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

\( 48 q - 432 q^{9} - 1200 q^{25} + 3792 q^{49} - 672 q^{57} + 1952 q^{65} - 320 q^{73} + 3888 q^{81} + 352 q^{89} - 3168 q^{97}+O(q^{100}) \)

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	Twist minimal	Largest	Maximal	Minimal twist	Inner twists	Rank*	Traces				Coefficient ring index	Sato-Tate	$q$-expansion
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
768.4.d.a	$768$	$4$	768.d	8.b	$2$	$2$	$2$	$45.313$	\(\Q(\sqrt{-1}) \)	None					96.4.a.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(-72\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-3 i q^{3}+14 i q^{5}-36 q^{7}-9 q^{9}+\cdots\)
768.4.d.b	$768$	$4$	768.d	8.b	$2$	$2$	$2$	$45.313$	\(\Q(\sqrt{-1}) \)	None					24.4.a.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(-48\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+3 i q^{3}+14 i q^{5}-24 q^{7}-9 q^{9}+\cdots\)
768.4.d.c	$768$	$4$	768.d	8.b	$2$	$2$	$2$	$45.313$	\(\Q(\sqrt{-1}) \)	None					6.4.a.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(-32\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-3 i q^{3}+6 i q^{5}-16 q^{7}-9 q^{9}+\cdots\)
768.4.d.d	$768$	$4$	768.d	8.b	$2$	$2$	$2$	$45.313$	\(\Q(\sqrt{-1}) \)	None					96.4.a.b	$2$	$0$	\(0\)	\(0\)	\(0\)	\(-24\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-3 i q^{3}+2 i q^{5}-12 q^{7}-9 q^{9}+\cdots\)
768.4.d.e	$768$	$4$	768.d	8.b	$2$	$2$	$2$	$45.313$	\(\Q(\sqrt{-1}) \)	None					384.4.a.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(-20\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+3 i q^{3}+8 i q^{5}-10 q^{7}-9 q^{9}+\cdots\)
768.4.d.f	$768$	$4$	768.d	8.b	$2$	$2$	$2$	$45.313$	\(\Q(\sqrt{-1}) \)	None					384.4.a.b	$2$	$0$	\(0\)	\(0\)	\(0\)	\(-20\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-3 i q^{3}+4 i q^{5}-10 q^{7}-9 q^{9}+\cdots\)
768.4.d.g	$768$	$4$	768.d	8.b	$2$	$2$	$2$	$45.313$	\(\Q(\sqrt{-1}) \)	None					12.4.a.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(-16\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+3 i q^{3}+18 i q^{5}-8 q^{7}-9 q^{9}+\cdots\)
768.4.d.h	$768$	$4$	768.d	8.b	$2$	$2$	$2$	$45.313$	\(\Q(\sqrt{-1}) \)	None					96.4.a.c	$2$	$0$	\(0\)	\(0\)	\(0\)	\(-8\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-3 i q^{3}+10 i q^{5}-4 q^{7}-9 q^{9}+\cdots\)
768.4.d.i	$768$	$4$	768.d	8.b	$2$	$2$	$2$	$45.313$	\(\Q(\sqrt{-1}) \)	None					96.4.a.c	$2$	$0$	\(0\)	\(0\)	\(0\)	\(8\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+3 i q^{3}+10 i q^{5}+4 q^{7}-9 q^{9}+\cdots\)
768.4.d.j	$768$	$4$	768.d	8.b	$2$	$2$	$2$	$45.313$	\(\Q(\sqrt{-1}) \)	None					12.4.a.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(16\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-3 i q^{3}+18 i q^{5}+8 q^{7}-9 q^{9}+\cdots\)
