Space of modular forms of level 8550, weight 2, and trivial character

• Label: 8550.2.a
• Level: $8550$
• Weight: $2$
• Character orbit: 8550.a
• Rep. character: $\chi_{8550}(1,\cdot)$
• Character field: $\Q$
• Dimension: $143$
• Newform subspaces: $76$
• Sturm bound: $3600$
• Trace bound: $23$

Defining parameters

Level:	\( N \)	\(=\)	\( 8550 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8550.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 76 \)
Sturm bound:			\(3600\)
Trace bound:			\(23\)
Distinguishing \(T_p\):			\(7\), \(11\), \(13\), \(17\), \(23\), \(53\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(8550))\).

	Total	New	Old
Modular forms	1848	143	1705
Cusp forms	1753	143	1610
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.

\(2\)	\(3\)	\(5\)	\(19\)	Fricke	Dim.
\(+\)	\(+\)	\(+\)	\(+\)	\(+\)	\(7\)
\(+\)	\(+\)	\(+\)	\(-\)	\(-\)	\(6\)
\(+\)	\(+\)	\(-\)	\(+\)	\(-\)	\(7\)
\(+\)	\(+\)	\(-\)	\(-\)	\(+\)	\(9\)
\(+\)	\(-\)	\(+\)	\(+\)	\(-\)	\(12\)
\(+\)	\(-\)	\(+\)	\(-\)	\(+\)	\(8\)
\(+\)	\(-\)	\(-\)	\(+\)	\(+\)	\(10\)
\(+\)	\(-\)	\(-\)	\(-\)	\(-\)	\(12\)
\(-\)	\(+\)	\(+\)	\(+\)	\(-\)	\(7\)
\(-\)	\(+\)	\(+\)	\(-\)	\(+\)	\(6\)
\(-\)	\(+\)	\(-\)	\(+\)	\(+\)	\(7\)
\(-\)	\(+\)	\(-\)	\(-\)	\(-\)	\(9\)
\(-\)	\(-\)	\(+\)	\(+\)	\(+\)	\(8\)
\(-\)	\(-\)	\(+\)	\(-\)	\(-\)	\(13\)
\(-\)	\(-\)	\(-\)	\(+\)	\(-\)	\(13\)
\(-\)	\(-\)	\(-\)	\(-\)	\(+\)	\(9\)
Plus space				\(+\)	\(64\)
Minus space				\(-\)	\(79\)

Trace form

\( 143 q + q^{2} + 143 q^{4} - 2 q^{7} + q^{8} + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(8550))\) 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	3	5	19
8550.2.a.a	$8550$	$2$	8550.a	1.a	$1$	$1$	$1$	$68.272$	\(\Q\)	None	✓	$1$	$1$	\(-1\)	\(0\)	\(0\)	\(-4\)		$+$	$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}-4q^{7}-q^{8}-4q^{11}+4q^{14}+\cdots\)
8550.2.a.b	$8550$	$2$	8550.a	1.a	$1$	$1$	$1$	$68.272$	\(\Q\)	None	✓	$1$	$0$	\(-1\)	\(0\)	\(0\)	\(-4\)		$+$	$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}-4q^{7}-q^{8}-2q^{13}+4q^{14}+\cdots\)
8550.2.a.c	$8550$	$2$	8550.a	1.a	$1$	$1$	$1$	$68.272$	\(\Q\)	None	✓	$1$	$0$	\(-1\)	\(0\)	\(0\)	\(-4\)		$+$	$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}-4q^{7}-q^{8}+q^{11}+4q^{14}+\cdots\)
8550.2.a.d	$8550$	$2$	8550.a	1.a	$1$	$1$	$1$	$68.272$	\(\Q\)	None	✓	$1$	$1$	\(-1\)	\(0\)	\(0\)	\(-2\)		$+$	$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}-2q^{7}-q^{8}-3q^{11}-2q^{13}+\cdots\)
8550.2.a.e	$8550$	$2$	8550.a	1.a	$1$	$1$	$1$	$68.272$	\(\Q\)	None	✓	$1$	$1$	\(-1\)	\(0\)	\(0\)	\(-2\)		$+$	$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}-2q^{7}-q^{8}-2q^{13}+2q^{14}+\cdots\)
8550.2.a.f	$8550$	$2$	8550.a	1.a	$1$	$1$	$1$	$68.272$	\(\Q\)	None	✓	$1$	$0$	\(-1\)	\(0\)	\(0\)	\(-2\)		$+$	$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}-2q^{7}-q^{8}+3q^{11}-6q^{13}+\cdots\)
