Embedded newform 847.2.u.a.23.2

• Label: 847.2.u.a.23.2
• 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			23.2
Character	\(\chi\)	\(=\)	847.23
Dual form			847.2.u.a.221.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.54654 - 2.17181i) q^{2} +(0.114913 + 0.199036i) q^{3} +(-1.67084 + 4.82758i) q^{4} +(0.314136 - 0.299528i) q^{5} +(0.254550 - 0.557387i) q^{6} +(2.59071 + 0.536841i) q^{7} +(7.95224 - 2.33499i) q^{8} +(1.47359 - 2.55233i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1720 q - 11 q^{2} - 20 q^{3} + 73 q^{4} - 7 q^{5} + 8 q^{6} - 24 q^{7} - 32 q^{8} - 844 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.54654	−	2.17181i	−1.09357	−	1.53570i	−0.818701	−	0.574220i	\(-0.805305\pi\)
							−0.274868	−	0.961482i	\(-0.588634\pi\)
\(3\)	0.114913	+	0.199036i	0.0663453	+	0.114913i	0.897290	−	0.441442i	\(-0.145533\pi\)
							−0.830945	+	0.556355i	\(0.812199\pi\)
\(5\)	0.314136	−	0.299528i	0.140486	−	0.133953i	−0.616578	−	0.787294i	\(-0.711482\pi\)
							0.757064	+	0.653341i	\(0.226633\pi\)
\(7\)	2.59071	+	0.536841i	0.979198	+	0.202907i
							\(11\)	−1.05119	+	3.14563i	−0.316947	+	0.948443i
\(13\)	−1.09433	−	1.26293i	−0.303513	−	0.350273i								0.583420	−	0.812170i	\(-0.301714\pi\)
							−0.886933	+	0.461898i	\(0.847169\pi\)
\(17\)	−0.0754516	+	1.58392i	−0.0182997	+	0.384158i	0.971075	+	0.238773i	\(0.0767452\pi\)
							−0.989375	+	0.145385i	\(0.953558\pi\)
\(19\)	0.317891	+	6.67336i	0.0729293	+	1.53097i	0.678155	+	0.734919i	\(0.262780\pi\)
							−0.605225	+	0.796054i	\(0.706917\pi\)
\(23\)	3.02706	+	2.38051i	0.631186	+	0.496370i	0.881742	−	0.471732i	\(-0.156371\pi\)
							−0.250556	+	0.968102i	\(0.580613\pi\)
\(29\)	−1.89232	+	1.21612i	−0.351395	+	0.225828i	−0.704416	−	0.709788i	\(-0.748791\pi\)
							0.353021	+	0.935615i	\(0.385154\pi\)
\(31\)	−1.27898	−	0.246504i	−0.229712	−	0.0442734i	0.0730960	−	0.997325i	\(-0.476712\pi\)
							−0.302808	+	0.953051i	\(0.597924\pi\)
\(37\)	1.37802	+	3.98152i	0.226544	+	0.654557i	0.999791	+	0.0204516i	\(0.00651039\pi\)
							−0.773246	+	0.634106i	\(0.781368\pi\)
\(41\)	4.67514	−	10.2371i	0.730134	−	1.59877i	−0.0690004	−	0.997617i	\(-0.521981\pi\)
							0.799134	−	0.601153i	\(-0.205292\pi\)
\(43\)	−10.5594	+	3.10053i	−1.61030	+	0.472826i	−0.958388	−	0.285470i	\(-0.907850\pi\)
							−0.651910	+	0.758296i	\(0.726032\pi\)
\(47\)	10.7980	+	1.03108i	1.57505	+	0.150399i	0.845668	−	0.533709i	\(-0.179202\pi\)
							0.729381	+	0.684108i	\(0.239808\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	847.2.u.a.23.2	✓	1720
7.4	even	3	inner	847.2.u.a.144.85	yes	1720
121.100	even	11	inner	847.2.u.a.100.85	yes	1720
847.221	even	33	inner	847.2.u.a.221.2	yes	1720

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
847.2.u.a.23.2	✓	1720	1.1	even	1	trivial
847.2.u.a.100.85	yes	1720	121.100	even	11	inner
847.2.u.a.144.85	yes	1720	7.4	even	3	inner
847.2.u.a.221.2	yes	1720	847.221	even	33	inner