Space of modular forms of level 5520, weight 2, and trivial character

• Label: 5520.2.a
• Level: $5520$
• Weight: $2$
• Character orbit: 5520.a
• Rep. character: $\chi_{5520}(1,\cdot)$
• Character field: $\Q$
• Dimension: $88$
• Newform subspaces: $55$
• Sturm bound: $2304$
• Trace bound: $17$

Defining parameters

Level:	\( N \)	\(=\)	\( 5520 = 2^{4} \cdot 3 \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5520.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 55 \)
Sturm bound:			\(2304\)
Trace bound:			\(17\)
Distinguishing \(T_p\):			\(7\), \(11\), \(13\), \(17\), \(19\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(5520))\).

	Total	New	Old
Modular forms	1176	88	1088
Cusp forms	1129	88	1041
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.

\(2\)	\(3\)	\(5\)	\(23\)	Fricke		Total				Cusp				Eisenstein
						All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)	\(+\)	\(+\)		\(68\)	\(6\)	\(62\)		\(66\)	\(6\)	\(60\)		\(2\)	\(0\)	\(2\)
\(+\)	\(+\)	\(+\)	\(-\)	\(-\)		\(74\)	\(4\)	\(70\)		\(71\)	\(4\)	\(67\)		\(3\)	\(0\)	\(3\)
\(+\)	\(+\)	\(-\)	\(+\)	\(-\)		\(75\)	\(4\)	\(71\)		\(72\)	\(4\)	\(68\)		\(3\)	\(0\)	\(3\)
\(+\)	\(+\)	\(-\)	\(-\)	\(+\)		\(77\)	\(6\)	\(71\)		\(74\)	\(6\)	\(68\)		\(3\)	\(0\)	\(3\)
\(+\)	\(-\)	\(+\)	\(+\)	\(-\)		\(77\)	\(8\)	\(69\)		\(74\)	\(8\)	\(66\)		\(3\)	\(0\)	\(3\)
\(+\)	\(-\)	\(+\)	\(-\)	\(+\)		\(71\)	\(4\)	\(67\)		\(68\)	\(4\)	\(64\)		\(3\)	\(0\)	\(3\)
\(+\)	\(-\)	\(-\)	\(+\)	\(+\)		\(74\)	\(4\)	\(70\)		\(71\)	\(4\)	\(67\)		\(3\)	\(0\)	\(3\)
\(+\)	\(-\)	\(-\)	\(-\)	\(-\)		\(72\)	\(8\)	\(64\)		\(69\)	\(8\)	\(61\)		\(3\)	\(0\)	\(3\)
\(-\)	\(+\)	\(+\)	\(+\)	\(-\)		\(77\)	\(6\)	\(71\)		\(74\)	\(6\)	\(68\)		\(3\)	\(0\)	\(3\)
\(-\)	\(+\)	\(+\)	\(-\)	\(+\)		\(71\)	\(5\)	\(66\)		\(68\)	\(5\)	\(63\)		\(3\)	\(0\)	\(3\)
\(-\)	\(+\)	\(-\)	\(+\)	\(+\)		\(74\)	\(5\)	\(69\)		\(71\)	\(5\)	\(66\)		\(3\)	\(0\)	\(3\)
\(-\)	\(+\)	\(-\)	\(-\)	\(-\)		\(72\)	\(6\)	\(66\)		\(69\)	\(6\)	\(63\)		\(3\)	\(0\)	\(3\)
\(-\)	\(-\)	\(+\)	\(+\)	\(+\)		\(72\)	\(5\)	\(67\)		\(69\)	\(5\)	\(64\)		\(3\)	\(0\)	\(3\)
\(-\)	\(-\)	\(+\)	\(-\)	\(-\)		\(78\)	\(6\)	\(72\)		\(75\)	\(6\)	\(69\)		\(3\)	\(0\)	\(3\)
\(-\)	\(-\)	\(-\)	\(+\)	\(-\)		\(71\)	\(6\)	\(65\)		\(68\)	\(6\)	\(62\)		\(3\)	\(0\)	\(3\)
\(-\)	\(-\)	\(-\)	\(-\)	\(+\)		\(73\)	\(5\)	\(68\)		\(70\)	\(5\)	\(65\)		\(3\)	\(0\)	\(3\)
Plus space				\(+\)		\(580\)	\(40\)	\(540\)		\(557\)	\(40\)	\(517\)		\(23\)	\(0\)	\(23\)
Minus space				\(-\)		\(596\)	\(48\)	\(548\)		\(572\)	\(48\)	\(524\)		\(24\)	\(0\)	\(24\)

Trace form

88 * q + 4 * q^3 + 8 * q^7 + 88 * q^9 + 8 * q^19 + 88 * q^25 + 4 * q^27 + 8 * q^31 + 8 * q^39 + 8 * q^43 + 104 * q^49 - 32 * q^53 - 16 * q^55 + 16 * q^57 - 32 * q^59 - 32 * q^61 + 8 * q^63 + 8 * q^67 - 32 * q^71 + 16 * q^73 + 4 * q^75 - 32 * q^77 + 8 * q^79 + 88 * q^81 - 24 * q^87 - 64 * q^91 + 16 * q^97

