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

• Label: 1764.2.j
• Level: $1764$
• Weight: $2$
• Character orbit: 1764.j
• Rep. character: $\chi_{1764}(589,\cdot)$
• Character field: $\Q(\zeta_{3})$
• Dimension: $82$
• Newform subspaces: $9$
• Sturm bound: $672$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 1764 = 2^{2} \cdot 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1764.j (of order \(3\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 9 \)
Character field:			\(\Q(\zeta_{3})\)
Newform subspaces:			\( 9 \)
Sturm bound:			\(672\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(5\)

Dimensions

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

	Total	New	Old
Modular forms	720	82	638
Cusp forms	624	82	542
Eisenstein series	96	0	96

Trace form

82 * q + q^5 + 6 * q^9 + q^11 - q^13 - 17 * q^15 - 16 * q^17 - 4 * q^19 + q^23 - 44 * q^25 - 15 * q^29 - q^31 + q^33 + 8 * q^37 + 17 * q^39 - 3 * q^41 + 5 * q^43 + q^45 - 3 * q^47 + 20 * q^51 + 28 * q^53 - 6 * q^55 + 42 * q^57 + 25 * q^59 - 13 * q^61 + 13 * q^65 - 7 * q^67 - 23 * q^69 + 44 * q^71 + 8 * q^73 - 34 * q^75 + 11 * q^79 + 14 * q^81 - 7 * q^83 - 6 * q^85 + 77 * q^87 - 48 * q^89 - 29 * q^93 - 13 * q^97 - 17 * q^99

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	Twist minimal	Largest	Maximal	Minimal twist	Inner twists	Rank*	Traces				Coefficient ring index	Sato-Tate	$q$-expansion
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
1764.2.j.a	$1764$	$2$	1764.j	9.c	$3$	$2$	$1$	$14.086$	\(\Q(\sqrt{-3}) \)	None					252.2.i.a	$2$	$1$	\(0\)	\(-3\)	\(2\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-1-\zeta_{6})q^{3}+2\zeta_{6}q^{5}+3\zeta_{6}q^{9}+\cdots\)
1764.2.j.b	$1764$	$2$	1764.j	9.c	$3$	$2$	$1$	$14.086$	\(\Q(\sqrt{-3}) \)	None					36.2.e.a	$2$	$0$	\(0\)	\(0\)	\(3\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(1-2\zeta_{6})q^{3}+3\zeta_{6}q^{5}-3q^{9}+(-3+\cdots)q^{11}+\cdots\)
1764.2.j.c	$1764$	$2$	1764.j	9.c	$3$	$2$	$1$	$14.086$	\(\Q(\sqrt{-3}) \)	None					252.2.i.a	$2$	$0$	\(0\)	\(3\)	\(-2\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(1+\zeta_{6})q^{3}-2\zeta_{6}q^{5}+3\zeta_{6}q^{9}+(-4+\cdots)q^{11}+\cdots\)
1764.2.j.d	$1764$	$2$	1764.j	9.c	$3$	$6$	$3$	$14.086$	6.0.309123.1	None					252.2.j.b	$2$	$0$	\(0\)	\(-2\)	\(-3\)	\(0\)	$3$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-1+\beta _{2}+\beta _{3}+\beta _{4})q^{3}+(\beta _{1}-\beta _{3}+\cdots)q^{5}+\cdots\)
1764.2.j.e	$1764$	$2$	1764.j	9.c	$3$	$6$	$3$	$14.086$	6.0.309123.1	None					252.2.j.a	$2$	$0$	\(0\)	\(2\)	\(1\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(1-\beta _{1}-\beta _{2})q^{3}+(-\beta _{1}-\beta _{2}-\beta _{3}+\cdots)q^{5}+\cdots\)
1764.2.j.f	$1764$	$2$	1764.j	9.c	$3$	$12$	$6$	$14.086$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None		✓			1764.2.j.f	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+\beta _{2}q^{3}+(-\beta _{8}+\beta _{11})q^{5}+(\beta _{3}-\beta _{4}+\cdots)q^{9}+\cdots\)
1764.2.j.g	$1764$	$2$	1764.j	9.c	$3$	$14$	$7$	$14.086$	\(\mathbb{Q}[x]/(x^{14} - \cdots)\)	None					252.2.i.b	$2$	$0$	\(0\)	\(-3\)	\(-2\)	\(0\)	$3^{5}$	$\mathrm{SU}(2)[C_{3}]$	\(q-\beta _{3}q^{3}+\beta _{5}q^{5}+(-\beta _{1}-\beta _{2}+\beta _{11}+\cdots)q^{9}+\cdots\)
1764.2.j.h	$1764$	$2$	1764.j	9.c	$3$	$14$	$7$	$14.086$	\(\mathbb{Q}[x]/(x^{14} - \cdots)\)	None					252.2.i.b	$2$	$0$	\(0\)	\(3\)	\(2\)	\(0\)	$3^{5}$	$\mathrm{SU}(2)[C_{3}]$	\(q+\beta _{3}q^{3}-\beta _{5}q^{5}+(-\beta _{1}-\beta _{2}+\beta _{11}+\cdots)q^{9}+\cdots\)
1764.2.j.i	$1764$	$2$	1764.j	9.c	$3$	$24$	$12$	$14.086$		None		✓	✓		1764.2.j.i	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$\mathrm{SU}(2)[C_{3}]$

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

\( S_{2}^{\mathrm{old}}(1764, [\chi]) \simeq \) \(S_{2}^{\mathrm{new}}(18, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(36, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(63, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(126, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(252, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(441, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(882, [\chi])\)\(^{\oplus 2}\)