Space of modular forms of level 1344, weight 4, and trivial character

• Label: 1344.4.a
• Level: $1344$
• Weight: $4$
• Character orbit: 1344.a
• Rep. character: $\chi_{1344}(1,\cdot)$
• Character field: $\Q$
• Dimension: $72$
• Newform subspaces: $48$
• Sturm bound: $1024$
• Trace bound: $11$

Defining parameters

Level:	\( N \)	\(=\)	\( 1344 = 2^{6} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1344.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 48 \)
Sturm bound:			\(1024\)
Trace bound:			\(11\)
Distinguishing \(T_p\):			\(5\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(1344))\).

	Total	New	Old
Modular forms	792	72	720
Cusp forms	744	72	672
Eisenstein series	48	0	48

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
\(+\)	\(+\)	\(+\)	\(+\)		\(102\)	\(10\)	\(92\)		\(96\)	\(10\)	\(86\)		\(6\)	\(0\)	\(6\)
\(+\)	\(+\)	\(-\)	\(-\)		\(98\)	\(9\)	\(89\)		\(92\)	\(9\)	\(83\)		\(6\)	\(0\)	\(6\)
\(+\)	\(-\)	\(+\)	\(-\)		\(96\)	\(8\)	\(88\)		\(90\)	\(8\)	\(82\)		\(6\)	\(0\)	\(6\)
\(+\)	\(-\)	\(-\)	\(+\)		\(100\)	\(11\)	\(89\)		\(94\)	\(11\)	\(83\)		\(6\)	\(0\)	\(6\)
\(-\)	\(+\)	\(+\)	\(-\)		\(96\)	\(8\)	\(88\)		\(90\)	\(8\)	\(82\)		\(6\)	\(0\)	\(6\)
\(-\)	\(+\)	\(-\)	\(+\)		\(100\)	\(9\)	\(91\)		\(94\)	\(9\)	\(85\)		\(6\)	\(0\)	\(6\)
\(-\)	\(-\)	\(+\)	\(+\)		\(102\)	\(10\)	\(92\)		\(96\)	\(10\)	\(86\)		\(6\)	\(0\)	\(6\)
\(-\)	\(-\)	\(-\)	\(-\)		\(98\)	\(7\)	\(91\)		\(92\)	\(7\)	\(85\)		\(6\)	\(0\)	\(6\)
Plus space			\(+\)		\(404\)	\(40\)	\(364\)		\(380\)	\(40\)	\(340\)		\(24\)	\(0\)	\(24\)
Minus space			\(-\)		\(388\)	\(32\)	\(356\)		\(364\)	\(32\)	\(332\)		\(24\)	\(0\)	\(24\)

Trace form

72 * q + 648 * q^9 - 208 * q^17 + 1976 * q^25 - 400 * q^29 - 16 * q^37 + 944 * q^41 + 3528 * q^49 + 752 * q^53 - 1824 * q^61 + 3072 * q^65 - 1056 * q^69 + 592 * q^73 + 1904 * q^77 + 5832 * q^81 + 480 * q^85 - 176 * q^89 + 3632 * q^97

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(1344))\) 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
1344.4.a.a	$1344$	$4$	1344.a	1.a	$1$	$1$	$1$	$79.299$	\(\Q\)	None	✓				42.4.a.a	$1$	$1$	\(0\)	\(-3\)	\(-18\)	\(-7\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-3q^{3}-18q^{5}-7q^{7}+9q^{9}-72q^{11}+\cdots\)
1344.4.a.b	$1344$	$4$	1344.a	1.a	$1$	$1$	$1$	$79.299$	\(\Q\)	None	✓				84.4.a.b	$1$	$0$	\(0\)	\(-3\)	\(-14\)	\(-7\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-3q^{3}-14q^{5}-7q^{7}+9q^{9}-4q^{11}+\cdots\)
1344.4.a.c	$1344$	$4$	1344.a	1.a	$1$	$1$	$1$	$79.299$	\(\Q\)	None	✓				672.4.a.b	$1$	$1$	\(0\)	\(-3\)	\(-6\)	\(-7\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-3q^{3}-6q^{5}-7q^{7}+9q^{9}-4q^{11}+\cdots\)
1344.4.a.d	$1344$	$4$	1344.a	1.a	$1$	$1$	$1$	$79.299$	\(\Q\)	None	✓				84.4.a.a	$1$	$1$	\(0\)	\(-3\)	\(-6\)	\(-7\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-3q^{3}-6q^{5}-7q^{7}+9q^{9}+6^{2}q^{11}+\cdots\)
