Space of modular forms of level 3600, weight 2, and trivial character

• Label: 3600.2.a
• Level: $3600$
• Weight: $2$
• Character orbit: 3600.a
• Rep. character: $\chi_{3600}(1,\cdot)$
• Character field: $\Q$
• Dimension: $46$
• Newform subspaces: $45$
• Sturm bound: $1440$
• Trace bound: $17$

Defining parameters

Level:	\( N \)	\(=\)	\( 3600 = 2^{4} \cdot 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3600.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 45 \)
Sturm bound:			\(1440\)
Trace bound:			\(17\)
Distinguishing \(T_p\):			\(7\), \(11\), \(13\), \(17\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(3600))\).

	Total	New	Old
Modular forms	792	49	743
Cusp forms	649	46	603
Eisenstein series	143	3	140

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
\(+\)	\(+\)	\(+\)	\(+\)		\(96\)	\(4\)	\(92\)		\(79\)	\(4\)	\(75\)		\(17\)	\(0\)	\(17\)
\(+\)	\(+\)	\(-\)	\(-\)		\(100\)	\(6\)	\(94\)		\(82\)	\(6\)	\(76\)		\(18\)	\(0\)	\(18\)
\(+\)	\(-\)	\(+\)	\(-\)		\(102\)	\(7\)	\(95\)		\(84\)	\(7\)	\(77\)		\(18\)	\(0\)	\(18\)
\(+\)	\(-\)	\(-\)	\(+\)		\(98\)	\(7\)	\(91\)		\(80\)	\(7\)	\(73\)		\(18\)	\(0\)	\(18\)
\(-\)	\(+\)	\(+\)	\(-\)		\(102\)	\(5\)	\(97\)		\(84\)	\(5\)	\(79\)		\(18\)	\(0\)	\(18\)
\(-\)	\(+\)	\(-\)	\(+\)		\(98\)	\(4\)	\(94\)		\(80\)	\(4\)	\(76\)		\(18\)	\(0\)	\(18\)
\(-\)	\(-\)	\(+\)	\(+\)		\(96\)	\(7\)	\(89\)		\(78\)	\(6\)	\(72\)		\(18\)	\(1\)	\(17\)
\(-\)	\(-\)	\(-\)	\(-\)		\(100\)	\(9\)	\(91\)		\(82\)	\(7\)	\(75\)		\(18\)	\(2\)	\(16\)
Plus space			\(+\)		\(388\)	\(22\)	\(366\)		\(317\)	\(21\)	\(296\)		\(71\)	\(1\)	\(70\)
Minus space			\(-\)		\(404\)	\(27\)	\(377\)		\(332\)	\(25\)	\(307\)		\(72\)	\(2\)	\(70\)

Trace form

46 * q + 2 * q^7 - 4 * q^11 - 2 * q^17 + 8 * q^19 + 6 * q^23 - 2 * q^29 - 4 * q^37 + 2 * q^41 + 2 * q^43 - 22 * q^47 + 42 * q^49 - 2 * q^53 - 28 * q^59 - 8 * q^61 - 18 * q^67 - 20 * q^71 + 20 * q^73 - 16 * q^77 + 20 * q^79 - 10 * q^83 - 2 * q^89 + 4 * q^91 + 12 * q^97

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(3600))\) 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
3600.2.a.a	$3600$	$2$	3600.a	1.a	$1$	$1$	$1$	$28.746$	\(\Q\)	None	✓				600.2.a.e	$1$	$0$	\(0\)	\(0\)	\(0\)	\(-5\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-5q^{7}-6q^{11}+3q^{13}-2q^{17}-q^{19}+\cdots\)
3600.2.a.b	$3600$	$2$	3600.a	1.a	$1$	$1$	$1$	$28.746$	\(\Q\)	\(\Q(\sqrt{-3}) \)	✓				225.2.a.c	$2$	$1$	\(0\)	\(0\)	\(0\)	\(-5\)		$-$	$+$	$-$	$+$	$1$	$N(\mathrm{U}(1))$	\(q-5q^{7}+5q^{13}+q^{19}+7q^{31}-10q^{37}+\cdots\)
3600.2.a.c	$3600$	$2$	3600.a	1.a	$1$	$1$	$1$	$28.746$	\(\Q\)	None	✓				360.2.f.b	$1$	$0$	\(0\)	\(0\)	\(0\)	\(-4\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-4q^{7}-4q^{11}-4q^{13}+6q^{17}+4q^{19}+\cdots\)
