Space of modular forms of level 2400, weight 4, and trivial character

• Label: 2400.4.a
• Level: $2400$
• Weight: $4$
• Character orbit: 2400.a
• Rep. character: $\chi_{2400}(1,\cdot)$
• Character field: $\Q$
• Dimension: $114$
• Newform subspaces: $54$
• Sturm bound: $1920$
• Trace bound: $13$

Defining parameters

Level:	\( N \)	\(=\)	\( 2400 = 2^{5} \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2400.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 54 \)
Sturm bound:			\(1920\)
Trace bound:			\(13\)
Distinguishing \(T_p\):			\(7\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(2400))\).

	Total	New	Old
Modular forms	1488	114	1374
Cusp forms	1392	114	1278
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		Total				Cusp				Eisenstein
					All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)	\(+\)		\(192\)	\(14\)	\(178\)		\(180\)	\(14\)	\(166\)		\(12\)	\(0\)	\(12\)
\(+\)	\(+\)	\(-\)	\(-\)		\(182\)	\(14\)	\(168\)		\(170\)	\(14\)	\(156\)		\(12\)	\(0\)	\(12\)
\(+\)	\(-\)	\(+\)	\(-\)		\(186\)	\(13\)	\(173\)		\(174\)	\(13\)	\(161\)		\(12\)	\(0\)	\(12\)
\(+\)	\(-\)	\(-\)	\(+\)		\(184\)	\(16\)	\(168\)		\(172\)	\(16\)	\(156\)		\(12\)	\(0\)	\(12\)
\(-\)	\(+\)	\(+\)	\(-\)		\(180\)	\(13\)	\(167\)		\(168\)	\(13\)	\(155\)		\(12\)	\(0\)	\(12\)
\(-\)	\(+\)	\(-\)	\(+\)		\(190\)	\(16\)	\(174\)		\(178\)	\(16\)	\(162\)		\(12\)	\(0\)	\(12\)
\(-\)	\(-\)	\(+\)	\(+\)		\(186\)	\(14\)	\(172\)		\(174\)	\(14\)	\(160\)		\(12\)	\(0\)	\(12\)
\(-\)	\(-\)	\(-\)	\(-\)		\(188\)	\(14\)	\(174\)		\(176\)	\(14\)	\(162\)		\(12\)	\(0\)	\(12\)
Plus space			\(+\)		\(752\)	\(60\)	\(692\)		\(704\)	\(60\)	\(644\)		\(48\)	\(0\)	\(48\)
Minus space			\(-\)		\(736\)	\(54\)	\(682\)		\(688\)	\(54\)	\(634\)		\(48\)	\(0\)	\(48\)

Trace form

114 * q + 1026 * q^9 - 20 * q^13 - 156 * q^17 - 120 * q^21 - 284 * q^29 - 24 * q^33 - 372 * q^37 + 1300 * q^41 + 5746 * q^49 - 1356 * q^53 + 168 * q^57 - 1636 * q^61 - 1644 * q^73 - 3136 * q^77 + 9234 * q^81 - 2940 * q^89 - 936 * q^93 - 1276 * q^97

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(2400))\) 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
2400.4.a.a	$2400$	$4$	2400.a	1.a	$1$	$1$	$1$	$141.605$	\(\Q\)	None	✓				480.4.a.c	$1$	$1$	\(0\)	\(-3\)	\(0\)	\(-32\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-3q^{3}-2^{5}q^{7}+9q^{9}+2^{6}q^{11}+6q^{13}+\cdots\)