1344.4.a.e	$1344$	$4$	1344.a	1.a	$1$	$1$	$1$	$79.299$	\(\Q\)	None	✓				168.4.a.c	$1$	$0$	\(0\)	\(-3\)	\(-4\)	\(7\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-3q^{3}-4q^{5}+7q^{7}+9q^{9}-26q^{11}+\cdots\)
1344.4.a.f	$1344$	$4$	1344.a	1.a	$1$	$1$	$1$	$79.299$	\(\Q\)	None	✓				42.4.a.b	$1$	$0$	\(0\)	\(-3\)	\(-2\)	\(-7\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-3q^{3}-2q^{5}-7q^{7}+9q^{9}+8q^{11}+\cdots\)
1344.4.a.g	$1344$	$4$	1344.a	1.a	$1$	$1$	$1$	$79.299$	\(\Q\)	None	✓				168.4.a.f	$1$	$0$	\(0\)	\(-3\)	\(2\)	\(-7\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-3q^{3}+2q^{5}-7q^{7}+9q^{9}-52q^{11}+\cdots\)
1344.4.a.h	$1344$	$4$	1344.a	1.a	$1$	$1$	$1$	$79.299$	\(\Q\)	None	✓				168.4.a.b	$1$	$1$	\(0\)	\(-3\)	\(2\)	\(-7\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-3q^{3}+2q^{5}-7q^{7}+9q^{9}+12q^{11}+\cdots\)
1344.4.a.i	$1344$	$4$	1344.a	1.a	$1$	$1$	$1$	$79.299$	\(\Q\)	None	✓				21.4.a.b	$1$	$0$	\(0\)	\(-3\)	\(4\)	\(7\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-3q^{3}+4q^{5}+7q^{7}+9q^{9}+62q^{11}+\cdots\)
1344.4.a.j	$1344$	$4$	1344.a	1.a	$1$	$1$	$1$	$79.299$	\(\Q\)	None	✓				168.4.a.a	$1$	$1$	\(0\)	\(-3\)	\(10\)	\(-7\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-3q^{3}+10q^{5}-7q^{7}+9q^{9}-12q^{11}+\cdots\)
1344.4.a.k	$1344$	$4$	1344.a	1.a	$1$	$1$	$1$	$79.299$	\(\Q\)	None	✓				168.4.a.e	$1$	$0$	\(0\)	\(-3\)	\(10\)	\(-7\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-3q^{3}+10q^{5}-7q^{7}+9q^{9}+52q^{11}+\cdots\)
1344.4.a.l	$1344$	$4$	1344.a	1.a	$1$	$1$	$1$	$79.299$	\(\Q\)	None	✓				168.4.a.d	$1$	$1$	\(0\)	\(-3\)	\(16\)	\(7\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-3q^{3}+2^{4}q^{5}+7q^{7}+9q^{9}+18q^{11}+\cdots\)
1344.4.a.m	$1344$	$4$	1344.a	1.a	$1$	$1$	$1$	$79.299$	\(\Q\)	None	✓				672.4.a.a	$1$	$0$	\(0\)	\(-3\)	\(18\)	\(-7\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-3q^{3}+18q^{5}-7q^{7}+9q^{9}-44q^{11}+\cdots\)
1344.4.a.n	$1344$	$4$	1344.a	1.a	$1$	$1$	$1$	$79.299$	\(\Q\)	None	✓				21.4.a.a	$1$	$1$	\(0\)	\(-3\)	\(18\)	\(-7\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-3q^{3}+18q^{5}-7q^{7}+9q^{9}-6^{2}q^{11}+\cdots\)
1344.4.a.o	$1344$	$4$	1344.a	1.a	$1$	$1$	$1$	$79.299$	\(\Q\)	None	✓				42.4.a.a	$1$	$0$	\(0\)	\(3\)	\(-18\)	\(7\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+3q^{3}-18q^{5}+7q^{7}+9q^{9}+72q^{11}+\cdots\)
1344.4.a.p	$1344$	$4$	1344.a	1.a	$1$	$1$	$1$	$79.299$	\(\Q\)	None	✓				84.4.a.b	$1$	$1$	\(0\)	\(3\)	\(-14\)	\(7\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3q^{3}-14q^{5}+7q^{7}+9q^{9}+4q^{11}+\cdots\)
1344.4.a.q	$1344$	$4$	1344.a	1.a	$1$	$1$	$1$	$79.299$	\(\Q\)	None	✓				84.4.a.a	$1$	$0$	\(0\)	\(3\)	\(-6\)	\(7\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+3q^{3}-6q^{5}+7q^{7}+9q^{9}-6^{2}q^{11}+\cdots\)
