Space of modular forms of level 9900, weight 2, and trivial character

• Label: 9900.2.a
• Level: $9900$
• Weight: $2$
• Character orbit: 9900.a
• Rep. character: $\chi_{9900}(1,\cdot)$
• Character field: $\Q$
• Dimension: $79$
• Newform subspaces: $54$
• Sturm bound: $4320$
• Trace bound: $17$

Defining parameters

Level:	\( N \)	\(=\)	\( 9900 = 2^{2} \cdot 3^{2} \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9900.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 54 \)
Sturm bound:			\(4320\)
Trace bound:			\(17\)
Distinguishing \(T_p\):			\(7\), \(13\), \(17\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(9900))\).

	Total	New	Old
Modular forms	2232	79	2153
Cusp forms	2089	79	2010
Eisenstein series	143	0	143

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

\(2\)	\(3\)	\(5\)	\(11\)	Fricke		Total				Cusp				Eisenstein
						All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)	\(+\)	\(+\)		\(132\)	\(0\)	\(132\)		\(121\)	\(0\)	\(121\)		\(11\)	\(0\)	\(11\)
\(+\)	\(+\)	\(+\)	\(-\)	\(-\)		\(150\)	\(0\)	\(150\)		\(138\)	\(0\)	\(138\)		\(12\)	\(0\)	\(12\)
\(+\)	\(+\)	\(-\)	\(+\)	\(-\)		\(144\)	\(0\)	\(144\)		\(132\)	\(0\)	\(132\)		\(12\)	\(0\)	\(12\)
\(+\)	\(+\)	\(-\)	\(-\)	\(+\)		\(138\)	\(0\)	\(138\)		\(126\)	\(0\)	\(126\)		\(12\)	\(0\)	\(12\)
\(+\)	\(-\)	\(+\)	\(+\)	\(-\)		\(141\)	\(0\)	\(141\)		\(129\)	\(0\)	\(129\)		\(12\)	\(0\)	\(12\)
\(+\)	\(-\)	\(+\)	\(-\)	\(+\)		\(141\)	\(0\)	\(141\)		\(129\)	\(0\)	\(129\)		\(12\)	\(0\)	\(12\)
\(+\)	\(-\)	\(-\)	\(+\)	\(+\)		\(141\)	\(0\)	\(141\)		\(129\)	\(0\)	\(129\)		\(12\)	\(0\)	\(12\)
\(+\)	\(-\)	\(-\)	\(-\)	\(-\)		\(141\)	\(0\)	\(141\)		\(129\)	\(0\)	\(129\)		\(12\)	\(0\)	\(12\)
\(-\)	\(+\)	\(+\)	\(+\)	\(-\)		\(138\)	\(7\)	\(131\)		\(132\)	\(7\)	\(125\)		\(6\)	\(0\)	\(6\)
\(-\)	\(+\)	\(+\)	\(-\)	\(+\)		\(138\)	\(7\)	\(131\)		\(132\)	\(7\)	\(125\)		\(6\)	\(0\)	\(6\)
\(-\)	\(+\)	\(-\)	\(+\)	\(+\)		\(138\)	\(9\)	\(129\)		\(132\)	\(9\)	\(123\)		\(6\)	\(0\)	\(6\)
\(-\)	\(+\)	\(-\)	\(-\)	\(-\)		\(138\)	\(9\)	\(129\)		\(132\)	\(9\)	\(123\)		\(6\)	\(0\)	\(6\)
\(-\)	\(-\)	\(+\)	\(+\)	\(+\)		\(138\)	\(11\)	\(127\)		\(132\)	\(11\)	\(121\)		\(6\)	\(0\)	\(6\)
\(-\)	\(-\)	\(+\)	\(-\)	\(-\)		\(138\)	\(12\)	\(126\)		\(132\)	\(12\)	\(120\)		\(6\)	\(0\)	\(6\)
\(-\)	\(-\)	\(-\)	\(+\)	\(-\)		\(138\)	\(13\)	\(125\)		\(132\)	\(13\)	\(119\)		\(6\)	\(0\)	\(6\)
\(-\)	\(-\)	\(-\)	\(-\)	\(+\)		\(138\)	\(11\)	\(127\)		\(132\)	\(11\)	\(121\)		\(6\)	\(0\)	\(6\)
Plus space				\(+\)		\(1104\)	\(38\)	\(1066\)		\(1033\)	\(38\)	\(995\)		\(71\)	\(0\)	\(71\)
Minus space				\(-\)		\(1128\)	\(41\)	\(1087\)		\(1056\)	\(41\)	\(1015\)		\(72\)	\(0\)	\(72\)

Trace form

79 * q + 2 * q^7 - q^11 + 4 * q^13 + 10 * q^17 - 4 * q^19 - 11 * q^23 - 24 * q^29 - 21 * q^31 + q^37 - 16 * q^41 + 6 * q^43 + 24 * q^47 + 87 * q^49 + 2 * q^53 + 3 * q^59 - 4 * q^61 - 7 * q^67 - 21 * q^71 - 4 * q^73 + 6 * q^77 + 26 * q^79 - 6 * q^83 - 49 * q^89 + 12 * q^91 + 23 * q^97

