Space of modular forms of level 4032, weight 2, and trivial character

• Label: 4032.2.a
• Level: $4032$
• Weight: $2$
• Character orbit: 4032.a
• Rep. character: $\chi_{4032}(1,\cdot)$
• Character field: $\Q$
• Dimension: $60$
• Newform subspaces: $50$
• Sturm bound: $1536$
• Trace bound: $13$

Defining parameters

Level:	\( N \)	\(=\)	\( 4032 = 2^{6} \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4032.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 50 \)
Sturm bound:			\(1536\)
Trace bound:			\(13\)
Distinguishing \(T_p\):			\(5\), \(11\), \(13\), \(17\), \(19\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(4032))\).

	Total	New	Old
Modular forms	816	60	756
Cusp forms	721	60	661
Eisenstein series	95	0	95

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

\(2\)	\(3\)	\(7\)	Fricke	Dim
\(+\)	\(+\)	\(+\)	\(+\)	\(6\)
\(+\)	\(+\)	\(-\)	\(-\)	\(8\)
\(+\)	\(-\)	\(+\)	\(-\)	\(9\)
\(+\)	\(-\)	\(-\)	\(+\)	\(8\)
\(-\)	\(+\)	\(+\)	\(-\)	\(6\)
\(-\)	\(+\)	\(-\)	\(+\)	\(4\)
\(-\)	\(-\)	\(+\)	\(+\)	\(9\)
\(-\)	\(-\)	\(-\)	\(-\)	\(10\)
Plus space			\(+\)	\(27\)
Minus space			\(-\)	\(33\)

Trace form

\( 60 q + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(4032))\) 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	7
4032.2.a.a	$4032$	$2$	4032.a	1.a	$1$	$1$	$1$	$32.196$	\(\Q\)	None	✓				56.2.a.b	$1$	$1$	\(0\)	\(0\)	\(-4\)	\(-1\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-4q^{5}-q^{7}+2q^{17}-2q^{19}+8q^{23}+\cdots\)
4032.2.a.b	$4032$	$2$	4032.a	1.a	$1$	$1$	$1$	$32.196$	\(\Q\)	None	✓				672.2.a.a	$1$	$0$	\(0\)	\(0\)	\(-4\)	\(-1\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-4q^{5}-q^{7}+2q^{11}+2q^{13}-4q^{19}+\cdots\)
4032.2.a.c	$4032$	$2$	4032.a	1.a	$1$	$1$	$1$	$32.196$	\(\Q\)	None	✓				672.2.a.a	$1$	$1$	\(0\)	\(0\)	\(-4\)	\(1\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-4q^{5}+q^{7}-2q^{11}+2q^{13}+4q^{19}+\cdots\)
4032.2.a.d	$4032$	$2$	4032.a	1.a	$1$	$1$	$1$	$32.196$	\(\Q\)	None	✓				56.2.a.b	$1$	$1$	\(0\)	\(0\)	\(-4\)	\(1\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-4q^{5}+q^{7}+2q^{17}+2q^{19}-8q^{23}+\cdots\)
4032.2.a.e	$4032$	$2$	4032.a	1.a	$1$	$1$	$1$	$32.196$	\(\Q\)	None	✓				42.2.a.a	$1$	$0$	\(0\)	\(0\)	\(-2\)	\(-1\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{5}-q^{7}-4q^{11}-6q^{13}-2q^{17}+\cdots\)
4032.2.a.f	$4032$	$2$	4032.a	1.a	$1$	$1$	$1$	$32.196$	\(\Q\)	None	✓				672.2.a.b	$1$	$1$	\(0\)	\(0\)	\(-2\)	\(-1\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{5}-q^{7}-4q^{11}+6q^{13}+2q^{17}+\cdots\)
4032.2.a.g	$4032$	$2$	4032.a	1.a	$1$	$1$	$1$	$32.196$	\(\Q\)	None	✓				504.2.a.a	$1$	$1$	\(0\)	\(0\)	\(-2\)	\(-1\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{5}-q^{7}-2q^{11}-2q^{13}+6q^{17}+\cdots\)