1344.4.a.r	$1344$	$4$	1344.a	1.a	$1$	$1$	$1$	$79.299$	\(\Q\)	None	✓				672.4.a.b	$1$	$1$	\(0\)	\(3\)	\(-6\)	\(7\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3q^{3}-6q^{5}+7q^{7}+9q^{9}+4q^{11}+\cdots\)
1344.4.a.s	$1344$	$4$	1344.a	1.a	$1$	$1$	$1$	$79.299$	\(\Q\)	None	✓				168.4.a.c	$1$	$1$	\(0\)	\(3\)	\(-4\)	\(-7\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3q^{3}-4q^{5}-7q^{7}+9q^{9}+26q^{11}+\cdots\)
1344.4.a.t	$1344$	$4$	1344.a	1.a	$1$	$1$	$1$	$79.299$	\(\Q\)	None	✓				42.4.a.b	$1$	$1$	\(0\)	\(3\)	\(-2\)	\(7\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3q^{3}-2q^{5}+7q^{7}+9q^{9}-8q^{11}+\cdots\)
1344.4.a.u	$1344$	$4$	1344.a	1.a	$1$	$1$	$1$	$79.299$	\(\Q\)	None	✓				168.4.a.b	$1$	$0$	\(0\)	\(3\)	\(2\)	\(7\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+3q^{3}+2q^{5}+7q^{7}+9q^{9}-12q^{11}+\cdots\)
1344.4.a.v	$1344$	$4$	1344.a	1.a	$1$	$1$	$1$	$79.299$	\(\Q\)	None	✓				168.4.a.f	$1$	$1$	\(0\)	\(3\)	\(2\)	\(7\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3q^{3}+2q^{5}+7q^{7}+9q^{9}+52q^{11}+\cdots\)
1344.4.a.w	$1344$	$4$	1344.a	1.a	$1$	$1$	$1$	$79.299$	\(\Q\)	None	✓				21.4.a.b	$1$	$1$	\(0\)	\(3\)	\(4\)	\(-7\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3q^{3}+4q^{5}-7q^{7}+9q^{9}-62q^{11}+\cdots\)
1344.4.a.x	$1344$	$4$	1344.a	1.a	$1$	$1$	$1$	$79.299$	\(\Q\)	None	✓				168.4.a.e	$1$	$1$	\(0\)	\(3\)	\(10\)	\(7\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3q^{3}+10q^{5}+7q^{7}+9q^{9}-52q^{11}+\cdots\)
1344.4.a.y	$1344$	$4$	1344.a	1.a	$1$	$1$	$1$	$79.299$	\(\Q\)	None	✓				168.4.a.a	$1$	$0$	\(0\)	\(3\)	\(10\)	\(7\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+3q^{3}+10q^{5}+7q^{7}+9q^{9}+12q^{11}+\cdots\)
1344.4.a.z	$1344$	$4$	1344.a	1.a	$1$	$1$	$1$	$79.299$	\(\Q\)	None	✓				168.4.a.d	$1$	$0$	\(0\)	\(3\)	\(16\)	\(-7\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+3q^{3}+2^{4}q^{5}-7q^{7}+9q^{9}-18q^{11}+\cdots\)
1344.4.a.ba	$1344$	$4$	1344.a	1.a	$1$	$1$	$1$	$79.299$	\(\Q\)	None	✓				21.4.a.a	$1$	$0$	\(0\)	\(3\)	\(18\)	\(7\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+3q^{3}+18q^{5}+7q^{7}+9q^{9}+6^{2}q^{11}+\cdots\)
1344.4.a.bb	$1344$	$4$	1344.a	1.a	$1$	$1$	$1$	$79.299$	\(\Q\)	None	✓				672.4.a.a	$1$	$0$	\(0\)	\(3\)	\(18\)	\(7\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+3q^{3}+18q^{5}+7q^{7}+9q^{9}+44q^{11}+\cdots\)
1344.4.a.bc	$1344$	$4$	1344.a	1.a	$1$	$2$	$2$	$79.299$	\(\Q(\sqrt{43}) \)	None	✓		✓		672.4.a.i	$1$	$1$	\(0\)	\(-6\)	\(-16\)	\(-14\)		$-$	$+$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q-3q^{3}+(-8+\beta )q^{5}-7q^{7}+9q^{9}+\cdots\)
1344.4.a.bd	$1344$	$4$	1344.a	1.a	$1$	$2$	$2$	$79.299$	\(\Q(\sqrt{177}) \)	None	✓				168.4.a.h	$1$	$1$	\(0\)	\(-6\)	\(-14\)	\(14\)		$+$	$+$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q-3q^{3}+(-7-\beta )q^{5}+7q^{7}+9q^{9}+\cdots\)
