Space of modular forms of level 8800, weight 2, and trivial character

• Label: 8800.2.a
• Level: $8800$
• Weight: $2$
• Character orbit: 8800.a
• Rep. character: $\chi_{8800}(1,\cdot)$
• Character field: $\Q$
• Dimension: $190$
• Newform subspaces: $62$
• Sturm bound: $2880$
• Trace bound: $13$

Defining parameters

Level:	\( N \)	\(=\)	\( 8800 = 2^{5} \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8800.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 62 \)
Sturm bound:			\(2880\)
Trace bound:			\(13\)
Distinguishing \(T_p\):			\(3\), \(7\), \(13\), \(17\), \(19\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(8800))\).

	Total	New	Old
Modular forms	1488	190	1298
Cusp forms	1393	190	1203
Eisenstein series	95	0	95

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

\(2\)	\(5\)	\(11\)	Fricke	Dim
\(+\)	\(+\)	\(+\)	\(+\)	\(21\)
\(+\)	\(+\)	\(-\)	\(-\)	\(24\)
\(+\)	\(-\)	\(+\)	\(-\)	\(26\)
\(+\)	\(-\)	\(-\)	\(+\)	\(24\)
\(-\)	\(+\)	\(+\)	\(-\)	\(24\)
\(-\)	\(+\)	\(-\)	\(+\)	\(21\)
\(-\)	\(-\)	\(+\)	\(+\)	\(24\)
\(-\)	\(-\)	\(-\)	\(-\)	\(26\)
Plus space			\(+\)	\(90\)
Minus space			\(-\)	\(100\)

Trace form

\( 190 q + 190 q^{9} + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(8800))\) 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	5	11
8800.2.a.a	$8800$	$2$	8800.a	1.a	$1$	$1$	$1$	$70.268$	\(\Q\)	None	✓				352.2.a.a	$1$	$1$	\(0\)	\(-3\)	\(0\)	\(0\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-3q^{3}+6q^{9}-q^{11}+6q^{13}+4q^{17}+\cdots\)
8800.2.a.b	$8800$	$2$	8800.a	1.a	$1$	$1$	$1$	$70.268$	\(\Q\)	None	✓				1760.2.b.a	$1$	$1$	\(0\)	\(-3\)	\(0\)	\(3\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-3q^{3}+3q^{7}+6q^{9}-q^{11}-4q^{13}+\cdots\)
8800.2.a.c	$8800$	$2$	8800.a	1.a	$1$	$1$	$1$	$70.268$	\(\Q\)	None	✓				1760.2.b.a	$1$	$1$	\(0\)	\(-3\)	\(0\)	\(3\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-3q^{3}+3q^{7}+6q^{9}+q^{11}+4q^{13}+\cdots\)
8800.2.a.d	$8800$	$2$	8800.a	1.a	$1$	$1$	$1$	$70.268$	\(\Q\)	None	✓				1760.2.a.b	$1$	$2$	\(0\)	\(-2\)	\(0\)	\(-4\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{3}-4q^{7}+q^{9}-q^{11}-4q^{13}+\cdots\)
8800.2.a.e	$8800$	$2$	8800.a	1.a	$1$	$1$	$1$	$70.268$	\(\Q\)	None	✓	✓			8800.2.a.e	$1$	$0$	\(0\)	\(-2\)	\(0\)	\(0\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{3}+q^{9}-q^{11}-q^{13}-4q^{17}+\cdots\)
8800.2.a.f	$8800$	$2$	8800.a	1.a	$1$	$1$	$1$	$70.268$	\(\Q\)	None	✓				1760.2.a.c	$1$	$0$	\(0\)	\(-2\)	\(0\)	\(0\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{3}+q^{9}+q^{11}-4q^{13}+4q^{17}+\cdots\)
8800.2.a.g	$8800$	$2$	8800.a	1.a	$1$	$1$	$1$	$70.268$	\(\Q\)	None	✓	✓			8800.2.a.e	$1$	$1$	\(0\)	\(-2\)	\(0\)	\(0\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{3}+q^{9}+q^{11}+q^{13}+4q^{17}+\cdots\)
8800.2.a.h	$8800$	$2$	8800.a	1.a	$1$	$1$	$1$	$70.268$	\(\Q\)	None	✓				1760.2.a.a	$1$	$0$	\(0\)	\(-2\)	\(0\)	\(2\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{3}+2q^{7}+q^{9}-q^{11}+2q^{13}+\cdots\)