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(9900))\) 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	11
9900.2.a.a	$9900$	$2$	9900.a	1.a	$1$	$1$	$1$	$79.052$	\(\Q\)	None	✓				3300.2.a.i	$1$	$1$	\(0\)	\(0\)	\(0\)	\(-5\)		$-$	$-$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-5q^{7}-q^{11}-4q^{13}-5q^{17}+7q^{19}+\cdots\)
9900.2.a.b	$9900$	$2$	9900.a	1.a	$1$	$1$	$1$	$79.052$	\(\Q\)	None	✓	✓			9900.2.a.b	$1$	$0$	\(0\)	\(0\)	\(0\)	\(-3\)		$-$	$+$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-3q^{7}-q^{11}+3q^{13}+8q^{17}+5q^{19}+\cdots\)
9900.2.a.c	$9900$	$2$	9900.a	1.a	$1$	$1$	$1$	$79.052$	\(\Q\)	None	✓				3300.2.a.a	$1$	$0$	\(0\)	\(0\)	\(0\)	\(-3\)		$-$	$-$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-3q^{7}+q^{11}-4q^{13}-q^{17}-7q^{19}+\cdots\)
9900.2.a.d	$9900$	$2$	9900.a	1.a	$1$	$1$	$1$	$79.052$	\(\Q\)	None	✓	✓			9900.2.a.b	$1$	$1$	\(0\)	\(0\)	\(0\)	\(-3\)		$-$	$+$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-3q^{7}+q^{11}+3q^{13}-8q^{17}+5q^{19}+\cdots\)
9900.2.a.e	$9900$	$2$	9900.a	1.a	$1$	$1$	$1$	$79.052$	\(\Q\)	None	✓				660.2.a.d	$1$	$1$	\(0\)	\(0\)	\(0\)	\(-2\)		$-$	$-$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{7}-q^{11}-2q^{13}+2q^{19}+8q^{31}+\cdots\)
9900.2.a.f	$9900$	$2$	9900.a	1.a	$1$	$1$	$1$	$79.052$	\(\Q\)	None	✓				660.2.c.a	$1$	$0$	\(0\)	\(0\)	\(0\)	\(-2\)		$-$	$-$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{7}-q^{11}-2q^{13}+6q^{17}+4q^{19}+\cdots\)
9900.2.a.g	$9900$	$2$	9900.a	1.a	$1$	$1$	$1$	$79.052$	\(\Q\)	None	✓				132.2.a.a	$1$	$0$	\(0\)	\(0\)	\(0\)	\(-2\)		$-$	$-$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{7}+q^{11}-6q^{13}-4q^{17}-2q^{19}+\cdots\)
9900.2.a.h	$9900$	$2$	9900.a	1.a	$1$	$1$	$1$	$79.052$	\(\Q\)	None	✓				44.2.a.a	$1$	$0$	\(0\)	\(0\)	\(0\)	\(-2\)		$-$	$-$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{7}+q^{11}+4q^{13}+6q^{17}+8q^{19}+\cdots\)
9900.2.a.i	$9900$	$2$	9900.a	1.a	$1$	$1$	$1$	$79.052$	\(\Q\)	None	✓				220.2.b.a	$1$	$1$	\(0\)	\(0\)	\(0\)	\(-2\)		$-$	$-$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{7}+q^{11}+6q^{13}-2q^{17}-4q^{19}+\cdots\)
9900.2.a.j	$9900$	$2$	9900.a	1.a	$1$	$1$	$1$	$79.052$	\(\Q\)	None	✓				3300.2.a.e	$1$	$1$	\(0\)	\(0\)	\(0\)	\(-1\)		$-$	$-$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{7}+q^{11}-4q^{13}-3q^{17}+5q^{19}+\cdots\)
9900.2.a.k	$9900$	$2$	9900.a	1.a	$1$	$1$	$1$	$79.052$	\(\Q\)	None	✓				3300.2.a.d	$1$	$0$	\(0\)	\(0\)	\(0\)	\(-1\)		$-$	$-$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{7}+q^{11}+q^{13}+2q^{17}-5q^{19}+\cdots\)
9900.2.a.l	$9900$	$2$	9900.a	1.a	$1$	$1$	$1$	$79.052$	\(\Q\)	None	✓				660.2.c.b	$1$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{11}-4q^{13}-8q^{19}+4q^{23}-2q^{29}+\cdots\)
9900.2.a.m	$9900$	$2$	9900.a	1.a	$1$	$1$	$1$	$79.052$	\(\Q\)	None	✓				1980.2.c.a	$1$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$+$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{11}-2q^{13}+6q^{17}-2q^{19}-4q^{23}+\cdots\)
