Space of modular forms of level 1200, weight 4, and trivial character

• Label: 1200.4.a
• Level: $1200$
• Weight: $4$
• Character orbit: 1200.a
• Rep. character: $\chi_{1200}(1,\cdot)$
• Character field: $\Q$
• Dimension: $57$
• Newform subspaces: $47$
• Sturm bound: $960$
• Trace bound: $11$

Defining parameters

Level:	\( N \)	\(=\)	\( 1200 = 2^{4} \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1200.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 47 \)
Sturm bound:			\(960\)
Trace bound:			\(11\)
Distinguishing \(T_p\):			\(7\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(1200))\).

	Total	New	Old
Modular forms	756	57	699
Cusp forms	684	57	627
Eisenstein series	72	0	72

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

\(2\)	\(3\)	\(5\)	Fricke		Total				Cusp				Eisenstein
					All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)	\(+\)		\(96\)	\(8\)	\(88\)		\(87\)	\(8\)	\(79\)		\(9\)	\(0\)	\(9\)
\(+\)	\(+\)	\(-\)	\(-\)		\(93\)	\(7\)	\(86\)		\(84\)	\(7\)	\(77\)		\(9\)	\(0\)	\(9\)
\(+\)	\(-\)	\(+\)	\(-\)		\(93\)	\(5\)	\(88\)		\(84\)	\(5\)	\(79\)		\(9\)	\(0\)	\(9\)
\(+\)	\(-\)	\(-\)	\(+\)		\(96\)	\(9\)	\(87\)		\(87\)	\(9\)	\(78\)		\(9\)	\(0\)	\(9\)
\(-\)	\(+\)	\(+\)	\(-\)		\(93\)	\(7\)	\(86\)		\(84\)	\(7\)	\(77\)		\(9\)	\(0\)	\(9\)
\(-\)	\(+\)	\(-\)	\(+\)		\(96\)	\(7\)	\(89\)		\(87\)	\(7\)	\(80\)		\(9\)	\(0\)	\(9\)
\(-\)	\(-\)	\(+\)	\(+\)		\(96\)	\(7\)	\(89\)		\(87\)	\(7\)	\(80\)		\(9\)	\(0\)	\(9\)
\(-\)	\(-\)	\(-\)	\(-\)		\(93\)	\(7\)	\(86\)		\(84\)	\(7\)	\(77\)		\(9\)	\(0\)	\(9\)
Plus space			\(+\)		\(384\)	\(31\)	\(353\)		\(348\)	\(31\)	\(317\)		\(36\)	\(0\)	\(36\)
Minus space			\(-\)		\(372\)	\(26\)	\(346\)		\(336\)	\(26\)	\(310\)		\(36\)	\(0\)	\(36\)

Trace form

57 * q - 3 * q^3 + 32 * q^7 + 513 * q^9 - 20 * q^11 + 46 * q^13 + 26 * q^17 + 192 * q^19 - 80 * q^23 - 27 * q^27 - 58 * q^29 - 84 * q^31 + 12 * q^33 - 74 * q^37 + 54 * q^39 - 414 * q^41 - 836 * q^43 - 168 * q^47 + 2713 * q^49 + 678 * q^51 + 1054 * q^53 - 84 * q^57 - 628 * q^59 + 734 * q^61 + 288 * q^63 + 1140 * q^67 + 264 * q^69 - 1512 * q^71 + 970 * q^73 + 768 * q^77 - 880 * q^79 + 4617 * q^81 - 3644 * q^83 - 162 * q^87 + 1426 * q^89 + 2468 * q^91 - 456 * q^93 + 802 * q^97 - 180 * q^99

