\)	\(57\)		\(58\)	\(9\)	\(49\)		\(8\)	\(0\)	\(8\)
\(-\)	\(-\)	\(-\)	\(-\)		\(68\)	\(7\)	\(61\)		\(60\)	\(7\)	\(53\)		\(8\)	\(0\)	\(8\)
Plus space			\(+\)		\(272\)	\(27\)	\(245\)		\(240\)	\(27\)	\(213\)		\(32\)	\(0\)	\(32\)
Minus space			\(-\)		\(264\)	\(24\)	\(240\)		\(232\)	\(24\)	\(208\)		\(32\)	\(0\)	\(32\)

Trace form

51 * q + 2 * q^2 + 204 * q^4 + 8 * q^8 + 32 * q^10 - 56 * q^11 + 68 * q^13 + 816 * q^16 - 162 * q^17 - 198 * q^19 - 56 * q^22 - 28 * q^23 + 909 * q^25 - 152 * q^26 + 530 * q^29 + 692 * q^31 + 32 * q^32 + 148 * q^34 + 1170 * q^37 - 260 * q^38 + 128 * q^40 + 798 * q^41 + 4 * q^43 - 224 * q^44 - 32 * q^46 - 180 * q^47 - 314 * q^50 + 272 * q^52 - 182 * q^53 - 80 * q^55 + 396 * q^58 - 1986 * q^59 + 1112 * q^61 + 472 * q^62 + 3264 * q^64 - 792 * q^65 + 816 * q^67 - 648 * q^68 + 2568 * q^71 + 1342 * q^73 + 1956 * q^74 - 792 * q^76 - 956 * q^79 + 1236 * q^82 + 78 * q^83 - 1436 * q^85 + 616 * q^86 - 224 * q^88 + 1650 * q^89 - 112 * q^92 - 792 * q^94 - 764 * q^95 - 902 * q^97

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(882))\) 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
882.4.a.a	$882$	$4$	882.a	1.a	$1$	$1$	$1$	$52.040$	\(\Q\)	None	✓				126.4.a.e	$1$	$1$	\(-2\)	\(0\)	\(-22\)	\(0\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+4q^{4}-22q^{5}-8q^{8}+44q^{10}+\cdots\)
882.4.a.b	$882$	$4$	882.a	1.a	$1$	$1$	$1$	$52.040$	\(\Q\)	None	✓				14.4.a.b	$1$	$0$	\(-2\)	\(0\)	\(-12\)	\(0\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+4q^{4}-12q^{5}-8q^{8}+24q^{10}+\cdots\)
882.4.a.c	$882$	$4$	882.a	1.a	$1$	$1$	$1$	$52.040$	\(\Q\)	None	✓				14.4.c.a	$1$	$0$	\(-2\)	\(0\)	\(-9\)	\(0\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+4q^{4}-9q^{5}-8q^{8}+18q^{10}+\cdots\)
882.4.a.d	$882$	$4$	882.a	1.a	$1$	$1$	$1$	$52.040$	\(\Q\)	None	✓				42.4.a.b	$1$	$0$	\(-2\)	\(0\)	\(2\)	\(0\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+4q^{4}+2q^{5}-8q^{8}-4q^{10}+\cdots\)
882.4.a.e	$882$	$4$	882.a	1.a	$1$	$1$	$1$	$52.040$	\(\Q\)	None	✓				126.4.a.b	$1$	$1$	\(-2\)	\(0\)	\(6\)	\(0\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+4q^{4}+6q^{5}-8q^{8}-12q^{10}+\cdots\)
882.4.a.f	$882$	$4$	882.a	1.a	$1$	$1$	$1$	$52.040$	\(\Q\)	None	✓				14.4.c.a	$1$	$1$	\(-2\)	\(0\)	\(9\)	\(0\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+4q^{4}+9q^{5}-8q^{8}-18q^{10}+\cdots\)
882.4.a.g	$882$	$4$	882.a	1.a	$1$	$1$	$1$	$52.040$	\(\Q\)	None	✓				42.4.a.a	$1$	$0$	\(-2\)	\(0\)	\(18\)	\(0\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+4q^{4}+18q^{5}-8q^{8}-6^{2}q^{10}+\cdots\)
882.4.a.h	$882$	$4$	882.a	1.a	$1$	$1$	$1$	$52.040$	\(\Q\)	None	✓				42.4.e.b	$1$	$1$	\(2\)	\(0\)	\(-15\)	\(0\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+4q^{4}-15q^{5}+8q^{8}-30q^{10}+\cdots\)
882.4.a.i	$882$	$4$	882.a	1.a	$1$	$1$	$1$	$52.040$	\(\Q\)	None	✓				14.4.a.a	$1$	$1$	\(2\)	\(0\)	\(-14\)	\(0\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+4q^{4}-14q^{5}+8q^{8}-28q^{10}+\cdots\)
882.4.a.j	$882$	$4$	882.a	1.a	$1$	$1$	$1$	$52.040$	\(\Q\)	None	✓				294.4.a.b	$1$	$1$	\(2\)	\(0\)	\(-8\)	\(0\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+4q^{4}-8q^{5}+8q^{8}-2^{4}q^{10}+\cdots\)