9900.2.a.n	$9900$	$2$	9900.a	1.a	$1$	$1$	$1$	$79.052$	\(\Q\)	None	✓				220.2.a.b	$1$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{11}-4q^{17}-4q^{19}+6q^{23}-2q^{29}+\cdots\)
9900.2.a.o	$9900$	$2$	9900.a	1.a	$1$	$1$	$1$	$79.052$	\(\Q\)	None	✓				1980.2.c.a	$1$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$+$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{11}+2q^{13}-6q^{17}-2q^{19}+4q^{23}+\cdots\)
9900.2.a.p	$9900$	$2$	9900.a	1.a	$1$	$1$	$1$	$79.052$	\(\Q\)	None	✓				660.2.a.b	$1$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{11}+4q^{13}-2q^{17}+2q^{19}-8q^{23}+\cdots\)
9900.2.a.q	$9900$	$2$	9900.a	1.a	$1$	$1$	$1$	$79.052$	\(\Q\)	None	✓				660.2.c.b	$1$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{11}+4q^{13}-8q^{19}-4q^{23}-2q^{29}+\cdots\)
9900.2.a.r	$9900$	$2$	9900.a	1.a	$1$	$1$	$1$	$79.052$	\(\Q\)	None	✓				1980.2.c.a	$1$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$+$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{11}-2q^{13}-6q^{17}-2q^{19}+4q^{23}+\cdots\)
9900.2.a.s	$9900$	$2$	9900.a	1.a	$1$	$1$	$1$	$79.052$	\(\Q\)	None	✓				1980.2.c.a	$1$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$+$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{11}+2q^{13}+6q^{17}-2q^{19}-4q^{23}+\cdots\)
9900.2.a.t	$9900$	$2$	9900.a	1.a	$1$	$1$	$1$	$79.052$	\(\Q\)	None	✓				3300.2.a.d	$1$	$1$	\(0\)	\(0\)	\(0\)	\(1\)		$-$	$-$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{7}+q^{11}-q^{13}-2q^{17}-5q^{19}+\cdots\)
9900.2.a.u	$9900$	$2$	9900.a	1.a	$1$	$1$	$1$	$79.052$	\(\Q\)	None	✓				3300.2.a.e	$1$	$0$	\(0\)	\(0\)	\(0\)	\(1\)		$-$	$-$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{7}+q^{11}+4q^{13}+3q^{17}+5q^{19}+\cdots\)
9900.2.a.v	$9900$	$2$	9900.a	1.a	$1$	$1$	$1$	$79.052$	\(\Q\)	None	✓				660.2.c.a	$1$	$0$	\(0\)	\(0\)	\(0\)	\(2\)		$-$	$-$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{7}-q^{11}+2q^{13}-6q^{17}+4q^{19}+\cdots\)
9900.2.a.w	$9900$	$2$	9900.a	1.a	$1$	$1$	$1$	$79.052$	\(\Q\)	None	✓				132.2.a.b	$1$	$1$	\(0\)	\(0\)	\(0\)	\(2\)		$-$	$-$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{7}-q^{11}+2q^{13}+4q^{17}-6q^{19}+\cdots\)
9900.2.a.x	$9900$	$2$	9900.a	1.a	$1$	$1$	$1$	$79.052$	\(\Q\)	None	✓				220.2.b.a	$1$	$1$	\(0\)	\(0\)	\(0\)	\(2\)		$-$	$-$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{7}+q^{11}-6q^{13}+2q^{17}-4q^{19}+\cdots\)
9900.2.a.y	$9900$	$2$	9900.a	1.a	$1$	$1$	$1$	$79.052$	\(\Q\)	None	✓				660.2.a.a	$1$	$0$	\(0\)	\(0\)	\(0\)	\(2\)		$-$	$-$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{7}+q^{11}-2q^{13}+8q^{17}-2q^{19}+\cdots\)
9900.2.a.z	$9900$	$2$	9900.a	1.a	$1$	$1$	$1$	$79.052$	\(\Q\)	None	✓	✓			9900.2.a.b	$1$	$1$	\(0\)	\(0\)	\(0\)	\(3\)		$-$	$+$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+3q^{7}-q^{11}-3q^{13}-8q^{17}+5q^{19}+\cdots\)
9900.2.a.ba	$9900$	$2$	9900.a	1.a	$1$	$1$	$1$	$79.052$	\(\Q\)	None	✓	✓			9900.2.a.b	$1$	$0$	\(0\)	\(0\)	\(0\)	\(3\)		$-$	$+$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3q^{7}+q^{11}-3q^{13}+8q^{17}+5q^{19}+\cdots\)
