Space of modular forms of level 768, weight 6, and Character 768.d

• Label: 768.6.d
• Level: $768$
• Weight: $6$
• Character orbit: 768.d
• Rep. character: $\chi_{768}(385,\cdot)$
• Character field: $\Q$
• Dimension: $80$
• Newform subspaces: $28$
• Sturm bound: $768$
• Trace bound: $15$

Defining parameters

Level:	\( N \)	\(=\)	\( 768 = 2^{8} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	768.d (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 8 \)
Character field:			\(\Q\)
Newform subspaces:			\( 28 \)
Sturm bound:			\(768\)
Trace bound:			\(15\)
Distinguishing \(T_p\):			\(5\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(768, [\chi])\).

	Total	New	Old
Modular forms	664	80	584
Cusp forms	616	80	536
Eisenstein series	48	0	48

Trace form

\( 80 q - 6480 q^{9} + O(q^{10}) \)

Decomposition of \(S_{6}^{\mathrm{new}}(768, [\chi])\) into newform subspaces

Label	Level	Weight	Char	Prim	Char order	Dim	Rel. Dim	$A$	Field	CM	Self-dual	Inner twists	Rank*	Traces				Coefficient ring index	Sato-Tate	$q$-expansion
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
768.6.d.a	$768$	$6$	768.d	8.b	$2$	$2$	$2$	$123.175$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(-480\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+9iq^{3}+34iq^{5}-240q^{7}-3^{4}q^{9}+\cdots\)
768.6.d.b	$768$	$6$	768.d	8.b	$2$	$2$	$2$	$123.175$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(-360\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+9iq^{3}+86iq^{5}-180q^{7}-3^{4}q^{9}+\cdots\)
768.6.d.c	$768$	$6$	768.d	8.b	$2$	$2$	$2$	$123.175$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(-352\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-9iq^{3}+66iq^{5}-176q^{7}-3^{4}q^{9}+\cdots\)
768.6.d.d	$768$	$6$	768.d	8.b	$2$	$2$	$2$	$123.175$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(-288\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+9iq^{3}+94iq^{5}-12^{2}q^{7}-3^{4}q^{9}+\cdots\)
768.6.d.e	$768$	$6$	768.d	8.b	$2$	$2$	$2$	$123.175$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(-244\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+9iq^{3}+20iq^{5}-122q^{7}-3^{4}q^{9}+\cdots\)
768.6.d.f	$768$	$6$	768.d	8.b	$2$	$2$	$2$	$123.175$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(-240\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-9iq^{3}+38iq^{5}-120q^{7}-3^{4}q^{9}+\cdots\)
768.6.d.g	$768$	$6$	768.d	8.b	$2$	$2$	$2$	$123.175$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(-200\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-9iq^{3}+14iq^{5}-10^{2}q^{7}-3^{4}q^{9}+\cdots\)
768.6.d.h	$768$	$6$	768.d	8.b	$2$	$2$	$2$	$123.175$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(-80\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+9iq^{3}+6iq^{5}-40q^{7}-3^{4}q^{9}+\cdots\)
768.6.d.i	$768$	$6$	768.d	8.b	$2$	$2$	$2$	$123.175$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(-72\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-9iq^{3}+26iq^{5}-6^{2}q^{7}-3^{4}q^{9}+\cdots\)
768.6.d.j	$768$	$6$	768.d	8.b	$2$	$2$	$2$	$123.175$	\(\Q(\sqrt{-1}) \)	None		$2$	$1$	\(0\)	\(0\)	\(0\)	\(72\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+9iq^{3}+26iq^{5}+6^{2}q^{7}-3^{4}q^{9}+\cdots\)
768.6.d.k	$768$	$6$	768.d	8.b	$2$	$2$	$2$	$123.175$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(80\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-9iq^{3}+6iq^{5}+40q^{7}-3^{4}q^{9}+\cdots\)
768.6.d.l	$768$	$6$	768.d	8.b	$2$	$2$	$2$	$123.175$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(200\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+9iq^{3}+14iq^{5}+10^{2}q^{7}-3^{4}q^{9}+\cdots\)
768.6.d.m	$768$	$6$	768.d	8.b	$2$	$2$	$2$	$123.175$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(240\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+9iq^{3}+38iq^{5}+120q^{7}-3^{4}q^{9}+\cdots\)
