Space of modular forms of level 768, weight 2, and Character 768.k

• Label: 768.2.k
• Level: $768$
• Weight: $2$
• Character orbit: 768.k
• Rep. character: $\chi_{768}(191,\cdot)$
• Character field: $\Q(\zeta_{4})$
• Dimension: $64$
• Newform subspaces: $8$
• Sturm bound: $256$
• Trace bound: $21$

Defining parameters

Level:	\( N \)	\(=\)	\( 768 = 2^{8} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	768.k (of order \(4\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 48 \)
Character field:			\(\Q(i)\)
Newform subspaces:			\( 8 \)
Sturm bound:			\(256\)
Trace bound:			\(21\)
Distinguishing \(T_p\):			\(5\), \(11\), \(13\), \(37\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(768, [\chi])\).

	Total	New	Old
Modular forms	304	64	240
Cusp forms	208	64	144
Eisenstein series	96	0	96

Trace form

\( 64 q + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(768, [\chi])\) 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
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
768.2.k.a	$768$	$2$	768.k	48.k	$4$	$4$	$2$	$6.133$	\(\Q(\zeta_{8})\)	None		$4$	$1$	\(0\)	\(-4\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+(-1-\zeta_{8}^{2}+\zeta_{8}^{3})q^{3}+2\zeta_{8}q^{5}+\cdots\)
768.2.k.b	$768$	$2$	768.k	48.k	$4$	$4$	$2$	$6.133$	\(\Q(\zeta_{8})\)	None		$4$	$0$	\(0\)	\(-4\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+(-1-\zeta_{8}^{2}-\zeta_{8}^{3})q^{3}+2\zeta_{8}q^{5}+\cdots\)
768.2.k.c	$768$	$2$	768.k	48.k	$4$	$4$	$2$	$6.133$	\(\Q(\zeta_{8})\)	None		$4$	$0$	\(0\)	\(4\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+(1+\zeta_{8}^{2}-\zeta_{8}^{3})q^{3}+2\zeta_{8}q^{5}+(\zeta_{8}+\cdots)q^{7}+\cdots\)
768.2.k.d	$768$	$2$	768.k	48.k	$4$	$4$	$2$	$6.133$	\(\Q(\zeta_{8})\)	None		$4$	$0$	\(0\)	\(4\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+(1+\zeta_{8}^{2}+\zeta_{8}^{3})q^{3}+2\zeta_{8}q^{5}+(\zeta_{8}+\cdots)q^{7}+\cdots\)
768.2.k.e	$768$	$2$	768.k	48.k	$4$	$8$	$4$	$6.133$	\(\Q(\zeta_{24})\)	\(\Q(\sqrt{-3}) \)		$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{10}$	$\mathrm{U}(1)[D_{4}]$	\(q+\zeta_{24}^{3}q^{3}+(-\zeta_{24}+\zeta_{24}^{3}-\zeta_{24}^{4}+\cdots)q^{7}+\cdots\)
768.2.k.f	$768$	$2$	768.k	48.k	$4$	$8$	$4$	$6.133$	\(\Q(\zeta_{24})\)	\(\Q(\sqrt{-3}) \)		$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{10}$	$\mathrm{U}(1)[D_{4}]$	\(q+\zeta_{24}^{3}q^{3}+(\zeta_{24}-\zeta_{24}^{3}+\zeta_{24}^{4}+\cdots)q^{7}+\cdots\)
768.2.k.g	$768$	$2$	768.k	48.k	$4$	$16$	$8$	$6.133$	16.0.\(\cdots\).1	None		$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{18}$	$\mathrm{SU}(2)[C_{4}]$	\(q+\beta _{4}q^{3}+(\beta _{1}+\beta _{6}+\beta _{8})q^{5}+(\beta _{3}+\beta _{7}+\cdots)q^{7}+\cdots\)
768.2.k.h	$768$	$2$	768.k	48.k	$4$	$16$	$8$	$6.133$	16.0.\(\cdots\).1	None		$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{18}$	$\mathrm{SU}(2)[C_{4}]$	\(q-\beta _{14}q^{3}+(\beta _{1}+\beta _{6}+\beta _{8})q^{5}+(-\beta _{3}+\cdots)q^{7}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(768, [\chi])\) into lower level spaces

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