Space of modular forms of level 1344, weight 2, and trivial character

• Label: 1344.2.a
• Level: $1344$
• Weight: $2$
• Character orbit: 1344.a
• Rep. character: $\chi_{1344}(1,\cdot)$
• Character field: $\Q$
• Dimension: $24$
• Newform subspaces: $22$
• Sturm bound: $512$
• Trace bound: $13$

Defining parameters

Level:	\( N \)	\(=\)	\( 1344 = 2^{6} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1344.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 22 \)
Sturm bound:			\(512\)
Trace bound:			\(13\)
Distinguishing \(T_p\):			\(5\), \(11\), \(13\), \(19\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(1344))\).

	Total	New	Old
Modular forms	280	24	256
Cusp forms	233	24	209
Eisenstein series	47	0	47

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
\(+\)	\(+\)	\(+\)	\(+\)		\(32\)	\(2\)	\(30\)		\(27\)	\(2\)	\(25\)		\(5\)	\(0\)	\(5\)
\(+\)	\(+\)	\(-\)	\(-\)		\(36\)	\(3\)	\(33\)		\(30\)	\(3\)	\(27\)		\(6\)	\(0\)	\(6\)
\(+\)	\(-\)	\(+\)	\(-\)		\(38\)	\(4\)	\(34\)		\(32\)	\(4\)	\(28\)		\(6\)	\(0\)	\(6\)
\(+\)	\(-\)	\(-\)	\(+\)		\(34\)	\(1\)	\(33\)		\(28\)	\(1\)	\(27\)		\(6\)	\(0\)	\(6\)
\(-\)	\(+\)	\(+\)	\(-\)		\(38\)	\(4\)	\(34\)		\(32\)	\(4\)	\(28\)		\(6\)	\(0\)	\(6\)
\(-\)	\(+\)	\(-\)	\(+\)		\(34\)	\(3\)	\(31\)		\(28\)	\(3\)	\(25\)		\(6\)	\(0\)	\(6\)
\(-\)	\(-\)	\(+\)	\(+\)		\(32\)	\(2\)	\(30\)		\(26\)	\(2\)	\(24\)		\(6\)	\(0\)	\(6\)
\(-\)	\(-\)	\(-\)	\(-\)		\(36\)	\(5\)	\(31\)		\(30\)	\(5\)	\(25\)		\(6\)	\(0\)	\(6\)
Plus space			\(+\)		\(132\)	\(8\)	\(124\)		\(109\)	\(8\)	\(101\)		\(23\)	\(0\)	\(23\)
Minus space			\(-\)		\(148\)	\(16\)	\(132\)		\(124\)	\(16\)	\(108\)		\(24\)	\(0\)	\(24\)