8550.2.a.g	$8550$	$2$	8550.a	1.a	$1$	$1$	$1$	$68.272$	\(\Q\)	None	✓	$1$	$0$	\(-1\)	\(0\)	\(0\)	\(-2\)		$+$	$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}-2q^{7}-q^{8}+6q^{11}+2q^{14}+\cdots\)
8550.2.a.h	$8550$	$2$	8550.a	1.a	$1$	$1$	$1$	$68.272$	\(\Q\)	None	✓	$1$	$1$	\(-1\)	\(0\)	\(0\)	\(0\)		$+$	$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}-q^{8}-q^{11}-4q^{13}+q^{16}+\cdots\)
8550.2.a.i	$8550$	$2$	8550.a	1.a	$1$	$1$	$1$	$68.272$	\(\Q\)	None	✓	$1$	$1$	\(-1\)	\(0\)	\(0\)	\(0\)		$+$	$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}-q^{8}+2q^{11}+4q^{13}+\cdots\)
8550.2.a.j	$8550$	$2$	8550.a	1.a	$1$	$1$	$1$	$68.272$	\(\Q\)	None	✓	$1$	$1$	\(-1\)	\(0\)	\(0\)	\(0\)		$+$	$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}-q^{8}+4q^{11}-4q^{13}+\cdots\)
8550.2.a.k	$8550$	$2$	8550.a	1.a	$1$	$1$	$1$	$68.272$	\(\Q\)	None	✓	$1$	$1$	\(-1\)	\(0\)	\(0\)	\(0\)		$+$	$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}-q^{8}+4q^{11}+2q^{13}+\cdots\)
8550.2.a.l	$8550$	$2$	8550.a	1.a	$1$	$1$	$1$	$68.272$	\(\Q\)	None	✓	$1$	$0$	\(-1\)	\(0\)	\(0\)	\(1\)		$+$	$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}+q^{7}-q^{8}+3q^{13}-q^{14}+\cdots\)
8550.2.a.m	$8550$	$2$	8550.a	1.a	$1$	$1$	$1$	$68.272$	\(\Q\)	None	✓	$1$	$1$	\(-1\)	\(0\)	\(0\)	\(1\)		$+$	$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}+q^{7}-q^{8}+6q^{11}-5q^{13}+\cdots\)
8550.2.a.n	$8550$	$2$	8550.a	1.a	$1$	$1$	$1$	$68.272$	\(\Q\)	None	✓	$1$	$1$	\(-1\)	\(0\)	\(0\)	\(2\)		$+$	$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}+2q^{7}-q^{8}-4q^{11}+6q^{13}+\cdots\)
8550.2.a.o	$8550$	$2$	8550.a	1.a	$1$	$1$	$1$	$68.272$	\(\Q\)	None	✓	$1$	$1$	\(-1\)	\(0\)	\(0\)	\(2\)		$+$	$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}+2q^{7}-q^{8}+2q^{11}-2q^{14}+\cdots\)
8550.2.a.p	$8550$	$2$	8550.a	1.a	$1$	$1$	$1$	$68.272$	\(\Q\)	None	✓	$1$	$0$	\(-1\)	\(0\)	\(0\)	\(2\)		$+$	$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}+2q^{7}-q^{8}+2q^{11}+4q^{13}+\cdots\)
8550.2.a.q	$8550$	$2$	8550.a	1.a	$1$	$1$	$1$	$68.272$	\(\Q\)	None	✓	$1$	$1$	\(-1\)	\(0\)	\(0\)	\(4\)		$+$	$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}+4q^{7}-q^{8}-6q^{11}-4q^{14}+\cdots\)
8550.2.a.r	$8550$	$2$	8550.a	1.a	$1$	$1$	$1$	$68.272$	\(\Q\)	None	✓	$1$	$0$	\(-1\)	\(0\)	\(0\)	\(4\)		$+$	$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}+4q^{7}-q^{8}+4q^{11}+6q^{13}+\cdots\)
8550.2.a.s	$8550$	$2$	8550.a	1.a	$1$	$1$	$1$	$68.272$	\(\Q\)	None	✓	$1$	$0$	\(1\)	\(0\)	\(0\)	\(-4\)		$-$	$-$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}-4q^{7}+q^{8}+6q^{13}-4q^{14}+\cdots\)
8550.2.a.t	$8550$	$2$	8550.a	1.a	$1$	$1$	$1$	$68.272$	\(\Q\)	None	✓	$1$	$1$	\(1\)	\(0\)	\(0\)	\(-4\)		$-$	$-$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}-4q^{7}+q^{8}+4q^{11}+2q^{13}+\cdots\)
