Space of modular forms of level 8820, weight 2, and trivial character

• Label: 8820.2.a
• Level: $8820$
• Weight: $2$
• Character orbit: 8820.a
• Rep. character: $\chi_{8820}(1,\cdot)$
• Character field: $\Q$
• Dimension: $69$
• Newform subspaces: $45$
• Sturm bound: $4032$
• Trace bound: $17$

Defining parameters

Level:	\( N \)	\(=\)	\( 8820 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8820.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 45 \)
Sturm bound:			\(4032\)
Trace bound:			\(17\)
Distinguishing \(T_p\):			\(11\), \(13\), \(17\), \(31\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(8820))\).

	Total	New	Old
Modular forms	2112	69	2043
Cusp forms	1921	69	1852
Eisenstein series	191	0	191

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
\(-\)	\(+\)	\(+\)	\(+\)	\(-\)	\(7\)
\(-\)	\(+\)	\(+\)	\(-\)	\(+\)	\(7\)
\(-\)	\(+\)	\(-\)	\(+\)	\(+\)	\(7\)
\(-\)	\(+\)	\(-\)	\(-\)	\(-\)	\(7\)
\(-\)	\(-\)	\(+\)	\(+\)	\(+\)	\(9\)
\(-\)	\(-\)	\(+\)	\(-\)	\(-\)	\(12\)
\(-\)	\(-\)	\(-\)	\(+\)	\(-\)	\(11\)
\(-\)	\(-\)	\(-\)	\(-\)	\(+\)	\(9\)
Plus space				\(+\)	\(32\)
Minus space				\(-\)	\(37\)

Trace form

69 * q - q^5 + 8 * q^11 + 2 * q^13 + 6 * q^17 - 4 * q^19 + 6 * q^23 + 69 * q^25 - 2 * q^29 - 38 * q^37 + 2 * q^41 - 38 * q^43 - 26 * q^47 + 6 * q^53 + 8 * q^55 + 14 * q^61 + 10 * q^65 - 58 * q^67 - 20 * q^71 - 18 * q^73 - 24 * q^79 - 26 * q^83 - 10 * q^85 - 6 * q^89 - 4 * q^95 - 6 * q^97

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(8820))\) 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	7
8820.2.a.a	$8820$	$2$	8820.a	1.a	$1$	$1$	$1$	$70.428$	\(\Q\)	None	✓				140.2.i.a	$1$	$0$	\(0\)	\(0\)	\(-1\)	\(0\)		$-$	$-$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{5}-6q^{11}-2q^{13}-6q^{17}-8q^{19}+\cdots\)
8820.2.a.b	$8820$	$2$	8820.a	1.a	$1$	$1$	$1$	$70.428$	\(\Q\)	None	✓				420.2.a.c	$1$	$0$	\(0\)	\(0\)	\(-1\)	\(0\)		$-$	$-$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{5}-6q^{11}+4q^{13}+6q^{17}-2q^{19}+\cdots\)
8820.2.a.c	$8820$	$2$	8820.a	1.a	$1$	$1$	$1$	$70.428$	\(\Q\)	None	✓				1260.2.a.b	$1$	$1$	\(0\)	\(0\)	\(-1\)	\(0\)		$-$	$+$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{5}-4q^{11}+2q^{17}+6q^{19}-6q^{23}+\cdots\)
8820.2.a.d	$8820$	$2$	8820.a	1.a	$1$	$1$	$1$	$70.428$	\(\Q\)	None	✓	✓			8820.2.a.d	$1$	$1$	\(0\)	\(0\)	\(-1\)	\(0\)		$-$	$+$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{5}-4q^{11}+4q^{13}-2q^{17}-4q^{23}+\cdots\)
8820.2.a.e	$8820$	$2$	8820.a	1.a	$1$	$1$	$1$	$70.428$	\(\Q\)	None	✓				2940.2.a.e	$1$	$0$	\(0\)	\(0\)	\(-1\)	\(0\)		$-$	$-$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{5}-2q^{11}-4q^{13}-2q^{17}-2q^{23}+\cdots\)
8820.2.a.f	$8820$	$2$	8820.a	1.a	$1$	$1$	$1$	$70.428$	\(\Q\)	None	✓				420.2.a.a	$1$	$0$	\(0\)	\(0\)	\(-1\)	\(0\)		$-$	$-$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{5}-2q^{11}-4q^{13}+6q^{17}-6q^{19}+\cdots\)
