Space of modular forms of level 5200, weight 2, and trivial character

• Label: 5200.2.a
• Level: $5200$
• Weight: $2$
• Character orbit: 5200.a
• Rep. character: $\chi_{5200}(1,\cdot)$
• Character field: $\Q$
• Dimension: $114$
• Newform subspaces: $66$
• Sturm bound: $1680$
• Trace bound: $11$

Defining parameters

Level:	\( N \)	\(=\)	\( 5200 = 2^{4} \cdot 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5200.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 66 \)
Sturm bound:			\(1680\)
Trace bound:			\(11\)
Distinguishing \(T_p\):			\(3\), \(7\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(5200))\).

	Total	New	Old
Modular forms	876	114	762
Cusp forms	805	114	691
Eisenstein series	71	0	71

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

\(2\)	\(5\)	\(13\)	Fricke		Total				Cusp				Eisenstein
					All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)	\(+\)		\(105\)	\(12\)	\(93\)		\(97\)	\(12\)	\(85\)		\(8\)	\(0\)	\(8\)
\(+\)	\(+\)	\(-\)	\(-\)		\(111\)	\(15\)	\(96\)		\(102\)	\(15\)	\(87\)		\(9\)	\(0\)	\(9\)
\(+\)	\(-\)	\(+\)	\(-\)		\(113\)	\(16\)	\(97\)		\(104\)	\(16\)	\(88\)		\(9\)	\(0\)	\(9\)
\(+\)	\(-\)	\(-\)	\(+\)		\(107\)	\(14\)	\(93\)		\(98\)	\(14\)	\(84\)		\(9\)	\(0\)	\(9\)
\(-\)	\(+\)	\(+\)	\(-\)		\(114\)	\(15\)	\(99\)		\(105\)	\(15\)	\(90\)		\(9\)	\(0\)	\(9\)
\(-\)	\(+\)	\(-\)	\(+\)		\(108\)	\(12\)	\(96\)		\(99\)	\(12\)	\(87\)		\(9\)	\(0\)	\(9\)
\(-\)	\(-\)	\(+\)	\(+\)		\(106\)	\(14\)	\(92\)		\(97\)	\(14\)	\(83\)		\(9\)	\(0\)	\(9\)
\(-\)	\(-\)	\(-\)	\(-\)		\(112\)	\(16\)	\(96\)		\(103\)	\(16\)	\(87\)		\(9\)	\(0\)	\(9\)
Plus space			\(+\)		\(426\)	\(52\)	\(374\)		\(391\)	\(52\)	\(339\)		\(35\)	\(0\)	\(35\)
Minus space			\(-\)		\(450\)	\(62\)	\(388\)		\(414\)	\(62\)	\(352\)		\(36\)	\(0\)	\(36\)

Trace form

114 * q - 2 * q^7 + 110 * q^9 - 10 * q^11 - 4 * q^17 + 2 * q^19 + 8 * q^21 - 16 * q^23 - 12 * q^27 + 16 * q^29 + 14 * q^31 + 8 * q^33 + 8 * q^37 + 4 * q^39 + 4 * q^41 + 4 * q^43 - 10 * q^47 + 122 * q^49 - 12 * q^51 + 4 * q^53 - 18 * q^59 - 20 * q^61 - 66 * q^63 - 2 * q^67 - 8 * q^69 + 22 * q^71 - 4 * q^73 + 4 * q^77 - 12 * q^79 + 98 * q^81 - 2 * q^83 + 64 * q^87 - 12 * q^89 - 6 * q^91 + 16 * q^93 - 20 * q^97 - 2 * q^99

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(5200))\) 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	13
5200.2.a.a	$5200$	$2$	5200.a	1.a	$1$	$1$	$1$	$41.522$	\(\Q\)	None	✓				2600.2.a.a	$1$	$0$	\(0\)	\(-3\)	\(0\)	\(-2\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-3q^{3}-2q^{7}+6q^{9}+2q^{11}-q^{13}+\cdots\)
