Space of modular forms of level 1088, weight 2, and trivial character

• Label: 1088.2.a
• Level: $1088$
• Weight: $2$
• Character orbit: 1088.a
• Rep. character: $\chi_{1088}(1,\cdot)$
• Character field: $\Q$
• Dimension: $32$
• Newform subspaces: $22$
• Sturm bound: $288$
• Trace bound: $9$

Defining parameters

Level:	\( N \)	\(=\)	\( 1088 = 2^{6} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1088.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 22 \)
Sturm bound:			\(288\)
Trace bound:			\(9\)
Distinguishing \(T_p\):			\(3\), \(5\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(1088))\).

	Total	New	Old
Modular forms	156	32	124
Cusp forms	133	32	101
Eisenstein series	23	0	23

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

\(2\)	\(17\)	Fricke		Total				Cusp				Eisenstein
				All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)		\(34\)	\(6\)	\(28\)		\(29\)	\(6\)	\(23\)		\(5\)	\(0\)	\(5\)
\(+\)	\(-\)	\(-\)		\(42\)	\(10\)	\(32\)		\(36\)	\(10\)	\(26\)		\(6\)	\(0\)	\(6\)
\(-\)	\(+\)	\(-\)		\(44\)	\(10\)	\(34\)		\(38\)	\(10\)	\(28\)		\(6\)	\(0\)	\(6\)
\(-\)	\(-\)	\(+\)		\(36\)	\(6\)	\(30\)		\(30\)	\(6\)	\(24\)		\(6\)	\(0\)	\(6\)
Plus space		\(+\)		\(70\)	\(12\)	\(58\)		\(59\)	\(12\)	\(47\)		\(11\)	\(0\)	\(11\)
Minus space		\(-\)		\(86\)	\(20\)	\(66\)		\(74\)	\(20\)	\(54\)		\(12\)	\(0\)	\(12\)

