Space of modular forms of level 5400, weight 2, and trivial character

• Label: 5400.2.a
• Level: $5400$
• Weight: $2$
• Character orbit: 5400.a
• Rep. character: $\chi_{5400}(1,\cdot)$
• Character field: $\Q$
• Dimension: $76$
• Newform subspaces: $60$
• Sturm bound: $2160$
• Trace bound: $17$

Defining parameters

Level:	\( N \)	\(=\)	\( 5400 = 2^{3} \cdot 3^{3} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5400.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 60 \)
Sturm bound:			\(2160\)
Trace bound:			\(17\)
Distinguishing \(T_p\):			\(7\), \(11\), \(13\), \(17\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(5400))\).

	Total	New	Old
Modular forms	1152	76	1076
Cusp forms	1009	76	933
Eisenstein series	143	0	143

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
\(+\)	\(+\)	\(+\)	\(+\)		\(138\)	\(10\)	\(128\)		\(121\)	\(10\)	\(111\)		\(17\)	\(0\)	\(17\)
\(+\)	\(+\)	\(-\)	\(-\)		\(147\)	\(10\)	\(137\)		\(129\)	\(10\)	\(119\)		\(18\)	\(0\)	\(18\)
\(+\)	\(-\)	\(+\)	\(-\)		\(147\)	\(8\)	\(139\)		\(129\)	\(8\)	\(121\)		\(18\)	\(0\)	\(18\)
\(+\)	\(-\)	\(-\)	\(+\)		\(144\)	\(10\)	\(134\)		\(126\)	\(10\)	\(116\)		\(18\)	\(0\)	\(18\)
\(-\)	\(+\)	\(+\)	\(-\)		\(150\)	\(11\)	\(139\)		\(132\)	\(11\)	\(121\)		\(18\)	\(0\)	\(18\)
\(-\)	\(+\)	\(-\)	\(+\)		\(141\)	\(8\)	\(133\)		\(123\)	\(8\)	\(115\)		\(18\)	\(0\)	\(18\)
\(-\)	\(-\)	\(+\)	\(+\)		\(141\)	\(7\)	\(134\)		\(123\)	\(7\)	\(116\)		\(18\)	\(0\)	\(18\)
\(-\)	\(-\)	\(-\)	\(-\)		\(144\)	\(12\)	\(132\)		\(126\)	\(12\)	\(114\)		\(18\)	\(0\)	\(18\)
Plus space			\(+\)		\(564\)	\(35\)	\(529\)		\(493\)	\(35\)	\(458\)		\(71\)	\(0\)	\(71\)
Minus space			\(-\)		\(588\)	\(41\)	\(547\)		\(516\)	\(41\)	\(475\)		\(72\)	\(0\)	\(72\)

Trace form

\( 76 q + 4 q^{7} - 6 q^{13} + 2 q^{19} - 10 q^{31} + 26 q^{37} - 20 q^{43} + 68 q^{49} + 22 q^{61} - 6 q^{67} - 8 q^{73} + 10 q^{79} - 18 q^{91} - 20 q^{97}+O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(5400))\) 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
5400.2.a.a	$5400$	$2$	5400.a	1.a	$1$	$1$	$1$	$43.119$	\(\Q\)	None	✓	✓			5400.2.a.a	$1$	$0$	\(0\)	\(0\)	\(0\)	\(-4\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-4q^{7}-4q^{11}+q^{13}-8q^{17}-8q^{19}+\cdots\)
5400.2.a.b	$5400$	$2$	5400.a	1.a	$1$	$1$	$1$	$43.119$	\(\Q\)	None	✓				1080.2.f.b	$1$	$1$	\(0\)	\(0\)	\(0\)	\(-4\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-4q^{7}-q^{11}+q^{13}+3q^{17}+2q^{19}+\cdots\)
5400.2.a.c	$5400$	$2$	5400.a	1.a	$1$	$1$	$1$	$43.119$	\(\Q\)	None	✓				1080.2.f.b	$1$	$0$	\(0\)	\(0\)	\(0\)	\(-4\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-4q^{7}+q^{11}+q^{13}-3q^{17}+2q^{19}+\cdots\)
5400.2.a.d	$5400$	$2$	5400.a	1.a	$1$	$1$	$1$	$43.119$	\(\Q\)	None	✓	✓			5400.2.a.a	$1$	$1$	\(0\)	\(0\)	\(0\)	\(-4\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-4q^{7}+4q^{11}+q^{13}+8q^{17}-8q^{19}+\cdots\)