5200.2.a.b	$5200$	$2$	5200.a	1.a	$1$	$1$	$1$	$41.522$	\(\Q\)	None	✓				650.2.a.a	$1$	$0$	\(0\)	\(-3\)	\(0\)	\(0\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-3q^{3}+6q^{9}+3q^{11}+q^{13}-7q^{17}+\cdots\)
5200.2.a.c	$5200$	$2$	5200.a	1.a	$1$	$1$	$1$	$41.522$	\(\Q\)	None	✓				26.2.a.b	$1$	$1$	\(0\)	\(-3\)	\(0\)	\(1\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-3q^{3}+q^{7}+6q^{9}+2q^{11}+q^{13}+\cdots\)
5200.2.a.d	$5200$	$2$	5200.a	1.a	$1$	$1$	$1$	$41.522$	\(\Q\)	None	✓				65.2.a.a	$1$	$1$	\(0\)	\(-2\)	\(0\)	\(-4\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{3}-4q^{7}+q^{9}-2q^{11}+q^{13}+\cdots\)
5200.2.a.e	$5200$	$2$	5200.a	1.a	$1$	$1$	$1$	$41.522$	\(\Q\)	None	✓				130.2.a.a	$1$	$0$	\(0\)	\(-2\)	\(0\)	\(-4\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{3}-4q^{7}+q^{9}+6q^{11}-q^{13}+\cdots\)
5200.2.a.f	$5200$	$2$	5200.a	1.a	$1$	$1$	$1$	$41.522$	\(\Q\)	None	✓				2600.2.a.b	$1$	$1$	\(0\)	\(-2\)	\(0\)	\(-3\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{3}-3q^{7}+q^{9}-3q^{11}+q^{13}+\cdots\)
5200.2.a.g	$5200$	$2$	5200.a	1.a	$1$	$1$	$1$	$41.522$	\(\Q\)	None	✓				650.2.a.b	$1$	$0$	\(0\)	\(-2\)	\(0\)	\(-1\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{3}-q^{7}+q^{9}-3q^{11}-q^{13}+\cdots\)
5200.2.a.h	$5200$	$2$	5200.a	1.a	$1$	$1$	$1$	$41.522$	\(\Q\)	None	✓				2600.2.a.d	$1$	$1$	\(0\)	\(-2\)	\(0\)	\(3\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{3}+3q^{7}+q^{9}-q^{11}+q^{13}+\cdots\)
5200.2.a.i	$5200$	$2$	5200.a	1.a	$1$	$1$	$1$	$41.522$	\(\Q\)	None	✓				650.2.a.f	$1$	$0$	\(0\)	\(-2\)	\(0\)	\(5\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{3}+5q^{7}+q^{9}+3q^{11}-q^{13}+\cdots\)
5200.2.a.j	$5200$	$2$	5200.a	1.a	$1$	$1$	$1$	$41.522$	\(\Q\)	None	✓				650.2.a.e	$1$	$1$	\(0\)	\(-1\)	\(0\)	\(-4\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-4q^{7}-2q^{9}-q^{11}+q^{13}+\cdots\)
5200.2.a.k	$5200$	$2$	5200.a	1.a	$1$	$1$	$1$	$41.522$	\(\Q\)	None	✓				325.2.a.a	$1$	$1$	\(0\)	\(-1\)	\(0\)	\(-2\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-2q^{7}-2q^{9}-2q^{11}-q^{13}+\cdots\)
5200.2.a.l	$5200$	$2$	5200.a	1.a	$1$	$1$	$1$	$41.522$	\(\Q\)	None	✓				2600.2.a.f	$1$	$1$	\(0\)	\(-1\)	\(0\)	\(0\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-2q^{9}+2q^{11}-q^{13}+2q^{17}+\cdots\)
5200.2.a.m	$5200$	$2$	5200.a	1.a	$1$	$1$	$1$	$41.522$	\(\Q\)	None	✓				1300.2.a.b	$1$	$1$	\(0\)	\(-1\)	\(0\)	\(2\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+2q^{7}-2q^{9}+2q^{11}+q^{13}+\cdots\)
5200.2.a.n	$5200$	$2$	5200.a	1.a	$1$	$1$	$1$	$41.522$	\(\Q\)	None	✓				325.2.a.b	$1$	$0$	\(0\)	\(-1\)	\(0\)	\(4\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+4q^{7}-2q^{9}+6q^{11}+q^{13}+\cdots\)