882.4.a.k	$882$	$4$	882.a	1.a	$1$	$1$	$1$	$52.040$	\(\Q\)	None	✓				14.4.c.b	$1$	$0$	\(2\)	\(0\)	\(-7\)	\(0\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+4q^{4}-7q^{5}+8q^{8}-14q^{10}+\cdots\)
882.4.a.l	$882$	$4$	882.a	1.a	$1$	$1$	$1$	$52.040$	\(\Q\)	None	✓				42.4.e.a	$1$	$1$	\(2\)	\(0\)	\(-6\)	\(0\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+4q^{4}-6q^{5}+8q^{8}-12q^{10}+\cdots\)
882.4.a.m	$882$	$4$	882.a	1.a	$1$	$1$	$1$	$52.040$	\(\Q\)	None	✓				126.4.a.b	$1$	$0$	\(2\)	\(0\)	\(-6\)	\(0\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+4q^{4}-6q^{5}+8q^{8}-12q^{10}+\cdots\)
882.4.a.n	$882$	$4$	882.a	1.a	$1$	$1$	$1$	$52.040$	\(\Q\)	None	✓				6.4.a.a	$1$	$1$	\(2\)	\(0\)	\(6\)	\(0\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+4q^{4}+6q^{5}+8q^{8}+12q^{10}+\cdots\)
882.4.a.o	$882$	$4$	882.a	1.a	$1$	$1$	$1$	$52.040$	\(\Q\)	None	✓				42.4.e.a	$1$	$0$	\(2\)	\(0\)	\(6\)	\(0\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+4q^{4}+6q^{5}+8q^{8}+12q^{10}+\cdots\)
882.4.a.p	$882$	$4$	882.a	1.a	$1$	$1$	$1$	$52.040$	\(\Q\)	None	✓				14.4.c.b	$1$	$1$	\(2\)	\(0\)	\(7\)	\(0\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+4q^{4}+7q^{5}+8q^{8}+14q^{10}+\cdots\)
882.4.a.q	$882$	$4$	882.a	1.a	$1$	$1$	$1$	$52.040$	\(\Q\)	None	✓				294.4.a.b	$1$	$1$	\(2\)	\(0\)	\(8\)	\(0\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+4q^{4}+8q^{5}+8q^{8}+2^{4}q^{10}+\cdots\)
882.4.a.r	$882$	$4$	882.a	1.a	$1$	$1$	$1$	$52.040$	\(\Q\)	None	✓				42.4.e.b	$1$	$0$	\(2\)	\(0\)	\(15\)	\(0\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+4q^{4}+15q^{5}+8q^{8}+30q^{10}+\cdots\)
882.4.a.s	$882$	$4$	882.a	1.a	$1$	$1$	$1$	$52.040$	\(\Q\)	None	✓				126.4.a.e	$1$	$0$	\(2\)	\(0\)	\(22\)	\(0\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+4q^{4}+22q^{5}+8q^{8}+44q^{10}+\cdots\)
882.4.a.t	$882$	$4$	882.a	1.a	$1$	$2$	$2$	$52.040$	\(\Q(\sqrt{2}) \)	None	✓				294.4.a.l	$1$	$1$	\(-4\)	\(0\)	\(-12\)	\(0\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+4q^{4}+(-6+\beta )q^{5}-8q^{8}+\cdots\)
882.4.a.u	$882$	$4$	882.a	1.a	$1$	$2$	$2$	$52.040$	\(\Q(\sqrt{193}) \)	None	✓				126.4.g.e	$1$	$1$	\(-4\)	\(0\)	\(-7\)	\(0\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+4q^{4}+(-3-\beta )q^{5}-8q^{8}+\cdots\)
882.4.a.v	$882$	$4$	882.a	1.a	$1$	$2$	$2$	$52.040$	\(\Q(\sqrt{1345}) \)	None	✓				42.4.e.c	$1$	$1$	\(-4\)	\(0\)	\(-5\)	\(0\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+4q^{4}+(-2-\beta )q^{5}-8q^{8}+\cdots\)
882.4.a.w	$882$	$4$	882.a	1.a	$1$	$2$	$2$	$52.040$	\(\Q(\sqrt{22}) \)	None	✓				98.4.a.h	$2$	$0$	\(-4\)	\(0\)	\(0\)	\(0\)		$+$	$-$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q-2q^{2}+4q^{4}+\beta q^{5}-8q^{8}-2\beta q^{10}+\cdots\)
882.4.a.x	$882$	$4$	882.a	1.a	$1$	$2$	$2$	$52.040$	\(\Q(\sqrt{58}) \)	None	✓	✓			882.4.a.x	$2$	$1$	\(-4\)	\(0\)	\(0\)	\(0\)		$+$	$+$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q-2q^{2}+4q^{4}+\beta q^{5}-8q^{8}-2\beta q^{10}+\cdots\)
882.4.a.y	$882$	$4$	882.a	1.a	$1$	$2$	$2$	$52.040$	\(\Q(\sqrt{2}) \)	None	✓	✓			882.4.a.y	$2$	$0$	\(-4\)	\(0\)	\(0\)	\(0\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+4q^{4}+5\beta q^{5}-8q^{8}-10\beta q^{10}+\cdots\)