8800.2.a.i	$8800$	$2$	8800.a	1.a	$1$	$1$	$1$	$70.268$	\(\Q\)	None	✓				352.2.a.c	$1$	$1$	\(0\)	\(-1\)	\(0\)	\(-4\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-4q^{7}-2q^{9}+q^{11}+2q^{13}+\cdots\)
8800.2.a.j	$8800$	$2$	8800.a	1.a	$1$	$1$	$1$	$70.268$	\(\Q\)	None	✓				1760.2.a.e	$1$	$1$	\(0\)	\(-1\)	\(0\)	\(1\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+q^{7}-2q^{9}+q^{11}+2q^{13}+\cdots\)
8800.2.a.k	$8800$	$2$	8800.a	1.a	$1$	$1$	$1$	$70.268$	\(\Q\)	None	✓				1760.2.a.f	$1$	$1$	\(0\)	\(-1\)	\(0\)	\(3\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+3q^{7}-2q^{9}-q^{11}+2q^{13}+\cdots\)
8800.2.a.l	$8800$	$2$	8800.a	1.a	$1$	$1$	$1$	$70.268$	\(\Q\)	None	✓				1760.2.a.d	$1$	$1$	\(0\)	\(-1\)	\(0\)	\(3\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+3q^{7}-2q^{9}+q^{11}+2q^{13}+\cdots\)
8800.2.a.m	$8800$	$2$	8800.a	1.a	$1$	$1$	$1$	$70.268$	\(\Q\)	None	✓				352.2.a.b	$1$	$0$	\(0\)	\(-1\)	\(0\)	\(4\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+4q^{7}-2q^{9}+q^{11}+2q^{13}+\cdots\)
8800.2.a.n	$8800$	$2$	8800.a	1.a	$1$	$1$	$1$	$70.268$	\(\Q\)	None	✓				1760.2.a.g	$1$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-3q^{9}-q^{11}+2q^{13}+2q^{17}+4q^{19}+\cdots\)
8800.2.a.o	$8800$	$2$	8800.a	1.a	$1$	$1$	$1$	$70.268$	\(\Q\)	None	✓				1760.2.a.g	$1$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-3q^{9}+q^{11}+2q^{13}+2q^{17}-4q^{19}+\cdots\)
8800.2.a.p	$8800$	$2$	8800.a	1.a	$1$	$1$	$1$	$70.268$	\(\Q\)	None	✓				352.2.a.b	$1$	$0$	\(0\)	\(1\)	\(0\)	\(-4\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-4q^{7}-2q^{9}-q^{11}+2q^{13}+\cdots\)
8800.2.a.q	$8800$	$2$	8800.a	1.a	$1$	$1$	$1$	$70.268$	\(\Q\)	None	✓				1760.2.a.d	$1$	$0$	\(0\)	\(1\)	\(0\)	\(-3\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-3q^{7}-2q^{9}-q^{11}+2q^{13}+\cdots\)
8800.2.a.r	$8800$	$2$	8800.a	1.a	$1$	$1$	$1$	$70.268$	\(\Q\)	None	✓				1760.2.a.f	$1$	$0$	\(0\)	\(1\)	\(0\)	\(-3\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-3q^{7}-2q^{9}+q^{11}+2q^{13}+\cdots\)
8800.2.a.s	$8800$	$2$	8800.a	1.a	$1$	$1$	$1$	$70.268$	\(\Q\)	None	✓				1760.2.a.e	$1$	$0$	\(0\)	\(1\)	\(0\)	\(-1\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-q^{7}-2q^{9}-q^{11}+2q^{13}+\cdots\)
8800.2.a.t	$8800$	$2$	8800.a	1.a	$1$	$1$	$1$	$70.268$	\(\Q\)	None	✓				352.2.a.c	$1$	$0$	\(0\)	\(1\)	\(0\)	\(4\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+4q^{7}-2q^{9}-q^{11}+2q^{13}+\cdots\)
8800.2.a.u	$8800$	$2$	8800.a	1.a	$1$	$1$	$1$	$70.268$	\(\Q\)	None	✓				1760.2.a.a	$1$	$0$	\(0\)	\(2\)	\(0\)	\(-2\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{3}-2q^{7}+q^{9}+q^{11}+2q^{13}+\cdots\)
8800.2.a.v	$8800$	$2$	8800.a	1.a	$1$	$1$	$1$	$70.268$	\(\Q\)	None	✓				1760.2.a.c	$1$	$1$	\(0\)	\(2\)	\(0\)	\(0\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{3}+q^{9}-q^{11}-4q^{13}+4q^{17}+\cdots\)