3600.2.a.d	$3600$	$2$	3600.a	1.a	$1$	$1$	$1$	$28.746$	\(\Q\)	None	✓				60.2.d.a	$1$	$0$	\(0\)	\(0\)	\(0\)	\(-4\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-4q^{7}-4q^{11}-4q^{17}+4q^{23}+6q^{29}+\cdots\)
3600.2.a.e	$3600$	$2$	3600.a	1.a	$1$	$1$	$1$	$28.746$	\(\Q\)	\(\Q(\sqrt{-3}) \)	✓				36.2.a.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(-4\)		$-$	$+$	$+$	$-$	$1$	$N(\mathrm{U}(1))$	\(q-4q^{7}-2q^{13}-8q^{19}+4q^{31}+10q^{37}+\cdots\)
3600.2.a.f	$3600$	$2$	3600.a	1.a	$1$	$1$	$1$	$28.746$	\(\Q\)	None	✓				30.2.a.a	$1$	$1$	\(0\)	\(0\)	\(0\)	\(-4\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-4q^{7}-2q^{13}+6q^{17}+4q^{19}+6q^{29}+\cdots\)
3600.2.a.g	$3600$	$2$	3600.a	1.a	$1$	$1$	$1$	$28.746$	\(\Q\)	None	✓				360.2.f.b	$1$	$0$	\(0\)	\(0\)	\(0\)	\(-4\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-4q^{7}+4q^{11}-4q^{13}-6q^{17}+4q^{19}+\cdots\)
3600.2.a.h	$3600$	$2$	3600.a	1.a	$1$	$1$	$1$	$28.746$	\(\Q\)	None	✓				40.2.a.a	$1$	$0$	\(0\)	\(0\)	\(0\)	\(-4\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-4q^{7}+4q^{11}+2q^{13}+2q^{17}-4q^{19}+\cdots\)
3600.2.a.i	$3600$	$2$	3600.a	1.a	$1$	$1$	$1$	$28.746$	\(\Q\)	None	✓				600.2.a.b	$1$	$1$	\(0\)	\(0\)	\(0\)	\(-3\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-3q^{7}+2q^{11}-3q^{13}-6q^{17}+7q^{19}+\cdots\)
3600.2.a.j	$3600$	$2$	3600.a	1.a	$1$	$1$	$1$	$28.746$	\(\Q\)	None	✓				75.2.a.a	$1$	$1$	\(0\)	\(0\)	\(0\)	\(-3\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-3q^{7}+2q^{11}-q^{13}+2q^{17}+5q^{19}+\cdots\)
3600.2.a.k	$3600$	$2$	3600.a	1.a	$1$	$1$	$1$	$28.746$	\(\Q\)	None	✓				40.2.c.a	$1$	$1$	\(0\)	\(0\)	\(0\)	\(-2\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{7}-4q^{11}+4q^{13}+4q^{19}-2q^{23}+\cdots\)
3600.2.a.l	$3600$	$2$	3600.a	1.a	$1$	$1$	$1$	$28.746$	\(\Q\)	None	✓				50.2.a.a	$1$	$0$	\(0\)	\(0\)	\(0\)	\(-2\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{7}-3q^{11}-4q^{13}+3q^{17}-5q^{19}+\cdots\)
3600.2.a.m	$3600$	$2$	3600.a	1.a	$1$	$1$	$1$	$28.746$	\(\Q\)	None	✓				200.2.a.a	$1$	$1$	\(0\)	\(0\)	\(0\)	\(-2\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{7}+q^{11}+4q^{13}-5q^{17}-q^{19}+\cdots\)
3600.2.a.n	$3600$	$2$	3600.a	1.a	$1$	$1$	$1$	$28.746$	\(\Q\)	None	✓				120.2.f.a	$1$	$1$	\(0\)	\(0\)	\(0\)	\(-2\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{7}+2q^{11}-2q^{13}+6q^{17}-8q^{19}+\cdots\)
3600.2.a.o	$3600$	$2$	3600.a	1.a	$1$	$1$	$1$	$28.746$	\(\Q\)	None	✓				30.2.c.a	$1$	$0$	\(0\)	\(0\)	\(0\)	\(-2\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{7}+2q^{11}+6q^{13}-2q^{17}-4q^{23}+\cdots\)
3600.2.a.p	$3600$	$2$	3600.a	1.a	$1$	$1$	$1$	$28.746$	\(\Q\)	None	✓				1800.2.a.k	$1$	$1$	\(0\)	\(0\)	\(0\)	\(-1\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{7}-4q^{11}-q^{13}+4q^{17}-q^{19}+\cdots\)