5400.2.a.e	$5400$	$2$	5400.a	1.a	$1$	$1$	$1$	$43.119$	\(\Q\)	None	✓				216.2.a.b	$1$	$0$	\(0\)	\(0\)	\(0\)	\(-3\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-3q^{7}-5q^{11}-4q^{13}-8q^{17}+2q^{19}+\cdots\)
5400.2.a.f	$5400$	$2$	5400.a	1.a	$1$	$1$	$1$	$43.119$	\(\Q\)	None	✓	✓			5400.2.a.f	$1$	$1$	\(0\)	\(0\)	\(0\)	\(-3\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-3q^{7}-4q^{11}+6q^{13}-6q^{17}-q^{19}+\cdots\)
5400.2.a.g	$5400$	$2$	5400.a	1.a	$1$	$1$	$1$	$43.119$	\(\Q\)	None	✓	✓			5400.2.a.f	$1$	$0$	\(0\)	\(0\)	\(0\)	\(-3\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-3q^{7}+4q^{11}+6q^{13}+6q^{17}-q^{19}+\cdots\)
5400.2.a.h	$5400$	$2$	5400.a	1.a	$1$	$1$	$1$	$43.119$	\(\Q\)	None	✓				216.2.a.b	$1$	$0$	\(0\)	\(0\)	\(0\)	\(-3\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-3q^{7}+5q^{11}-4q^{13}+8q^{17}+2q^{19}+\cdots\)
5400.2.a.i	$5400$	$2$	5400.a	1.a	$1$	$1$	$1$	$43.119$	\(\Q\)	None	✓	✓			5400.2.a.i	$1$	$0$	\(0\)	\(0\)	\(0\)	\(-2\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{7}-5q^{11}-q^{13}-2q^{17}-8q^{19}+\cdots\)
5400.2.a.j	$5400$	$2$	5400.a	1.a	$1$	$1$	$1$	$43.119$	\(\Q\)	None	✓				1080.2.a.e	$1$	$1$	\(0\)	\(0\)	\(0\)	\(-2\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{7}-4q^{11}+2q^{13}+5q^{17}-5q^{19}+\cdots\)
5400.2.a.k	$5400$	$2$	5400.a	1.a	$1$	$1$	$1$	$43.119$	\(\Q\)	None	✓	✓			5400.2.a.k	$1$	$1$	\(0\)	\(0\)	\(0\)	\(-2\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{7}-3q^{11}-3q^{13}+2q^{17}+4q^{19}+\cdots\)
5400.2.a.l	$5400$	$2$	5400.a	1.a	$1$	$1$	$1$	$43.119$	\(\Q\)	None	✓				1080.2.f.a	$1$	$1$	\(0\)	\(0\)	\(0\)	\(-2\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{7}-2q^{11}+2q^{13}+3q^{17}-q^{19}+\cdots\)
5400.2.a.m	$5400$	$2$	5400.a	1.a	$1$	$1$	$1$	$43.119$	\(\Q\)	None	✓				1080.2.a.f	$1$	$0$	\(0\)	\(0\)	\(0\)	\(-2\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{7}-q^{11}-q^{13}-q^{17}+4q^{19}+\cdots\)
5400.2.a.n	$5400$	$2$	5400.a	1.a	$1$	$1$	$1$	$43.119$	\(\Q\)	None	✓				1080.2.a.f	$1$	$1$	\(0\)	\(0\)	\(0\)	\(-2\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{7}+q^{11}-q^{13}+q^{17}+4q^{19}+\cdots\)
5400.2.a.o	$5400$	$2$	5400.a	1.a	$1$	$1$	$1$	$43.119$	\(\Q\)	None	✓				1080.2.f.a	$1$	$1$	\(0\)	\(0\)	\(0\)	\(-2\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{7}+2q^{11}+2q^{13}-3q^{17}-q^{19}+\cdots\)
5400.2.a.p	$5400$	$2$	5400.a	1.a	$1$	$1$	$1$	$43.119$	\(\Q\)	None	✓	✓			5400.2.a.k	$1$	$1$	\(0\)	\(0\)	\(0\)	\(-2\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{7}+3q^{11}-3q^{13}-2q^{17}+4q^{19}+\cdots\)
5400.2.a.q	$5400$	$2$	5400.a	1.a	$1$	$1$	$1$	$43.119$	\(\Q\)	None	✓				1080.2.a.e	$1$	$1$	\(0\)	\(0\)	\(0\)	\(-2\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{7}+4q^{11}+2q^{13}-5q^{17}-5q^{19}+\cdots\)
5400.2.a.r	$5400$	$2$	5400.a	1.a	$1$	$1$	$1$	$43.119$	\(\Q\)	None	✓	✓			5400.2.a.i	$1$	$0$	\(0\)	\(0\)	\(0\)	\(-2\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{7}+5q^{11}-q^{13}+2q^{17}-8q^{19}+\cdots\)
