Space of modular forms of level 1350, weight 4, and trivial character

• Label: 1350.4.a
• Level: $1350$
• Weight: $4$
• Character orbit: 1350.a
• Rep. character: $\chi_{1350}(1,\cdot)$
• Character field: $\Q$
• Dimension: $76$
• Newform subspaces: $48$
• Sturm bound: $1080$
• Trace bound: $11$

Defining parameters

Level:	\( N \)	\(=\)	\( 1350 = 2 \cdot 3^{3} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1350.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 48 \)
Sturm bound:			\(1080\)
Trace bound:			\(11\)
Distinguishing \(T_p\):			\(7\), \(11\), \(17\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(1350))\).

	Total	New	Old
Modular forms	846	76	770
Cusp forms	774	76	698
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
\(+\)	\(+\)	\(+\)	\(+\)		\(111\)	\(9\)	\(102\)		\(102\)	\(9\)	\(93\)		\(9\)	\(0\)	\(9\)
\(+\)	\(+\)	\(-\)	\(-\)		\(102\)	\(9\)	\(93\)		\(93\)	\(9\)	\(84\)		\(9\)	\(0\)	\(9\)
\(+\)	\(-\)	\(+\)	\(-\)		\(102\)	\(9\)	\(93\)		\(93\)	\(9\)	\(84\)		\(9\)	\(0\)	\(9\)
\(+\)	\(-\)	\(-\)	\(+\)		\(108\)	\(11\)	\(97\)		\(99\)	\(11\)	\(88\)		\(9\)	\(0\)	\(9\)
\(-\)	\(+\)	\(+\)	\(-\)		\(105\)	\(8\)	\(97\)		\(96\)	\(8\)	\(88\)		\(9\)	\(0\)	\(9\)
\(-\)	\(+\)	\(-\)	\(+\)		\(105\)	\(11\)	\(94\)		\(96\)	\(11\)	\(85\)		\(9\)	\(0\)	\(9\)
\(-\)	\(-\)	\(+\)	\(+\)		\(105\)	\(10\)	\(95\)		\(96\)	\(10\)	\(86\)		\(9\)	\(0\)	\(9\)
\(-\)	\(-\)	\(-\)	\(-\)		\(108\)	\(9\)	\(99\)		\(99\)	\(9\)	\(90\)		\(9\)	\(0\)	\(9\)
Plus space			\(+\)		\(429\)	\(41\)	\(388\)		\(393\)	\(41\)	\(352\)		\(36\)	\(0\)	\(36\)
Minus space			\(-\)		\(417\)	\(35\)	\(382\)		\(381\)	\(35\)	\(346\)		\(36\)	\(0\)	\(36\)

Trace form

76 * q + 304 * q^4 + 56 * q^7 + 26 * q^13 + 1216 * q^16 - 310 * q^19 - 12 * q^22 + 224 * q^28 + 146 * q^31 + 480 * q^34 - 1078 * q^37 - 1900 * q^43 - 408 * q^46 + 3252 * q^49 + 104 * q^52 + 408 * q^58 + 2102 * q^61 + 4864 * q^64 + 1130 * q^67 + 4232 * q^73 - 1240 * q^76 - 1702 * q^79 - 216 * q^82 - 48 * q^88 - 2426 * q^91 - 3672 * q^94 + 1436 * q^97

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(1350))\) 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
1350.4.a.a	$1350$	$4$	1350.a	1.a	$1$	$1$	$1$	$79.653$	\(\Q\)	None	✓				54.4.a.b	$1$	$0$	\(-2\)	\(0\)	\(0\)	\(-29\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+4q^{4}-29q^{7}-8q^{8}-57q^{11}+\cdots\)
