Space of modular forms of level 1800, weight 4, and trivial character

• Label: 1800.4.a
• Level: $1800$
• Weight: $4$
• Character orbit: 1800.a
• Rep. character: $\chi_{1800}(1,\cdot)$
• Character field: $\Q$
• Dimension: $71$
• Newform subspaces: $49$
• Sturm bound: $1440$
• Trace bound: $17$

Defining parameters

Level:	\( N \)	\(=\)	\( 1800 = 2^{3} \cdot 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1800.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 49 \)
Sturm bound:			\(1440\)
Trace bound:			\(17\)
Distinguishing \(T_p\):			\(7\), \(11\), \(17\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(1800))\).

	Total	New	Old
Modular forms	1128	71	1057
Cusp forms	1032	71	961
Eisenstein series	96	0	96

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	Dim
\(+\)	\(+\)	\(+\)	\(+\)	\(7\)
\(+\)	\(+\)	\(-\)	\(-\)	\(7\)
\(+\)	\(-\)	\(+\)	\(-\)	\(9\)
\(+\)	\(-\)	\(-\)	\(+\)	\(12\)
\(-\)	\(+\)	\(+\)	\(-\)	\(7\)
\(-\)	\(+\)	\(-\)	\(+\)	\(7\)
\(-\)	\(-\)	\(+\)	\(+\)	\(11\)
\(-\)	\(-\)	\(-\)	\(-\)	\(11\)
Plus space			\(+\)	\(37\)
Minus space			\(-\)	\(34\)

Trace form

71 * q - 12 * q^7 - 22 * q^11 - 2 * q^13 - 46 * q^17 + 46 * q^19 + 28 * q^23 + 50 * q^29 - 100 * q^31 - 178 * q^37 - 16 * q^41 - 400 * q^43 + 52 * q^47 + 3527 * q^49 - 594 * q^53 - 252 * q^59 - 1446 * q^61 - 304 * q^67 - 136 * q^71 + 634 * q^73 + 32 * q^77 + 1172 * q^79 + 1040 * q^83 - 2772 * q^89 + 2768 * q^91 - 862 * q^97

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(1800))\) 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
1800.4.a.a	$1800$	$4$	1800.a	1.a	$1$	$1$	$1$	$106.203$	\(\Q\)	None	✓				360.4.a.g	$1$	$0$	\(0\)	\(0\)	\(0\)	\(-34\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-34q^{7}-18q^{11}-12q^{13}-106q^{17}+\cdots\)
1800.4.a.b	$1800$	$4$	1800.a	1.a	$1$	$1$	$1$	$106.203$	\(\Q\)	None	✓				360.4.a.g	$1$	$1$	\(0\)	\(0\)	\(0\)	\(-34\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-34q^{7}+18q^{11}-12q^{13}+106q^{17}+\cdots\)
1800.4.a.c	$1800$	$4$	1800.a	1.a	$1$	$1$	$1$	$106.203$	\(\Q\)	None	✓				200.4.a.b	$1$	$0$	\(0\)	\(0\)	\(0\)	\(-26\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-26q^{7}+59q^{11}-28q^{13}+5q^{17}+\cdots\)
1800.4.a.d	$1800$	$4$	1800.a	1.a	$1$	$1$	$1$	$106.203$	\(\Q\)	None	✓				8.4.a.a	$1$	$1$	\(0\)	\(0\)	\(0\)	\(-24\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-24q^{7}+44q^{11}-22q^{13}+50q^{17}+\cdots\)
1800.4.a.e	$1800$	$4$	1800.a	1.a	$1$	$1$	$1$	$106.203$	\(\Q\)	None	✓				120.4.a.e	$1$	$0$	\(0\)	\(0\)	\(0\)	\(-20\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-20q^{7}-2^{4}q^{11}-58q^{13}+38q^{17}+\cdots\)
1800.4.a.f	$1800$	$4$	1800.a	1.a	$1$	$1$	$1$	$106.203$	\(\Q\)	None	✓				120.4.a.b	$1$	$0$	\(0\)	\(0\)	\(0\)	\(-20\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-20q^{7}+56q^{11}+86q^{13}-106q^{17}+\cdots\)
