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

• Label: 1764.3.c
• Level: $1764$
• Weight: $3$
• Character orbit: 1764.c
• Rep. character: $\chi_{1764}(197,\cdot)$
• Character field: $\Q$
• Dimension: $28$
• Newform subspaces: $8$
• 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.c (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 3 \)
Character field:			\(\Q\)
Newform subspaces:			\( 8 \)
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	720	28	692
Cusp forms	624	28	596
Eisenstein series	96	0	96

Trace form

\( 28 q + 8 q^{13} + 40 q^{19} - 228 q^{25} - 40 q^{31} + 4 q^{37} + 172 q^{43} + 104 q^{55} + 272 q^{61} + 60 q^{67} - 144 q^{73} - 596 q^{79} + 160 q^{85} - 96 q^{97}+O(q^{100}) \)

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	Twist minimal	Largest	Maximal	Minimal twist	Inner twists	Rank*	Traces				Coefficient ring index	Sato-Tate	$q$-expansion
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
1764.3.c.a	$1764$	$3$	1764.c	3.b	$2$	$2$	$2$	$48.066$	\(\Q(\sqrt{-2}) \)	None					252.3.bk.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$3$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta q^{5}-5\beta q^{11}-23q^{13}+4\beta q^{17}+\cdots\)
1764.3.c.b	$1764$	$3$	1764.c	3.b	$2$	$2$	$2$	$48.066$	\(\Q(\sqrt{-2}) \)	None		✓			1764.3.c.b	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+4\beta q^{5}+\beta q^{11}-2^{4}q^{13}-12\beta q^{17}+\cdots\)
1764.3.c.c	$1764$	$3$	1764.c	3.b	$2$	$2$	$2$	$48.066$	\(\Q(\sqrt{-2}) \)	None		✓			1764.3.c.b	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+4\beta q^{5}-\beta q^{11}+2^{4}q^{13}-12\beta q^{17}+\cdots\)
1764.3.c.d	$1764$	$3$	1764.c	3.b	$2$	$2$	$2$	$48.066$	\(\Q(\sqrt{-2}) \)	None					252.3.bk.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$3$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta q^{5}+5\beta q^{11}+23q^{13}+4\beta q^{17}+\cdots\)
1764.3.c.e	$1764$	$3$	1764.c	3.b	$2$	$4$	$4$	$48.066$	\(\Q(\sqrt{-2}, \sqrt{-23})\)	None					252.3.bk.b	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}\cdot 3$	$\mathrm{SU}(2)[C_{2}]$	\(q+(\beta _{1}-\beta _{3})q^{5}+(2\beta _{1}+\beta _{3})q^{11}+(-14+\cdots)q^{13}+\cdots\)
1764.3.c.f	$1764$	$3$	1764.c	3.b	$2$	$4$	$4$	$48.066$	\(\Q(\sqrt{-2}, \sqrt{7})\)	None					252.3.c.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}\cdot 3$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{1}q^{5}+(-2\beta _{1}-\beta _{2})q^{11}+(2-2\beta _{3})q^{13}+\cdots\)
1764.3.c.g	$1764$	$3$	1764.c	3.b	$2$	$4$	$4$	$48.066$	\(\Q(\sqrt{-2}, \sqrt{-23})\)	None					252.3.bk.b	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}\cdot 3$	$\mathrm{SU}(2)[C_{2}]$	\(q+(\beta _{1}-\beta _{3})q^{5}+(-2\beta _{1}-\beta _{3})q^{11}+\cdots\)
1764.3.c.h	$1764$	$3$	1764.c	3.b	$2$	$8$	$8$	$48.066$	\(\Q(i, \sqrt{2}, \sqrt{43})\)	None		✓	✓		1764.3.c.h	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{8}\cdot 3^{4}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{2}q^{5}+\beta _{5}q^{11}+3\beta _{3}q^{13}+(-\beta _{1}+\cdots)q^{17}+\cdots\)

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

\( S_{3}^{\mathrm{old}}(1764, [\chi]) \simeq \) \(S_{3}^{\mathrm{new}}(12, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(18, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(21, [\chi])\)\(^{\oplus 12}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(42, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(63, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(84, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(126, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(147, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(252, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(294, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(441, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(588, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(882, [\chi])\)\(^{\oplus 2}\)