Space of modular forms of level 2352, weight 4, and trivial character

• Label: 2352.4.a
• Level: $2352$
• Weight: $4$
• Character orbit: 2352.a
• Rep. character: $\chi_{2352}(1,\cdot)$
• Character field: $\Q$
• Dimension: $123$
• Newform subspaces: $70$
• Sturm bound: $1792$
• Trace bound: $11$

Defining parameters

Level:	\( N \)	\(=\)	\( 2352 = 2^{4} \cdot 3 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2352.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 70 \)
Sturm bound:			\(1792\)
Trace bound:			\(11\)
Distinguishing \(T_p\):			\(5\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(2352))\).

	Total	New	Old
Modular forms	1392	123	1269
Cusp forms	1296	123	1173
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.

\(2\)	\(3\)	\(7\)	Fricke		Total				Cusp				Eisenstein
					All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)	\(+\)		\(176\)	\(15\)	\(161\)		\(164\)	\(15\)	\(149\)		\(12\)	\(0\)	\(12\)
\(+\)	\(+\)	\(-\)	\(-\)		\(173\)	\(15\)	\(158\)		\(161\)	\(15\)	\(146\)		\(12\)	\(0\)	\(12\)
\(+\)	\(-\)	\(+\)	\(-\)		\(172\)	\(13\)	\(159\)		\(160\)	\(13\)	\(147\)		\(12\)	\(0\)	\(12\)
\(+\)	\(-\)	\(-\)	\(+\)		\(175\)	\(18\)	\(157\)		\(163\)	\(18\)	\(145\)		\(12\)	\(0\)	\(12\)
\(-\)	\(+\)	\(+\)	\(-\)		\(172\)	\(16\)	\(156\)		\(160\)	\(16\)	\(144\)		\(12\)	\(0\)	\(12\)
\(-\)	\(+\)	\(-\)	\(+\)		\(175\)	\(15\)	\(160\)		\(163\)	\(15\)	\(148\)		\(12\)	\(0\)	\(12\)
\(-\)	\(-\)	\(+\)	\(+\)		\(176\)	\(16\)	\(160\)		\(164\)	\(16\)	\(148\)		\(12\)	\(0\)	\(12\)
\(-\)	\(-\)	\(-\)	\(-\)		\(173\)	\(15\)	\(158\)		\(161\)	\(15\)	\(146\)		\(12\)	\(0\)	\(12\)
Plus space			\(+\)		\(702\)	\(64\)	\(638\)		\(654\)	\(64\)	\(590\)		\(48\)	\(0\)	\(48\)
Minus space			\(-\)		\(690\)	\(59\)	\(631\)		\(642\)	\(59\)	\(583\)		\(48\)	\(0\)	\(48\)

Trace form

123 * q + 3 * q^3 + 2 * q^5 + 1107 * q^9 + 20 * q^11 - 46 * q^13 + 30 * q^15 - 26 * q^17 + 36 * q^19 + 248 * q^23 + 3013 * q^25 + 27 * q^27 + 58 * q^29 - 288 * q^31 + 12 * q^33 + 74 * q^37 - 54 * q^39 - 178 * q^41 + 320 * q^43 + 18 * q^45 - 168 * q^47 - 678 * q^51 - 1054 * q^53 - 2048 * q^55 + 84 * q^57 - 628 * q^59 - 734 * q^61 - 68 * q^65 - 96 * q^67 - 264 * q^69 - 32 * q^71 + 14 * q^73 - 27 * q^75 + 1684 * q^79 + 9963 * q^81 - 964 * q^83 - 1228 * q^85 + 162 * q^87 + 1646 * q^89 - 456 * q^93 + 4448 * q^95 + 918 * q^97 + 180 * q^99

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(2352))\) 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	7