1800.4.a.g	$1800$	$4$	1800.a	1.a	$1$	$1$	$1$	$106.203$	\(\Q\)	None	✓				600.4.a.g	$1$	$0$	\(0\)	\(0\)	\(0\)	\(-19\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-19q^{7}-22q^{11}+q^{13}+58q^{17}+\cdots\)
1800.4.a.h	$1800$	$4$	1800.a	1.a	$1$	$1$	$1$	$106.203$	\(\Q\)	None	✓				40.4.a.b	$1$	$0$	\(0\)	\(0\)	\(0\)	\(-16\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2^{4}q^{7}-6^{2}q^{11}+42q^{13}-110q^{17}+\cdots\)
1800.4.a.i	$1800$	$4$	1800.a	1.a	$1$	$1$	$1$	$106.203$	\(\Q\)	None	✓				120.4.f.b	$1$	$1$	\(0\)	\(0\)	\(0\)	\(-10\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-10q^{7}+14q^{11}+82q^{13}+18q^{17}+\cdots\)
1800.4.a.j	$1800$	$4$	1800.a	1.a	$1$	$1$	$1$	$106.203$	\(\Q\)	None	✓				120.4.f.a	$1$	$0$	\(0\)	\(0\)	\(0\)	\(-10\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-10q^{7}+46q^{11}+34q^{13}+66q^{17}+\cdots\)
1800.4.a.k	$1800$	$4$	1800.a	1.a	$1$	$1$	$1$	$106.203$	\(\Q\)	None	✓				120.4.a.f	$1$	$1$	\(0\)	\(0\)	\(0\)	\(-8\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-8q^{7}-20q^{11}-22q^{13}-14q^{17}+\cdots\)
1800.4.a.l	$1800$	$4$	1800.a	1.a	$1$	$1$	$1$	$106.203$	\(\Q\)	None	✓				200.4.a.e	$1$	$1$	\(0\)	\(0\)	\(0\)	\(-6\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-6q^{7}+19q^{11}+12q^{13}+75q^{17}+\cdots\)
1800.4.a.m	$1800$	$4$	1800.a	1.a	$1$	$1$	$1$	$106.203$	\(\Q\)	None	✓				600.4.a.e	$1$	$1$	\(0\)	\(0\)	\(0\)	\(-5\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-5q^{7}-14q^{11}-q^{13}+46q^{17}+\cdots\)
1800.4.a.n	$1800$	$4$	1800.a	1.a	$1$	$1$	$1$	$106.203$	\(\Q\)	None	✓				120.4.a.a	$1$	$1$	\(0\)	\(0\)	\(0\)	\(-4\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-4q^{7}-72q^{11}+6q^{13}+38q^{17}+\cdots\)
1800.4.a.o	$1800$	$4$	1800.a	1.a	$1$	$1$	$1$	$106.203$	\(\Q\)	None	✓				120.4.f.c	$1$	$0$	\(0\)	\(0\)	\(0\)	\(-4\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-4q^{7}+28q^{11}+2^{4}q^{13}+108q^{17}+\cdots\)
1800.4.a.p	$1800$	$4$	1800.a	1.a	$1$	$1$	$1$	$106.203$	\(\Q\)	None	✓				200.4.a.c	$1$	$0$	\(0\)	\(0\)	\(0\)	\(-2\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{7}-39q^{11}-84q^{13}-61q^{17}+\cdots\)
1800.4.a.q	$1800$	$4$	1800.a	1.a	$1$	$1$	$1$	$106.203$	\(\Q\)	None	✓				360.4.a.d	$1$	$1$	\(0\)	\(0\)	\(0\)	\(-2\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{7}-34q^{11}+68q^{13}+38q^{17}+\cdots\)
1800.4.a.r	$1800$	$4$	1800.a	1.a	$1$	$1$	$1$	$106.203$	\(\Q\)	None	✓				360.4.a.d	$1$	$0$	\(0\)	\(0\)	\(0\)	\(-2\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{7}+34q^{11}+68q^{13}-38q^{17}+\cdots\)
1800.4.a.s	$1800$	$4$	1800.a	1.a	$1$	$1$	$1$	$106.203$	\(\Q\)	None	✓				120.4.a.d	$1$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-4q^{11}-54q^{13}+114q^{17}+44q^{19}+\cdots\)
1800.4.a.t	$1800$	$4$	1800.a	1.a	$1$	$1$	$1$	$106.203$	\(\Q\)	None	✓				200.4.a.c	$1$	$0$	\(0\)	\(0\)	\(0\)	\(2\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{7}-39q^{11}+84q^{13}+61q^{17}+\cdots\)
