Space of modular forms of level 1764, weight 2, and Character 1764.k

• Label: 1764.2.k
• Level: $1764$
• Weight: $2$
• Character orbit: 1764.k
• Rep. character: $\chi_{1764}(361,\cdot)$
• Character field: $\Q(\zeta_{3})$
• Dimension: $34$
• Newform subspaces: $13$
• Sturm bound: $672$
• Trace bound: $13$

Defining parameters

Level:	\( N \)	\(=\)	\( 1764 = 2^{2} \cdot 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1764.k (of order \(3\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 7 \)
Character field:			\(\Q(\zeta_{3})\)
Newform subspaces:			\( 13 \)
Sturm bound:			\(672\)
Trace bound:			\(13\)
Distinguishing \(T_p\):			\(5\), \(11\), \(13\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(1764, [\chi])\).

	Total	New	Old
Modular forms	768	34	734
Cusp forms	576	34	542
Eisenstein series	192	0	192

Trace form

\( 34 q - q^{5} + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(1764, [\chi])\) 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
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
1764.2.k.a	$1764$	$2$	1764.k	7.c	$3$	$2$	$1$	$14.086$	\(\Q(\sqrt{-3}) \)	None		$2$	$0$	\(0\)	\(0\)	\(-4\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q-4\zeta_{6}q^{5}+(2-2\zeta_{6})q^{11}+6q^{13}+\cdots\)
1764.2.k.b	$1764$	$2$	1764.k	7.c	$3$	$2$	$1$	$14.086$	\(\Q(\sqrt{-3}) \)	None		$2$	$0$	\(0\)	\(0\)	\(-3\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q-3\zeta_{6}q^{5}+(-3+3\zeta_{6})q^{11}-2q^{13}+\cdots\)
1764.2.k.c	$1764$	$2$	1764.k	7.c	$3$	$2$	$1$	$14.086$	\(\Q(\sqrt{-3}) \)	None		$2$	$0$	\(0\)	\(0\)	\(-2\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q-2\zeta_{6}q^{5}+(2-2\zeta_{6})q^{11}+4q^{13}+\cdots\)
1764.2.k.d	$1764$	$2$	1764.k	7.c	$3$	$2$	$1$	$14.086$	\(\Q(\sqrt{-3}) \)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-6+6\zeta_{6})q^{11}-2q^{13}-4\zeta_{6}q^{19}+\cdots\)
1764.2.k.e	$1764$	$2$	1764.k	7.c	$3$	$2$	$1$	$14.086$	\(\Q(\sqrt{-3}) \)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-6+6\zeta_{6})q^{11}+2q^{13}+4\zeta_{6}q^{19}+\cdots\)
1764.2.k.f	$1764$	$2$	1764.k	7.c	$3$	$2$	$1$	$14.086$	\(\Q(\sqrt{-3}) \)	\(\Q(\sqrt{-3}) \)		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{U}(1)[D_{3}]$	\(q-5q^{13}-\zeta_{6}q^{19}+(5-5\zeta_{6})q^{25}+\cdots\)
1764.2.k.g	$1764$	$2$	1764.k	7.c	$3$	$2$	$1$	$14.086$	\(\Q(\sqrt{-3}) \)	\(\Q(\sqrt{-3}) \)		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{U}(1)[D_{3}]$	\(q-2q^{13}+8\zeta_{6}q^{19}+(5-5\zeta_{6})q^{25}+\cdots\)
1764.2.k.h	$1764$	$2$	1764.k	7.c	$3$	$2$	$1$	$14.086$	\(\Q(\sqrt{-3}) \)	\(\Q(\sqrt{-3}) \)		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{U}(1)[D_{3}]$	\(q+2q^{13}-8\zeta_{6}q^{19}+(5-5\zeta_{6})q^{25}+\cdots\)
1764.2.k.i	$1764$	$2$	1764.k	7.c	$3$	$2$	$1$	$14.086$	\(\Q(\sqrt{-3}) \)	None		$2$	$0$	\(0\)	\(0\)	\(2\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+2\zeta_{6}q^{5}+(2-2\zeta_{6})q^{11}-4q^{13}+\cdots\)
1764.2.k.j	$1764$	$2$	1764.k	7.c	$3$	$2$	$1$	$14.086$	\(\Q(\sqrt{-3}) \)	None		$2$	$0$	\(0\)	\(0\)	\(2\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+2\zeta_{6}q^{5}+(2-2\zeta_{6})q^{11}+3q^{13}+\cdots\)
1764.2.k.k	$1764$	$2$	1764.k	7.c	$3$	$2$	$1$	$14.086$	\(\Q(\sqrt{-3}) \)	None		$2$	$0$	\(0\)	\(0\)	\(4\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+4\zeta_{6}q^{5}+(2-2\zeta_{6})q^{11}-6q^{13}+\cdots\)
1764.2.k.l	$1764$	$2$	1764.k	7.c	$3$	$4$	$2$	$14.086$	\(\Q(\sqrt{2}, \sqrt{-3})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+\beta _{1}q^{5}-4\beta _{2}q^{11}-3\beta _{3}q^{13}+(-\beta _{1}+\cdots)q^{17}+\cdots\)
1764.2.k.m	$1764$	$2$	1764.k	7.c	$3$	$8$	$4$	$14.086$	8.0.\(\cdots\).4	None		$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{4}$	$\mathrm{SU}(2)[C_{3}]$	\(q-\beta _{7}q^{5}-\beta _{5}q^{11}-3\beta _{6}q^{13}+\beta _{3}q^{17}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(1764, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(1764, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(21, [\chi])\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(28, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(42, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(49, [\chi])\)\(^{\oplus 9}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(63, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(84, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(98, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(126, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(147, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(196, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(252, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(294, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(441, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(588, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(882, [\chi])\)\(^{\oplus 2}\)