Space of modular forms of level 9000, weight 2, and trivial character

• Label: 9000.2.a
• Level: $9000$
• Weight: $2$
• Character orbit: 9000.a
• Rep. character: $\chi_{9000}(1,\cdot)$
• Character field: $\Q$
• Dimension: $120$
• Newform subspaces: $34$
• Sturm bound: $3600$
• Trace bound: $17$

Defining parameters

Level:	\( N \)	\(=\)	\( 9000 = 2^{3} \cdot 3^{2} \cdot 5^{3} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9000.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 34 \)
Sturm bound:			\(3600\)
Trace bound:			\(17\)
Distinguishing \(T_p\):			\(7\), \(11\), \(13\), \(17\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(9000))\).

	Total	New	Old
Modular forms	1880	120	1760
Cusp forms	1721	120	1601
Eisenstein series	159	0	159

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
\(+\)	\(+\)	\(+\)	\(+\)		\(230\)	\(10\)	\(220\)		\(211\)	\(10\)	\(201\)		\(19\)	\(0\)	\(19\)
\(+\)	\(+\)	\(-\)	\(-\)		\(240\)	\(14\)	\(226\)		\(220\)	\(14\)	\(206\)		\(20\)	\(0\)	\(20\)
\(+\)	\(-\)	\(+\)	\(-\)		\(240\)	\(20\)	\(220\)		\(220\)	\(20\)	\(200\)		\(20\)	\(0\)	\(20\)
\(+\)	\(-\)	\(-\)	\(+\)		\(230\)	\(16\)	\(214\)		\(210\)	\(16\)	\(194\)		\(20\)	\(0\)	\(20\)
\(-\)	\(+\)	\(+\)	\(-\)		\(240\)	\(14\)	\(226\)		\(220\)	\(14\)	\(206\)		\(20\)	\(0\)	\(20\)
\(-\)	\(+\)	\(-\)	\(+\)		\(230\)	\(10\)	\(220\)		\(210\)	\(10\)	\(200\)		\(20\)	\(0\)	\(20\)
\(-\)	\(-\)	\(+\)	\(+\)		\(230\)	\(18\)	\(212\)		\(210\)	\(18\)	\(192\)		\(20\)	\(0\)	\(20\)
\(-\)	\(-\)	\(-\)	\(-\)		\(240\)	\(18\)	\(222\)		\(220\)	\(18\)	\(202\)		\(20\)	\(0\)	\(20\)
Plus space			\(+\)		\(920\)	\(54\)	\(866\)		\(841\)	\(54\)	\(787\)		\(79\)	\(0\)	\(79\)
Minus space			\(-\)		\(960\)	\(66\)	\(894\)		\(880\)	\(66\)	\(814\)		\(80\)	\(0\)	\(80\)