5200.2.a.o	$5200$	$2$	5200.a	1.a	$1$	$1$	$1$	$41.522$	\(\Q\)	None	✓				520.2.d.a	$1$	$1$	\(0\)	\(0\)	\(0\)	\(-4\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-4q^{7}-3q^{9}+2q^{11}+q^{13}+4q^{17}+\cdots\)
5200.2.a.p	$5200$	$2$	5200.a	1.a	$1$	$1$	$1$	$41.522$	\(\Q\)	None	✓				1300.2.a.c	$1$	$0$	\(0\)	\(0\)	\(0\)	\(-3\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-3q^{7}-3q^{9}-3q^{11}-q^{13}+q^{17}+\cdots\)
5200.2.a.q	$5200$	$2$	5200.a	1.a	$1$	$1$	$1$	$41.522$	\(\Q\)	None	✓				52.2.a.a	$1$	$1$	\(0\)	\(0\)	\(0\)	\(-2\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{7}-3q^{9}+2q^{11}+q^{13}-6q^{17}+\cdots\)
5200.2.a.r	$5200$	$2$	5200.a	1.a	$1$	$1$	$1$	$41.522$	\(\Q\)	None	✓				130.2.a.b	$1$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-3q^{9}-q^{13}-2q^{17}+8q^{19}-4q^{23}+\cdots\)
5200.2.a.s	$5200$	$2$	5200.a	1.a	$1$	$1$	$1$	$41.522$	\(\Q\)	None	✓				520.2.a.a	$1$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-3q^{9}+4q^{11}+q^{13}+6q^{17}-4q^{19}+\cdots\)
5200.2.a.t	$5200$	$2$	5200.a	1.a	$1$	$1$	$1$	$41.522$	\(\Q\)	None	✓				1300.2.a.c	$1$	$0$	\(0\)	\(0\)	\(0\)	\(3\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3q^{7}-3q^{9}-3q^{11}+q^{13}-q^{17}+\cdots\)
5200.2.a.u	$5200$	$2$	5200.a	1.a	$1$	$1$	$1$	$41.522$	\(\Q\)	None	✓				520.2.d.a	$1$	$0$	\(0\)	\(0\)	\(0\)	\(4\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+4q^{7}-3q^{9}+2q^{11}-q^{13}-4q^{17}+\cdots\)
5200.2.a.v	$5200$	$2$	5200.a	1.a	$1$	$1$	$1$	$41.522$	\(\Q\)	None	✓				325.2.a.b	$1$	$0$	\(0\)	\(1\)	\(0\)	\(-4\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-4q^{7}-2q^{9}+6q^{11}-q^{13}+\cdots\)
5200.2.a.w	$5200$	$2$	5200.a	1.a	$1$	$1$	$1$	$41.522$	\(\Q\)	None	✓				1300.2.a.b	$1$	$1$	\(0\)	\(1\)	\(0\)	\(-2\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-2q^{7}-2q^{9}+2q^{11}-q^{13}+\cdots\)
5200.2.a.x	$5200$	$2$	5200.a	1.a	$1$	$1$	$1$	$41.522$	\(\Q\)	None	✓				26.2.a.a	$1$	$0$	\(0\)	\(1\)	\(0\)	\(-1\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-q^{7}-2q^{9}-6q^{11}-q^{13}+\cdots\)
5200.2.a.y	$5200$	$2$	5200.a	1.a	$1$	$1$	$1$	$41.522$	\(\Q\)	None	✓				2600.2.a.f	$1$	$1$	\(0\)	\(1\)	\(0\)	\(0\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-2q^{9}+2q^{11}+q^{13}-2q^{17}+\cdots\)
5200.2.a.z	$5200$	$2$	5200.a	1.a	$1$	$1$	$1$	$41.522$	\(\Q\)	None	✓				325.2.a.a	$1$	$1$	\(0\)	\(1\)	\(0\)	\(2\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+2q^{7}-2q^{9}-2q^{11}+q^{13}+\cdots\)
