Space of modular forms of level 9920, weight 2, and trivial character

• Label: 9920.2.a
• Level: $9920$
• Weight: $2$
• Character orbit: 9920.a
• Rep. character: $\chi_{9920}(1,\cdot)$
• Character field: $\Q$
• Dimension: $240$
• Newform subspaces: $80$
• Sturm bound: $3072$
• Trace bound: $13$

Defining parameters

Level:	\( N \)	\(=\)	\( 9920 = 2^{6} \cdot 5 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9920.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 80 \)
Sturm bound:			\(3072\)
Trace bound:			\(13\)
Distinguishing \(T_p\):			\(3\), \(7\), \(11\), \(13\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(9920))\).

	Total	New	Old
Modular forms	1560	240	1320
Cusp forms	1513	240	1273
Eisenstein series	47	0	47

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

\(2\)	\(5\)	\(31\)	Fricke		Total				Cusp				Eisenstein
					All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)	\(+\)		\(183\)	\(28\)	\(155\)		\(178\)	\(28\)	\(150\)		\(5\)	\(0\)	\(5\)
\(+\)	\(+\)	\(-\)	\(-\)		\(207\)	\(34\)	\(173\)		\(201\)	\(34\)	\(167\)		\(6\)	\(0\)	\(6\)
\(+\)	\(-\)	\(+\)	\(-\)		\(195\)	\(32\)	\(163\)		\(189\)	\(32\)	\(157\)		\(6\)	\(0\)	\(6\)
\(+\)	\(-\)	\(-\)	\(+\)		\(195\)	\(26\)	\(169\)		\(189\)	\(26\)	\(163\)		\(6\)	\(0\)	\(6\)
\(-\)	\(+\)	\(+\)	\(-\)		\(201\)	\(32\)	\(169\)		\(195\)	\(32\)	\(163\)		\(6\)	\(0\)	\(6\)
\(-\)	\(+\)	\(-\)	\(+\)		\(189\)	\(26\)	\(163\)		\(183\)	\(26\)	\(157\)		\(6\)	\(0\)	\(6\)
\(-\)	\(-\)	\(+\)	\(+\)		\(189\)	\(28\)	\(161\)		\(183\)	\(28\)	\(155\)		\(6\)	\(0\)	\(6\)
\(-\)	\(-\)	\(-\)	\(-\)		\(201\)	\(34\)	\(167\)		\(195\)	\(34\)	\(161\)		\(6\)	\(0\)	\(6\)
Plus space			\(+\)		\(756\)	\(108\)	\(648\)		\(733\)	\(108\)	\(625\)		\(23\)	\(0\)	\(23\)
Minus space			\(-\)		\(804\)	\(132\)	\(672\)		\(780\)	\(132\)	\(648\)		\(24\)	\(0\)	\(24\)

Trace form

\( 240 q + 240 q^{9} + 240 q^{25} + 32 q^{29} + 32 q^{33} + 32 q^{37} + 32 q^{41} + 240 q^{49} + 32 q^{53} + 32 q^{57} + 32 q^{77} + 304 q^{81} + 32 q^{89}+O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(9920))\) 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	31
9920.2.a.a	$9920$	$2$	9920.a	1.a	$1$	$1$	$1$	$79.212$	\(\Q\)	None	✓				620.2.a.a	$1$	$1$	\(0\)	\(-3\)	\(-1\)	\(2\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-3q^{3}-q^{5}+2q^{7}+6q^{9}+2q^{11}+\cdots\)
9920.2.a.b	$9920$	$2$	9920.a	1.a	$1$	$1$	$1$	$79.212$	\(\Q\)	None	✓				1240.2.a.g	$1$	$1$	\(0\)	\(-3\)	\(-1\)	\(2\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-3q^{3}-q^{5}+2q^{7}+6q^{9}+2q^{11}+\cdots\)
9920.2.a.c	$9920$	$2$	9920.a	1.a	$1$	$1$	$1$	$79.212$	\(\Q\)	None	✓				1240.2.a.e	$1$	$2$	\(0\)	\(-2\)	\(1\)	\(-4\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{3}+q^{5}-4q^{7}+q^{9}-4q^{11}+\cdots\)
