Embedded newform 847.2.u.a.221.18

• Label: 847.2.u.a.221.18
• Level: $847$
• Weight: $2$
• Character: 847.221
• 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			221.18
Character	\(\chi\)	\(=\)	847.221
Dual form			847.2.u.a.23.18

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.06888 + 1.50103i) q^{2} +(-1.66808 + 2.88919i) q^{3} +(-0.456452 - 1.31883i) q^{4} +(1.17953 + 1.12468i) q^{5} +(-2.55379 - 5.59203i) q^{6} +(-0.367949 - 2.62004i) q^{7} +(-1.06864 - 0.313782i) q^{8} +(-4.06496 - 7.04072i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1720 q - 11 q^{2} - 20 q^{3} + 73 q^{4} - 7 q^{5} + 8 q^{6} - 24 q^{7} - 32 q^{8} - 844 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(122\)	\(365\)
\(\chi(n)\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{4}{11}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.06888	+	1.50103i	−0.755811	+	1.06139i	0.240074	+	0.970755i	\(0.422828\pi\)
							−0.995884	+	0.0906329i	\(0.971111\pi\)
\(3\)	−1.66808	+	2.88919i	−0.963065	+	1.66808i	−0.248338	+	0.968674i	\(0.579884\pi\)
							−0.714727	+	0.699404i	\(0.753449\pi\)
\(5\)	1.17953	+	1.12468i	0.527504	+	0.502974i	0.906374	−	0.422476i	\(-0.138839\pi\)
							−0.378871	+	0.925450i	\(0.623687\pi\)
\(7\)	−0.367949	−	2.62004i	−0.139072	−	0.990282i
							\(11\)	−1.76068	+	2.81070i	−0.530864	+	0.847457i
\(13\)	−2.34623	+	2.70769i	−0.650727	+	0.750979i								−0.981233	−	0.192825i	\(-0.938235\pi\)
							0.330506	+	0.943804i	\(0.392781\pi\)
\(17\)	−0.306316	−	6.43037i	−0.0742926	−	1.55959i	−0.661368	−	0.750062i	\(-0.730024\pi\)
							0.587075	−	0.809532i	\(-0.300279\pi\)
\(19\)	−0.158775	+	3.33309i	−0.0364254	+	0.764664i	0.904586	+	0.426290i	\(0.140180\pi\)
							−0.941012	+	0.338374i	\(0.890123\pi\)
\(23\)	2.00093	−	1.57355i	0.417223	−	0.328108i	−0.387381	−	0.921920i	\(-0.626620\pi\)
							0.804604	+	0.593812i	\(0.202378\pi\)
\(29\)	−0.0575142	−	0.0369622i	−0.0106801	−	0.00686370i	0.535290	−	0.844668i	\(-0.320202\pi\)
							−0.545970	+	0.837805i	\(0.683839\pi\)
\(31\)	−2.24096	+	0.431910i	−0.402489	+	0.0775733i	−0.386478	−	0.922298i	\(-0.626309\pi\)
							−0.0160104	+	0.999872i	\(0.505096\pi\)
\(37\)	0.437928	−	1.26531i	0.0719950	−	0.208016i	−0.903262	−	0.429089i	\(-0.858835\pi\)
							0.975257	+	0.221073i	\(0.0709559\pi\)
\(41\)	1.63665	+	3.58376i	0.255601	+	0.559689i	0.993316	−	0.115424i	\(-0.0368225\pi\)
							−0.737715	+	0.675112i	\(0.764095\pi\)
\(43\)	−9.77387	−	2.86987i	−1.49050	−	0.437651i	−0.567800	−	0.823167i	\(-0.692205\pi\)
							−0.922701	+	0.385516i	\(0.874023\pi\)
\(47\)	13.4083	−	1.28034i	1.95580	−	0.186757i	0.960122	−	0.279583i	\(-0.0901961\pi\)
							0.995682	+	0.0928261i	\(0.0295901\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	847.2.u.a.221.18	yes	1720
7.2	even	3	inner	847.2.u.a.100.69	yes	1720
121.23	even	11	inner	847.2.u.a.144.69	yes	1720
847.23	even	33	inner	847.2.u.a.23.18	✓	1720

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
847.2.u.a.23.18	✓	1720	847.23	even	33	inner
847.2.u.a.100.69	yes	1720	7.2	even	3	inner
847.2.u.a.144.69	yes	1720	121.23	even	11	inner
847.2.u.a.221.18	yes	1720	1.1	even	1	trivial