2400.4.a.b	$2400$	$4$	2400.a	1.a	$1$	$1$	$1$	$141.605$	\(\Q\)	None	✓				480.4.a.a	$1$	$1$	\(0\)	\(-3\)	\(0\)	\(-12\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-3q^{3}-12q^{7}+9q^{9}-20q^{11}+58q^{13}+\cdots\)
2400.4.a.c	$2400$	$4$	2400.a	1.a	$1$	$1$	$1$	$141.605$	\(\Q\)	None	✓				96.4.a.b	$1$	$0$	\(0\)	\(-3\)	\(0\)	\(-12\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-3q^{3}-12q^{7}+9q^{9}+60q^{11}+42q^{13}+\cdots\)
2400.4.a.d	$2400$	$4$	2400.a	1.a	$1$	$1$	$1$	$141.605$	\(\Q\)	None	✓				480.4.a.d	$1$	$0$	\(0\)	\(-3\)	\(0\)	\(-4\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-3q^{3}-4q^{7}+9q^{9}-40q^{11}+90q^{13}+\cdots\)
2400.4.a.e	$2400$	$4$	2400.a	1.a	$1$	$1$	$1$	$141.605$	\(\Q\)	None	✓				96.4.a.c	$1$	$0$	\(0\)	\(-3\)	\(0\)	\(-4\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-3q^{3}-4q^{7}+9q^{9}-20q^{11}-70q^{13}+\cdots\)
2400.4.a.f	$2400$	$4$	2400.a	1.a	$1$	$1$	$1$	$141.605$	\(\Q\)	None	✓				480.4.a.b	$1$	$1$	\(0\)	\(-3\)	\(0\)	\(8\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-3q^{3}+8q^{7}+9q^{9}+4q^{11}+6q^{13}+\cdots\)
2400.4.a.g	$2400$	$4$	2400.a	1.a	$1$	$1$	$1$	$141.605$	\(\Q\)	None	✓				480.4.a.e	$1$	$1$	\(0\)	\(-3\)	\(0\)	\(12\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-3q^{3}+12q^{7}+9q^{9}-24q^{11}-38q^{13}+\cdots\)
2400.4.a.h	$2400$	$4$	2400.a	1.a	$1$	$1$	$1$	$141.605$	\(\Q\)	None	✓				480.4.a.f	$1$	$0$	\(0\)	\(-3\)	\(0\)	\(16\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-3q^{3}+2^{4}q^{7}+9q^{9}-24q^{11}+14q^{13}+\cdots\)
2400.4.a.i	$2400$	$4$	2400.a	1.a	$1$	$1$	$1$	$141.605$	\(\Q\)	None	✓				480.4.f.a	$1$	$0$	\(0\)	\(-3\)	\(0\)	\(18\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-3q^{3}+18q^{7}+9q^{9}-30q^{11}+38q^{13}+\cdots\)
2400.4.a.j	$2400$	$4$	2400.a	1.a	$1$	$1$	$1$	$141.605$	\(\Q\)	None	✓				480.4.f.a	$1$	$1$	\(0\)	\(-3\)	\(0\)	\(18\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-3q^{3}+18q^{7}+9q^{9}+30q^{11}-38q^{13}+\cdots\)
2400.4.a.k	$2400$	$4$	2400.a	1.a	$1$	$1$	$1$	$141.605$	\(\Q\)	None	✓				96.4.a.a	$1$	$1$	\(0\)	\(-3\)	\(0\)	\(36\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-3q^{3}+6^{2}q^{7}+9q^{9}-6^{2}q^{11}-54q^{13}+\cdots\)
2400.4.a.l	$2400$	$4$	2400.a	1.a	$1$	$1$	$1$	$141.605$	\(\Q\)	None	✓				96.4.a.a	$1$	$0$	\(0\)	\(3\)	\(0\)	\(-36\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+3q^{3}-6^{2}q^{7}+9q^{9}+6^{2}q^{11}-54q^{13}+\cdots\)