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(5520))\) 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	3	5	23
5520.2.a.a	$5520$	$2$	5520.a	1.a	$1$	$1$	$1$	$44.077$	\(\Q\)	None	✓				345.2.a.e	$1$	$1$	\(0\)	\(-1\)	\(-1\)	\(-4\)		$-$	$+$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-q^{5}-4q^{7}+q^{9}-4q^{11}+6q^{13}+\cdots\)
5520.2.a.b	$5520$	$2$	5520.a	1.a	$1$	$1$	$1$	$44.077$	\(\Q\)	None	✓				345.2.a.b	$1$	$1$	\(0\)	\(-1\)	\(-1\)	\(-4\)		$-$	$+$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-q^{5}-4q^{7}+q^{9}+4q^{11}-2q^{13}+\cdots\)
5520.2.a.c	$5520$	$2$	5520.a	1.a	$1$	$1$	$1$	$44.077$	\(\Q\)	None	✓				2760.2.a.i	$1$	$0$	\(0\)	\(-1\)	\(-1\)	\(-3\)		$+$	$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-q^{5}-3q^{7}+q^{9}+6q^{11}-2q^{13}+\cdots\)
5520.2.a.d	$5520$	$2$	5520.a	1.a	$1$	$1$	$1$	$44.077$	\(\Q\)	None	✓				690.2.a.i	$1$	$0$	\(0\)	\(-1\)	\(-1\)	\(0\)		$-$	$+$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-q^{5}+q^{9}-2q^{11}+q^{15}+6q^{17}+\cdots\)
5520.2.a.e	$5520$	$2$	5520.a	1.a	$1$	$1$	$1$	$44.077$	\(\Q\)	None	✓				2760.2.a.h	$1$	$0$	\(0\)	\(-1\)	\(-1\)	\(0\)		$+$	$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-q^{5}+q^{9}-2q^{13}+q^{15}-6q^{17}+\cdots\)
5520.2.a.f	$5520$	$2$	5520.a	1.a	$1$	$1$	$1$	$44.077$	\(\Q\)	None	✓				690.2.a.e	$1$	$0$	\(0\)	\(-1\)	\(-1\)	\(0\)		$-$	$+$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-q^{5}+q^{9}+4q^{11}-6q^{13}+\cdots\)
5520.2.a.g	$5520$	$2$	5520.a	1.a	$1$	$1$	$1$	$44.077$	\(\Q\)	None	✓				1380.2.a.c	$1$	$1$	\(0\)	\(-1\)	\(-1\)	\(1\)		$-$	$+$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-q^{5}+q^{7}+q^{9}-4q^{13}+q^{15}+\cdots\)
5520.2.a.h	$5520$	$2$	5520.a	1.a	$1$	$1$	$1$	$44.077$	\(\Q\)	None	✓				345.2.a.d	$1$	$0$	\(0\)	\(-1\)	\(-1\)	\(3\)		$-$	$+$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-q^{5}+3q^{7}+q^{9}+4q^{11}+q^{15}+\cdots\)
5520.2.a.i	$5520$	$2$	5520.a	1.a	$1$	$1$	$1$	$44.077$	\(\Q\)	None	✓				690.2.a.k	$1$	$0$	\(0\)	\(-1\)	\(1\)	\(0\)		$-$	$+$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+q^{5}+q^{9}-4q^{11}-2q^{13}+\cdots\)
5520.2.a.j	$5520$	$2$	5520.a	1.a	$1$	$1$	$1$	$44.077$	\(\Q\)	None	✓				690.2.a.f	$1$	$1$	\(0\)	\(-1\)	\(1\)	\(0\)		$-$	$+$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+q^{5}+q^{9}-4q^{11}-2q^{13}+\cdots\)
5520.2.a.k	$5520$	$2$	5520.a	1.a	$1$	$1$	$1$	$44.077$	\(\Q\)	None	✓				2760.2.a.k	$1$	$1$	\(0\)	\(-1\)	\(1\)	\(0\)		$+$	$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+q^{5}+q^{9}-2q^{11}-q^{15}+6q^{17}+\cdots\)
5520.2.a.l	$5520$	$2$	5520.a	1.a	$1$	$1$	$1$	$44.077$	\(\Q\)	None	✓				690.2.a.j	$1$	$0$	\(0\)	\(-1\)	\(1\)	\(0\)		$-$	$+$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+q^{5}+q^{9}+2q^{11}+4q^{13}+\cdots\)
5520.2.a.m	$5520$	$2$	5520.a	1.a	$1$	$1$	$1$	$44.077$	\(\Q\)	None	✓				1380.2.a.e	$1$	$1$	\(0\)	\(-1\)	\(1\)	\(3\)		$-$	$+$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+q^{5}+3q^{7}+q^{9}+2q^{11}-2q^{13}+\cdots\)
