Space of modular forms of level 1088, weight 4, and trivial character

• Label: 1088.4.a
• Level: $1088$
• Weight: $4$
• Character orbit: 1088.a
• Rep. character: $\chi_{1088}(1,\cdot)$
• Character field: $\Q$
• Dimension: $96$
• Newform subspaces: $35$
• Sturm bound: $576$
• Trace bound: $7$

Defining parameters

Level:	\( N \)	\(=\)	\( 1088 = 2^{6} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1088.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 35 \)
Sturm bound:			\(576\)
Trace bound:			\(7\)
Distinguishing \(T_p\):			\(3\), \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(1088))\).

	Total	New	Old
Modular forms	444	96	348
Cusp forms	420	96	324
Eisenstein series	24	0	24

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	Dim
\(+\)	\(+\)	$+$	\(26\)
\(+\)	\(-\)	$-$	\(22\)
\(-\)	\(+\)	$-$	\(22\)
\(-\)	\(-\)	$+$	\(26\)
Plus space		\(+\)	\(52\)
Minus space		\(-\)	\(44\)

Trace form

\( 96 q + 864 q^{9} + O(q^{10}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(1088))\) into newform subspaces

Label	Level	Weight	Char	Prim	Char order	Dim	Rel. Dim	$A$	Field	CM	Self-dual	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
1088.4.a.a	$1088$	$4$	1088.a	1.a	$1$	$1$	$1$	$64.194$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(-8\)	\(-6\)	\(28\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-8q^{3}-6q^{5}+28q^{7}+37q^{9}-24q^{11}+\cdots\)
1088.4.a.b	$1088$	$4$	1088.a	1.a	$1$	$1$	$1$	$64.194$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(-6\)	\(-18\)	\(-2\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-6q^{3}-18q^{5}-2q^{7}+9q^{9}-26q^{11}+\cdots\)
1088.4.a.c	$1088$	$4$	1088.a	1.a	$1$	$1$	$1$	$64.194$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(-4\)	\(-8\)	\(-14\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-4q^{3}-8q^{5}-14q^{7}-11q^{9}-8q^{11}+\cdots\)
1088.4.a.d	$1088$	$4$	1088.a	1.a	$1$	$1$	$1$	$64.194$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(-2\)	\(-16\)	\(-24\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{3}-2^{4}q^{5}-24q^{7}-23q^{9}+62q^{11}+\cdots\)
1088.4.a.e	$1088$	$4$	1088.a	1.a	$1$	$1$	$1$	$64.194$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(-2\)	\(8\)	\(12\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{3}+8q^{5}+12q^{7}-23q^{9}-10q^{11}+\cdots\)
1088.4.a.f	$1088$	$4$	1088.a	1.a	$1$	$1$	$1$	$64.194$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(-2\)	\(18\)	\(10\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{3}+18q^{5}+10q^{7}-23q^{9}-6q^{11}+\cdots\)
1088.4.a.g	$1088$	$4$	1088.a	1.a	$1$	$1$	$1$	$64.194$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(2\)	\(-16\)	\(24\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{3}-2^{4}q^{5}+24q^{7}-23q^{9}-62q^{11}+\cdots\)
1088.4.a.h	$1088$	$4$	1088.a	1.a	$1$	$1$	$1$	$64.194$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(2\)	\(8\)	\(-12\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{3}+8q^{5}-12q^{7}-23q^{9}+10q^{11}+\cdots\)
1088.4.a.i	$1088$	$4$	1088.a	1.a	$1$	$1$	$1$	$64.194$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(2\)	\(18\)	\(-10\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{3}+18q^{5}-10q^{7}-23q^{9}+6q^{11}+\cdots\)
1088.4.a.j	$1088$	$4$	1088.a	1.a	$1$	$1$	$1$	$64.194$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(4\)	\(-8\)	\(14\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+4q^{3}-8q^{5}+14q^{7}-11q^{9}+8q^{11}+\cdots\)
