Space of modular forms of level 1728, weight 4, and trivial character

• Label: 1728.4.a
• Level: $1728$
• Weight: $4$
• Character orbit: 1728.a
• Rep. character: $\chi_{1728}(1,\cdot)$
• Character field: $\Q$
• Dimension: $96$
• Newform subspaces: $58$
• Sturm bound: $1152$
• Trace bound: $11$

Defining parameters

Level:	\( N \)	\(=\)	\( 1728 = 2^{6} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1728.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 58 \)
Sturm bound:			\(1152\)
Trace bound:			\(11\)
Distinguishing \(T_p\):			\(5\), \(7\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(1728))\).

	Total	New	Old
Modular forms	900	96	804
Cusp forms	828	96	732
Eisenstein series	72	0	72

The following table gives the dimensions of the cuspidal new subspaces with specified eigenvalues for the Atkin-Lehner operators and the Fricke involution.

\(2\)	\(3\)	Fricke		Total				Cusp				Eisenstein
				All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)		\(228\)	\(25\)	\(203\)		\(210\)	\(25\)	\(185\)		\(18\)	\(0\)	\(18\)
\(+\)	\(-\)	\(-\)		\(222\)	\(23\)	\(199\)		\(204\)	\(23\)	\(181\)		\(18\)	\(0\)	\(18\)
\(-\)	\(+\)	\(-\)		\(222\)	\(23\)	\(199\)		\(204\)	\(23\)	\(181\)		\(18\)	\(0\)	\(18\)
\(-\)	\(-\)	\(+\)		\(228\)	\(25\)	\(203\)		\(210\)	\(25\)	\(185\)		\(18\)	\(0\)	\(18\)
Plus space		\(+\)		\(456\)	\(50\)	\(406\)		\(420\)	\(50\)	\(370\)		\(36\)	\(0\)	\(36\)
Minus space		\(-\)		\(444\)	\(46\)	\(398\)		\(408\)	\(46\)	\(362\)		\(36\)	\(0\)	\(36\)

Trace form

\( 96 q + 72 q^{13} + 2400 q^{25} - 504 q^{37} + 4704 q^{49} - 1992 q^{61} + 240 q^{85} - 1488 q^{97}+O(q^{100}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(1728))\) 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
1728.4.a.a	$1728$	$4$	1728.a	1.a	$1$	$1$	$1$	$101.955$	\(\Q\)	None	✓				864.4.a.a	$1$	$1$	\(0\)	\(0\)	\(-19\)	\(-13\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-19q^{5}-13q^{7}-65q^{11}+56q^{13}+\cdots\)
1728.4.a.b	$1728$	$4$	1728.a	1.a	$1$	$1$	$1$	$101.955$	\(\Q\)	None	✓				864.4.a.a	$1$	$0$	\(0\)	\(0\)	\(-19\)	\(13\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-19q^{5}+13q^{7}+65q^{11}+56q^{13}+\cdots\)
1728.4.a.c	$1728$	$4$	1728.a	1.a	$1$	$1$	$1$	$101.955$	\(\Q\)	None	✓				27.4.a.a	$1$	$1$	\(0\)	\(0\)	\(-15\)	\(-25\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-15q^{5}-5^{2}q^{7}+15q^{11}-20q^{13}+\cdots\)
1728.4.a.d	$1728$	$4$	1728.a	1.a	$1$	$1$	$1$	$101.955$	\(\Q\)	None	✓				27.4.a.a	$1$	$1$	\(0\)	\(0\)	\(-15\)	\(25\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-15q^{5}+5^{2}q^{7}-15q^{11}-20q^{13}+\cdots\)
1728.4.a.e	$1728$	$4$	1728.a	1.a	$1$	$1$	$1$	$101.955$	\(\Q\)	None	✓				54.4.a.a	$1$	$0$	\(0\)	\(0\)	\(-12\)	\(-7\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-12q^{5}-7q^{7}-60q^{11}+79q^{13}+\cdots\)