882.4.a.z	$882$	$4$	882.a	1.a	$1$	$2$	$2$	$52.040$	\(\Q(\sqrt{1345}) \)	None	✓				42.4.e.c	$1$	$0$	\(-4\)	\(0\)	\(5\)	\(0\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+4q^{4}+(3-\beta )q^{5}-8q^{8}+(-6+\cdots)q^{10}+\cdots\)
882.4.a.ba	$882$	$4$	882.a	1.a	$1$	$2$	$2$	$52.040$	\(\Q(\sqrt{193}) \)	None	✓				126.4.g.e	$1$	$0$	\(-4\)	\(0\)	\(7\)	\(0\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+4q^{4}+(4-\beta )q^{5}-8q^{8}+(-8+\cdots)q^{10}+\cdots\)
882.4.a.bb	$882$	$4$	882.a	1.a	$1$	$2$	$2$	$52.040$	\(\Q(\sqrt{2}) \)	None	✓				294.4.a.l	$1$	$1$	\(-4\)	\(0\)	\(12\)	\(0\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+4q^{4}+(6+\beta )q^{5}-8q^{8}+(-12+\cdots)q^{10}+\cdots\)
882.4.a.bc	$882$	$4$	882.a	1.a	$1$	$2$	$2$	$52.040$	\(\Q(\sqrt{2}) \)	None	✓				294.4.a.j	$1$	$0$	\(4\)	\(0\)	\(-12\)	\(0\)		$-$	$-$	$+$	$+$	$7$	$\mathrm{SU}(2)$	\(q+2q^{2}+4q^{4}+(-6+\beta )q^{5}+8q^{8}+\cdots\)
882.4.a.bd	$882$	$4$	882.a	1.a	$1$	$2$	$2$	$52.040$	\(\Q(\sqrt{193}) \)	None	✓				126.4.g.e	$1$	$1$	\(4\)	\(0\)	\(-7\)	\(0\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+4q^{4}+(-3-\beta )q^{5}+8q^{8}+\cdots\)
882.4.a.be	$882$	$4$	882.a	1.a	$1$	$2$	$2$	$52.040$	\(\Q(\sqrt{2}) \)	None	✓	✓			882.4.a.y	$2$	$1$	\(4\)	\(0\)	\(0\)	\(0\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+4q^{4}+5\beta q^{5}+8q^{8}+10\beta q^{10}+\cdots\)
882.4.a.bf	$882$	$4$	882.a	1.a	$1$	$2$	$2$	$52.040$	\(\Q(\sqrt{58}) \)	None	✓	✓			882.4.a.x	$2$	$0$	\(4\)	\(0\)	\(0\)	\(0\)		$-$	$+$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q+2q^{2}+4q^{4}+\beta q^{5}+8q^{8}+2\beta q^{10}+\cdots\)
882.4.a.bg	$882$	$4$	882.a	1.a	$1$	$2$	$2$	$52.040$	\(\Q(\sqrt{2}) \)	None	✓				98.4.a.g	$2$	$0$	\(4\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+4q^{4}+14\beta q^{5}+8q^{8}+28\beta q^{10}+\cdots\)
882.4.a.bh	$882$	$4$	882.a	1.a	$1$	$2$	$2$	$52.040$	\(\Q(\sqrt{193}) \)	None	✓				126.4.g.e	$1$	$0$	\(4\)	\(0\)	\(7\)	\(0\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+4q^{4}+(4-\beta )q^{5}+8q^{8}+(8+\cdots)q^{10}+\cdots\)
882.4.a.bi	$882$	$4$	882.a	1.a	$1$	$2$	$2$	$52.040$	\(\Q(\sqrt{2}) \)	None	✓				294.4.a.j	$1$	$0$	\(4\)	\(0\)	\(12\)	\(0\)		$-$	$-$	$+$	$+$	$7$	$\mathrm{SU}(2)$	\(q+2q^{2}+4q^{4}+(6+\beta )q^{5}+8q^{8}+(12+\cdots)q^{10}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(882))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(882)) \simeq \) \(S_{4}^{\mathrm{new}}(\Gamma_0(6))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(7))\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(9))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(18))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(21))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(42))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(49))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(63))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(98))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(126))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(147))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(294))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(441))\)\(^{\oplus 2}\)