5200.2.a.ba	$5200$	$2$	5200.a	1.a	$1$	$1$	$1$	$41.522$	\(\Q\)	None	✓				650.2.a.e	$1$	$1$	\(0\)	\(1\)	\(0\)	\(4\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+4q^{7}-2q^{9}-q^{11}-q^{13}+\cdots\)
5200.2.a.bb	$5200$	$2$	5200.a	1.a	$1$	$1$	$1$	$41.522$	\(\Q\)	None	✓				104.2.a.a	$1$	$0$	\(0\)	\(1\)	\(0\)	\(5\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+5q^{7}-2q^{9}+2q^{11}+q^{13}+\cdots\)
5200.2.a.bc	$5200$	$2$	5200.a	1.a	$1$	$1$	$1$	$41.522$	\(\Q\)	None	✓				650.2.a.f	$1$	$0$	\(0\)	\(2\)	\(0\)	\(-5\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{3}-5q^{7}+q^{9}+3q^{11}+q^{13}+\cdots\)
5200.2.a.bd	$5200$	$2$	5200.a	1.a	$1$	$1$	$1$	$41.522$	\(\Q\)	None	✓				130.2.a.c	$1$	$1$	\(0\)	\(2\)	\(0\)	\(-4\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{3}-4q^{7}+q^{9}+2q^{11}+q^{13}+\cdots\)
5200.2.a.be	$5200$	$2$	5200.a	1.a	$1$	$1$	$1$	$41.522$	\(\Q\)	None	✓				2600.2.a.d	$1$	$1$	\(0\)	\(2\)	\(0\)	\(-3\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{3}-3q^{7}+q^{9}-q^{11}-q^{13}+\cdots\)
5200.2.a.bf	$5200$	$2$	5200.a	1.a	$1$	$1$	$1$	$41.522$	\(\Q\)	None	✓				520.2.a.b	$1$	$1$	\(0\)	\(2\)	\(0\)	\(0\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{3}+q^{9}-2q^{11}-q^{13}-2q^{17}+\cdots\)
5200.2.a.bg	$5200$	$2$	5200.a	1.a	$1$	$1$	$1$	$41.522$	\(\Q\)	None	✓				650.2.a.b	$1$	$0$	\(0\)	\(2\)	\(0\)	\(1\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{3}+q^{7}+q^{9}-3q^{11}+q^{13}+\cdots\)
5200.2.a.bh	$5200$	$2$	5200.a	1.a	$1$	$1$	$1$	$41.522$	\(\Q\)	None	✓				260.2.a.a	$1$	$1$	\(0\)	\(2\)	\(0\)	\(2\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{3}+2q^{7}+q^{9}-4q^{11}+q^{13}+\cdots\)
5200.2.a.bi	$5200$	$2$	5200.a	1.a	$1$	$1$	$1$	$41.522$	\(\Q\)	None	✓				2600.2.a.b	$1$	$1$	\(0\)	\(2\)	\(0\)	\(3\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{3}+3q^{7}+q^{9}-3q^{11}-q^{13}+\cdots\)
5200.2.a.bj	$5200$	$2$	5200.a	1.a	$1$	$1$	$1$	$41.522$	\(\Q\)	None	✓				650.2.a.a	$1$	$0$	\(0\)	\(3\)	\(0\)	\(0\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3q^{3}+6q^{9}+3q^{11}-q^{13}+7q^{17}+\cdots\)
5200.2.a.bk	$5200$	$2$	5200.a	1.a	$1$	$1$	$1$	$41.522$	\(\Q\)	None	✓				2600.2.a.a	$1$	$0$	\(0\)	\(3\)	\(0\)	\(2\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3q^{3}+2q^{7}+6q^{9}+2q^{11}+q^{13}+\cdots\)
5200.2.a.bl	$5200$	$2$	5200.a	1.a	$1$	$2$	$2$	$41.522$	\(\Q(\sqrt{2}) \)	None	✓				520.2.a.c	$1$	$0$	\(0\)	\(-4\)	\(0\)	\(-4\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-2+\beta )q^{3}-2q^{7}+(3-4\beta )q^{9}+\cdots\)
5200.2.a.bm	$5200$	$2$	5200.a	1.a	$1$	$2$	$2$	$41.522$	\(\Q(\sqrt{73}) \)	None	✓				2600.2.a.n	$1$	$1$	\(0\)	\(-2\)	\(0\)	\(0\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-2q^{9}+\beta q^{11}-q^{13}+\beta q^{17}+\cdots\)
