Space of modular forms of level 2304, weight 4, and trivial character

• Label: 2304.4.a
• Level: $2304$
• Weight: $4$
• Character orbit: 2304.a
• Rep. character: $\chi_{2304}(1,\cdot)$
• Character field: $\Q$
• Dimension: $118$
• Newform subspaces: $56$
• Sturm bound: $1536$
• Trace bound: $19$

Defining parameters

Level:	\( N \)	\(=\)	\( 2304 = 2^{8} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2304.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 56 \)
Sturm bound:			\(1536\)
Trace bound:			\(19\)
Distinguishing \(T_p\):			\(5\), \(7\), \(11\), \(13\), \(17\), \(19\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(2304))\).

	Total	New	Old
Modular forms	1200	122	1078
Cusp forms	1104	118	986
Eisenstein series	96	4	92

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
\(+\)	\(+\)	\(+\)		\(304\)	\(26\)	\(278\)		\(280\)	\(26\)	\(254\)		\(24\)	\(0\)	\(24\)
\(+\)	\(-\)	\(-\)		\(296\)	\(36\)	\(260\)		\(272\)	\(34\)	\(238\)		\(24\)	\(2\)	\(22\)
\(-\)	\(+\)	\(-\)		\(296\)	\(22\)	\(274\)		\(272\)	\(22\)	\(250\)		\(24\)	\(0\)	\(24\)
\(-\)	\(-\)	\(+\)		\(304\)	\(38\)	\(266\)		\(280\)	\(36\)	\(244\)		\(24\)	\(2\)	\(22\)
Plus space		\(+\)		\(608\)	\(64\)	\(544\)		\(560\)	\(62\)	\(498\)		\(48\)	\(2\)	\(46\)
Minus space		\(-\)		\(592\)	\(58\)	\(534\)		\(544\)	\(56\)	\(488\)		\(48\)	\(2\)	\(46\)

Trace form

\( 118 q + 4 q^{17} + 2754 q^{25} - 4 q^{41} + 3750 q^{49} - 1472 q^{65} + 276 q^{73} + 172 q^{89} + 3164 q^{97}+O(q^{100}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(2304))\) 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
2304.4.a.a	$2304$	$4$	2304.a	1.a	$1$	$1$	$1$	$135.940$	\(\Q\)	\(\Q(\sqrt{-1}) \)	✓				1152.4.d.c	$2$	$1$	\(0\)	\(0\)	\(-22\)	\(0\)		$-$	$+$	$-$	$1$	$N(\mathrm{U}(1))$	\(q-22q^{5}-92q^{13}+104q^{17}+359q^{25}+\cdots\)
2304.4.a.b	$2304$	$4$	2304.a	1.a	$1$	$1$	$1$	$135.940$	\(\Q\)	\(\Q(\sqrt{-1}) \)	✓				1152.4.d.c	$2$	$0$	\(0\)	\(0\)	\(-22\)	\(0\)		$+$	$+$	$+$	$1$	$N(\mathrm{U}(1))$	\(q-22q^{5}+92q^{13}-104q^{17}+359q^{25}+\cdots\)
2304.4.a.c	$2304$	$4$	2304.a	1.a	$1$	$1$	$1$	$135.940$	\(\Q\)	None	✓				128.4.b.a	$1$	$1$	\(0\)	\(0\)	\(-12\)	\(-32\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-12q^{5}-2^{5}q^{7}-8q^{11}-20q^{13}+\cdots\)
2304.4.a.d	$2304$	$4$	2304.a	1.a	$1$	$1$	$1$	$135.940$	\(\Q\)	None	✓				128.4.b.a	$1$	$1$	\(0\)	\(0\)	\(-12\)	\(32\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-12q^{5}+2^{5}q^{7}+8q^{11}-20q^{13}+\cdots\)
2304.4.a.e	$2304$	$4$	2304.a	1.a	$1$	$1$	$1$	$135.940$	\(\Q\)	None	✓				384.4.d.a	$1$	$0$	\(0\)	\(0\)	\(-8\)	\(-12\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-8q^{5}-12q^{7}-12q^{11}-20q^{13}+\cdots\)