2352.4.a.a	$2352$	$4$	2352.a	1.a	$1$	$1$	$1$	$138.772$	\(\Q\)	None	✓				42.4.a.a	$1$	$0$	\(0\)	\(-3\)	\(-18\)	\(0\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-3q^{3}-18q^{5}+9q^{9}+72q^{11}+34q^{13}+\cdots\)
2352.4.a.b	$2352$	$4$	2352.a	1.a	$1$	$1$	$1$	$138.772$	\(\Q\)	None	✓				147.4.a.d	$1$	$0$	\(0\)	\(-3\)	\(-12\)	\(0\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-3q^{3}-12q^{5}+9q^{9}-20q^{11}+84q^{13}+\cdots\)
2352.4.a.c	$2352$	$4$	2352.a	1.a	$1$	$1$	$1$	$138.772$	\(\Q\)	None	✓				168.4.q.a	$1$	$1$	\(0\)	\(-3\)	\(-7\)	\(0\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-3q^{3}-7q^{5}+9q^{9}-7q^{11}+52q^{13}+\cdots\)
2352.4.a.d	$2352$	$4$	2352.a	1.a	$1$	$1$	$1$	$138.772$	\(\Q\)	None	✓				84.4.a.a	$1$	$2$	\(0\)	\(-3\)	\(-6\)	\(0\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-3q^{3}-6q^{5}+9q^{9}-6^{2}q^{11}-62q^{13}+\cdots\)
2352.4.a.e	$2352$	$4$	2352.a	1.a	$1$	$1$	$1$	$138.772$	\(\Q\)	None	✓				6.4.a.a	$1$	$0$	\(0\)	\(-3\)	\(-6\)	\(0\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-3q^{3}-6q^{5}+9q^{9}-12q^{11}-38q^{13}+\cdots\)
2352.4.a.f	$2352$	$4$	2352.a	1.a	$1$	$1$	$1$	$138.772$	\(\Q\)	None	✓				42.4.e.a	$1$	$1$	\(0\)	\(-3\)	\(-6\)	\(0\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-3q^{3}-6q^{5}+9q^{9}+30q^{11}+53q^{13}+\cdots\)
2352.4.a.g	$2352$	$4$	2352.a	1.a	$1$	$1$	$1$	$138.772$	\(\Q\)	None	✓				588.4.a.b	$1$	$0$	\(0\)	\(-3\)	\(-4\)	\(0\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-3q^{3}-4q^{5}+9q^{9}+20q^{11}+4q^{13}+\cdots\)
2352.4.a.h	$2352$	$4$	2352.a	1.a	$1$	$1$	$1$	$138.772$	\(\Q\)	None	✓				168.4.a.c	$1$	$1$	\(0\)	\(-3\)	\(-4\)	\(0\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-3q^{3}-4q^{5}+9q^{9}+26q^{11}-2q^{13}+\cdots\)
2352.4.a.i	$2352$	$4$	2352.a	1.a	$1$	$1$	$1$	$138.772$	\(\Q\)	None	✓				21.4.e.a	$1$	$1$	\(0\)	\(-3\)	\(-3\)	\(0\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-3q^{3}-3q^{5}+9q^{9}+15q^{11}-2^{6}q^{13}+\cdots\)
2352.4.a.j	$2352$	$4$	2352.a	1.a	$1$	$1$	$1$	$138.772$	\(\Q\)	None	✓				168.4.q.b	$1$	$1$	\(0\)	\(-3\)	\(-2\)	\(0\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-3q^{3}-2q^{5}+9q^{9}+18q^{11}-33q^{13}+\cdots\)
2352.4.a.k	$2352$	$4$	2352.a	1.a	$1$	$1$	$1$	$138.772$	\(\Q\)	None	✓				168.4.a.b	$1$	$1$	\(0\)	\(-3\)	\(2\)	\(0\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-3q^{3}+2q^{5}+9q^{9}-12q^{11}+66q^{13}+\cdots\)
2352.4.a.l	$2352$	$4$	2352.a	1.a	$1$	$1$	$1$	$138.772$	\(\Q\)	None	✓				21.4.a.b	$1$	$0$	\(0\)	\(-3\)	\(4\)	\(0\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-3q^{3}+4q^{5}+9q^{9}-62q^{11}+62q^{13}+\cdots\)