1728.4.a.f	$1728$	$4$	1728.a	1.a	$1$	$1$	$1$	$101.955$	\(\Q\)	None	✓				54.4.a.a	$1$	$0$	\(0\)	\(0\)	\(-12\)	\(7\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-12q^{5}+7q^{7}+60q^{11}+79q^{13}+\cdots\)
1728.4.a.g	$1728$	$4$	1728.a	1.a	$1$	$1$	$1$	$101.955$	\(\Q\)	None	✓				108.4.a.a	$1$	$0$	\(0\)	\(0\)	\(-9\)	\(-1\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-9q^{5}-q^{7}-63q^{11}+28q^{13}+72q^{17}+\cdots\)
1728.4.a.h	$1728$	$4$	1728.a	1.a	$1$	$1$	$1$	$101.955$	\(\Q\)	None	✓				108.4.a.a	$1$	$0$	\(0\)	\(0\)	\(-9\)	\(1\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-9q^{5}+q^{7}+63q^{11}+28q^{13}+72q^{17}+\cdots\)
1728.4.a.i	$1728$	$4$	1728.a	1.a	$1$	$1$	$1$	$101.955$	\(\Q\)	None	✓				216.4.a.a	$1$	$1$	\(0\)	\(0\)	\(-4\)	\(-3\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-4q^{5}-3q^{7}+28q^{11}+11q^{13}+\cdots\)
1728.4.a.j	$1728$	$4$	1728.a	1.a	$1$	$1$	$1$	$101.955$	\(\Q\)	None	✓				216.4.a.a	$1$	$1$	\(0\)	\(0\)	\(-4\)	\(3\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-4q^{5}+3q^{7}-28q^{11}+11q^{13}+\cdots\)
1728.4.a.k	$1728$	$4$	1728.a	1.a	$1$	$1$	$1$	$101.955$	\(\Q\)	None	✓				54.4.a.b	$1$	$2$	\(0\)	\(0\)	\(-3\)	\(-29\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-3q^{5}-29q^{7}-57q^{11}-20q^{13}+\cdots\)
1728.4.a.l	$1728$	$4$	1728.a	1.a	$1$	$1$	$1$	$101.955$	\(\Q\)	None	✓				54.4.a.b	$1$	$0$	\(0\)	\(0\)	\(-3\)	\(29\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-3q^{5}+29q^{7}+57q^{11}-20q^{13}+\cdots\)
1728.4.a.m	$1728$	$4$	1728.a	1.a	$1$	$1$	$1$	$101.955$	\(\Q\)	None	✓				216.4.a.b	$1$	$0$	\(0\)	\(0\)	\(-1\)	\(-9\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{5}-9q^{7}+17q^{11}+44q^{13}+56q^{17}+\cdots\)
1728.4.a.n	$1728$	$4$	1728.a	1.a	$1$	$1$	$1$	$101.955$	\(\Q\)	None	✓				216.4.a.b	$1$	$0$	\(0\)	\(0\)	\(-1\)	\(9\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{5}+9q^{7}-17q^{11}+44q^{13}+56q^{17}+\cdots\)
1728.4.a.o	$1728$	$4$	1728.a	1.a	$1$	$1$	$1$	$101.955$	\(\Q\)	\(\Q(\sqrt{-3}) \)	✓				108.4.a.b	$2$	$1$	\(0\)	\(0\)	\(0\)	\(-37\)		$+$	$-$	$-$	$1$	$N(\mathrm{U}(1))$	\(q-37q^{7}+19q^{13}+163q^{19}-5^{3}q^{25}+\cdots\)
1728.4.a.p	$1728$	$4$	1728.a	1.a	$1$	$1$	$1$	$101.955$	\(\Q\)	\(\Q(\sqrt{-3}) \)	✓				108.4.a.c	$2$	$0$	\(0\)	\(0\)	\(0\)	\(-17\)		$-$	$-$	$+$	$1$	$N(\mathrm{U}(1))$	\(q-17q^{7}-89q^{13}+107q^{19}-5^{3}q^{25}+\cdots\)
1728.4.a.q	$1728$	$4$	1728.a	1.a	$1$	$1$	$1$	$101.955$	\(\Q\)	\(\Q(\sqrt{-3}) \)	✓				108.4.a.c	$2$	$0$	\(0\)	\(0\)	\(0\)	\(17\)		$+$	$+$	$+$	$1$	$N(\mathrm{U}(1))$	\(q+17q^{7}-89q^{13}-107q^{19}-5^{3}q^{25}+\cdots\)
1728.4.a.r	$1728$	$4$	1728.a	1.a	$1$	$1$	$1$	$101.955$	\(\Q\)	\(\Q(\sqrt{-3}) \)	✓				108.4.a.b	$2$	$1$	\(0\)	\(0\)	\(0\)	\(37\)		$-$	$+$	$-$	$1$	$N(\mathrm{U}(1))$	\(q+37q^{7}+19q^{13}-163q^{19}-5^{3}q^{25}+\cdots\)
