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	Dim.
\(+\)	\(+\)	\(+\)	\(+\)	\(8\)
\(+\)	\(+\)	\(-\)	\(-\)	\(7\)
\(+\)	\(-\)	\(+\)	\(-\)	\(5\)
\(+\)	\(-\)	\(-\)	\(+\)	\(9\)
\(-\)	\(+\)	\(+\)	\(-\)	\(7\)
\(-\)	\(+\)	\(-\)	\(+\)	\(7\)
\(-\)	\(-\)	\(+\)	\(+\)	\(7\)
\(-\)	\(-\)	\(-\)	\(-\)	\(7\)
Plus space			\(+\)	\(31\)
Minus space			\(-\)	\(26\)

Trace form

\( 57 q - 3 q^{3} + 32 q^{7} + 513 q^{9} + O(q^{10}) \)

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

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

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