2304.4.a.f	$2304$	$4$	2304.a	1.a	$1$	$1$	$1$	$135.940$	\(\Q\)	None	✓				384.4.d.a	$1$	$0$	\(0\)	\(0\)	\(-8\)	\(12\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-8q^{5}+12q^{7}+12q^{11}-20q^{13}+\cdots\)
2304.4.a.g	$2304$	$4$	2304.a	1.a	$1$	$1$	$1$	$135.940$	\(\Q\)	\(\Q(\sqrt{-1}) \)	✓				128.4.b.c	$2$	$0$	\(0\)	\(0\)	\(-4\)	\(0\)		$-$	$-$	$+$	$1$	$N(\mathrm{U}(1))$	\(q-4q^{5}-92q^{13}-94q^{17}-109q^{25}+\cdots\)
2304.4.a.h	$2304$	$4$	2304.a	1.a	$1$	$1$	$1$	$135.940$	\(\Q\)	\(\Q(\sqrt{-2}) \)	✓				64.4.b.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$+$	$1$	$N(\mathrm{U}(1))$	\(q-18q^{11}-90q^{17}-106q^{19}-5^{3}q^{25}+\cdots\)
2304.4.a.i	$2304$	$4$	2304.a	1.a	$1$	$1$	$1$	$135.940$	\(\Q\)	\(\Q(\sqrt{-2}) \)	✓				64.4.b.a	$2$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$+$	$-$	$-$	$1$	$N(\mathrm{U}(1))$	\(q+18q^{11}-90q^{17}+106q^{19}-5^{3}q^{25}+\cdots\)
2304.4.a.j	$2304$	$4$	2304.a	1.a	$1$	$1$	$1$	$135.940$	\(\Q\)	\(\Q(\sqrt{-1}) \)	✓				128.4.b.c	$2$	$1$	\(0\)	\(0\)	\(4\)	\(0\)		$+$	$-$	$-$	$1$	$N(\mathrm{U}(1))$	\(q+4q^{5}+92q^{13}-94q^{17}-109q^{25}+\cdots\)
2304.4.a.k	$2304$	$4$	2304.a	1.a	$1$	$1$	$1$	$135.940$	\(\Q\)	None	✓				384.4.d.a	$1$	$1$	\(0\)	\(0\)	\(8\)	\(-12\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+8q^{5}-12q^{7}+12q^{11}+20q^{13}+\cdots\)
2304.4.a.l	$2304$	$4$	2304.a	1.a	$1$	$1$	$1$	$135.940$	\(\Q\)	None	✓				384.4.d.a	$1$	$1$	\(0\)	\(0\)	\(8\)	\(12\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+8q^{5}+12q^{7}-12q^{11}+20q^{13}+\cdots\)
2304.4.a.m	$2304$	$4$	2304.a	1.a	$1$	$1$	$1$	$135.940$	\(\Q\)	None	✓				128.4.b.a	$1$	$0$	\(0\)	\(0\)	\(12\)	\(-32\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+12q^{5}-2^{5}q^{7}+8q^{11}+20q^{13}+\cdots\)
2304.4.a.n	$2304$	$4$	2304.a	1.a	$1$	$1$	$1$	$135.940$	\(\Q\)	None	✓				128.4.b.a	$1$	$0$	\(0\)	\(0\)	\(12\)	\(32\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+12q^{5}+2^{5}q^{7}-8q^{11}+20q^{13}+\cdots\)
2304.4.a.o	$2304$	$4$	2304.a	1.a	$1$	$1$	$1$	$135.940$	\(\Q\)	\(\Q(\sqrt{-1}) \)	✓				1152.4.d.c	$2$	$1$	\(0\)	\(0\)	\(22\)	\(0\)		$-$	$+$	$-$	$1$	$N(\mathrm{U}(1))$	\(q+22q^{5}-92q^{13}-104q^{17}+359q^{25}+\cdots\)
2304.4.a.p	$2304$	$4$	2304.a	1.a	$1$	$1$	$1$	$135.940$	\(\Q\)	\(\Q(\sqrt{-1}) \)	✓				1152.4.d.c	$2$	$0$	\(0\)	\(0\)	\(22\)	\(0\)		$+$	$+$	$+$	$1$	$N(\mathrm{U}(1))$	\(q+22q^{5}+92q^{13}+104q^{17}+359q^{25}+\cdots\)
2304.4.a.q	$2304$	$4$	2304.a	1.a	$1$	$2$	$2$	$135.940$	\(\Q(\sqrt{7}) \)	None	✓				1152.4.d.k	$2$	$0$	\(0\)	\(0\)	\(-12\)	\(0\)		$+$	$+$	$+$	$2^{3}$	$\mathrm{SU}(2)$	\(q-6q^{5}-\beta q^{7}-2\beta q^{11}-20q^{13}+\cdots\)
2304.4.a.r	$2304$	$4$	2304.a	1.a	$1$	$2$	$2$	$135.940$	\(\Q(\sqrt{7}) \)	None	✓				1152.4.d.k	$2$	$1$	\(0\)	\(0\)	\(-12\)	\(0\)		$-$	$+$	$-$	$2^{3}$	$\mathrm{SU}(2)$	\(q-6q^{5}-\beta q^{7}-2\beta q^{11}+20q^{13}+\cdots\)
