Space of modular forms of level 3200, weight 2, and trivial character

• Label: 3200.2.a
• Level: $3200$
• Weight: $2$
• Character orbit: 3200.a
• Rep. character: $\chi_{3200}(1,\cdot)$
• Character field: $\Q$
• Dimension: $76$
• Newform subspaces: $48$
• Sturm bound: $960$
• Trace bound: $13$

Defining parameters

Level:	\( N \)	\(=\)	\( 3200 = 2^{7} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3200.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 48 \)
Sturm bound:			\(960\)
Trace bound:			\(13\)
Distinguishing \(T_p\):			\(3\), \(7\), \(11\), \(13\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(3200))\).

	Total	New	Old
Modular forms	528	76	452
Cusp forms	433	76	357
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.

\(2\)	\(5\)	Fricke		Total				Cusp				Eisenstein
				All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)		\(126\)	\(15\)	\(111\)		\(103\)	\(15\)	\(88\)		\(23\)	\(0\)	\(23\)
\(+\)	\(-\)	\(-\)		\(138\)	\(22\)	\(116\)		\(114\)	\(22\)	\(92\)		\(24\)	\(0\)	\(24\)
\(-\)	\(+\)	\(-\)		\(138\)	\(21\)	\(117\)		\(114\)	\(21\)	\(93\)		\(24\)	\(0\)	\(24\)
\(-\)	\(-\)	\(+\)		\(126\)	\(18\)	\(108\)		\(102\)	\(18\)	\(84\)		\(24\)	\(0\)	\(24\)
Plus space		\(+\)		\(252\)	\(33\)	\(219\)		\(205\)	\(33\)	\(172\)		\(47\)	\(0\)	\(47\)
Minus space		\(-\)		\(276\)	\(43\)	\(233\)		\(228\)	\(43\)	\(185\)		\(48\)	\(0\)	\(48\)

Trace form

\( 76 q + 76 q^{9} + 8 q^{17} - 16 q^{33} - 8 q^{41} + 108 q^{49} + 16 q^{57} + 8 q^{73} + 92 q^{81} + 8 q^{89} + 72 q^{97}+O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(3200))\) 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
3200.2.a.a	$3200$	$2$	3200.a	1.a	$1$	$1$	$1$	$25.552$	\(\Q\)	None	✓	✓			3200.2.a.a	$1$	$2$	\(0\)	\(-3\)	\(0\)	\(-4\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-3q^{3}-4q^{7}+6q^{9}-3q^{11}-2q^{13}+\cdots\)
3200.2.a.b	$3200$	$2$	3200.a	1.a	$1$	$1$	$1$	$25.552$	\(\Q\)	None	✓	✓			3200.2.a.a	$1$	$0$	\(0\)	\(-3\)	\(0\)	\(-4\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-3q^{3}-4q^{7}+6q^{9}+3q^{11}+2q^{13}+\cdots\)
3200.2.a.c	$3200$	$2$	3200.a	1.a	$1$	$1$	$1$	$25.552$	\(\Q\)	None	✓	✓			3200.2.a.a	$1$	$1$	\(0\)	\(-3\)	\(0\)	\(4\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-3q^{3}+4q^{7}+6q^{9}-3q^{11}+2q^{13}+\cdots\)
3200.2.a.d	$3200$	$2$	3200.a	1.a	$1$	$1$	$1$	$25.552$	\(\Q\)	None	✓	✓			3200.2.a.a	$1$	$0$	\(0\)	\(-3\)	\(0\)	\(4\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-3q^{3}+4q^{7}+6q^{9}+3q^{11}-2q^{13}+\cdots\)
3200.2.a.e	$3200$	$2$	3200.a	1.a	$1$	$1$	$1$	$25.552$	\(\Q\)	None	✓				128.2.a.a	$1$	$1$	\(0\)	\(-2\)	\(0\)	\(-4\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{3}-4q^{7}+q^{9}-2q^{11}+2q^{13}+\cdots\)