9920.2.a.d	$9920$	$2$	9920.a	1.a	$1$	$1$	$1$	$79.212$	\(\Q\)	None	✓				310.2.a.b	$1$	$0$	\(0\)	\(-2\)	\(1\)	\(0\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{3}+q^{5}+q^{9}-2q^{11}-2q^{15}+\cdots\)
9920.2.a.e	$9920$	$2$	9920.a	1.a	$1$	$1$	$1$	$79.212$	\(\Q\)	None	✓				1240.2.a.f	$1$	$1$	\(0\)	\(-2\)	\(1\)	\(0\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{3}+q^{5}+q^{9}-4q^{13}-2q^{15}+\cdots\)
9920.2.a.f	$9920$	$2$	9920.a	1.a	$1$	$1$	$1$	$79.212$	\(\Q\)	None	✓				4960.2.a.a	$1$	$1$	\(0\)	\(-2\)	\(1\)	\(0\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{3}+q^{5}+q^{9}+6q^{11}+4q^{13}+\cdots\)
9920.2.a.g	$9920$	$2$	9920.a	1.a	$1$	$1$	$1$	$79.212$	\(\Q\)	None	✓				155.2.a.b	$1$	$1$	\(0\)	\(-2\)	\(1\)	\(4\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{3}+q^{5}+4q^{7}+q^{9}-4q^{11}+\cdots\)
9920.2.a.h	$9920$	$2$	9920.a	1.a	$1$	$1$	$1$	$79.212$	\(\Q\)	None	✓				310.2.a.a	$1$	$1$	\(0\)	\(-2\)	\(1\)	\(4\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{3}+q^{5}+4q^{7}+q^{9}+4q^{13}+\cdots\)
9920.2.a.i	$9920$	$2$	9920.a	1.a	$1$	$1$	$1$	$79.212$	\(\Q\)	None	✓				1240.2.a.b	$1$	$1$	\(0\)	\(-1\)	\(-1\)	\(-2\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-q^{5}-2q^{7}-2q^{9}-2q^{11}+\cdots\)
9920.2.a.j	$9920$	$2$	9920.a	1.a	$1$	$1$	$1$	$79.212$	\(\Q\)	None	✓				155.2.a.a	$1$	$0$	\(0\)	\(-1\)	\(-1\)	\(2\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-q^{5}+2q^{7}-2q^{9}+2q^{11}+\cdots\)
9920.2.a.k	$9920$	$2$	9920.a	1.a	$1$	$1$	$1$	$79.212$	\(\Q\)	None	✓				620.2.a.c	$1$	$1$	\(0\)	\(-1\)	\(1\)	\(-4\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+q^{5}-4q^{7}-2q^{9}-2q^{13}+\cdots\)
9920.2.a.l	$9920$	$2$	9920.a	1.a	$1$	$1$	$1$	$79.212$	\(\Q\)	None	✓				155.2.a.c	$1$	$0$	\(0\)	\(-1\)	\(1\)	\(0\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+q^{5}-2q^{9}-4q^{11}+6q^{13}+\cdots\)
9920.2.a.m	$9920$	$2$	9920.a	1.a	$1$	$1$	$1$	$79.212$	\(\Q\)	None	✓				1240.2.a.a	$1$	$1$	\(0\)	\(-1\)	\(1\)	\(0\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+q^{5}-2q^{9}+2q^{13}-q^{15}+\cdots\)
9920.2.a.n	$9920$	$2$	9920.a	1.a	$1$	$1$	$1$	$79.212$	\(\Q\)	None	✓				4960.2.a.d	$1$	$1$	\(0\)	\(0\)	\(-1\)	\(-2\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{5}-2q^{7}-3q^{9}-4q^{11}+4q^{19}+\cdots\)
9920.2.a.o	$9920$	$2$	9920.a	1.a	$1$	$1$	$1$	$79.212$	\(\Q\)	None	✓				620.2.a.b	$1$	$1$	\(0\)	\(0\)	\(-1\)	\(-2\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{5}-2q^{7}-3q^{9}+4q^{11}+4q^{13}+\cdots\)
9920.2.a.p	$9920$	$2$	9920.a	1.a	$1$	$1$	$1$	$79.212$	\(\Q\)	None	✓				1240.2.a.d	$1$	$0$	\(0\)	\(0\)	\(-1\)	\(0\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{5}-3q^{9}-4q^{11}-2q^{13}+6q^{17}+\cdots\)