8550.2.a.u	$8550$	$2$	8550.a	1.a	$1$	$1$	$1$	$68.272$	\(\Q\)	None	✓	$1$	$1$	\(1\)	\(0\)	\(0\)	\(-3\)		$-$	$-$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}-3q^{7}+q^{8}-2q^{11}+q^{13}+\cdots\)
8550.2.a.v	$8550$	$2$	8550.a	1.a	$1$	$1$	$1$	$68.272$	\(\Q\)	None	✓	$1$	$0$	\(1\)	\(0\)	\(0\)	\(-2\)		$-$	$-$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}-2q^{7}+q^{8}-6q^{11}+4q^{13}+\cdots\)
8550.2.a.w	$8550$	$2$	8550.a	1.a	$1$	$1$	$1$	$68.272$	\(\Q\)	None	✓	$1$	$0$	\(1\)	\(0\)	\(0\)	\(-2\)		$-$	$-$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}-2q^{7}+q^{8}+4q^{11}-6q^{13}+\cdots\)
8550.2.a.x	$8550$	$2$	8550.a	1.a	$1$	$1$	$1$	$68.272$	\(\Q\)	None	✓	$1$	$1$	\(1\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}+q^{8}-4q^{11}-2q^{13}+\cdots\)
8550.2.a.y	$8550$	$2$	8550.a	1.a	$1$	$1$	$1$	$68.272$	\(\Q\)	None	✓	$1$	$0$	\(1\)	\(0\)	\(0\)	\(0\)		$-$	$+$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}+q^{8}-2q^{11}+4q^{13}+\cdots\)
8550.2.a.z	$8550$	$2$	8550.a	1.a	$1$	$1$	$1$	$68.272$	\(\Q\)	None	✓	$1$	$1$	\(1\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}+q^{8}-q^{11}+4q^{13}+q^{16}+\cdots\)
8550.2.a.ba	$8550$	$2$	8550.a	1.a	$1$	$1$	$1$	$68.272$	\(\Q\)	None	✓	$1$	$1$	\(1\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}+q^{8}+4q^{11}-2q^{13}+\cdots\)
8550.2.a.bb	$8550$	$2$	8550.a	1.a	$1$	$1$	$1$	$68.272$	\(\Q\)	None	✓	$1$	$0$	\(1\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}+q^{8}+4q^{11}-2q^{13}+\cdots\)
8550.2.a.bc	$8550$	$2$	8550.a	1.a	$1$	$1$	$1$	$68.272$	\(\Q\)	None	✓	$1$	$0$	\(1\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}+q^{8}+4q^{11}+4q^{13}+\cdots\)
8550.2.a.bd	$8550$	$2$	8550.a	1.a	$1$	$1$	$1$	$68.272$	\(\Q\)	None	✓	$1$	$0$	\(1\)	\(0\)	\(0\)	\(1\)		$-$	$-$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}+q^{7}+q^{8}+q^{13}+q^{14}+\cdots\)
8550.2.a.be	$8550$	$2$	8550.a	1.a	$1$	$1$	$1$	$68.272$	\(\Q\)	None	✓	$1$	$1$	\(1\)	\(0\)	\(0\)	\(2\)		$-$	$-$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}+2q^{7}+q^{8}-3q^{11}+2q^{13}+\cdots\)
8550.2.a.bf	$8550$	$2$	8550.a	1.a	$1$	$1$	$1$	$68.272$	\(\Q\)	None	✓	$1$	$1$	\(1\)	\(0\)	\(0\)	\(2\)		$-$	$-$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}+2q^{7}+q^{8}-2q^{11}-4q^{13}+\cdots\)
8550.2.a.bg	$8550$	$2$	8550.a	1.a	$1$	$1$	$1$	$68.272$	\(\Q\)	None	✓	$1$	$1$	\(1\)	\(0\)	\(0\)	\(2\)		$-$	$+$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}+2q^{7}+q^{8}-2q^{11}+4q^{13}+\cdots\)
8550.2.a.bh	$8550$	$2$	8550.a	1.a	$1$	$1$	$1$	$68.272$	\(\Q\)	None	✓	$1$	$0$	\(1\)	\(0\)	\(0\)	\(2\)		$-$	$-$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}+2q^{7}+q^{8}-6q^{13}+2q^{14}+\cdots\)