2400.4.a.m	$2400$	$4$	2400.a	1.a	$1$	$1$	$1$	$141.605$	\(\Q\)	None	✓				480.4.f.a	$1$	$0$	\(0\)	\(3\)	\(0\)	\(-18\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+3q^{3}-18q^{7}+9q^{9}-30q^{11}-38q^{13}+\cdots\)
2400.4.a.n	$2400$	$4$	2400.a	1.a	$1$	$1$	$1$	$141.605$	\(\Q\)	None	✓				480.4.f.a	$1$	$1$	\(0\)	\(3\)	\(0\)	\(-18\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3q^{3}-18q^{7}+9q^{9}+30q^{11}+38q^{13}+\cdots\)
2400.4.a.o	$2400$	$4$	2400.a	1.a	$1$	$1$	$1$	$141.605$	\(\Q\)	None	✓				480.4.a.f	$1$	$1$	\(0\)	\(3\)	\(0\)	\(-16\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3q^{3}-2^{4}q^{7}+9q^{9}+24q^{11}+14q^{13}+\cdots\)
2400.4.a.p	$2400$	$4$	2400.a	1.a	$1$	$1$	$1$	$141.605$	\(\Q\)	None	✓				480.4.a.e	$1$	$1$	\(0\)	\(3\)	\(0\)	\(-12\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3q^{3}-12q^{7}+9q^{9}+24q^{11}-38q^{13}+\cdots\)
2400.4.a.q	$2400$	$4$	2400.a	1.a	$1$	$1$	$1$	$141.605$	\(\Q\)	None	✓				480.4.a.b	$1$	$1$	\(0\)	\(3\)	\(0\)	\(-8\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3q^{3}-8q^{7}+9q^{9}-4q^{11}+6q^{13}+\cdots\)
2400.4.a.r	$2400$	$4$	2400.a	1.a	$1$	$1$	$1$	$141.605$	\(\Q\)	None	✓				96.4.a.c	$1$	$0$	\(0\)	\(3\)	\(0\)	\(4\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+3q^{3}+4q^{7}+9q^{9}+20q^{11}-70q^{13}+\cdots\)
2400.4.a.s	$2400$	$4$	2400.a	1.a	$1$	$1$	$1$	$141.605$	\(\Q\)	None	✓				480.4.a.d	$1$	$0$	\(0\)	\(3\)	\(0\)	\(4\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+3q^{3}+4q^{7}+9q^{9}+40q^{11}+90q^{13}+\cdots\)
2400.4.a.t	$2400$	$4$	2400.a	1.a	$1$	$1$	$1$	$141.605$	\(\Q\)	None	✓				96.4.a.b	$1$	$1$	\(0\)	\(3\)	\(0\)	\(12\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3q^{3}+12q^{7}+9q^{9}-60q^{11}+42q^{13}+\cdots\)
2400.4.a.u	$2400$	$4$	2400.a	1.a	$1$	$1$	$1$	$141.605$	\(\Q\)	None	✓				480.4.a.a	$1$	$0$	\(0\)	\(3\)	\(0\)	\(12\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+3q^{3}+12q^{7}+9q^{9}+20q^{11}+58q^{13}+\cdots\)
2400.4.a.v	$2400$	$4$	2400.a	1.a	$1$	$1$	$1$	$141.605$	\(\Q\)	None	✓				480.4.a.c	$1$	$1$	\(0\)	\(3\)	\(0\)	\(32\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3q^{3}+2^{5}q^{7}+9q^{9}-2^{6}q^{11}+6q^{13}+\cdots\)
2400.4.a.w	$2400$	$4$	2400.a	1.a	$1$	$2$	$2$	$141.605$	\(\Q(\sqrt{89}) \)	None	✓				480.4.f.c	$2$	$0$	\(0\)	\(-6\)	\(0\)	\(-44\)		$-$	$+$	$-$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q-3q^{3}-22q^{7}+9q^{9}-\beta q^{11}+\beta q^{13}+\cdots\)