3200.2.a.f	$3200$	$2$	3200.a	1.a	$1$	$1$	$1$	$25.552$	\(\Q\)	None	✓				640.2.a.a	$1$	$1$	\(0\)	\(-2\)	\(0\)	\(0\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{3}+q^{9}+2q^{11}-2q^{13}-6q^{17}+\cdots\)
3200.2.a.g	$3200$	$2$	3200.a	1.a	$1$	$1$	$1$	$25.552$	\(\Q\)	None	✓				640.2.a.a	$1$	$0$	\(0\)	\(-2\)	\(0\)	\(0\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{3}+q^{9}+2q^{11}+2q^{13}-6q^{17}+\cdots\)
3200.2.a.h	$3200$	$2$	3200.a	1.a	$1$	$1$	$1$	$25.552$	\(\Q\)	None	✓				128.2.a.a	$1$	$1$	\(0\)	\(-2\)	\(0\)	\(4\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{3}+4q^{7}+q^{9}-2q^{11}-2q^{13}+\cdots\)
3200.2.a.i	$3200$	$2$	3200.a	1.a	$1$	$1$	$1$	$25.552$	\(\Q\)	None	✓	✓			3200.2.a.i	$1$	$0$	\(0\)	\(-1\)	\(0\)	\(0\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-2q^{9}-q^{11}-2q^{13}-3q^{17}+\cdots\)
3200.2.a.j	$3200$	$2$	3200.a	1.a	$1$	$1$	$1$	$25.552$	\(\Q\)	None	✓	✓			3200.2.a.i	$1$	$1$	\(0\)	\(-1\)	\(0\)	\(0\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-2q^{9}-q^{11}+2q^{13}-3q^{17}+\cdots\)
3200.2.a.k	$3200$	$2$	3200.a	1.a	$1$	$1$	$1$	$25.552$	\(\Q\)	None	✓	✓			3200.2.a.i	$1$	$1$	\(0\)	\(-1\)	\(0\)	\(0\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-2q^{9}+q^{11}-2q^{13}+3q^{17}+\cdots\)
3200.2.a.l	$3200$	$2$	3200.a	1.a	$1$	$1$	$1$	$25.552$	\(\Q\)	None	✓	✓			3200.2.a.i	$1$	$1$	\(0\)	\(-1\)	\(0\)	\(0\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-2q^{9}+q^{11}+2q^{13}+3q^{17}+\cdots\)
3200.2.a.m	$3200$	$2$	3200.a	1.a	$1$	$1$	$1$	$25.552$	\(\Q\)	None	✓				640.2.a.c	$1$	$0$	\(0\)	\(0\)	\(0\)	\(-2\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{7}-3q^{9}-6q^{11}+2q^{13}+6q^{17}+\cdots\)
3200.2.a.n	$3200$	$2$	3200.a	1.a	$1$	$1$	$1$	$25.552$	\(\Q\)	None	✓				640.2.a.c	$1$	$0$	\(0\)	\(0\)	\(0\)	\(-2\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{7}-3q^{9}+6q^{11}-2q^{13}+6q^{17}+\cdots\)
3200.2.a.o	$3200$	$2$	3200.a	1.a	$1$	$1$	$1$	$25.552$	\(\Q\)	None	✓				640.2.a.c	$1$	$1$	\(0\)	\(0\)	\(0\)	\(2\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{7}-3q^{9}-6q^{11}-2q^{13}+6q^{17}+\cdots\)
3200.2.a.p	$3200$	$2$	3200.a	1.a	$1$	$1$	$1$	$25.552$	\(\Q\)	None	✓				640.2.a.c	$1$	$0$	\(0\)	\(0\)	\(0\)	\(2\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{7}-3q^{9}+6q^{11}+2q^{13}+6q^{17}+\cdots\)
3200.2.a.q	$3200$	$2$	3200.a	1.a	$1$	$1$	$1$	$25.552$	\(\Q\)	None	✓	✓			3200.2.a.i	$1$	$1$	\(0\)	\(1\)	\(0\)	\(0\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-2q^{9}-q^{11}-2q^{13}+3q^{17}+\cdots\)