8800.2.a.w	$8800$	$2$	8800.a	1.a	$1$	$1$	$1$	$70.268$	\(\Q\)	None	✓	✓			8800.2.a.e	$1$	$0$	\(0\)	\(2\)	\(0\)	\(0\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{3}+q^{9}-q^{11}+q^{13}+4q^{17}+\cdots\)
8800.2.a.x	$8800$	$2$	8800.a	1.a	$1$	$1$	$1$	$70.268$	\(\Q\)	None	✓	✓			8800.2.a.e	$1$	$1$	\(0\)	\(2\)	\(0\)	\(0\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{3}+q^{9}+q^{11}-q^{13}-4q^{17}+\cdots\)
8800.2.a.y	$8800$	$2$	8800.a	1.a	$1$	$1$	$1$	$70.268$	\(\Q\)	None	✓				1760.2.a.b	$1$	$1$	\(0\)	\(2\)	\(0\)	\(4\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{3}+4q^{7}+q^{9}+q^{11}-4q^{13}+\cdots\)
8800.2.a.z	$8800$	$2$	8800.a	1.a	$1$	$1$	$1$	$70.268$	\(\Q\)	None	✓				1760.2.b.a	$1$	$0$	\(0\)	\(3\)	\(0\)	\(-3\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3q^{3}-3q^{7}+6q^{9}-q^{11}+4q^{13}+\cdots\)
8800.2.a.ba	$8800$	$2$	8800.a	1.a	$1$	$1$	$1$	$70.268$	\(\Q\)	None	✓				1760.2.b.a	$1$	$0$	\(0\)	\(3\)	\(0\)	\(-3\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3q^{3}-3q^{7}+6q^{9}+q^{11}-4q^{13}+\cdots\)
8800.2.a.bb	$8800$	$2$	8800.a	1.a	$1$	$1$	$1$	$70.268$	\(\Q\)	None	✓				352.2.a.a	$1$	$0$	\(0\)	\(3\)	\(0\)	\(0\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3q^{3}+6q^{9}+q^{11}+6q^{13}+4q^{17}+\cdots\)
8800.2.a.bc	$8800$	$2$	8800.a	1.a	$1$	$2$	$2$	$70.268$	\(\Q(\sqrt{5}) \)	None	✓				1760.2.a.o	$1$	$1$	\(0\)	\(-2\)	\(0\)	\(-4\)		$-$	$+$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q+(-1-\beta )q^{3}-2q^{7}+(3+2\beta )q^{9}+\cdots\)
8800.2.a.bd	$8800$	$2$	8800.a	1.a	$1$	$2$	$2$	$70.268$	\(\Q(\sqrt{17}) \)	None	✓				352.2.a.g	$1$	$1$	\(0\)	\(-1\)	\(0\)	\(0\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta q^{3}+(1+\beta )q^{9}+q^{11}-2q^{13}+\cdots\)
8800.2.a.be	$8800$	$2$	8800.a	1.a	$1$	$2$	$2$	$70.268$	\(\Q(\sqrt{17}) \)	None	✓				352.2.a.g	$1$	$1$	\(0\)	\(1\)	\(0\)	\(0\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{3}+(1+\beta )q^{9}-q^{11}-2q^{13}+\cdots\)
8800.2.a.bf	$8800$	$2$	8800.a	1.a	$1$	$2$	$2$	$70.268$	\(\Q(\sqrt{5}) \)	None	✓				1760.2.a.o	$1$	$0$	\(0\)	\(2\)	\(0\)	\(4\)		$-$	$+$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{3}+2q^{7}+(3+2\beta )q^{9}-q^{11}+\cdots\)
8800.2.a.bg	$8800$	$2$	8800.a	1.a	$1$	$3$	$3$	$70.268$	3.3.229.1	None	✓				1760.2.a.q	$1$	$1$	\(0\)	\(-2\)	\(0\)	\(4\)		$+$	$+$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q+(-1-\beta _{2})q^{3}+(1-\beta _{1})q^{7}+(2-\beta _{1}+\cdots)q^{9}+\cdots\)
8800.2.a.bh	$8800$	$2$	8800.a	1.a	$1$	$3$	$3$	$70.268$	3.3.229.1	None	✓				1760.2.a.r	$1$	$0$	\(0\)	\(-1\)	\(0\)	\(7\)		$+$	$+$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+\beta _{2}q^{3}+(2-\beta _{1})q^{7}+(1-\beta _{1})q^{9}+\cdots\)
8800.2.a.bi	$8800$	$2$	8800.a	1.a	$1$	$3$	$3$	$70.268$	3.3.229.1	None	✓				1760.2.a.r	$1$	$1$	\(0\)	\(1\)	\(0\)	\(-7\)		$+$	$+$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q-\beta _{2}q^{3}+(-2+\beta _{1})q^{7}+(1-\beta _{1})q^{9}+\cdots\)
