Space of modular forms of level 768, weight 4, and trivial character

• Label: 768.4.a
• Level: $768$
• Weight: $4$
• Character orbit: 768.a
• Rep. character: $\chi_{768}(1,\cdot)$
• Character field: $\Q$
• Dimension: $48$
• Newform subspaces: $22$
• Sturm bound: $512$
• Trace bound: $11$

Defining parameters

Level:	\( N \)	\(=\)	\( 768 = 2^{8} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	768.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 22 \)
Sturm bound:			\(512\)
Trace bound:			\(11\)
Distinguishing \(T_p\):			\(5\), \(7\), \(11\), \(19\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(768))\).

	Total	New	Old
Modular forms	408	48	360
Cusp forms	360	48	312
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.

\(2\)	\(3\)	Fricke		Total				Cusp				Eisenstein
				All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)		\(104\)	\(12\)	\(92\)		\(92\)	\(12\)	\(80\)		\(12\)	\(0\)	\(12\)
\(+\)	\(-\)	\(-\)		\(100\)	\(10\)	\(90\)		\(88\)	\(10\)	\(78\)		\(12\)	\(0\)	\(12\)
\(-\)	\(+\)	\(-\)		\(100\)	\(12\)	\(88\)		\(88\)	\(12\)	\(76\)		\(12\)	\(0\)	\(12\)
\(-\)	\(-\)	\(+\)		\(104\)	\(14\)	\(90\)		\(92\)	\(14\)	\(78\)		\(12\)	\(0\)	\(12\)
Plus space		\(+\)		\(208\)	\(26\)	\(182\)		\(184\)	\(26\)	\(158\)		\(24\)	\(0\)	\(24\)
Minus space		\(-\)		\(200\)	\(22\)	\(178\)		\(176\)	\(22\)	\(154\)		\(24\)	\(0\)	\(24\)