1350.4.a.b	$1350$	$4$	1350.a	1.a	$1$	$1$	$1$	$79.653$	\(\Q\)	None	✓	✓			1350.4.a.b	$1$	$0$	\(-2\)	\(0\)	\(0\)	\(-23\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+4q^{4}-23q^{7}-8q^{8}-30q^{11}+\cdots\)
1350.4.a.c	$1350$	$4$	1350.a	1.a	$1$	$1$	$1$	$79.653$	\(\Q\)	None	✓	✓			1350.4.a.c	$1$	$0$	\(-2\)	\(0\)	\(0\)	\(-19\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+4q^{4}-19q^{7}-8q^{8}-12q^{11}+\cdots\)
1350.4.a.d	$1350$	$4$	1350.a	1.a	$1$	$1$	$1$	$79.653$	\(\Q\)	None	✓				270.4.c.a	$1$	$1$	\(-2\)	\(0\)	\(0\)	\(-14\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+4q^{4}-14q^{7}-8q^{8}-22q^{11}+\cdots\)
1350.4.a.e	$1350$	$4$	1350.a	1.a	$1$	$1$	$1$	$79.653$	\(\Q\)	None	✓				270.4.a.f	$1$	$0$	\(-2\)	\(0\)	\(0\)	\(-14\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+4q^{4}-14q^{7}-8q^{8}-3q^{11}+\cdots\)
1350.4.a.f	$1350$	$4$	1350.a	1.a	$1$	$1$	$1$	$79.653$	\(\Q\)	None	✓				270.4.a.b	$1$	$1$	\(-2\)	\(0\)	\(0\)	\(-8\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+4q^{4}-8q^{7}-8q^{8}+18q^{11}+\cdots\)
1350.4.a.g	$1350$	$4$	1350.a	1.a	$1$	$1$	$1$	$79.653$	\(\Q\)	None	✓				270.4.a.e	$1$	$0$	\(-2\)	\(0\)	\(0\)	\(4\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+4q^{4}+4q^{7}-8q^{8}+42q^{11}+\cdots\)
1350.4.a.h	$1350$	$4$	1350.a	1.a	$1$	$1$	$1$	$79.653$	\(\Q\)	None	✓				54.4.a.a	$1$	$0$	\(-2\)	\(0\)	\(0\)	\(7\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+4q^{4}+7q^{7}-8q^{8}+60q^{11}+\cdots\)
1350.4.a.i	$1350$	$4$	1350.a	1.a	$1$	$1$	$1$	$79.653$	\(\Q\)	None	✓				270.4.a.d	$1$	$1$	\(-2\)	\(0\)	\(0\)	\(13\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+4q^{4}+13q^{7}-8q^{8}-30q^{11}+\cdots\)
1350.4.a.j	$1350$	$4$	1350.a	1.a	$1$	$1$	$1$	$79.653$	\(\Q\)	None	✓				270.4.c.a	$1$	$1$	\(-2\)	\(0\)	\(0\)	\(14\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+4q^{4}+14q^{7}-8q^{8}+22q^{11}+\cdots\)
1350.4.a.k	$1350$	$4$	1350.a	1.a	$1$	$1$	$1$	$79.653$	\(\Q\)	None	✓	✓			1350.4.a.c	$1$	$0$	\(-2\)	\(0\)	\(0\)	\(19\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+4q^{4}+19q^{7}-8q^{8}+12q^{11}+\cdots\)
1350.4.a.l	$1350$	$4$	1350.a	1.a	$1$	$1$	$1$	$79.653$	\(\Q\)	None	✓				270.4.a.c	$1$	$1$	\(-2\)	\(0\)	\(0\)	\(22\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+4q^{4}+22q^{7}-8q^{8}-12q^{11}+\cdots\)