5520.2.a.n	$5520$	$2$	5520.a	1.a	$1$	$1$	$1$	$44.077$	\(\Q\)	None	✓				2760.2.a.j	$1$	$0$	\(0\)	\(-1\)	\(1\)	\(3\)		$+$	$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+q^{5}+3q^{7}+q^{9}+4q^{11}-q^{15}+\cdots\)
5520.2.a.o	$5520$	$2$	5520.a	1.a	$1$	$1$	$1$	$44.077$	\(\Q\)	None	✓				1380.2.a.d	$1$	$0$	\(0\)	\(-1\)	\(1\)	\(4\)		$-$	$+$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+q^{5}+4q^{7}+q^{9}+2q^{13}-q^{15}+\cdots\)
5520.2.a.p	$5520$	$2$	5520.a	1.a	$1$	$1$	$1$	$44.077$	\(\Q\)	None	✓				345.2.a.a	$1$	$0$	\(0\)	\(-1\)	\(1\)	\(5\)		$-$	$+$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+q^{5}+5q^{7}+q^{9}+2q^{11}-6q^{13}+\cdots\)
5520.2.a.q	$5520$	$2$	5520.a	1.a	$1$	$1$	$1$	$44.077$	\(\Q\)	None	✓				2760.2.a.d	$1$	$0$	\(0\)	\(1\)	\(-1\)	\(-4\)		$+$	$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-q^{5}-4q^{7}+q^{9}-4q^{11}+6q^{13}+\cdots\)
5520.2.a.r	$5520$	$2$	5520.a	1.a	$1$	$1$	$1$	$44.077$	\(\Q\)	None	✓				690.2.a.b	$1$	$1$	\(0\)	\(1\)	\(-1\)	\(-4\)		$-$	$-$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-q^{5}-4q^{7}+q^{9}+2q^{11}-q^{15}+\cdots\)
5520.2.a.s	$5520$	$2$	5520.a	1.a	$1$	$1$	$1$	$44.077$	\(\Q\)	None	✓				2760.2.a.c	$1$	$0$	\(0\)	\(1\)	\(-1\)	\(-4\)		$+$	$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-q^{5}-4q^{7}+q^{9}+6q^{11}-4q^{13}+\cdots\)
5520.2.a.t	$5520$	$2$	5520.a	1.a	$1$	$1$	$1$	$44.077$	\(\Q\)	None	✓				2760.2.a.b	$1$	$1$	\(0\)	\(1\)	\(-1\)	\(-3\)		$+$	$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-q^{5}-3q^{7}+q^{9}+2q^{11}+2q^{13}+\cdots\)
5520.2.a.u	$5520$	$2$	5520.a	1.a	$1$	$1$	$1$	$44.077$	\(\Q\)	None	✓				345.2.a.c	$1$	$1$	\(0\)	\(1\)	\(-1\)	\(-1\)		$-$	$-$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-q^{5}-q^{7}+q^{9}-4q^{11}-q^{15}+\cdots\)
5520.2.a.v	$5520$	$2$	5520.a	1.a	$1$	$1$	$1$	$44.077$	\(\Q\)	None	✓				2760.2.a.a	$1$	$1$	\(0\)	\(1\)	\(-1\)	\(0\)		$+$	$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-q^{5}+q^{9}-4q^{11}+2q^{13}+\cdots\)
5520.2.a.w	$5520$	$2$	5520.a	1.a	$1$	$1$	$1$	$44.077$	\(\Q\)	None	✓				1380.2.a.b	$1$	$0$	\(0\)	\(1\)	\(-1\)	\(0\)		$-$	$-$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-q^{5}+q^{9}-6q^{13}-q^{15}+2q^{17}+\cdots\)
5520.2.a.x	$5520$	$2$	5520.a	1.a	$1$	$1$	$1$	$44.077$	\(\Q\)	None	✓				690.2.a.h	$1$	$0$	\(0\)	\(1\)	\(-1\)	\(0\)		$-$	$-$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-q^{5}+q^{9}+6q^{13}-q^{15}+2q^{17}+\cdots\)
5520.2.a.y	$5520$	$2$	5520.a	1.a	$1$	$1$	$1$	$44.077$	\(\Q\)	None	✓				690.2.a.a	$1$	$0$	\(0\)	\(1\)	\(-1\)	\(2\)		$-$	$-$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-q^{5}+2q^{7}+q^{9}-6q^{11}-2q^{13}+\cdots\)
5520.2.a.z	$5520$	$2$	5520.a	1.a	$1$	$1$	$1$	$44.077$	\(\Q\)	None	✓				690.2.a.g	$1$	$1$	\(0\)	\(1\)	\(-1\)	\(2\)		$-$	$-$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-q^{5}+2q^{7}+q^{9}+2q^{11}-6q^{13}+\cdots\)