5200.2.a.bn	$5200$	$2$	5200.a	1.a	$1$	$2$	$2$	$41.522$	\(\Q(\sqrt{3}) \)	None	✓				520.2.a.e	$1$	$1$	\(0\)	\(-2\)	\(0\)	\(0\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta )q^{3}+2\beta q^{7}+(1-2\beta )q^{9}+\cdots\)
5200.2.a.bo	$5200$	$2$	5200.a	1.a	$1$	$2$	$2$	$41.522$	\(\Q(\sqrt{3}) \)	None	✓				520.2.a.d	$1$	$1$	\(0\)	\(-2\)	\(0\)	\(0\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta )q^{3}-2\beta q^{7}+(1-2\beta )q^{9}+\cdots\)
5200.2.a.bp	$5200$	$2$	5200.a	1.a	$1$	$2$	$2$	$41.522$	\(\Q(\sqrt{33}) \)	None	✓				1300.2.a.g	$1$	$0$	\(0\)	\(-1\)	\(0\)	\(1\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta q^{3}+(1-\beta )q^{7}+(5+\beta )q^{9}+(1+\beta )q^{11}+\cdots\)
5200.2.a.bq	$5200$	$2$	5200.a	1.a	$1$	$2$	$2$	$41.522$	\(\Q(\sqrt{10}) \)	None	✓				2600.2.a.r	$1$	$0$	\(0\)	\(0\)	\(0\)	\(-2\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-1-\beta )q^{7}-3q^{9}+(-3-\beta )q^{11}+\cdots\)
5200.2.a.br	$5200$	$2$	5200.a	1.a	$1$	$2$	$2$	$41.522$	\(\Q(\sqrt{2}) \)	None	✓				325.2.a.f	$1$	$1$	\(0\)	\(0\)	\(0\)	\(-2\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2\beta q^{3}+(-1-\beta )q^{7}+5q^{9}+(-5+\cdots)q^{11}+\cdots\)
5200.2.a.bs	$5200$	$2$	5200.a	1.a	$1$	$2$	$2$	$41.522$	\(\Q(\sqrt{10}) \)	None	✓				2600.2.a.r	$1$	$0$	\(0\)	\(0\)	\(0\)	\(2\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1-\beta )q^{7}-3q^{9}+(-3+\beta )q^{11}+\cdots\)
5200.2.a.bt	$5200$	$2$	5200.a	1.a	$1$	$2$	$2$	$41.522$	\(\Q(\sqrt{2}) \)	None	✓				325.2.a.f	$1$	$1$	\(0\)	\(0\)	\(0\)	\(2\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2\beta q^{3}+(1-\beta )q^{7}+5q^{9}+(-5+\beta )q^{11}+\cdots\)
5200.2.a.bu	$5200$	$2$	5200.a	1.a	$1$	$2$	$2$	$41.522$	\(\Q(\sqrt{2}) \)	None	✓				65.2.a.b	$1$	$1$	\(0\)	\(0\)	\(0\)	\(4\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{3}+(2-2\beta )q^{7}-q^{9}+(-2+\beta )q^{11}+\cdots\)
5200.2.a.bv	$5200$	$2$	5200.a	1.a	$1$	$2$	$2$	$41.522$	\(\Q(\sqrt{6}) \)	None	✓				520.2.a.f	$1$	$0$	\(0\)	\(0\)	\(0\)	\(4\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{3}+2q^{7}+3q^{9}+(2-\beta )q^{11}+\cdots\)
5200.2.a.bw	$5200$	$2$	5200.a	1.a	$1$	$2$	$2$	$41.522$	\(\Q(\sqrt{17}) \)	None	✓				104.2.a.b	$1$	$1$	\(0\)	\(1\)	\(0\)	\(-1\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{3}-\beta q^{7}+(1+\beta )q^{9}+2\beta q^{11}+\cdots\)
5200.2.a.bx	$5200$	$2$	5200.a	1.a	$1$	$2$	$2$	$41.522$	\(\Q(\sqrt{33}) \)	None	✓				1300.2.a.g	$1$	$0$	\(0\)	\(1\)	\(0\)	\(-1\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{3}+(-1+\beta )q^{7}+(5+\beta )q^{9}+(1+\cdots)q^{11}+\cdots\)