2352.4.a.m	$2352$	$4$	2352.a	1.a	$1$	$1$	$1$	$138.772$	\(\Q\)	None	✓				294.4.a.b	$1$	$0$	\(0\)	\(-3\)	\(8\)	\(0\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-3q^{3}+8q^{5}+9q^{9}-40q^{11}+4q^{13}+\cdots\)
2352.4.a.n	$2352$	$4$	2352.a	1.a	$1$	$1$	$1$	$138.772$	\(\Q\)	None	✓				168.4.a.a	$1$	$1$	\(0\)	\(-3\)	\(10\)	\(0\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-3q^{3}+10q^{5}+9q^{9}+12q^{11}-30q^{13}+\cdots\)
2352.4.a.o	$2352$	$4$	2352.a	1.a	$1$	$1$	$1$	$138.772$	\(\Q\)	None	✓				168.4.q.c	$1$	$1$	\(0\)	\(-3\)	\(11\)	\(0\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-3q^{3}+11q^{5}+9q^{9}-39q^{11}+2^{5}q^{13}+\cdots\)
2352.4.a.p	$2352$	$4$	2352.a	1.a	$1$	$1$	$1$	$138.772$	\(\Q\)	None	✓				1176.4.a.b	$1$	$1$	\(0\)	\(-3\)	\(12\)	\(0\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-3q^{3}+12q^{5}+9q^{9}+60q^{11}+44q^{13}+\cdots\)
2352.4.a.q	$2352$	$4$	2352.a	1.a	$1$	$1$	$1$	$138.772$	\(\Q\)	None	✓				42.4.e.b	$1$	$0$	\(0\)	\(-3\)	\(15\)	\(0\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-3q^{3}+15q^{5}+9q^{9}+9q^{11}+88q^{13}+\cdots\)
2352.4.a.r	$2352$	$4$	2352.a	1.a	$1$	$1$	$1$	$138.772$	\(\Q\)	None	✓				21.4.a.a	$1$	$0$	\(0\)	\(-3\)	\(18\)	\(0\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-3q^{3}+18q^{5}+9q^{9}+6^{2}q^{11}+34q^{13}+\cdots\)
2352.4.a.s	$2352$	$4$	2352.a	1.a	$1$	$1$	$1$	$138.772$	\(\Q\)	None	✓				147.4.a.f	$1$	$0$	\(0\)	\(-3\)	\(18\)	\(0\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-3q^{3}+18q^{5}+9q^{9}+50q^{11}-6^{2}q^{13}+\cdots\)
2352.4.a.t	$2352$	$4$	2352.a	1.a	$1$	$1$	$1$	$138.772$	\(\Q\)	None	✓				147.4.a.f	$1$	$1$	\(0\)	\(3\)	\(-18\)	\(0\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3q^{3}-18q^{5}+9q^{9}+50q^{11}+6^{2}q^{13}+\cdots\)
2352.4.a.u	$2352$	$4$	2352.a	1.a	$1$	$1$	$1$	$138.772$	\(\Q\)	None	✓				42.4.e.b	$1$	$0$	\(0\)	\(3\)	\(-15\)	\(0\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+3q^{3}-15q^{5}+9q^{9}+9q^{11}-88q^{13}+\cdots\)
2352.4.a.v	$2352$	$4$	2352.a	1.a	$1$	$1$	$1$	$138.772$	\(\Q\)	None	✓				84.4.a.b	$1$	$1$	\(0\)	\(3\)	\(-14\)	\(0\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3q^{3}-14q^{5}+9q^{9}-4q^{11}-54q^{13}+\cdots\)
2352.4.a.w	$2352$	$4$	2352.a	1.a	$1$	$1$	$1$	$138.772$	\(\Q\)	None	✓				24.4.a.a	$1$	$0$	\(0\)	\(3\)	\(-14\)	\(0\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+3q^{3}-14q^{5}+9q^{9}+28q^{11}+74q^{13}+\cdots\)
2352.4.a.x	$2352$	$4$	2352.a	1.a	$1$	$1$	$1$	$138.772$	\(\Q\)	None	✓				1176.4.a.b	$1$	$0$	\(0\)	\(3\)	\(-12\)	\(0\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+3q^{3}-12q^{5}+9q^{9}+60q^{11}-44q^{13}+\cdots\)