9920.2.a.q	$9920$	$2$	9920.a	1.a	$1$	$1$	$1$	$79.212$	\(\Q\)	None	✓				1240.2.a.d	$1$	$0$	\(0\)	\(0\)	\(-1\)	\(0\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{5}-3q^{9}+4q^{11}-2q^{13}+6q^{17}+\cdots\)
9920.2.a.r	$9920$	$2$	9920.a	1.a	$1$	$1$	$1$	$79.212$	\(\Q\)	None	✓				620.2.a.b	$1$	$1$	\(0\)	\(0\)	\(-1\)	\(2\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{5}+2q^{7}-3q^{9}-4q^{11}+4q^{13}+\cdots\)
9920.2.a.s	$9920$	$2$	9920.a	1.a	$1$	$1$	$1$	$79.212$	\(\Q\)	None	✓				4960.2.a.d	$1$	$0$	\(0\)	\(0\)	\(-1\)	\(2\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{5}+2q^{7}-3q^{9}+4q^{11}-4q^{19}+\cdots\)
9920.2.a.t	$9920$	$2$	9920.a	1.a	$1$	$1$	$1$	$79.212$	\(\Q\)	None	✓				4960.2.a.b	$1$	$1$	\(0\)	\(0\)	\(1\)	\(-4\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{5}-4q^{7}-3q^{9}+2q^{11}+6q^{13}+\cdots\)
9920.2.a.u	$9920$	$2$	9920.a	1.a	$1$	$1$	$1$	$79.212$	\(\Q\)	None	✓				1240.2.a.c	$1$	$1$	\(0\)	\(0\)	\(1\)	\(0\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{5}-3q^{9}-2q^{11}-2q^{13}+8q^{19}+\cdots\)
9920.2.a.v	$9920$	$2$	9920.a	1.a	$1$	$1$	$1$	$79.212$	\(\Q\)	None	✓				1240.2.a.c	$1$	$1$	\(0\)	\(0\)	\(1\)	\(0\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{5}-3q^{9}+2q^{11}-2q^{13}-8q^{19}+\cdots\)
9920.2.a.w	$9920$	$2$	9920.a	1.a	$1$	$1$	$1$	$79.212$	\(\Q\)	None	✓				4960.2.a.b	$1$	$0$	\(0\)	\(0\)	\(1\)	\(4\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{5}+4q^{7}-3q^{9}-2q^{11}+6q^{13}+\cdots\)
9920.2.a.x	$9920$	$2$	9920.a	1.a	$1$	$1$	$1$	$79.212$	\(\Q\)	None	✓				155.2.a.a	$1$	$0$	\(0\)	\(1\)	\(-1\)	\(-2\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-q^{5}-2q^{7}-2q^{9}-2q^{11}+\cdots\)
9920.2.a.y	$9920$	$2$	9920.a	1.a	$1$	$1$	$1$	$79.212$	\(\Q\)	None	✓				1240.2.a.b	$1$	$1$	\(0\)	\(1\)	\(-1\)	\(2\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-q^{5}+2q^{7}-2q^{9}+2q^{11}+\cdots\)
9920.2.a.z	$9920$	$2$	9920.a	1.a	$1$	$1$	$1$	$79.212$	\(\Q\)	None	✓				1240.2.a.a	$1$	$1$	\(0\)	\(1\)	\(1\)	\(0\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+q^{5}-2q^{9}+2q^{13}+q^{15}+\cdots\)
9920.2.a.ba	$9920$	$2$	9920.a	1.a	$1$	$1$	$1$	$79.212$	\(\Q\)	None	✓				155.2.a.c	$1$	$0$	\(0\)	\(1\)	\(1\)	\(0\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+q^{5}-2q^{9}+4q^{11}+6q^{13}+\cdots\)
9920.2.a.bb	$9920$	$2$	9920.a	1.a	$1$	$1$	$1$	$79.212$	\(\Q\)	None	✓				620.2.a.c	$1$	$1$	\(0\)	\(1\)	\(1\)	\(4\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+q^{5}+4q^{7}-2q^{9}-2q^{13}+\cdots\)
9920.2.a.bc	$9920$	$2$	9920.a	1.a	$1$	$1$	$1$	$79.212$	\(\Q\)	None	✓				310.2.a.a	$1$	$1$	\(0\)	\(2\)	\(1\)	\(-4\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{3}+q^{5}-4q^{7}+q^{9}+4q^{13}+\cdots\)
9920.2.a.bd	$9920$	$2$	9920.a	1.a	$1$	$1$	$1$	$79.212$	\(\Q\)	None	✓				155.2.a.b	$1$	$1$	\(0\)	\(2\)	\(1\)	\(-4\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{3}+q^{5}-4q^{7}+q^{9}+4q^{11}+\cdots\)
