Space of modular forms of level 768, weight 2, and Character 768.f

• Label: 768.2.f
• Level: $768$
• Weight: $2$
• Character orbit: 768.f
• Rep. character: $\chi_{768}(383,\cdot)$
• Character field: $\Q$
• Dimension: $28$
• Newform subspaces: $7$
• Sturm bound: $256$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 768 = 2^{8} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	768.f (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 24 \)
Character field:			\(\Q\)
Newform subspaces:			\( 7 \)
Sturm bound:			\(256\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(5\), \(19\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(768, [\chi])\).

	Total	New	Old
Modular forms	152	36	116
Cusp forms	104	28	76
Eisenstein series	48	8	40

Trace form

\( 28 q + 4 q^{9} + 20 q^{25} - 16 q^{33} + 20 q^{49} - 8 q^{57} + 40 q^{73} - 4 q^{81} - 40 q^{97}+O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(768, [\chi])\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
768.2.f.a	$768$	$2$	768.f	24.f	$2$	$4$	$4$	$6.133$	\(\Q(\zeta_{8})\)	None					96.2.c.a	$4$	$0$	\(0\)	\(-4\)	\(0\)	\(0\)	$2^{3}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-\beta_{2}-1)q^{3}-\beta_{3} q^{5}+\beta_1 q^{7}+\cdots\)
768.2.f.b	$768$	$2$	768.f	24.f	$2$	$4$	$4$	$6.133$	\(\Q(i, \sqrt{5})\)	None					384.2.c.a	$2$	$0$	\(0\)	\(-2\)	\(-4\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-1+\beta _{3})q^{3}+(-2-\beta _{1}+\beta _{3})q^{5}+\cdots\)
768.2.f.c	$768$	$2$	768.f	24.f	$2$	$4$	$4$	$6.133$	\(\Q(i, \sqrt{5})\)	None					384.2.c.a	$2$	$0$	\(0\)	\(-2\)	\(4\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-1-\beta _{1})q^{3}+(2+\beta _{1}-\beta _{3})q^{5}+\cdots\)
768.2.f.d	$768$	$2$	768.f	24.f	$2$	$4$	$4$	$6.133$	\(\Q(\zeta_{12})\)	\(\Q(\sqrt{-3}) \)					48.2.c.a	$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{4}$	$\mathrm{U}(1)[D_{2}]$	\(q+\beta_1 q^{3}+\beta_{2} q^{7}+3 q^{9}+\beta_{3} q^{13}+\cdots\)
768.2.f.e	$768$	$2$	768.f	24.f	$2$	$4$	$4$	$6.133$	\(\Q(i, \sqrt{5})\)	None					384.2.c.a	$2$	$0$	\(0\)	\(2\)	\(-4\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(1-\beta _{3})q^{3}+(-2-\beta _{1}+\beta _{3})q^{5}+\cdots\)
768.2.f.f	$768$	$2$	768.f	24.f	$2$	$4$	$4$	$6.133$	\(\Q(i, \sqrt{5})\)	None					384.2.c.a	$2$	$0$	\(0\)	\(2\)	\(4\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(1+\beta _{1})q^{3}+(2+\beta _{1}-\beta _{3})q^{5}+(\beta _{1}+\cdots)q^{7}+\cdots\)
768.2.f.g	$768$	$2$	768.f	24.f	$2$	$4$	$4$	$6.133$	\(\Q(\zeta_{8})\)	None					96.2.c.a	$4$	$0$	\(0\)	\(4\)	\(0\)	\(0\)	$2^{3}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(\beta_{2}+1)q^{3}-\beta_{3} q^{5}-\beta_1 q^{7}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(768, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(768, [\chi]) \simeq \) \(S_{2}^{\mathrm{new}}(24, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(96, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(192, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(384, [\chi])\)\(^{\oplus 2}\)