Space of modular forms of level 4200, weight 2, and trivial character

• Label: 4200.2.a
• Level: $4200$
• Weight: $2$
• Character orbit: 4200.a
• Rep. character: $\chi_{4200}(1,\cdot)$
• Character field: $\Q$
• Dimension: $58$
• Newform subspaces: $43$
• Sturm bound: $1920$
• Trace bound: $19$

Defining parameters

Level:	\( N \)	\(=\)	\( 4200 = 2^{3} \cdot 3 \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4200.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 43 \)
Sturm bound:			\(1920\)
Trace bound:			\(19\)
Distinguishing \(T_p\):			\(11\), \(13\), \(17\), \(19\), \(23\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(4200))\).

	Total	New	Old
Modular forms	1008	58	950
Cusp forms	913	58	855
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.

\(2\)	\(3\)	\(5\)	\(7\)	Fricke	Dim.
\(+\)	\(+\)	\(+\)	\(+\)	\(+\)	\(2\)
\(+\)	\(+\)	\(+\)	\(-\)	\(-\)	\(4\)
\(+\)	\(+\)	\(-\)	\(+\)	\(-\)	\(5\)
\(+\)	\(+\)	\(-\)	\(-\)	\(+\)	\(3\)
\(+\)	\(-\)	\(+\)	\(+\)	\(-\)	\(4\)
\(+\)	\(-\)	\(+\)	\(-\)	\(+\)	\(2\)
\(+\)	\(-\)	\(-\)	\(+\)	\(+\)	\(3\)
\(+\)	\(-\)	\(-\)	\(-\)	\(-\)	\(5\)
\(-\)	\(+\)	\(+\)	\(+\)	\(-\)	\(4\)
\(-\)	\(+\)	\(+\)	\(-\)	\(+\)	\(3\)
\(-\)	\(+\)	\(-\)	\(+\)	\(+\)	\(4\)
\(-\)	\(+\)	\(-\)	\(-\)	\(-\)	\(4\)
\(-\)	\(-\)	\(+\)	\(+\)	\(+\)	\(3\)
\(-\)	\(-\)	\(+\)	\(-\)	\(-\)	\(4\)
\(-\)	\(-\)	\(-\)	\(+\)	\(-\)	\(4\)
\(-\)	\(-\)	\(-\)	\(-\)	\(+\)	\(4\)
Plus space				\(+\)	\(24\)
Minus space				\(-\)	\(34\)