8550.2.a.bi	$8550$	$2$	8550.a	1.a	$1$	$1$	$1$	$68.272$	\(\Q\)	None	✓	$1$	$0$	\(1\)	\(0\)	\(0\)	\(2\)		$-$	$-$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}+2q^{7}+q^{8}+3q^{11}+6q^{13}+\cdots\)
8550.2.a.bj	$8550$	$2$	8550.a	1.a	$1$	$1$	$1$	$68.272$	\(\Q\)	None	✓	$1$	$0$	\(1\)	\(0\)	\(0\)	\(4\)		$-$	$-$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}+4q^{7}+q^{8}+4q^{13}+4q^{14}+\cdots\)
8550.2.a.bk	$8550$	$2$	8550.a	1.a	$1$	$1$	$1$	$68.272$	\(\Q\)	None	✓	$1$	$0$	\(1\)	\(0\)	\(0\)	\(4\)		$-$	$-$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}+4q^{7}+q^{8}+q^{11}+4q^{14}+\cdots\)
8550.2.a.bl	$8550$	$2$	8550.a	1.a	$1$	$1$	$1$	$68.272$	\(\Q\)	None	✓	$1$	$0$	\(1\)	\(0\)	\(0\)	\(4\)		$-$	$+$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}+4q^{7}+q^{8}+6q^{11}+4q^{14}+\cdots\)
8550.2.a.bm	$8550$	$2$	8550.a	1.a	$1$	$1$	$1$	$68.272$	\(\Q\)	None	✓	$1$	$0$	\(1\)	\(0\)	\(0\)	\(5\)		$-$	$-$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}+5q^{7}+q^{8}+4q^{11}+q^{13}+\cdots\)
8550.2.a.bn	$8550$	$2$	8550.a	1.a	$1$	$2$	$2$	$68.272$	\(\Q(\sqrt{2}) \)	None	✓	$1$	$1$	\(-2\)	\(0\)	\(0\)	\(-6\)		$+$	$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}+(-3+\beta )q^{7}-q^{8}-\beta q^{11}+\cdots\)
8550.2.a.bo	$8550$	$2$	8550.a	1.a	$1$	$2$	$2$	$68.272$	\(\Q(\sqrt{3}) \)	None	✓	$1$	$0$	\(-2\)	\(0\)	\(0\)	\(-4\)		$+$	$+$	$+$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}-2q^{7}-q^{8}+\beta q^{11}+(-2+\cdots)q^{13}+\cdots\)
8550.2.a.bp	$8550$	$2$	8550.a	1.a	$1$	$2$	$2$	$68.272$	\(\Q(\sqrt{17}) \)	None	✓	$1$	$1$	\(-2\)	\(0\)	\(0\)	\(-2\)		$+$	$+$	$+$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}+(-1-\beta )q^{7}-q^{8}+2q^{11}+\cdots\)
8550.2.a.bq	$8550$	$2$	8550.a	1.a	$1$	$2$	$2$	$68.272$	\(\Q(\sqrt{10}) \)	None	✓	$1$	$0$	\(-2\)	\(0\)	\(0\)	\(0\)		$+$	$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}+\beta q^{7}-q^{8}+(-1-\beta )q^{11}+\cdots\)
8550.2.a.br	$8550$	$2$	8550.a	1.a	$1$	$2$	$2$	$68.272$	\(\Q(\sqrt{17}) \)	None	✓	$1$	$0$	\(-2\)	\(0\)	\(0\)	\(1\)		$+$	$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}+\beta q^{7}-q^{8}-4q^{11}+(2+\cdots)q^{13}+\cdots\)
8550.2.a.bs	$8550$	$2$	8550.a	1.a	$1$	$2$	$2$	$68.272$	\(\Q(\sqrt{3}) \)	None	✓	$1$	$1$	\(-2\)	\(0\)	\(0\)	\(2\)		$+$	$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}+(1+\beta )q^{7}-q^{8}+(-2+\cdots)q^{11}+\cdots\)
8550.2.a.bt	$8550$	$2$	8550.a	1.a	$1$	$2$	$2$	$68.272$	\(\Q(\sqrt{3}) \)	None	✓	$1$	$0$	\(-2\)	\(0\)	\(0\)	\(2\)		$+$	$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}+(1+\beta )q^{7}-q^{8}-3\beta q^{11}+\cdots\)
8550.2.a.bu	$8550$	$2$	8550.a	1.a	$1$	$2$	$2$	$68.272$	\(\Q(\sqrt{6}) \)	None	✓	$1$	$1$	\(-2\)	\(0\)	\(0\)	\(4\)		$+$	$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}+(2+\beta )q^{7}-q^{8}+(-1+\cdots)q^{11}+\cdots\)