1728.4.a.s	$1728$	$4$	1728.a	1.a	$1$	$1$	$1$	$101.955$	\(\Q\)	None	✓				216.4.a.b	$1$	$1$	\(0\)	\(0\)	\(1\)	\(-9\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{5}-9q^{7}-17q^{11}+44q^{13}-56q^{17}+\cdots\)
1728.4.a.t	$1728$	$4$	1728.a	1.a	$1$	$1$	$1$	$101.955$	\(\Q\)	None	✓				216.4.a.b	$1$	$1$	\(0\)	\(0\)	\(1\)	\(9\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{5}+9q^{7}+17q^{11}+44q^{13}-56q^{17}+\cdots\)
1728.4.a.u	$1728$	$4$	1728.a	1.a	$1$	$1$	$1$	$101.955$	\(\Q\)	None	✓				54.4.a.b	$1$	$1$	\(0\)	\(0\)	\(3\)	\(-29\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3q^{5}-29q^{7}+57q^{11}-20q^{13}+\cdots\)
1728.4.a.v	$1728$	$4$	1728.a	1.a	$1$	$1$	$1$	$101.955$	\(\Q\)	None	✓				54.4.a.b	$1$	$1$	\(0\)	\(0\)	\(3\)	\(29\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3q^{5}+29q^{7}-57q^{11}-20q^{13}+\cdots\)
1728.4.a.w	$1728$	$4$	1728.a	1.a	$1$	$1$	$1$	$101.955$	\(\Q\)	None	✓				216.4.a.a	$1$	$1$	\(0\)	\(0\)	\(4\)	\(-3\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+4q^{5}-3q^{7}-28q^{11}+11q^{13}+\cdots\)
1728.4.a.x	$1728$	$4$	1728.a	1.a	$1$	$1$	$1$	$101.955$	\(\Q\)	None	✓				216.4.a.a	$1$	$1$	\(0\)	\(0\)	\(4\)	\(3\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+4q^{5}+3q^{7}+28q^{11}+11q^{13}+\cdots\)
1728.4.a.y	$1728$	$4$	1728.a	1.a	$1$	$1$	$1$	$101.955$	\(\Q\)	None	✓				108.4.a.a	$1$	$1$	\(0\)	\(0\)	\(9\)	\(-1\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+9q^{5}-q^{7}+63q^{11}+28q^{13}-72q^{17}+\cdots\)
1728.4.a.z	$1728$	$4$	1728.a	1.a	$1$	$1$	$1$	$101.955$	\(\Q\)	None	✓				108.4.a.a	$1$	$1$	\(0\)	\(0\)	\(9\)	\(1\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+9q^{5}+q^{7}-63q^{11}+28q^{13}-72q^{17}+\cdots\)
1728.4.a.ba	$1728$	$4$	1728.a	1.a	$1$	$1$	$1$	$101.955$	\(\Q\)	None	✓				54.4.a.a	$1$	$0$	\(0\)	\(0\)	\(12\)	\(-7\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+12q^{5}-7q^{7}+60q^{11}+79q^{13}+\cdots\)
1728.4.a.bb	$1728$	$4$	1728.a	1.a	$1$	$1$	$1$	$101.955$	\(\Q\)	None	✓				54.4.a.a	$1$	$0$	\(0\)	\(0\)	\(12\)	\(7\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+12q^{5}+7q^{7}-60q^{11}+79q^{13}+\cdots\)
1728.4.a.bc	$1728$	$4$	1728.a	1.a	$1$	$1$	$1$	$101.955$	\(\Q\)	None	✓				27.4.a.a	$1$	$0$	\(0\)	\(0\)	\(15\)	\(-25\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+15q^{5}-5^{2}q^{7}-15q^{11}-20q^{13}+\cdots\)
1728.4.a.bd	$1728$	$4$	1728.a	1.a	$1$	$1$	$1$	$101.955$	\(\Q\)	None	✓				27.4.a.a	$1$	$0$	\(0\)	\(0\)	\(15\)	\(25\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+15q^{5}+5^{2}q^{7}+15q^{11}-20q^{13}+\cdots\)
1728.4.a.be	$1728$	$4$	1728.a	1.a	$1$	$1$	$1$	$101.955$	\(\Q\)	None	✓				864.4.a.a	$1$	$0$	\(0\)	\(0\)	\(19\)	\(-13\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+19q^{5}-13q^{7}+65q^{11}+56q^{13}+\cdots\)
