Embedded newform 847.2.u.a.23.16

• Label: 847.2.u.a.23.16
• 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			23.16
Character	\(\chi\)	\(=\)	847.23
Dual form			847.2.u.a.221.16

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.16442 - 1.63519i) q^{2} +(1.21406 + 2.10281i) q^{3} +(-0.663855 + 1.91808i) q^{4} +(2.54551 - 2.42714i) q^{5} +(2.02483 - 4.43376i) q^{6} +(-2.62336 + 0.343509i) q^{7} +(0.0572360 - 0.0168060i) q^{8} +(-1.44787 + 2.50779i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1720 q - 11 q^{2} - 20 q^{3} + 73 q^{4} - 7 q^{5} + 8 q^{6} - 24 q^{7} - 32 q^{8} - 844 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.16442	−	1.63519i	−0.823366	−	1.15626i	−0.985532	−	0.169492i	\(-0.945787\pi\)
							0.162166	−	0.986764i	\(-0.448152\pi\)
\(3\)	1.21406	+	2.10281i	0.700937	+	1.21406i	0.968138	+	0.250417i	\(0.0805678\pi\)
							−0.267201	+	0.963641i	\(0.586099\pi\)
\(5\)	2.54551	−	2.42714i	1.13839	−	1.08545i	0.142978	−	0.989726i	\(-0.454332\pi\)
							0.995408	−	0.0957229i	\(-0.0305163\pi\)
\(7\)	−2.62336	+	0.343509i	−0.991536	+	0.129834i
							\(11\)	3.10483	+	1.16619i	0.936143	+	0.351620i
\(13\)	3.51957	+	4.06180i	0.976154	+	1.12654i								0.991945	+	0.126667i	\(0.0404278\pi\)
							−0.0157913	+	0.999875i	\(0.505027\pi\)
\(17\)	0.145923	−	3.06330i	0.0353915	−	0.742959i	−0.909525	−	0.415648i	\(-0.863555\pi\)
							0.944917	−	0.327310i	\(-0.106142\pi\)
\(19\)	0.348800	+	7.32221i	0.0800202	+	1.67983i	0.580638	+	0.814162i	\(0.302803\pi\)
							−0.500617	+	0.865669i	\(0.666894\pi\)
\(23\)	−3.38412	−	2.66130i	−0.705638	−	0.554920i	0.199572	−	0.979883i	\(-0.436045\pi\)
							−0.905211	+	0.424963i	\(0.860287\pi\)
\(29\)	0.861747	−	0.553811i	0.160022	−	0.102840i	−0.458179	−	0.888860i	\(-0.651498\pi\)
							0.618201	+	0.786020i	\(0.287862\pi\)
\(31\)	4.73115	+	0.911855i	0.849740	+	0.163774i	0.595488	−	0.803364i	\(-0.296959\pi\)
							0.254252	+	0.967138i	\(0.418171\pi\)
\(37\)	0.597163	+	1.72539i	0.0981730	+	0.283652i	0.983384	−	0.181537i	\(-0.0581073\pi\)
							−0.885211	+	0.465190i	\(0.845986\pi\)
\(41\)	3.86892	−	8.47176i	0.604224	−	1.32307i	−0.322231	−	0.946661i	\(-0.604433\pi\)
							0.926455	−	0.376405i	\(-0.122840\pi\)
\(43\)	−9.17129	+	2.69293i	−1.39861	+	0.410669i	−0.892206	−	0.451629i	\(-0.850843\pi\)
							−0.506403	+	0.862297i	\(0.669025\pi\)
\(47\)	4.23694	+	0.404578i	0.618021	+	0.0590138i	0.399371	−	0.916790i	\(-0.369229\pi\)
							0.218650	+	0.975803i	\(0.429835\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	847.2.u.a.23.16	✓	1720
7.4	even	3	inner	847.2.u.a.144.71	yes	1720
121.100	even	11	inner	847.2.u.a.100.71	yes	1720
847.221	even	33	inner	847.2.u.a.221.16	yes	1720

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
847.2.u.a.23.16	✓	1720	1.1	even	1	trivial
847.2.u.a.100.71	yes	1720	121.100	even	11	inner
847.2.u.a.144.71	yes	1720	7.4	even	3	inner
847.2.u.a.221.16	yes	1720	847.221	even	33	inner