8800.2.a.bj	$8800$	$2$	8800.a	1.a	$1$	$3$	$3$	$70.268$	3.3.229.1	None	✓				1760.2.a.q	$1$	$1$	\(0\)	\(2\)	\(0\)	\(-4\)		$-$	$+$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q+(1+\beta _{2})q^{3}+(-1+\beta _{1})q^{7}+(2-\beta _{1}+\cdots)q^{9}+\cdots\)
8800.2.a.bk	$8800$	$2$	8800.a	1.a	$1$	$4$	$4$	$70.268$	4.4.8112.1	None	✓	✓			8800.2.a.bk	$1$	$0$	\(0\)	\(-4\)	\(0\)	\(4\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{3}+(1-\beta _{1}+\beta _{3})q^{7}+\cdots\)
8800.2.a.bl	$8800$	$2$	8800.a	1.a	$1$	$4$	$4$	$70.268$	4.4.8112.1	None	✓	✓			8800.2.a.bk	$1$	$1$	\(0\)	\(-4\)	\(0\)	\(4\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{3}+(1-\beta _{1}+\beta _{3})q^{7}+\cdots\)
8800.2.a.bm	$8800$	$2$	8800.a	1.a	$1$	$4$	$4$	$70.268$	4.4.18736.1	None	✓	✓			8800.2.a.bm	$1$	$1$	\(0\)	\(-2\)	\(0\)	\(4\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{3}+(1-\beta _{3})q^{7}+(1-\beta _{1}+\cdots)q^{9}+\cdots\)
8800.2.a.bn	$8800$	$2$	8800.a	1.a	$1$	$4$	$4$	$70.268$	4.4.18736.1	None	✓	✓			8800.2.a.bm	$1$	$1$	\(0\)	\(-2\)	\(0\)	\(4\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{3}+(1-\beta _{3})q^{7}+(1-\beta _{1}+\cdots)q^{9}+\cdots\)
8800.2.a.bo	$8800$	$2$	8800.a	1.a	$1$	$4$	$4$	$70.268$	4.4.18736.1	None	✓	✓			8800.2.a.bm	$1$	$1$	\(0\)	\(2\)	\(0\)	\(-4\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{3}+(-1+\beta _{3})q^{7}+(1-\beta _{1}+\cdots)q^{9}+\cdots\)
8800.2.a.bp	$8800$	$2$	8800.a	1.a	$1$	$4$	$4$	$70.268$	4.4.18736.1	None	✓	✓			8800.2.a.bm	$1$	$1$	\(0\)	\(2\)	\(0\)	\(-4\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{3}+(-1+\beta _{3})q^{7}+(1-\beta _{1}+\cdots)q^{9}+\cdots\)
8800.2.a.bq	$8800$	$2$	8800.a	1.a	$1$	$4$	$4$	$70.268$	4.4.8112.1	None	✓	✓			8800.2.a.bk	$1$	$0$	\(0\)	\(4\)	\(0\)	\(-4\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta _{1})q^{3}+(-1-\beta _{1}+\beta _{3})q^{7}+\cdots\)
8800.2.a.br	$8800$	$2$	8800.a	1.a	$1$	$4$	$4$	$70.268$	4.4.8112.1	None	✓	✓			8800.2.a.bk	$1$	$1$	\(0\)	\(4\)	\(0\)	\(-4\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta _{1})q^{3}+(-1-\beta _{1}+\beta _{3})q^{7}+\cdots\)
8800.2.a.bs	$8800$	$2$	8800.a	1.a	$1$	$5$	$5$	$70.268$	5.5.5060976.1	None	✓	✓			8800.2.a.bs	$1$	$1$	\(0\)	\(-2\)	\(0\)	\(4\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}+(1+\beta _{3})q^{7}+(1+\beta _{2})q^{9}+\cdots\)
8800.2.a.bt	$8800$	$2$	8800.a	1.a	$1$	$5$	$5$	$70.268$	5.5.5060976.1	None	✓	✓			8800.2.a.bs	$1$	$0$	\(0\)	\(-2\)	\(0\)	\(4\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}+(1+\beta _{3})q^{7}+(1+\beta _{2})q^{9}+\cdots\)
8800.2.a.bu	$8800$	$2$	8800.a	1.a	$1$	$5$	$5$	$70.268$	5.5.792644.1	None	✓				1760.2.a.u	$1$	$0$	\(0\)	\(0\)	\(0\)	\(-2\)		$+$	$+$	$-$	$-$	$2^{3}$	$\mathrm{SU}(2)$	\(q-\beta _{3}q^{3}-\beta _{4}q^{7}+(2-\beta _{1})q^{9}+q^{11}+\cdots\)