2304.4.a.s	$2304$	$4$	2304.a	1.a	$1$	$2$	$2$	$135.940$	\(\Q(\sqrt{13}) \)	None	✓				384.4.d.c	$1$	$1$	\(0\)	\(0\)	\(-8\)	\(-16\)		$+$	$-$	$-$	$2^{3}$	$\mathrm{SU}(2)$	\(q+(-4-\beta )q^{5}+(-8-\beta )q^{7}+(-4+\cdots)q^{11}+\cdots\)
2304.4.a.t	$2304$	$4$	2304.a	1.a	$1$	$2$	$2$	$135.940$	\(\Q(\sqrt{13}) \)	None	✓				384.4.d.c	$1$	$1$	\(0\)	\(0\)	\(-8\)	\(16\)		$+$	$-$	$-$	$2^{3}$	$\mathrm{SU}(2)$	\(q+(-4-\beta )q^{5}+(8+\beta )q^{7}+(4-4\beta )q^{11}+\cdots\)
2304.4.a.u	$2304$	$4$	2304.a	1.a	$1$	$2$	$2$	$135.940$	\(\Q(\sqrt{2}) \)	\(\Q(\sqrt{-6}) \)	✓				72.4.d.a	$4$	$1$	\(0\)	\(0\)	\(0\)	\(-68\)		$-$	$+$	$-$	$2$	$N(\mathrm{U}(1))$	\(q+7\beta q^{5}-34q^{7}+2\beta q^{11}+267q^{25}+\cdots\)
2304.4.a.v	$2304$	$4$	2304.a	1.a	$1$	$2$	$2$	$135.940$	\(\Q(\sqrt{7}) \)	None	✓				8.4.b.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(-16\)		$-$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q+2\beta q^{5}-8q^{7}-3\beta q^{11}-10\beta q^{13}+\cdots\)
2304.4.a.w	$2304$	$4$	2304.a	1.a	$1$	$2$	$2$	$135.940$	\(\Q(\sqrt{3}) \)	None	✓				576.4.d.b	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$+$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q+4\beta q^{5}-\beta q^{7}-48q^{11}+6\beta q^{13}+\cdots\)
2304.4.a.x	$2304$	$4$	2304.a	1.a	$1$	$2$	$2$	$135.940$	\(\Q(\sqrt{3}) \)	None	✓				192.4.d.c	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q+\beta q^{5}+7\beta q^{7}-48q^{11}-12\beta q^{13}+\cdots\)
2304.4.a.y	$2304$	$4$	2304.a	1.a	$1$	$2$	$2$	$135.940$	\(\Q(\sqrt{11}) \)	None	✓				192.4.d.b	$2$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$+$	$-$	$-$	$2\cdot 3$	$\mathrm{SU}(2)$	\(q+\beta q^{5}+\beta q^{7}-48q^{11}-4\beta q^{13}+\cdots\)
2304.4.a.z	$2304$	$4$	2304.a	1.a	$1$	$2$	$2$	$135.940$	\(\Q(\sqrt{3}) \)	None	✓				576.4.d.b	$2$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q+4\beta q^{5}-\beta q^{7}-48q^{11}-6\beta q^{13}+\cdots\)
2304.4.a.ba	$2304$	$4$	2304.a	1.a	$1$	$2$	$2$	$135.940$	\(\Q(\sqrt{3}) \)	None	✓				64.4.b.b	$2$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$+$	$-$	$-$	$2^{3}$	$\mathrm{SU}(2)$	\(q+\beta q^{5}-2\beta q^{7}-42q^{11}+3\beta q^{13}+\cdots\)
2304.4.a.bb	$2304$	$4$	2304.a	1.a	$1$	$2$	$2$	$135.940$	\(\Q(\sqrt{2}) \)	None	✓				384.4.d.d	$2$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$+$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+\beta q^{5}-5\beta q^{7}-20q^{11}-14\beta q^{13}+\cdots\)
2304.4.a.bc	$2304$	$4$	2304.a	1.a	$1$	$2$	$2$	$135.940$	\(\Q(\sqrt{3}) \)	\(\Q(\sqrt{-3}) \)	✓				576.4.d.d	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$+$	$+$	$+$	$2\cdot 3^{2}$	$N(\mathrm{U}(1))$	\(q+\beta q^{7}+2\beta q^{13}-56q^{19}-5^{3}q^{25}+\cdots\)