2400.4.a.x	$2400$	$4$	2400.a	1.a	$1$	$2$	$2$	$141.605$	\(\Q(\sqrt{41}) \)	None	✓				480.4.a.m	$1$	$0$	\(0\)	\(-6\)	\(0\)	\(-12\)		$+$	$+$	$+$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q-3q^{3}+(-6-\beta )q^{7}+9q^{9}+(-12+\cdots)q^{11}+\cdots\)
2400.4.a.y	$2400$	$4$	2400.a	1.a	$1$	$2$	$2$	$141.605$	\(\Q(\sqrt{89}) \)	None	✓				480.4.a.o	$1$	$1$	\(0\)	\(-6\)	\(0\)	\(-12\)		$-$	$+$	$+$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q-3q^{3}+(-6-\beta )q^{7}+9q^{9}+(12-2\beta )q^{11}+\cdots\)
2400.4.a.z	$2400$	$4$	2400.a	1.a	$1$	$2$	$2$	$141.605$	\(\Q(\sqrt{201}) \)	None	✓				480.4.a.n	$1$	$0$	\(0\)	\(-6\)	\(0\)	\(-4\)		$+$	$+$	$+$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q-3q^{3}+(-2-\beta )q^{7}+9q^{9}+20q^{11}+\cdots\)
2400.4.a.ba	$2400$	$4$	2400.a	1.a	$1$	$2$	$2$	$141.605$	\(\Q(\sqrt{201}) \)	None	✓				480.4.a.n	$1$	$0$	\(0\)	\(6\)	\(0\)	\(4\)		$-$	$-$	$+$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q+3q^{3}+(2+\beta )q^{7}+9q^{9}-20q^{11}+\cdots\)
2400.4.a.bb	$2400$	$4$	2400.a	1.a	$1$	$2$	$2$	$141.605$	\(\Q(\sqrt{89}) \)	None	✓				480.4.a.o	$1$	$0$	\(0\)	\(6\)	\(0\)	\(12\)		$-$	$-$	$+$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q+3q^{3}+(6+\beta )q^{7}+9q^{9}+(-12+2\beta )q^{11}+\cdots\)
2400.4.a.bc	$2400$	$4$	2400.a	1.a	$1$	$2$	$2$	$141.605$	\(\Q(\sqrt{41}) \)	None	✓				480.4.a.m	$1$	$1$	\(0\)	\(6\)	\(0\)	\(12\)		$+$	$-$	$+$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q+3q^{3}+(6+\beta )q^{7}+9q^{9}+(12+4\beta )q^{11}+\cdots\)
2400.4.a.bd	$2400$	$4$	2400.a	1.a	$1$	$2$	$2$	$141.605$	\(\Q(\sqrt{89}) \)	None	✓				480.4.f.c	$2$	$0$	\(0\)	\(6\)	\(0\)	\(44\)		$+$	$-$	$-$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q+3q^{3}+22q^{7}+9q^{9}-\beta q^{11}-\beta q^{13}+\cdots\)
2400.4.a.be	$2400$	$4$	2400.a	1.a	$1$	$3$	$3$	$141.605$	3.3.23109.1	None	✓				480.4.f.d	$1$	$1$	\(0\)	\(-9\)	\(0\)	\(-32\)		$+$	$+$	$-$	$-$	$2^{3}$	$\mathrm{SU}(2)$	\(q-3q^{3}+(-11+\beta _{1})q^{7}+9q^{9}+(-17+\cdots)q^{11}+\cdots\)
2400.4.a.bf	$2400$	$4$	2400.a	1.a	$1$	$3$	$3$	$141.605$	3.3.23109.1	None	✓				480.4.f.d	$1$	$0$	\(0\)	\(-9\)	\(0\)	\(-32\)		$-$	$+$	$-$	$+$	$2^{3}$	$\mathrm{SU}(2)$	\(q-3q^{3}+(-11+\beta _{1})q^{7}+9q^{9}+(17+\cdots)q^{11}+\cdots\)
2400.4.a.bg	$2400$	$4$	2400.a	1.a	$1$	$3$	$3$	$141.605$	3.3.304244.1	None	✓	✓			2400.4.a.bg	$1$	$1$	\(0\)	\(-9\)	\(0\)	\(-9\)		$+$	$+$	$-$	$-$	$2^{3}$	$\mathrm{SU}(2)$	\(q-3q^{3}+(-3-\beta _{1})q^{7}+9q^{9}+(-1+\cdots)q^{11}+\cdots\)