1800.4.a.u	$1800$	$4$	1800.a	1.a	$1$	$1$	$1$	$106.203$	\(\Q\)	None	✓				120.4.f.c	$1$	$1$	\(0\)	\(0\)	\(0\)	\(4\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+4q^{7}+28q^{11}-2^{4}q^{13}-108q^{17}+\cdots\)
1800.4.a.v	$1800$	$4$	1800.a	1.a	$1$	$1$	$1$	$106.203$	\(\Q\)	None	✓				600.4.a.e	$1$	$1$	\(0\)	\(0\)	\(0\)	\(5\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+5q^{7}-14q^{11}+q^{13}-46q^{17}+\cdots\)
1800.4.a.w	$1800$	$4$	1800.a	1.a	$1$	$1$	$1$	$106.203$	\(\Q\)	None	✓				200.4.a.e	$1$	$1$	\(0\)	\(0\)	\(0\)	\(6\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+6q^{7}+19q^{11}-12q^{13}-75q^{17}+\cdots\)
1800.4.a.x	$1800$	$4$	1800.a	1.a	$1$	$1$	$1$	$106.203$	\(\Q\)	None	✓				120.4.f.b	$1$	$0$	\(0\)	\(0\)	\(0\)	\(10\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+10q^{7}+14q^{11}-82q^{13}-18q^{17}+\cdots\)
1800.4.a.y	$1800$	$4$	1800.a	1.a	$1$	$1$	$1$	$106.203$	\(\Q\)	None	✓				120.4.f.a	$1$	$1$	\(0\)	\(0\)	\(0\)	\(10\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+10q^{7}+46q^{11}-34q^{13}-66q^{17}+\cdots\)
1800.4.a.z	$1800$	$4$	1800.a	1.a	$1$	$1$	$1$	$106.203$	\(\Q\)	None	✓				72.4.a.a	$1$	$0$	\(0\)	\(0\)	\(0\)	\(12\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+12q^{7}-2^{6}q^{11}-58q^{13}+2^{5}q^{17}+\cdots\)
1800.4.a.ba	$1800$	$4$	1800.a	1.a	$1$	$1$	$1$	$106.203$	\(\Q\)	None	✓				72.4.a.a	$1$	$1$	\(0\)	\(0\)	\(0\)	\(12\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+12q^{7}+2^{6}q^{11}-58q^{13}-2^{5}q^{17}+\cdots\)
1800.4.a.bb	$1800$	$4$	1800.a	1.a	$1$	$1$	$1$	$106.203$	\(\Q\)	None	✓				120.4.a.c	$1$	$1$	\(0\)	\(0\)	\(0\)	\(16\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2^{4}q^{7}+28q^{11}+26q^{13}-62q^{17}+\cdots\)
1800.4.a.bc	$1800$	$4$	1800.a	1.a	$1$	$1$	$1$	$106.203$	\(\Q\)	None	✓				360.4.a.a	$1$	$1$	\(0\)	\(0\)	\(0\)	\(18\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+18q^{7}-34q^{11}-12q^{13}-102q^{17}+\cdots\)
1800.4.a.bd	$1800$	$4$	1800.a	1.a	$1$	$1$	$1$	$106.203$	\(\Q\)	None	✓				40.4.a.c	$1$	$1$	\(0\)	\(0\)	\(0\)	\(18\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+18q^{7}+2^{4}q^{11}+6q^{13}-6q^{17}+\cdots\)
1800.4.a.be	$1800$	$4$	1800.a	1.a	$1$	$1$	$1$	$106.203$	\(\Q\)	None	✓				360.4.a.a	$1$	$0$	\(0\)	\(0\)	\(0\)	\(18\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+18q^{7}+34q^{11}-12q^{13}+102q^{17}+\cdots\)
1800.4.a.bf	$1800$	$4$	1800.a	1.a	$1$	$1$	$1$	$106.203$	\(\Q\)	None	✓				600.4.a.g	$1$	$0$	\(0\)	\(0\)	\(0\)	\(19\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+19q^{7}-22q^{11}-q^{13}-58q^{17}+\cdots\)
1800.4.a.bg	$1800$	$4$	1800.a	1.a	$1$	$1$	$1$	$106.203$	\(\Q\)	None	✓				24.4.a.a	$1$	$0$	\(0\)	\(0\)	\(0\)	\(24\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+24q^{7}+28q^{11}+74q^{13}+82q^{17}+\cdots\)
1800.4.a.bh	$1800$	$4$	1800.a	1.a	$1$	$1$	$1$	$106.203$	\(\Q\)	None	✓				200.4.a.b	$1$	$0$	\(0\)	\(0\)	\(0\)	\(26\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+26q^{7}+59q^{11}+28q^{13}-5q^{17}+\cdots\)