9920.2.a.be	$9920$	$2$	9920.a	1.a	$1$	$1$	$1$	$79.212$	\(\Q\)	None	✓				4960.2.a.a	$1$	$0$	\(0\)	\(2\)	\(1\)	\(0\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{3}+q^{5}+q^{9}-6q^{11}+4q^{13}+\cdots\)
9920.2.a.bf	$9920$	$2$	9920.a	1.a	$1$	$1$	$1$	$79.212$	\(\Q\)	None	✓				1240.2.a.f	$1$	$1$	\(0\)	\(2\)	\(1\)	\(0\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{3}+q^{5}+q^{9}-4q^{13}+2q^{15}+\cdots\)
9920.2.a.bg	$9920$	$2$	9920.a	1.a	$1$	$1$	$1$	$79.212$	\(\Q\)	None	✓				310.2.a.b	$1$	$0$	\(0\)	\(2\)	\(1\)	\(0\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{3}+q^{5}+q^{9}+2q^{11}+2q^{15}+\cdots\)
9920.2.a.bh	$9920$	$2$	9920.a	1.a	$1$	$1$	$1$	$79.212$	\(\Q\)	None	✓				1240.2.a.e	$1$	$0$	\(0\)	\(2\)	\(1\)	\(4\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{3}+q^{5}+4q^{7}+q^{9}+4q^{11}+\cdots\)
9920.2.a.bi	$9920$	$2$	9920.a	1.a	$1$	$1$	$1$	$79.212$	\(\Q\)	None	✓				620.2.a.a	$1$	$1$	\(0\)	\(3\)	\(-1\)	\(-2\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+3q^{3}-q^{5}-2q^{7}+6q^{9}-2q^{11}+\cdots\)
9920.2.a.bj	$9920$	$2$	9920.a	1.a	$1$	$1$	$1$	$79.212$	\(\Q\)	None	✓				1240.2.a.g	$1$	$1$	\(0\)	\(3\)	\(-1\)	\(-2\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+3q^{3}-q^{5}-2q^{7}+6q^{9}-2q^{11}+\cdots\)
9920.2.a.bk	$9920$	$2$	9920.a	1.a	$1$	$2$	$2$	$79.212$	\(\Q(\sqrt{5}) \)	None	✓				4960.2.a.g	$1$	$1$	\(0\)	\(-2\)	\(-2\)	\(0\)		$+$	$+$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q+(-1-\beta )q^{3}-q^{5}+(3+2\beta )q^{9}-2\beta q^{11}+\cdots\)
9920.2.a.bl	$9920$	$2$	9920.a	1.a	$1$	$2$	$2$	$79.212$	\(\Q(\sqrt{3}) \)	None	✓				310.2.a.c	$1$	$0$	\(0\)	\(-2\)	\(2\)	\(0\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta )q^{3}+q^{5}+2\beta q^{7}+(1-2\beta )q^{9}+\cdots\)
9920.2.a.bm	$9920$	$2$	9920.a	1.a	$1$	$2$	$2$	$79.212$	\(\Q(\sqrt{17}) \)	None	✓				1240.2.a.h	$1$	$1$	\(0\)	\(-1\)	\(2\)	\(0\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta q^{3}+q^{5}+(1+\beta )q^{9}+(-2+2\beta )q^{11}+\cdots\)
9920.2.a.bn	$9920$	$2$	9920.a	1.a	$1$	$2$	$2$	$79.212$	\(\Q(\sqrt{2}) \)	None	✓				1240.2.a.i	$1$	$1$	\(0\)	\(0\)	\(-2\)	\(-4\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{3}-q^{5}-2q^{7}-q^{9}+(2+\beta )q^{11}+\cdots\)
9920.2.a.bo	$9920$	$2$	9920.a	1.a	$1$	$2$	$2$	$79.212$	\(\Q(\sqrt{6}) \)	None	✓				310.2.a.d	$1$	$1$	\(0\)	\(0\)	\(-2\)	\(-4\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{3}-q^{5}-2q^{7}+3q^{9}+(-2+\beta )q^{11}+\cdots\)
9920.2.a.bp	$9920$	$2$	9920.a	1.a	$1$	$2$	$2$	$79.212$	\(\Q(\sqrt{2}) \)	None	✓				1240.2.a.i	$1$	$1$	\(0\)	\(0\)	\(-2\)	\(4\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{3}-q^{5}+2q^{7}-q^{9}+(-2+\beta )q^{11}+\cdots\)