1350.4.a.m	$1350$	$4$	1350.a	1.a	$1$	$1$	$1$	$79.653$	\(\Q\)	None	✓	✓			1350.4.a.b	$1$	$0$	\(-2\)	\(0\)	\(0\)	\(23\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+4q^{4}+23q^{7}-8q^{8}+30q^{11}+\cdots\)
1350.4.a.n	$1350$	$4$	1350.a	1.a	$1$	$1$	$1$	$79.653$	\(\Q\)	None	✓				270.4.a.a	$1$	$0$	\(-2\)	\(0\)	\(0\)	\(34\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+4q^{4}+34q^{7}-8q^{8}-48q^{11}+\cdots\)
1350.4.a.o	$1350$	$4$	1350.a	1.a	$1$	$1$	$1$	$79.653$	\(\Q\)	None	✓				54.4.a.b	$1$	$0$	\(2\)	\(0\)	\(0\)	\(-29\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+4q^{4}-29q^{7}+8q^{8}+57q^{11}+\cdots\)
1350.4.a.p	$1350$	$4$	1350.a	1.a	$1$	$1$	$1$	$79.653$	\(\Q\)	None	✓	✓			1350.4.a.b	$1$	$1$	\(2\)	\(0\)	\(0\)	\(-23\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+4q^{4}-23q^{7}+8q^{8}+30q^{11}+\cdots\)
1350.4.a.q	$1350$	$4$	1350.a	1.a	$1$	$1$	$1$	$79.653$	\(\Q\)	None	✓	✓			1350.4.a.c	$1$	$1$	\(2\)	\(0\)	\(0\)	\(-19\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+4q^{4}-19q^{7}+8q^{8}+12q^{11}+\cdots\)
1350.4.a.r	$1350$	$4$	1350.a	1.a	$1$	$1$	$1$	$79.653$	\(\Q\)	None	✓				270.4.a.f	$1$	$1$	\(2\)	\(0\)	\(0\)	\(-14\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+4q^{4}-14q^{7}+8q^{8}+3q^{11}+\cdots\)
1350.4.a.s	$1350$	$4$	1350.a	1.a	$1$	$1$	$1$	$79.653$	\(\Q\)	None	✓				270.4.c.a	$1$	$1$	\(2\)	\(0\)	\(0\)	\(-14\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+4q^{4}-14q^{7}+8q^{8}+22q^{11}+\cdots\)
1350.4.a.t	$1350$	$4$	1350.a	1.a	$1$	$1$	$1$	$79.653$	\(\Q\)	None	✓				270.4.a.b	$1$	$1$	\(2\)	\(0\)	\(0\)	\(-8\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+4q^{4}-8q^{7}+8q^{8}-18q^{11}+\cdots\)
1350.4.a.u	$1350$	$4$	1350.a	1.a	$1$	$1$	$1$	$79.653$	\(\Q\)	None	✓				270.4.a.e	$1$	$0$	\(2\)	\(0\)	\(0\)	\(4\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+4q^{4}+4q^{7}+8q^{8}-42q^{11}+\cdots\)
1350.4.a.v	$1350$	$4$	1350.a	1.a	$1$	$1$	$1$	$79.653$	\(\Q\)	None	✓				54.4.a.a	$1$	$1$	\(2\)	\(0\)	\(0\)	\(7\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+4q^{4}+7q^{7}+8q^{8}-60q^{11}+\cdots\)
1350.4.a.w	$1350$	$4$	1350.a	1.a	$1$	$1$	$1$	$79.653$	\(\Q\)	None	✓				270.4.a.d	$1$	$0$	\(2\)	\(0\)	\(0\)	\(13\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+4q^{4}+13q^{7}+8q^{8}+30q^{11}+\cdots\)
