Space of modular forms of level 768, weight 2, and trivial character

• Label: 768.2.a
• Level: $768$
• Weight: $2$
• Character orbit: 768.a
• Rep. character: $\chi_{768}(1,\cdot)$
• Character field: $\Q$
• Dimension: $16$
• Newform subspaces: $12$
• Sturm bound: $256$
• Trace bound: $11$

Defining parameters

Level:	\( N \)	\(=\)	\( 768 = 2^{8} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	768.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 12 \)
Sturm bound:			\(256\)
Trace bound:			\(11\)
Distinguishing \(T_p\):			\(5\), \(7\), \(11\), \(19\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(768))\).

	Total	New	Old
Modular forms	152	16	136
Cusp forms	105	16	89
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.

\(2\)	\(3\)	Fricke	Dim
\(+\)	\(+\)	\(+\)	\(4\)
\(+\)	\(-\)	\(-\)	\(6\)
\(-\)	\(+\)	\(-\)	\(4\)
\(-\)	\(-\)	\(+\)	\(2\)
Plus space		\(+\)	\(6\)
Minus space		\(-\)	\(10\)

Trace form

\( 16 q + 16 q^{9} + 16 q^{25} + 48 q^{49} + 32 q^{57} - 32 q^{65} + 16 q^{81} - 32 q^{89} + 32 q^{97}+O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(768))\) 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
768.2.a.a	$768$	$2$	768.a	1.a	$1$	$1$	$1$	$6.133$	\(\Q\)	None	✓				24.2.d.a	$1$	$1$	\(0\)	\(-1\)	\(-2\)	\(2\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-2q^{5}+2q^{7}+q^{9}-4q^{13}+\cdots\)
768.2.a.b	$768$	$2$	768.a	1.a	$1$	$1$	$1$	$6.133$	\(\Q\)	None	✓				384.2.d.a	$1$	$1$	\(0\)	\(-1\)	\(0\)	\(-4\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-4q^{7}+q^{9}+4q^{11}+4q^{13}+\cdots\)
768.2.a.c	$768$	$2$	768.a	1.a	$1$	$1$	$1$	$6.133$	\(\Q\)	None	✓				384.2.d.a	$1$	$0$	\(0\)	\(-1\)	\(0\)	\(4\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+4q^{7}+q^{9}+4q^{11}-4q^{13}+\cdots\)
768.2.a.d	$768$	$2$	768.a	1.a	$1$	$1$	$1$	$6.133$	\(\Q\)	None	✓				24.2.d.a	$1$	$0$	\(0\)	\(-1\)	\(2\)	\(-2\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+2q^{5}-2q^{7}+q^{9}+4q^{13}+\cdots\)
768.2.a.e	$768$	$2$	768.a	1.a	$1$	$1$	$1$	$6.133$	\(\Q\)	None	✓				24.2.d.a	$1$	$1$	\(0\)	\(1\)	\(-2\)	\(-2\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-2q^{5}-2q^{7}+q^{9}-4q^{13}+\cdots\)
768.2.a.f	$768$	$2$	768.a	1.a	$1$	$1$	$1$	$6.133$	\(\Q\)	None	✓				384.2.d.a	$1$	$1$	\(0\)	\(1\)	\(0\)	\(-4\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-4q^{7}+q^{9}-4q^{11}-4q^{13}+\cdots\)
768.2.a.g	$768$	$2$	768.a	1.a	$1$	$1$	$1$	$6.133$	\(\Q\)	None	✓				384.2.d.a	$1$	$0$	\(0\)	\(1\)	\(0\)	\(4\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+4q^{7}+q^{9}-4q^{11}+4q^{13}+\cdots\)
768.2.a.h	$768$	$2$	768.a	1.a	$1$	$1$	$1$	$6.133$	\(\Q\)	None	✓				24.2.d.a	$1$	$0$	\(0\)	\(1\)	\(2\)	\(2\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+2q^{5}+2q^{7}+q^{9}+4q^{13}+\cdots\)
768.2.a.i	$768$	$2$	768.a	1.a	$1$	$2$	$2$	$6.133$	\(\Q(\sqrt{2}) \)	None	✓		✓		384.2.d.c	$2$	$1$	\(0\)	\(-2\)	\(0\)	\(0\)		$+$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q-q^{3}+\beta q^{5}-\beta q^{7}+q^{9}-4q^{11}+\cdots\)
768.2.a.j	$768$	$2$	768.a	1.a	$1$	$2$	$2$	$6.133$	\(\Q(\sqrt{3}) \)	None	✓		✓		192.2.d.a	$2$	$0$	\(0\)	\(-2\)	\(0\)	\(0\)		$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q-q^{3}+\beta q^{5}+\beta q^{7}+q^{9}-\beta q^{15}+\cdots\)
768.2.a.k	$768$	$2$	768.a	1.a	$1$	$2$	$2$	$6.133$	\(\Q(\sqrt{3}) \)	None	✓				192.2.d.a	$2$	$0$	\(0\)	\(2\)	\(0\)	\(0\)		$+$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+q^{3}+\beta q^{5}-\beta q^{7}+q^{9}+\beta q^{15}+\cdots\)
768.2.a.l	$768$	$2$	768.a	1.a	$1$	$2$	$2$	$6.133$	\(\Q(\sqrt{2}) \)	None	✓				384.2.d.c	$2$	$0$	\(0\)	\(2\)	\(0\)	\(0\)		$+$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+q^{3}+\beta q^{5}+\beta q^{7}+q^{9}+4q^{11}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(768))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(768)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(24))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(32))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(48))\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(64))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(96))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(128))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(192))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(256))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(384))\)\(^{\oplus 2}\)