Embedded newform 847.2.u.a.23.19

• Label: 847.2.u.a.23.19
• 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			23.19
Character	\(\chi\)	\(=\)	847.23
Dual form			847.2.u.a.221.19

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.05144 - 1.47654i) q^{2} +(-0.136956 - 0.237214i) q^{3} +(-0.420511 + 1.21499i) q^{4} +(1.03151 - 0.983547i) q^{5} +(-0.206256 + 0.451637i) q^{6} +(1.54090 - 2.15073i) q^{7} +(-1.24233 + 0.364781i) q^{8} +(1.46249 - 2.53310i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1720 q - 11 q^{2} - 20 q^{3} + 73 q^{4} - 7 q^{5} + 8 q^{6} - 24 q^{7} - 32 q^{8} - 844 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.05144	−	1.47654i	−0.743480	−	1.04407i	−0.997050	−	0.0767549i	\(-0.975544\pi\)
							0.253570	−	0.967317i	\(-0.418395\pi\)
\(3\)	−0.136956	−	0.237214i	−0.0790714	−	0.136956i	0.823778	−	0.566912i	\(-0.191862\pi\)
							−0.902850	+	0.429957i	\(0.858529\pi\)
\(5\)	1.03151	−	0.983547i	0.461307	−	0.439856i	−0.423442	−	0.905923i	\(-0.639178\pi\)
							0.884749	+	0.466068i	\(0.154330\pi\)
\(7\)	1.54090	−	2.15073i	0.582405	−	0.812899i
							\(11\)	−2.73514	−	1.87590i	−0.824676	−	0.565606i
\(13\)	2.60992	+	3.01201i	0.723861	+	0.835380i								0.991766	−	0.128065i	\(-0.0408766\pi\)
							−0.267904	+	0.963445i	\(0.586331\pi\)
\(17\)	0.104273	−	2.18895i	0.0252898	−	0.530899i	−0.950305	−	0.311321i	\(-0.899229\pi\)
							0.975595	−	0.219579i	\(-0.0704683\pi\)
\(19\)	−0.260228	−	5.46286i	−0.0597004	−	1.25327i	−0.807711	−	0.589579i	\(-0.799294\pi\)
							0.748011	−	0.663687i	\(-0.231009\pi\)
\(23\)	0.269814	+	0.212184i	0.0562601	+	0.0442434i	0.645893	−	0.763428i	\(-0.276485\pi\)
							−0.589633	+	0.807671i	\(0.700728\pi\)
\(29\)	3.83581	−	2.46513i	0.712292	−	0.457762i	−0.133656	−	0.991028i	\(-0.542672\pi\)
							0.845948	+	0.533265i	\(0.179035\pi\)
\(31\)	0.915139	+	0.176379i	0.164364	+	0.0316785i	0.270769	−	0.962644i	\(-0.412722\pi\)
							−0.106405	+	0.994323i	\(0.533934\pi\)
\(37\)	0.433801	+	1.25339i	0.0713165	+	0.206055i	0.975026	−	0.222091i	\(-0.0712881\pi\)
							−0.903710	+	0.428146i	\(0.859167\pi\)
\(41\)	2.96325	−	6.48861i	0.462782	−	1.01335i	−0.524063	−	0.851680i	\(-0.675584\pi\)
							0.986845	−	0.161671i	\(-0.0516884\pi\)
\(43\)	−9.28073	+	2.72507i	−1.41530	+	0.415569i	−0.897909	−	0.440181i	\(-0.854914\pi\)
							−0.517390	+	0.855750i	\(0.673096\pi\)
\(47\)	−3.15061	−	0.300847i	−0.459564	−	0.0438831i	−0.137293	−	0.990530i	\(-0.543840\pi\)
							−0.322271	+	0.946647i	\(0.604446\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	847.2.u.a.23.19	✓	1720
7.4	even	3	inner	847.2.u.a.144.68	yes	1720
121.100	even	11	inner	847.2.u.a.100.68	yes	1720
847.221	even	33	inner	847.2.u.a.221.19	yes	1720

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
847.2.u.a.23.19	✓	1720	1.1	even	1	trivial
847.2.u.a.100.68	yes	1720	121.100	even	11	inner
847.2.u.a.144.68	yes	1720	7.4	even	3	inner
847.2.u.a.221.19	yes	1720	847.221	even	33	inner