Trace form

\( 58 q + 58 q^{9} + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(4200))\) 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	7
4200.2.a.a	$4200$	$2$	4200.a	1.a	$1$	$1$	$1$	$33.537$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(-1\)	\(0\)	\(-1\)		$+$	$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-q^{7}+q^{9}-3q^{11}-2q^{13}+\cdots\)
4200.2.a.b	$4200$	$2$	4200.a	1.a	$1$	$1$	$1$	$33.537$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(-1\)	\(0\)	\(-1\)		$-$	$+$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-q^{7}+q^{9}-q^{11}+4q^{13}+2q^{17}+\cdots\)
4200.2.a.c	$4200$	$2$	4200.a	1.a	$1$	$1$	$1$	$33.537$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(-1\)	\(0\)	\(-1\)		$+$	$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-q^{7}+q^{9}-2q^{13}-2q^{17}+\cdots\)
4200.2.a.d	$4200$	$2$	4200.a	1.a	$1$	$1$	$1$	$33.537$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(-1\)	\(0\)	\(1\)		$+$	$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+q^{7}+q^{9}-5q^{11}-4q^{13}+\cdots\)
4200.2.a.e	$4200$	$2$	4200.a	1.a	$1$	$1$	$1$	$33.537$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(-1\)	\(0\)	\(1\)		$+$	$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+q^{7}+q^{9}-4q^{11}+6q^{13}+\cdots\)
4200.2.a.f	$4200$	$2$	4200.a	1.a	$1$	$1$	$1$	$33.537$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(-1\)	\(0\)	\(1\)		$+$	$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+q^{7}+q^{9}-3q^{11}-2q^{13}+\cdots\)
4200.2.a.g	$4200$	$2$	4200.a	1.a	$1$	$1$	$1$	$33.537$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(-1\)	\(0\)	\(1\)		$-$	$+$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+q^{7}+q^{9}-2q^{11}-5q^{13}+\cdots\)
4200.2.a.h	$4200$	$2$	4200.a	1.a	$1$	$1$	$1$	$33.537$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(-1\)	\(0\)	\(1\)		$-$	$+$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+q^{7}+q^{9}-6q^{13}+2q^{17}+\cdots\)
4200.2.a.i	$4200$	$2$	4200.a	1.a	$1$	$1$	$1$	$33.537$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(-1\)	\(0\)	\(1\)		$-$	$+$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+q^{7}+q^{9}+2q^{13}-6q^{17}+\cdots\)
4200.2.a.j	$4200$	$2$	4200.a	1.a	$1$	$1$	$1$	$33.537$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(-1\)	\(0\)	\(1\)		$+$	$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+q^{7}+q^{9}+2q^{11}-2q^{13}+\cdots\)
4200.2.a.k	$4200$	$2$	4200.a	1.a	$1$	$1$	$1$	$33.537$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(-1\)	\(0\)	\(1\)		$+$	$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+q^{7}+q^{9}+2q^{11}-2q^{13}+\cdots\)
4200.2.a.l	$4200$	$2$	4200.a	1.a	$1$	$1$	$1$	$33.537$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(-1\)	\(0\)	\(1\)		$+$	$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+q^{7}+q^{9}+2q^{11}+3q^{13}+\cdots\)
4200.2.a.m	$4200$	$2$	4200.a	1.a	$1$	$1$	$1$	$33.537$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(-1\)	\(0\)	\(1\)		$-$	$+$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+q^{7}+q^{9}+3q^{11}+2q^{17}+\cdots\)
4200.2.a.n	$4200$	$2$	4200.a	1.a	$1$	$1$	$1$	$33.537$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(-1\)	\(0\)	\(1\)		$+$	$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+q^{7}+q^{9}+4q^{11}+2q^{13}+\cdots\)
4200.2.a.o	$4200$	$2$	4200.a	1.a	$1$	$1$	$1$	$33.537$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(-1\)	\(0\)	\(1\)		$-$	$+$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+q^{7}+q^{9}+5q^{11}+2q^{13}+\cdots\)
4200.2.a.p	$4200$	$2$	4200.a	1.a	$1$	$1$	$1$	$33.537$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(1\)	\(0\)	\(-1\)		$-$	$-$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-q^{7}+q^{9}-5q^{11}+4q^{13}+\cdots\)
4200.2.a.q	$4200$	$2$	4200.a	1.a	$1$	$1$	$1$	$33.537$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(1\)	\(0\)	\(-1\)		$+$	$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-q^{7}+q^{9}-4q^{11}-2q^{13}+\cdots\)