2304.4.a.bd	$2304$	$4$	2304.a	1.a	$1$	$2$	$2$	$135.940$	\(\Q(\sqrt{3}) \)	\(\Q(\sqrt{-3}) \)	✓				576.4.d.d	$4$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$+$	$-$	$2\cdot 3^{2}$	$N(\mathrm{U}(1))$	\(q-\beta q^{7}+2\beta q^{13}+56q^{19}-5^{3}q^{25}+\cdots\)
2304.4.a.be	$2304$	$4$	2304.a	1.a	$1$	$2$	$2$	$135.940$	\(\Q(\sqrt{3}) \)	None	✓				192.4.d.a	$2$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$+$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+3\beta q^{5}+\beta q^{7}+2^{4}\beta q^{13}+90q^{17}+\cdots\)
2304.4.a.bf	$2304$	$4$	2304.a	1.a	$1$	$2$	$2$	$135.940$	\(\Q(\sqrt{2}) \)	\(\Q(\sqrt{-2}) \)	✓				128.4.b.b	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$+$	$2$	$N(\mathrm{U}(1))$	\(q+5^{2}\beta q^{11}+90q^{17}+45\beta q^{19}-5^{3}q^{25}+\cdots\)
2304.4.a.bg	$2304$	$4$	2304.a	1.a	$1$	$2$	$2$	$135.940$	\(\Q(\sqrt{3}) \)	None	✓				192.4.d.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q+3\beta q^{5}-\beta q^{7}+2^{4}\beta q^{13}+90q^{17}+\cdots\)
2304.4.a.bh	$2304$	$4$	2304.a	1.a	$1$	$2$	$2$	$135.940$	\(\Q(\sqrt{2}) \)	None	✓				384.4.d.d	$2$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$+$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+\beta q^{5}+5\beta q^{7}+20q^{11}-14\beta q^{13}+\cdots\)
2304.4.a.bi	$2304$	$4$	2304.a	1.a	$1$	$2$	$2$	$135.940$	\(\Q(\sqrt{3}) \)	None	✓				64.4.b.b	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$+$	$2^{3}$	$\mathrm{SU}(2)$	\(q+\beta q^{5}+2\beta q^{7}+42q^{11}+3\beta q^{13}+\cdots\)
2304.4.a.bj	$2304$	$4$	2304.a	1.a	$1$	$2$	$2$	$135.940$	\(\Q(\sqrt{3}) \)	None	✓				576.4.d.b	$2$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q+4\beta q^{5}+\beta q^{7}+48q^{11}+6\beta q^{13}+\cdots\)
2304.4.a.bk	$2304$	$4$	2304.a	1.a	$1$	$2$	$2$	$135.940$	\(\Q(\sqrt{3}) \)	None	✓				192.4.d.c	$2$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$+$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+\beta q^{5}-7\beta q^{7}+48q^{11}-12\beta q^{13}+\cdots\)
2304.4.a.bl	$2304$	$4$	2304.a	1.a	$1$	$2$	$2$	$135.940$	\(\Q(\sqrt{11}) \)	None	✓				192.4.d.b	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$+$	$2\cdot 3$	$\mathrm{SU}(2)$	\(q+\beta q^{5}-\beta q^{7}+48q^{11}-4\beta q^{13}+\cdots\)
2304.4.a.bm	$2304$	$4$	2304.a	1.a	$1$	$2$	$2$	$135.940$	\(\Q(\sqrt{3}) \)	None	✓				576.4.d.b	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$+$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q+4\beta q^{5}+\beta q^{7}+48q^{11}-6\beta q^{13}+\cdots\)
2304.4.a.bn	$2304$	$4$	2304.a	1.a	$1$	$2$	$2$	$135.940$	\(\Q(\sqrt{7}) \)	None	✓				8.4.b.a	$2$	$1$	\(0\)	\(0\)	\(0\)	\(16\)		$+$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+2\beta q^{5}+8q^{7}+3\beta q^{11}-10\beta q^{13}+\cdots\)