1728.4.a.bf	$1728$	$4$	1728.a	1.a	$1$	$1$	$1$	$101.955$	\(\Q\)	None	✓				864.4.a.a	$1$	$1$	\(0\)	\(0\)	\(19\)	\(13\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+19q^{5}+13q^{7}-65q^{11}+56q^{13}+\cdots\)
1728.4.a.bg	$1728$	$4$	1728.a	1.a	$1$	$2$	$2$	$101.955$	\(\Q(\sqrt{33}) \)	None	✓				216.4.a.e	$1$	$1$	\(0\)	\(0\)	\(-8\)	\(-24\)		$-$	$+$	$-$	$2\cdot 3$	$\mathrm{SU}(2)$	\(q+(-4-\beta )q^{5}+(-12-\beta )q^{7}+q^{11}+\cdots\)
1728.4.a.bh	$1728$	$4$	1728.a	1.a	$1$	$2$	$2$	$101.955$	\(\Q(\sqrt{33}) \)	None	✓				216.4.a.e	$1$	$1$	\(0\)	\(0\)	\(-8\)	\(24\)		$+$	$-$	$-$	$2\cdot 3$	$\mathrm{SU}(2)$	\(q+(-4-\beta )q^{5}+(12+\beta )q^{7}-q^{11}+\cdots\)
1728.4.a.bi	$1728$	$4$	1728.a	1.a	$1$	$2$	$2$	$101.955$	\(\Q(\sqrt{5}) \)	None	✓				216.4.a.f	$1$	$0$	\(0\)	\(0\)	\(-4\)	\(-6\)		$+$	$+$	$+$	$2^{2}\cdot 3$	$\mathrm{SU}(2)$	\(q+(-2-\beta )q^{5}+(-3+2\beta )q^{7}+(-26+\cdots)q^{11}+\cdots\)
1728.4.a.bj	$1728$	$4$	1728.a	1.a	$1$	$2$	$2$	$101.955$	\(\Q(\sqrt{5}) \)	None	✓				216.4.a.f	$1$	$0$	\(0\)	\(0\)	\(-4\)	\(6\)		$-$	$-$	$+$	$2^{2}\cdot 3$	$\mathrm{SU}(2)$	\(q+(-2-\beta )q^{5}+(3-2\beta )q^{7}+(26-3\beta )q^{11}+\cdots\)
1728.4.a.bk	$1728$	$4$	1728.a	1.a	$1$	$2$	$2$	$101.955$	\(\Q(\sqrt{2}) \)	None	✓				27.4.a.c	$2$	$1$	\(0\)	\(0\)	\(0\)	\(-22\)		$-$	$+$	$-$	$2^{2}\cdot 3$	$\mathrm{SU}(2)$	\(q+\beta q^{5}-11q^{7}+\beta q^{11}-29q^{13}+\cdots\)
1728.4.a.bl	$1728$	$4$	1728.a	1.a	$1$	$2$	$2$	$101.955$	\(\Q(\sqrt{37}) \)	None	✓				864.4.a.e	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$+$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q-\beta q^{5}-5\beta q^{7}-43q^{11}-52q^{13}+\cdots\)
1728.4.a.bm	$1728$	$4$	1728.a	1.a	$1$	$2$	$2$	$101.955$	\(\Q(\sqrt{73}) \)	None	✓				864.4.a.f	$2$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$+$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q-\beta q^{5}+\beta q^{7}-q^{11}+8q^{13}-12\beta q^{17}+\cdots\)
1728.4.a.bn	$1728$	$4$	1728.a	1.a	$1$	$2$	$2$	$101.955$	\(\Q(\sqrt{73}) \)	None	✓				864.4.a.f	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$+$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q-\beta q^{5}-\beta q^{7}+q^{11}+8q^{13}-12\beta q^{17}+\cdots\)
1728.4.a.bo	$1728$	$4$	1728.a	1.a	$1$	$2$	$2$	$101.955$	\(\Q(\sqrt{37}) \)	None	✓				864.4.a.e	$2$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$+$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q-\beta q^{5}+5\beta q^{7}+43q^{11}-52q^{13}+\cdots\)
1728.4.a.bp	$1728$	$4$	1728.a	1.a	$1$	$2$	$2$	$101.955$	\(\Q(\sqrt{2}) \)	None	✓				27.4.a.c	$2$	$1$	\(0\)	\(0\)	\(0\)	\(22\)		$+$	$-$	$-$	$2^{2}\cdot 3$	$\mathrm{SU}(2)$	\(q+\beta q^{5}+11q^{7}-\beta q^{11}-29q^{13}+\cdots\)
