Space of modular forms of level 1764, weight 1, and Character 1764.g

• Label: 1764.1.g
• Level: $1764$
• Weight: $1$
• Character orbit: 1764.g
• Rep. character: $\chi_{1764}(883,\cdot)$
• Character field: $\Q$
• Dimension: $7$
• Newform subspaces: $4$
• Sturm bound: $336$
• Trace bound: $2$

Defining parameters

Level:	\( N \)	\(=\)	\( 1764 = 2^{2} \cdot 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	1764.g (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 4 \)
Character field:			\(\Q\)
Newform subspaces:			\( 4 \)
Sturm bound:			\(336\)
Trace bound:			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{1}(1764, [\chi])\).

	Total	New	Old
Modular forms	42	12	30
Cusp forms	10	7	3
Eisenstein series	32	5	27

The following table gives the dimensions of subspaces with specified projective image type.

	\(D_n\)	\(A_4\)	\(S_4\)	\(A_5\)
Dimension	7	0	0	0

Trace form

\( 7 q + q^{2} + 3 q^{4} + q^{8} + O(q^{10}) \)

Decomposition of \(S_{1}^{\mathrm{new}}(1764, [\chi])\) into newform subspaces

Label	Level	Weight	Char	Prim	Char order	Dim	Rel. Dim	$A$	Field	Image	CM	RM	Self-dual	Inner twists	Rank*	Traces				Coefficient ring index	Sato-Tate	$q$-expansion
																$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
1764.1.g.a	$1764$	$1$	1764.g	4.b	$2$	$1$	$1$	$0.880$	\(\Q\)	$D_{2}$	\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{-7}) \)	\(\Q(\sqrt{7}) \)	✓	$4$	$0$	\(1\)	\(0\)	\(0\)	\(0\)	$1$		\(q+q^{2}+q^{4}+q^{8}+q^{16}-q^{25}+2q^{29}+\cdots\)
1764.1.g.b	$1764$	$1$	1764.g	4.b	$2$	$2$	$2$	$0.880$	\(\Q(\sqrt{2}) \)	$D_{4}$	\(\Q(\sqrt{-1}) \)	None	✓	$4$	$0$	\(-2\)	\(0\)	\(0\)	\(0\)	$1$		\(q-q^{2}+q^{4}-\beta q^{5}-q^{8}+\beta q^{10}-\beta q^{13}+\cdots\)
1764.1.g.c	$1764$	$1$	1764.g	4.b	$2$	$2$	$2$	$0.880$	\(\Q(\sqrt{-1}) \)	$D_{2}$	\(\Q(\sqrt{-7}) \), \(\Q(\sqrt{-21}) \)	\(\Q(\sqrt{3}) \)		$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$		\(q-iq^{2}-q^{4}+iq^{8}+iq^{11}+q^{16}+\cdots\)
1764.1.g.d	$1764$	$1$	1764.g	4.b	$2$	$2$	$2$	$0.880$	\(\Q(\sqrt{2}) \)	$D_{4}$	\(\Q(\sqrt{-1}) \)	None	✓	$4$	$0$	\(2\)	\(0\)	\(0\)	\(0\)	$1$		\(q+q^{2}+q^{4}-\beta q^{5}+q^{8}-\beta q^{10}+\beta q^{13}+\cdots\)

Decomposition of \(S_{1}^{\mathrm{old}}(1764, [\chi])\) into lower level spaces

\( S_{1}^{\mathrm{old}}(1764, [\chi]) \cong \) \(S_{1}^{\mathrm{new}}(196, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{1}^{\mathrm{new}}(588, [\chi])\)\(^{\oplus 2}\)