Space of modular forms of level 768, weight 3, and Character 768.e

• Label: 768.3.e
• Level: $768$
• Weight: $3$
• Character orbit: 768.e
• Rep. character: $\chi_{768}(257,\cdot)$
• Character field: $\Q$
• Dimension: $60$
• Newform subspaces: $16$
• Sturm bound: $384$
• Trace bound: $21$

Defining parameters

Level:	\( N \)	\(=\)	\( 768 = 2^{8} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	768.e (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 3 \)
Character field:			\(\Q\)
Newform subspaces:			\( 16 \)
Sturm bound:			\(384\)
Trace bound:			\(21\)
Distinguishing \(T_p\):			\(5\), \(7\), \(11\), \(19\), \(37\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{3}(768, [\chi])\).

	Total	New	Old
Modular forms	280	68	212
Cusp forms	232	60	172
Eisenstein series	48	8	40

Trace form

\( 60 q + 4 q^{9} - 212 q^{25} - 40 q^{33} + 180 q^{49} - 32 q^{57} + 328 q^{73} - 4 q^{81} - 456 q^{97}+O(q^{100}) \)

Decomposition of \(S_{3}^{\mathrm{new}}(768, [\chi])\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
768.3.e.a	$768$	$3$	768.e	3.b	$2$	$2$	$2$	$20.926$	\(\Q(\sqrt{3}) \)	\(\Q(\sqrt{-3}) \)	✓				192.3.h.a	$4$	$0$	\(0\)	\(-6\)	\(0\)	\(0\)	$2^{3}$	$\mathrm{U}(1)[D_{2}]$	\(q-3q^{3}+\beta q^{7}+9q^{9}+\beta q^{13}-26q^{19}+\cdots\)
768.3.e.b	$768$	$3$	768.e	3.b	$2$	$2$	$2$	$20.926$	\(\Q(\sqrt{-2}) \)	\(\Q(\sqrt{-2}) \)					192.3.h.b	$4$	$0$	\(0\)	\(-2\)	\(0\)	\(0\)	$2$	$\mathrm{U}(1)[D_{2}]$	\(q+(-1-\beta )q^{3}+(-7+2\beta )q^{9}+6\beta q^{11}+\cdots\)
768.3.e.c	$768$	$3$	768.e	3.b	$2$	$2$	$2$	$20.926$	\(\Q(\sqrt{-1}) \)	\(\Q(\sqrt{-6}) \)					24.3.h.a	$4$	$0$	\(0\)	\(0\)	\(0\)	\(-20\)	$1$	$\mathrm{U}(1)[D_{2}]$	\(q+3 i q^{3}+2 i q^{5}-10 q^{7}-9 q^{9}+\cdots\)
768.3.e.d	$768$	$3$	768.e	3.b	$2$	$2$	$2$	$20.926$	\(\Q(\sqrt{-1}) \)	\(\Q(\sqrt{-6}) \)					24.3.h.a	$4$	$0$	\(0\)	\(0\)	\(0\)	\(20\)	$1$	$\mathrm{U}(1)[D_{2}]$	\(q-3 i q^{3}+2 i q^{5}+10 q^{7}-9 q^{9}+\cdots\)
768.3.e.e	$768$	$3$	768.e	3.b	$2$	$2$	$2$	$20.926$	\(\Q(\sqrt{-2}) \)	\(\Q(\sqrt{-2}) \)					192.3.h.b	$4$	$0$	\(0\)	\(2\)	\(0\)	\(0\)	$2$	$\mathrm{U}(1)[D_{2}]$	\(q+(1+\beta )q^{3}+(-7+2\beta )q^{9}-6\beta q^{11}+\cdots\)
768.3.e.f	$768$	$3$	768.e	3.b	$2$	$2$	$2$	$20.926$	\(\Q(\sqrt{3}) \)	\(\Q(\sqrt{-3}) \)	✓				192.3.h.a	$4$	$0$	\(0\)	\(6\)	\(0\)	\(0\)	$2^{3}$	$\mathrm{U}(1)[D_{2}]$	\(q+3q^{3}+\beta q^{7}+9q^{9}-\beta q^{13}+26q^{19}+\cdots\)
768.3.e.g	$768$	$3$	768.e	3.b	$2$	$4$	$4$	$20.926$	\(\Q(\sqrt{3}, \sqrt{-5})\)	None					192.3.h.c	$4$	$0$	\(0\)	\(-8\)	\(0\)	\(0\)	$2^{4}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-2-\beta _{1})q^{3}+\beta _{2}q^{5}-\beta _{3}q^{7}+(-1+\cdots)q^{9}+\cdots\)