1350.4.a.x	$1350$	$4$	1350.a	1.a	$1$	$1$	$1$	$79.653$	\(\Q\)	None	✓				270.4.c.a	$1$	$1$	\(2\)	\(0\)	\(0\)	\(14\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+4q^{4}+14q^{7}+8q^{8}-22q^{11}+\cdots\)
1350.4.a.y	$1350$	$4$	1350.a	1.a	$1$	$1$	$1$	$79.653$	\(\Q\)	None	✓	✓			1350.4.a.c	$1$	$1$	\(2\)	\(0\)	\(0\)	\(19\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+4q^{4}+19q^{7}+8q^{8}-12q^{11}+\cdots\)
1350.4.a.z	$1350$	$4$	1350.a	1.a	$1$	$1$	$1$	$79.653$	\(\Q\)	None	✓				270.4.a.c	$1$	$1$	\(2\)	\(0\)	\(0\)	\(22\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+4q^{4}+22q^{7}+8q^{8}+12q^{11}+\cdots\)
1350.4.a.ba	$1350$	$4$	1350.a	1.a	$1$	$1$	$1$	$79.653$	\(\Q\)	None	✓	✓			1350.4.a.b	$1$	$1$	\(2\)	\(0\)	\(0\)	\(23\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+4q^{4}+23q^{7}+8q^{8}-30q^{11}+\cdots\)
1350.4.a.bb	$1350$	$4$	1350.a	1.a	$1$	$1$	$1$	$79.653$	\(\Q\)	None	✓				270.4.a.a	$1$	$0$	\(2\)	\(0\)	\(0\)	\(34\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+4q^{4}+34q^{7}+8q^{8}+48q^{11}+\cdots\)
1350.4.a.bc	$1350$	$4$	1350.a	1.a	$1$	$2$	$2$	$79.653$	\(\Q(\sqrt{6}) \)	None	✓	✓			1350.4.a.bc	$1$	$1$	\(-4\)	\(0\)	\(0\)	\(-20\)		$+$	$+$	$-$	$-$	$2\cdot 3$	$\mathrm{SU}(2)$	\(q-2q^{2}+4q^{4}+(-10+\beta )q^{7}-8q^{8}+\cdots\)
1350.4.a.bd	$1350$	$4$	1350.a	1.a	$1$	$2$	$2$	$79.653$	\(\Q(\sqrt{209}) \)	None	✓	✓			1350.4.a.bd	$1$	$1$	\(-4\)	\(0\)	\(0\)	\(-13\)		$+$	$-$	$+$	$-$	$3$	$\mathrm{SU}(2)$	\(q-2q^{2}+4q^{4}+(-6-\beta )q^{7}-8q^{8}+\cdots\)
1350.4.a.be	$1350$	$4$	1350.a	1.a	$1$	$2$	$2$	$79.653$	\(\Q(\sqrt{21}) \)	None	✓	✓			1350.4.a.be	$1$	$1$	\(-4\)	\(0\)	\(0\)	\(-10\)		$+$	$-$	$+$	$-$	$2\cdot 3$	$\mathrm{SU}(2)$	\(q-2q^{2}+4q^{4}+(-5-\beta )q^{7}-8q^{8}+\cdots\)
1350.4.a.bf	$1350$	$4$	1350.a	1.a	$1$	$2$	$2$	$79.653$	\(\Q(\sqrt{401}) \)	None	✓				270.4.a.m	$1$	$1$	\(-4\)	\(0\)	\(0\)	\(-1\)		$+$	$-$	$+$	$-$	$3$	$\mathrm{SU}(2)$	\(q-2q^{2}+4q^{4}+(-1-\beta )q^{7}-8q^{8}+\cdots\)
1350.4.a.bg	$1350$	$4$	1350.a	1.a	$1$	$2$	$2$	$79.653$	\(\Q(\sqrt{21}) \)	None	✓	✓			1350.4.a.be	$1$	$0$	\(-4\)	\(0\)	\(0\)	\(10\)		$+$	$-$	$-$	$+$	$2\cdot 3$	$\mathrm{SU}(2)$	\(q-2q^{2}+4q^{4}+(5+\beta )q^{7}-8q^{8}+(-15+\cdots)q^{11}+\cdots\)