2304.4.a.bo	$2304$	$4$	2304.a	1.a	$1$	$2$	$2$	$135.940$	\(\Q(\sqrt{2}) \)	\(\Q(\sqrt{-6}) \)	✓				72.4.d.a	$4$	$0$	\(0\)	\(0\)	\(0\)	\(68\)		$+$	$+$	$+$	$2$	$N(\mathrm{U}(1))$	\(q+7\beta q^{5}+34q^{7}-2\beta q^{11}+267q^{25}+\cdots\)
2304.4.a.bp	$2304$	$4$	2304.a	1.a	$1$	$2$	$2$	$135.940$	\(\Q(\sqrt{13}) \)	None	✓				384.4.d.c	$1$	$0$	\(0\)	\(0\)	\(8\)	\(-16\)		$-$	$-$	$+$	$2^{3}$	$\mathrm{SU}(2)$	\(q+(4+\beta )q^{5}+(-8-\beta )q^{7}+(4-4\beta )q^{11}+\cdots\)
2304.4.a.bq	$2304$	$4$	2304.a	1.a	$1$	$2$	$2$	$135.940$	\(\Q(\sqrt{13}) \)	None	✓				384.4.d.c	$1$	$0$	\(0\)	\(0\)	\(8\)	\(16\)		$-$	$-$	$+$	$2^{3}$	$\mathrm{SU}(2)$	\(q+(4+\beta )q^{5}+(8+\beta )q^{7}+(-4+4\beta )q^{11}+\cdots\)
2304.4.a.br	$2304$	$4$	2304.a	1.a	$1$	$2$	$2$	$135.940$	\(\Q(\sqrt{7}) \)	None	✓				1152.4.d.k	$2$	$0$	\(0\)	\(0\)	\(12\)	\(0\)		$+$	$+$	$+$	$2^{3}$	$\mathrm{SU}(2)$	\(q+6q^{5}-\beta q^{7}+2\beta q^{11}-20q^{13}+\cdots\)
2304.4.a.bs	$2304$	$4$	2304.a	1.a	$1$	$2$	$2$	$135.940$	\(\Q(\sqrt{7}) \)	None	✓				1152.4.d.k	$2$	$1$	\(0\)	\(0\)	\(12\)	\(0\)		$-$	$+$	$-$	$2^{3}$	$\mathrm{SU}(2)$	\(q+6q^{5}-\beta q^{7}+2\beta q^{11}+20q^{13}+\cdots\)
2304.4.a.bt	$2304$	$4$	2304.a	1.a	$1$	$3$	$3$	$135.940$	3.3.1436.1	None	✓				24.4.d.a	$1$	$1$	\(0\)	\(0\)	\(-10\)	\(-14\)		$+$	$-$	$-$	$2^{5}$	$\mathrm{SU}(2)$	\(q+(-3-\beta _{2})q^{5}+(-5+\beta _{1})q^{7}+(-2\beta _{1}+\cdots)q^{11}+\cdots\)
2304.4.a.bu	$2304$	$4$	2304.a	1.a	$1$	$3$	$3$	$135.940$	3.3.1436.1	None	✓				24.4.d.a	$1$	$0$	\(0\)	\(0\)	\(-10\)	\(14\)		$-$	$-$	$+$	$2^{5}$	$\mathrm{SU}(2)$	\(q+(-3-\beta _{2})q^{5}+(5-\beta _{1})q^{7}+(2\beta _{1}+\cdots)q^{11}+\cdots\)
2304.4.a.bv	$2304$	$4$	2304.a	1.a	$1$	$3$	$3$	$135.940$	3.3.1436.1	None	✓				24.4.d.a	$1$	$1$	\(0\)	\(0\)	\(10\)	\(-14\)		$+$	$-$	$-$	$2^{5}$	$\mathrm{SU}(2)$	\(q+(3+\beta _{2})q^{5}+(-5+\beta _{1})q^{7}+(2\beta _{1}+\cdots)q^{11}+\cdots\)
2304.4.a.bw	$2304$	$4$	2304.a	1.a	$1$	$3$	$3$	$135.940$	3.3.1436.1	None	✓				24.4.d.a	$1$	$0$	\(0\)	\(0\)	\(10\)	\(14\)		$-$	$-$	$+$	$2^{5}$	$\mathrm{SU}(2)$	\(q+(3+\beta _{2})q^{5}+(5-\beta _{1})q^{7}+(-2\beta _{1}+\cdots)q^{11}+\cdots\)
2304.4.a.bx	$2304$	$4$	2304.a	1.a	$1$	$4$	$4$	$135.940$	\(\Q(\sqrt{10}, \sqrt{22})\)	None	✓				72.4.d.c	$4$	$0$	\(0\)	\(0\)	\(0\)	\(-40\)		$+$	$+$	$+$	$2^{9}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{5}-10q^{7}+6\beta _{1}q^{11}-\beta _{3}q^{13}+\cdots\)
2304.4.a.by	$2304$	$4$	2304.a	1.a	$1$	$4$	$4$	$135.940$	4.4.9792.1	None	✓				384.4.d.f	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$+$	$2^{10}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{5}+(-\beta _{1}+\beta _{3})q^{7}+(-12-\beta _{2}+\cdots)q^{11}+\cdots\)
2304.4.a.bz	$2304$	$4$	2304.a	1.a	$1$	$4$	$4$	$135.940$	\(\Q(\sqrt{2}, \sqrt{5})\)	None	✓		✓		128.4.b.e	$4$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$+$	$-$	$-$	$2^{10}$	$\mathrm{SU}(2)$	\(q+\beta _{3}q^{5}-\beta _{2}q^{7}+7\beta _{1}q^{11}-\beta _{3}q^{13}+\cdots\)