9920.2.a.bq	$9920$	$2$	9920.a	1.a	$1$	$2$	$2$	$79.212$	\(\Q(\sqrt{6}) \)	None	✓				310.2.a.d	$1$	$1$	\(0\)	\(0\)	\(-2\)	\(4\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{3}-q^{5}+2q^{7}+3q^{9}+(2+\beta )q^{11}+\cdots\)
9920.2.a.br	$9920$	$2$	9920.a	1.a	$1$	$2$	$2$	$79.212$	\(\Q(\sqrt{17}) \)	None	✓				1240.2.a.h	$1$	$1$	\(0\)	\(1\)	\(2\)	\(0\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{3}+q^{5}+(1+\beta )q^{9}+(2-2\beta )q^{11}+\cdots\)
9920.2.a.bs	$9920$	$2$	9920.a	1.a	$1$	$2$	$2$	$79.212$	\(\Q(\sqrt{5}) \)	None	✓				4960.2.a.g	$1$	$0$	\(0\)	\(2\)	\(-2\)	\(0\)		$+$	$+$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{3}-q^{5}+(3+2\beta )q^{9}+2\beta q^{11}+\cdots\)
9920.2.a.bt	$9920$	$2$	9920.a	1.a	$1$	$2$	$2$	$79.212$	\(\Q(\sqrt{3}) \)	None	✓				310.2.a.c	$1$	$0$	\(0\)	\(2\)	\(2\)	\(0\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{3}+q^{5}+2\beta q^{7}+(1+2\beta )q^{9}+\cdots\)
9920.2.a.bu	$9920$	$2$	9920.a	1.a	$1$	$3$	$3$	$79.212$	3.3.148.1	None	✓				1240.2.a.j	$1$	$0$	\(0\)	\(-3\)	\(-3\)	\(2\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-1-\beta _{2})q^{3}-q^{5}+(1-\beta _{1}-\beta _{2})q^{7}+\cdots\)
9920.2.a.bv	$9920$	$2$	9920.a	1.a	$1$	$3$	$3$	$79.212$	3.3.756.1	None	✓				620.2.a.d	$1$	$0$	\(0\)	\(-3\)	\(3\)	\(0\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{3}+q^{5}-\beta _{2}q^{7}+(2-2\beta _{1}+\cdots)q^{9}+\cdots\)
9920.2.a.bw	$9920$	$2$	9920.a	1.a	$1$	$3$	$3$	$79.212$	3.3.148.1	None	✓				310.2.a.e	$1$	$0$	\(0\)	\(-2\)	\(-3\)	\(0\)		$+$	$+$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{3}-q^{5}+(\beta _{1}-\beta _{2})q^{7}+\cdots\)
9920.2.a.bx	$9920$	$2$	9920.a	1.a	$1$	$3$	$3$	$79.212$	3.3.148.1	None	✓				310.2.a.e	$1$	$0$	\(0\)	\(2\)	\(-3\)	\(0\)		$-$	$+$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{3}-q^{5}+(-\beta _{1}+\beta _{2})q^{7}+\cdots\)
9920.2.a.by	$9920$	$2$	9920.a	1.a	$1$	$3$	$3$	$79.212$	3.3.148.1	None	✓				1240.2.a.j	$1$	$0$	\(0\)	\(3\)	\(-3\)	\(-2\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta _{2})q^{3}-q^{5}+(-1+\beta _{1}+\beta _{2})q^{7}+\cdots\)
9920.2.a.bz	$9920$	$2$	9920.a	1.a	$1$	$3$	$3$	$79.212$	3.3.756.1	None	✓				620.2.a.d	$1$	$0$	\(0\)	\(3\)	\(3\)	\(0\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{3}+q^{5}+\beta _{2}q^{7}+(2-2\beta _{1}+\cdots)q^{9}+\cdots\)
9920.2.a.ca	$9920$	$2$	9920.a	1.a	$1$	$4$	$4$	$79.212$	4.4.25492.1	None	✓				620.2.a.e	$1$	$0$	\(0\)	\(-3\)	\(-4\)	\(8\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-1-\beta _{2})q^{3}-q^{5}+(2-\beta _{2}-\beta _{3})q^{7}+\cdots\)
9920.2.a.cb	$9920$	$2$	9920.a	1.a	$1$	$4$	$4$	$79.212$	4.4.8468.1	None	✓				155.2.a.e	$1$	$1$	\(0\)	\(-1\)	\(-4\)	\(2\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}-q^{5}+(1-\beta _{1}+2\beta _{2}-\beta _{3})q^{7}+\cdots\)