4032.2.a.h	$4032$	$2$	4032.a	1.a	$1$	$1$	$1$	$32.196$	\(\Q\)	None	✓				21.2.a.a	$1$	$0$	\(0\)	\(0\)	\(-2\)	\(-1\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{5}-q^{7}+4q^{11}+2q^{13}+6q^{17}+\cdots\)
4032.2.a.i	$4032$	$2$	4032.a	1.a	$1$	$1$	$1$	$32.196$	\(\Q\)	None	✓				504.2.a.d	$1$	$0$	\(0\)	\(0\)	\(-2\)	\(-1\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{5}-q^{7}+6q^{11}+6q^{13}-2q^{17}+\cdots\)
4032.2.a.j	$4032$	$2$	4032.a	1.a	$1$	$1$	$1$	$32.196$	\(\Q\)	None	✓				504.2.a.d	$1$	$0$	\(0\)	\(0\)	\(-2\)	\(1\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{5}+q^{7}-6q^{11}+6q^{13}-2q^{17}+\cdots\)
4032.2.a.k	$4032$	$2$	4032.a	1.a	$1$	$1$	$1$	$32.196$	\(\Q\)	None	✓				21.2.a.a	$1$	$0$	\(0\)	\(0\)	\(-2\)	\(1\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{5}+q^{7}-4q^{11}+2q^{13}+6q^{17}+\cdots\)
4032.2.a.l	$4032$	$2$	4032.a	1.a	$1$	$1$	$1$	$32.196$	\(\Q\)	None	✓				504.2.a.a	$1$	$1$	\(0\)	\(0\)	\(-2\)	\(1\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{5}+q^{7}+2q^{11}-2q^{13}+6q^{17}+\cdots\)
4032.2.a.m	$4032$	$2$	4032.a	1.a	$1$	$1$	$1$	$32.196$	\(\Q\)	None	✓				42.2.a.a	$1$	$0$	\(0\)	\(0\)	\(-2\)	\(1\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{5}+q^{7}+4q^{11}-6q^{13}-2q^{17}+\cdots\)
4032.2.a.n	$4032$	$2$	4032.a	1.a	$1$	$1$	$1$	$32.196$	\(\Q\)	None	✓				672.2.a.b	$1$	$0$	\(0\)	\(0\)	\(-2\)	\(1\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{5}+q^{7}+4q^{11}+6q^{13}+2q^{17}+\cdots\)
4032.2.a.o	$4032$	$2$	4032.a	1.a	$1$	$1$	$1$	$32.196$	\(\Q\)	None	✓				2016.2.a.c	$1$	$0$	\(0\)	\(0\)	\(0\)	\(-1\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{7}-4q^{11}-2q^{13}-4q^{17}+4q^{23}+\cdots\)
4032.2.a.p	$4032$	$2$	4032.a	1.a	$1$	$1$	$1$	$32.196$	\(\Q\)	None	✓				224.2.a.a	$1$	$1$	\(0\)	\(0\)	\(0\)	\(-1\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{7}-4q^{11}+4q^{13}+2q^{17}+6q^{19}+\cdots\)
4032.2.a.q	$4032$	$2$	4032.a	1.a	$1$	$1$	$1$	$32.196$	\(\Q\)	None	✓				672.2.a.c	$1$	$0$	\(0\)	\(0\)	\(0\)	\(-1\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{7}-2q^{11}+2q^{13}-4q^{17}+4q^{19}+\cdots\)
4032.2.a.r	$4032$	$2$	4032.a	1.a	$1$	$1$	$1$	$32.196$	\(\Q\)	None	✓				14.2.a.a	$1$	$1$	\(0\)	\(0\)	\(0\)	\(-1\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{7}+4q^{13}-6q^{17}+2q^{19}-5q^{25}+\cdots\)
4032.2.a.s	$4032$	$2$	4032.a	1.a	$1$	$1$	$1$	$32.196$	\(\Q\)	None	✓				2016.2.a.c	$1$	$0$	\(0\)	\(0\)	\(0\)	\(-1\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{7}+4q^{11}-2q^{13}+4q^{17}-4q^{23}+\cdots\)
4032.2.a.t	$4032$	$2$	4032.a	1.a	$1$	$1$	$1$	$32.196$	\(\Q\)	None	✓				84.2.a.b	$1$	$1$	\(0\)	\(0\)	\(0\)	\(-1\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{7}+6q^{11}-2q^{13}-4q^{19}-6q^{23}+\cdots\)
4032.2.a.u	$4032$	$2$	4032.a	1.a	$1$	$1$	$1$	$32.196$	\(\Q\)	None	✓				84.2.a.b	$1$	$1$	\(0\)	\(0\)	\(0\)	\(1\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{7}-6q^{11}-2q^{13}+4q^{19}+6q^{23}+\cdots\)