5520.2.a.ba	$5520$	$2$	5520.a	1.a	$1$	$1$	$1$	$44.077$	\(\Q\)	None	✓				1380.2.a.a	$1$	$0$	\(0\)	\(1\)	\(-1\)	\(5\)		$-$	$-$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-q^{5}+5q^{7}+q^{9}+4q^{13}-q^{15}+\cdots\)
5520.2.a.bb	$5520$	$2$	5520.a	1.a	$1$	$1$	$1$	$44.077$	\(\Q\)	None	✓				690.2.a.d	$1$	$1$	\(0\)	\(1\)	\(1\)	\(-4\)		$-$	$-$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+q^{5}-4q^{7}+q^{9}-2q^{11}+4q^{13}+\cdots\)
5520.2.a.bc	$5520$	$2$	5520.a	1.a	$1$	$1$	$1$	$44.077$	\(\Q\)	None	✓				345.2.a.f	$1$	$1$	\(0\)	\(1\)	\(1\)	\(-3\)		$-$	$-$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+q^{5}-3q^{7}+q^{9}-2q^{11}-2q^{13}+\cdots\)
5520.2.a.bd	$5520$	$2$	5520.a	1.a	$1$	$1$	$1$	$44.077$	\(\Q\)	None	✓				2760.2.a.g	$1$	$1$	\(0\)	\(1\)	\(1\)	\(-2\)		$+$	$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+q^{5}-2q^{7}+q^{9}+2q^{11}+2q^{13}+\cdots\)
5520.2.a.be	$5520$	$2$	5520.a	1.a	$1$	$1$	$1$	$44.077$	\(\Q\)	None	✓				2760.2.a.f	$1$	$1$	\(0\)	\(1\)	\(1\)	\(-1\)		$+$	$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+q^{5}-q^{7}+q^{9}-4q^{13}+q^{15}+\cdots\)
5520.2.a.bf	$5520$	$2$	5520.a	1.a	$1$	$1$	$1$	$44.077$	\(\Q\)	None	✓				2760.2.a.e	$1$	$0$	\(0\)	\(1\)	\(1\)	\(0\)		$+$	$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+q^{5}+q^{9}+4q^{11}-2q^{13}+\cdots\)
5520.2.a.bg	$5520$	$2$	5520.a	1.a	$1$	$1$	$1$	$44.077$	\(\Q\)	None	✓				690.2.a.c	$1$	$1$	\(0\)	\(1\)	\(1\)	\(2\)		$-$	$-$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+q^{5}+2q^{7}+q^{9}-2q^{11}-2q^{13}+\cdots\)
5520.2.a.bh	$5520$	$2$	5520.a	1.a	$1$	$2$	$2$	$44.077$	\(\Q(\sqrt{17}) \)	None	✓		✓		2760.2.a.p	$1$	$0$	\(0\)	\(-2\)	\(-2\)	\(1\)		$+$	$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-q^{5}+\beta q^{7}+q^{9}+2\beta q^{11}+\cdots\)
5520.2.a.bi	$5520$	$2$	5520.a	1.a	$1$	$2$	$2$	$44.077$	\(\Q(\sqrt{6}) \)	None	✓		✓		345.2.a.i	$1$	$1$	\(0\)	\(-2\)	\(-2\)	\(2\)		$-$	$+$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-q^{5}+q^{7}+q^{9}-\beta q^{11}+(2+\cdots)q^{13}+\cdots\)
5520.2.a.bj	$5520$	$2$	5520.a	1.a	$1$	$2$	$2$	$44.077$	\(\Q(\sqrt{15}) \)	None	✓		✓		1380.2.a.i	$1$	$0$	\(0\)	\(-2\)	\(2\)	\(-6\)		$-$	$+$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+q^{5}-3q^{7}+q^{9}+(-1-\beta )q^{11}+\cdots\)
5520.2.a.bk	$5520$	$2$	5520.a	1.a	$1$	$2$	$2$	$44.077$	\(\Q(\sqrt{2}) \)	None	✓				2760.2.a.q	$1$	$1$	\(0\)	\(-2\)	\(2\)	\(2\)		$+$	$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+q^{5}+q^{7}+q^{9}+(2+3\beta )q^{11}+\cdots\)
5520.2.a.bl	$5520$	$2$	5520.a	1.a	$1$	$2$	$2$	$44.077$	\(\Q(\sqrt{6}) \)	None	✓		✓		1380.2.a.f	$1$	$1$	\(0\)	\(2\)	\(-2\)	\(-2\)		$-$	$-$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-q^{5}-q^{7}+q^{9}+(2+\beta )q^{11}+\cdots\)
5520.2.a.bm	$5520$	$2$	5520.a	1.a	$1$	$2$	$2$	$44.077$	\(\Q(\sqrt{2}) \)	None	✓		✓		345.2.a.h	$1$	$0$	\(0\)	\(2\)	\(-2\)	\(2\)		$-$	$-$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-q^{5}+(1+2\beta )q^{7}+q^{9}+(4+\beta )q^{11}+\cdots\)
