Embedded newform 927.2.bg.a.77.56

• Label: 927.2.bg.a.77.56
• 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			77.56
Character	\(\chi\)	\(=\)	927.77
Dual form			927.2.bg.a.614.56

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.224368 - 0.0949667i) q^{2} +(0.463899 + 1.66877i) q^{3} +(-1.35095 + 1.39321i) q^{4} +(-2.45168 + 1.73551i) q^{5} +(0.262562 + 0.330363i) q^{6} +(-1.87506 + 0.726402i) q^{7} +(-0.346825 + 0.895257i) q^{8} +(-2.56960 + 1.54828i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3264 q - 48 q^{2} - 37 q^{3} - 118 q^{4} - 48 q^{5} - 37 q^{6} - 17 q^{7} - 23 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.224368	−	0.0949667i	0.158652	−	0.0671516i	−0.309011	−	0.951058i	\(-0.599998\pi\)
							0.467663	+	0.883907i	\(0.345096\pi\)
\(3\)	0.463899	+	1.66877i	0.267832	+	0.963466i
							\(5\)	−2.45168	+	1.73551i	−1.09643	+	0.776144i	−0.976258	−	0.216610i	\(-0.930500\pi\)
−0.120168	+	0.992754i	\(0.538343\pi\)
\(7\)	−1.87506	+	0.726402i	−0.708706	+	0.274554i	−0.688494	−	0.725242i	\(-0.741728\pi\)
							−0.0202114	+	0.999796i	\(0.506434\pi\)
\(11\)	−0.633228	+	0.838529i	−0.190925	+	0.252826i	−0.883404	−	0.468612i	\(-0.844754\pi\)
							0.692478	+	0.721439i	\(0.256519\pi\)
\(13\)	−0.223727	+	1.44127i	−0.0620507	+	0.399738i	0.936716	+	0.350090i	\(0.113849\pi\)
							−0.998767	+	0.0496475i	\(0.984190\pi\)
\(17\)	7.33080	+	2.58327i	1.77798	+	0.626536i	0.999998	−	0.00175056i	\(-0.000557222\pi\)
							0.777982	+	0.628287i	\(0.216243\pi\)
\(19\)	2.06890	−	0.127605i	0.474638	−	0.0292747i	0.177524	−	0.984117i	\(-0.443191\pi\)
							0.297114	+	0.954842i	\(0.403976\pi\)
\(23\)	−1.08629	−	8.77268i	−0.226508	−	1.82923i	−0.498453	−	0.866917i	\(-0.666098\pi\)
							0.271945	−	0.962313i	\(-0.412333\pi\)
\(29\)	−2.37330	+	5.15852i	−0.440710	+	0.957913i	0.551597	+	0.834111i	\(0.314019\pi\)
							−0.992307	+	0.123802i	\(0.960491\pi\)
\(31\)	2.18092	−	0.0671934i	0.391705	−	0.0120683i	0.165654	−	0.986184i	\(-0.447026\pi\)
							0.226051	+	0.974116i	\(0.427419\pi\)
\(37\)	6.41628	+	7.03832i	1.05483	+	1.15709i	0.987217	+	0.159381i	\(0.0509498\pi\)
							0.0676121	+	0.997712i	\(0.478462\pi\)
\(41\)	−11.9962	−	1.11161i	−1.87349	−	0.173604i	−0.905458	−	0.424435i	\(-0.860473\pi\)
							−0.968031	+	0.250831i	\(0.919296\pi\)
\(43\)	0.824176	+	2.59075i	0.125686	+	0.395085i	0.995085	−	0.0990250i	\(-0.0315724\pi\)
							−0.869399	+	0.494110i	\(0.835494\pi\)
\(47\)	0.502415			0.0732848			0.0366424	−	0.999328i	\(-0.488334\pi\)
							0.0366424	+	0.999328i	\(0.488334\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	927.2.bg.a.77.56	✓	3264
9.2	odd	6		927.2.bm.a.695.56	yes	3264
103.99	odd	102		927.2.bm.a.923.56	yes	3264
927.614	even	102	inner	927.2.bg.a.614.56	yes	3264

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
927.2.bg.a.77.56	✓	3264	1.1	even	1	trivial
927.2.bg.a.614.56	yes	3264	927.614	even	102	inner
927.2.bm.a.695.56	yes	3264	9.2	odd	6
927.2.bm.a.923.56	yes	3264	103.99	odd	102