Trace form

\( 48 q + 432 q^{9} + 1200 q^{25} + 912 q^{49} - 672 q^{57} + 1952 q^{65} + 2048 q^{73} + 3888 q^{81} - 352 q^{89} + 3168 q^{97}+O(q^{100}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(768))\) 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
768.4.a.a	$768$	$4$	768.a	1.a	$1$	$1$	$1$	$45.313$	\(\Q\)	None	✓				384.4.d.a	$1$	$0$	\(0\)	\(-3\)	\(-8\)	\(12\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-3q^{3}-8q^{5}+12q^{7}+9q^{9}+12q^{11}+\cdots\)
768.4.a.b	$768$	$4$	768.a	1.a	$1$	$1$	$1$	$45.313$	\(\Q\)	None	✓				384.4.d.a	$1$	$1$	\(0\)	\(-3\)	\(8\)	\(-12\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-3q^{3}+8q^{5}-12q^{7}+9q^{9}+12q^{11}+\cdots\)
768.4.a.c	$768$	$4$	768.a	1.a	$1$	$1$	$1$	$45.313$	\(\Q\)	None	✓				384.4.d.a	$1$	$1$	\(0\)	\(3\)	\(-8\)	\(-12\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3q^{3}-8q^{5}-12q^{7}+9q^{9}-12q^{11}+\cdots\)
768.4.a.d	$768$	$4$	768.a	1.a	$1$	$1$	$1$	$45.313$	\(\Q\)	None	✓				384.4.d.a	$1$	$0$	\(0\)	\(3\)	\(8\)	\(12\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+3q^{3}+8q^{5}+12q^{7}+9q^{9}-12q^{11}+\cdots\)
768.4.a.e	$768$	$4$	768.a	1.a	$1$	$2$	$2$	$45.313$	\(\Q(\sqrt{13}) \)	None	✓				384.4.d.c	$1$	$1$	\(0\)	\(-6\)	\(-8\)	\(-16\)		$-$	$+$	$-$	$2^{3}$	$\mathrm{SU}(2)$	\(q-3q^{3}+(-4-\beta )q^{5}+(-8-\beta )q^{7}+\cdots\)
768.4.a.f	$768$	$4$	768.a	1.a	$1$	$2$	$2$	$45.313$	\(\Q(\sqrt{3}) \)	None	✓				192.4.d.c	$2$	$0$	\(0\)	\(-6\)	\(0\)	\(0\)		$+$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q-3q^{3}+\beta q^{5}+7\beta q^{7}+9q^{9}-48q^{11}+\cdots\)
768.4.a.g	$768$	$4$	768.a	1.a	$1$	$2$	$2$	$45.313$	\(\Q(\sqrt{3}) \)	None	✓				192.4.d.a	$2$	$1$	\(0\)	\(-6\)	\(0\)	\(0\)		$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q-3q^{3}+3\beta q^{5}+\beta q^{7}+9q^{9}-2^{4}\beta q^{13}+\cdots\)
768.4.a.h	$768$	$4$	768.a	1.a	$1$	$2$	$2$	$45.313$	\(\Q(\sqrt{2}) \)	None	✓				384.4.d.d	$2$	$0$	\(0\)	\(-6\)	\(0\)	\(0\)		$+$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q-3q^{3}+\beta q^{5}+5\beta q^{7}+9q^{9}+20q^{11}+\cdots\)
768.4.a.i	$768$	$4$	768.a	1.a	$1$	$2$	$2$	$45.313$	\(\Q(\sqrt{11}) \)	None	✓				192.4.d.b	$2$	$0$	\(0\)	\(-6\)	\(0\)	\(0\)		$+$	$+$	$+$	$2\cdot 3$	$\mathrm{SU}(2)$	\(q-3q^{3}+\beta q^{5}-\beta q^{7}+9q^{9}+48q^{11}+\cdots\)
768.4.a.j	$768$	$4$	768.a	1.a	$1$	$2$	$2$	$45.313$	\(\Q(\sqrt{13}) \)	None	✓				384.4.d.c	$1$	$0$	\(0\)	\(-6\)	\(8\)	\(16\)		$+$	$+$	$+$	$2^{3}$	$\mathrm{SU}(2)$	\(q-3q^{3}+(4+\beta )q^{5}+(8+\beta )q^{7}+9q^{9}+\cdots\)
768.4.a.k	$768$	$4$	768.a	1.a	$1$	$2$	$2$	$45.313$	\(\Q(\sqrt{13}) \)	None	✓				384.4.d.c	$1$	$0$	\(0\)	\(6\)	\(-8\)	\(16\)		$-$	$-$	$+$	$2^{3}$	$\mathrm{SU}(2)$	\(q+3q^{3}+(-4-\beta )q^{5}+(8+\beta )q^{7}+9q^{9}+\cdots\)
768.4.a.l	$768$	$4$	768.a	1.a	$1$	$2$	$2$	$45.313$	\(\Q(\sqrt{11}) \)	None	✓				192.4.d.b	$2$	$0$	\(0\)	\(6\)	\(0\)	\(0\)		$-$	$-$	$+$	$2\cdot 3$	$\mathrm{SU}(2)$	\(q+3q^{3}+\beta q^{5}+\beta q^{7}+9q^{9}-48q^{11}+\cdots\)
768.4.a.m	$768$	$4$	768.a	1.a	$1$	$2$	$2$	$45.313$	\(\Q(\sqrt{2}) \)	None	✓				384.4.d.d	$2$	$1$	\(0\)	\(6\)	\(0\)	\(0\)		$+$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+3q^{3}+\beta q^{5}-5\beta q^{7}+9q^{9}-20q^{11}+\cdots\)