9920.2.a.cc	$9920$	$2$	9920.a	1.a	$1$	$4$	$4$	$79.212$	4.4.112820.1	None	✓				1240.2.a.k	$1$	$0$	\(0\)	\(-1\)	\(4\)	\(-4\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}+q^{5}+(-1+\beta _{1}-\beta _{3})q^{7}+\cdots\)
9920.2.a.cd	$9920$	$2$	9920.a	1.a	$1$	$4$	$4$	$79.212$	4.4.20308.1	None	✓				155.2.a.d	$1$	$1$	\(0\)	\(-1\)	\(4\)	\(0\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{3}q^{3}+q^{5}-\beta _{2}q^{7}+(2-\beta _{1}-2\beta _{2}+\cdots)q^{9}+\cdots\)
9920.2.a.ce	$9920$	$2$	9920.a	1.a	$1$	$4$	$4$	$79.212$	4.4.6224.1	None	✓				4960.2.a.i	$1$	$0$	\(0\)	\(0\)	\(-4\)	\(0\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}-q^{5}+(\beta _{1}+\beta _{2})q^{7}+(\beta _{1}+\beta _{2}+\cdots)q^{9}+\cdots\)
9920.2.a.cf	$9920$	$2$	9920.a	1.a	$1$	$4$	$4$	$79.212$	4.4.6224.1	None	✓				4960.2.a.i	$1$	$1$	\(0\)	\(0\)	\(-4\)	\(0\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}-q^{5}+(-\beta _{1}-\beta _{2})q^{7}+(\beta _{1}+\cdots)q^{9}+\cdots\)
9920.2.a.cg	$9920$	$2$	9920.a	1.a	$1$	$4$	$4$	$79.212$	4.4.8468.1	None	✓				155.2.a.e	$1$	$1$	\(0\)	\(1\)	\(-4\)	\(-2\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}-q^{5}+(-1+\beta _{1}-2\beta _{2}+\beta _{3})q^{7}+\cdots\)
9920.2.a.ch	$9920$	$2$	9920.a	1.a	$1$	$4$	$4$	$79.212$	4.4.20308.1	None	✓				155.2.a.d	$1$	$1$	\(0\)	\(1\)	\(4\)	\(0\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{3}q^{3}+q^{5}+\beta _{2}q^{7}+(2-\beta _{1}-2\beta _{2}+\cdots)q^{9}+\cdots\)
9920.2.a.ci	$9920$	$2$	9920.a	1.a	$1$	$4$	$4$	$79.212$	4.4.112820.1	None	✓				1240.2.a.k	$1$	$0$	\(0\)	\(1\)	\(4\)	\(4\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}+q^{5}+(1-\beta _{1}+\beta _{3})q^{7}+(2+\cdots)q^{9}+\cdots\)
9920.2.a.cj	$9920$	$2$	9920.a	1.a	$1$	$4$	$4$	$79.212$	4.4.25492.1	None	✓				620.2.a.e	$1$	$0$	\(0\)	\(3\)	\(-4\)	\(-8\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta _{2})q^{3}-q^{5}+(-2+\beta _{2}+\beta _{3})q^{7}+\cdots\)
9920.2.a.ck	$9920$	$2$	9920.a	1.a	$1$	$6$	$6$	$79.212$	6.6.50574608.1	None	✓				4960.2.a.l	$1$	$1$	\(0\)	\(-4\)	\(6\)	\(-4\)		$+$	$-$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{3}+q^{5}+(-1+\beta _{5})q^{7}+\cdots\)
9920.2.a.cl	$9920$	$2$	9920.a	1.a	$1$	$6$	$6$	$79.212$	6.6.197463312.1	None	✓				4960.2.a.k	$1$	$1$	\(0\)	\(-4\)	\(6\)	\(0\)		$-$	$-$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{3}+q^{5}-\beta _{4}q^{7}+(2-\beta _{1}+\cdots)q^{9}+\cdots\)
9920.2.a.cm	$9920$	$2$	9920.a	1.a	$1$	$6$	$6$	$79.212$	6.6.473125168.1	None	✓				1240.2.a.l	$1$	$0$	\(0\)	\(-3\)	\(6\)	\(4\)		$-$	$-$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}+q^{5}+(-\beta _{2}-\beta _{4})q^{7}+(\beta _{1}+\cdots)q^{9}+\cdots\)
