Embedded newform 927.2.bg.a.77.69

• Label: 927.2.bg.a.77.69
• Level: $927$
• Weight: $2$
• Character: 927.77
• Analytic conductor: $7.402$
• Analytic rank: $0$
• Dimension: $3264$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 927 = 3^{2} \cdot 103 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	927.bg (of order \(102\), degree \(32\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.40213226737\)
Analytic rank:	\(0\)
Dimension:	\(3264\)
Relative dimension:	\(102\) over \(\Q(\zeta_{102})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{102}]$

Embedding invariants

Embedding label			77.69
Character	\(\chi\)	\(=\)	927.77
Dual form			927.2.bg.a.614.69

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.976116 - 0.413154i) q^{2} +(1.67824 - 0.428381i) q^{3} +(-0.610162 + 0.629251i) q^{4} +(-0.817756 + 0.578878i) q^{5} +(1.46117 - 1.11152i) q^{6} +(-1.34671 + 0.521719i) q^{7} +(-1.10141 + 2.84306i) q^{8} +(2.63298 - 1.43785i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3264 q - 48 q^{2} - 37 q^{3} - 118 q^{4} - 48 q^{5} - 37 q^{6} - 17 q^{7} - 23 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/927\mathbb{Z}\right)^\times\).

\(n\)	\(722\)	\(829\)
\(\chi(n)\)	\(e\left(\frac{5}{6}\right)\)	\(e\left(\frac{65}{102}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.976116	−	0.413154i	0.690218	−	0.292144i	−0.0162715	−	0.999868i	\(-0.505180\pi\)
							0.706490	+	0.707723i	\(0.250278\pi\)
\(3\)	1.67824	−	0.428381i	0.968932	−	0.247326i
							\(5\)	−0.817756	+	0.578878i	−0.365712	+	0.258882i	−0.745943	−	0.666009i	\(-0.768001\pi\)
0.380232	+	0.924891i	\(0.375844\pi\)
\(7\)	−1.34671	+	0.521719i	−0.509009	+	0.197191i	−0.602031	−	0.798473i	\(-0.705642\pi\)
							0.0930215	+	0.995664i	\(0.470347\pi\)
\(11\)	0.132718	−	0.175747i	0.0400159	−	0.0529896i	−0.777569	−	0.628797i	\(-0.783548\pi\)
							0.817585	+	0.575808i	\(0.195312\pi\)
\(13\)	−0.873194	+	5.62521i	−0.242180	+	1.56015i	0.485501	+	0.874236i	\(0.338637\pi\)
							−0.727681	+	0.685915i	\(0.759402\pi\)
\(17\)	2.01474	+	0.709966i	0.488646	+	0.172192i	0.565332	−	0.824863i	\(-0.308748\pi\)
							−0.0766868	+	0.997055i	\(0.524434\pi\)
\(19\)	8.36074	−	0.515673i	1.91809	−	0.118304i	0.941990	−	0.335641i	\(-0.108953\pi\)
							0.976096	+	0.217338i	\(0.0697373\pi\)
\(23\)	0.580163	+	4.68527i	0.120972	+	0.976947i	0.924368	+	0.381502i	\(0.124593\pi\)
							−0.803396	+	0.595445i	\(0.796976\pi\)
\(29\)	−0.339475	+	0.737872i	−0.0630389	+	0.137019i	−0.937398	−	0.348260i	\(-0.886772\pi\)
							0.874359	+	0.485280i	\(0.161282\pi\)
\(31\)	3.08155	−	0.0949415i	0.553462	−	0.0170520i	0.247140	−	0.968980i	\(-0.420509\pi\)
							0.306322	+	0.951928i	\(0.400901\pi\)
\(37\)	−3.34559	−	3.66994i	−0.550012	−	0.603335i	0.399628	−	0.916677i	\(-0.369139\pi\)
							−0.949640	+	0.313343i	\(0.898551\pi\)
\(41\)	−4.50812	−	0.417739i	−0.704050	−	0.0652398i	−0.265715	−	0.964052i	\(-0.585608\pi\)
							−0.438335	+	0.898812i	\(0.644432\pi\)
\(43\)	−1.56834	−	4.92998i	−0.239169	−	0.751814i	−0.995251	−	0.0973374i	\(-0.968967\pi\)
							0.756082	−	0.654477i	\(-0.227111\pi\)
\(47\)	2.47978			0.361712			0.180856	−	0.983510i	\(-0.442113\pi\)
							0.180856	+	0.983510i	\(0.442113\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	927.2.bg.a.77.69	✓	3264
9.2	odd	6		927.2.bm.a.695.69	yes	3264
103.99	odd	102		927.2.bm.a.923.69	yes	3264
927.614	even	102	inner	927.2.bg.a.614.69	yes	3264

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
927.2.bg.a.77.69	✓	3264	1.1	even	1	trivial
927.2.bg.a.614.69	yes	3264	927.614	even	102	inner
927.2.bm.a.695.69	yes	3264	9.2	odd	6
927.2.bm.a.923.69	yes	3264	103.99	odd	102