9900.2.a.bb	$9900$	$2$	9900.a	1.a	$1$	$1$	$1$	$79.052$	\(\Q\)	None	✓				3300.2.a.a	$1$	$1$	\(0\)	\(0\)	\(0\)	\(3\)		$-$	$-$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+3q^{7}+q^{11}+4q^{13}+q^{17}-7q^{19}+\cdots\)
9900.2.a.bc	$9900$	$2$	9900.a	1.a	$1$	$1$	$1$	$79.052$	\(\Q\)	None	✓				660.2.a.c	$1$	$0$	\(0\)	\(0\)	\(0\)	\(4\)		$-$	$-$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+4q^{7}+q^{11}+4q^{13}-6q^{17}+2q^{19}+\cdots\)
9900.2.a.bd	$9900$	$2$	9900.a	1.a	$1$	$1$	$1$	$79.052$	\(\Q\)	None	✓				220.2.a.a	$1$	$0$	\(0\)	\(0\)	\(0\)	\(4\)		$-$	$-$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+4q^{7}+q^{11}+4q^{13}-4q^{19}-6q^{23}+\cdots\)
9900.2.a.be	$9900$	$2$	9900.a	1.a	$1$	$1$	$1$	$79.052$	\(\Q\)	None	✓				3300.2.a.i	$1$	$0$	\(0\)	\(0\)	\(0\)	\(5\)		$-$	$-$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+5q^{7}-q^{11}+4q^{13}+5q^{17}+7q^{19}+\cdots\)
9900.2.a.bf	$9900$	$2$	9900.a	1.a	$1$	$2$	$2$	$79.052$	\(\Q(\sqrt{145}) \)	None	✓				3300.2.a.s	$1$	$0$	\(0\)	\(0\)	\(0\)	\(-6\)		$-$	$-$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-3q^{7}-q^{11}+(-1+\beta )q^{13}+(2-\beta )q^{17}+\cdots\)
9900.2.a.bg	$9900$	$2$	9900.a	1.a	$1$	$2$	$2$	$79.052$	\(\Q(\sqrt{13}) \)	None	✓				1100.2.a.f	$1$	$1$	\(0\)	\(0\)	\(0\)	\(-5\)		$-$	$-$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-2-\beta )q^{7}-q^{11}+(1+2\beta )q^{13}+\cdots\)
9900.2.a.bh	$9900$	$2$	9900.a	1.a	$1$	$2$	$2$	$79.052$	\(\Q(\sqrt{21}) \)	None	✓				1100.2.a.g	$1$	$1$	\(0\)	\(0\)	\(0\)	\(-5\)		$-$	$-$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-2-\beta )q^{7}+q^{11}-q^{13}+(1+\beta )q^{17}+\cdots\)
9900.2.a.bi	$9900$	$2$	9900.a	1.a	$1$	$2$	$2$	$79.052$	\(\Q(\sqrt{10}) \)	None	✓				1980.2.c.f	$1$	$1$	\(0\)	\(0\)	\(0\)	\(-4\)		$-$	$+$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-2+\beta )q^{7}-q^{11}+(2+\beta )q^{13}+\cdots\)
9900.2.a.bj	$9900$	$2$	9900.a	1.a	$1$	$2$	$2$	$79.052$	\(\Q(\sqrt{10}) \)	None	✓				1980.2.c.f	$1$	$0$	\(0\)	\(0\)	\(0\)	\(-4\)		$-$	$+$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-2+\beta )q^{7}+q^{11}+(2+\beta )q^{13}+\cdots\)
9900.2.a.bk	$9900$	$2$	9900.a	1.a	$1$	$2$	$2$	$79.052$	\(\Q(\sqrt{7}) \)	None	✓				1980.2.a.i	$1$	$0$	\(0\)	\(0\)	\(0\)	\(-2\)		$-$	$+$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta )q^{7}-q^{11}+(-1+\beta )q^{13}+\cdots\)
9900.2.a.bl	$9900$	$2$	9900.a	1.a	$1$	$2$	$2$	$79.052$	\(\Q(\sqrt{13}) \)	None	✓				660.2.a.e	$1$	$1$	\(0\)	\(0\)	\(0\)	\(-2\)		$-$	$-$	$+$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q+(-1-\beta )q^{7}-q^{11}+(1+\beta )q^{13}+\cdots\)
9900.2.a.bm	$9900$	$2$	9900.a	1.a	$1$	$2$	$2$	$79.052$	\(\Q(\sqrt{6}) \)	None	✓	✓			9900.2.a.bm	$1$	$1$	\(0\)	\(0\)	\(0\)	\(-2\)		$-$	$+$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta )q^{7}-q^{11}+2q^{13}+(3+\beta )q^{17}+\cdots\)
9900.2.a.bn	$9900$	$2$	9900.a	1.a	$1$	$2$	$2$	$79.052$	\(\Q(\sqrt{13}) \)	None	✓				660.2.a.f	$1$	$0$	\(0\)	\(0\)	\(0\)	\(-2\)		$-$	$-$	$+$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(-1-\beta )q^{7}+q^{11}+(-3+\beta )q^{13}+\cdots\)