8550.2.a.bv	$8550$	$2$	8550.a	1.a	$1$	$2$	$2$	$68.272$	\(\Q(\sqrt{6}) \)	None	✓	$1$	$1$	\(2\)	\(0\)	\(0\)	\(-4\)		$-$	$-$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}+(-2+\beta )q^{7}+q^{8}+(-1+\cdots)q^{11}+\cdots\)
8550.2.a.bw	$8550$	$2$	8550.a	1.a	$1$	$2$	$2$	$68.272$	\(\Q(\sqrt{3}) \)	None	✓	$1$	$1$	\(2\)	\(0\)	\(0\)	\(-4\)		$-$	$+$	$+$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}-2q^{7}+q^{8}+\beta q^{11}+(-2+\cdots)q^{13}+\cdots\)
8550.2.a.bx	$8550$	$2$	8550.a	1.a	$1$	$2$	$2$	$68.272$	\(\Q(\sqrt{17}) \)	None	✓	$1$	$0$	\(2\)	\(0\)	\(0\)	\(-2\)		$-$	$+$	$+$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}+(-1-\beta )q^{7}+q^{8}-2q^{11}+\cdots\)
8550.2.a.by	$8550$	$2$	8550.a	1.a	$1$	$2$	$2$	$68.272$	\(\Q(\sqrt{3}) \)	None	✓	$1$	$1$	\(2\)	\(0\)	\(0\)	\(-2\)		$-$	$-$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}+(-1+\beta )q^{7}+q^{8}+(-2+\cdots)q^{11}+\cdots\)
8550.2.a.bz	$8550$	$2$	8550.a	1.a	$1$	$2$	$2$	$68.272$	\(\Q(\sqrt{3}) \)	None	✓	$1$	$0$	\(2\)	\(0\)	\(0\)	\(-2\)		$-$	$-$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}+(-1+\beta )q^{7}+q^{8}+3\beta q^{11}+\cdots\)
8550.2.a.ca	$8550$	$2$	8550.a	1.a	$1$	$2$	$2$	$68.272$	\(\Q(\sqrt{10}) \)	None	✓	$1$	$0$	\(2\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}+\beta q^{7}+q^{8}+(-1+\beta )q^{11}+\cdots\)
8550.2.a.cb	$8550$	$2$	8550.a	1.a	$1$	$2$	$2$	$68.272$	\(\Q(\sqrt{2}) \)	None	✓	$1$	$0$	\(2\)	\(0\)	\(0\)	\(6\)		$-$	$-$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}+(3+\beta )q^{7}+q^{8}+\beta q^{11}+\cdots\)
8550.2.a.cc	$8550$	$2$	8550.a	1.a	$1$	$3$	$3$	$68.272$	3.3.788.1	None	✓	$1$	$0$	\(-3\)	\(0\)	\(0\)	\(-2\)		$+$	$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}+(-1+\beta _{1})q^{7}-q^{8}+(1+\cdots)q^{11}+\cdots\)
8550.2.a.cd	$8550$	$2$	8550.a	1.a	$1$	$3$	$3$	$68.272$	3.3.564.1	None	✓	$1$	$0$	\(-3\)	\(0\)	\(0\)	\(-2\)		$+$	$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}+(-\beta _{1}+\beta _{2})q^{7}-q^{8}+\cdots\)
8550.2.a.ce	$8550$	$2$	8550.a	1.a	$1$	$3$	$3$	$68.272$	3.3.148.1	None	✓	$1$	$0$	\(-3\)	\(0\)	\(0\)	\(0\)		$+$	$-$	$-$	$-$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}+\beta _{2}q^{7}-q^{8}+(-3-\beta _{1}+\cdots)q^{11}+\cdots\)
8550.2.a.cf	$8550$	$2$	8550.a	1.a	$1$	$3$	$3$	$68.272$	3.3.148.1	None	✓	$1$	$1$	\(-3\)	\(0\)	\(0\)	\(0\)		$+$	$-$	$-$	$+$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}+\beta _{2}q^{7}-q^{8}+(1-\beta _{1}+\cdots)q^{11}+\cdots\)
8550.2.a.cg	$8550$	$2$	8550.a	1.a	$1$	$3$	$3$	$68.272$	3.3.564.1	None	✓	$1$	$1$	\(-3\)	\(0\)	\(0\)	\(2\)		$+$	$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}+(\beta _{1}-\beta _{2})q^{7}-q^{8}+(-2+\cdots)q^{11}+\cdots\)
8550.2.a.ch	$8550$	$2$	8550.a	1.a	$1$	$3$	$3$	$68.272$	3.3.788.1	None	✓	$1$	$1$	\(-3\)	\(0\)	\(0\)	\(2\)		$+$	$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}+(1-\beta _{1})q^{7}-q^{8}+(-1+\cdots)q^{11}+\cdots\)