2400.4.a.bh	$2400$	$4$	2400.a	1.a	$1$	$3$	$3$	$141.605$	3.3.304244.1	None	✓	✓			2400.4.a.bg	$1$	$1$	\(0\)	\(-9\)	\(0\)	\(-9\)		$-$	$+$	$+$	$-$	$2^{3}$	$\mathrm{SU}(2)$	\(q-3q^{3}+(-3-\beta _{1})q^{7}+9q^{9}+(1-\beta _{2})q^{11}+\cdots\)
2400.4.a.bi	$2400$	$4$	2400.a	1.a	$1$	$3$	$3$	$141.605$	3.3.115636.1	None	✓	✓			2400.4.a.bi	$1$	$0$	\(0\)	\(-9\)	\(0\)	\(3\)		$-$	$+$	$-$	$+$	$2^{3}$	$\mathrm{SU}(2)$	\(q-3q^{3}+(1-\beta _{2})q^{7}+9q^{9}+(-15+\cdots)q^{11}+\cdots\)
2400.4.a.bj	$2400$	$4$	2400.a	1.a	$1$	$3$	$3$	$141.605$	3.3.115636.1	None	✓	✓			2400.4.a.bi	$1$	$1$	\(0\)	\(-9\)	\(0\)	\(3\)		$-$	$+$	$+$	$-$	$2^{3}$	$\mathrm{SU}(2)$	\(q-3q^{3}+(1-\beta _{2})q^{7}+9q^{9}+(15-\beta _{1}+\cdots)q^{11}+\cdots\)
2400.4.a.bk	$2400$	$4$	2400.a	1.a	$1$	$3$	$3$	$141.605$	3.3.32340.1	None	✓	✓			2400.4.a.bk	$1$	$1$	\(0\)	\(-9\)	\(0\)	\(7\)		$+$	$+$	$-$	$-$	$2^{3}$	$\mathrm{SU}(2)$	\(q-3q^{3}+(2-\beta _{2})q^{7}+9q^{9}+(1-3\beta _{1}+\cdots)q^{11}+\cdots\)
2400.4.a.bl	$2400$	$4$	2400.a	1.a	$1$	$3$	$3$	$141.605$	3.3.32340.1	None	✓	✓			2400.4.a.bk	$1$	$0$	\(0\)	\(-9\)	\(0\)	\(7\)		$+$	$+$	$+$	$+$	$2^{3}$	$\mathrm{SU}(2)$	\(q-3q^{3}+(2-\beta _{2})q^{7}+9q^{9}+(-1+3\beta _{1}+\cdots)q^{11}+\cdots\)
2400.4.a.bm	$2400$	$4$	2400.a	1.a	$1$	$3$	$3$	$141.605$	3.3.28724.1	None	✓	✓			2400.4.a.bm	$1$	$0$	\(0\)	\(-9\)	\(0\)	\(19\)		$+$	$+$	$+$	$+$	$2^{3}$	$\mathrm{SU}(2)$	\(q-3q^{3}+(6+\beta _{1}-\beta _{2})q^{7}+9q^{9}+(-2^{4}+\cdots)q^{11}+\cdots\)
2400.4.a.bn	$2400$	$4$	2400.a	1.a	$1$	$3$	$3$	$141.605$	3.3.28724.1	None	✓	✓			2400.4.a.bm	$1$	$0$	\(0\)	\(-9\)	\(0\)	\(19\)		$-$	$+$	$-$	$+$	$2^{3}$	$\mathrm{SU}(2)$	\(q-3q^{3}+(6+\beta _{1}-\beta _{2})q^{7}+9q^{9}+(2^{4}+\cdots)q^{11}+\cdots\)
2400.4.a.bo	$2400$	$4$	2400.a	1.a	$1$	$3$	$3$	$141.605$	3.3.28724.1	None	✓	✓			2400.4.a.bm	$1$	$1$	\(0\)	\(9\)	\(0\)	\(-19\)		$-$	$-$	$-$	$-$	$2^{3}$	$\mathrm{SU}(2)$	\(q+3q^{3}+(-6-\beta _{1}+\beta _{2})q^{7}+9q^{9}+\cdots\)
2400.4.a.bp	$2400$	$4$	2400.a	1.a	$1$	$3$	$3$	$141.605$	3.3.28724.1	None	✓	✓			2400.4.a.bm	$1$	$1$	\(0\)	\(9\)	\(0\)	\(-19\)		$+$	$-$	$+$	$-$	$2^{3}$	$\mathrm{SU}(2)$	\(q+3q^{3}+(-6-\beta _{1}+\beta _{2})q^{7}+9q^{9}+\cdots\)
2400.4.a.bq	$2400$	$4$	2400.a	1.a	$1$	$3$	$3$	$141.605$	3.3.32340.1	None	✓	✓			2400.4.a.bk	$1$	$1$	\(0\)	\(9\)	\(0\)	\(-7\)		$-$	$-$	$-$	$-$	$2^{3}$	$\mathrm{SU}(2)$	\(q+3q^{3}+(-2+\beta _{2})q^{7}+9q^{9}+(-1+\cdots)q^{11}+\cdots\)
