Embedded newform 927.2.bg.a.77.23

• Label: 927.2.bg.a.77.23
• 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			77.23
Character	\(\chi\)	\(=\)	927.77
Dual form			927.2.bg.a.614.23

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.67225 + 0.707803i) q^{2} +(-0.370899 + 1.69187i) q^{3} +(0.903169 - 0.931424i) q^{4} +(-3.53084 + 2.49943i) q^{5} +(-0.577276 - 3.09176i) q^{6} +(2.58693 - 1.00218i) q^{7} +(0.460880 - 1.18967i) q^{8} +(-2.72487 - 1.25503i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3264 q - 48 q^{2} - 37 q^{3} - 118 q^{4} - 48 q^{5} - 37 q^{6} - 17 q^{7} - 23 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.67225	+	0.707803i	−1.18246	+	0.500492i	−0.890077	−	0.455809i	\(-0.849350\pi\)
							−0.292382	+	0.956302i	\(0.594448\pi\)
\(3\)	−0.370899	+	1.69187i	−0.214139	+	0.976803i
							\(5\)	−3.53084	+	2.49943i	−1.57904	+	1.11778i	−0.643009	+	0.765859i	\(0.722314\pi\)
−0.936030	+	0.351920i	\(0.885529\pi\)
\(7\)	2.58693	−	1.00218i	0.977769	−	0.378790i	0.181273	−	0.983433i	\(-0.441978\pi\)
							0.796497	+	0.604643i	\(0.206684\pi\)
\(11\)	1.88858	−	2.50088i	0.569428	−	0.754045i	−0.418625	−	0.908159i	\(-0.637488\pi\)
							0.988053	+	0.154114i	\(0.0492524\pi\)
\(13\)	−0.0356270	+	0.229513i	−0.00988114	+	0.0636554i	−0.992593	−	0.121484i	\(-0.961235\pi\)
							0.982712	+	0.185140i	\(0.0592738\pi\)
\(17\)	5.05390	+	1.78093i	1.22575	+	0.431938i	0.865496	−	0.500916i	\(-0.167003\pi\)
							0.360255	+	0.932854i	\(0.382690\pi\)
\(19\)	6.05189	−	0.373267i	1.38840	−	0.0856334i	0.649967	−	0.759963i	\(-0.274783\pi\)
							0.738431	+	0.674329i	\(0.235567\pi\)
\(23\)	0.585071	+	4.72491i	0.121996	+	0.985212i	0.922530	+	0.385925i	\(0.126118\pi\)
							−0.800535	+	0.599287i	\(0.795451\pi\)
\(29\)	1.20607	−	2.62147i	0.223961	−	0.486795i	−0.763272	−	0.646078i	\(-0.776408\pi\)
							0.987233	+	0.159282i	\(0.0509180\pi\)
\(31\)	10.3706	−	0.319513i	1.86261	−	0.0573863i	0.920121	−	0.391634i	\(-0.128090\pi\)
							0.942485	+	0.334248i	\(0.108482\pi\)
\(37\)	−4.93324	−	5.41150i	−0.811019	−	0.889645i	0.184632	−	0.982808i	\(-0.440891\pi\)
							−0.995651	+	0.0931624i	\(0.970302\pi\)
\(41\)	−4.47237	−	0.414426i	−0.698466	−	0.0647224i	−0.262826	−	0.964843i	\(-0.584655\pi\)
							−0.435640	+	0.900121i	\(0.643478\pi\)
\(43\)	−0.890749	−	2.80002i	−0.135838	−	0.426999i	0.860799	−	0.508946i	\(-0.169965\pi\)
							−0.996637	+	0.0819471i	\(0.973886\pi\)
\(47\)	−2.89687			−0.422552			−0.211276	−	0.977426i	\(-0.567762\pi\)
							−0.211276	+	0.977426i	\(0.567762\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	927.2.bg.a.77.23	✓	3264
9.2	odd	6		927.2.bm.a.695.23	yes	3264
103.99	odd	102		927.2.bm.a.923.23	yes	3264
927.614	even	102	inner	927.2.bg.a.614.23	yes	3264

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
927.2.bg.a.77.23	✓	3264	1.1	even	1	trivial
927.2.bg.a.614.23	yes	3264	927.614	even	102	inner
927.2.bm.a.695.23	yes	3264	9.2	odd	6
927.2.bm.a.923.23	yes	3264	103.99	odd	102