2352.4.a.y	$2352$	$4$	2352.a	1.a	$1$	$1$	$1$	$138.772$	\(\Q\)	None	✓				168.4.q.c	$1$	$1$	\(0\)	\(3\)	\(-11\)	\(0\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3q^{3}-11q^{5}+9q^{9}-39q^{11}-2^{5}q^{13}+\cdots\)
2352.4.a.z	$2352$	$4$	2352.a	1.a	$1$	$1$	$1$	$138.772$	\(\Q\)	None	✓				294.4.a.b	$1$	$1$	\(0\)	\(3\)	\(-8\)	\(0\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3q^{3}-8q^{5}+9q^{9}-40q^{11}-4q^{13}+\cdots\)
2352.4.a.ba	$2352$	$4$	2352.a	1.a	$1$	$1$	$1$	$138.772$	\(\Q\)	None	✓				42.4.a.b	$1$	$1$	\(0\)	\(3\)	\(-2\)	\(0\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3q^{3}-2q^{5}+9q^{9}+8q^{11}+42q^{13}+\cdots\)
2352.4.a.bb	$2352$	$4$	2352.a	1.a	$1$	$1$	$1$	$138.772$	\(\Q\)	None	✓				168.4.a.f	$1$	$0$	\(0\)	\(3\)	\(2\)	\(0\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+3q^{3}+2q^{5}+9q^{9}-52q^{11}-86q^{13}+\cdots\)
2352.4.a.bc	$2352$	$4$	2352.a	1.a	$1$	$1$	$1$	$138.772$	\(\Q\)	None	✓				168.4.q.b	$1$	$1$	\(0\)	\(3\)	\(2\)	\(0\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3q^{3}+2q^{5}+9q^{9}+18q^{11}+33q^{13}+\cdots\)
2352.4.a.bd	$2352$	$4$	2352.a	1.a	$1$	$1$	$1$	$138.772$	\(\Q\)	None	✓				21.4.e.a	$1$	$1$	\(0\)	\(3\)	\(3\)	\(0\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3q^{3}+3q^{5}+9q^{9}+15q^{11}+2^{6}q^{13}+\cdots\)
2352.4.a.be	$2352$	$4$	2352.a	1.a	$1$	$1$	$1$	$138.772$	\(\Q\)	None	✓				588.4.a.b	$1$	$1$	\(0\)	\(3\)	\(4\)	\(0\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3q^{3}+4q^{5}+9q^{9}+20q^{11}-4q^{13}+\cdots\)
2352.4.a.bf	$2352$	$4$	2352.a	1.a	$1$	$1$	$1$	$138.772$	\(\Q\)	None	✓				42.4.e.a	$1$	$1$	\(0\)	\(3\)	\(6\)	\(0\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3q^{3}+6q^{5}+9q^{9}+30q^{11}-53q^{13}+\cdots\)
2352.4.a.bg	$2352$	$4$	2352.a	1.a	$1$	$1$	$1$	$138.772$	\(\Q\)	None	✓				168.4.q.a	$1$	$1$	\(0\)	\(3\)	\(7\)	\(0\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3q^{3}+7q^{5}+9q^{9}-7q^{11}-52q^{13}+\cdots\)
2352.4.a.bh	$2352$	$4$	2352.a	1.a	$1$	$1$	$1$	$138.772$	\(\Q\)	None	✓				168.4.a.e	$1$	$0$	\(0\)	\(3\)	\(10\)	\(0\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+3q^{3}+10q^{5}+9q^{9}+52q^{11}+10q^{13}+\cdots\)
2352.4.a.bi	$2352$	$4$	2352.a	1.a	$1$	$1$	$1$	$138.772$	\(\Q\)	None	✓				147.4.a.d	$1$	$1$	\(0\)	\(3\)	\(12\)	\(0\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3q^{3}+12q^{5}+9q^{9}-20q^{11}-84q^{13}+\cdots\)