Trace form

\( 120 q - 2 q^{11} - 2 q^{19} - 6 q^{29} + 4 q^{31} + 8 q^{41} + 120 q^{49} - 50 q^{59} - 6 q^{61} - 16 q^{71} + 96 q^{79} - 4 q^{89} + 92 q^{91}+O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(9000))\) 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
9000.2.a.a	$9000$	$2$	9000.a	1.a	$1$	$2$	$2$	$71.865$	\(\Q(\sqrt{5}) \)	None	✓				3000.2.a.d	$1$	$1$	\(0\)	\(0\)	\(0\)	\(-5\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-2-\beta )q^{7}+q^{11}+(1+2\beta )q^{13}+\cdots\)
9000.2.a.b	$9000$	$2$	9000.a	1.a	$1$	$2$	$2$	$71.865$	\(\Q(\sqrt{5}) \)	None	✓				3000.2.a.c	$1$	$1$	\(0\)	\(0\)	\(0\)	\(-4\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{7}-2q^{11}+(-2-2\beta )q^{13}+(2+\cdots)q^{17}+\cdots\)
9000.2.a.c	$9000$	$2$	9000.a	1.a	$1$	$2$	$2$	$71.865$	\(\Q(\sqrt{5}) \)	None	✓				1000.2.a.b	$1$	$0$	\(0\)	\(0\)	\(0\)	\(-4\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-1-2\beta )q^{7}+(3+2\beta )q^{11}+(4-\beta )q^{13}+\cdots\)
9000.2.a.d	$9000$	$2$	9000.a	1.a	$1$	$2$	$2$	$71.865$	\(\Q(\sqrt{5}) \)	None	✓	✓			9000.2.a.d	$1$	$1$	\(0\)	\(0\)	\(0\)	\(-2\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2\beta q^{7}+(-2+2\beta )q^{11}+2\beta q^{13}+\cdots\)
9000.2.a.e	$9000$	$2$	9000.a	1.a	$1$	$2$	$2$	$71.865$	\(\Q(\sqrt{5}) \)	None	✓	✓			9000.2.a.d	$1$	$1$	\(0\)	\(0\)	\(0\)	\(-2\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2\beta q^{7}+(2-2\beta )q^{11}+2\beta q^{13}+(1+\cdots)q^{17}+\cdots\)
9000.2.a.f	$9000$	$2$	9000.a	1.a	$1$	$2$	$2$	$71.865$	\(\Q(\sqrt{5}) \)	None	✓				1000.2.a.a	$1$	$0$	\(0\)	\(0\)	\(0\)	\(-1\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta q^{7}+(-2+2\beta )q^{11}+(-2+2\beta )q^{13}+\cdots\)
9000.2.a.g	$9000$	$2$	9000.a	1.a	$1$	$2$	$2$	$71.865$	\(\Q(\sqrt{5}) \)	None	✓				3000.2.a.a	$1$	$0$	\(0\)	\(0\)	\(0\)	\(-1\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta q^{7}+3q^{11}+3q^{13}-3\beta q^{17}+\cdots\)
9000.2.a.h	$9000$	$2$	9000.a	1.a	$1$	$2$	$2$	$71.865$	\(\Q(\sqrt{5}) \)	None	✓				3000.2.a.b	$1$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2\beta q^{11}+(-4+2\beta )q^{13}+(6-3\beta )q^{17}+\cdots\)
9000.2.a.i	$9000$	$2$	9000.a	1.a	$1$	$2$	$2$	$71.865$	\(\Q(\sqrt{5}) \)	None	✓				3000.2.a.b	$1$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2\beta q^{11}+(4-2\beta )q^{13}+(-6+3\beta )q^{17}+\cdots\)
9000.2.a.j	$9000$	$2$	9000.a	1.a	$1$	$2$	$2$	$71.865$	\(\Q(\sqrt{5}) \)	None	✓				1000.2.a.a	$1$	$1$	\(0\)	\(0\)	\(0\)	\(1\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{7}+(-2+2\beta )q^{11}+(2-2\beta )q^{13}+\cdots\)
9000.2.a.k	$9000$	$2$	9000.a	1.a	$1$	$2$	$2$	$71.865$	\(\Q(\sqrt{5}) \)	None	✓				3000.2.a.a	$1$	$1$	\(0\)	\(0\)	\(0\)	\(1\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{7}+3q^{11}-3q^{13}+3\beta q^{17}+\cdots\)
9000.2.a.l	$9000$	$2$	9000.a	1.a	$1$	$2$	$2$	$71.865$	\(\Q(\sqrt{5}) \)	None	✓	✓			9000.2.a.d	$1$	$1$	\(0\)	\(0\)	\(0\)	\(2\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2\beta q^{7}+(-2+2\beta )q^{11}-2\beta q^{13}+\cdots\)
9000.2.a.m	$9000$	$2$	9000.a	1.a	$1$	$2$	$2$	$71.865$	\(\Q(\sqrt{5}) \)	None	✓	✓			9000.2.a.d	$1$	$1$	\(0\)	\(0\)	\(0\)	\(2\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2\beta q^{7}+(2-2\beta )q^{11}-2\beta q^{13}+(-1+\cdots)q^{17}+\cdots\)
9000.2.a.n	$9000$	$2$	9000.a	1.a	$1$	$2$	$2$	$71.865$	\(\Q(\sqrt{5}) \)	None	✓				3000.2.a.c	$1$	$1$	\(0\)	\(0\)	\(0\)	\(4\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{7}-2q^{11}+(2+2\beta )q^{13}+(-2+\cdots)q^{17}+\cdots\)