1800.4.a.bi	$1800$	$4$	1800.a	1.a	$1$	$1$	$1$	$106.203$	\(\Q\)	None	✓				40.4.a.a	$1$	$0$	\(0\)	\(0\)	\(0\)	\(34\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+34q^{7}-2^{4}q^{11}-58q^{13}-70q^{17}+\cdots\)
1800.4.a.bj	$1800$	$4$	1800.a	1.a	$1$	$2$	$2$	$106.203$	\(\Q(\sqrt{181}) \)	None	✓				600.4.a.r	$1$	$1$	\(0\)	\(0\)	\(0\)	\(-6\)		$-$	$-$	$-$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q+(-3-\beta )q^{7}+(-4-\beta )q^{11}+(-7+\cdots)q^{13}+\cdots\)
1800.4.a.bk	$1800$	$4$	1800.a	1.a	$1$	$2$	$2$	$106.203$	\(\Q(\sqrt{6}) \)	None	✓				40.4.c.a	$1$	$1$	\(0\)	\(0\)	\(0\)	\(-4\)		$-$	$-$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(-2+3\beta )q^{7}+(-20-8\beta )q^{11}+\cdots\)
1800.4.a.bl	$1800$	$4$	1800.a	1.a	$1$	$2$	$2$	$106.203$	\(\Q(\sqrt{129}) \)	None	✓				120.4.f.d	$1$	$1$	\(0\)	\(0\)	\(0\)	\(-2\)		$-$	$-$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(-1-3\beta )q^{7}+(-37+\beta )q^{11}+(7^{2}+\cdots)q^{13}+\cdots\)
1800.4.a.bm	$1800$	$4$	1800.a	1.a	$1$	$2$	$2$	$106.203$	\(\Q(\sqrt{109}) \)	None	✓		✓		600.4.a.s	$1$	$0$	\(0\)	\(0\)	\(0\)	\(-2\)		$-$	$-$	$+$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q+(-1-\beta )q^{7}+(8-3\beta )q^{11}+(-41+\cdots)q^{13}+\cdots\)
1800.4.a.bn	$1800$	$4$	1800.a	1.a	$1$	$2$	$2$	$106.203$	\(\Q(\sqrt{129}) \)	None	✓				120.4.f.d	$1$	$0$	\(0\)	\(0\)	\(0\)	\(2\)		$+$	$-$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q+(1+3\beta )q^{7}+(-37+\beta )q^{11}+(-7^{2}+\cdots)q^{13}+\cdots\)
1800.4.a.bo	$1800$	$4$	1800.a	1.a	$1$	$2$	$2$	$106.203$	\(\Q(\sqrt{109}) \)	None	✓				600.4.a.s	$1$	$0$	\(0\)	\(0\)	\(0\)	\(2\)		$+$	$-$	$-$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{7}+(8-3\beta )q^{11}+(41+2\beta )q^{13}+\cdots\)
1800.4.a.bp	$1800$	$4$	1800.a	1.a	$1$	$2$	$2$	$106.203$	\(\Q(\sqrt{6}) \)	None	✓				40.4.c.a	$1$	$0$	\(0\)	\(0\)	\(0\)	\(4\)		$+$	$-$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q+(2+3\beta )q^{7}+(-20+8\beta )q^{11}+(52+\cdots)q^{13}+\cdots\)
1800.4.a.bq	$1800$	$4$	1800.a	1.a	$1$	$2$	$2$	$106.203$	\(\Q(\sqrt{181}) \)	None	✓		✓		600.4.a.r	$1$	$1$	\(0\)	\(0\)	\(0\)	\(6\)		$+$	$-$	$+$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q+(3+\beta )q^{7}+(-4-\beta )q^{11}+(7-2\beta )q^{13}+\cdots\)
1800.4.a.br	$1800$	$4$	1800.a	1.a	$1$	$3$	$3$	$106.203$	3.3.121909.1	None	✓	✓			1800.4.a.br	$1$	$1$	\(0\)	\(0\)	\(0\)	\(-9\)		$+$	$+$	$-$	$-$	$2^{3}$	$\mathrm{SU}(2)$	\(q+(-3+\beta _{1})q^{7}+(-3+\beta _{1}+\beta _{2})q^{11}+\cdots\)
1800.4.a.bs	$1800$	$4$	1800.a	1.a	$1$	$3$	$3$	$106.203$	3.3.121909.1	None	✓	✓			1800.4.a.br	$1$	$0$	\(0\)	\(0\)	\(0\)	\(-9\)		$-$	$+$	$-$	$+$	$2^{3}$	$\mathrm{SU}(2)$	\(q+(-3+\beta _{1})q^{7}+(3-\beta _{1}-\beta _{2})q^{11}+\cdots\)
1800.4.a.bt	$1800$	$4$	1800.a	1.a	$1$	$3$	$3$	$106.203$	3.3.121909.1	None	✓	✓	✓		1800.4.a.br	$1$	$1$	\(0\)	\(0\)	\(0\)	\(9\)		$-$	$+$	$+$	$-$	$2^{3}$	$\mathrm{SU}(2)$	\(q+(3-\beta _{1})q^{7}+(-3+\beta _{1}+\beta _{2})q^{11}+\cdots\)