768.6.d.n	$768$	$6$	768.d	8.b	$2$	$2$	$2$	$123.175$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(244\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-9iq^{3}+20iq^{5}+122q^{7}-3^{4}q^{9}+\cdots\)
768.6.d.o	$768$	$6$	768.d	8.b	$2$	$2$	$2$	$123.175$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(288\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-9iq^{3}+94iq^{5}+12^{2}q^{7}-3^{4}q^{9}+\cdots\)
768.6.d.p	$768$	$6$	768.d	8.b	$2$	$2$	$2$	$123.175$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(352\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+9iq^{3}+66iq^{5}+176q^{7}-3^{4}q^{9}+\cdots\)
768.6.d.q	$768$	$6$	768.d	8.b	$2$	$2$	$2$	$123.175$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(360\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-9iq^{3}+86iq^{5}+180q^{7}-3^{4}q^{9}+\cdots\)
768.6.d.r	$768$	$6$	768.d	8.b	$2$	$2$	$2$	$123.175$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(480\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-9iq^{3}+34iq^{5}+240q^{7}-3^{4}q^{9}+\cdots\)
768.6.d.s	$768$	$6$	768.d	8.b	$2$	$4$	$4$	$123.175$	\(\Q(i, \sqrt{31})\)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(-240\)	$2^{10}$	$\mathrm{SU}(2)[C_{2}]$	\(q-9\beta _{1}q^{3}+(18\beta _{1}+\beta _{3})q^{5}+(-60+\cdots)q^{7}+\cdots\)
768.6.d.t	$768$	$6$	768.d	8.b	$2$	$4$	$4$	$123.175$	\(\Q(i, \sqrt{34})\)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(-184\)	$2^{7}$	$\mathrm{SU}(2)[C_{2}]$	\(q+9\beta _{1}q^{3}+(20\beta _{1}+\beta _{2})q^{5}+(-46+\cdots)q^{7}+\cdots\)
768.6.d.u	$768$	$6$	768.d	8.b	$2$	$4$	$4$	$123.175$	\(\Q(i, \sqrt{7})\)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(-136\)	$2^{6}$	$\mathrm{SU}(2)[C_{2}]$	\(q-9\beta _{1}q^{3}+(-4\beta _{1}+5\beta _{3})q^{5}+(-34+\cdots)q^{7}+\cdots\)
768.6.d.v	$768$	$6$	768.d	8.b	$2$	$4$	$4$	$123.175$	\(\Q(i, \sqrt{231})\)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(-136\)	$2^{6}$	$\mathrm{SU}(2)[C_{2}]$	\(q+9\beta _{1}q^{3}+(-40\beta _{1}-\beta _{3})q^{5}+(-34+\cdots)q^{7}+\cdots\)
768.6.d.w	$768$	$6$	768.d	8.b	$2$	$4$	$4$	$123.175$	\(\Q(i, \sqrt{231})\)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(136\)	$2^{6}$	$\mathrm{SU}(2)[C_{2}]$	\(q-9\beta _{1}q^{3}+(-40\beta _{1}-\beta _{3})q^{5}+(34+\cdots)q^{7}+\cdots\)
768.6.d.x	$768$	$6$	768.d	8.b	$2$	$4$	$4$	$123.175$	\(\Q(i, \sqrt{7})\)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(136\)	$2^{6}$	$\mathrm{SU}(2)[C_{2}]$	\(q+9\beta _{1}q^{3}+(-4\beta _{1}+5\beta _{3})q^{5}+(34+\cdots)q^{7}+\cdots\)
768.6.d.y	$768$	$6$	768.d	8.b	$2$	$4$	$4$	$123.175$	\(\Q(i, \sqrt{34})\)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(184\)	$2^{7}$	$\mathrm{SU}(2)[C_{2}]$	\(q-9\beta _{1}q^{3}+(20\beta _{1}+\beta _{2})q^{5}+(46-3\beta _{3})q^{7}+\cdots\)
768.6.d.z	$768$	$6$	768.d	8.b	$2$	$4$	$4$	$123.175$	\(\Q(i, \sqrt{31})\)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(240\)	$2^{10}$	$\mathrm{SU}(2)[C_{2}]$	\(q+9\beta _{1}q^{3}+(18\beta _{1}+\beta _{3})q^{5}+(60+\beta _{2}+\cdots)q^{7}+\cdots\)
768.6.d.ba	$768$	$6$	768.d	8.b	$2$	$6$	$6$	$123.175$	6.0.\(\cdots\).1	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(-60\)	$2^{17}$	$\mathrm{SU}(2)[C_{2}]$	\(q-9\beta _{1}q^{3}+(3\beta _{1}+\beta _{4})q^{5}+(-10-\beta _{2}+\cdots)q^{7}+\cdots\)
768.6.d.bb	$768$	$6$	768.d	8.b	$2$	$6$	$6$	$123.175$	6.0.\(\cdots\).1	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(60\)	$2^{17}$	$\mathrm{SU}(2)[C_{2}]$	\(q+9\beta _{1}q^{3}+(3\beta _{1}+\beta _{4})q^{5}+(10+\beta _{2}+\cdots)q^{7}+\cdots\)

Decomposition of \(S_{6}^{\mathrm{old}}(768, [\chi])\) into lower level spaces

\( S_{6}^{\mathrm{old}}(768, [\chi]) \cong \)