8800.2.a.bv	$8800$	$2$	8800.a	1.a	$1$	$5$	$5$	$70.268$	5.5.792644.1	None	✓				1760.2.a.u	$1$	$0$	\(0\)	\(0\)	\(0\)	\(2\)		$-$	$+$	$+$	$-$	$2^{3}$	$\mathrm{SU}(2)$	\(q+\beta _{3}q^{3}+\beta _{4}q^{7}+(2-\beta _{1})q^{9}-q^{11}+\cdots\)
8800.2.a.bw	$8800$	$2$	8800.a	1.a	$1$	$5$	$5$	$70.268$	5.5.5060976.1	None	✓	✓	✓		8800.2.a.bs	$1$	$1$	\(0\)	\(2\)	\(0\)	\(-4\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}+(-1-\beta _{3})q^{7}+(1+\beta _{2})q^{9}+\cdots\)
8800.2.a.bx	$8800$	$2$	8800.a	1.a	$1$	$5$	$5$	$70.268$	5.5.5060976.1	None	✓	✓			8800.2.a.bs	$1$	$0$	\(0\)	\(2\)	\(0\)	\(-4\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}+(-1-\beta _{3})q^{7}+(1+\beta _{2})q^{9}+\cdots\)
8800.2.a.by	$8800$	$2$	8800.a	1.a	$1$	$6$	$6$	$70.268$	6.6.726559536.1	None	✓	✓	✓		8800.2.a.by	$1$	$0$	\(0\)	\(-2\)	\(0\)	\(4\)		$-$	$+$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}+(1+\beta _{4})q^{7}+(2+\beta _{1}+\beta _{2}+\cdots)q^{9}+\cdots\)
8800.2.a.bz	$8800$	$2$	8800.a	1.a	$1$	$6$	$6$	$70.268$	6.6.726559536.1	None	✓	✓			8800.2.a.by	$1$	$0$	\(0\)	\(-2\)	\(0\)	\(4\)		$-$	$-$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}+(1+\beta _{4})q^{7}+(2+\beta _{1}+\beta _{2}+\cdots)q^{9}+\cdots\)
8800.2.a.ca	$8800$	$2$	8800.a	1.a	$1$	$6$	$6$	$70.268$	6.6.726559536.1	None	✓	✓			8800.2.a.by	$1$	$0$	\(0\)	\(2\)	\(0\)	\(-4\)		$+$	$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}+(-1-\beta _{4})q^{7}+(2+\beta _{1}+\beta _{2}+\cdots)q^{9}+\cdots\)
8800.2.a.cb	$8800$	$2$	8800.a	1.a	$1$	$6$	$6$	$70.268$	6.6.726559536.1	None	✓	✓	✓		8800.2.a.by	$1$	$0$	\(0\)	\(2\)	\(0\)	\(-4\)		$+$	$+$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}+(-1-\beta _{4})q^{7}+(2+\beta _{1}+\beta _{2}+\cdots)q^{9}+\cdots\)
8800.2.a.cc	$8800$	$2$	8800.a	1.a	$1$	$7$	$7$	$70.268$	\(\mathbb{Q}[x]/(x^{7} - \cdots)\)	None	✓				1760.2.b.e	$1$	$1$	\(0\)	\(-1\)	\(0\)	\(-7\)		$-$	$-$	$+$	$+$	$2^{4}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}+(-1+\beta _{3})q^{7}+(1-\beta _{1}+\beta _{5}+\cdots)q^{9}+\cdots\)
8800.2.a.cd	$8800$	$2$	8800.a	1.a	$1$	$7$	$7$	$70.268$	\(\mathbb{Q}[x]/(x^{7} - \cdots)\)	None	✓				1760.2.b.e	$1$	$1$	\(0\)	\(-1\)	\(0\)	\(-7\)		$+$	$-$	$-$	$+$	$2^{4}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}+(-1+\beta _{3})q^{7}+(1-\beta _{1}+\beta _{5}+\cdots)q^{9}+\cdots\)
8800.2.a.ce	$8800$	$2$	8800.a	1.a	$1$	$7$	$7$	$70.268$	\(\mathbb{Q}[x]/(x^{7} - \cdots)\)	None	✓				1760.2.b.c	$1$	$0$	\(0\)	\(0\)	\(0\)	\(-4\)		$+$	$-$	$+$	$-$	$2^{3}$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}+(-1-\beta _{5})q^{7}+\beta _{4}q^{9}-q^{11}+\cdots\)
8800.2.a.cf	$8800$	$2$	8800.a	1.a	$1$	$7$	$7$	$70.268$	\(\mathbb{Q}[x]/(x^{7} - \cdots)\)	None	✓				1760.2.b.c	$1$	$1$	\(0\)	\(0\)	\(0\)	\(-4\)		$+$	$-$	$-$	$+$	$2^{3}$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}+(-1-\beta _{5})q^{7}+\beta _{4}q^{9}+q^{11}+\cdots\)