768.3.e.h	$768$	$3$	768.e	3.b	$2$	$4$	$4$	$20.926$	\(\Q(i, \sqrt{5})\)	None					384.3.h.e	$4$	$0$	\(0\)	\(-8\)	\(0\)	\(0\)	$2^{6}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-2+\beta _{1})q^{3}-\beta _{2}q^{5}+\beta _{3}q^{7}+(-1+\cdots)q^{9}+\cdots\)
768.3.e.i	$768$	$3$	768.e	3.b	$2$	$4$	$4$	$20.926$	\(\Q(\sqrt{-2}, \sqrt{7})\)	None					24.3.h.c	$4$	$0$	\(0\)	\(0\)	\(0\)	\(-16\)	$2^{3}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{3}q^{3}-2\beta _{1}q^{5}-4q^{7}+(5+\beta _{2}+\cdots)q^{9}+\cdots\)
768.3.e.j	$768$	$3$	768.e	3.b	$2$	$4$	$4$	$20.926$	\(\Q(i, \sqrt{6})\)	\(\Q(\sqrt{-6}) \)					384.3.h.a	$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{5}$	$\mathrm{U}(1)[D_{2}]$	\(q+3\beta _{1}q^{3}+\beta _{2}q^{5}-\beta _{3}q^{7}-9q^{9}+\cdots\)
768.3.e.k	$768$	$3$	768.e	3.b	$2$	$4$	$4$	$20.926$	\(\Q(\zeta_{8})\)	\(\Q(\sqrt{-2}) \)					384.3.h.b	$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{5}$	$\mathrm{U}(1)[D_{2}]$	\(q+(\beta_{2}+\beta_1)q^{3}+(\beta_{3}+7)q^{9}-7\beta_1 q^{11}+\cdots\)
768.3.e.l	$768$	$3$	768.e	3.b	$2$	$4$	$4$	$20.926$	\(\Q(\sqrt{-2}, \sqrt{7})\)	None					24.3.h.c	$4$	$0$	\(0\)	\(0\)	\(0\)	\(16\)	$2^{3}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{3}q^{3}+2\beta _{1}q^{5}+4q^{7}+(5+\beta _{2}+\cdots)q^{9}+\cdots\)
768.3.e.m	$768$	$3$	768.e	3.b	$2$	$4$	$4$	$20.926$	\(\Q(i, \sqrt{5})\)	None					384.3.h.e	$4$	$0$	\(0\)	\(8\)	\(0\)	\(0\)	$2^{6}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(2+\beta _{1})q^{3}-\beta _{2}q^{5}+\beta _{3}q^{7}+(-1+\cdots)q^{9}+\cdots\)
768.3.e.n	$768$	$3$	768.e	3.b	$2$	$4$	$4$	$20.926$	\(\Q(\sqrt{3}, \sqrt{-5})\)	None					192.3.h.c	$4$	$0$	\(0\)	\(8\)	\(0\)	\(0\)	$2^{4}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(2+\beta _{1})q^{3}+\beta _{2}q^{5}+\beta _{3}q^{7}+(-1+\cdots)q^{9}+\cdots\)
768.3.e.o	$768$	$3$	768.e	3.b	$2$	$8$	$8$	$20.926$	8.0.\(\cdots\).2	None					384.3.h.g	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{12}\cdot 3$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{1}q^{3}+\beta _{3}q^{5}+(-\beta _{1}-\beta _{5})q^{7}+\cdots\)
768.3.e.p	$768$	$3$	768.e	3.b	$2$	$8$	$8$	$20.926$	8.0.\(\cdots\).2	None					384.3.h.g	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{12}\cdot 3$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{1}q^{3}+\beta _{3}q^{5}+(\beta _{1}+\beta _{4})q^{7}+(1+\cdots)q^{9}+\cdots\)

Decomposition of \(S_{3}^{\mathrm{old}}(768, [\chi])\) into lower level spaces

\( S_{3}^{\mathrm{old}}(768, [\chi]) \simeq \) \(S_{3}^{\mathrm{new}}(12, [\chi])\)\(^{\oplus 7}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(24, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(48, [\chi])\)\(^{\oplus 5}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(96, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(192, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(384, [\chi])\)\(^{\oplus 2}\)