Space of modular forms of level 768, weight 4, and Character 768.d

• Label: 768.4.d
• Level: $768$
• Weight: $4$
• Character orbit: 768.d
• Rep. character: $\chi_{768}(385,\cdot)$
• Character field: $\Q$
• Dimension: $48$
• Newform subspaces: $20$
• Sturm bound: $512$
• Trace bound: $15$

Defining parameters

Level:	\( N \)	\(=\)	\( 768 = 2^{8} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	768.d (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 8 \)
Character field:			\(\Q\)
Newform subspaces:			\( 20 \)
Sturm bound:			\(512\)
Trace bound:			\(15\)
Distinguishing \(T_p\):			\(5\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(768, [\chi])\).

	Total	New	Old
Modular forms	408	48	360
Cusp forms	360	48	312
Eisenstein series	48	0	48

Trace form

\( 48 q - 432 q^{9} + O(q^{10}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(768, [\chi])\) 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
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
768.4.d.a	$768$	$4$	768.d	8.b	$2$	$2$	$2$	$45.313$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(-72\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-3iq^{3}+14iq^{5}-6^{2}q^{7}-9q^{9}+\cdots\)
768.4.d.b	$768$	$4$	768.d	8.b	$2$	$2$	$2$	$45.313$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(-48\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+3iq^{3}+14iq^{5}-24q^{7}-9q^{9}+\cdots\)
768.4.d.c	$768$	$4$	768.d	8.b	$2$	$2$	$2$	$45.313$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(-32\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-3iq^{3}+6iq^{5}-2^{4}q^{7}-9q^{9}-12iq^{11}+\cdots\)
768.4.d.d	$768$	$4$	768.d	8.b	$2$	$2$	$2$	$45.313$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(-24\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-3iq^{3}+2iq^{5}-12q^{7}-9q^{9}+60iq^{11}+\cdots\)
768.4.d.e	$768$	$4$	768.d	8.b	$2$	$2$	$2$	$45.313$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(-20\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+3iq^{3}+8iq^{5}-10q^{7}-9q^{9}+68iq^{11}+\cdots\)
768.4.d.f	$768$	$4$	768.d	8.b	$2$	$2$	$2$	$45.313$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(-20\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-3iq^{3}+4iq^{5}-10q^{7}-9q^{9}+4iq^{11}+\cdots\)
768.4.d.g	$768$	$4$	768.d	8.b	$2$	$2$	$2$	$45.313$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(-16\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+3iq^{3}+18iq^{5}-8q^{7}-9q^{9}-6^{2}iq^{11}+\cdots\)
768.4.d.h	$768$	$4$	768.d	8.b	$2$	$2$	$2$	$45.313$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(-8\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-3iq^{3}+10iq^{5}-4q^{7}-9q^{9}-20iq^{11}+\cdots\)
768.4.d.i	$768$	$4$	768.d	8.b	$2$	$2$	$2$	$45.313$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(8\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+3iq^{3}+10iq^{5}+4q^{7}-9q^{9}+20iq^{11}+\cdots\)
768.4.d.j	$768$	$4$	768.d	8.b	$2$	$2$	$2$	$45.313$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(16\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-3iq^{3}+18iq^{5}+8q^{7}-9q^{9}+6^{2}iq^{11}+\cdots\)
768.4.d.k	$768$	$4$	768.d	8.b	$2$	$2$	$2$	$45.313$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(20\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+3iq^{3}+4iq^{5}+10q^{7}-9q^{9}-4iq^{11}+\cdots\)
768.4.d.l	$768$	$4$	768.d	8.b	$2$	$2$	$2$	$45.313$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(20\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-3iq^{3}+8iq^{5}+10q^{7}-9q^{9}-68iq^{11}+\cdots\)
768.4.d.m	$768$	$4$	768.d	8.b	$2$	$2$	$2$	$45.313$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(24\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+3iq^{3}+2iq^{5}+12q^{7}-9q^{9}-60iq^{11}+\cdots\)
768.4.d.n	$768$	$4$	768.d	8.b	$2$	$2$	$2$	$45.313$	\(\Q(\sqrt{-1}) \)	None		$2$	$1$	\(0\)	\(0\)	\(0\)	\(32\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+3iq^{3}+6iq^{5}+2^{4}q^{7}-9q^{9}+12iq^{11}+\cdots\)
768.4.d.o	$768$	$4$	768.d	8.b	$2$	$2$	$2$	$45.313$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(48\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-3iq^{3}+14iq^{5}+24q^{7}-9q^{9}+\cdots\)
768.4.d.p	$768$	$4$	768.d	8.b	$2$	$2$	$2$	$45.313$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(72\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+3iq^{3}+14iq^{5}+6^{2}q^{7}-9q^{9}+\cdots\)
768.4.d.q	$768$	$4$	768.d	8.b	$2$	$4$	$4$	$45.313$	\(\Q(i, \sqrt{15})\)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(-8\)	$2^{6}$	$\mathrm{SU}(2)[C_{2}]$	\(q+3\beta _{1}q^{3}+(-4\beta _{1}+\beta _{3})q^{5}+(-2+\cdots)q^{7}+\cdots\)
768.4.d.r	$768$	$4$	768.d	8.b	$2$	$4$	$4$	$45.313$	\(\Q(i, \sqrt{7})\)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(-8\)	$2^{6}$	$\mathrm{SU}(2)[C_{2}]$	\(q+3\beta _{1}q^{3}+(-8\beta _{1}-\beta _{3})q^{5}+(-2+\cdots)q^{7}+\cdots\)
768.4.d.s	$768$	$4$	768.d	8.b	$2$	$4$	$4$	$45.313$	\(\Q(i, \sqrt{7})\)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(8\)	$2^{6}$	$\mathrm{SU}(2)[C_{2}]$	\(q-3\beta _{1}q^{3}+(-8\beta _{1}-\beta _{3})q^{5}+(2-3\beta _{2}+\cdots)q^{7}+\cdots\)
768.4.d.t	$768$	$4$	768.d	8.b	$2$	$4$	$4$	$45.313$	\(\Q(i, \sqrt{15})\)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(8\)	$2^{6}$	$\mathrm{SU}(2)[C_{2}]$	\(q-3\beta _{1}q^{3}+(-4\beta _{1}+\beta _{3})q^{5}+(2-\beta _{2}+\cdots)q^{7}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(768, [\chi])\) into lower level spaces

\( S_{4}^{\mathrm{old}}(768, [\chi]) \cong \)