8820.2.a.g	$8820$	$2$	8820.a	1.a	$1$	$1$	$1$	$70.428$	\(\Q\)	None	✓				20.2.a.a	$1$	$0$	\(0\)	\(0\)	\(-1\)	\(0\)		$-$	$-$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{5}-2q^{13}-6q^{17}+4q^{19}-6q^{23}+\cdots\)
8820.2.a.h	$8820$	$2$	8820.a	1.a	$1$	$1$	$1$	$70.428$	\(\Q\)	None	✓				1260.2.a.d	$1$	$1$	\(0\)	\(0\)	\(-1\)	\(0\)		$-$	$+$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{5}+4q^{13}-6q^{17}-2q^{19}+6q^{23}+\cdots\)
8820.2.a.i	$8820$	$2$	8820.a	1.a	$1$	$1$	$1$	$70.428$	\(\Q\)	None	✓				980.2.a.d	$1$	$0$	\(0\)	\(0\)	\(-1\)	\(0\)		$-$	$-$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{5}+q^{11}+5q^{13}+q^{17}+6q^{19}+\cdots\)
8820.2.a.j	$8820$	$2$	8820.a	1.a	$1$	$1$	$1$	$70.428$	\(\Q\)	None	✓				420.2.q.a	$1$	$1$	\(0\)	\(0\)	\(-1\)	\(0\)		$-$	$-$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{5}+2q^{11}+q^{13}+4q^{17}-q^{19}+\cdots\)
8820.2.a.k	$8820$	$2$	8820.a	1.a	$1$	$1$	$1$	$70.428$	\(\Q\)	None	✓				140.2.i.b	$1$	$0$	\(0\)	\(0\)	\(-1\)	\(0\)		$-$	$-$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{5}+2q^{11}+6q^{13}+2q^{17}+9q^{23}+\cdots\)
8820.2.a.l	$8820$	$2$	8820.a	1.a	$1$	$1$	$1$	$70.428$	\(\Q\)	None	✓				420.2.q.b	$1$	$0$	\(0\)	\(0\)	\(-1\)	\(0\)		$-$	$-$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{5}+4q^{11}-7q^{13}-6q^{17}-3q^{19}+\cdots\)
8820.2.a.m	$8820$	$2$	8820.a	1.a	$1$	$1$	$1$	$70.428$	\(\Q\)	None	✓	✓			8820.2.a.d	$1$	$1$	\(0\)	\(0\)	\(-1\)	\(0\)		$-$	$+$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{5}+4q^{11}-4q^{13}-2q^{17}+4q^{23}+\cdots\)
8820.2.a.n	$8820$	$2$	8820.a	1.a	$1$	$1$	$1$	$70.428$	\(\Q\)	None	✓				140.2.a.b	$1$	$0$	\(0\)	\(0\)	\(-1\)	\(0\)		$-$	$-$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{5}+5q^{11}+3q^{13}-q^{17}-6q^{19}+\cdots\)
8820.2.a.o	$8820$	$2$	8820.a	1.a	$1$	$1$	$1$	$70.428$	\(\Q\)	None	✓				2940.2.a.a	$1$	$0$	\(0\)	\(0\)	\(-1\)	\(0\)		$-$	$-$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{5}+6q^{11}+6q^{17}-4q^{19}-6q^{23}+\cdots\)
8820.2.a.p	$8820$	$2$	8820.a	1.a	$1$	$1$	$1$	$70.428$	\(\Q\)	None	✓				140.2.i.a	$1$	$0$	\(0\)	\(0\)	\(1\)	\(0\)		$-$	$-$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{5}-6q^{11}+2q^{13}+6q^{17}+8q^{19}+\cdots\)
8820.2.a.q	$8820$	$2$	8820.a	1.a	$1$	$1$	$1$	$70.428$	\(\Q\)	None	✓	✓			8820.2.a.d	$1$	$0$	\(0\)	\(0\)	\(1\)	\(0\)		$-$	$+$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{5}-4q^{11}-4q^{13}+2q^{17}-4q^{23}+\cdots\)
8820.2.a.r	$8820$	$2$	8820.a	1.a	$1$	$1$	$1$	$70.428$	\(\Q\)	None	✓				140.2.a.a	$1$	$1$	\(0\)	\(0\)	\(1\)	\(0\)		$-$	$-$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{5}-3q^{11}+q^{13}-3q^{17}-2q^{19}+\cdots\)
8820.2.a.s	$8820$	$2$	8820.a	1.a	$1$	$1$	$1$	$70.428$	\(\Q\)	None	✓				420.2.a.d	$1$	$1$	\(0\)	\(0\)	\(1\)	\(0\)		$-$	$-$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{5}-2q^{11}-4q^{13}+2q^{17}+2q^{19}+\cdots\)