4032.2.a.v	$4032$	$2$	4032.a	1.a	$1$	$1$	$1$	$32.196$	\(\Q\)	None	✓				2016.2.a.c	$1$	$1$	\(0\)	\(0\)	\(0\)	\(1\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{7}-4q^{11}-2q^{13}+4q^{17}+4q^{23}+\cdots\)
4032.2.a.w	$4032$	$2$	4032.a	1.a	$1$	$1$	$1$	$32.196$	\(\Q\)	None	✓				14.2.a.a	$1$	$1$	\(0\)	\(0\)	\(0\)	\(1\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{7}+4q^{13}-6q^{17}-2q^{19}-5q^{25}+\cdots\)
4032.2.a.x	$4032$	$2$	4032.a	1.a	$1$	$1$	$1$	$32.196$	\(\Q\)	None	✓				672.2.a.c	$1$	$1$	\(0\)	\(0\)	\(0\)	\(1\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{7}+2q^{11}+2q^{13}-4q^{17}-4q^{19}+\cdots\)
4032.2.a.y	$4032$	$2$	4032.a	1.a	$1$	$1$	$1$	$32.196$	\(\Q\)	None	✓				2016.2.a.c	$1$	$1$	\(0\)	\(0\)	\(0\)	\(1\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{7}+4q^{11}-2q^{13}-4q^{17}-4q^{23}+\cdots\)
4032.2.a.z	$4032$	$2$	4032.a	1.a	$1$	$1$	$1$	$32.196$	\(\Q\)	None	✓				224.2.a.a	$1$	$0$	\(0\)	\(0\)	\(0\)	\(1\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{7}+4q^{11}+4q^{13}+2q^{17}-6q^{19}+\cdots\)
4032.2.a.ba	$4032$	$2$	4032.a	1.a	$1$	$1$	$1$	$32.196$	\(\Q\)	None	✓				504.2.a.d	$1$	$0$	\(0\)	\(0\)	\(2\)	\(-1\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{5}-q^{7}-6q^{11}+6q^{13}+2q^{17}+\cdots\)
4032.2.a.bb	$4032$	$2$	4032.a	1.a	$1$	$1$	$1$	$32.196$	\(\Q\)	None	✓				56.2.a.a	$1$	$0$	\(0\)	\(0\)	\(2\)	\(-1\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{5}-q^{7}-4q^{11}-2q^{13}+6q^{17}+\cdots\)
4032.2.a.bc	$4032$	$2$	4032.a	1.a	$1$	$1$	$1$	$32.196$	\(\Q\)	None	✓				168.2.a.a	$1$	$1$	\(0\)	\(0\)	\(2\)	\(-1\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{5}-q^{7}-6q^{13}+2q^{17}+4q^{19}+\cdots\)
4032.2.a.bd	$4032$	$2$	4032.a	1.a	$1$	$1$	$1$	$32.196$	\(\Q\)	None	✓				672.2.a.d	$1$	$1$	\(0\)	\(0\)	\(2\)	\(-1\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{5}-q^{7}-2q^{13}-2q^{17}-4q^{19}+\cdots\)
4032.2.a.be	$4032$	$2$	4032.a	1.a	$1$	$1$	$1$	$32.196$	\(\Q\)	None	✓				168.2.a.b	$1$	$0$	\(0\)	\(0\)	\(2\)	\(-1\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{5}-q^{7}+2q^{13}-6q^{17}+4q^{19}+\cdots\)
4032.2.a.bf	$4032$	$2$	4032.a	1.a	$1$	$1$	$1$	$32.196$	\(\Q\)	None	✓				504.2.a.a	$1$	$1$	\(0\)	\(0\)	\(2\)	\(-1\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{5}-q^{7}+2q^{11}-2q^{13}-6q^{17}+\cdots\)
4032.2.a.bg	$4032$	$2$	4032.a	1.a	$1$	$1$	$1$	$32.196$	\(\Q\)	None	✓				504.2.a.a	$1$	$1$	\(0\)	\(0\)	\(2\)	\(1\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{5}+q^{7}-2q^{11}-2q^{13}-6q^{17}+\cdots\)
4032.2.a.bh	$4032$	$2$	4032.a	1.a	$1$	$1$	$1$	$32.196$	\(\Q\)	None	✓				168.2.a.a	$1$	$1$	\(0\)	\(0\)	\(2\)	\(1\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{5}+q^{7}-6q^{13}+2q^{17}-4q^{19}+\cdots\)