1728.4.a.bq	$1728$	$4$	1728.a	1.a	$1$	$2$	$2$	$101.955$	\(\Q(\sqrt{5}) \)	None	✓				216.4.a.f	$1$	$0$	\(0\)	\(0\)	\(4\)	\(-6\)		$+$	$+$	$+$	$2^{2}\cdot 3$	$\mathrm{SU}(2)$	\(q+(2+\beta )q^{5}+(-3+2\beta )q^{7}+(26-3\beta )q^{11}+\cdots\)
1728.4.a.br	$1728$	$4$	1728.a	1.a	$1$	$2$	$2$	$101.955$	\(\Q(\sqrt{5}) \)	None	✓				216.4.a.f	$1$	$0$	\(0\)	\(0\)	\(4\)	\(6\)		$-$	$-$	$+$	$2^{2}\cdot 3$	$\mathrm{SU}(2)$	\(q+(2+\beta )q^{5}+(3-2\beta )q^{7}+(-26+3\beta )q^{11}+\cdots\)
1728.4.a.bs	$1728$	$4$	1728.a	1.a	$1$	$2$	$2$	$101.955$	\(\Q(\sqrt{33}) \)	None	✓				216.4.a.e	$1$	$0$	\(0\)	\(0\)	\(8\)	\(-24\)		$-$	$-$	$+$	$2\cdot 3$	$\mathrm{SU}(2)$	\(q+(4+\beta )q^{5}+(-12-\beta )q^{7}-q^{11}+\cdots\)
1728.4.a.bt	$1728$	$4$	1728.a	1.a	$1$	$2$	$2$	$101.955$	\(\Q(\sqrt{33}) \)	None	✓				216.4.a.e	$1$	$0$	\(0\)	\(0\)	\(8\)	\(24\)		$+$	$+$	$+$	$2\cdot 3$	$\mathrm{SU}(2)$	\(q+(4+\beta )q^{5}+(12+\beta )q^{7}+q^{11}+(-2^{4}+\cdots)q^{13}+\cdots\)
1728.4.a.bu	$1728$	$4$	1728.a	1.a	$1$	$3$	$3$	$101.955$	3.3.2708.1	None	✓				864.4.a.i	$1$	$1$	\(0\)	\(0\)	\(-10\)	\(-19\)		$+$	$-$	$-$	$2^{3}\cdot 3$	$\mathrm{SU}(2)$	\(q+(-3-\beta _{1})q^{5}+(-6-\beta _{2})q^{7}+(13+\cdots)q^{11}+\cdots\)
1728.4.a.bv	$1728$	$4$	1728.a	1.a	$1$	$3$	$3$	$101.955$	3.3.2708.1	None	✓				864.4.a.i	$1$	$0$	\(0\)	\(0\)	\(-10\)	\(19\)		$+$	$+$	$+$	$2^{3}\cdot 3$	$\mathrm{SU}(2)$	\(q+(-3-\beta _{1})q^{5}+(6+\beta _{2})q^{7}+(-13+\cdots)q^{11}+\cdots\)
1728.4.a.bw	$1728$	$4$	1728.a	1.a	$1$	$3$	$3$	$101.955$	3.3.148.1	None	✓				864.4.a.k	$1$	$1$	\(0\)	\(0\)	\(-6\)	\(-9\)		$-$	$+$	$-$	$2^{3}\cdot 3^{2}$	$\mathrm{SU}(2)$	\(q+(-2+\beta _{2})q^{5}+(-3-\beta _{1}+\beta _{2})q^{7}+\cdots\)
1728.4.a.bx	$1728$	$4$	1728.a	1.a	$1$	$3$	$3$	$101.955$	3.3.148.1	None	✓				864.4.a.k	$1$	$0$	\(0\)	\(0\)	\(-6\)	\(9\)		$-$	$-$	$+$	$2^{3}\cdot 3^{2}$	$\mathrm{SU}(2)$	\(q+(-2+\beta _{2})q^{5}+(3+\beta _{1}-\beta _{2})q^{7}+\cdots\)
1728.4.a.by	$1728$	$4$	1728.a	1.a	$1$	$3$	$3$	$101.955$	3.3.229.1	None	✓				864.4.a.m	$1$	$1$	\(0\)	\(0\)	\(-3\)	\(-3\)		$-$	$+$	$-$	$2^{3}\cdot 3^{2}$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{5}+(-1+\beta _{2})q^{7}+(-7+\cdots)q^{11}+\cdots\)
1728.4.a.bz	$1728$	$4$	1728.a	1.a	$1$	$3$	$3$	$101.955$	3.3.229.1	None	✓				864.4.a.m	$1$	$0$	\(0\)	\(0\)	\(-3\)	\(3\)		$-$	$-$	$+$	$2^{3}\cdot 3^{2}$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{5}+(1-\beta _{2})q^{7}+(7-3\beta _{1}+\cdots)q^{11}+\cdots\)
1728.4.a.ca	$1728$	$4$	1728.a	1.a	$1$	$3$	$3$	$101.955$	3.3.229.1	None	✓				864.4.a.m	$1$	$0$	\(0\)	\(0\)	\(3\)	\(-3\)		$-$	$-$	$+$	$2^{3}\cdot 3^{2}$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{5}+(-1+\beta _{2})q^{7}+(7-3\beta _{1}+\cdots)q^{11}+\cdots\)