3200.2.a.r	$3200$	$2$	3200.a	1.a	$1$	$1$	$1$	$25.552$	\(\Q\)	None	✓	✓			3200.2.a.i	$1$	$0$	\(0\)	\(1\)	\(0\)	\(0\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-2q^{9}-q^{11}+2q^{13}+3q^{17}+\cdots\)
3200.2.a.s	$3200$	$2$	3200.a	1.a	$1$	$1$	$1$	$25.552$	\(\Q\)	None	✓	✓			3200.2.a.i	$1$	$1$	\(0\)	\(1\)	\(0\)	\(0\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-2q^{9}+q^{11}-2q^{13}-3q^{17}+\cdots\)
3200.2.a.t	$3200$	$2$	3200.a	1.a	$1$	$1$	$1$	$25.552$	\(\Q\)	None	✓	✓			3200.2.a.i	$1$	$1$	\(0\)	\(1\)	\(0\)	\(0\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-2q^{9}+q^{11}+2q^{13}-3q^{17}+\cdots\)
3200.2.a.u	$3200$	$2$	3200.a	1.a	$1$	$1$	$1$	$25.552$	\(\Q\)	None	✓				128.2.a.a	$1$	$1$	\(0\)	\(2\)	\(0\)	\(-4\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{3}-4q^{7}+q^{9}+2q^{11}-2q^{13}+\cdots\)
3200.2.a.v	$3200$	$2$	3200.a	1.a	$1$	$1$	$1$	$25.552$	\(\Q\)	None	✓				640.2.a.a	$1$	$1$	\(0\)	\(2\)	\(0\)	\(0\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{3}+q^{9}-2q^{11}-2q^{13}-6q^{17}+\cdots\)
3200.2.a.w	$3200$	$2$	3200.a	1.a	$1$	$1$	$1$	$25.552$	\(\Q\)	None	✓				640.2.a.a	$1$	$1$	\(0\)	\(2\)	\(0\)	\(0\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{3}+q^{9}-2q^{11}+2q^{13}-6q^{17}+\cdots\)
3200.2.a.x	$3200$	$2$	3200.a	1.a	$1$	$1$	$1$	$25.552$	\(\Q\)	None	✓				128.2.a.a	$1$	$0$	\(0\)	\(2\)	\(0\)	\(4\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{3}+4q^{7}+q^{9}+2q^{11}+2q^{13}+\cdots\)
3200.2.a.y	$3200$	$2$	3200.a	1.a	$1$	$1$	$1$	$25.552$	\(\Q\)	None	✓	✓			3200.2.a.a	$1$	$1$	\(0\)	\(3\)	\(0\)	\(-4\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+3q^{3}-4q^{7}+6q^{9}-3q^{11}-2q^{13}+\cdots\)
3200.2.a.z	$3200$	$2$	3200.a	1.a	$1$	$1$	$1$	$25.552$	\(\Q\)	None	✓	✓			3200.2.a.a	$1$	$0$	\(0\)	\(3\)	\(0\)	\(-4\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3q^{3}-4q^{7}+6q^{9}+3q^{11}+2q^{13}+\cdots\)
3200.2.a.ba	$3200$	$2$	3200.a	1.a	$1$	$1$	$1$	$25.552$	\(\Q\)	None	✓	✓			3200.2.a.a	$1$	$0$	\(0\)	\(3\)	\(0\)	\(4\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3q^{3}+4q^{7}+6q^{9}-3q^{11}+2q^{13}+\cdots\)
3200.2.a.bb	$3200$	$2$	3200.a	1.a	$1$	$1$	$1$	$25.552$	\(\Q\)	None	✓	✓			3200.2.a.a	$1$	$0$	\(0\)	\(3\)	\(0\)	\(4\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3q^{3}+4q^{7}+6q^{9}+3q^{11}-2q^{13}+\cdots\)
3200.2.a.bc	$3200$	$2$	3200.a	1.a	$1$	$2$	$2$	$25.552$	\(\Q(\sqrt{2}) \)	None	✓	✓			3200.2.a.bc	$1$	$0$	\(0\)	\(-2\)	\(0\)	\(-4\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta )q^{3}+(-2-2\beta )q^{7}-2\beta q^{9}+\cdots\)
3200.2.a.bd	$3200$	$2$	3200.a	1.a	$1$	$2$	$2$	$25.552$	\(\Q(\sqrt{2}) \)	None	✓	✓			3200.2.a.bc	$1$	$0$	\(0\)	\(-2\)	\(0\)	\(-4\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta )q^{3}+(-2-2\beta )q^{7}-2\beta q^{9}+\cdots\)