1350.4.a.bh	$1350$	$4$	1350.a	1.a	$1$	$2$	$2$	$79.653$	\(\Q(\sqrt{209}) \)	None	✓	✓			1350.4.a.bd	$1$	$1$	\(-4\)	\(0\)	\(0\)	\(13\)		$+$	$+$	$-$	$-$	$3$	$\mathrm{SU}(2)$	\(q-2q^{2}+4q^{4}+(7-\beta )q^{7}-8q^{8}+(-10+\cdots)q^{11}+\cdots\)
1350.4.a.bi	$1350$	$4$	1350.a	1.a	$1$	$2$	$2$	$79.653$	\(\Q(\sqrt{6}) \)	None	✓	✓	✓		1350.4.a.bc	$1$	$0$	\(-4\)	\(0\)	\(0\)	\(20\)		$+$	$+$	$+$	$+$	$2\cdot 3$	$\mathrm{SU}(2)$	\(q-2q^{2}+4q^{4}+(10+\beta )q^{7}-8q^{8}+\cdots\)
1350.4.a.bj	$1350$	$4$	1350.a	1.a	$1$	$2$	$2$	$79.653$	\(\Q(\sqrt{6}) \)	None	✓	✓			1350.4.a.bc	$1$	$1$	\(4\)	\(0\)	\(0\)	\(-20\)		$-$	$-$	$-$	$-$	$2\cdot 3$	$\mathrm{SU}(2)$	\(q+2q^{2}+4q^{4}+(-10+\beta )q^{7}+8q^{8}+\cdots\)
1350.4.a.bk	$1350$	$4$	1350.a	1.a	$1$	$2$	$2$	$79.653$	\(\Q(\sqrt{209}) \)	None	✓	✓			1350.4.a.bd	$1$	$0$	\(4\)	\(0\)	\(0\)	\(-13\)		$-$	$-$	$+$	$+$	$3$	$\mathrm{SU}(2)$	\(q+2q^{2}+4q^{4}+(-6-\beta )q^{7}+8q^{8}+\cdots\)
1350.4.a.bl	$1350$	$4$	1350.a	1.a	$1$	$2$	$2$	$79.653$	\(\Q(\sqrt{21}) \)	None	✓	✓	✓		1350.4.a.be	$1$	$1$	\(4\)	\(0\)	\(0\)	\(-10\)		$-$	$+$	$+$	$-$	$2\cdot 3$	$\mathrm{SU}(2)$	\(q+2q^{2}+4q^{4}+(-5-\beta )q^{7}+8q^{8}+\cdots\)
1350.4.a.bm	$1350$	$4$	1350.a	1.a	$1$	$2$	$2$	$79.653$	\(\Q(\sqrt{401}) \)	None	✓				270.4.a.m	$1$	$0$	\(4\)	\(0\)	\(0\)	\(-1\)		$-$	$-$	$+$	$+$	$3$	$\mathrm{SU}(2)$	\(q+2q^{2}+4q^{4}+(-1-\beta )q^{7}+8q^{8}+\cdots\)
1350.4.a.bn	$1350$	$4$	1350.a	1.a	$1$	$2$	$2$	$79.653$	\(\Q(\sqrt{21}) \)	None	✓	✓			1350.4.a.be	$1$	$0$	\(4\)	\(0\)	\(0\)	\(10\)		$-$	$+$	$-$	$+$	$2\cdot 3$	$\mathrm{SU}(2)$	\(q+2q^{2}+4q^{4}+(5+\beta )q^{7}+8q^{8}+(15+\cdots)q^{11}+\cdots\)
1350.4.a.bo	$1350$	$4$	1350.a	1.a	$1$	$2$	$2$	$79.653$	\(\Q(\sqrt{209}) \)	None	✓	✓			1350.4.a.bd	$1$	$0$	\(4\)	\(0\)	\(0\)	\(13\)		$-$	$+$	$-$	$+$	$3$	$\mathrm{SU}(2)$	\(q+2q^{2}+4q^{4}+(7-\beta )q^{7}+8q^{8}+(10+\cdots)q^{11}+\cdots\)
1350.4.a.bp	$1350$	$4$	1350.a	1.a	$1$	$2$	$2$	$79.653$	\(\Q(\sqrt{6}) \)	None	✓	✓			1350.4.a.bc	$1$	$0$	\(4\)	\(0\)	\(0\)	\(20\)		$-$	$-$	$+$	$+$	$2\cdot 3$	$\mathrm{SU}(2)$	\(q+2q^{2}+4q^{4}+(10+\beta )q^{7}+8q^{8}+\cdots\)
1350.4.a.bq	$1350$	$4$	1350.a	1.a	$1$	$3$	$3$	$79.653$	3.3.46616.1	None	✓				270.4.c.c	$1$	$0$	\(-6\)	\(0\)	\(0\)	\(-22\)		$+$	$-$	$-$	$+$	$3$	$\mathrm{SU}(2)$	\(q-2q^{2}+4q^{4}+(-7+\beta _{2})q^{7}-8q^{8}+\cdots\)
1350.4.a.br	$1350$	$4$	1350.a	1.a	$1$	$3$	$3$	$79.653$	3.3.46616.1	None	✓		✓		270.4.c.c	$1$	$1$	\(-6\)	\(0\)	\(0\)	\(22\)		$+$	$+$	$-$	$-$	$3$	$\mathrm{SU}(2)$	\(q-2q^{2}+4q^{4}+(7-\beta _{2})q^{7}-8q^{8}+\cdots\)