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(1200))\) 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
1200.4.a.a	$1200$	$4$	1200.a	1.a	$1$	$1$	$1$	$70.802$	\(\Q\)	None	✓				60.4.a.a	$1$	$1$	\(0\)	\(-3\)	\(0\)	\(-28\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-3q^{3}-28q^{7}+9q^{9}+24q^{11}+70q^{13}+\cdots\)
1200.4.a.b	$1200$	$4$	1200.a	1.a	$1$	$1$	$1$	$70.802$	\(\Q\)	None	✓				6.4.a.a	$1$	$1$	\(0\)	\(-3\)	\(0\)	\(-16\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-3q^{3}-2^{4}q^{7}+9q^{9}-12q^{11}-38q^{13}+\cdots\)
1200.4.a.c	$1200$	$4$	1200.a	1.a	$1$	$1$	$1$	$70.802$	\(\Q\)	None	✓				120.4.a.c	$1$	$0$	\(0\)	\(-3\)	\(0\)	\(-16\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-3q^{3}-2^{4}q^{7}+9q^{9}+28q^{11}+26q^{13}+\cdots\)
1200.4.a.d	$1200$	$4$	1200.a	1.a	$1$	$1$	$1$	$70.802$	\(\Q\)	None	✓				300.4.a.a	$1$	$1$	\(0\)	\(-3\)	\(0\)	\(-13\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-3q^{3}-13q^{7}+9q^{9}-6q^{11}-5q^{13}+\cdots\)
1200.4.a.e	$1200$	$4$	1200.a	1.a	$1$	$1$	$1$	$70.802$	\(\Q\)	None	✓				120.4.f.b	$1$	$1$	\(0\)	\(-3\)	\(0\)	\(-10\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-3q^{3}-10q^{7}+9q^{9}+14q^{11}-82q^{13}+\cdots\)
1200.4.a.f	$1200$	$4$	1200.a	1.a	$1$	$1$	$1$	$70.802$	\(\Q\)	None	✓				120.4.f.a	$1$	$1$	\(0\)	\(-3\)	\(0\)	\(-10\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-3q^{3}-10q^{7}+9q^{9}+46q^{11}-34q^{13}+\cdots\)
1200.4.a.g	$1200$	$4$	1200.a	1.a	$1$	$1$	$1$	$70.802$	\(\Q\)	None	✓				600.4.a.e	$1$	$1$	\(0\)	\(-3\)	\(0\)	\(-5\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-3q^{3}-5q^{7}+9q^{9}-14q^{11}+q^{13}+\cdots\)
1200.4.a.h	$1200$	$4$	1200.a	1.a	$1$	$1$	$1$	$70.802$	\(\Q\)	None	✓				30.4.c.a	$1$	$0$	\(0\)	\(-3\)	\(0\)	\(-2\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-3q^{3}-2q^{7}+9q^{9}-70q^{11}-54q^{13}+\cdots\)
1200.4.a.i	$1200$	$4$	1200.a	1.a	$1$	$1$	$1$	$70.802$	\(\Q\)	None	✓				150.4.a.a	$1$	$1$	\(0\)	\(-3\)	\(0\)	\(-1\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-3q^{3}-q^{7}+9q^{9}-42q^{11}+67q^{13}+\cdots\)
1200.4.a.j	$1200$	$4$	1200.a	1.a	$1$	$1$	$1$	$70.802$	\(\Q\)	None	✓				120.4.a.d	$1$	$0$	\(0\)	\(-3\)	\(0\)	\(0\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-3q^{3}+9q^{9}-4q^{11}-54q^{13}-114q^{17}+\cdots\)
1200.4.a.k	$1200$	$4$	1200.a	1.a	$1$	$1$	$1$	$70.802$	\(\Q\)	None	✓				120.4.a.a	$1$	$0$	\(0\)	\(-3\)	\(0\)	\(4\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-3q^{3}+4q^{7}+9q^{9}-72q^{11}+6q^{13}+\cdots\)
1200.4.a.l	$1200$	$4$	1200.a	1.a	$1$	$1$	$1$	$70.802$	\(\Q\)	None	✓				120.4.f.c	$1$	$1$	\(0\)	\(-3\)	\(0\)	\(4\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-3q^{3}+4q^{7}+9q^{9}+28q^{11}+2^{4}q^{13}+\cdots\)
1200.4.a.m	$1200$	$4$	1200.a	1.a	$1$	$1$	$1$	$70.802$	\(\Q\)	None	✓				300.4.a.c	$1$	$0$	\(0\)	\(-3\)	\(0\)	\(7\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-3q^{3}+7q^{7}+9q^{9}+54q^{11}-55q^{13}+\cdots\)
1200.4.a.n	$1200$	$4$	1200.a	1.a	$1$	$1$	$1$	$70.802$	\(\Q\)	None	✓				600.4.a.g	$1$	$1$	\(0\)	\(-3\)	\(0\)	\(19\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-3q^{3}+19q^{7}+9q^{9}-22q^{11}+q^{13}+\cdots\)