768.4.d.k	$768$	$4$	768.d	8.b	$2$	$2$	$2$	$45.313$	\(\Q(\sqrt{-1}) \)	None					384.4.a.b	$2$	$0$	\(0\)	\(0\)	\(0\)	\(20\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+3 i q^{3}+4 i q^{5}+10 q^{7}-9 q^{9}+\cdots\)
768.4.d.l	$768$	$4$	768.d	8.b	$2$	$2$	$2$	$45.313$	\(\Q(\sqrt{-1}) \)	None					384.4.a.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(20\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-3 i q^{3}+8 i q^{5}+10 q^{7}-9 q^{9}+\cdots\)
768.4.d.m	$768$	$4$	768.d	8.b	$2$	$2$	$2$	$45.313$	\(\Q(\sqrt{-1}) \)	None					96.4.a.b	$2$	$0$	\(0\)	\(0\)	\(0\)	\(24\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+3 i q^{3}+2 i q^{5}+12 q^{7}-9 q^{9}+\cdots\)
768.4.d.n	$768$	$4$	768.d	8.b	$2$	$2$	$2$	$45.313$	\(\Q(\sqrt{-1}) \)	None					6.4.a.a	$2$	$1$	\(0\)	\(0\)	\(0\)	\(32\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+3 i q^{3}+6 i q^{5}+16 q^{7}-9 q^{9}+\cdots\)
768.4.d.o	$768$	$4$	768.d	8.b	$2$	$2$	$2$	$45.313$	\(\Q(\sqrt{-1}) \)	None					24.4.a.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(48\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-3 i q^{3}+14 i q^{5}+24 q^{7}-9 q^{9}+\cdots\)
768.4.d.p	$768$	$4$	768.d	8.b	$2$	$2$	$2$	$45.313$	\(\Q(\sqrt{-1}) \)	None					96.4.a.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(72\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+3 i q^{3}+14 i q^{5}+36 q^{7}-9 q^{9}+\cdots\)
768.4.d.q	$768$	$4$	768.d	8.b	$2$	$4$	$4$	$45.313$	\(\Q(i, \sqrt{15})\)	None					384.4.a.j	$2$	$0$	\(0\)	\(0\)	\(0\)	\(-8\)	$2^{6}$	$\mathrm{SU}(2)[C_{2}]$	\(q+3\beta _{1}q^{3}+(-4\beta _{1}+\beta _{3})q^{5}+(-2+\cdots)q^{7}+\cdots\)
768.4.d.r	$768$	$4$	768.d	8.b	$2$	$4$	$4$	$45.313$	\(\Q(i, \sqrt{7})\)	None					384.4.a.i	$2$	$0$	\(0\)	\(0\)	\(0\)	\(-8\)	$2^{6}$	$\mathrm{SU}(2)[C_{2}]$	\(q+3\beta _{1}q^{3}+(-8\beta _{1}-\beta _{3})q^{5}+(-2+\cdots)q^{7}+\cdots\)
768.4.d.s	$768$	$4$	768.d	8.b	$2$	$4$	$4$	$45.313$	\(\Q(i, \sqrt{7})\)	None					384.4.a.i	$2$	$0$	\(0\)	\(0\)	\(0\)	\(8\)	$2^{6}$	$\mathrm{SU}(2)[C_{2}]$	\(q-3\beta _{1}q^{3}+(-8\beta _{1}-\beta _{3})q^{5}+(2-3\beta _{2}+\cdots)q^{7}+\cdots\)
768.4.d.t	$768$	$4$	768.d	8.b	$2$	$4$	$4$	$45.313$	\(\Q(i, \sqrt{15})\)	None					384.4.a.j	$2$	$0$	\(0\)	\(0\)	\(0\)	\(8\)	$2^{6}$	$\mathrm{SU}(2)[C_{2}]$	\(q-3\beta _{1}q^{3}+(-4\beta _{1}+\beta _{3})q^{5}+(2-\beta _{2}+\cdots)q^{7}+\cdots\)

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

\( S_{4}^{\mathrm{old}}(768, [\chi]) \simeq \) \(S_{4}^{\mathrm{new}}(8, [\chi])\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(24, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(32, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(64, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(96, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(128, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(192, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(256, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(384, [\chi])\)\(^{\oplus 2}\)