2304.4.a.ca	$2304$	$4$	2304.a	1.a	$1$	$4$	$4$	$135.940$	\(\Q(\zeta_{24})^+\)	\(\Q(\sqrt{-6}) \)	✓				1152.4.d.l	$8$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$+$	$-$	$2^{5}\cdot 3^{2}$	$N(\mathrm{U}(1))$	\(q+\beta _{3}q^{5}-\beta _{2}q^{7}-\beta _{1}q^{11}-17q^{25}+\cdots\)
2304.4.a.cb	$2304$	$4$	2304.a	1.a	$1$	$4$	$4$	$135.940$	4.4.9792.1	None	✓				384.4.d.f	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$+$	$2^{10}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{5}+(\beta _{1}-\beta _{3})q^{7}+(12+\beta _{2})q^{11}+\cdots\)
2304.4.a.cc	$2304$	$4$	2304.a	1.a	$1$	$4$	$4$	$135.940$	\(\Q(\sqrt{10}, \sqrt{22})\)	None	✓				72.4.d.c	$4$	$1$	\(0\)	\(0\)	\(0\)	\(40\)		$-$	$+$	$-$	$2^{9}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{5}+10q^{7}-6\beta _{1}q^{11}-\beta _{3}q^{13}+\cdots\)
2304.4.a.cd	$2304$	$4$	2304.a	1.a	$1$	$8$	$8$	$135.940$	8.8.\(\cdots\).7	None	✓		✓		1152.4.d.q	$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$+$	$+$	$+$	$2^{26}$	$\mathrm{SU}(2)$	\(q-\beta _{2}q^{5}-\beta _{3}q^{7}-\beta _{5}q^{11}-\beta _{6}q^{13}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(2304))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(2304)) \simeq \) \(S_{4}^{\mathrm{new}}(\Gamma_0(6))\)\(^{\oplus 16}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(8))\)\(^{\oplus 18}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(9))\)\(^{\oplus 9}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(12))\)\(^{\oplus 14}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(16))\)\(^{\oplus 15}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(18))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(24))\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(32))\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(36))\)\(^{\oplus 7}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(48))\)\(^{\oplus 10}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(64))\)\(^{\oplus 9}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(72))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(96))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(128))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(144))\)\(^{\oplus 5}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(192))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(256))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(288))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(384))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(576))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(768))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(1152))\)\(^{\oplus 2}\)