2352.4.a.bj	$2352$	$4$	2352.a	1.a	$1$	$1$	$1$	$138.772$	\(\Q\)	None	✓				168.4.a.d	$1$	$0$	\(0\)	\(3\)	\(16\)	\(0\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+3q^{3}+2^{4}q^{5}+9q^{9}+18q^{11}+54q^{13}+\cdots\)
2352.4.a.bk	$2352$	$4$	2352.a	1.a	$1$	$1$	$1$	$138.772$	\(\Q\)	None	✓				12.4.a.a	$1$	$1$	\(0\)	\(3\)	\(18\)	\(0\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3q^{3}+18q^{5}+9q^{9}-6^{2}q^{11}+10q^{13}+\cdots\)
2352.4.a.bl	$2352$	$4$	2352.a	1.a	$1$	$2$	$2$	$138.772$	\(\Q(\sqrt{2}) \)	None	✓				147.4.a.j	$1$	$1$	\(0\)	\(-6\)	\(-20\)	\(0\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-3q^{3}+(-10+7\beta )q^{5}+9q^{9}+(10+\cdots)q^{11}+\cdots\)
2352.4.a.bm	$2352$	$4$	2352.a	1.a	$1$	$2$	$2$	$138.772$	\(\Q(\sqrt{137}) \)	None	✓				1176.4.a.s	$1$	$1$	\(0\)	\(-6\)	\(-18\)	\(0\)		$+$	$+$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q-3q^{3}+(-9-\beta )q^{5}+9q^{9}+(-29+\cdots)q^{11}+\cdots\)
2352.4.a.bn	$2352$	$4$	2352.a	1.a	$1$	$2$	$2$	$138.772$	\(\Q(\sqrt{2}) \)	None	✓				294.4.a.j	$1$	$1$	\(0\)	\(-6\)	\(-12\)	\(0\)		$-$	$+$	$+$	$-$	$7$	$\mathrm{SU}(2)$	\(q-3q^{3}+(-6+\beta )q^{5}+9q^{9}+(2+6\beta )q^{11}+\cdots\)
2352.4.a.bo	$2352$	$4$	2352.a	1.a	$1$	$2$	$2$	$138.772$	\(\Q(\sqrt{505}) \)	None	✓				168.4.q.d	$1$	$1$	\(0\)	\(-6\)	\(-9\)	\(0\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-3q^{3}+(-4-\beta )q^{5}+9q^{9}-5\beta q^{11}+\cdots\)
2352.4.a.bp	$2352$	$4$	2352.a	1.a	$1$	$2$	$2$	$138.772$	\(\Q(\sqrt{57}) \)	None	✓				84.4.i.b	$1$	$0$	\(0\)	\(-6\)	\(3\)	\(0\)		$-$	$+$	$-$	$+$	$3$	$\mathrm{SU}(2)$	\(q-3q^{3}+(2+\beta )q^{5}+9q^{9}+(26+\beta )q^{11}+\cdots\)
2352.4.a.bq	$2352$	$4$	2352.a	1.a	$1$	$2$	$2$	$138.772$	\(\Q(\sqrt{1345}) \)	None	✓				42.4.e.c	$1$	$1$	\(0\)	\(-6\)	\(5\)	\(0\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-3q^{3}+(3-\beta )q^{5}+9q^{9}+(-33-\beta )q^{11}+\cdots\)
2352.4.a.br	$2352$	$4$	2352.a	1.a	$1$	$2$	$2$	$138.772$	\(\Q(\sqrt{337}) \)	None	✓				168.4.a.g	$1$	$1$	\(0\)	\(-6\)	\(6\)	\(0\)		$+$	$+$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q-3q^{3}+(3+\beta )q^{5}+9q^{9}+(-13-\beta )q^{11}+\cdots\)
2352.4.a.bs	$2352$	$4$	2352.a	1.a	$1$	$2$	$2$	$138.772$	\(\Q(\sqrt{113}) \)	None	✓				1176.4.a.q	$1$	$1$	\(0\)	\(-6\)	\(6\)	\(0\)		$+$	$+$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q-3q^{3}+(3+\beta )q^{5}+9q^{9}+(-1-3\beta )q^{11}+\cdots\)
2352.4.a.bt	$2352$	$4$	2352.a	1.a	$1$	$2$	$2$	$138.772$	\(\Q(\sqrt{193}) \)	None	✓				84.4.i.a	$1$	$1$	\(0\)	\(-6\)	\(11\)	\(0\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-3q^{3}+(6-\beta )q^{5}+9q^{9}+(6-7\beta )q^{11}+\cdots\)
