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

• Label: 1764.2.b
• Level: $1764$
• Weight: $2$
• Character orbit: 1764.b
• Rep. character: $\chi_{1764}(1567,\cdot)$
• Character field: $\Q$
• Dimension: $96$
• Newform subspaces: $14$
• Sturm bound: $672$
• Trace bound: $50$

Defining parameters

Level:	\( N \)	\(=\)	\( 1764 = 2^{2} \cdot 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1764.b (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 28 \)
Character field:			\(\Q\)
Newform subspaces:			\( 14 \)
Sturm bound:			\(672\)
Trace bound:			\(50\)
Distinguishing \(T_p\):			\(5\), \(11\), \(19\), \(29\), \(53\)

Dimensions

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

	Total	New	Old
Modular forms	368	104	264
Cusp forms	304	96	208
Eisenstein series	64	8	56

Trace form

\( 96 q - 4 q^{2} + 4 q^{4} - 4 q^{8} + O(q^{10}) \)

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.b.a	$1764$	$2$	1764.b	28.d	$2$	$4$	$4$	$14.086$	\(\Q(\zeta_{12})\)	None		$4$	$0$	\(-4\)	\(0\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-1+\zeta_{12})q^{2}-2\zeta_{12}q^{4}-\zeta_{12}^{2}q^{5}+\cdots\)
1764.2.b.b	$1764$	$2$	1764.b	28.d	$2$	$4$	$4$	$14.086$	\(\Q(\sqrt{-3}, \sqrt{-7})\)	None		$4$	$0$	\(-2\)	\(0\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{2}q^{2}+(-2-\beta _{2})q^{4}+\beta _{3}q^{5}+(2+\cdots)q^{8}+\cdots\)
1764.2.b.c	$1764$	$2$	1764.b	28.d	$2$	$4$	$4$	$14.086$	\(\Q(\sqrt{-2}, \sqrt{-3})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{3}q^{2}+(1-\beta _{2})q^{4}+(\beta _{1}-\beta _{3})q^{5}+\cdots\)
1764.2.b.d	$1764$	$2$	1764.b	28.d	$2$	$4$	$4$	$14.086$	\(\Q(\sqrt{-2}, \sqrt{-3})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{3}q^{2}+(1-\beta _{2})q^{4}+(\beta _{1}-\beta _{3})q^{5}+\cdots\)
1764.2.b.e	$1764$	$2$	1764.b	28.d	$2$	$4$	$4$	$14.086$	4.0.2048.2	\(\Q(\sqrt{-1}) \)		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{U}(1)[D_{2}]$	\(q+\beta _{2}q^{2}+2q^{4}+(\beta _{1}+2\beta _{3})q^{5}+2\beta _{2}q^{8}+\cdots\)
1764.2.b.f	$1764$	$2$	1764.b	28.d	$2$	$4$	$4$	$14.086$	4.0.2048.2	\(\Q(\sqrt{-1}) \)		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$7$	$\mathrm{U}(1)[D_{2}]$	\(q+\beta _{2}q^{2}+2q^{4}+\beta _{1}q^{5}+2\beta _{2}q^{8}+\cdots\)
1764.2.b.g	$1764$	$2$	1764.b	28.d	$2$	$4$	$4$	$14.086$	4.0.2048.2	\(\Q(\sqrt{-1}) \)		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{U}(1)[D_{2}]$	\(q-\beta _{2}q^{2}+2q^{4}+(\beta _{1}+2\beta _{3})q^{5}-2\beta _{2}q^{8}+\cdots\)
1764.2.b.h	$1764$	$2$	1764.b	28.d	$2$	$4$	$4$	$14.086$	\(\Q(\sqrt{-3}, \sqrt{-7})\)	None		$4$	$0$	\(2\)	\(0\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{2}q^{2}+(-2-\beta _{2})q^{4}+\beta _{3}q^{5}+(-2+\cdots)q^{8}+\cdots\)
1764.2.b.i	$1764$	$2$	1764.b	28.d	$2$	$8$	$8$	$14.086$	8.0.562828176.1	None		$2$	$0$	\(2\)	\(0\)	\(0\)	\(0\)	$2^{4}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{1}q^{2}+\beta _{2}q^{4}+(\beta _{2}-\beta _{5})q^{5}+(\beta _{3}+\cdots)q^{8}+\cdots\)
1764.2.b.j	$1764$	$2$	1764.b	28.d	$2$	$8$	$8$	$14.086$	8.0.562828176.1	None		$2$	$0$	\(2\)	\(0\)	\(0\)	\(0\)	$2^{4}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{4}q^{2}+\beta _{6}q^{4}+(\beta _{2}-\beta _{6})q^{5}+(-1+\cdots)q^{8}+\cdots\)
1764.2.b.k	$1764$	$2$	1764.b	28.d	$2$	$8$	$8$	$14.086$	8.0.\(\cdots\).10	None		$4$	$0$	\(4\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+(1+\beta _{3})q^{2}+(1+\beta _{3}+\beta _{5}-\beta _{6})q^{4}+\cdots\)
1764.2.b.l	$1764$	$2$	1764.b	28.d	$2$	$12$	$12$	$14.086$	12.0.\(\cdots\).1	None		$2$	$0$	\(-4\)	\(0\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{1}q^{2}+(\beta _{5}+\beta _{6})q^{4}+(1-\beta _{1}+\beta _{6}+\cdots)q^{5}+\cdots\)
1764.2.b.m	$1764$	$2$	1764.b	28.d	$2$	$12$	$12$	$14.086$	12.0.\(\cdots\).1	None		$2$	$0$	\(-4\)	\(0\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{4}q^{2}+(-\beta _{2}-\beta _{11})q^{4}+(-1+\beta _{3}+\cdots)q^{5}+\cdots\)
1764.2.b.n	$1764$	$2$	1764.b	28.d	$2$	$16$	$16$	$14.086$	\(\mathbb{Q}[x]/(x^{16} + \cdots)\)	None		$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{5}\cdot 7^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{7}q^{2}+(-1+\beta _{1}-\beta _{10})q^{4}+\beta _{12}q^{5}+\cdots\)

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

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