8550.2.a.ci	$8550$	$2$	8550.a	1.a	$1$	$3$	$3$	$68.272$	3.3.993.1	None	✓	$1$	$0$	\(-3\)	\(0\)	\(0\)	\(2\)		$+$	$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}+(1-\beta _{2})q^{7}-q^{8}-2\beta _{1}q^{11}+\cdots\)
8550.2.a.cj	$8550$	$2$	8550.a	1.a	$1$	$3$	$3$	$68.272$	3.3.257.1	None	✓	$1$	$0$	\(-3\)	\(0\)	\(0\)	\(2\)		$+$	$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}+(1-\beta _{1}+2\beta _{2})q^{7}-q^{8}+\cdots\)
8550.2.a.ck	$8550$	$2$	8550.a	1.a	$1$	$3$	$3$	$68.272$	3.3.568.1	None	✓	$1$	$0$	\(-3\)	\(0\)	\(0\)	\(4\)		$+$	$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}+(1+\beta _{1})q^{7}-q^{8}+(\beta _{1}+\cdots)q^{11}+\cdots\)
8550.2.a.cl	$8550$	$2$	8550.a	1.a	$1$	$3$	$3$	$68.272$	3.3.568.1	None	✓	$1$	$1$	\(3\)	\(0\)	\(0\)	\(-4\)		$-$	$-$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}+(-1-\beta _{1})q^{7}+q^{8}+(\beta _{1}+\cdots)q^{11}+\cdots\)
8550.2.a.cm	$8550$	$2$	8550.a	1.a	$1$	$3$	$3$	$68.272$	3.3.788.1	None	✓	$1$	$1$	\(3\)	\(0\)	\(0\)	\(-2\)		$-$	$+$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}+(-1+\beta _{1})q^{7}+q^{8}+(-1+\cdots)q^{11}+\cdots\)
8550.2.a.cn	$8550$	$2$	8550.a	1.a	$1$	$3$	$3$	$68.272$	3.3.564.1	None	✓	$1$	$1$	\(3\)	\(0\)	\(0\)	\(-2\)		$-$	$+$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}+(-\beta _{1}+\beta _{2})q^{7}+q^{8}+\cdots\)
8550.2.a.co	$8550$	$2$	8550.a	1.a	$1$	$3$	$3$	$68.272$	3.3.257.1	None	✓	$1$	$0$	\(3\)	\(0\)	\(0\)	\(-2\)		$-$	$-$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}+(-1+\beta _{1}-2\beta _{2})q^{7}+\cdots\)
8550.2.a.cp	$8550$	$2$	8550.a	1.a	$1$	$3$	$3$	$68.272$	3.3.993.1	None	✓	$1$	$0$	\(3\)	\(0\)	\(0\)	\(-2\)		$-$	$-$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}+(-1+\beta _{2})q^{7}+q^{8}-2\beta _{1}q^{11}+\cdots\)
8550.2.a.cq	$8550$	$2$	8550.a	1.a	$1$	$3$	$3$	$68.272$	3.3.148.1	None	✓	$1$	$1$	\(3\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$-$	$-$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}-\beta _{2}q^{7}+q^{8}+(-3-\beta _{1}+\cdots)q^{11}+\cdots\)
8550.2.a.cr	$8550$	$2$	8550.a	1.a	$1$	$3$	$3$	$68.272$	3.3.148.1	None	✓	$1$	$0$	\(3\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$-$	$+$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}-\beta _{2}q^{7}+q^{8}+(1-\beta _{1}+\cdots)q^{11}+\cdots\)
8550.2.a.cs	$8550$	$2$	8550.a	1.a	$1$	$3$	$3$	$68.272$	3.3.564.1	None	✓	$1$	$0$	\(3\)	\(0\)	\(0\)	\(2\)		$-$	$+$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}+(\beta _{1}-\beta _{2})q^{7}+q^{8}+(2+\cdots)q^{11}+\cdots\)
8550.2.a.ct	$8550$	$2$	8550.a	1.a	$1$	$3$	$3$	$68.272$	3.3.788.1	None	✓	$1$	$0$	\(3\)	\(0\)	\(0\)	\(2\)		$-$	$+$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}+(1-\beta _{1})q^{7}+q^{8}+(1+\beta _{1}+\cdots)q^{11}+\cdots\)