Trace form

\( 32 q + 32 q^{9} + 16 q^{13} + 16 q^{21} + 32 q^{25} - 16 q^{29} + 48 q^{45} + 32 q^{49} + 32 q^{53} + 16 q^{61} + 48 q^{69} - 16 q^{77} - 32 q^{89} + 16 q^{93} - 32 q^{97}+O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(1088))\) 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	17
1088.2.a.a	$1088$	$2$	1088.a	1.a	$1$	$1$	$1$	$8.688$	\(\Q\)	None	✓				544.2.a.b	$1$	$2$	\(0\)	\(-2\)	\(-4\)	\(-4\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{3}-4q^{5}-4q^{7}+q^{9}-2q^{11}+\cdots\)
1088.2.a.b	$1088$	$2$	1088.a	1.a	$1$	$1$	$1$	$8.688$	\(\Q\)	None	✓				544.2.a.a	$1$	$1$	\(0\)	\(-2\)	\(-2\)	\(2\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{3}-2q^{5}+2q^{7}+q^{9}+2q^{11}+\cdots\)
1088.2.a.c	$1088$	$2$	1088.a	1.a	$1$	$1$	$1$	$8.688$	\(\Q\)	None	✓				136.2.a.b	$1$	$1$	\(0\)	\(-2\)	\(0\)	\(0\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{3}+q^{9}-2q^{11}+6q^{13}-q^{17}+\cdots\)
1088.2.a.d	$1088$	$2$	1088.a	1.a	$1$	$1$	$1$	$8.688$	\(\Q\)	None	✓				34.2.a.a	$1$	$0$	\(0\)	\(-2\)	\(0\)	\(4\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{3}+4q^{7}+q^{9}+6q^{11}-2q^{13}+\cdots\)
1088.2.a.e	$1088$	$2$	1088.a	1.a	$1$	$1$	$1$	$8.688$	\(\Q\)	None	✓				136.2.a.a	$1$	$1$	\(0\)	\(-2\)	\(2\)	\(2\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{3}+2q^{5}+2q^{7}+q^{9}-6q^{11}+\cdots\)
1088.2.a.f	$1088$	$2$	1088.a	1.a	$1$	$1$	$1$	$8.688$	\(\Q\)	None	✓				544.2.a.c	$1$	$1$	\(0\)	\(0\)	\(0\)	\(-2\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{7}-3q^{9}+4q^{11}-2q^{13}-q^{17}+\cdots\)
1088.2.a.g	$1088$	$2$	1088.a	1.a	$1$	$1$	$1$	$8.688$	\(\Q\)	None	✓				544.2.a.c	$1$	$1$	\(0\)	\(0\)	\(0\)	\(2\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{7}-3q^{9}-4q^{11}-2q^{13}-q^{17}+\cdots\)
1088.2.a.h	$1088$	$2$	1088.a	1.a	$1$	$1$	$1$	$8.688$	\(\Q\)	None	✓				17.2.a.a	$1$	$1$	\(0\)	\(0\)	\(2\)	\(-4\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{5}-4q^{7}-3q^{9}+2q^{13}+q^{17}+\cdots\)
1088.2.a.i	$1088$	$2$	1088.a	1.a	$1$	$1$	$1$	$8.688$	\(\Q\)	None	✓				17.2.a.a	$1$	$0$	\(0\)	\(0\)	\(2\)	\(4\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{5}+4q^{7}-3q^{9}+2q^{13}+q^{17}+\cdots\)
1088.2.a.j	$1088$	$2$	1088.a	1.a	$1$	$1$	$1$	$8.688$	\(\Q\)	None	✓				544.2.a.b	$1$	$0$	\(0\)	\(2\)	\(-4\)	\(4\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{3}-4q^{5}+4q^{7}+q^{9}+2q^{11}+\cdots\)
1088.2.a.k	$1088$	$2$	1088.a	1.a	$1$	$1$	$1$	$8.688$	\(\Q\)	None	✓				544.2.a.a	$1$	$1$	\(0\)	\(2\)	\(-2\)	\(-2\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{3}-2q^{5}-2q^{7}+q^{9}-2q^{11}+\cdots\)
1088.2.a.l	$1088$	$2$	1088.a	1.a	$1$	$1$	$1$	$8.688$	\(\Q\)	None	✓				34.2.a.a	$1$	$1$	\(0\)	\(2\)	\(0\)	\(-4\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{3}-4q^{7}+q^{9}-6q^{11}-2q^{13}+\cdots\)
1088.2.a.m	$1088$	$2$	1088.a	1.a	$1$	$1$	$1$	$8.688$	\(\Q\)	None	✓				136.2.a.b	$1$	$0$	\(0\)	\(2\)	\(0\)	\(0\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{3}+q^{9}+2q^{11}+6q^{13}-q^{17}+\cdots\)
1088.2.a.n	$1088$	$2$	1088.a	1.a	$1$	$1$	$1$	$8.688$	\(\Q\)	None	✓				136.2.a.a	$1$	$0$	\(0\)	\(2\)	\(2\)	\(-2\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{3}+2q^{5}-2q^{7}+q^{9}+6q^{11}+\cdots\)
1088.2.a.o	$1088$	$2$	1088.a	1.a	$1$	$2$	$2$	$8.688$	\(\Q(\sqrt{5}) \)	None	✓		✓		136.2.a.c	$1$	$1$	\(0\)	\(-2\)	\(-4\)	\(-2\)		$-$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q+(-1-\beta )q^{3}-2q^{5}+(-1-\beta )q^{7}+\cdots\)
1088.2.a.p	$1088$	$2$	1088.a	1.a	$1$	$2$	$2$	$8.688$	\(\Q(\sqrt{3}) \)	None	✓		✓		68.2.a.a	$1$	$1$	\(0\)	\(-2\)	\(0\)	\(-2\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta )q^{3}-2\beta q^{5}+(-1+\beta )q^{7}+\cdots\)
1088.2.a.q	$1088$	$2$	1088.a	1.a	$1$	$2$	$2$	$8.688$	\(\Q(\sqrt{2}) \)	None	✓				544.2.a.g	$2$	$0$	\(0\)	\(0\)	\(4\)	\(0\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{3}+2q^{5}+3\beta q^{7}-q^{9}+\beta q^{11}+\cdots\)
1088.2.a.r	$1088$	$2$	1088.a	1.a	$1$	$2$	$2$	$8.688$	\(\Q(\sqrt{10}) \)	None	✓				544.2.a.h	$2$	$0$	\(0\)	\(0\)	\(4\)	\(0\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{3}+2q^{5}-\beta q^{7}+7q^{9}+\beta q^{11}+\cdots\)
1088.2.a.s	$1088$	$2$	1088.a	1.a	$1$	$2$	$2$	$8.688$	\(\Q(\sqrt{5}) \)	None	✓				136.2.a.c	$1$	$0$	\(0\)	\(2\)	\(-4\)	\(2\)		$+$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{3}-2q^{5}+(1+\beta )q^{7}+(3+2\beta )q^{9}+\cdots\)
1088.2.a.t	$1088$	$2$	1088.a	1.a	$1$	$2$	$2$	$8.688$	\(\Q(\sqrt{3}) \)	None	✓				68.2.a.a	$1$	$0$	\(0\)	\(2\)	\(0\)	\(2\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{3}+2\beta q^{5}+(1+\beta )q^{7}+(1+\cdots)q^{9}+\cdots\)
1088.2.a.u	$1088$	$2$	1088.a	1.a	$1$	$3$	$3$	$8.688$	3.3.148.1	None	✓				544.2.a.i	$1$	$0$	\(0\)	\(-2\)	\(2\)	\(2\)		$+$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{3}+(\beta _{1}+\beta _{2})q^{5}+(1-\beta _{2})q^{7}+\cdots\)
1088.2.a.v	$1088$	$2$	1088.a	1.a	$1$	$3$	$3$	$8.688$	3.3.148.1	None	✓				544.2.a.i	$1$	$0$	\(0\)	\(2\)	\(2\)	\(-2\)		$+$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{3}+(\beta _{1}+\beta _{2})q^{5}+(-1+\beta _{2})q^{7}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(1088))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(1088)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(17))\)\(^{\oplus 7}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(32))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(34))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(64))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(68))\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(136))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(272))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(544))\)\(^{\oplus 2}\)