Space of modular forms of level 88, weight 4, and trivial character

• Label: 88.4.a
• Level: $88$
• Weight: $4$
• Character orbit: 88.a
• Rep. character: $\chi_{88}(1,\cdot)$
• Character field: $\Q$
• Dimension: $7$
• Newform subspaces: $4$
• Sturm bound: $48$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 88 = 2^{3} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	88.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 4 \)
Sturm bound:			\(48\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(88))\).

	Total	New	Old
Modular forms	40	7	33
Cusp forms	32	7	25
Eisenstein series	8	0	8

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

\(2\)	\(11\)	Fricke		Total				Cusp				Eisenstein
				All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)		\(11\)	\(1\)	\(10\)		\(9\)	\(1\)	\(8\)		\(2\)	\(0\)	\(2\)
\(+\)	\(-\)	\(-\)		\(10\)	\(2\)	\(8\)		\(8\)	\(2\)	\(6\)		\(2\)	\(0\)	\(2\)
\(-\)	\(+\)	\(-\)		\(9\)	\(1\)	\(8\)		\(7\)	\(1\)	\(6\)		\(2\)	\(0\)	\(2\)
\(-\)	\(-\)	\(+\)		\(10\)	\(3\)	\(7\)		\(8\)	\(3\)	\(5\)		\(2\)	\(0\)	\(2\)
Plus space		\(+\)		\(21\)	\(4\)	\(17\)		\(17\)	\(4\)	\(13\)		\(4\)	\(0\)	\(4\)
Minus space		\(-\)		\(19\)	\(3\)	\(16\)		\(15\)	\(3\)	\(12\)		\(4\)	\(0\)	\(4\)

Trace form

7 * q + 6 * q^3 + 4 * q^5 - 36 * q^7 + 73 * q^9 + 33 * q^11 - 18 * q^13 - 94 * q^15 + 86 * q^17 - 180 * q^19 + 316 * q^21 + 66 * q^23 + 287 * q^25 - 282 * q^27 - 362 * q^29 + 250 * q^31 - 66 * q^33 - 236 * q^35 + 192 * q^37 - 352 * q^39 + 250 * q^41 - 704 * q^43 - 54 * q^45 - 608 * q^47 - 9 * q^49 + 468 * q^51 - 934 * q^53 + 864 * q^57 + 1170 * q^59 - 674 * q^61 + 1904 * q^63 + 668 * q^65 - 718 * q^67 - 1274 * q^69 + 374 * q^71 + 290 * q^73 + 1500 * q^75 - 308 * q^77 + 820 * q^79 + 623 * q^81 + 568 * q^83 - 3360 * q^85 - 760 * q^87 + 1540 * q^89 - 320 * q^91 - 3994 * q^93 + 824 * q^95 - 1672 * q^97 + 891 * q^99

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(88))\) 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	11
88.4.a.a	$88$	$4$	88.a	1.a	$1$	$1$	$1$	$5.192$	\(\Q\)	None	✓	✓	✓	✓	88.4.a.a	$1$	$1$	\(0\)	\(-1\)	\(-7\)	\(-6\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-7q^{5}-6q^{7}-26q^{9}-11q^{11}+\cdots\)
88.4.a.b	$88$	$4$	88.a	1.a	$1$	$1$	$1$	$5.192$	\(\Q\)	None	✓	✓	✓	✓	88.4.a.b	$1$	$0$	\(0\)	\(7\)	\(9\)	\(2\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+7q^{3}+9q^{5}+2q^{7}+22q^{9}-11q^{11}+\cdots\)
88.4.a.c	$88$	$4$	88.a	1.a	$1$	$2$	$2$	$5.192$	\(\Q(\sqrt{5}) \)	None	✓	✓	✓	✓	88.4.a.c	$1$	$1$	\(0\)	\(-2\)	\(-6\)	\(-56\)		$+$	$-$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q+(-1-\beta )q^{3}+(-3+4\beta )q^{5}+(-28+\cdots)q^{7}+\cdots\)
88.4.a.d	$88$	$4$	88.a	1.a	$1$	$3$	$3$	$5.192$	3.3.11109.1	None	✓	✓	✓	✓	88.4.a.d	$1$	$0$	\(0\)	\(2\)	\(8\)	\(24\)		$-$	$-$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{3}+(3+\beta _{2})q^{5}+(8-\beta _{1}+\cdots)q^{7}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(88))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(88)) \simeq \) \(S_{4}^{\mathrm{new}}(\Gamma_0(8))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(11))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(22))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(44))\)\(^{\oplus 2}\)