Embedded newform 847.2.u.a.23.4

• Label: 847.2.u.a.23.4
• Level: $847$
• Weight: $2$
• Character: 847.23
• Analytic conductor: $6.763$
• Analytic rank: $0$
• Dimension: $1720$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 847 = 7 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	847.u (of order \(33\), degree \(20\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(6.76332905120\)
Analytic rank:	\(0\)
Dimension:	\(1720\)
Relative dimension:	\(86\) over \(\Q(\zeta_{33})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{33}]$

Embedding invariants

Embedding label			23.4
Character	\(\chi\)	\(=\)	847.23
Dual form			847.2.u.a.221.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.54361 - 2.16770i) q^{2} +(1.30997 + 2.26894i) q^{3} +(-1.66206 + 4.80220i) q^{4} +(0.519449 - 0.495293i) q^{5} +(2.89630 - 6.34201i) q^{6} +(2.54979 - 0.706084i) q^{7} +(7.86862 - 2.31044i) q^{8} +(-1.93206 + 3.34643i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1720 q - 11 q^{2} - 20 q^{3} + 73 q^{4} - 7 q^{5} + 8 q^{6} - 24 q^{7} - 32 q^{8} - 844 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/847\mathbb{Z}\right)^\times\).

\(n\)	\(122\)	\(365\)
\(\chi(n)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{7}{11}\right)\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

(See \(a_n\) instead)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.54361	−	2.16770i	−1.09150	−	1.53280i	−0.821733	−	0.569873i	\(-0.806993\pi\)
							−0.269768	−	0.962925i	\(-0.586947\pi\)
\(3\)	1.30997	+	2.26894i	0.756314	+	1.30997i	0.944719	+	0.327882i	\(0.106335\pi\)
							−0.188405	+	0.982091i	\(0.560332\pi\)
\(5\)	0.519449	−	0.495293i	0.232305	−	0.221502i	−0.564982	−	0.825104i	\(-0.691117\pi\)
							0.797286	+	0.603602i	\(0.206268\pi\)
\(7\)	2.54979	−	0.706084i	0.963731	−	0.266875i
							\(11\)	1.99045	−	2.65294i	0.600145	−	0.799891i
\(13\)	2.30920	+	2.66496i	0.640457	+	0.739127i								0.979456	−	0.201660i	\(-0.0646337\pi\)
							−0.338999	+	0.940787i	\(0.610088\pi\)
\(17\)	0.367664	−	7.71821i	0.0891715	−	1.87194i	−0.304264	−	0.952588i	\(-0.598410\pi\)
							0.393435	−	0.919352i	\(-0.371287\pi\)
\(19\)	−0.187773	−	3.94184i	−0.0430780	−	0.904320i	−0.912206	−	0.409732i	\(-0.865622\pi\)
							0.869128	−	0.494587i	\(-0.164681\pi\)
\(23\)	−1.84403	−	1.45016i	−0.384506	−	0.302379i	0.407132	−	0.913369i	\(-0.366529\pi\)
							−0.791638	+	0.610990i	\(0.790771\pi\)
\(29\)	−0.778233	+	0.500140i	−0.144514	+	0.0928737i	−0.610900	−	0.791708i	\(-0.709192\pi\)
							0.466386	+	0.884581i	\(0.345556\pi\)
\(31\)	−0.354442	−	0.0683131i	−0.0636597	−	0.0122694i	0.157322	−	0.987547i	\(-0.449714\pi\)
							−0.220982	+	0.975278i	\(0.570926\pi\)
\(37\)	1.31082	+	3.78737i	0.215498	+	0.622640i	0.999997	+	0.00251021i	\(0.000799024\pi\)
							−0.784499	+	0.620130i	\(0.787080\pi\)
\(41\)	−4.04342	+	8.85385i	−0.631475	+	1.38274i	0.275396	+	0.961331i	\(0.411191\pi\)
							−0.906871	+	0.421408i	\(0.861536\pi\)
\(43\)	8.61722	−	2.53024i	1.31411	−	0.385859i	0.451748	−	0.892145i	\(-0.350800\pi\)
							0.862365	+	0.506287i	\(0.168982\pi\)
\(47\)	−3.22508	−	0.307958i	−0.470426	−	0.0449202i	−0.142849	−	0.989745i	\(-0.545626\pi\)
							−0.327578	+	0.944824i	\(0.606232\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	847.2.u.a.23.4	✓	1720
7.4	even	3	inner	847.2.u.a.144.83	yes	1720
121.100	even	11	inner	847.2.u.a.100.83	yes	1720
847.221	even	33	inner	847.2.u.a.221.4	yes	1720

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
847.2.u.a.23.4	✓	1720	1.1	even	1	trivial
847.2.u.a.100.83	yes	1720	121.100	even	11	inner
847.2.u.a.144.83	yes	1720	7.4	even	3	inner
847.2.u.a.221.4	yes	1720	847.221	even	33	inner