1088.4.a.k	$1088$	$4$	1088.a	1.a	$1$	$1$	$1$	$64.194$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(6\)	\(-18\)	\(2\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+6q^{3}-18q^{5}+2q^{7}+9q^{9}+26q^{11}+\cdots\)
1088.4.a.l	$1088$	$4$	1088.a	1.a	$1$	$1$	$1$	$64.194$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(8\)	\(-6\)	\(-28\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+8q^{3}-6q^{5}-28q^{7}+37q^{9}+24q^{11}+\cdots\)
1088.4.a.m	$1088$	$4$	1088.a	1.a	$1$	$2$	$2$	$64.194$	\(\Q(\sqrt{13}) \)	None	✓	$1$	$1$	\(0\)	\(-6\)	\(4\)	\(-6\)		$+$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(-3-\beta )q^{3}+(2+4\beta )q^{5}+(-3-\beta )q^{7}+\cdots\)
1088.4.a.n	$1088$	$4$	1088.a	1.a	$1$	$2$	$2$	$64.194$	\(\Q(\sqrt{3}) \)	None	✓	$1$	$1$	\(0\)	\(-4\)	\(12\)	\(-36\)		$+$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(-2+\beta )q^{3}+(6-2\beta )q^{5}+(-18+\cdots)q^{7}+\cdots\)
1088.4.a.o	$1088$	$4$	1088.a	1.a	$1$	$2$	$2$	$64.194$	\(\Q(\sqrt{34}) \)	None	✓	$2$	$1$	\(0\)	\(0\)	\(20\)	\(0\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{3}+10q^{5}-\beta q^{7}+7q^{9}-7\beta q^{11}+\cdots\)
1088.4.a.p	$1088$	$4$	1088.a	1.a	$1$	$2$	$2$	$64.194$	\(\Q(\sqrt{3}) \)	None	✓	$1$	$0$	\(0\)	\(4\)	\(12\)	\(36\)		$-$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q+(2+\beta )q^{3}+(6+2\beta )q^{5}+(18+3\beta )q^{7}+\cdots\)
1088.4.a.q	$1088$	$4$	1088.a	1.a	$1$	$2$	$2$	$64.194$	\(\Q(\sqrt{13}) \)	None	✓	$1$	$0$	\(0\)	\(6\)	\(4\)	\(6\)		$-$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q+(3+\beta )q^{3}+(2+4\beta )q^{5}+(3+\beta )q^{7}+\cdots\)
1088.4.a.r	$1088$	$4$	1088.a	1.a	$1$	$3$	$3$	$64.194$	3.3.6420.1	None	✓	$1$	$0$	\(0\)	\(-14\)	\(18\)	\(14\)		$-$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q+(-5+\beta _{1})q^{3}+(7-2\beta _{1}+\beta _{2})q^{5}+\cdots\)
1088.4.a.s	$1088$	$4$	1088.a	1.a	$1$	$3$	$3$	$64.194$	3.3.1556.1	None	✓	$1$	$1$	\(0\)	\(-8\)	\(-2\)	\(12\)		$+$	$-$	$-$	$2^{3}$	$\mathrm{SU}(2)$	\(q+(-3+\beta _{2})q^{3}+(-1+\beta _{1})q^{5}+(4+\cdots)q^{7}+\cdots\)
1088.4.a.t	$1088$	$4$	1088.a	1.a	$1$	$3$	$3$	$64.194$	3.3.1524.1	None	✓	$1$	$1$	\(0\)	\(-4\)	\(-26\)	\(8\)		$+$	$-$	$-$	$2^{3}$	$\mathrm{SU}(2)$	\(q+(-1-\beta _{2})q^{3}+(-9+\beta _{1})q^{5}+(2+\cdots)q^{7}+\cdots\)
1088.4.a.u	$1088$	$4$	1088.a	1.a	$1$	$3$	$3$	$64.194$	3.3.8396.1	None	✓	$1$	$1$	\(0\)	\(-4\)	\(8\)	\(2\)		$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{2})q^{3}+(3-\beta _{1}+2\beta _{2})q^{5}+\cdots\)
1088.4.a.v	$1088$	$4$	1088.a	1.a	$1$	$3$	$3$	$64.194$	3.3.2636.1	None	✓	$1$	$0$	\(0\)	\(-4\)	\(8\)	\(22\)		$+$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1}-\beta _{2})q^{3}+(3-\beta _{2})q^{5}+\cdots\)
1088.4.a.w	$1088$	$4$	1088.a	1.a	$1$	$3$	$3$	$64.194$	3.3.1524.1	None	✓	$1$	$0$	\(0\)	\(4\)	\(-26\)	\(-8\)		$-$	$-$	$+$	$2^{3}$	$\mathrm{SU}(2)$	\(q+(1+\beta _{2})q^{3}+(-9+\beta _{1})q^{5}+(-2+\cdots)q^{7}+\cdots\)
1088.4.a.x	$1088$	$4$	1088.a	1.a	$1$	$3$	$3$	$64.194$	3.3.2636.1	None	✓	$1$	$1$	\(0\)	\(4\)	\(8\)	\(-22\)		$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1}+\beta _{2})q^{3}+(3-\beta _{2})q^{5}+(-8+\cdots)q^{7}+\cdots\)
1088.4.a.y	$1088$	$4$	1088.a	1.a	$1$	$3$	$3$	$64.194$	3.3.8396.1	None	✓	$1$	$0$	\(0\)	\(4\)	\(8\)	\(-2\)		$+$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q+(1-\beta _{2})q^{3}+(3-\beta _{1}+2\beta _{2})q^{5}+(-2+\cdots)q^{7}+\cdots\)