1728.4.a.cb	$1728$	$4$	1728.a	1.a	$1$	$3$	$3$	$101.955$	3.3.229.1	None	✓				864.4.a.m	$1$	$1$	\(0\)	\(0\)	\(3\)	\(3\)		$-$	$+$	$-$	$2^{3}\cdot 3^{2}$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{5}+(1-\beta _{2})q^{7}+(-7+3\beta _{1}+\cdots)q^{11}+\cdots\)
1728.4.a.cc	$1728$	$4$	1728.a	1.a	$1$	$3$	$3$	$101.955$	3.3.148.1	None	✓				864.4.a.k	$1$	$1$	\(0\)	\(0\)	\(6\)	\(-9\)		$-$	$+$	$-$	$2^{3}\cdot 3^{2}$	$\mathrm{SU}(2)$	\(q+(2-\beta _{2})q^{5}+(-3-\beta _{1}+\beta _{2})q^{7}+\cdots\)
1728.4.a.cd	$1728$	$4$	1728.a	1.a	$1$	$3$	$3$	$101.955$	3.3.148.1	None	✓				864.4.a.k	$1$	$0$	\(0\)	\(0\)	\(6\)	\(9\)		$-$	$-$	$+$	$2^{3}\cdot 3^{2}$	$\mathrm{SU}(2)$	\(q+(2-\beta _{2})q^{5}+(3+\beta _{1}-\beta _{2})q^{7}+(6+\cdots)q^{11}+\cdots\)
1728.4.a.ce	$1728$	$4$	1728.a	1.a	$1$	$3$	$3$	$101.955$	3.3.2708.1	None	✓				864.4.a.i	$1$	$1$	\(0\)	\(0\)	\(10\)	\(-19\)		$+$	$-$	$-$	$2^{3}\cdot 3$	$\mathrm{SU}(2)$	\(q+(3+\beta _{1})q^{5}+(-6-\beta _{2})q^{7}+(-13+\cdots)q^{11}+\cdots\)
1728.4.a.cf	$1728$	$4$	1728.a	1.a	$1$	$3$	$3$	$101.955$	3.3.2708.1	None	✓				864.4.a.i	$1$	$0$	\(0\)	\(0\)	\(10\)	\(19\)		$+$	$+$	$+$	$2^{3}\cdot 3$	$\mathrm{SU}(2)$	\(q+(3+\beta _{1})q^{5}+(6+\beta _{2})q^{7}+(13+\beta _{1}+\cdots)q^{11}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(1728))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(1728)) \simeq \) \(S_{4}^{\mathrm{new}}(\Gamma_0(6))\)\(^{\oplus 18}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(8))\)\(^{\oplus 16}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(9))\)\(^{\oplus 14}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(12))\)\(^{\oplus 15}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(16))\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(18))\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(24))\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(27))\)\(^{\oplus 7}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(32))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(36))\)\(^{\oplus 10}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(48))\)\(^{\oplus 9}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(54))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(64))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(72))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(96))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(108))\)\(^{\oplus 5}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(144))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(192))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(216))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(288))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(432))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(576))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(864))\)\(^{\oplus 2}\)