Space of modular forms of level 888, weight 2, and trivial character

• Label: 888.2.a
• Level: $888$
• Weight: $2$
• Character orbit: 888.a
• Rep. character: $\chi_{888}(1,\cdot)$
• Character field: $\Q$
• Dimension: $18$
• Newform subspaces: $10$
• Sturm bound: $304$
• Trace bound: $7$

Defining parameters

Level:	\( N \)	\(=\)	\( 888 = 2^{3} \cdot 3 \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	888.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 10 \)
Sturm bound:			\(304\)
Trace bound:			\(7\)
Distinguishing \(T_p\):			\(5\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(888))\).

	Total	New	Old
Modular forms	160	18	142
Cusp forms	145	18	127
Eisenstein series	15	0	15

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

\(2\)	\(3\)	\(37\)	Fricke		Total				Cusp				Eisenstein
					All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)	\(+\)		\(12\)	\(2\)	\(10\)		\(11\)	\(2\)	\(9\)		\(1\)	\(0\)	\(1\)
\(+\)	\(+\)	\(-\)	\(-\)		\(28\)	\(3\)	\(25\)		\(26\)	\(3\)	\(23\)		\(2\)	\(0\)	\(2\)
\(+\)	\(-\)	\(+\)	\(-\)		\(23\)	\(3\)	\(20\)		\(21\)	\(3\)	\(18\)		\(2\)	\(0\)	\(2\)
\(+\)	\(-\)	\(-\)	\(+\)		\(17\)	\(1\)	\(16\)		\(15\)	\(1\)	\(14\)		\(2\)	\(0\)	\(2\)
\(-\)	\(+\)	\(+\)	\(-\)		\(20\)	\(3\)	\(17\)		\(18\)	\(3\)	\(15\)		\(2\)	\(0\)	\(2\)
\(-\)	\(+\)	\(-\)	\(+\)		\(20\)	\(1\)	\(19\)		\(18\)	\(1\)	\(17\)		\(2\)	\(0\)	\(2\)
\(-\)	\(-\)	\(+\)	\(+\)		\(25\)	\(2\)	\(23\)		\(23\)	\(2\)	\(21\)		\(2\)	\(0\)	\(2\)
\(-\)	\(-\)	\(-\)	\(-\)		\(15\)	\(3\)	\(12\)		\(13\)	\(3\)	\(10\)		\(2\)	\(0\)	\(2\)
Plus space			\(+\)		\(74\)	\(6\)	\(68\)		\(67\)	\(6\)	\(61\)		\(7\)	\(0\)	\(7\)
Minus space			\(-\)		\(86\)	\(12\)	\(74\)		\(78\)	\(12\)	\(66\)		\(8\)	\(0\)	\(8\)

Trace form

18 * q - 4 * q^7 + 18 * q^9 - 4 * q^13 - 4 * q^15 - 8 * q^17 + 12 * q^19 + 16 * q^23 + 30 * q^25 - 8 * q^29 - 12 * q^31 + 12 * q^33 + 24 * q^35 - 2 * q^37 - 4 * q^41 + 4 * q^43 + 16 * q^47 + 22 * q^49 + 4 * q^51 + 20 * q^53 + 40 * q^55 - 4 * q^57 + 24 * q^59 + 4 * q^61 - 4 * q^63 - 24 * q^65 - 8 * q^69 - 8 * q^71 - 16 * q^73 - 16 * q^75 + 16 * q^77 + 20 * q^79 + 18 * q^81 + 8 * q^83 + 8 * q^85 + 20 * q^87 + 24 * q^89 - 16 * q^91 - 12 * q^93 + 16 * q^95 - 4 * q^97

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(888))\) 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	37
888.2.a.a	$888$	$2$	888.a	1.a	$1$	$1$	$1$	$7.091$	\(\Q\)	None	✓	✓			888.2.a.a	$1$	$0$	\(0\)	\(-1\)	\(-4\)	\(-1\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-4q^{5}-q^{7}+q^{9}-3q^{11}-5q^{13}+\cdots\)
888.2.a.b	$888$	$2$	888.a	1.a	$1$	$1$	$1$	$7.091$	\(\Q\)	None	✓	✓	✓	✓	888.2.a.b	$1$	$1$	\(0\)	\(-1\)	\(0\)	\(0\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+q^{9}-6q^{13}-4q^{17}+4q^{19}+\cdots\)
888.2.a.c	$888$	$2$	888.a	1.a	$1$	$1$	$1$	$7.091$	\(\Q\)	None	✓	✓	✓	✓	888.2.a.c	$1$	$1$	\(0\)	\(1\)	\(-2\)	\(0\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-2q^{5}+q^{9}-4q^{11}-2q^{13}+\cdots\)
888.2.a.d	$888$	$2$	888.a	1.a	$1$	$1$	$1$	$7.091$	\(\Q\)	None	✓	✓			888.2.a.d	$1$	$0$	\(0\)	\(1\)	\(4\)	\(0\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+4q^{5}+q^{9}+2q^{13}+4q^{15}+\cdots\)
888.2.a.e	$888$	$2$	888.a	1.a	$1$	$2$	$2$	$7.091$	\(\Q(\sqrt{3}) \)	None	✓	✓	✓	✓	888.2.a.e	$1$	$1$	\(0\)	\(-2\)	\(2\)	\(-4\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+(1+\beta )q^{5}-2q^{7}+q^{9}+(-2+\cdots)q^{11}+\cdots\)
888.2.a.f	$888$	$2$	888.a	1.a	$1$	$2$	$2$	$7.091$	\(\Q(\sqrt{5}) \)	None	✓	✓	✓		888.2.a.f	$1$	$0$	\(0\)	\(-2\)	\(2\)	\(8\)		$+$	$+$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q-q^{3}+(1+\beta )q^{5}+4q^{7}+q^{9}+(2+2\beta )q^{11}+\cdots\)
888.2.a.g	$888$	$2$	888.a	1.a	$1$	$2$	$2$	$7.091$	\(\Q(\sqrt{2}) \)	None	✓	✓	✓	✓	888.2.a.g	$1$	$1$	\(0\)	\(2\)	\(-4\)	\(-4\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+(-2+\beta )q^{5}+(-2-2\beta )q^{7}+\cdots\)
888.2.a.h	$888$	$2$	888.a	1.a	$1$	$2$	$2$	$7.091$	\(\Q(\sqrt{41}) \)	None	✓	✓	✓		888.2.a.h	$1$	$0$	\(0\)	\(2\)	\(0\)	\(3\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+(1+\beta )q^{7}+q^{9}+(1+\beta )q^{11}+\cdots\)
888.2.a.i	$888$	$2$	888.a	1.a	$1$	$3$	$3$	$7.091$	3.3.316.1	None	✓	✓	✓	✓	888.2.a.i	$1$	$0$	\(0\)	\(-3\)	\(2\)	\(-5\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+(1-\beta _{1}-\beta _{2})q^{5}+(-1-2\beta _{1}+\cdots)q^{7}+\cdots\)
888.2.a.j	$888$	$2$	888.a	1.a	$1$	$3$	$3$	$7.091$	3.3.568.1	None	✓	✓	✓	✓	888.2.a.j	$1$	$0$	\(0\)	\(3\)	\(0\)	\(-1\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+(\beta _{1}+\beta _{2})q^{5}+\beta _{2}q^{7}+q^{9}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(888))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(888)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(24))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(37))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(74))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(111))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(148))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(222))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(296))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(444))\)\(^{\oplus 2}\)