5520.2.a.bn	$5520$	$2$	5520.a	1.a	$1$	$2$	$2$	$44.077$	\(\Q(\sqrt{17}) \)	None	✓		✓		2760.2.a.m	$1$	$1$	\(0\)	\(2\)	\(-2\)	\(3\)		$+$	$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-q^{5}+(1+\beta )q^{7}+q^{9}-2q^{11}+\cdots\)
5520.2.a.bo	$5520$	$2$	5520.a	1.a	$1$	$2$	$2$	$44.077$	\(\Q(\sqrt{11}) \)	None	✓				2760.2.a.l	$1$	$0$	\(0\)	\(2\)	\(-2\)	\(6\)		$+$	$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-q^{5}+3q^{7}+q^{9}+(1+\beta )q^{11}+\cdots\)
5520.2.a.bp	$5520$	$2$	5520.a	1.a	$1$	$2$	$2$	$44.077$	\(\Q(\sqrt{3}) \)	None	✓		✓		1380.2.a.h	$1$	$1$	\(0\)	\(2\)	\(2\)	\(-2\)		$-$	$-$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+q^{5}+(-1+2\beta )q^{7}+q^{9}+(1+\cdots)q^{11}+\cdots\)
5520.2.a.bq	$5520$	$2$	5520.a	1.a	$1$	$2$	$2$	$44.077$	\(\Q(\sqrt{33}) \)	None	✓		✓		2760.2.a.o	$1$	$1$	\(0\)	\(2\)	\(2\)	\(-1\)		$+$	$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+q^{5}-\beta q^{7}+q^{9}-4q^{11}+(-2+\cdots)q^{13}+\cdots\)
5520.2.a.br	$5520$	$2$	5520.a	1.a	$1$	$2$	$2$	$44.077$	\(\Q(\sqrt{17}) \)	None	✓				1380.2.a.g	$1$	$0$	\(0\)	\(2\)	\(2\)	\(-1\)		$-$	$-$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+q^{5}-\beta q^{7}+q^{9}-2\beta q^{11}+\cdots\)
5520.2.a.bs	$5520$	$2$	5520.a	1.a	$1$	$2$	$2$	$44.077$	\(\Q(\sqrt{17}) \)	None	✓				690.2.a.l	$1$	$0$	\(0\)	\(2\)	\(2\)	\(2\)		$-$	$-$	$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q+q^{3}+q^{5}+(1+\beta )q^{7}+q^{9}+(-1+\cdots)q^{11}+\cdots\)
5520.2.a.bt	$5520$	$2$	5520.a	1.a	$1$	$2$	$2$	$44.077$	\(\Q(\sqrt{2}) \)	None	✓				2760.2.a.n	$1$	$0$	\(0\)	\(2\)	\(2\)	\(2\)		$+$	$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+q^{5}+(1+2\beta )q^{7}+q^{9}-\beta q^{11}+\cdots\)
5520.2.a.bu	$5520$	$2$	5520.a	1.a	$1$	$2$	$2$	$44.077$	\(\Q(\sqrt{3}) \)	None	✓				345.2.a.g	$1$	$0$	\(0\)	\(2\)	\(2\)	\(6\)		$-$	$-$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+q^{5}+3q^{7}+q^{9}+(1+\beta )q^{11}+\cdots\)
5520.2.a.bv	$5520$	$2$	5520.a	1.a	$1$	$3$	$3$	$44.077$	3.3.3144.1	None	✓		✓		1380.2.a.j	$1$	$0$	\(0\)	\(-3\)	\(-3\)	\(-2\)		$-$	$+$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-q^{5}+(-1+\beta _{1})q^{7}+q^{9}+(-1+\cdots)q^{11}+\cdots\)
5520.2.a.bw	$5520$	$2$	5520.a	1.a	$1$	$3$	$3$	$44.077$	3.3.1436.1	None	✓				2760.2.a.s	$1$	$1$	\(0\)	\(-3\)	\(-3\)	\(2\)		$+$	$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-q^{5}+(1-\beta _{2})q^{7}+q^{9}+(-1+\cdots)q^{11}+\cdots\)
5520.2.a.bx	$5520$	$2$	5520.a	1.a	$1$	$3$	$3$	$44.077$	3.3.316.1	None	✓				2760.2.a.r	$1$	$1$	\(0\)	\(-3\)	\(-3\)	\(6\)		$+$	$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-q^{5}+(2-\beta _{2})q^{7}+q^{9}+(-1+\cdots)q^{11}+\cdots\)
5520.2.a.by	$5520$	$2$	5520.a	1.a	$1$	$3$	$3$	$44.077$	3.3.316.1	None	✓		✓		345.2.a.j	$1$	$1$	\(0\)	\(-3\)	\(3\)	\(-6\)		$-$	$+$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+q^{5}+(-2-\beta _{2})q^{7}+q^{9}+(1+\cdots)q^{11}+\cdots\)