Trace form

\( 24 q + 24 q^{9} + 16 q^{17} + 40 q^{25} + 16 q^{29} + 16 q^{37} + 16 q^{41} + 24 q^{49} + 16 q^{53} + 32 q^{61} + 32 q^{69} - 16 q^{73} + 16 q^{77} + 24 q^{81} + 32 q^{85} - 16 q^{89} + 16 q^{97}+O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(1344))\) 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
1344.2.a.a	$1344$	$2$	1344.a	1.a	$1$	$1$	$1$	$10.732$	\(\Q\)	None	✓				84.2.a.a	$1$	$1$	\(0\)	\(-1\)	\(-4\)	\(1\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-4q^{5}+q^{7}+q^{9}+2q^{11}+6q^{13}+\cdots\)
1344.2.a.b	$1344$	$2$	1344.a	1.a	$1$	$1$	$1$	$10.732$	\(\Q\)	None	✓				168.2.a.a	$1$	$0$	\(0\)	\(-1\)	\(-2\)	\(-1\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-2q^{5}-q^{7}+q^{9}-6q^{13}+2q^{15}+\cdots\)
1344.2.a.c	$1344$	$2$	1344.a	1.a	$1$	$1$	$1$	$10.732$	\(\Q\)	None	✓				168.2.a.b	$1$	$1$	\(0\)	\(-1\)	\(-2\)	\(-1\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-2q^{5}-q^{7}+q^{9}+2q^{13}+2q^{15}+\cdots\)
1344.2.a.d	$1344$	$2$	1344.a	1.a	$1$	$1$	$1$	$10.732$	\(\Q\)	None	✓				672.2.a.d	$1$	$1$	\(0\)	\(-1\)	\(-2\)	\(1\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-2q^{5}+q^{7}+q^{9}-2q^{13}+2q^{15}+\cdots\)
1344.2.a.e	$1344$	$2$	1344.a	1.a	$1$	$1$	$1$	$10.732$	\(\Q\)	None	✓				672.2.a.c	$1$	$0$	\(0\)	\(-1\)	\(0\)	\(1\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+q^{7}+q^{9}-2q^{11}+2q^{13}+\cdots\)
1344.2.a.f	$1344$	$2$	1344.a	1.a	$1$	$1$	$1$	$10.732$	\(\Q\)	None	✓				84.2.a.b	$1$	$0$	\(0\)	\(-1\)	\(0\)	\(1\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+q^{7}+q^{9}+6q^{11}-2q^{13}+\cdots\)
1344.2.a.g	$1344$	$2$	1344.a	1.a	$1$	$1$	$1$	$10.732$	\(\Q\)	None	✓				21.2.a.a	$1$	$1$	\(0\)	\(-1\)	\(2\)	\(-1\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+2q^{5}-q^{7}+q^{9}-4q^{11}+2q^{13}+\cdots\)
1344.2.a.h	$1344$	$2$	1344.a	1.a	$1$	$1$	$1$	$10.732$	\(\Q\)	None	✓				672.2.a.b	$1$	$0$	\(0\)	\(-1\)	\(2\)	\(-1\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+2q^{5}-q^{7}+q^{9}+4q^{11}+6q^{13}+\cdots\)
1344.2.a.i	$1344$	$2$	1344.a	1.a	$1$	$1$	$1$	$10.732$	\(\Q\)	None	✓				42.2.a.a	$1$	$1$	\(0\)	\(-1\)	\(2\)	\(1\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+2q^{5}+q^{7}+q^{9}-4q^{11}-6q^{13}+\cdots\)
1344.2.a.j	$1344$	$2$	1344.a	1.a	$1$	$1$	$1$	$10.732$	\(\Q\)	None	✓				672.2.a.a	$1$	$0$	\(0\)	\(-1\)	\(4\)	\(1\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+4q^{5}+q^{7}+q^{9}+2q^{11}+2q^{13}+\cdots\)
1344.2.a.k	$1344$	$2$	1344.a	1.a	$1$	$1$	$1$	$10.732$	\(\Q\)	None	✓				84.2.a.a	$1$	$0$	\(0\)	\(1\)	\(-4\)	\(-1\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-4q^{5}-q^{7}+q^{9}-2q^{11}+6q^{13}+\cdots\)
1344.2.a.l	$1344$	$2$	1344.a	1.a	$1$	$1$	$1$	$10.732$	\(\Q\)	None	✓				672.2.a.d	$1$	$1$	\(0\)	\(1\)	\(-2\)	\(-1\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-2q^{5}-q^{7}+q^{9}-2q^{13}-2q^{15}+\cdots\)
1344.2.a.m	$1344$	$2$	1344.a	1.a	$1$	$1$	$1$	$10.732$	\(\Q\)	None	✓		✓	✓	168.2.a.a	$1$	$1$	\(0\)	\(1\)	\(-2\)	\(1\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-2q^{5}+q^{7}+q^{9}-6q^{13}-2q^{15}+\cdots\)
1344.2.a.n	$1344$	$2$	1344.a	1.a	$1$	$1$	$1$	$10.732$	\(\Q\)	None	✓				168.2.a.b	$1$	$0$	\(0\)	\(1\)	\(-2\)	\(1\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-2q^{5}+q^{7}+q^{9}+2q^{13}-2q^{15}+\cdots\)
1344.2.a.o	$1344$	$2$	1344.a	1.a	$1$	$1$	$1$	$10.732$	\(\Q\)	None	✓				84.2.a.b	$1$	$1$	\(0\)	\(1\)	\(0\)	\(-1\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-q^{7}+q^{9}-6q^{11}-2q^{13}+\cdots\)
1344.2.a.p	$1344$	$2$	1344.a	1.a	$1$	$1$	$1$	$10.732$	\(\Q\)	None	✓				672.2.a.c	$1$	$0$	\(0\)	\(1\)	\(0\)	\(-1\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-q^{7}+q^{9}+2q^{11}+2q^{13}+\cdots\)
1344.2.a.q	$1344$	$2$	1344.a	1.a	$1$	$1$	$1$	$10.732$	\(\Q\)	None	✓				42.2.a.a	$1$	$0$	\(0\)	\(1\)	\(2\)	\(-1\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+2q^{5}-q^{7}+q^{9}+4q^{11}-6q^{13}+\cdots\)
1344.2.a.r	$1344$	$2$	1344.a	1.a	$1$	$1$	$1$	$10.732$	\(\Q\)	None	✓				672.2.a.b	$1$	$0$	\(0\)	\(1\)	\(2\)	\(1\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+2q^{5}+q^{7}+q^{9}-4q^{11}+6q^{13}+\cdots\)
1344.2.a.s	$1344$	$2$	1344.a	1.a	$1$	$1$	$1$	$10.732$	\(\Q\)	None	✓				21.2.a.a	$1$	$0$	\(0\)	\(1\)	\(2\)	\(1\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+2q^{5}+q^{7}+q^{9}+4q^{11}+2q^{13}+\cdots\)
1344.2.a.t	$1344$	$2$	1344.a	1.a	$1$	$1$	$1$	$10.732$	\(\Q\)	None	✓				672.2.a.a	$1$	$0$	\(0\)	\(1\)	\(4\)	\(-1\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+4q^{5}-q^{7}+q^{9}-2q^{11}+2q^{13}+\cdots\)
1344.2.a.u	$1344$	$2$	1344.a	1.a	$1$	$2$	$2$	$10.732$	\(\Q(\sqrt{3}) \)	None	✓		✓		672.2.a.i	$1$	$0$	\(0\)	\(-2\)	\(0\)	\(-2\)		$-$	$+$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q-q^{3}+\beta q^{5}-q^{7}+q^{9}+(-2-\beta )q^{11}+\cdots\)
1344.2.a.v	$1344$	$2$	1344.a	1.a	$1$	$2$	$2$	$10.732$	\(\Q(\sqrt{3}) \)	None	✓		✓		672.2.a.i	$1$	$0$	\(0\)	\(2\)	\(0\)	\(2\)		$-$	$-$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+q^{3}+\beta q^{5}+q^{7}+q^{9}+(2+\beta )q^{11}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(1344))\) into lower level spaces

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