2352.4.a.bu	$2352$	$4$	2352.a	1.a	$1$	$2$	$2$	$138.772$	\(\Q(\sqrt{2}) \)	None	✓				294.4.a.l	$1$	$1$	\(0\)	\(-6\)	\(12\)	\(0\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-3q^{3}+(6+\beta )q^{5}+9q^{9}+(-2-6\beta )q^{11}+\cdots\)
2352.4.a.bv	$2352$	$4$	2352.a	1.a	$1$	$2$	$2$	$138.772$	\(\Q(\sqrt{177}) \)	None	✓				168.4.a.h	$1$	$0$	\(0\)	\(6\)	\(-14\)	\(0\)		$+$	$-$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q+3q^{3}+(-7-\beta )q^{5}+9q^{9}+(-9+\cdots)q^{11}+\cdots\)
2352.4.a.bw	$2352$	$4$	2352.a	1.a	$1$	$2$	$2$	$138.772$	\(\Q(\sqrt{2}) \)	None	✓				294.4.a.l	$1$	$0$	\(0\)	\(6\)	\(-12\)	\(0\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+3q^{3}+(-6+\beta )q^{5}+9q^{9}+(-2+\cdots)q^{11}+\cdots\)
2352.4.a.bx	$2352$	$4$	2352.a	1.a	$1$	$2$	$2$	$138.772$	\(\Q(\sqrt{193}) \)	None	✓				84.4.i.a	$1$	$1$	\(0\)	\(6\)	\(-11\)	\(0\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3q^{3}+(-5-\beta )q^{5}+9q^{9}+(-1+\cdots)q^{11}+\cdots\)
2352.4.a.by	$2352$	$4$	2352.a	1.a	$1$	$2$	$2$	$138.772$	\(\Q(\sqrt{113}) \)	None	✓				1176.4.a.q	$1$	$0$	\(0\)	\(6\)	\(-6\)	\(0\)		$+$	$-$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q+3q^{3}+(-3-\beta )q^{5}+9q^{9}+(-1+\cdots)q^{11}+\cdots\)
2352.4.a.bz	$2352$	$4$	2352.a	1.a	$1$	$2$	$2$	$138.772$	\(\Q(\sqrt{57}) \)	None	✓				21.4.a.c	$1$	$1$	\(0\)	\(6\)	\(-6\)	\(0\)		$-$	$-$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+3q^{3}+(-3-\beta )q^{5}+9q^{9}+(3-5\beta )q^{11}+\cdots\)
2352.4.a.ca	$2352$	$4$	2352.a	1.a	$1$	$2$	$2$	$138.772$	\(\Q(\sqrt{1345}) \)	None	✓				42.4.e.c	$1$	$1$	\(0\)	\(6\)	\(-5\)	\(0\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3q^{3}+(-2-\beta )q^{5}+9q^{9}+(-34+\cdots)q^{11}+\cdots\)
2352.4.a.cb	$2352$	$4$	2352.a	1.a	$1$	$2$	$2$	$138.772$	\(\Q(\sqrt{57}) \)	None	✓				84.4.i.b	$1$	$0$	\(0\)	\(6\)	\(-3\)	\(0\)		$-$	$-$	$+$	$+$	$3$	$\mathrm{SU}(2)$	\(q+3q^{3}+(-2-\beta )q^{5}+9q^{9}+(26+\beta )q^{11}+\cdots\)
2352.4.a.cc	$2352$	$4$	2352.a	1.a	$1$	$2$	$2$	$138.772$	\(\Q(\sqrt{505}) \)	None	✓				168.4.q.d	$1$	$1$	\(0\)	\(6\)	\(9\)	\(0\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3q^{3}+(5-\beta )q^{5}+9q^{9}+(-5+5\beta )q^{11}+\cdots\)
2352.4.a.cd	$2352$	$4$	2352.a	1.a	$1$	$2$	$2$	$138.772$	\(\Q(\sqrt{2}) \)	None	✓				294.4.a.j	$1$	$0$	\(0\)	\(6\)	\(12\)	\(0\)		$-$	$-$	$+$	$+$	$7$	$\mathrm{SU}(2)$	\(q+3q^{3}+(6+\beta )q^{5}+9q^{9}+(2-6\beta )q^{11}+\cdots\)