4032.2.a.bi	$4032$	$2$	4032.a	1.a	$1$	$1$	$1$	$32.196$	\(\Q\)	None	✓				672.2.a.d	$1$	$0$	\(0\)	\(0\)	\(2\)	\(1\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{5}+q^{7}-2q^{13}-2q^{17}+4q^{19}+\cdots\)
4032.2.a.bj	$4032$	$2$	4032.a	1.a	$1$	$1$	$1$	$32.196$	\(\Q\)	None	✓				168.2.a.b	$1$	$0$	\(0\)	\(0\)	\(2\)	\(1\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{5}+q^{7}+2q^{13}-6q^{17}-4q^{19}+\cdots\)
4032.2.a.bk	$4032$	$2$	4032.a	1.a	$1$	$1$	$1$	$32.196$	\(\Q\)	None	✓				56.2.a.a	$1$	$0$	\(0\)	\(0\)	\(2\)	\(1\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{5}+q^{7}+4q^{11}-2q^{13}+6q^{17}+\cdots\)
4032.2.a.bl	$4032$	$2$	4032.a	1.a	$1$	$1$	$1$	$32.196$	\(\Q\)	None	✓				504.2.a.d	$1$	$0$	\(0\)	\(0\)	\(2\)	\(1\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{5}+q^{7}+6q^{11}+6q^{13}+2q^{17}+\cdots\)
4032.2.a.bm	$4032$	$2$	4032.a	1.a	$1$	$1$	$1$	$32.196$	\(\Q\)	None	✓				84.2.a.a	$1$	$0$	\(0\)	\(0\)	\(4\)	\(-1\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+4q^{5}-q^{7}+2q^{11}+6q^{13}+4q^{17}+\cdots\)
4032.2.a.bn	$4032$	$2$	4032.a	1.a	$1$	$1$	$1$	$32.196$	\(\Q\)	None	✓				84.2.a.a	$1$	$0$	\(0\)	\(0\)	\(4\)	\(1\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+4q^{5}+q^{7}-2q^{11}+6q^{13}+4q^{17}+\cdots\)
4032.2.a.bo	$4032$	$2$	4032.a	1.a	$1$	$2$	$2$	$32.196$	\(\Q(\sqrt{5}) \)	None	✓				2016.2.a.p	$1$	$1$	\(0\)	\(0\)	\(-2\)	\(-2\)		$+$	$+$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q+(-1-\beta )q^{5}-q^{7}+(1+\beta )q^{11}-2\beta q^{13}+\cdots\)
4032.2.a.bp	$4032$	$2$	4032.a	1.a	$1$	$2$	$2$	$32.196$	\(\Q(\sqrt{5}) \)	None	✓				2016.2.a.p	$1$	$0$	\(0\)	\(0\)	\(-2\)	\(2\)		$+$	$+$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(-1-\beta )q^{5}+q^{7}+(-1-\beta )q^{11}+\cdots\)
4032.2.a.bq	$4032$	$2$	4032.a	1.a	$1$	$2$	$2$	$32.196$	\(\Q(\sqrt{3}) \)	None	✓		✓		63.2.a.b	$2$	$0$	\(0\)	\(0\)	\(0\)	\(-2\)		$-$	$+$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q+\beta q^{5}-q^{7}+\beta q^{11}-2q^{13}+\beta q^{17}+\cdots\)
4032.2.a.br	$4032$	$2$	4032.a	1.a	$1$	$2$	$2$	$32.196$	\(\Q(\sqrt{3}) \)	None	✓		✓		672.2.a.i	$1$	$1$	\(0\)	\(0\)	\(0\)	\(-2\)		$-$	$-$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q+\beta q^{5}-q^{7}+(2-\beta )q^{11}-2q^{13}+\cdots\)
4032.2.a.bs	$4032$	$2$	4032.a	1.a	$1$	$2$	$2$	$32.196$	\(\Q(\sqrt{3}) \)	None	✓		✓		672.2.a.i	$1$	$0$	\(0\)	\(0\)	\(0\)	\(2\)		$-$	$-$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+\beta q^{5}+q^{7}+(-2+\beta )q^{11}-2q^{13}+\cdots\)
4032.2.a.bt	$4032$	$2$	4032.a	1.a	$1$	$2$	$2$	$32.196$	\(\Q(\sqrt{3}) \)	None	✓				63.2.a.b	$2$	$0$	\(0\)	\(0\)	\(0\)	\(2\)		$+$	$+$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+\beta q^{5}+q^{7}-\beta q^{11}-2q^{13}+\beta q^{17}+\cdots\)