3200.2.a.be	$3200$	$2$	3200.a	1.a	$1$	$2$	$2$	$25.552$	\(\Q(\sqrt{5}) \)	None	✓				640.2.a.i	$1$	$1$	\(0\)	\(-2\)	\(0\)	\(-2\)		$+$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q+(-1-\beta )q^{3}+(-1+\beta )q^{7}+(3+2\beta )q^{9}+\cdots\)
3200.2.a.bf	$3200$	$2$	3200.a	1.a	$1$	$2$	$2$	$25.552$	\(\Q(\sqrt{5}) \)	None	✓				640.2.a.i	$1$	$0$	\(0\)	\(-2\)	\(0\)	\(2\)		$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(-1-\beta )q^{3}+(1-\beta )q^{7}+(3+2\beta )q^{9}+\cdots\)
3200.2.a.bg	$3200$	$2$	3200.a	1.a	$1$	$2$	$2$	$25.552$	\(\Q(\sqrt{2}) \)	None	✓	✓			3200.2.a.bc	$1$	$1$	\(0\)	\(-2\)	\(0\)	\(4\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta )q^{3}+(2+2\beta )q^{7}-2\beta q^{9}+\cdots\)
3200.2.a.bh	$3200$	$2$	3200.a	1.a	$1$	$2$	$2$	$25.552$	\(\Q(\sqrt{2}) \)	None	✓	✓			3200.2.a.bc	$1$	$0$	\(0\)	\(-2\)	\(0\)	\(4\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta )q^{3}+(2+2\beta )q^{7}-2\beta q^{9}+\cdots\)
3200.2.a.bi	$3200$	$2$	3200.a	1.a	$1$	$2$	$2$	$25.552$	\(\Q(\sqrt{2}) \)	None	✓	✓			3200.2.a.bc	$1$	$1$	\(0\)	\(2\)	\(0\)	\(-4\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{3}+(-2+2\beta )q^{7}+2\beta q^{9}+\cdots\)
3200.2.a.bj	$3200$	$2$	3200.a	1.a	$1$	$2$	$2$	$25.552$	\(\Q(\sqrt{2}) \)	None	✓	✓			3200.2.a.bc	$1$	$0$	\(0\)	\(2\)	\(0\)	\(-4\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{3}+(-2+2\beta )q^{7}+2\beta q^{9}+\cdots\)
3200.2.a.bk	$3200$	$2$	3200.a	1.a	$1$	$2$	$2$	$25.552$	\(\Q(\sqrt{5}) \)	None	✓				640.2.a.i	$1$	$0$	\(0\)	\(2\)	\(0\)	\(-2\)		$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{3}+(-1+\beta )q^{7}+(3+2\beta )q^{9}+\cdots\)
3200.2.a.bl	$3200$	$2$	3200.a	1.a	$1$	$2$	$2$	$25.552$	\(\Q(\sqrt{5}) \)	None	✓				640.2.a.i	$1$	$0$	\(0\)	\(2\)	\(0\)	\(2\)		$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{3}+(1-\beta )q^{7}+(3+2\beta )q^{9}+\cdots\)
3200.2.a.bm	$3200$	$2$	3200.a	1.a	$1$	$2$	$2$	$25.552$	\(\Q(\sqrt{2}) \)	None	✓	✓			3200.2.a.bc	$1$	$0$	\(0\)	\(2\)	\(0\)	\(4\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{3}+(2-2\beta )q^{7}+2\beta q^{9}+(-1+\cdots)q^{11}+\cdots\)
3200.2.a.bn	$3200$	$2$	3200.a	1.a	$1$	$2$	$2$	$25.552$	\(\Q(\sqrt{2}) \)	None	✓	✓			3200.2.a.bc	$1$	$0$	\(0\)	\(2\)	\(0\)	\(4\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{3}+(2-2\beta )q^{7}+2\beta q^{9}+(1+\cdots)q^{11}+\cdots\)
3200.2.a.bo	$3200$	$2$	3200.a	1.a	$1$	$3$	$3$	$25.552$	3.3.148.1	None	✓				640.2.c.a	$1$	$1$	\(0\)	\(-2\)	\(0\)	\(-4\)		$-$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{3}+(-1-\beta _{2})q^{7}+(1+\cdots)q^{9}+\cdots\)