2400.4.a.br	$2400$	$4$	2400.a	1.a	$1$	$3$	$3$	$141.605$	3.3.32340.1	None	✓	✓			2400.4.a.bk	$1$	$0$	\(0\)	\(9\)	\(0\)	\(-7\)		$-$	$-$	$+$	$+$	$2^{3}$	$\mathrm{SU}(2)$	\(q+3q^{3}+(-2+\beta _{2})q^{7}+9q^{9}+(1-3\beta _{1}+\cdots)q^{11}+\cdots\)
2400.4.a.bs	$2400$	$4$	2400.a	1.a	$1$	$3$	$3$	$141.605$	3.3.115636.1	None	✓	✓			2400.4.a.bi	$1$	$1$	\(0\)	\(9\)	\(0\)	\(-3\)		$+$	$-$	$+$	$-$	$2^{3}$	$\mathrm{SU}(2)$	\(q+3q^{3}+(-1+\beta _{2})q^{7}+9q^{9}+(-15+\cdots)q^{11}+\cdots\)
2400.4.a.bt	$2400$	$4$	2400.a	1.a	$1$	$3$	$3$	$141.605$	3.3.115636.1	None	✓	✓			2400.4.a.bi	$1$	$0$	\(0\)	\(9\)	\(0\)	\(-3\)		$+$	$-$	$-$	$+$	$2^{3}$	$\mathrm{SU}(2)$	\(q+3q^{3}+(-1+\beta _{2})q^{7}+9q^{9}+(15+\cdots)q^{11}+\cdots\)
2400.4.a.bu	$2400$	$4$	2400.a	1.a	$1$	$3$	$3$	$141.605$	3.3.304244.1	None	✓	✓			2400.4.a.bg	$1$	$0$	\(0\)	\(9\)	\(0\)	\(9\)		$-$	$-$	$+$	$+$	$2^{3}$	$\mathrm{SU}(2)$	\(q+3q^{3}+(3+\beta _{1})q^{7}+9q^{9}+(-1+\beta _{2})q^{11}+\cdots\)
2400.4.a.bv	$2400$	$4$	2400.a	1.a	$1$	$3$	$3$	$141.605$	3.3.304244.1	None	✓	✓			2400.4.a.bg	$1$	$0$	\(0\)	\(9\)	\(0\)	\(9\)		$+$	$-$	$-$	$+$	$2^{3}$	$\mathrm{SU}(2)$	\(q+3q^{3}+(3+\beta _{1})q^{7}+9q^{9}+(1-\beta _{2})q^{11}+\cdots\)
2400.4.a.bw	$2400$	$4$	2400.a	1.a	$1$	$3$	$3$	$141.605$	3.3.23109.1	None	✓				480.4.f.d	$1$	$1$	\(0\)	\(9\)	\(0\)	\(32\)		$-$	$-$	$-$	$-$	$2^{3}$	$\mathrm{SU}(2)$	\(q+3q^{3}+(11-\beta _{1})q^{7}+9q^{9}+(-17+\cdots)q^{11}+\cdots\)
2400.4.a.bx	$2400$	$4$	2400.a	1.a	$1$	$3$	$3$	$141.605$	3.3.23109.1	None	✓				480.4.f.d	$1$	$0$	\(0\)	\(9\)	\(0\)	\(32\)		$+$	$-$	$-$	$+$	$2^{3}$	$\mathrm{SU}(2)$	\(q+3q^{3}+(11-\beta _{1})q^{7}+9q^{9}+(17-2\beta _{1}+\cdots)q^{11}+\cdots\)
2400.4.a.by	$2400$	$4$	2400.a	1.a	$1$	$4$	$4$	$141.605$	\(\Q(\sqrt{138 -6 \sqrt{209}})\)	None	✓		✓		480.4.f.g	$2$	$1$	\(0\)	\(-12\)	\(0\)	\(12\)		$+$	$+$	$-$	$-$	$2^{6}$	$\mathrm{SU}(2)$	\(q-3q^{3}+(3+\beta _{2})q^{7}+9q^{9}+(-\beta _{1}+\cdots)q^{11}+\cdots\)
2400.4.a.bz	$2400$	$4$	2400.a	1.a	$1$	$4$	$4$	$141.605$	\(\Q(\sqrt{21}, \sqrt{141})\)	None	✓		✓		480.4.f.f	$2$	$0$	\(0\)	\(-12\)	\(0\)	\(60\)		$-$	$+$	$-$	$+$	$2^{6}$	$\mathrm{SU}(2)$	\(q-3q^{3}+(15-\beta _{3})q^{7}+9q^{9}+(4\beta _{1}+\cdots)q^{11}+\cdots\)
2400.4.a.ca	$2400$	$4$	2400.a	1.a	$1$	$4$	$4$	$141.605$	\(\Q(\sqrt{21}, \sqrt{141})\)	None	✓		✓		480.4.f.f	$2$	$0$	\(0\)	\(12\)	\(0\)	\(-60\)		$+$	$-$	$-$	$+$	$2^{6}$	$\mathrm{SU}(2)$	\(q+3q^{3}+(-15+\beta _{3})q^{7}+9q^{9}+(4\beta _{1}+\cdots)q^{11}+\cdots\)
2400.4.a.cb	$2400$	$4$	2400.a	1.a	$1$	$4$	$4$	$141.605$	\(\Q(\sqrt{138 -6 \sqrt{209}})\)	None	✓		✓		480.4.f.g	$2$	$1$	\(0\)	\(12\)	\(0\)	\(-12\)		$-$	$-$	$-$	$-$	$2^{6}$	$\mathrm{SU}(2)$	\(q+3q^{3}+(-3-\beta _{2})q^{7}+9q^{9}+(-\beta _{1}+\cdots)q^{11}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(2400))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(2400)) \simeq \) \(S_{4}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 24}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(6))\)\(^{\oplus 15}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(8))\)\(^{\oplus 18}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(10))\)\(^{\oplus 20}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(12))\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(16))\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 16}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(24))\)\(^{\oplus 9}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(25))\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(30))\)\(^{\oplus 10}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(32))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(40))\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(48))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(50))\)\(^{\oplus 10}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(60))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(75))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(80))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(96))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(100))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(120))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(150))\)\(^{\oplus 5}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(160))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(200))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(240))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(300))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(400))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(480))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(600))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(800))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(1200))\)\(^{\oplus 2}\)