4200.2.a.r	$4200$	$2$	4200.a	1.a	$1$	$1$	$1$	$33.537$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(1\)	\(0\)	\(-1\)		$-$	$-$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-q^{7}+q^{9}-3q^{11}+2q^{13}+\cdots\)
4200.2.a.s	$4200$	$2$	4200.a	1.a	$1$	$1$	$1$	$33.537$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(1\)	\(0\)	\(-1\)		$+$	$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-q^{7}+q^{9}-2q^{11}+5q^{13}+\cdots\)
4200.2.a.t	$4200$	$2$	4200.a	1.a	$1$	$1$	$1$	$33.537$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(1\)	\(0\)	\(-1\)		$-$	$-$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-q^{7}+q^{9}-6q^{13}+2q^{17}+\cdots\)
4200.2.a.u	$4200$	$2$	4200.a	1.a	$1$	$1$	$1$	$33.537$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(1\)	\(0\)	\(-1\)		$-$	$-$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-q^{7}+q^{9}+2q^{13}-6q^{17}+\cdots\)
4200.2.a.v	$4200$	$2$	4200.a	1.a	$1$	$1$	$1$	$33.537$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(1\)	\(0\)	\(-1\)		$-$	$-$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-q^{7}+q^{9}+2q^{11}-3q^{13}+\cdots\)
4200.2.a.w	$4200$	$2$	4200.a	1.a	$1$	$1$	$1$	$33.537$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(1\)	\(0\)	\(-1\)		$-$	$-$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-q^{7}+q^{9}+2q^{11}+2q^{13}+\cdots\)
4200.2.a.x	$4200$	$2$	4200.a	1.a	$1$	$1$	$1$	$33.537$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(1\)	\(0\)	\(-1\)		$-$	$-$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-q^{7}+q^{9}+2q^{11}+2q^{13}+\cdots\)
4200.2.a.y	$4200$	$2$	4200.a	1.a	$1$	$1$	$1$	$33.537$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(1\)	\(0\)	\(-1\)		$+$	$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-q^{7}+q^{9}+3q^{11}-2q^{17}+\cdots\)
4200.2.a.z	$4200$	$2$	4200.a	1.a	$1$	$1$	$1$	$33.537$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(1\)	\(0\)	\(-1\)		$+$	$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-q^{7}+q^{9}+4q^{11}+2q^{13}+\cdots\)
4200.2.a.ba	$4200$	$2$	4200.a	1.a	$1$	$1$	$1$	$33.537$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(1\)	\(0\)	\(-1\)		$+$	$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-q^{7}+q^{9}+5q^{11}-2q^{13}+\cdots\)
4200.2.a.bb	$4200$	$2$	4200.a	1.a	$1$	$1$	$1$	$33.537$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(1\)	\(0\)	\(1\)		$+$	$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+q^{7}+q^{9}-4q^{11}+2q^{13}+\cdots\)
4200.2.a.bc	$4200$	$2$	4200.a	1.a	$1$	$1$	$1$	$33.537$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(1\)	\(0\)	\(1\)		$-$	$-$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+q^{7}+q^{9}-3q^{11}+2q^{13}+\cdots\)
4200.2.a.bd	$4200$	$2$	4200.a	1.a	$1$	$1$	$1$	$33.537$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(1\)	\(0\)	\(1\)		$+$	$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+q^{7}+q^{9}-q^{11}-4q^{13}-2q^{17}+\cdots\)
4200.2.a.be	$4200$	$2$	4200.a	1.a	$1$	$1$	$1$	$33.537$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(1\)	\(0\)	\(1\)		$-$	$-$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+q^{7}+q^{9}-2q^{13}-2q^{17}+\cdots\)
4200.2.a.bf	$4200$	$2$	4200.a	1.a	$1$	$1$	$1$	$33.537$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(1\)	\(0\)	\(1\)		$-$	$-$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+q^{7}+q^{9}+4q^{11}+2q^{13}+\cdots\)
4200.2.a.bg	$4200$	$2$	4200.a	1.a	$1$	$2$	$2$	$33.537$	\(\Q(\sqrt{2}) \)	None	✓	$1$	$0$	\(0\)	\(-2\)	\(0\)	\(-2\)		$-$	$+$	$+$	$+$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q-q^{3}-q^{7}+q^{9}+\beta q^{11}-2q^{13}+\cdots\)
4200.2.a.bh	$4200$	$2$	4200.a	1.a	$1$	$2$	$2$	$33.537$	\(\Q(\sqrt{73}) \)	None	✓	$1$	$0$	\(0\)	\(-2\)	\(0\)	\(-2\)		$+$	$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-q^{7}+q^{9}+\beta q^{11}+(-1-\beta )q^{13}+\cdots\)
4200.2.a.bi	$4200$	$2$	4200.a	1.a	$1$	$2$	$2$	$33.537$	\(\Q(\sqrt{17}) \)	None	✓	$1$	$0$	\(0\)	\(-2\)	\(0\)	\(-2\)		$-$	$+$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-q^{7}+q^{9}+(1+\beta )q^{11}+(-2+\cdots)q^{13}+\cdots\)
