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

• Label: 768.4.c
• Level: $768$
• Weight: $4$
• Character orbit: 768.c
• Rep. character: $\chi_{768}(767,\cdot)$
• Character field: $\Q$
• Dimension: $92$
• Newform subspaces: $22$
• Sturm bound: $512$
• Trace bound: $25$

Defining parameters

Level:	\( N \)	\(=\)	\( 768 = 2^{8} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	768.c (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 12 \)
Character field:			\(\Q\)
Newform subspaces:			\( 22 \)
Sturm bound:			\(512\)
Trace bound:			\(25\)
Distinguishing \(T_p\):			\(5\), \(7\), \(11\), \(13\), \(23\)

Dimensions

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

	Total	New	Old
Modular forms	408	100	308
Cusp forms	360	92	268
Eisenstein series	48	8	40

Trace form

\( 92 q + 4 q^{9} + O(q^{10}) \)

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.c.a	$768$	$4$	768.c	12.b	$2$	$2$	$2$	$45.313$	\(\Q(\sqrt{-2}) \)	\(\Q(\sqrt{-2}) \)		$4$	$1$	\(0\)	\(-10\)	\(0\)	\(0\)	$1$	$\mathrm{U}(1)[D_{2}]$	\(q+(-5-\beta )q^{3}+(23+10\beta )q^{9}-18q^{11}+\cdots\)
768.4.c.b	$768$	$4$	768.c	12.b	$2$	$2$	$2$	$45.313$	\(\Q(\sqrt{-2}) \)	None		$2$	$0$	\(0\)	\(-6\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-3+3\beta )q^{3}+4\beta q^{5}-12\beta q^{7}+\cdots\)
768.4.c.c	$768$	$4$	768.c	12.b	$2$	$2$	$2$	$45.313$	\(\Q(\sqrt{-2}) \)	None		$2$	$0$	\(0\)	\(-6\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-3-3\beta )q^{3}+4\beta q^{5}-12\beta q^{7}+\cdots\)
768.4.c.d	$768$	$4$	768.c	12.b	$2$	$2$	$2$	$45.313$	\(\Q(\sqrt{-26}) \)	None		$2$	$0$	\(0\)	\(-2\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-1-\beta )q^{3}+2\beta q^{5}-2\beta q^{7}+(-5^{2}+\cdots)q^{9}+\cdots\)
768.4.c.e	$768$	$4$	768.c	12.b	$2$	$2$	$2$	$45.313$	\(\Q(\sqrt{-26}) \)	None		$2$	$0$	\(0\)	\(-2\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-1+\beta )q^{3}+2\beta q^{5}-2\beta q^{7}+(-5^{2}+\cdots)q^{9}+\cdots\)
768.4.c.f	$768$	$4$	768.c	12.b	$2$	$2$	$2$	$45.313$	\(\Q(\sqrt{-26}) \)	None		$2$	$1$	\(0\)	\(2\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+(1+\beta )q^{3}+2\beta q^{5}+2\beta q^{7}+(-5^{2}+\cdots)q^{9}+\cdots\)
768.4.c.g	$768$	$4$	768.c	12.b	$2$	$2$	$2$	$45.313$	\(\Q(\sqrt{-26}) \)	None		$2$	$0$	\(0\)	\(2\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+(1-\beta )q^{3}+2\beta q^{5}+2\beta q^{7}+(-5^{2}+\cdots)q^{9}+\cdots\)
768.4.c.h	$768$	$4$	768.c	12.b	$2$	$2$	$2$	$45.313$	\(\Q(\sqrt{-2}) \)	None		$2$	$0$	\(0\)	\(6\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+(3-3\beta )q^{3}+4\beta q^{5}+12\beta q^{7}+(-9+\cdots)q^{9}+\cdots\)
768.4.c.i	$768$	$4$	768.c	12.b	$2$	$2$	$2$	$45.313$	\(\Q(\sqrt{-2}) \)	None		$2$	$0$	\(0\)	\(6\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+(3+3\beta )q^{3}+4\beta q^{5}+12\beta q^{7}+(-9+\cdots)q^{9}+\cdots\)
768.4.c.j	$768$	$4$	768.c	12.b	$2$	$2$	$2$	$45.313$	\(\Q(\sqrt{-2}) \)	\(\Q(\sqrt{-2}) \)		$4$	$0$	\(0\)	\(10\)	\(0\)	\(0\)	$1$	$\mathrm{U}(1)[D_{2}]$	\(q+(5+\beta )q^{3}+(23+10\beta )q^{9}+18q^{11}+\cdots\)
768.4.c.k	$768$	$4$	768.c	12.b	$2$	$4$	$4$	$45.313$	\(\Q(\zeta_{12})\)	\(\Q(\sqrt{-3}) \)		$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{6}\cdot 3^{2}$	$\mathrm{U}(1)[D_{2}]$	\(q+\zeta_{12}^{2}q^{3}-5\zeta_{12}q^{7}-3^{3}q^{9}-3\zeta_{12}^{3}q^{13}+\cdots\)
