Space of modular forms of level 1764, weight 3, and Character 1764.bk

• Label: 1764.3.bk
• Level: $1764$
• Weight: $3$
• Character orbit: 1764.bk
• Rep. character: $\chi_{1764}(557,\cdot)$
• Character field: $\Q(\zeta_{6})$
• Dimension: $52$
• Newform subspaces: $7$
• Sturm bound: $1008$
• Trace bound: $13$

Defining parameters

Level:	\( N \)	\(=\)	\( 1764 = 2^{2} \cdot 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1764.bk (of order \(6\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 21 \)
Character field:			\(\Q(\zeta_{6})\)
Newform subspaces:			\( 7 \)
Sturm bound:			\(1008\)
Trace bound:			\(13\)
Distinguishing \(T_p\):			\(5\), \(13\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{3}(1764, [\chi])\).

	Total	New	Old
Modular forms	1440	52	1388
Cusp forms	1248	52	1196
Eisenstein series	192	0	192

Trace form

\( 52 q + O(q^{10}) \)

Decomposition of \(S_{3}^{\mathrm{new}}(1764, [\chi])\) 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
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
1764.3.bk.a	$1764$	$3$	1764.bk	21.h	$6$	$4$	$2$	$48.066$	\(\Q(\sqrt{-2}, \sqrt{-3})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$3^{2}$	$\mathrm{SU}(2)[C_{6}]$	\(q+\beta _{1}q^{5}+(5\beta _{1}-5\beta _{3})q^{11}-23q^{13}+\cdots\)
1764.3.bk.b	$1764$	$3$	1764.bk	21.h	$6$	$4$	$2$	$48.066$	\(\Q(\sqrt{-2}, \sqrt{-3})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{6}]$	\(q+4\beta _{1}q^{5}+(-\beta _{1}+\beta _{3})q^{11}-2^{4}q^{13}+\cdots\)
1764.3.bk.c	$1764$	$3$	1764.bk	21.h	$6$	$4$	$2$	$48.066$	\(\Q(\sqrt{-2}, \sqrt{-3})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{6}]$	\(q+4\beta _{1}q^{5}+(\beta _{1}-\beta _{3})q^{11}+2^{4}q^{13}+\cdots\)
1764.3.bk.d	$1764$	$3$	1764.bk	21.h	$6$	$8$	$4$	$48.066$	8.0.\(\cdots\).5	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{4}\cdot 3^{2}$	$\mathrm{SU}(2)[C_{6}]$	\(q+\beta _{1}q^{5}+(2\beta _{5}-\beta _{7})q^{11}+(-2-2\beta _{3}+\cdots)q^{13}+\cdots\)
1764.3.bk.e	$1764$	$3$	1764.bk	21.h	$6$	$8$	$4$	$48.066$	8.0.\(\cdots\).5	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{4}\cdot 3^{2}$	$\mathrm{SU}(2)[C_{6}]$	\(q+\beta _{1}q^{5}+(-2\beta _{5}+\beta _{7})q^{11}+(2+2\beta _{3}+\cdots)q^{13}+\cdots\)
1764.3.bk.f	$1764$	$3$	1764.bk	21.h	$6$	$8$	$4$	$48.066$	8.0.\(\cdots\).1	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{4}\cdot 3^{2}$	$\mathrm{SU}(2)[C_{6}]$	\(q+(-\beta _{3}-\beta _{5})q^{5}+(2\beta _{4}-\beta _{6})q^{11}+\cdots\)
1764.3.bk.g	$1764$	$3$	1764.bk	21.h	$6$	$16$	$8$	$48.066$	16.0.\(\cdots\).1	None		$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{16}\cdot 3^{8}$	$\mathrm{SU}(2)[C_{6}]$	\(q+(-\beta _{3}-\beta _{4}-\beta _{6}+\beta _{7})q^{5}-\beta _{13}q^{11}+\cdots\)

Decomposition of \(S_{3}^{\mathrm{old}}(1764, [\chi])\) into lower level spaces

\( S_{3}^{\mathrm{old}}(1764, [\chi]) \cong \) \(S_{3}^{\mathrm{new}}(21, [\chi])\)\(^{\oplus 12}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(42, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(63, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(84, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(126, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(147, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(252, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(294, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(441, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(588, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(882, [\chi])\)\(^{\oplus 2}\)