5400.2.a.s	$5400$	$2$	5400.a	1.a	$1$	$1$	$1$	$43.119$	\(\Q\)	None	✓	✓			5400.2.a.s	$1$	$0$	\(0\)	\(0\)	\(0\)	\(-1\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{7}-6q^{11}-6q^{13}+2q^{17}+7q^{19}+\cdots\)
5400.2.a.t	$5400$	$2$	5400.a	1.a	$1$	$1$	$1$	$43.119$	\(\Q\)	None	✓	✓			5400.2.a.t	$1$	$0$	\(0\)	\(0\)	\(0\)	\(-1\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{7}-2q^{11}-2q^{13}+2q^{17}+q^{19}+\cdots\)
5400.2.a.u	$5400$	$2$	5400.a	1.a	$1$	$1$	$1$	$43.119$	\(\Q\)	None	✓	✓			5400.2.a.t	$1$	$1$	\(0\)	\(0\)	\(0\)	\(-1\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{7}+2q^{11}-2q^{13}-2q^{17}+q^{19}+\cdots\)
5400.2.a.v	$5400$	$2$	5400.a	1.a	$1$	$1$	$1$	$43.119$	\(\Q\)	None	✓	✓			5400.2.a.s	$1$	$1$	\(0\)	\(0\)	\(0\)	\(-1\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{7}+6q^{11}-6q^{13}-2q^{17}+7q^{19}+\cdots\)
5400.2.a.w	$5400$	$2$	5400.a	1.a	$1$	$1$	$1$	$43.119$	\(\Q\)	None	✓				1080.2.a.d	$1$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{11}+3q^{17}-q^{19}+3q^{23}-4q^{29}+\cdots\)
5400.2.a.x	$5400$	$2$	5400.a	1.a	$1$	$1$	$1$	$43.119$	\(\Q\)	None	✓				1080.2.a.d	$1$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{11}-3q^{17}-q^{19}-3q^{23}+4q^{29}+\cdots\)
5400.2.a.y	$5400$	$2$	5400.a	1.a	$1$	$1$	$1$	$43.119$	\(\Q\)	None	✓	✓			5400.2.a.s	$1$	$1$	\(0\)	\(0\)	\(0\)	\(1\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{7}-6q^{11}+6q^{13}-2q^{17}+7q^{19}+\cdots\)
5400.2.a.z	$5400$	$2$	5400.a	1.a	$1$	$1$	$1$	$43.119$	\(\Q\)	None	✓	✓			5400.2.a.t	$1$	$1$	\(0\)	\(0\)	\(0\)	\(1\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{7}-2q^{11}+2q^{13}-2q^{17}+q^{19}+\cdots\)
5400.2.a.ba	$5400$	$2$	5400.a	1.a	$1$	$1$	$1$	$43.119$	\(\Q\)	None	✓				1080.2.a.c	$1$	$0$	\(0\)	\(0\)	\(0\)	\(1\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{7}-2q^{11}+5q^{13}+4q^{17}-5q^{19}+\cdots\)
5400.2.a.bb	$5400$	$2$	5400.a	1.a	$1$	$1$	$1$	$43.119$	\(\Q\)	None	✓	✓			5400.2.a.t	$1$	$0$	\(0\)	\(0\)	\(0\)	\(1\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{7}+2q^{11}+2q^{13}+2q^{17}+q^{19}+\cdots\)
5400.2.a.bc	$5400$	$2$	5400.a	1.a	$1$	$1$	$1$	$43.119$	\(\Q\)	None	✓				1080.2.a.c	$1$	$1$	\(0\)	\(0\)	\(0\)	\(1\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{7}+2q^{11}+5q^{13}-4q^{17}-5q^{19}+\cdots\)
5400.2.a.bd	$5400$	$2$	5400.a	1.a	$1$	$1$	$1$	$43.119$	\(\Q\)	None	✓	✓			5400.2.a.s	$1$	$0$	\(0\)	\(0\)	\(0\)	\(1\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{7}+6q^{11}+6q^{13}+2q^{17}+7q^{19}+\cdots\)
5400.2.a.be	$5400$	$2$	5400.a	1.a	$1$	$1$	$1$	$43.119$	\(\Q\)	None	✓	✓			5400.2.a.i	$1$	$1$	\(0\)	\(0\)	\(0\)	\(2\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{7}-5q^{11}+q^{13}+2q^{17}-8q^{19}+\cdots\)
5400.2.a.bf	$5400$	$2$	5400.a	1.a	$1$	$1$	$1$	$43.119$	\(\Q\)	None	✓	✓			5400.2.a.k	$1$	$0$	\(0\)	\(0\)	\(0\)	\(2\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{7}-3q^{11}+3q^{13}-2q^{17}+4q^{19}+\cdots\)