1200.4.a.o	$1200$	$4$	1200.a	1.a	$1$	$1$	$1$	$70.802$	\(\Q\)	None	✓				15.4.a.b	$1$	$1$	\(0\)	\(-3\)	\(0\)	\(20\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-3q^{3}+20q^{7}+9q^{9}+24q^{11}-74q^{13}+\cdots\)
1200.4.a.p	$1200$	$4$	1200.a	1.a	$1$	$1$	$1$	$70.802$	\(\Q\)	None	✓				120.4.a.b	$1$	$0$	\(0\)	\(-3\)	\(0\)	\(20\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-3q^{3}+20q^{7}+9q^{9}+56q^{11}+86q^{13}+\cdots\)
1200.4.a.q	$1200$	$4$	1200.a	1.a	$1$	$1$	$1$	$70.802$	\(\Q\)	None	✓				60.4.d.a	$1$	$0$	\(0\)	\(-3\)	\(0\)	\(22\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-3q^{3}+22q^{7}+9q^{9}+14q^{11}+30q^{13}+\cdots\)
1200.4.a.r	$1200$	$4$	1200.a	1.a	$1$	$1$	$1$	$70.802$	\(\Q\)	None	✓				150.4.a.c	$1$	$1$	\(0\)	\(-3\)	\(0\)	\(23\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-3q^{3}+23q^{7}+9q^{9}+30q^{11}-29q^{13}+\cdots\)
1200.4.a.s	$1200$	$4$	1200.a	1.a	$1$	$1$	$1$	$70.802$	\(\Q\)	None	✓				60.4.a.b	$1$	$1$	\(0\)	\(-3\)	\(0\)	\(32\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-3q^{3}+2^{5}q^{7}+9q^{9}-6^{2}q^{11}+10q^{13}+\cdots\)
1200.4.a.t	$1200$	$4$	1200.a	1.a	$1$	$1$	$1$	$70.802$	\(\Q\)	None	✓				15.4.a.a	$1$	$0$	\(0\)	\(3\)	\(0\)	\(-24\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+3q^{3}-24q^{7}+9q^{9}-52q^{11}-22q^{13}+\cdots\)
1200.4.a.u	$1200$	$4$	1200.a	1.a	$1$	$1$	$1$	$70.802$	\(\Q\)	None	✓				24.4.a.a	$1$	$1$	\(0\)	\(3\)	\(0\)	\(-24\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3q^{3}-24q^{7}+9q^{9}+28q^{11}+74q^{13}+\cdots\)
1200.4.a.v	$1200$	$4$	1200.a	1.a	$1$	$1$	$1$	$70.802$	\(\Q\)	None	✓				150.4.a.c	$1$	$1$	\(0\)	\(3\)	\(0\)	\(-23\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3q^{3}-23q^{7}+9q^{9}+30q^{11}+29q^{13}+\cdots\)
1200.4.a.w	$1200$	$4$	1200.a	1.a	$1$	$1$	$1$	$70.802$	\(\Q\)	None	✓				60.4.d.a	$1$	$1$	\(0\)	\(3\)	\(0\)	\(-22\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3q^{3}-22q^{7}+9q^{9}+14q^{11}-30q^{13}+\cdots\)
1200.4.a.x	$1200$	$4$	1200.a	1.a	$1$	$1$	$1$	$70.802$	\(\Q\)	None	✓				600.4.a.g	$1$	$1$	\(0\)	\(3\)	\(0\)	\(-19\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3q^{3}-19q^{7}+9q^{9}-22q^{11}-q^{13}+\cdots\)
1200.4.a.y	$1200$	$4$	1200.a	1.a	$1$	$1$	$1$	$70.802$	\(\Q\)	None	✓				300.4.a.c	$1$	$0$	\(0\)	\(3\)	\(0\)	\(-7\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+3q^{3}-7q^{7}+9q^{9}+54q^{11}+55q^{13}+\cdots\)
1200.4.a.z	$1200$	$4$	1200.a	1.a	$1$	$1$	$1$	$70.802$	\(\Q\)	None	✓				120.4.f.c	$1$	$0$	\(0\)	\(3\)	\(0\)	\(-4\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+3q^{3}-4q^{7}+9q^{9}+28q^{11}-2^{4}q^{13}+\cdots\)
1200.4.a.ba	$1200$	$4$	1200.a	1.a	$1$	$1$	$1$	$70.802$	\(\Q\)	None	✓				30.4.a.b	$1$	$0$	\(0\)	\(3\)	\(0\)	\(-4\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+3q^{3}-4q^{7}+9q^{9}+48q^{11}-2q^{13}+\cdots\)
1200.4.a.bb	$1200$	$4$	1200.a	1.a	$1$	$1$	$1$	$70.802$	\(\Q\)	None	✓				150.4.a.a	$1$	$1$	\(0\)	\(3\)	\(0\)	\(1\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3q^{3}+q^{7}+9q^{9}-42q^{11}-67q^{13}+\cdots\)