5200.2.a.by	$5200$	$2$	5200.a	1.a	$1$	$2$	$2$	$41.522$	\(\Q(\sqrt{73}) \)	None	✓				2600.2.a.n	$1$	$1$	\(0\)	\(2\)	\(0\)	\(0\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-2q^{9}+(1-\beta )q^{11}+q^{13}+(-1+\cdots)q^{17}+\cdots\)
5200.2.a.bz	$5200$	$2$	5200.a	1.a	$1$	$2$	$2$	$41.522$	\(\Q(\sqrt{5}) \)	None	✓				520.2.a.g	$1$	$0$	\(0\)	\(2\)	\(0\)	\(0\)		$+$	$+$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{3}+(3+2\beta )q^{9}+(-3+\beta )q^{11}+\cdots\)
5200.2.a.ca	$5200$	$2$	5200.a	1.a	$1$	$2$	$2$	$41.522$	\(\Q(\sqrt{3}) \)	None	✓				65.2.a.c	$1$	$0$	\(0\)	\(2\)	\(0\)	\(4\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{3}+2q^{7}+(1+2\beta )q^{9}+(3+\cdots)q^{11}+\cdots\)
5200.2.a.cb	$5200$	$2$	5200.a	1.a	$1$	$3$	$3$	$41.522$	3.3.148.1	None	✓				65.2.b.a	$1$	$1$	\(0\)	\(-4\)	\(0\)	\(-2\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1-\beta _{1})q^{3}+(-1+\beta _{1}+\beta _{2})q^{7}+\cdots\)
5200.2.a.cc	$5200$	$2$	5200.a	1.a	$1$	$3$	$3$	$41.522$	3.3.148.1	None	✓				520.2.d.b	$1$	$0$	\(0\)	\(-2\)	\(0\)	\(2\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{3}+(1-\beta _{1}+\beta _{2})q^{7}+\cdots\)
5200.2.a.cd	$5200$	$2$	5200.a	1.a	$1$	$3$	$3$	$41.522$	3.3.148.1	None	✓				260.2.c.a	$1$	$0$	\(0\)	\(0\)	\(0\)	\(-4\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{2}q^{3}+(-1-\beta _{1}-\beta _{2})q^{7}+(-\beta _{1}+\cdots)q^{9}+\cdots\)
5200.2.a.ce	$5200$	$2$	5200.a	1.a	$1$	$3$	$3$	$41.522$	3.3.940.1	None	✓				130.2.b.a	$1$	$1$	\(0\)	\(0\)	\(0\)	\(-2\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}+(-1-\beta _{1}-\beta _{2})q^{7}+(2+\beta _{1}+\cdots)q^{9}+\cdots\)
5200.2.a.cf	$5200$	$2$	5200.a	1.a	$1$	$3$	$3$	$41.522$	3.3.940.1	None	✓				130.2.b.a	$1$	$0$	\(0\)	\(0\)	\(0\)	\(2\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}+(1+\beta _{1}+\beta _{2})q^{7}+(2+\beta _{1}+\cdots)q^{9}+\cdots\)
5200.2.a.cg	$5200$	$2$	5200.a	1.a	$1$	$3$	$3$	$41.522$	3.3.148.1	None	✓				260.2.c.a	$1$	$1$	\(0\)	\(0\)	\(0\)	\(4\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{2}q^{3}+(1+\beta _{1}+\beta _{2})q^{7}+(-\beta _{1}+\cdots)q^{9}+\cdots\)
5200.2.a.ch	$5200$	$2$	5200.a	1.a	$1$	$3$	$3$	$41.522$	3.3.148.1	None	✓				520.2.d.b	$1$	$1$	\(0\)	\(2\)	\(0\)	\(-2\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{3}+(-1+\beta _{1}-\beta _{2})q^{7}+\cdots\)
5200.2.a.ci	$5200$	$2$	5200.a	1.a	$1$	$3$	$3$	$41.522$	3.3.564.1	None	✓		✓		260.2.a.b	$1$	$0$	\(0\)	\(2\)	\(0\)	\(-2\)		$-$	$+$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(1+\beta _{2})q^{3}+(-1-\beta _{1})q^{7}+(4+\beta _{1}+\cdots)q^{9}+\cdots\)