768.4.c.l	$768$	$4$	768.c	12.b	$2$	$4$	$4$	$45.313$	\(\Q(\zeta_{8})\)	\(\Q(\sqrt{-2}) \)		$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{3}$	$\mathrm{U}(1)[D_{2}]$	\(q+(2\zeta_{8}-3\zeta_{8}^{2})q^{3}+(-23+5\zeta_{8}^{3})q^{9}+\cdots\)
768.4.c.m	$768$	$4$	768.c	12.b	$2$	$4$	$4$	$45.313$	\(\Q(\sqrt{-6}, \sqrt{-14})\)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{4}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(\beta _{1}+\beta _{3})q^{3}+(-2\beta _{1}-\beta _{2})q^{5}+(3\beta _{1}+\cdots)q^{7}+\cdots\)
768.4.c.n	$768$	$4$	768.c	12.b	$2$	$4$	$4$	$45.313$	\(\Q(\sqrt{-6}, \sqrt{-14})\)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{4}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{3}q^{3}+(-2\beta _{1}-\beta _{2})q^{5}+(3\beta _{1}-2\beta _{2}+\cdots)q^{7}+\cdots\)
768.4.c.o	$768$	$4$	768.c	12.b	$2$	$4$	$4$	$45.313$	\(\Q(\sqrt{-6}, \sqrt{-14})\)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{4}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-\beta _{1}-\beta _{3})q^{3}+(-2\beta _{1}-\beta _{2})q^{5}+\cdots\)
768.4.c.p	$768$	$4$	768.c	12.b	$2$	$4$	$4$	$45.313$	\(\Q(\sqrt{-6}, \sqrt{-14})\)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{4}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{3}q^{3}+(-2\beta _{1}-\beta _{2})q^{5}+(-3\beta _{1}+\cdots)q^{7}+\cdots\)
768.4.c.q	$768$	$4$	768.c	12.b	$2$	$4$	$4$	$45.313$	\(\Q(\sqrt{-2}, \sqrt{3})\)	\(\Q(\sqrt{-6}) \)		$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{3}\cdot 3^{2}$	$\mathrm{U}(1)[D_{2}]$	\(q-\beta _{3}q^{3}-7\beta _{1}q^{5}+\beta _{2}q^{7}+3^{3}q^{9}+\cdots\)
768.4.c.r	$768$	$4$	768.c	12.b	$2$	$4$	$4$	$45.313$	\(\Q(\zeta_{12})\)	\(\Q(\sqrt{-6}) \)		$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{4}\cdot 3^{2}$	$\mathrm{U}(1)[D_{2}]$	\(q+\zeta_{12}^{2}q^{3}-\zeta_{12}^{3}q^{5}+17\zeta_{12}q^{7}+\cdots\)
768.4.c.s	$768$	$4$	768.c	12.b	$2$	$8$	$8$	$45.313$	8.0.\(\cdots\).4	None		$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{18}\cdot 3^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-\beta _{1}-\beta _{3})q^{3}+\beta _{4}q^{5}-\beta _{6}q^{7}+\cdots\)
768.4.c.t	$768$	$4$	768.c	12.b	$2$	$8$	$8$	$45.313$	8.0.1731891456.1	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{12}\cdot 3^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{4}q^{3}+(3\beta _{2}+\beta _{4}+\beta _{6})q^{5}+\beta _{7}q^{7}+\cdots\)
768.4.c.u	$768$	$4$	768.c	12.b	$2$	$8$	$8$	$45.313$	8.0.1731891456.1	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{12}\cdot 3^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{4}q^{3}+(3\beta _{2}+\beta _{4}+\beta _{6})q^{5}-\beta _{7}q^{7}+\cdots\)
768.4.c.v	$768$	$4$	768.c	12.b	$2$	$16$	$16$	$45.313$	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)	None		$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{50}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{4}q^{3}+\beta _{9}q^{5}+\beta _{6}q^{7}+(6-3\beta _{7}+\cdots)q^{9}+\cdots\)

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

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