1350.4.a.bs	$1350$	$4$	1350.a	1.a	$1$	$3$	$3$	$79.653$	3.3.46616.1	None	✓		✓		270.4.c.c	$1$	$1$	\(6\)	\(0\)	\(0\)	\(-22\)		$-$	$-$	$-$	$-$	$3$	$\mathrm{SU}(2)$	\(q+2q^{2}+4q^{4}+(-7+\beta _{2})q^{7}+8q^{8}+\cdots\)
1350.4.a.bt	$1350$	$4$	1350.a	1.a	$1$	$3$	$3$	$79.653$	3.3.46616.1	None	✓				270.4.c.c	$1$	$0$	\(6\)	\(0\)	\(0\)	\(22\)		$-$	$+$	$-$	$+$	$3$	$\mathrm{SU}(2)$	\(q+2q^{2}+4q^{4}+(7-\beta _{2})q^{7}+8q^{8}+\cdots\)
1350.4.a.bu	$1350$	$4$	1350.a	1.a	$1$	$4$	$4$	$79.653$	4.4.29021904.1	None	✓		✓		270.4.c.e	$2$	$0$	\(-8\)	\(0\)	\(0\)	\(0\)		$+$	$-$	$-$	$+$	$2\cdot 3^{2}\cdot 5^{2}$	$\mathrm{SU}(2)$	\(q-2q^{2}+4q^{4}+\beta _{1}q^{7}-8q^{8}-\beta _{1}q^{11}+\cdots\)
1350.4.a.bv	$1350$	$4$	1350.a	1.a	$1$	$4$	$4$	$79.653$	4.4.29021904.1	None	✓		✓		270.4.c.e	$2$	$0$	\(8\)	\(0\)	\(0\)	\(0\)		$-$	$+$	$-$	$+$	$2\cdot 3^{2}\cdot 5^{2}$	$\mathrm{SU}(2)$	\(q+2q^{2}+4q^{4}+\beta _{1}q^{7}+8q^{8}+\beta _{1}q^{11}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(1350))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(1350)) \simeq \) \(S_{4}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 16}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(6))\)\(^{\oplus 9}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(9))\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(10))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(18))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(25))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(27))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(30))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(45))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(50))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(54))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(75))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(90))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(135))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(150))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(225))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(270))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(450))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(675))\)\(^{\oplus 2}\)