3600.2.a.q	$3600$	$2$	3600.a	1.a	$1$	$1$	$1$	$28.746$	\(\Q\)	\(\Q(\sqrt{-3}) \)	✓				900.2.a.d	$2$	$0$	\(0\)	\(0\)	\(0\)	\(-1\)		$-$	$+$	$+$	$-$	$1$	$N(\mathrm{U}(1))$	\(q-q^{7}+7q^{13}+7q^{19}-11q^{31}+10q^{37}+\cdots\)
3600.2.a.r	$3600$	$2$	3600.a	1.a	$1$	$1$	$1$	$28.746$	\(\Q\)	None	✓				1800.2.a.k	$1$	$1$	\(0\)	\(0\)	\(0\)	\(-1\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{7}+4q^{11}-q^{13}-4q^{17}-q^{19}+\cdots\)
3600.2.a.s	$3600$	$2$	3600.a	1.a	$1$	$1$	$1$	$28.746$	\(\Q\)	None	✓				300.2.a.b	$1$	$1$	\(0\)	\(0\)	\(0\)	\(-1\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{7}+6q^{11}-5q^{13}-6q^{17}-5q^{19}+\cdots\)
3600.2.a.t	$3600$	$2$	3600.a	1.a	$1$	$1$	$1$	$28.746$	\(\Q\)	None	✓				120.2.a.b	$1$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-4q^{11}-6q^{13}-6q^{17}+4q^{19}+\cdots\)
3600.2.a.u	$3600$	$2$	3600.a	1.a	$1$	$1$	$1$	$28.746$	\(\Q\)	None	✓				15.2.a.a	$1$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-4q^{11}+2q^{13}+2q^{17}-4q^{19}+\cdots\)
3600.2.a.v	$3600$	$2$	3600.a	1.a	$1$	$1$	$1$	$28.746$	\(\Q\)	None	✓				24.2.a.a	$1$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+4q^{11}+2q^{13}+2q^{17}+4q^{19}+\cdots\)
3600.2.a.w	$3600$	$2$	3600.a	1.a	$1$	$1$	$1$	$28.746$	\(\Q\)	None	✓				1800.2.a.k	$1$	$0$	\(0\)	\(0\)	\(0\)	\(1\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{7}-4q^{11}+q^{13}-4q^{17}-q^{19}+\cdots\)
3600.2.a.x	$3600$	$2$	3600.a	1.a	$1$	$1$	$1$	$28.746$	\(\Q\)	\(\Q(\sqrt{-3}) \)	✓				900.2.a.d	$2$	$1$	\(0\)	\(0\)	\(0\)	\(1\)		$-$	$+$	$-$	$+$	$1$	$N(\mathrm{U}(1))$	\(q+q^{7}-7q^{13}+7q^{19}-11q^{31}-10q^{37}+\cdots\)
3600.2.a.y	$3600$	$2$	3600.a	1.a	$1$	$1$	$1$	$28.746$	\(\Q\)	None	✓				1800.2.a.k	$1$	$0$	\(0\)	\(0\)	\(0\)	\(1\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{7}+4q^{11}+q^{13}+4q^{17}-q^{19}+\cdots\)
3600.2.a.z	$3600$	$2$	3600.a	1.a	$1$	$1$	$1$	$28.746$	\(\Q\)	None	✓				300.2.a.b	$1$	$0$	\(0\)	\(0\)	\(0\)	\(1\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{7}+6q^{11}+5q^{13}+6q^{17}-5q^{19}+\cdots\)
3600.2.a.ba	$3600$	$2$	3600.a	1.a	$1$	$1$	$1$	$28.746$	\(\Q\)	None	✓				90.2.a.a	$1$	$0$	\(0\)	\(0\)	\(0\)	\(2\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{7}-6q^{11}+4q^{13}+6q^{17}+4q^{19}+\cdots\)
3600.2.a.bb	$3600$	$2$	3600.a	1.a	$1$	$1$	$1$	$28.746$	\(\Q\)	None	✓				40.2.c.a	$1$	$1$	\(0\)	\(0\)	\(0\)	\(2\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{7}-4q^{11}-4q^{13}+4q^{19}+2q^{23}+\cdots\)
3600.2.a.bc	$3600$	$2$	3600.a	1.a	$1$	$1$	$1$	$28.746$	\(\Q\)	None	✓				50.2.a.a	$1$	$1$	\(0\)	\(0\)	\(0\)	\(2\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{7}-3q^{11}+4q^{13}-3q^{17}-5q^{19}+\cdots\)