5400.2.a.bg	$5400$	$2$	5400.a	1.a	$1$	$1$	$1$	$43.119$	\(\Q\)	None	✓				1080.2.f.a	$1$	$1$	\(0\)	\(0\)	\(0\)	\(2\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{7}-2q^{11}-2q^{13}-3q^{17}-q^{19}+\cdots\)
5400.2.a.bh	$5400$	$2$	5400.a	1.a	$1$	$1$	$1$	$43.119$	\(\Q\)	None	✓				1080.2.a.b	$1$	$0$	\(0\)	\(0\)	\(0\)	\(2\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{7}+6q^{13}-7q^{17}+7q^{19}-7q^{23}+\cdots\)
5400.2.a.bi	$5400$	$2$	5400.a	1.a	$1$	$1$	$1$	$43.119$	\(\Q\)	None	✓				1080.2.a.b	$1$	$0$	\(0\)	\(0\)	\(0\)	\(2\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{7}+6q^{13}+7q^{17}+7q^{19}+7q^{23}+\cdots\)
5400.2.a.bj	$5400$	$2$	5400.a	1.a	$1$	$1$	$1$	$43.119$	\(\Q\)	None	✓				1080.2.f.a	$1$	$1$	\(0\)	\(0\)	\(0\)	\(2\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{7}+2q^{11}-2q^{13}+3q^{17}-q^{19}+\cdots\)
5400.2.a.bk	$5400$	$2$	5400.a	1.a	$1$	$1$	$1$	$43.119$	\(\Q\)	None	✓	✓			5400.2.a.k	$1$	$0$	\(0\)	\(0\)	\(0\)	\(2\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{7}+3q^{11}+3q^{13}+2q^{17}+4q^{19}+\cdots\)
5400.2.a.bl	$5400$	$2$	5400.a	1.a	$1$	$1$	$1$	$43.119$	\(\Q\)	None	✓	✓			5400.2.a.i	$1$	$1$	\(0\)	\(0\)	\(0\)	\(2\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{7}+5q^{11}+q^{13}-2q^{17}-8q^{19}+\cdots\)
5400.2.a.bm	$5400$	$2$	5400.a	1.a	$1$	$1$	$1$	$43.119$	\(\Q\)	None	✓	✓			5400.2.a.f	$1$	$0$	\(0\)	\(0\)	\(0\)	\(3\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3q^{7}-4q^{11}-6q^{13}+6q^{17}-q^{19}+\cdots\)
5400.2.a.bn	$5400$	$2$	5400.a	1.a	$1$	$1$	$1$	$43.119$	\(\Q\)	None	✓				216.2.a.a	$1$	$1$	\(0\)	\(0\)	\(0\)	\(3\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+3q^{7}-4q^{11}-q^{13}-4q^{17}-q^{19}+\cdots\)
5400.2.a.bo	$5400$	$2$	5400.a	1.a	$1$	$1$	$1$	$43.119$	\(\Q\)	None	✓	✓			5400.2.a.f	$1$	$1$	\(0\)	\(0\)	\(0\)	\(3\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+3q^{7}+4q^{11}-6q^{13}-6q^{17}-q^{19}+\cdots\)
5400.2.a.bp	$5400$	$2$	5400.a	1.a	$1$	$1$	$1$	$43.119$	\(\Q\)	None	✓				216.2.a.a	$1$	$0$	\(0\)	\(0\)	\(0\)	\(3\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3q^{7}+4q^{11}-q^{13}+4q^{17}-q^{19}+\cdots\)
5400.2.a.bq	$5400$	$2$	5400.a	1.a	$1$	$1$	$1$	$43.119$	\(\Q\)	None	✓	✓			5400.2.a.a	$1$	$1$	\(0\)	\(0\)	\(0\)	\(4\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+4q^{7}-4q^{11}-q^{13}+8q^{17}-8q^{19}+\cdots\)
5400.2.a.br	$5400$	$2$	5400.a	1.a	$1$	$1$	$1$	$43.119$	\(\Q\)	None	✓				1080.2.a.a	$1$	$0$	\(0\)	\(0\)	\(0\)	\(4\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+4q^{7}-2q^{11}-4q^{13}+q^{17}-5q^{19}+\cdots\)
5400.2.a.bs	$5400$	$2$	5400.a	1.a	$1$	$1$	$1$	$43.119$	\(\Q\)	None	✓				1080.2.f.b	$1$	$1$	\(0\)	\(0\)	\(0\)	\(4\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+4q^{7}-q^{11}-q^{13}-3q^{17}+2q^{19}+\cdots\)
