Space of modular forms of level 768, weight 3, and Character 768.l

• Label: 768.3.l
• Level: $768$
• Weight: $3$
• Character orbit: 768.l
• Rep. character: $\chi_{768}(319,\cdot)$
• Character field: $\Q(\zeta_{4})$
• Dimension: $64$
• Newform subspaces: $6$
• Sturm bound: $384$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 768 = 2^{8} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	768.l (of order \(4\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 16 \)
Character field:			\(\Q(i)\)
Newform subspaces:			\( 6 \)
Sturm bound:			\(384\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{3}(768, [\chi])\).

	Total	New	Old
Modular forms	560	64	496
Cusp forms	464	64	400
Eisenstein series	96	0	96

Trace form

\( 64 q + 448 q^{49} + 384 q^{65} - 576 q^{81}+O(q^{100}) \)

Decomposition of \(S_{3}^{\mathrm{new}}(768, [\chi])\) 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				Coefficient ring index	Sato-Tate	$q$-expansion
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
768.3.l.a	$768$	$3$	768.l	16.f	$4$	$8$	$4$	$20.926$	\(\Q(\zeta_{24})\)	None		✓			768.3.l.a	$4$	$0$	\(0\)	\(0\)	\(-16\)	\(0\)	$2^{10}$	$\mathrm{SU}(2)[C_{4}]$	\(q+\beta_{3} q^{3}+(2\beta_{2}-2)q^{5}+(\beta_{4}-\beta_{3}+\beta_1)q^{7}+\cdots\)
768.3.l.b	$768$	$3$	768.l	16.f	$4$	$8$	$4$	$20.926$	\(\Q(\zeta_{24})\)	None		✓			768.3.l.b	$4$	$0$	\(0\)	\(0\)	\(-8\)	\(0\)	$2^{6}$	$\mathrm{SU}(2)[C_{4}]$	\(q-\beta_1 q^{3}+(-\beta_{5}-\beta_{2}-1)q^{5}+\cdots\)
768.3.l.c	$768$	$3$	768.l	16.f	$4$	$8$	$4$	$20.926$	\(\Q(\zeta_{24})\)	None		✓			768.3.l.b	$4$	$0$	\(0\)	\(0\)	\(8\)	\(0\)	$2^{6}$	$\mathrm{SU}(2)[C_{4}]$	\(q-\beta_1 q^{3}+(\beta_{5}+\beta_{2}+1)q^{5}+(5\beta_{4}+2\beta_{3}-2\beta_1)q^{7}+\cdots\)
768.3.l.d	$768$	$3$	768.l	16.f	$4$	$8$	$4$	$20.926$	\(\Q(\zeta_{24})\)	None		✓			768.3.l.a	$4$	$0$	\(0\)	\(0\)	\(16\)	\(0\)	$2^{10}$	$\mathrm{SU}(2)[C_{4}]$	\(q-\beta_1 q^{3}+(2\beta_{2}+2)q^{5}+(-\beta_{4}+\beta_{3}-\beta_1)q^{7}+\cdots\)
768.3.l.e	$768$	$3$	768.l	16.f	$4$	$16$	$8$	$20.926$	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)	None		✓			768.3.l.e	$4$	$0$	\(0\)	\(0\)	\(-8\)	\(0\)	$2^{24}$	$\mathrm{SU}(2)[C_{4}]$	\(q+\beta _{8}q^{3}+(-1+\beta _{10})q^{5}+(-2\beta _{7}+\cdots)q^{7}+\cdots\)
768.3.l.f	$768$	$3$	768.l	16.f	$4$	$16$	$8$	$20.926$	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)	None		✓			768.3.l.e	$4$	$0$	\(0\)	\(0\)	\(8\)	\(0\)	$2^{24}$	$\mathrm{SU}(2)[C_{4}]$	\(q+\beta _{7}q^{3}+(1+\beta _{1}-\beta _{6})q^{5}+(2\beta _{7}+2\beta _{8}+\cdots)q^{7}+\cdots\)

Decomposition of \(S_{3}^{\mathrm{old}}(768, [\chi])\) into lower level spaces

\( S_{3}^{\mathrm{old}}(768, [\chi]) \simeq \) \(S_{3}^{\mathrm{new}}(16, [\chi])\)\(^{\oplus 10}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(48, [\chi])\)\(^{\oplus 5}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(64, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(128, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(192, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(256, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(384, [\chi])\)\(^{\oplus 2}\)