1344.4.a.be	$1344$	$4$	1344.a	1.a	$1$	$2$	$2$	$79.299$	\(\Q(\sqrt{137}) \)	None	✓				672.4.a.h	$1$	$1$	\(0\)	\(-6\)	\(-10\)	\(14\)		$+$	$+$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q-3q^{3}+(-5-\beta )q^{5}+7q^{7}+9q^{9}+\cdots\)
1344.4.a.bf	$1344$	$4$	1344.a	1.a	$1$	$2$	$2$	$79.299$	\(\Q(\sqrt{11}) \)	None	✓				672.4.a.g	$1$	$0$	\(0\)	\(-6\)	\(-8\)	\(-14\)		$+$	$+$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q-3q^{3}+(-4+\beta )q^{5}-7q^{7}+9q^{9}+\cdots\)
1344.4.a.bg	$1344$	$4$	1344.a	1.a	$1$	$2$	$2$	$79.299$	\(\Q(\sqrt{57}) \)	None	✓				21.4.a.c	$1$	$1$	\(0\)	\(-6\)	\(-6\)	\(14\)		$+$	$+$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q-3q^{3}+(-3-\beta )q^{5}+7q^{7}+9q^{9}+\cdots\)
1344.4.a.bh	$1344$	$4$	1344.a	1.a	$1$	$2$	$2$	$79.299$	\(\Q(\sqrt{37}) \)	None	✓				672.4.a.f	$1$	$0$	\(0\)	\(-6\)	\(4\)	\(14\)		$-$	$+$	$-$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q-3q^{3}+(2+\beta )q^{5}+7q^{7}+9q^{9}+(24+\cdots)q^{11}+\cdots\)
1344.4.a.bi	$1344$	$4$	1344.a	1.a	$1$	$2$	$2$	$79.299$	\(\Q(\sqrt{337}) \)	None	✓				168.4.a.g	$1$	$0$	\(0\)	\(-6\)	\(6\)	\(14\)		$-$	$+$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q-3q^{3}+(3+\beta )q^{5}+7q^{7}+9q^{9}+(13+\cdots)q^{11}+\cdots\)
1344.4.a.bj	$1344$	$4$	1344.a	1.a	$1$	$2$	$2$	$79.299$	\(\Q(\sqrt{17}) \)	None	✓				672.4.a.e	$1$	$1$	\(0\)	\(-6\)	\(10\)	\(14\)		$+$	$+$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q-3q^{3}+(5-\beta )q^{5}+7q^{7}+9q^{9}+(11+\cdots)q^{11}+\cdots\)
1344.4.a.bk	$1344$	$4$	1344.a	1.a	$1$	$2$	$2$	$79.299$	\(\Q(\sqrt{43}) \)	None	✓		✓		672.4.a.i	$1$	$1$	\(0\)	\(6\)	\(-16\)	\(14\)		$-$	$-$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+3q^{3}+(-8+\beta )q^{5}+7q^{7}+9q^{9}+\cdots\)
1344.4.a.bl	$1344$	$4$	1344.a	1.a	$1$	$2$	$2$	$79.299$	\(\Q(\sqrt{177}) \)	None	✓				168.4.a.h	$1$	$0$	\(0\)	\(6\)	\(-14\)	\(-14\)		$-$	$-$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q+3q^{3}+(-7-\beta )q^{5}-7q^{7}+9q^{9}+\cdots\)
1344.4.a.bm	$1344$	$4$	1344.a	1.a	$1$	$2$	$2$	$79.299$	\(\Q(\sqrt{137}) \)	None	✓				672.4.a.h	$1$	$1$	\(0\)	\(6\)	\(-10\)	\(-14\)		$+$	$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q+3q^{3}+(-5-\beta )q^{5}-7q^{7}+9q^{9}+\cdots\)
1344.4.a.bn	$1344$	$4$	1344.a	1.a	$1$	$2$	$2$	$79.299$	\(\Q(\sqrt{11}) \)	None	✓				672.4.a.g	$1$	$0$	\(0\)	\(6\)	\(-8\)	\(14\)		$+$	$-$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q+3q^{3}+(-4+\beta )q^{5}+7q^{7}+9q^{9}+\cdots\)
1344.4.a.bo	$1344$	$4$	1344.a	1.a	$1$	$2$	$2$	$79.299$	\(\Q(\sqrt{57}) \)	None	✓				21.4.a.c	$1$	$0$	\(0\)	\(6\)	\(-6\)	\(-14\)		$-$	$-$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q+3q^{3}+(-3-\beta )q^{5}-7q^{7}+9q^{9}+\cdots\)
1344.4.a.bp	$1344$	$4$	1344.a	1.a	$1$	$2$	$2$	$79.299$	\(\Q(\sqrt{37}) \)	None	✓				672.4.a.f	$1$	$0$	\(0\)	\(6\)	\(4\)	\(-14\)		$-$	$-$	$+$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q+3q^{3}+(2+\beta )q^{5}-7q^{7}+9q^{9}+(-24+\cdots)q^{11}+\cdots\)