1200.4.a.bc	$1200$	$4$	1200.a	1.a	$1$	$1$	$1$	$70.802$	\(\Q\)	None	✓				30.4.c.a	$1$	$1$	\(0\)	\(3\)	\(0\)	\(2\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3q^{3}+2q^{7}+9q^{9}-70q^{11}+54q^{13}+\cdots\)
1200.4.a.bd	$1200$	$4$	1200.a	1.a	$1$	$1$	$1$	$70.802$	\(\Q\)	None	✓				600.4.a.e	$1$	$1$	\(0\)	\(3\)	\(0\)	\(5\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3q^{3}+5q^{7}+9q^{9}-14q^{11}-q^{13}+\cdots\)
1200.4.a.be	$1200$	$4$	1200.a	1.a	$1$	$1$	$1$	$70.802$	\(\Q\)	None	✓				12.4.a.a	$1$	$0$	\(0\)	\(3\)	\(0\)	\(8\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+3q^{3}+8q^{7}+9q^{9}-6^{2}q^{11}+10q^{13}+\cdots\)
1200.4.a.bf	$1200$	$4$	1200.a	1.a	$1$	$1$	$1$	$70.802$	\(\Q\)	None	✓				120.4.a.f	$1$	$1$	\(0\)	\(3\)	\(0\)	\(8\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3q^{3}+8q^{7}+9q^{9}-20q^{11}-22q^{13}+\cdots\)
1200.4.a.bg	$1200$	$4$	1200.a	1.a	$1$	$1$	$1$	$70.802$	\(\Q\)	None	✓				120.4.f.b	$1$	$0$	\(0\)	\(3\)	\(0\)	\(10\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+3q^{3}+10q^{7}+9q^{9}+14q^{11}+82q^{13}+\cdots\)
1200.4.a.bh	$1200$	$4$	1200.a	1.a	$1$	$1$	$1$	$70.802$	\(\Q\)	None	✓				120.4.f.a	$1$	$0$	\(0\)	\(3\)	\(0\)	\(10\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+3q^{3}+10q^{7}+9q^{9}+46q^{11}+34q^{13}+\cdots\)
1200.4.a.bi	$1200$	$4$	1200.a	1.a	$1$	$1$	$1$	$70.802$	\(\Q\)	None	✓				300.4.a.a	$1$	$1$	\(0\)	\(3\)	\(0\)	\(13\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3q^{3}+13q^{7}+9q^{9}-6q^{11}+5q^{13}+\cdots\)
1200.4.a.bj	$1200$	$4$	1200.a	1.a	$1$	$1$	$1$	$70.802$	\(\Q\)	None	✓				120.4.a.e	$1$	$1$	\(0\)	\(3\)	\(0\)	\(20\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3q^{3}+20q^{7}+9q^{9}-2^{4}q^{11}-58q^{13}+\cdots\)
1200.4.a.bk	$1200$	$4$	1200.a	1.a	$1$	$1$	$1$	$70.802$	\(\Q\)	None	✓				30.4.a.a	$1$	$0$	\(0\)	\(3\)	\(0\)	\(32\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+3q^{3}+2^{5}q^{7}+9q^{9}+60q^{11}+34q^{13}+\cdots\)
1200.4.a.bl	$1200$	$4$	1200.a	1.a	$1$	$2$	$2$	$70.802$	\(\Q(\sqrt{19}) \)	None	✓				75.4.a.d	$1$	$0$	\(0\)	\(-6\)	\(0\)	\(-26\)		$-$	$+$	$-$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q-3q^{3}+(-13+\beta )q^{7}+9q^{9}+(-14+\cdots)q^{11}+\cdots\)
1200.4.a.bm	$1200$	$4$	1200.a	1.a	$1$	$2$	$2$	$70.802$	\(\Q(\sqrt{181}) \)	None	✓				600.4.a.r	$1$	$0$	\(0\)	\(-6\)	\(0\)	\(-6\)		$+$	$+$	$+$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q-3q^{3}+(-3-\beta )q^{7}+9q^{9}+(-4+\cdots)q^{11}+\cdots\)
1200.4.a.bn	$1200$	$4$	1200.a	1.a	$1$	$2$	$2$	$70.802$	\(\Q(\sqrt{41}) \)	None	✓				15.4.b.a	$1$	$0$	\(0\)	\(-6\)	\(0\)	\(-6\)		$-$	$+$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q-3q^{3}+(-3-3\beta )q^{7}+9q^{9}+(21+\cdots)q^{11}+\cdots\)
1200.4.a.bo	$1200$	$4$	1200.a	1.a	$1$	$2$	$2$	$70.802$	\(\Q(\sqrt{129}) \)	None	✓		✓		120.4.f.d	$1$	$1$	\(0\)	\(-6\)	\(0\)	\(2\)		$+$	$+$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q-3q^{3}+(1+3\beta )q^{7}+9q^{9}+(-37+\cdots)q^{11}+\cdots\)