1800.4.a.bu	$1800$	$4$	1800.a	1.a	$1$	$3$	$3$	$106.203$	3.3.121909.1	None	✓	✓	✓		1800.4.a.br	$1$	$0$	\(0\)	\(0\)	\(0\)	\(9\)		$+$	$+$	$+$	$+$	$2^{3}$	$\mathrm{SU}(2)$	\(q+(3-\beta _{1})q^{7}+(3-\beta _{1}-\beta _{2})q^{11}+(5+\cdots)q^{13}+\cdots\)
1800.4.a.bv	$1800$	$4$	1800.a	1.a	$1$	$4$	$4$	$106.203$	\(\Q(\sqrt{6}, \sqrt{46})\)	None	✓		✓		360.4.f.f	$2$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$+$	$+$	$-$	$-$	$2^{7}$	$\mathrm{SU}(2)$	\(q+(-\beta _{1}+\beta _{2})q^{7}+(\beta _{1}-2\beta _{2})q^{11}+\cdots\)
1800.4.a.bw	$1800$	$4$	1800.a	1.a	$1$	$4$	$4$	$106.203$	\(\Q(\sqrt{6}, \sqrt{46})\)	None	✓		✓		360.4.f.f	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$+$	$-$	$+$	$2^{7}$	$\mathrm{SU}(2)$	\(q+(-\beta _{1}+\beta _{2})q^{7}+(-\beta _{1}+2\beta _{2})q^{11}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(1800))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(1800)) \simeq \) \(S_{4}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 24}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(6))\)\(^{\oplus 18}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(8))\)\(^{\oplus 9}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(9))\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(10))\)\(^{\oplus 18}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(12))\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 16}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(18))\)\(^{\oplus 9}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(24))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(25))\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(30))\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(36))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(40))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(45))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(50))\)\(^{\oplus 9}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(60))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(72))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(75))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(90))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(100))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(120))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(150))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(180))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(200))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(225))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(300))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(360))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(450))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(600))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(900))\)\(^{\oplus 2}\)