8820.2.a.t	$8820$	$2$	8820.a	1.a	$1$	$1$	$1$	$70.428$	\(\Q\)	None	✓				2940.2.a.e	$1$	$1$	\(0\)	\(0\)	\(1\)	\(0\)		$-$	$-$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{5}-2q^{11}+4q^{13}+2q^{17}-2q^{23}+\cdots\)
8820.2.a.u	$8820$	$2$	8820.a	1.a	$1$	$1$	$1$	$70.428$	\(\Q\)	None	✓				1260.2.a.d	$1$	$0$	\(0\)	\(0\)	\(1\)	\(0\)		$-$	$+$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{5}+4q^{13}+6q^{17}-2q^{19}-6q^{23}+\cdots\)
8820.2.a.v	$8820$	$2$	8820.a	1.a	$1$	$1$	$1$	$70.428$	\(\Q\)	None	✓				980.2.a.d	$1$	$1$	\(0\)	\(0\)	\(1\)	\(0\)		$-$	$-$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{5}+q^{11}-5q^{13}-q^{17}-6q^{19}+\cdots\)
8820.2.a.w	$8820$	$2$	8820.a	1.a	$1$	$1$	$1$	$70.428$	\(\Q\)	None	✓				140.2.i.b	$1$	$0$	\(0\)	\(0\)	\(1\)	\(0\)		$-$	$-$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{5}+2q^{11}-6q^{13}-2q^{17}+9q^{23}+\cdots\)
8820.2.a.x	$8820$	$2$	8820.a	1.a	$1$	$1$	$1$	$70.428$	\(\Q\)	None	✓				420.2.a.b	$1$	$1$	\(0\)	\(0\)	\(1\)	\(0\)		$-$	$-$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{5}+2q^{11}-4q^{13}+2q^{17}-2q^{19}+\cdots\)
8820.2.a.y	$8820$	$2$	8820.a	1.a	$1$	$1$	$1$	$70.428$	\(\Q\)	None	✓				420.2.q.a	$1$	$1$	\(0\)	\(0\)	\(1\)	\(0\)		$-$	$-$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{5}+2q^{11}-q^{13}-4q^{17}+q^{19}+\cdots\)
8820.2.a.z	$8820$	$2$	8820.a	1.a	$1$	$1$	$1$	$70.428$	\(\Q\)	None	✓				1260.2.a.b	$1$	$0$	\(0\)	\(0\)	\(1\)	\(0\)		$-$	$+$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{5}+4q^{11}-2q^{17}+6q^{19}+6q^{23}+\cdots\)
8820.2.a.ba	$8820$	$2$	8820.a	1.a	$1$	$1$	$1$	$70.428$	\(\Q\)	None	✓	✓			8820.2.a.d	$1$	$0$	\(0\)	\(0\)	\(1\)	\(0\)		$-$	$+$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{5}+4q^{11}+4q^{13}+2q^{17}+4q^{23}+\cdots\)
8820.2.a.bb	$8820$	$2$	8820.a	1.a	$1$	$1$	$1$	$70.428$	\(\Q\)	None	✓				420.2.q.b	$1$	$0$	\(0\)	\(0\)	\(1\)	\(0\)		$-$	$-$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{5}+4q^{11}+7q^{13}+6q^{17}+3q^{19}+\cdots\)
8820.2.a.bc	$8820$	$2$	8820.a	1.a	$1$	$1$	$1$	$70.428$	\(\Q\)	None	✓				2940.2.a.a	$1$	$1$	\(0\)	\(0\)	\(1\)	\(0\)		$-$	$-$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{5}+6q^{11}-6q^{17}+4q^{19}-6q^{23}+\cdots\)
8820.2.a.bd	$8820$	$2$	8820.a	1.a	$1$	$2$	$2$	$70.428$	\(\Q(\sqrt{2}) \)	None	✓				2940.2.a.n	$1$	$1$	\(0\)	\(0\)	\(-2\)	\(0\)		$-$	$-$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{5}+(-2+2\beta )q^{11}-2\beta q^{17}-\beta q^{19}+\cdots\)
8820.2.a.be	$8820$	$2$	8820.a	1.a	$1$	$2$	$2$	$70.428$	\(\Q(\sqrt{7}) \)	None	✓				420.2.q.c	$1$	$1$	\(0\)	\(0\)	\(-2\)	\(0\)		$-$	$-$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{5}+(-1+\beta )q^{11}-\beta q^{13}+(-1+\cdots)q^{17}+\cdots\)
8820.2.a.bf	$8820$	$2$	8820.a	1.a	$1$	$2$	$2$	$70.428$	\(\Q(\sqrt{2}) \)	None	✓		✓		420.2.q.d	$1$	$0$	\(0\)	\(0\)	\(-2\)	\(0\)		$-$	$-$	$+$	$-$	$-$	$3$	$\mathrm{SU}(2)$	\(q-q^{5}+\beta q^{11}+(1-\beta )q^{13}+\beta q^{17}+\cdots\)