1344.4.a.bq	$1344$	$4$	1344.a	1.a	$1$	$2$	$2$	$79.299$	\(\Q(\sqrt{337}) \)	None	✓				168.4.a.g	$1$	$1$	\(0\)	\(6\)	\(6\)	\(-14\)		$+$	$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q+3q^{3}+(3+\beta )q^{5}-7q^{7}+9q^{9}+(-13+\cdots)q^{11}+\cdots\)
1344.4.a.br	$1344$	$4$	1344.a	1.a	$1$	$2$	$2$	$79.299$	\(\Q(\sqrt{17}) \)	None	✓				672.4.a.e	$1$	$1$	\(0\)	\(6\)	\(10\)	\(-14\)		$+$	$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q+3q^{3}+(5-\beta )q^{5}-7q^{7}+9q^{9}+(-11+\cdots)q^{11}+\cdots\)
1344.4.a.bs	$1344$	$4$	1344.a	1.a	$1$	$3$	$3$	$79.299$	3.3.37341.1	None	✓		✓		672.4.a.p	$1$	$0$	\(0\)	\(-9\)	\(-6\)	\(21\)		$-$	$+$	$-$	$+$	$2^{4}$	$\mathrm{SU}(2)$	\(q-3q^{3}+(-2+\beta _{1})q^{5}+7q^{7}+9q^{9}+\cdots\)
1344.4.a.bt	$1344$	$4$	1344.a	1.a	$1$	$3$	$3$	$79.299$	3.3.22700.1	None	✓		✓		672.4.a.o	$1$	$0$	\(0\)	\(-9\)	\(10\)	\(-21\)		$+$	$+$	$+$	$+$	$2^{4}$	$\mathrm{SU}(2)$	\(q-3q^{3}+(3+\beta _{2})q^{5}-7q^{7}+9q^{9}+\cdots\)
1344.4.a.bu	$1344$	$4$	1344.a	1.a	$1$	$3$	$3$	$79.299$	3.3.37341.1	None	✓		✓		672.4.a.p	$1$	$0$	\(0\)	\(9\)	\(-6\)	\(-21\)		$-$	$-$	$+$	$+$	$2^{4}$	$\mathrm{SU}(2)$	\(q+3q^{3}+(-2+\beta _{1})q^{5}-7q^{7}+9q^{9}+\cdots\)
1344.4.a.bv	$1344$	$4$	1344.a	1.a	$1$	$3$	$3$	$79.299$	3.3.22700.1	None	✓		✓		672.4.a.o	$1$	$0$	\(0\)	\(9\)	\(10\)	\(21\)		$+$	$-$	$-$	$+$	$2^{4}$	$\mathrm{SU}(2)$	\(q+3q^{3}+(3+\beta _{2})q^{5}+7q^{7}+9q^{9}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(1344))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(1344)) \simeq \) \(S_{4}^{\mathrm{new}}(\Gamma_0(6))\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(7))\)\(^{\oplus 14}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(8))\)\(^{\oplus 16}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(12))\)\(^{\oplus 10}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(16))\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(21))\)\(^{\oplus 7}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(24))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(28))\)\(^{\oplus 10}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(32))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(42))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(48))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(56))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(64))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(84))\)\(^{\oplus 5}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(96))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(112))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(168))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(192))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(224))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(336))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(448))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(672))\)\(^{\oplus 2}\)