2352.4.a.ce	$2352$	$4$	2352.a	1.a	$1$	$2$	$2$	$138.772$	\(\Q(\sqrt{137}) \)	None	✓				1176.4.a.s	$1$	$0$	\(0\)	\(6\)	\(18\)	\(0\)		$+$	$-$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q+3q^{3}+(9-\beta )q^{5}+9q^{9}+(-29-3\beta )q^{11}+\cdots\)
2352.4.a.cf	$2352$	$4$	2352.a	1.a	$1$	$2$	$2$	$138.772$	\(\Q(\sqrt{2}) \)	None	✓				147.4.a.j	$1$	$0$	\(0\)	\(6\)	\(20\)	\(0\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+3q^{3}+(10+7\beta )q^{5}+9q^{9}+(10-24\beta )q^{11}+\cdots\)
2352.4.a.cg	$2352$	$4$	2352.a	1.a	$1$	$3$	$3$	$138.772$	3.3.57516.1	None	✓		✓		21.4.e.b	$1$	$0$	\(0\)	\(-9\)	\(-11\)	\(0\)		$-$	$+$	$-$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q-3q^{3}+(-4-\beta _{2})q^{5}+9q^{9}+(-11+\cdots)q^{11}+\cdots\)
2352.4.a.ch	$2352$	$4$	2352.a	1.a	$1$	$3$	$3$	$138.772$	3.3.58461.1	None	✓				168.4.q.e	$1$	$0$	\(0\)	\(-9\)	\(-11\)	\(0\)		$+$	$+$	$+$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q-3q^{3}+(-4-\beta _{1})q^{5}+9q^{9}+(6-\beta _{1}+\cdots)q^{11}+\cdots\)
2352.4.a.ci	$2352$	$4$	2352.a	1.a	$1$	$3$	$3$	$138.772$	3.3.57516.1	None	✓				21.4.e.b	$1$	$0$	\(0\)	\(9\)	\(11\)	\(0\)		$-$	$-$	$+$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q+3q^{3}+(4+\beta _{2})q^{5}+9q^{9}+(-11+\cdots)q^{11}+\cdots\)
2352.4.a.cj	$2352$	$4$	2352.a	1.a	$1$	$3$	$3$	$138.772$	3.3.58461.1	None	✓				168.4.q.e	$1$	$0$	\(0\)	\(9\)	\(11\)	\(0\)		$+$	$-$	$-$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q+3q^{3}+(4+\beta _{1})q^{5}+9q^{9}+(6-\beta _{1}+\cdots)q^{11}+\cdots\)
2352.4.a.ck	$2352$	$4$	2352.a	1.a	$1$	$4$	$4$	$138.772$	4.4.145408.2	None	✓				1176.4.a.bb	$1$	$0$	\(0\)	\(-12\)	\(-8\)	\(0\)		$+$	$+$	$+$	$+$	$2^{3}$	$\mathrm{SU}(2)$	\(q-3q^{3}+(-2+\beta _{2}-3\beta _{3})q^{5}+9q^{9}+\cdots\)
2352.4.a.cl	$2352$	$4$	2352.a	1.a	$1$	$4$	$4$	$138.772$	4.4.136768.1	None	✓		✓		588.4.a.j	$1$	$1$	\(0\)	\(-12\)	\(0\)	\(0\)		$-$	$+$	$+$	$-$	$2^{4}\cdot 7$	$\mathrm{SU}(2)$	\(q-3q^{3}+\beta _{1}q^{5}+9q^{9}+(-3\beta _{1}+\beta _{3})q^{11}+\cdots\)
2352.4.a.cm	$2352$	$4$	2352.a	1.a	$1$	$4$	$4$	$138.772$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓				168.4.q.f	$1$	$0$	\(0\)	\(-12\)	\(4\)	\(0\)		$+$	$+$	$+$	$+$	$2^{3}\cdot 7$	$\mathrm{SU}(2)$	\(q-3q^{3}+(1+\beta _{1})q^{5}+9q^{9}+(-3-2\beta _{1}+\cdots)q^{11}+\cdots\)
2352.4.a.cn	$2352$	$4$	2352.a	1.a	$1$	$4$	$4$	$138.772$	4.4.391168.1	None	✓				1176.4.a.z	$1$	$0$	\(0\)	\(-12\)	\(8\)	\(0\)		$+$	$+$	$+$	$+$	$2^{3}\cdot 7$	$\mathrm{SU}(2)$	\(q-3q^{3}+(2+\beta _{1})q^{5}+9q^{9}+(-10+\cdots)q^{11}+\cdots\)
2352.4.a.co	$2352$	$4$	2352.a	1.a	$1$	$4$	$4$	$138.772$	4.4.391168.1	None	✓				1176.4.a.z	$1$	$1$	\(0\)	\(12\)	\(-8\)	\(0\)		$+$	$-$	$+$	$-$	$2^{3}\cdot 7$	$\mathrm{SU}(2)$	\(q+3q^{3}+(-2-\beta _{1})q^{5}+9q^{9}+(-10+\cdots)q^{11}+\cdots\)