9000.2.a.o	$9000$	$2$	9000.a	1.a	$1$	$2$	$2$	$71.865$	\(\Q(\sqrt{5}) \)	None	✓				1000.2.a.b	$1$	$0$	\(0\)	\(0\)	\(0\)	\(4\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1+2\beta )q^{7}+(3+2\beta )q^{11}+(-4+\beta )q^{13}+\cdots\)
9000.2.a.p	$9000$	$2$	9000.a	1.a	$1$	$2$	$2$	$71.865$	\(\Q(\sqrt{5}) \)	None	✓				3000.2.a.d	$1$	$0$	\(0\)	\(0\)	\(0\)	\(5\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(3-\beta )q^{7}+q^{11}+(-3+2\beta )q^{13}+\cdots\)
9000.2.a.q	$9000$	$2$	9000.a	1.a	$1$	$4$	$4$	$71.865$	4.4.10025.1	None	✓				1000.2.a.f	$1$	$1$	\(0\)	\(0\)	\(0\)	\(-9\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-2+\beta _{2}+\beta _{3})q^{7}+(-1-\beta _{1})q^{11}+\cdots\)
9000.2.a.r	$9000$	$2$	9000.a	1.a	$1$	$4$	$4$	$71.865$	4.4.4400.1	None	✓				1000.2.a.e	$1$	$0$	\(0\)	\(0\)	\(0\)	\(-6\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-2-\beta _{2}+\beta _{3})q^{7}+2\beta _{3}q^{11}+(2\beta _{1}+\cdots)q^{13}+\cdots\)
9000.2.a.s	$9000$	$2$	9000.a	1.a	$1$	$4$	$4$	$71.865$	4.4.13025.1	None	✓				3000.2.a.i	$1$	$0$	\(0\)	\(0\)	\(0\)	\(-3\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{7}+(-1+\beta _{1}-\beta _{2}-\beta _{3})q^{11}+\cdots\)
9000.2.a.t	$9000$	$2$	9000.a	1.a	$1$	$4$	$4$	$71.865$	4.4.47025.1	None	✓				3000.2.a.j	$1$	$1$	\(0\)	\(0\)	\(0\)	\(-3\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1-\beta _{1}+\beta _{2})q^{7}+(\beta _{1}-2\beta _{2})q^{11}+\cdots\)
9000.2.a.u	$9000$	$2$	9000.a	1.a	$1$	$4$	$4$	$71.865$	4.4.13025.1	None	✓				3000.2.a.k	$1$	$1$	\(0\)	\(0\)	\(0\)	\(-1\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1}+\beta _{3})q^{7}+(-\beta _{1}+\beta _{2}+\cdots)q^{11}+\cdots\)
9000.2.a.v	$9000$	$2$	9000.a	1.a	$1$	$4$	$4$	$71.865$	4.4.36025.1	None	✓				3000.2.a.l	$1$	$0$	\(0\)	\(0\)	\(0\)	\(-1\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{7}+(1+2\beta _{2}+\beta _{3})q^{11}+(1+2\beta _{2}+\cdots)q^{13}+\cdots\)
9000.2.a.w	$9000$	$2$	9000.a	1.a	$1$	$4$	$4$	$71.865$	4.4.36025.1	None	✓				3000.2.a.l	$1$	$0$	\(0\)	\(0\)	\(0\)	\(1\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{7}+(1+2\beta _{2}+\beta _{3})q^{11}+(-1+\cdots)q^{13}+\cdots\)
9000.2.a.x	$9000$	$2$	9000.a	1.a	$1$	$4$	$4$	$71.865$	4.4.13025.1	None	✓				3000.2.a.k	$1$	$0$	\(0\)	\(0\)	\(0\)	\(1\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1}-\beta _{2}+\beta _{3})q^{7}+(\beta _{1}-\beta _{3})q^{11}+\cdots\)
9000.2.a.y	$9000$	$2$	9000.a	1.a	$1$	$4$	$4$	$71.865$	4.4.47025.1	None	✓				3000.2.a.j	$1$	$0$	\(0\)	\(0\)	\(0\)	\(3\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta _{1}-\beta _{2})q^{7}+(\beta _{1}-2\beta _{2})q^{11}+\cdots\)
9000.2.a.z	$9000$	$2$	9000.a	1.a	$1$	$4$	$4$	$71.865$	4.4.13025.1	None	✓				3000.2.a.i	$1$	$0$	\(0\)	\(0\)	\(0\)	\(3\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{7}+(-1+\beta _{1}-\beta _{2}-\beta _{3})q^{11}+\cdots\)
9000.2.a.ba	$9000$	$2$	9000.a	1.a	$1$	$4$	$4$	$71.865$	4.4.4400.1	None	✓				1000.2.a.e	$1$	$1$	\(0\)	\(0\)	\(0\)	\(6\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(2+\beta _{2}+\beta _{3})q^{7}-2\beta _{3}q^{11}+(2\beta _{1}+\cdots)q^{13}+\cdots\)
9000.2.a.bb	$9000$	$2$	9000.a	1.a	$1$	$4$	$4$	$71.865$	4.4.10025.1	None	✓				1000.2.a.f	$1$	$1$	\(0\)	\(0\)	\(0\)	\(9\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(2+\beta _{3})q^{7}+(-2+\beta _{1}-\beta _{2})q^{11}+\cdots\)
9000.2.a.bc	$9000$	$2$	9000.a	1.a	$1$	$6$	$6$	$71.865$	6.6.239482000.1	None	✓	✓	✓		9000.2.a.bc	$1$	$1$	\(0\)	\(0\)	\(0\)	\(-4\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{7}+(\beta _{3}-\beta _{5})q^{11}+(-\beta _{2}+\cdots)q^{13}+\cdots\)