4032.2.a.bu	$4032$	$2$	4032.a	1.a	$1$	$2$	$2$	$32.196$	\(\Q(\sqrt{5}) \)	None	✓				2016.2.a.p	$1$	$1$	\(0\)	\(0\)	\(2\)	\(-2\)		$+$	$+$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{5}-q^{7}+(-1-\beta )q^{11}-2\beta q^{13}+\cdots\)
4032.2.a.bv	$4032$	$2$	4032.a	1.a	$1$	$2$	$2$	$32.196$	\(\Q(\sqrt{5}) \)	None	✓		✓		224.2.a.c	$1$	$0$	\(0\)	\(0\)	\(2\)	\(-2\)		$+$	$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{5}-q^{7}+(2+2\beta )q^{11}+(-3+\cdots)q^{13}+\cdots\)
4032.2.a.bw	$4032$	$2$	4032.a	1.a	$1$	$2$	$2$	$32.196$	\(\Q(\sqrt{5}) \)	None	✓		✓		224.2.a.c	$1$	$1$	\(0\)	\(0\)	\(2\)	\(2\)		$+$	$-$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{5}+q^{7}+(-2-2\beta )q^{11}+\cdots\)
4032.2.a.bx	$4032$	$2$	4032.a	1.a	$1$	$2$	$2$	$32.196$	\(\Q(\sqrt{5}) \)	None	✓				2016.2.a.p	$1$	$0$	\(0\)	\(0\)	\(2\)	\(2\)		$+$	$+$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{5}+q^{7}+(1+\beta )q^{11}-2\beta q^{13}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(4032))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(4032)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 18}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(21))\)\(^{\oplus 14}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(24))\)\(^{\oplus 16}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(32))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(36))\)\(^{\oplus 10}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(42))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(48))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(56))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(63))\)\(^{\oplus 7}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(64))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(72))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(84))\)\(^{\oplus 10}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(96))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(112))\)\(^{\oplus 9}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(126))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(144))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(168))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(192))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(224))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(252))\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(288))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(336))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(448))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(504))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(576))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(672))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1008))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1344))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(2016))\)\(^{\oplus 2}\)