1200.4.a.bp	$1200$	$4$	1200.a	1.a	$1$	$2$	$2$	$70.802$	\(\Q(\sqrt{109}) \)	None	✓				600.4.a.s	$1$	$0$	\(0\)	\(-6\)	\(0\)	\(2\)		$+$	$+$	$+$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q-3q^{3}+(1+\beta )q^{7}+9q^{9}+(8-3\beta )q^{11}+\cdots\)
1200.4.a.bq	$1200$	$4$	1200.a	1.a	$1$	$2$	$2$	$70.802$	\(\Q(\sqrt{129}) \)	None	✓				120.4.f.d	$1$	$0$	\(0\)	\(6\)	\(0\)	\(-2\)		$+$	$-$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q+3q^{3}+(-1-3\beta )q^{7}+9q^{9}+(-37+\cdots)q^{11}+\cdots\)
1200.4.a.br	$1200$	$4$	1200.a	1.a	$1$	$2$	$2$	$70.802$	\(\Q(\sqrt{109}) \)	None	✓				600.4.a.s	$1$	$0$	\(0\)	\(6\)	\(0\)	\(-2\)		$+$	$-$	$-$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q+3q^{3}+(-1-\beta )q^{7}+9q^{9}+(8-3\beta )q^{11}+\cdots\)
1200.4.a.bs	$1200$	$4$	1200.a	1.a	$1$	$2$	$2$	$70.802$	\(\Q(\sqrt{181}) \)	None	✓				600.4.a.r	$1$	$0$	\(0\)	\(6\)	\(0\)	\(6\)		$+$	$-$	$-$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q+3q^{3}+(3+\beta )q^{7}+9q^{9}+(-4-\beta )q^{11}+\cdots\)
1200.4.a.bt	$1200$	$4$	1200.a	1.a	$1$	$2$	$2$	$70.802$	\(\Q(\sqrt{41}) \)	None	✓		✓		15.4.b.a	$1$	$1$	\(0\)	\(6\)	\(0\)	\(6\)		$-$	$-$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+3q^{3}+(3+3\beta )q^{7}+9q^{9}+(21-3\beta )q^{11}+\cdots\)
1200.4.a.bu	$1200$	$4$	1200.a	1.a	$1$	$2$	$2$	$70.802$	\(\Q(\sqrt{19}) \)	None	✓		✓		75.4.a.d	$1$	$0$	\(0\)	\(6\)	\(0\)	\(26\)		$-$	$-$	$+$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q+3q^{3}+(13+\beta )q^{7}+9q^{9}+(-14+\cdots)q^{11}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(1200))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(1200)) \simeq \) \(S_{4}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 20}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(6))\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(8))\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(10))\)\(^{\oplus 16}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(12))\)\(^{\oplus 9}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 10}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(16))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(24))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(25))\)\(^{\oplus 10}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(30))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(40))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(48))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(50))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(60))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(75))\)\(^{\oplus 5}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(80))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(100))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(120))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(150))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(200))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(240))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(300))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(400))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(600))\)\(^{\oplus 2}\)