5520.2.a.bz	$5520$	$2$	5520.a	1.a	$1$	$3$	$3$	$44.077$	3.3.568.1	None	✓		✓		2760.2.a.u	$1$	$1$	\(0\)	\(-3\)	\(3\)	\(-2\)		$+$	$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+q^{5}+(-1+2\beta _{1}+\beta _{2})q^{7}+\cdots\)
5520.2.a.ca	$5520$	$2$	5520.a	1.a	$1$	$3$	$3$	$44.077$	3.3.229.1	None	✓		✓		2760.2.a.t	$1$	$0$	\(0\)	\(-3\)	\(3\)	\(-1\)		$+$	$+$	$-$	$+$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q-q^{3}+q^{5}+\beta _{1}q^{7}+q^{9}+(1-\beta _{2})q^{13}+\cdots\)
5520.2.a.cb	$5520$	$2$	5520.a	1.a	$1$	$4$	$4$	$44.077$	4.4.54764.1	None	✓		✓		2760.2.a.v	$1$	$0$	\(0\)	\(4\)	\(-4\)	\(0\)		$+$	$-$	$+$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q+q^{3}-q^{5}+\beta _{1}q^{7}+q^{9}+\beta _{3}q^{11}+\cdots\)
5520.2.a.cc	$5520$	$2$	5520.a	1.a	$1$	$5$	$5$	$44.077$	5.5.20087896.1	None	✓		✓		2760.2.a.w	$1$	$0$	\(0\)	\(5\)	\(5\)	\(4\)		$+$	$-$	$-$	$-$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q+q^{3}+q^{5}+(1-\beta _{1})q^{7}+q^{9}+(1-\beta _{2}+\cdots)q^{11}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(5520))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(5520)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 10}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(23))\)\(^{\oplus 20}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(24))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(30))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(40))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(46))\)\(^{\oplus 16}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(48))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(69))\)\(^{\oplus 10}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(80))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(92))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(115))\)\(^{\oplus 10}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(120))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(138))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(184))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(230))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(240))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(276))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(345))\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(368))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(460))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(552))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(690))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(920))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1104))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1380))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1840))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(2760))\)\(^{\oplus 2}\)