8800.2.a.cg	$8800$	$2$	8800.a	1.a	$1$	$7$	$7$	$70.268$	\(\mathbb{Q}[x]/(x^{7} - \cdots)\)	None	✓				1760.2.b.c	$1$	$1$	\(0\)	\(0\)	\(0\)	\(4\)		$-$	$-$	$+$	$+$	$2^{3}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}+(1+\beta _{5})q^{7}+\beta _{4}q^{9}-q^{11}+\cdots\)
8800.2.a.ch	$8800$	$2$	8800.a	1.a	$1$	$7$	$7$	$70.268$	\(\mathbb{Q}[x]/(x^{7} - \cdots)\)	None	✓				1760.2.b.c	$1$	$0$	\(0\)	\(0\)	\(0\)	\(4\)		$-$	$-$	$-$	$-$	$2^{3}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}+(1+\beta _{5})q^{7}+\beta _{4}q^{9}+q^{11}+\cdots\)
8800.2.a.ci	$8800$	$2$	8800.a	1.a	$1$	$7$	$7$	$70.268$	\(\mathbb{Q}[x]/(x^{7} - \cdots)\)	None	✓				1760.2.b.e	$1$	$0$	\(0\)	\(1\)	\(0\)	\(7\)		$+$	$-$	$+$	$-$	$2^{4}$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}+(1-\beta _{3})q^{7}+(1-\beta _{1}+\beta _{5}+\cdots)q^{9}+\cdots\)
8800.2.a.cj	$8800$	$2$	8800.a	1.a	$1$	$7$	$7$	$70.268$	\(\mathbb{Q}[x]/(x^{7} - \cdots)\)	None	✓				1760.2.b.e	$1$	$0$	\(0\)	\(1\)	\(0\)	\(7\)		$-$	$-$	$-$	$-$	$2^{4}$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}+(1-\beta _{3})q^{7}+(1-\beta _{1}+\beta _{5}+\cdots)q^{9}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(8800))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(8800)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(11))\)\(^{\oplus 18}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 16}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(32))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(40))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(44))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(50))\)\(^{\oplus 10}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(55))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(80))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(88))\)\(^{\oplus 9}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(100))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(110))\)\(^{\oplus 10}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(160))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(176))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(200))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(220))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(275))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(352))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(400))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(440))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(550))\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(800))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(880))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1100))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1760))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(2200))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(4400))\)\(^{\oplus 2}\)