3600.2.a.bd	$3600$	$2$	3600.a	1.a	$1$	$1$	$1$	$28.746$	\(\Q\)	None	✓				360.2.a.c	$1$	$1$	\(0\)	\(0\)	\(0\)	\(2\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{7}-2q^{11}-4q^{13}+2q^{17}-4q^{19}+\cdots\)
3600.2.a.be	$3600$	$2$	3600.a	1.a	$1$	$1$	$1$	$28.746$	\(\Q\)	None	✓				20.2.a.a	$1$	$1$	\(0\)	\(0\)	\(0\)	\(2\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{7}-2q^{13}-6q^{17}+4q^{19}-6q^{23}+\cdots\)
3600.2.a.bf	$3600$	$2$	3600.a	1.a	$1$	$1$	$1$	$28.746$	\(\Q\)	None	✓				200.2.a.a	$1$	$0$	\(0\)	\(0\)	\(0\)	\(2\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{7}+q^{11}-4q^{13}+5q^{17}-q^{19}+\cdots\)
3600.2.a.bg	$3600$	$2$	3600.a	1.a	$1$	$1$	$1$	$28.746$	\(\Q\)	None	✓				30.2.c.a	$1$	$0$	\(0\)	\(0\)	\(0\)	\(2\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{7}+2q^{11}-6q^{13}+2q^{17}+4q^{23}+\cdots\)
3600.2.a.bh	$3600$	$2$	3600.a	1.a	$1$	$1$	$1$	$28.746$	\(\Q\)	None	✓				360.2.a.c	$1$	$1$	\(0\)	\(0\)	\(0\)	\(2\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{7}+2q^{11}-4q^{13}-2q^{17}-4q^{19}+\cdots\)
3600.2.a.bi	$3600$	$2$	3600.a	1.a	$1$	$1$	$1$	$28.746$	\(\Q\)	None	✓				120.2.f.a	$1$	$1$	\(0\)	\(0\)	\(0\)	\(2\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{7}+2q^{11}+2q^{13}-6q^{17}-8q^{19}+\cdots\)
3600.2.a.bj	$3600$	$2$	3600.a	1.a	$1$	$1$	$1$	$28.746$	\(\Q\)	None	✓				90.2.a.a	$1$	$0$	\(0\)	\(0\)	\(0\)	\(2\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{7}+6q^{11}+4q^{13}-6q^{17}+4q^{19}+\cdots\)
3600.2.a.bk	$3600$	$2$	3600.a	1.a	$1$	$1$	$1$	$28.746$	\(\Q\)	None	✓				75.2.a.a	$1$	$0$	\(0\)	\(0\)	\(0\)	\(3\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3q^{7}+2q^{11}+q^{13}-2q^{17}+5q^{19}+\cdots\)
3600.2.a.bl	$3600$	$2$	3600.a	1.a	$1$	$1$	$1$	$28.746$	\(\Q\)	None	✓				600.2.a.b	$1$	$0$	\(0\)	\(0\)	\(0\)	\(3\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3q^{7}+2q^{11}+3q^{13}+6q^{17}+7q^{19}+\cdots\)
3600.2.a.bm	$3600$	$2$	3600.a	1.a	$1$	$1$	$1$	$28.746$	\(\Q\)	None	✓				60.2.d.a	$1$	$0$	\(0\)	\(0\)	\(0\)	\(4\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+4q^{7}-4q^{11}+4q^{17}-4q^{23}+6q^{29}+\cdots\)
3600.2.a.bn	$3600$	$2$	3600.a	1.a	$1$	$1$	$1$	$28.746$	\(\Q\)	None	✓				360.2.f.b	$1$	$0$	\(0\)	\(0\)	\(0\)	\(4\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+4q^{7}-4q^{11}+4q^{13}-6q^{17}+4q^{19}+\cdots\)
3600.2.a.bo	$3600$	$2$	3600.a	1.a	$1$	$1$	$1$	$28.746$	\(\Q\)	None	✓				120.2.a.a	$1$	$0$	\(0\)	\(0\)	\(0\)	\(4\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+4q^{7}+6q^{13}-2q^{17}-4q^{19}+8q^{23}+\cdots\)
3600.2.a.bp	$3600$	$2$	3600.a	1.a	$1$	$1$	$1$	$28.746$	\(\Q\)	None	✓				360.2.f.b	$1$	$0$	\(0\)	\(0\)	\(0\)	\(4\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+4q^{7}+4q^{11}+4q^{13}+6q^{17}+4q^{19}+\cdots\)