9900.2.a.bo	$9900$	$2$	9900.a	1.a	$1$	$2$	$2$	$79.052$	\(\Q(\sqrt{7}) \)	None	✓				1980.2.a.i	$1$	$1$	\(0\)	\(0\)	\(0\)	\(-2\)		$-$	$+$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta )q^{7}+q^{11}+(-1+\beta )q^{13}+\cdots\)
9900.2.a.bp	$9900$	$2$	9900.a	1.a	$1$	$2$	$2$	$79.052$	\(\Q(\sqrt{5}) \)	None	✓				660.2.c.c	$1$	$1$	\(0\)	\(0\)	\(0\)	\(-2\)		$-$	$-$	$-$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q+(-1-\beta )q^{7}+q^{11}+(1+\beta )q^{13}+\cdots\)
9900.2.a.bq	$9900$	$2$	9900.a	1.a	$1$	$2$	$2$	$79.052$	\(\Q(\sqrt{6}) \)	None	✓	✓			9900.2.a.bm	$1$	$0$	\(0\)	\(0\)	\(0\)	\(-2\)		$-$	$+$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta )q^{7}+q^{11}+2q^{13}+(-3+\cdots)q^{17}+\cdots\)
9900.2.a.br	$9900$	$2$	9900.a	1.a	$1$	$2$	$2$	$79.052$	\(\Q(\sqrt{6}) \)	None	✓	✓			9900.2.a.bm	$1$	$0$	\(0\)	\(0\)	\(0\)	\(2\)		$-$	$+$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{7}-q^{11}-2q^{13}+(-3+\beta )q^{17}+\cdots\)
9900.2.a.bs	$9900$	$2$	9900.a	1.a	$1$	$2$	$2$	$79.052$	\(\Q(\sqrt{3}) \)	None	✓				1980.2.a.g	$1$	$0$	\(0\)	\(0\)	\(0\)	\(2\)		$-$	$+$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{7}-q^{11}+(1-3\beta )q^{13}+(3+\cdots)q^{17}+\cdots\)
9900.2.a.bt	$9900$	$2$	9900.a	1.a	$1$	$2$	$2$	$79.052$	\(\Q(\sqrt{6}) \)	None	✓	✓			9900.2.a.bm	$1$	$1$	\(0\)	\(0\)	\(0\)	\(2\)		$-$	$+$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{7}+q^{11}-2q^{13}+(3-\beta )q^{17}+\cdots\)
9900.2.a.bu	$9900$	$2$	9900.a	1.a	$1$	$2$	$2$	$79.052$	\(\Q(\sqrt{5}) \)	None	✓				660.2.c.c	$1$	$1$	\(0\)	\(0\)	\(0\)	\(2\)		$-$	$-$	$-$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{7}+q^{11}+(-1-\beta )q^{13}+\cdots\)
9900.2.a.bv	$9900$	$2$	9900.a	1.a	$1$	$2$	$2$	$79.052$	\(\Q(\sqrt{3}) \)	None	✓				1980.2.a.g	$1$	$1$	\(0\)	\(0\)	\(0\)	\(2\)		$-$	$+$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{7}+q^{11}+(1-3\beta )q^{13}+(-3+\cdots)q^{17}+\cdots\)
9900.2.a.bw	$9900$	$2$	9900.a	1.a	$1$	$2$	$2$	$79.052$	\(\Q(\sqrt{10}) \)	None	✓				1980.2.c.f	$1$	$1$	\(0\)	\(0\)	\(0\)	\(4\)		$-$	$+$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(2+\beta )q^{7}-q^{11}+(-2+\beta )q^{13}+\cdots\)
9900.2.a.bx	$9900$	$2$	9900.a	1.a	$1$	$2$	$2$	$79.052$	\(\Q(\sqrt{10}) \)	None	✓				1980.2.c.f	$1$	$0$	\(0\)	\(0\)	\(0\)	\(4\)		$-$	$+$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(2+\beta )q^{7}+q^{11}+(-2+\beta )q^{13}+\cdots\)
9900.2.a.by	$9900$	$2$	9900.a	1.a	$1$	$2$	$2$	$79.052$	\(\Q(\sqrt{13}) \)	None	✓				1100.2.a.f	$1$	$0$	\(0\)	\(0\)	\(0\)	\(5\)		$-$	$-$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(3-\beta )q^{7}-q^{11}+(-3+2\beta )q^{13}+\cdots\)
9900.2.a.bz	$9900$	$2$	9900.a	1.a	$1$	$2$	$2$	$79.052$	\(\Q(\sqrt{21}) \)	None	✓				1100.2.a.g	$1$	$0$	\(0\)	\(0\)	\(0\)	\(5\)		$-$	$-$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(3-\beta )q^{7}+q^{11}+q^{13}+(-2+\beta )q^{17}+\cdots\)