9920.2.a.cn	$9920$	$2$	9920.a	1.a	$1$	$6$	$6$	$79.212$	6.6.27554944.1	None	✓				4960.2.a.m	$1$	$0$	\(0\)	\(-2\)	\(-6\)	\(4\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}-q^{5}+(1-\beta _{4})q^{7}+(-1+\beta _{1}+\cdots)q^{9}+\cdots\)
9920.2.a.co	$9920$	$2$	9920.a	1.a	$1$	$6$	$6$	$79.212$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓				1240.2.a.m	$1$	$0$	\(0\)	\(-1\)	\(-6\)	\(2\)		$-$	$+$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}-q^{5}+\beta _{2}q^{7}+(4-\beta _{3}+\beta _{4}+\cdots)q^{9}+\cdots\)
9920.2.a.cp	$9920$	$2$	9920.a	1.a	$1$	$6$	$6$	$79.212$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓				1240.2.a.m	$1$	$0$	\(0\)	\(1\)	\(-6\)	\(-2\)		$+$	$+$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}-q^{5}-\beta _{2}q^{7}+(4-\beta _{3}+\beta _{4}+\cdots)q^{9}+\cdots\)
9920.2.a.cq	$9920$	$2$	9920.a	1.a	$1$	$6$	$6$	$79.212$	6.6.27554944.1	None	✓				4960.2.a.m	$1$	$1$	\(0\)	\(2\)	\(-6\)	\(-4\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}-q^{5}+(-1+\beta _{4})q^{7}+(-1+\cdots)q^{9}+\cdots\)
9920.2.a.cr	$9920$	$2$	9920.a	1.a	$1$	$6$	$6$	$79.212$	6.6.473125168.1	None	✓				1240.2.a.l	$1$	$0$	\(0\)	\(3\)	\(6\)	\(-4\)		$+$	$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}+q^{5}+(\beta _{2}+\beta _{4})q^{7}+(\beta _{1}-\beta _{2}+\cdots)q^{9}+\cdots\)
9920.2.a.cs	$9920$	$2$	9920.a	1.a	$1$	$6$	$6$	$79.212$	6.6.197463312.1	None	✓				4960.2.a.k	$1$	$0$	\(0\)	\(4\)	\(6\)	\(0\)		$-$	$-$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{3}+q^{5}+\beta _{4}q^{7}+(2-\beta _{1}+\cdots)q^{9}+\cdots\)
9920.2.a.ct	$9920$	$2$	9920.a	1.a	$1$	$6$	$6$	$79.212$	6.6.50574608.1	None	✓				4960.2.a.l	$1$	$0$	\(0\)	\(4\)	\(6\)	\(4\)		$+$	$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{3}+q^{5}+(1-\beta _{5})q^{7}+(1+\cdots)q^{9}+\cdots\)
9920.2.a.cu	$9920$	$2$	9920.a	1.a	$1$	$8$	$8$	$79.212$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓		✓		4960.2.a.q	$1$	$1$	\(0\)	\(-4\)	\(-8\)	\(0\)		$-$	$+$	$-$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{3}-q^{5}+\beta _{7}q^{7}+(2-\beta _{1}+\cdots)q^{9}+\cdots\)
9920.2.a.cv	$9920$	$2$	9920.a	1.a	$1$	$8$	$8$	$79.212$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓		✓		4960.2.a.r	$1$	$1$	\(0\)	\(-2\)	\(8\)	\(-8\)		$+$	$-$	$-$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q-\beta _{4}q^{3}+q^{5}+(-1+\beta _{2})q^{7}+(1-\beta _{5}+\cdots)q^{9}+\cdots\)
9920.2.a.cw	$9920$	$2$	9920.a	1.a	$1$	$8$	$8$	$79.212$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓		✓		4960.2.a.s	$1$	$1$	\(0\)	\(0\)	\(8\)	\(-8\)		$-$	$-$	$+$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q-\beta _{4}q^{3}+q^{5}+(-1-\beta _{5})q^{7}+(1+\beta _{3}+\cdots)q^{9}+\cdots\)
