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

• Label: 768.2.c
• Level: $768$
• Weight: $2$
• Character orbit: 768.c
• Rep. character: $\chi_{768}(767,\cdot)$
• Character field: $\Q$
• Dimension: $28$
• Newform subspaces: $10$
• Sturm bound: $256$
• Trace bound: $25$

Defining parameters

Level:	\( N \)	\(=\)	\( 768 = 2^{8} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	768.c (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 12 \)
Character field:			\(\Q\)
Newform subspaces:			\( 10 \)
Sturm bound:			\(256\)
Trace bound:			\(25\)
Distinguishing \(T_p\):			\(5\), \(7\), \(11\), \(13\)

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

\( 28 q + 4 q^{9} + O(q^{10}) \)

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.c.a	$768$	$2$	768.c	12.b	$2$	$2$	$2$	$6.133$	\(\Q(\sqrt{-2}) \)	None					384.2.f.b	$2$	$1$	\(0\)	\(-2\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-1+\beta )q^{3}+2\beta q^{5}-2\beta q^{7}+(-1+\cdots)q^{9}+\cdots\)
768.2.c.b	$768$	$2$	768.c	12.b	$2$	$2$	$2$	$6.133$	\(\Q(\sqrt{-2}) \)	None					384.2.f.b	$2$	$0$	\(0\)	\(-2\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-1-\beta )q^{3}+2\beta q^{5}-2\beta q^{7}+(-1+\cdots)q^{9}+\cdots\)
768.2.c.c	$768$	$2$	768.c	12.b	$2$	$2$	$2$	$6.133$	\(\Q(\sqrt{-2}) \)	\(\Q(\sqrt{-2}) \)					384.2.f.c	$4$	$0$	\(0\)	\(-2\)	\(0\)	\(0\)	$1$	$\mathrm{U}(1)[D_{2}]$	\(q+(-1-\beta )q^{3}+(-1+2\beta )q^{9}+6q^{11}+\cdots\)
768.2.c.d	$768$	$2$	768.c	12.b	$2$	$2$	$2$	$6.133$	\(\Q(\sqrt{-2}) \)	\(\Q(\sqrt{-2}) \)					384.2.f.c	$4$	$0$	\(0\)	\(2\)	\(0\)	\(0\)	$1$	$\mathrm{U}(1)[D_{2}]$	\(q+(1+\beta )q^{3}+(-1+2\beta )q^{9}-6q^{11}+\cdots\)
768.2.c.e	$768$	$2$	768.c	12.b	$2$	$2$	$2$	$6.133$	\(\Q(\sqrt{-2}) \)	None					384.2.f.b	$2$	$0$	\(0\)	\(2\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+(1-\beta )q^{3}+2\beta q^{5}+2\beta q^{7}+(-1+\cdots)q^{9}+\cdots\)
768.2.c.f	$768$	$2$	768.c	12.b	$2$	$2$	$2$	$6.133$	\(\Q(\sqrt{-2}) \)	None					384.2.f.b	$2$	$0$	\(0\)	\(2\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+(1+\beta )q^{3}+2\beta q^{5}+2\beta q^{7}+(-1+\cdots)q^{9}+\cdots\)
768.2.c.g	$768$	$2$	768.c	12.b	$2$	$4$	$4$	$6.133$	\(\Q(\zeta_{12})\)	\(\Q(\sqrt{-3}) \)					192.2.f.b	$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{6}$	$\mathrm{U}(1)[D_{2}]$	\(q+\beta_{2} q^{3}-\beta_1 q^{7}-3 q^{9}+\beta_{3} q^{13}+\cdots\)
768.2.c.h	$768$	$2$	768.c	12.b	$2$	$4$	$4$	$6.133$	\(\Q(\zeta_{8})\)	\(\Q(\sqrt{-2}) \)					24.2.f.a	$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{3}$	$\mathrm{U}(1)[D_{2}]$	\(q+\beta_{2} q^{3}+(\beta_{3}+1)q^{9}+(\beta_{2}+\beta_1)q^{11}+\cdots\)
768.2.c.i	$768$	$2$	768.c	12.b	$2$	$4$	$4$	$6.133$	\(\Q(\zeta_{12})\)	\(\Q(\sqrt{-6}) \)					192.2.f.a	$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{4}$	$\mathrm{U}(1)[D_{2}]$	\(q-\beta_1 q^{3}-\beta_{2} q^{5}+\beta_{3} q^{7}+3 q^{9}+\cdots\)
768.2.c.j	$768$	$2$	768.c	12.b	$2$	$4$	$4$	$6.133$	\(\Q(\sqrt{-2}, \sqrt{3})\)	\(\Q(\sqrt{-6}) \)					384.2.f.a	$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{3}$	$\mathrm{U}(1)[D_{2}]$	\(q-\beta _{2}q^{3}-\beta _{1}q^{5}-\beta _{3}q^{7}+3q^{9}-2\beta _{2}q^{11}+\cdots\)

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

\( S_{2}^{\mathrm{old}}(768, [\chi]) \simeq \) \(S_{2}^{\mathrm{new}}(48, [\chi])\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(96, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(192, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(384, [\chi])\)\(^{\oplus 2}\)