3600.2.a.bq	$3600$	$2$	3600.a	1.a	$1$	$1$	$1$	$28.746$	\(\Q\)	None	✓				600.2.a.e	$1$	$1$	\(0\)	\(0\)	\(0\)	\(5\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+5q^{7}-6q^{11}-3q^{13}+2q^{17}-q^{19}+\cdots\)
3600.2.a.br	$3600$	$2$	3600.a	1.a	$1$	$1$	$1$	$28.746$	\(\Q\)	\(\Q(\sqrt{-3}) \)	✓				225.2.a.c	$2$	$0$	\(0\)	\(0\)	\(0\)	\(5\)		$-$	$+$	$+$	$-$	$1$	$N(\mathrm{U}(1))$	\(q+5q^{7}-5q^{13}+q^{19}+7q^{31}+10q^{37}+\cdots\)
3600.2.a.bs	$3600$	$2$	3600.a	1.a	$1$	$2$	$2$	$28.746$	\(\Q(\sqrt{5}) \)	\(\Q(\sqrt{-15}) \)	✓		✓		45.2.b.a	$4$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$+$	$-$	$+$	$2^{2}$	$N(\mathrm{U}(1))$	\(q-\beta q^{17}-4q^{19}+2\beta q^{23}-8q^{31}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(3600))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(3600)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 20}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 18}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(24))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(30))\)\(^{\oplus 16}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(36))\)\(^{\oplus 9}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(40))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(45))\)\(^{\oplus 10}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(48))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(50))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(72))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(75))\)\(^{\oplus 10}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(80))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(90))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(100))\)\(^{\oplus 9}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(120))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(144))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(150))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(180))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(200))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(225))\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(240))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(300))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(360))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(400))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(450))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(600))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(720))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(900))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1200))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1800))\)\(^{\oplus 2}\)