9920.2.a.cx	$9920$	$2$	9920.a	1.a	$1$	$8$	$8$	$79.212$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓		✓		4960.2.a.s	$1$	$0$	\(0\)	\(0\)	\(8\)	\(8\)		$-$	$-$	$-$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q+\beta _{4}q^{3}+q^{5}+(1+\beta _{5})q^{7}+(1+\beta _{3}+\cdots)q^{9}+\cdots\)
9920.2.a.cy	$9920$	$2$	9920.a	1.a	$1$	$8$	$8$	$79.212$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓		✓		4960.2.a.r	$1$	$0$	\(0\)	\(2\)	\(8\)	\(8\)		$+$	$-$	$+$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q+\beta _{4}q^{3}+q^{5}+(1-\beta _{2})q^{7}+(1-\beta _{5}+\cdots)q^{9}+\cdots\)
9920.2.a.cz	$9920$	$2$	9920.a	1.a	$1$	$8$	$8$	$79.212$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓		✓		4960.2.a.q	$1$	$0$	\(0\)	\(4\)	\(-8\)	\(0\)		$-$	$+$	$+$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{3}-q^{5}-\beta _{7}q^{7}+(2-\beta _{1}+\cdots)q^{9}+\cdots\)
9920.2.a.da	$9920$	$2$	9920.a	1.a	$1$	$9$	$9$	$79.212$	\(\mathbb{Q}[x]/(x^{9} - \cdots)\)	None	✓		✓		4960.2.a.w	$1$	$1$	\(0\)	\(0\)	\(-9\)	\(-2\)		$+$	$+$	$+$	$+$	$2^{3}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}-q^{5}+\beta _{3}q^{7}+(1-\beta _{4}+\beta _{5}+\cdots)q^{9}+\cdots\)
9920.2.a.db	$9920$	$2$	9920.a	1.a	$1$	$9$	$9$	$79.212$	\(\mathbb{Q}[x]/(x^{9} - \cdots)\)	None	✓		✓		4960.2.a.w	$1$	$0$	\(0\)	\(0\)	\(-9\)	\(2\)		$+$	$+$	$-$	$-$	$2^{3}$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}-q^{5}-\beta _{3}q^{7}+(1-\beta _{4}+\beta _{5}+\cdots)q^{9}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(9920))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(9920)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 10}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(31))\)\(^{\oplus 14}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(32))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(40))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(62))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(64))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(80))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(124))\)\(^{\oplus 10}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(155))\)\(^{\oplus 7}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(160))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(248))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(310))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(320))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(496))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(620))\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(992))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1240))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1984))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(2480))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(4960))\)\(^{\oplus 2}\)