8550.2.a.cu	$8550$	$2$	8550.a	1.a	$1$	$4$	$4$	$68.272$	\(\Q(\zeta_{24})^+\)	None	✓	$2$	$0$	\(-4\)	\(0\)	\(0\)	\(0\)		$+$	$+$	$-$	$+$	$-$	$2^{3}$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}+\beta _{1}q^{7}-q^{8}-\beta _{3}q^{11}+\cdots\)
8550.2.a.cv	$8550$	$2$	8550.a	1.a	$1$	$4$	$4$	$68.272$	\(\Q(\zeta_{24})^+\)	None	✓	$2$	$1$	\(4\)	\(0\)	\(0\)	\(0\)		$-$	$+$	$-$	$+$	$+$	$2^{3}$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}+\beta _{1}q^{7}+q^{8}+\beta _{3}q^{11}+\cdots\)
8550.2.a.cw	$8550$	$2$	8550.a	1.a	$1$	$6$	$6$	$68.272$	6.6.3356224.1	None	✓	$2$	$1$	\(-6\)	\(0\)	\(0\)	\(0\)		$+$	$+$	$-$	$-$	$+$	$2^{5}$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}+\beta _{1}q^{7}-q^{8}+(-\beta _{1}+\beta _{4}+\cdots)q^{11}+\cdots\)
8550.2.a.cx	$8550$	$2$	8550.a	1.a	$1$	$6$	$6$	$68.272$	6.6.3356224.1	None	✓	$2$	$0$	\(6\)	\(0\)	\(0\)	\(0\)		$-$	$+$	$-$	$-$	$-$	$2^{5}$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}+\beta _{1}q^{7}+q^{8}+(\beta _{1}-\beta _{4}+\cdots)q^{11}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(8550))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(8550)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 16}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(19))\)\(^{\oplus 18}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(30))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(38))\)\(^{\oplus 9}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(45))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(50))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(57))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(75))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(90))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(95))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(114))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(150))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(171))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(190))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(225))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(285))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(342))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(450))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(475))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(570))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(855))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(950))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1425))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1710))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(2850))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(4275))\)\(^{\oplus 2}\)