9900.2.a.ca	$9900$	$2$	9900.a	1.a	$1$	$2$	$2$	$79.052$	\(\Q(\sqrt{145}) \)	None	✓				3300.2.a.s	$1$	$1$	\(0\)	\(0\)	\(0\)	\(6\)		$-$	$-$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+3q^{7}-q^{11}+\beta q^{13}+(-1-\beta )q^{17}+\cdots\)
9900.2.a.cb	$9900$	$2$	9900.a	1.a	$1$	$4$	$4$	$79.052$	\(\Q(\sqrt{3}, \sqrt{19})\)	None	✓		✓		220.2.b.b	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(\beta _{1}+\beta _{2})q^{7}-q^{11}+(\beta _{1}-\beta _{2})q^{13}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(9900))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(9900)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(11))\)\(^{\oplus 27}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 24}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(30))\)\(^{\oplus 16}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(33))\)\(^{\oplus 18}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(36))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(44))\)\(^{\oplus 9}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(45))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(50))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(55))\)\(^{\oplus 18}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(66))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(75))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(90))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(99))\)\(^{\oplus 9}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(100))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(110))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(132))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(150))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(165))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(180))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(198))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(220))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(225))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(275))\)\(^{\oplus 9}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(300))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(330))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(396))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(450))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(495))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(550))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(660))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(825))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(900))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(990))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1100))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1650))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1980))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(2475))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(3300))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(4950))\)\(^{\oplus 2}\)