2352.4.a.cp	$2352$	$4$	2352.a	1.a	$1$	$4$	$4$	$138.772$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓		✓		168.4.q.f	$1$	$0$	\(0\)	\(12\)	\(-4\)	\(0\)		$+$	$-$	$-$	$+$	$2^{3}\cdot 7$	$\mathrm{SU}(2)$	\(q+3q^{3}+(-1-\beta _{1})q^{5}+9q^{9}+(-3+\cdots)q^{11}+\cdots\)
2352.4.a.cq	$2352$	$4$	2352.a	1.a	$1$	$4$	$4$	$138.772$	4.4.136768.1	None	✓		✓		588.4.a.j	$1$	$0$	\(0\)	\(12\)	\(0\)	\(0\)		$-$	$-$	$+$	$+$	$2^{4}\cdot 7$	$\mathrm{SU}(2)$	\(q+3q^{3}-\beta _{1}q^{5}+9q^{9}+(-3\beta _{1}+\beta _{3})q^{11}+\cdots\)
2352.4.a.cr	$2352$	$4$	2352.a	1.a	$1$	$4$	$4$	$138.772$	4.4.145408.2	None	✓				1176.4.a.bb	$1$	$1$	\(0\)	\(12\)	\(8\)	\(0\)		$+$	$-$	$+$	$-$	$2^{3}$	$\mathrm{SU}(2)$	\(q+3q^{3}+(2+\beta _{2}+3\beta _{3})q^{5}+9q^{9}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(2352))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(2352)) \simeq \) \(S_{4}^{\mathrm{new}}(\Gamma_0(6))\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(7))\)\(^{\oplus 20}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(8))\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(12))\)\(^{\oplus 9}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 16}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(16))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(21))\)\(^{\oplus 10}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(24))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(28))\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(42))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(48))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(49))\)\(^{\oplus 10}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(56))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(84))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(98))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(112))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(147))\)\(^{\oplus 5}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(168))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(196))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(294))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(336))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(392))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(588))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(784))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(1176))\)\(^{\oplus 2}\)