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

• Label: 1764.2.e
• Level: $1764$
• Weight: $2$
• Character orbit: 1764.e
• Rep. character: $\chi_{1764}(1079,\cdot)$
• Character field: $\Q$
• Dimension: $82$
• Newform subspaces: $10$
• Sturm bound: $672$
• Trace bound: $25$

Defining parameters

Level:	\( N \)	\(=\)	\( 1764 = 2^{2} \cdot 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1764.e (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 12 \)
Character field:			\(\Q\)
Newform subspaces:			\( 10 \)
Sturm bound:			\(672\)
Trace bound:			\(25\)
Distinguishing \(T_p\):			\(5\), \(13\)

Dimensions

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

	Total	New	Old
Modular forms	368	82	286
Cusp forms	304	82	222
Eisenstein series	64	0	64

Trace form

\( 82 q - 4 q^{4} + O(q^{10}) \)

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.e.a	$1764$	$2$	1764.e	12.b	$2$	$2$	$2$	$14.086$	\(\Q(\sqrt{-2}) \)	\(\Q(\sqrt{-1}) \)		$4$	$1$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{U}(1)[D_{2}]$	\(q+\beta q^{2}-2q^{4}+3\beta q^{5}-2\beta q^{8}-6q^{10}+\cdots\)
1764.2.e.b	$1764$	$2$	1764.e	12.b	$2$	$2$	$2$	$14.086$	\(\Q(\sqrt{-2}) \)	\(\Q(\sqrt{-1}) \)		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{U}(1)[D_{2}]$	\(q+\beta q^{2}-2q^{4}+\beta q^{5}-2\beta q^{8}-2q^{10}+\cdots\)
1764.2.e.c	$1764$	$2$	1764.e	12.b	$2$	$2$	$2$	$14.086$	\(\Q(\sqrt{-2}) \)	\(\Q(\sqrt{-1}) \)		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{U}(1)[D_{2}]$	\(q-\beta q^{2}-2q^{4}+3\beta q^{5}+2\beta q^{8}+6q^{10}+\cdots\)
1764.2.e.d	$1764$	$2$	1764.e	12.b	$2$	$4$	$4$	$14.086$	\(\Q(\zeta_{8})\)	\(\Q(\sqrt{-1}) \)		$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{U}(1)[D_{2}]$	\(q-\zeta_{8}^{2}q^{2}-2q^{4}+2\zeta_{8}q^{5}+2\zeta_{8}^{2}q^{8}+\cdots\)
1764.2.e.e	$1764$	$2$	1764.e	12.b	$2$	$4$	$4$	$14.086$	\(\Q(\zeta_{8})\)	\(\Q(\sqrt{-1}) \)		$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{U}(1)[D_{2}]$	\(q+\zeta_{8}^{2}q^{2}-2q^{4}-\zeta_{8}q^{5}-2\zeta_{8}^{2}q^{8}+\cdots\)
1764.2.e.f	$1764$	$2$	1764.e	12.b	$2$	$8$	$8$	$14.086$	8.0.157351936.1	\(\Q(\sqrt{-7}) \)		$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}\cdot 3^{4}$	$\mathrm{U}(1)[D_{2}]$	\(q+\beta _{6}q^{2}-\beta _{1}q^{4}-\beta _{5}q^{8}+(\beta _{5}+\beta _{6}+\cdots)q^{11}+\cdots\)
1764.2.e.g	$1764$	$2$	1764.e	12.b	$2$	$12$	$12$	$14.086$	12.0.\(\cdots\).1	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{9}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{6}q^{2}+(1+\beta _{10})q^{4}-\beta _{4}q^{5}+(\beta _{6}+\cdots)q^{8}+\cdots\)
1764.2.e.h	$1764$	$2$	1764.e	12.b	$2$	$16$	$16$	$14.086$	\(\mathbb{Q}[x]/(x^{16} + \cdots)\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{10}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{1}q^{2}+\beta _{2}q^{4}-\beta _{5}q^{5}+(\beta _{3}-\beta _{5}+\cdots)q^{8}+\cdots\)
1764.2.e.i	$1764$	$2$	1764.e	12.b	$2$	$16$	$16$	$14.086$	\(\mathbb{Q}[x]/(x^{16} + \cdots)\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{10}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{1}q^{2}+\beta _{2}q^{4}-\beta _{5}q^{5}+(-\beta _{3}+\beta _{5}+\cdots)q^{8}+\cdots\)
1764.2.e.j	$1764$	$2$	1764.e	12.b	$2$	$16$	$16$	$14.086$	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)	None		$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{14}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{3}q^{2}+(1-\beta _{8})q^{4}+\beta _{2}q^{5}+(-\beta _{3}+\cdots)q^{8}+\cdots\)

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

\( S_{2}^{\mathrm{old}}(1764, [\chi]) \cong \)