3200.2.a.bp	$3200$	$2$	3200.a	1.a	$1$	$3$	$3$	$25.552$	3.3.148.1	None	✓				640.2.c.a	$1$	$1$	\(0\)	\(-2\)	\(0\)	\(-4\)		$-$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{3}+(-1-\beta _{2})q^{7}+(1+\cdots)q^{9}+\cdots\)
3200.2.a.bq	$3200$	$2$	3200.a	1.a	$1$	$3$	$3$	$25.552$	3.3.148.1	None	✓				640.2.c.a	$1$	$1$	\(0\)	\(-2\)	\(0\)	\(4\)		$-$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{3}+(1+\beta _{2})q^{7}+(1-\beta _{1}+\cdots)q^{9}+\cdots\)
3200.2.a.br	$3200$	$2$	3200.a	1.a	$1$	$3$	$3$	$25.552$	3.3.148.1	None	✓				640.2.c.a	$1$	$0$	\(0\)	\(-2\)	\(0\)	\(4\)		$+$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{3}+(1+\beta _{2})q^{7}+(1-\beta _{1}+\cdots)q^{9}+\cdots\)
3200.2.a.bs	$3200$	$2$	3200.a	1.a	$1$	$3$	$3$	$25.552$	3.3.148.1	None	✓				640.2.c.a	$1$	$0$	\(0\)	\(2\)	\(0\)	\(-4\)		$+$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{3}+(-1-\beta _{2})q^{7}+(1-\beta _{1}+\cdots)q^{9}+\cdots\)
3200.2.a.bt	$3200$	$2$	3200.a	1.a	$1$	$3$	$3$	$25.552$	3.3.148.1	None	✓				640.2.c.a	$1$	$1$	\(0\)	\(2\)	\(0\)	\(-4\)		$-$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{3}+(-1-\beta _{2})q^{7}+(1-\beta _{1}+\cdots)q^{9}+\cdots\)
3200.2.a.bu	$3200$	$2$	3200.a	1.a	$1$	$3$	$3$	$25.552$	3.3.148.1	None	✓				640.2.c.a	$1$	$0$	\(0\)	\(2\)	\(0\)	\(4\)		$+$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{3}+(1+\beta _{2})q^{7}+(1-\beta _{1}+\cdots)q^{9}+\cdots\)
3200.2.a.bv	$3200$	$2$	3200.a	1.a	$1$	$3$	$3$	$25.552$	3.3.148.1	None	✓				640.2.c.a	$1$	$0$	\(0\)	\(2\)	\(0\)	\(4\)		$+$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{3}+(1+\beta _{2})q^{7}+(1-\beta _{1}+\cdots)q^{9}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(3200))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(3200)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(32))\)\(^{\oplus 9}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(40))\)\(^{\oplus 10}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(50))\)\(^{\oplus 7}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(64))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(80))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(100))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(128))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(160))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(200))\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(320))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(400))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(640))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(800))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1600))\)\(^{\oplus 2}\)