768.4.a.n	$768$	$4$	768.a	1.a	$1$	$2$	$2$	$45.313$	\(\Q(\sqrt{3}) \)	None	✓				192.4.d.a	$2$	$1$	\(0\)	\(6\)	\(0\)	\(0\)		$+$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+3q^{3}+3\beta q^{5}-\beta q^{7}+9q^{9}-2^{4}\beta q^{13}+\cdots\)
768.4.a.o	$768$	$4$	768.a	1.a	$1$	$2$	$2$	$45.313$	\(\Q(\sqrt{3}) \)	None	✓				192.4.d.c	$2$	$0$	\(0\)	\(6\)	\(0\)	\(0\)		$-$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q+3q^{3}+\beta q^{5}-7\beta q^{7}+9q^{9}+48q^{11}+\cdots\)
768.4.a.p	$768$	$4$	768.a	1.a	$1$	$2$	$2$	$45.313$	\(\Q(\sqrt{13}) \)	None	✓				384.4.d.c	$1$	$1$	\(0\)	\(6\)	\(8\)	\(-16\)		$+$	$-$	$-$	$2^{3}$	$\mathrm{SU}(2)$	\(q+3q^{3}+(4+\beta )q^{5}+(-8-\beta )q^{7}+9q^{9}+\cdots\)
768.4.a.q	$768$	$4$	768.a	1.a	$1$	$3$	$3$	$45.313$	3.3.1436.1	None	✓				24.4.d.a	$1$	$1$	\(0\)	\(-9\)	\(-10\)	\(14\)		$-$	$+$	$-$	$2^{5}$	$\mathrm{SU}(2)$	\(q-3q^{3}+(-3-\beta _{2})q^{5}+(5-\beta _{1})q^{7}+\cdots\)
768.4.a.r	$768$	$4$	768.a	1.a	$1$	$3$	$3$	$45.313$	3.3.1436.1	None	✓		✓		24.4.d.a	$1$	$0$	\(0\)	\(-9\)	\(10\)	\(-14\)		$+$	$+$	$+$	$2^{5}$	$\mathrm{SU}(2)$	\(q-3q^{3}+(3+\beta _{2})q^{5}+(-5+\beta _{1})q^{7}+\cdots\)
768.4.a.s	$768$	$4$	768.a	1.a	$1$	$3$	$3$	$45.313$	3.3.1436.1	None	✓		✓		24.4.d.a	$1$	$1$	\(0\)	\(9\)	\(-10\)	\(-14\)		$+$	$-$	$-$	$2^{5}$	$\mathrm{SU}(2)$	\(q+3q^{3}+(-3-\beta _{2})q^{5}+(-5+\beta _{1}+\cdots)q^{7}+\cdots\)
768.4.a.t	$768$	$4$	768.a	1.a	$1$	$3$	$3$	$45.313$	3.3.1436.1	None	✓				24.4.d.a	$1$	$0$	\(0\)	\(9\)	\(10\)	\(14\)		$-$	$-$	$+$	$2^{5}$	$\mathrm{SU}(2)$	\(q+3q^{3}+(3+\beta _{2})q^{5}+(5-\beta _{1})q^{7}+9q^{9}+\cdots\)
768.4.a.u	$768$	$4$	768.a	1.a	$1$	$4$	$4$	$45.313$	4.4.9792.1	None	✓		✓		384.4.d.f	$2$	$1$	\(0\)	\(-12\)	\(0\)	\(0\)		$-$	$+$	$-$	$2^{10}$	$\mathrm{SU}(2)$	\(q-3q^{3}+\beta _{1}q^{5}+(-\beta _{1}+\beta _{3})q^{7}+9q^{9}+\cdots\)
768.4.a.v	$768$	$4$	768.a	1.a	$1$	$4$	$4$	$45.313$	4.4.9792.1	None	✓		✓		384.4.d.f	$2$	$0$	\(0\)	\(12\)	\(0\)	\(0\)		$-$	$-$	$+$	$2^{10}$	$\mathrm{SU}(2)$	\(q+3q^{3}+\beta _{1}q^{5}+(\beta _{1}-\beta _{3})q^{7}+9q^{9}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(768))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(768)) \simeq \) \(S_{4}^{\mathrm{new}}(\Gamma_0(6))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(8))\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(12))\)\(^{\oplus 7}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(16))\)\(^{\oplus 10}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(24))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(32))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(48))\)\(^{\oplus 5}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(64))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(96))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(128))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(192))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(256))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(384))\)\(^{\oplus 2}\)