1088.4.a.z	$1088$	$4$	1088.a	1.a	$1$	$3$	$3$	$64.194$	3.3.1556.1	None	✓	$1$	$0$	\(0\)	\(8\)	\(-2\)	\(-12\)		$-$	$-$	$+$	$2^{3}$	$\mathrm{SU}(2)$	\(q+(3-\beta _{2})q^{3}+(-1+\beta _{1})q^{5}+(-4+\cdots)q^{7}+\cdots\)
1088.4.a.ba	$1088$	$4$	1088.a	1.a	$1$	$3$	$3$	$64.194$	3.3.6420.1	None	✓	$1$	$0$	\(0\)	\(14\)	\(18\)	\(-14\)		$-$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q+(5-\beta _{1})q^{3}+(7-2\beta _{1}+\beta _{2})q^{5}+(-4+\cdots)q^{7}+\cdots\)
1088.4.a.bb	$1088$	$4$	1088.a	1.a	$1$	$4$	$4$	$64.194$	4.4.550476.1	None	✓	$1$	$1$	\(0\)	\(-2\)	\(-8\)	\(22\)		$-$	$+$	$-$	$2^{3}$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}+(-1+\beta _{1}+\beta _{2}+2\beta _{3})q^{5}+\cdots\)
1088.4.a.bc	$1088$	$4$	1088.a	1.a	$1$	$4$	$4$	$64.194$	4.4.1224468.1	None	✓	$1$	$1$	\(0\)	\(0\)	\(8\)	\(-32\)		$+$	$-$	$-$	$2^{3}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}+(2+\beta _{1}-\beta _{2})q^{5}+(-8-3\beta _{2}+\cdots)q^{7}+\cdots\)
1088.4.a.bd	$1088$	$4$	1088.a	1.a	$1$	$4$	$4$	$64.194$	4.4.1224468.1	None	✓	$1$	$1$	\(0\)	\(0\)	\(8\)	\(32\)		$+$	$-$	$-$	$2^{3}$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}+(2+\beta _{1}-\beta _{2})q^{5}+(8+3\beta _{2}+\cdots)q^{7}+\cdots\)
1088.4.a.be	$1088$	$4$	1088.a	1.a	$1$	$4$	$4$	$64.194$	4.4.550476.1	None	✓	$1$	$0$	\(0\)	\(2\)	\(-8\)	\(-22\)		$+$	$+$	$+$	$2^{3}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}+(-1+\beta _{1}+\beta _{2}+2\beta _{3})q^{5}+\cdots\)
1088.4.a.bf	$1088$	$4$	1088.a	1.a	$1$	$6$	$6$	$64.194$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓	$2$	$1$	\(0\)	\(0\)	\(-4\)	\(0\)		$-$	$+$	$-$	$2^{8}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}+(-1+\beta _{3})q^{5}-\beta _{4}q^{7}+(2^{4}+\cdots)q^{9}+\cdots\)
1088.4.a.bg	$1088$	$4$	1088.a	1.a	$1$	$7$	$7$	$64.194$	\(\mathbb{Q}[x]/(x^{7} - \cdots)\)	None	✓	$1$	$0$	\(0\)	\(0\)	\(0\)	\(-2\)		$+$	$+$	$+$	$2^{11}$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}+(-\beta _{1}-\beta _{3})q^{5}+(-\beta _{1}+\beta _{3}+\cdots)q^{7}+\cdots\)
1088.4.a.bh	$1088$	$4$	1088.a	1.a	$1$	$7$	$7$	$64.194$	\(\mathbb{Q}[x]/(x^{7} - \cdots)\)	None	✓	$1$	$0$	\(0\)	\(0\)	\(0\)	\(2\)		$+$	$+$	$+$	$2^{11}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}+(-\beta _{1}-\beta _{3})q^{5}+(\beta _{1}-\beta _{3}+\cdots)q^{7}+\cdots\)
1088.4.a.bi	$1088$	$4$	1088.a	1.a	$1$	$8$	$8$	$64.194$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓	$2$	$0$	\(0\)	\(0\)	\(-16\)	\(0\)		$-$	$-$	$+$	$2^{14}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}+(-2-\beta _{2})q^{5}+(-\beta _{1}-\beta _{6}+\cdots)q^{7}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(1088))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(1088)) \cong \) \(S_{4}^{\mathrm{new}}(\Gamma_0(8))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(16))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(17))\)\(^{\oplus 7}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(32))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(34))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(64))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(68))\)\(^{\oplus 5}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(136))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(272))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(544))\)\(^{\oplus 2}\)