8820.2.a.bg	$8820$	$2$	8820.a	1.a	$1$	$2$	$2$	$70.428$	\(\Q(\sqrt{2}) \)	None	✓				980.2.a.j	$1$	$1$	\(0\)	\(0\)	\(-2\)	\(0\)		$-$	$-$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{5}+(1+2\beta )q^{11}+(-5+\beta )q^{13}+\cdots\)
8820.2.a.bh	$8820$	$2$	8820.a	1.a	$1$	$2$	$2$	$70.428$	\(\Q(\sqrt{2}) \)	None	✓				2940.2.a.o	$1$	$1$	\(0\)	\(0\)	\(-2\)	\(0\)		$-$	$-$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{5}+2q^{11}+2\beta q^{13}+(-4+2\beta )q^{17}+\cdots\)
8820.2.a.bi	$8820$	$2$	8820.a	1.a	$1$	$2$	$2$	$70.428$	\(\Q(\sqrt{2}) \)	None	✓				2940.2.a.n	$1$	$0$	\(0\)	\(0\)	\(2\)	\(0\)		$-$	$-$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{5}+(-2+2\beta )q^{11}+2\beta q^{17}+\beta q^{19}+\cdots\)
8820.2.a.bj	$8820$	$2$	8820.a	1.a	$1$	$2$	$2$	$70.428$	\(\Q(\sqrt{7}) \)	None	✓		✓		420.2.q.c	$1$	$1$	\(0\)	\(0\)	\(2\)	\(0\)		$-$	$-$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{5}+(-1+\beta )q^{11}+\beta q^{13}+(1-\beta )q^{17}+\cdots\)
8820.2.a.bk	$8820$	$2$	8820.a	1.a	$1$	$2$	$2$	$70.428$	\(\Q(\sqrt{2}) \)	None	✓				420.2.q.d	$1$	$0$	\(0\)	\(0\)	\(2\)	\(0\)		$-$	$-$	$-$	$+$	$-$	$3$	$\mathrm{SU}(2)$	\(q+q^{5}+\beta q^{11}+(-1+\beta )q^{13}-\beta q^{17}+\cdots\)
8820.2.a.bl	$8820$	$2$	8820.a	1.a	$1$	$2$	$2$	$70.428$	\(\Q(\sqrt{2}) \)	None	✓				980.2.a.j	$1$	$0$	\(0\)	\(0\)	\(2\)	\(0\)		$-$	$-$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{5}+(1+2\beta )q^{11}+(5-\beta )q^{13}+(-5+\cdots)q^{17}+\cdots\)
8820.2.a.bm	$8820$	$2$	8820.a	1.a	$1$	$2$	$2$	$70.428$	\(\Q(\sqrt{2}) \)	None	✓				2940.2.a.o	$1$	$0$	\(0\)	\(0\)	\(2\)	\(0\)		$-$	$-$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{5}+2q^{11}-2\beta q^{13}+(4-2\beta )q^{17}+\cdots\)
8820.2.a.bn	$8820$	$2$	8820.a	1.a	$1$	$3$	$3$	$70.428$	3.3.404.1	None	✓				1260.2.s.g	$1$	$0$	\(0\)	\(0\)	\(-3\)	\(0\)		$-$	$+$	$+$	$+$	$-$	$3$	$\mathrm{SU}(2)$	\(q-q^{5}+(-1+\beta _{2})q^{11}+(-2+\beta _{2})q^{13}+\cdots\)
8820.2.a.bo	$8820$	$2$	8820.a	1.a	$1$	$3$	$3$	$70.428$	3.3.404.1	None	✓		✓		1260.2.s.g	$1$	$1$	\(0\)	\(0\)	\(-3\)	\(0\)		$-$	$+$	$+$	$-$	$+$	$3$	$\mathrm{SU}(2)$	\(q-q^{5}+(1-\beta _{2})q^{11}+(2-\beta _{2})q^{13}+\cdots\)
8820.2.a.bp	$8820$	$2$	8820.a	1.a	$1$	$3$	$3$	$70.428$	3.3.404.1	None	✓		✓		1260.2.s.g	$1$	$0$	\(0\)	\(0\)	\(3\)	\(0\)		$-$	$+$	$-$	$-$	$-$	$3$	$\mathrm{SU}(2)$	\(q+q^{5}+(-1+\beta _{2})q^{11}+(2-\beta _{2})q^{13}+\cdots\)
8820.2.a.bq	$8820$	$2$	8820.a	1.a	$1$	$3$	$3$	$70.428$	3.3.404.1	None	✓				1260.2.s.g	$1$	$1$	\(0\)	\(0\)	\(3\)	\(0\)		$-$	$+$	$-$	$+$	$+$	$3$	$\mathrm{SU}(2)$	\(q+q^{5}+(1-\beta _{2})q^{11}+(-2+\beta _{2})q^{13}+\cdots\)
8820.2.a.br	$8820$	$2$	8820.a	1.a	$1$	$4$	$4$	$70.428$	\(\Q(\sqrt{2}, \sqrt{5})\)	None	✓	✓	✓		8820.2.a.br	$2$	$0$	\(0\)	\(0\)	\(-4\)	\(0\)		$-$	$+$	$+$	$+$	$-$	$2^{3}$	$\mathrm{SU}(2)$	\(q-q^{5}+(\beta _{1}+\beta _{2})q^{11}+(-\beta _{1}+\beta _{2}+\cdots)q^{13}+\cdots\)
8820.2.a.bs	$8820$	$2$	8820.a	1.a	$1$	$4$	$4$	$70.428$	\(\Q(\sqrt{2}, \sqrt{5})\)	None	✓	✓	✓		8820.2.a.br	$2$	$1$	\(0\)	\(0\)	\(4\)	\(0\)		$-$	$+$	$-$	$+$	$+$	$2^{3}$	$\mathrm{SU}(2)$	\(q+q^{5}+(\beta _{1}+\beta _{2})q^{11}+(\beta _{1}-\beta _{2})q^{13}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(8820))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(8820)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 24}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 18}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 9}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(21))\)\(^{\oplus 24}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(30))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(35))\)\(^{\oplus 18}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(36))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(42))\)\(^{\oplus 16}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(45))\)\(^{\oplus 9}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(49))\)\(^{\oplus 18}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(63))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(70))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(84))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(90))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(98))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(105))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(126))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(140))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(147))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(180))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(196))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(210))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(245))\)\(^{\oplus 9}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(252))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(294))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(315))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(420))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(441))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(490))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(588))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(630))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(735))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(882))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(980))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1260))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1470))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1764))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(2205))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(2940))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(4410))\)\(^{\oplus 2}\)