5400.2.a.bt	$5400$	$2$	5400.a	1.a	$1$	$1$	$1$	$43.119$	\(\Q\)	None	✓				1080.2.f.b	$1$	$0$	\(0\)	\(0\)	\(0\)	\(4\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+4q^{7}+q^{11}-q^{13}+3q^{17}+2q^{19}+\cdots\)
5400.2.a.bu	$5400$	$2$	5400.a	1.a	$1$	$1$	$1$	$43.119$	\(\Q\)	None	✓				1080.2.a.a	$1$	$0$	\(0\)	\(0\)	\(0\)	\(4\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+4q^{7}+2q^{11}-4q^{13}-q^{17}-5q^{19}+\cdots\)
5400.2.a.bv	$5400$	$2$	5400.a	1.a	$1$	$1$	$1$	$43.119$	\(\Q\)	None	✓	✓			5400.2.a.a	$1$	$0$	\(0\)	\(0\)	\(0\)	\(4\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+4q^{7}+4q^{11}-q^{13}-8q^{17}-8q^{19}+\cdots\)
5400.2.a.bw	$5400$	$2$	5400.a	1.a	$1$	$2$	$2$	$43.119$	\(\Q(\sqrt{6}) \)	None	✓		✓		1080.2.f.e	$1$	$1$	\(0\)	\(0\)	\(0\)	\(-2\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{7}+(-2-\beta )q^{11}+(1+2\beta )q^{13}+\cdots\)
5400.2.a.bx	$5400$	$2$	5400.a	1.a	$1$	$2$	$2$	$43.119$	\(\Q(\sqrt{13}) \)	None	✓	✓			5400.2.a.bx	$1$	$1$	\(0\)	\(0\)	\(0\)	\(-2\)		$+$	$+$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q+(-1-\beta )q^{7}-q^{11}+(-2+\beta )q^{13}+\cdots\)
5400.2.a.by	$5400$	$2$	5400.a	1.a	$1$	$2$	$2$	$43.119$	\(\Q(\sqrt{13}) \)	None	✓	✓	✓		5400.2.a.bx	$1$	$1$	\(0\)	\(0\)	\(0\)	\(-2\)		$-$	$-$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q+(-1-\beta )q^{7}+q^{11}+(-2+\beta )q^{13}+\cdots\)
5400.2.a.bz	$5400$	$2$	5400.a	1.a	$1$	$2$	$2$	$43.119$	\(\Q(\sqrt{6}) \)	None	✓				1080.2.f.e	$1$	$0$	\(0\)	\(0\)	\(0\)	\(-2\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{7}+(2+\beta )q^{11}+(1+2\beta )q^{13}+2\beta q^{17}+\cdots\)
5400.2.a.ca	$5400$	$2$	5400.a	1.a	$1$	$2$	$2$	$43.119$	\(\Q(\sqrt{73}) \)	None	✓				1080.2.a.m	$1$	$1$	\(0\)	\(0\)	\(0\)	\(-1\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta q^{7}+(-1+\beta )q^{11}-3q^{13}+(-3+\cdots)q^{17}+\cdots\)
5400.2.a.cb	$5400$	$2$	5400.a	1.a	$1$	$2$	$2$	$43.119$	\(\Q(\sqrt{73}) \)	None	✓		✓		1080.2.a.m	$1$	$0$	\(0\)	\(0\)	\(0\)	\(-1\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta q^{7}+(1-\beta )q^{11}-3q^{13}+(3-\beta )q^{17}+\cdots\)
5400.2.a.cc	$5400$	$2$	5400.a	1.a	$1$	$2$	$2$	$43.119$	\(\Q(\sqrt{6}) \)	None	✓		✓		1080.2.f.e	$1$	$1$	\(0\)	\(0\)	\(0\)	\(2\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{7}+(-2-\beta )q^{11}+(-1-2\beta )q^{13}+\cdots\)
5400.2.a.cd	$5400$	$2$	5400.a	1.a	$1$	$2$	$2$	$43.119$	\(\Q(\sqrt{13}) \)	None	✓	✓			5400.2.a.bx	$1$	$0$	\(0\)	\(0\)	\(0\)	\(2\)		$-$	$-$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{7}-q^{11}+(2-\beta )q^{13}+(-1+\cdots)q^{17}+\cdots\)
5400.2.a.ce	$5400$	$2$	5400.a	1.a	$1$	$2$	$2$	$43.119$	\(\Q(\sqrt{13}) \)	None	✓	✓			5400.2.a.bx	$1$	$0$	\(0\)	\(0\)	\(0\)	\(2\)		$+$	$+$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{7}+q^{11}+(2-\beta )q^{13}+(1+\cdots)q^{17}+\cdots\)