4200.2.a.bj	$4200$	$2$	4200.a	1.a	$1$	$2$	$2$	$33.537$	\(\Q(\sqrt{5}) \)	None	✓	$1$	$0$	\(0\)	\(-2\)	\(0\)	\(2\)		$-$	$+$	$-$	$-$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q-q^{3}+q^{7}+q^{9}-2q^{11}-\beta q^{13}+\cdots\)
4200.2.a.bk	$4200$	$2$	4200.a	1.a	$1$	$2$	$2$	$33.537$	\(\Q(\sqrt{5}) \)	None	✓	$1$	$1$	\(0\)	\(2\)	\(0\)	\(-2\)		$+$	$-$	$-$	$+$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q+q^{3}-q^{7}+q^{9}-2q^{11}+\beta q^{13}+\cdots\)
4200.2.a.bl	$4200$	$2$	4200.a	1.a	$1$	$2$	$2$	$33.537$	\(\Q(\sqrt{73}) \)	None	✓	$1$	$0$	\(0\)	\(2\)	\(0\)	\(2\)		$-$	$-$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+q^{7}+q^{9}+\beta q^{11}+(1+\beta )q^{13}+\cdots\)
4200.2.a.bm	$4200$	$2$	4200.a	1.a	$1$	$2$	$2$	$33.537$	\(\Q(\sqrt{17}) \)	None	✓	$1$	$0$	\(0\)	\(2\)	\(0\)	\(2\)		$+$	$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+q^{7}+q^{9}+(1+\beta )q^{11}+(2-3\beta )q^{13}+\cdots\)
4200.2.a.bn	$4200$	$2$	4200.a	1.a	$1$	$3$	$3$	$33.537$	3.3.148.1	None	✓	$1$	$0$	\(0\)	\(-3\)	\(0\)	\(-3\)		$+$	$+$	$-$	$+$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q-q^{3}-q^{7}+q^{9}+(-1-\beta _{1}+\beta _{2})q^{11}+\cdots\)
4200.2.a.bo	$4200$	$2$	4200.a	1.a	$1$	$3$	$3$	$33.537$	3.3.148.1	None	✓	$1$	$1$	\(0\)	\(-3\)	\(0\)	\(-3\)		$-$	$+$	$-$	$+$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q-q^{3}-q^{7}+q^{9}+(1+\beta _{1}+\beta _{2})q^{11}+\cdots\)
4200.2.a.bp	$4200$	$2$	4200.a	1.a	$1$	$3$	$3$	$33.537$	3.3.148.1	None	✓	$1$	$1$	\(0\)	\(3\)	\(0\)	\(3\)		$-$	$-$	$-$	$-$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q+q^{3}+q^{7}+q^{9}+(-1-\beta _{1}+\beta _{2})q^{11}+\cdots\)
4200.2.a.bq	$4200$	$2$	4200.a	1.a	$1$	$3$	$3$	$33.537$	3.3.148.1	None	✓	$1$	$0$	\(0\)	\(3\)	\(0\)	\(3\)		$+$	$-$	$-$	$-$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q+q^{3}+q^{7}+q^{9}+(1+\beta _{1}+\beta _{2})q^{11}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(4200))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(4200)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 18}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 16}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 16}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(21))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(24))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(30))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(35))\)\(^{\oplus 16}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(40))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(42))\)\(^{\oplus 9}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(50))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(56))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(70))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(75))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(84))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(100))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(105))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(120))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(140))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(150))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(168))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(175))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(200))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(210))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(280))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(300))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(350))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(420))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(525))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(600))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(700))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(840))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1050))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1400))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(2100))\)\(^{\oplus 2}\)