9000.2.a.bd	$9000$	$2$	9000.a	1.a	$1$	$6$	$6$	$71.865$	6.6.239482000.1	None	✓	✓	✓		9000.2.a.bc	$1$	$1$	\(0\)	\(0\)	\(0\)	\(-4\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{7}+(-\beta _{3}+\beta _{5})q^{11}+\cdots\)
9000.2.a.be	$9000$	$2$	9000.a	1.a	$1$	$6$	$6$	$71.865$	6.6.239482000.1	None	✓	✓			9000.2.a.bc	$1$	$0$	\(0\)	\(0\)	\(0\)	\(4\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{7}+(\beta _{3}-\beta _{5})q^{11}+(\beta _{2}+\beta _{5})q^{13}+\cdots\)
9000.2.a.bf	$9000$	$2$	9000.a	1.a	$1$	$6$	$6$	$71.865$	6.6.239482000.1	None	✓	✓			9000.2.a.bc	$1$	$0$	\(0\)	\(0\)	\(0\)	\(4\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{7}+(-\beta _{3}+\beta _{5})q^{11}+(\beta _{2}+\cdots)q^{13}+\cdots\)
9000.2.a.bg	$9000$	$2$	9000.a	1.a	$1$	$8$	$8$	$71.865$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓	✓	✓		9000.2.a.bg	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$+$	$+$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q-\beta _{4}q^{7}+(-\beta _{2}+\beta _{7})q^{11}+(-\beta _{2}+\cdots)q^{13}+\cdots\)
9000.2.a.bh	$9000$	$2$	9000.a	1.a	$1$	$8$	$8$	$71.865$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓	✓	✓		9000.2.a.bg	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$+$	$+$	$-$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q-\beta _{4}q^{7}+(\beta _{2}-\beta _{7})q^{11}+(-\beta _{2}-\beta _{4}+\cdots)q^{13}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(9000))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(9000)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 24}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 18}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(24))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(30))\)\(^{\oplus 18}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(36))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(40))\)\(^{\oplus 9}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(45))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(50))\)\(^{\oplus 18}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(72))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(75))\)\(^{\oplus 16}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(90))\)\(^{\oplus 9}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(100))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(120))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(125))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(150))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(180))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(200))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(225))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(250))\)\(^{\oplus 9}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(300))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(360))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(375))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(450))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(500))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(600))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(750))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(900))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1000))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1125))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1500))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1800))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(2250))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(3000))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(4500))\)\(^{\oplus 2}\)