5400.2.a.cf	$5400$	$2$	5400.a	1.a	$1$	$2$	$2$	$43.119$	\(\Q(\sqrt{6}) \)	None	✓				1080.2.f.e	$1$	$0$	\(0\)	\(0\)	\(0\)	\(2\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{7}+(2-\beta )q^{11}+(-1+2\beta )q^{13}+\cdots\)
5400.2.a.cg	$5400$	$2$	5400.a	1.a	$1$	$4$	$4$	$43.119$	\(\Q(\sqrt{3}, \sqrt{19})\)	None	✓		✓		1080.2.f.g	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$+$	$+$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{7}+(-\beta _{1}-2\beta _{2})q^{11}+(\beta _{1}-\beta _{2}+\cdots)q^{13}+\cdots\)
5400.2.a.ch	$5400$	$2$	5400.a	1.a	$1$	$4$	$4$	$43.119$	\(\Q(\sqrt{3}, \sqrt{19})\)	None	✓		✓		1080.2.f.g	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{7}+(\beta _{1}+2\beta _{2})q^{11}+(\beta _{1}-\beta _{2}+\cdots)q^{13}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(5400))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(5400)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 24}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 16}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(24))\)\(^{\oplus 9}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(27))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(30))\)\(^{\oplus 18}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(36))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(40))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(45))\)\(^{\oplus 16}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(50))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(54))\)\(^{\oplus 9}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(72))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(75))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(90))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(100))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(108))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(120))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(135))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(150))\)\(^{\oplus 9}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(180))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(200))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(216))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(225))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(270))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(300))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(360))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(450))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(540))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(600))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(675))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(900))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1080))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1350))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1800))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(2700))\)\(^{\oplus 2}\)