5200.2.a.cj	$5200$	$2$	5200.a	1.a	$1$	$3$	$3$	$41.522$	3.3.148.1	None	✓				65.2.b.a	$1$	$0$	\(0\)	\(4\)	\(0\)	\(2\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta _{1})q^{3}+(1-\beta _{1}-\beta _{2})q^{7}+(3\beta _{1}+\cdots)q^{9}+\cdots\)
5200.2.a.ck	$5200$	$2$	5200.a	1.a	$1$	$4$	$4$	$41.522$	4.4.108664.1	None	✓		✓		2600.2.a.z	$1$	$0$	\(0\)	\(-2\)	\(0\)	\(-4\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{2}q^{3}+(-1-\beta _{3})q^{7}+(1-\beta _{2})q^{9}+\cdots\)
5200.2.a.cl	$5200$	$2$	5200.a	1.a	$1$	$4$	$4$	$41.522$	4.4.108664.1	None	✓				2600.2.a.z	$1$	$0$	\(0\)	\(2\)	\(0\)	\(4\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{2}q^{3}+(1-\beta _{3})q^{7}+(1-\beta _{2})q^{9}+\cdots\)
5200.2.a.cm	$5200$	$2$	5200.a	1.a	$1$	$5$	$5$	$41.522$	5.5.10592240.1	None	✓		✓		520.2.d.c	$1$	$1$	\(0\)	\(-2\)	\(0\)	\(0\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}+(-\beta _{2}-\beta _{3})q^{7}+(3+\beta _{2}+\cdots)q^{9}+\cdots\)
5200.2.a.cn	$5200$	$2$	5200.a	1.a	$1$	$5$	$5$	$41.522$	5.5.10592240.1	None	✓		✓		520.2.d.c	$1$	$0$	\(0\)	\(2\)	\(0\)	\(0\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}+(\beta _{2}+\beta _{3})q^{7}+(3+\beta _{2}+\beta _{3}+\cdots)q^{9}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(5200))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(5200)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(26))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(40))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(50))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(52))\)\(^{\oplus 9}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(65))\)\(^{\oplus 10}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(80))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(100))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(104))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(130))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(200))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(208))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(260))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(325))\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(400))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(520))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(650))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1040))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1300))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(2600))\)\(^{\oplus 2}\)