Embedded newform 927.2.bg.a.77.25

• Label: 927.2.bg.a.77.25
• 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			77.25
Character	\(\chi\)	\(=\)	927.77
Dual form			927.2.bg.a.614.25

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.44647 + 0.612237i) q^{2} +(0.991545 - 1.42015i) q^{3} +(0.325163 - 0.335336i) q^{4} +(-3.30392 + 2.33880i) q^{5} +(-0.564766 + 2.66127i) q^{6} +(-4.08749 + 1.58350i) q^{7} +(0.869772 - 2.24514i) q^{8} +(-1.03368 - 2.81629i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3264 q - 48 q^{2} - 37 q^{3} - 118 q^{4} - 48 q^{5} - 37 q^{6} - 17 q^{7} - 23 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.44647	+	0.612237i	−1.02281	+	0.432917i	−0.835562	−	0.549397i	\(-0.814858\pi\)
							−0.187244	+	0.982313i	\(0.559956\pi\)
\(3\)	0.991545	−	1.42015i	0.572469	−	0.819927i
							\(5\)	−3.30392	+	2.33880i	−1.47756	+	1.04594i	−0.493171	+	0.869933i	\(0.664162\pi\)
−0.984388	+	0.176010i	\(0.943681\pi\)
\(7\)	−4.08749	+	1.58350i	−1.54493	+	0.598508i	−0.974806	−	0.223054i	\(-0.928398\pi\)
							−0.570120	+	0.821561i	\(0.693103\pi\)
\(11\)	−2.63079	+	3.48374i	−0.793214	+	1.05039i	0.204229	+	0.978923i	\(0.434531\pi\)
							−0.997443	+	0.0714625i	\(0.977233\pi\)
\(13\)	−0.654337	+	4.21531i	−0.181480	+	1.16912i	0.705980	+	0.708232i	\(0.250507\pi\)
							−0.887460	+	0.460884i	\(0.847532\pi\)
\(17\)	−2.59528	−	0.914542i	−0.629448	−	0.221809i	−0.00141930	−	0.999999i	\(-0.500452\pi\)
							−0.628029	+	0.778190i	\(0.716138\pi\)
\(19\)	6.49886	−	0.400836i	1.49094	−	0.0919580i	0.704535	−	0.709669i	\(-0.251156\pi\)
							0.786405	+	0.617711i	\(0.211940\pi\)
\(23\)	−0.606167	−	4.89528i	−0.126395	−	1.02074i	−0.914328	−	0.404976i	\(-0.867280\pi\)
							0.787933	−	0.615761i	\(-0.211151\pi\)
\(29\)	0.997900	−	2.16900i	0.185305	−	0.402774i	−0.793210	−	0.608948i	\(-0.791592\pi\)
							0.978515	+	0.206174i	\(0.0661014\pi\)
\(31\)	1.94916	−	0.0600530i	0.350079	−	0.0107858i	0.144721	−	0.989473i	\(-0.453772\pi\)
							0.205359	+	0.978687i	\(0.434164\pi\)
\(37\)	2.94823	+	3.23406i	0.484687	+	0.531676i	0.931883	−	0.362760i	\(-0.118166\pi\)
							−0.447196	+	0.894436i	\(0.647577\pi\)
\(41\)	−4.77301	−	0.442284i	−0.745418	−	0.0690732i	−0.287148	−	0.957886i	\(-0.592707\pi\)
							−0.458270	+	0.888813i	\(0.651531\pi\)
\(43\)	−2.72680	−	8.57152i	−0.415832	−	1.30714i	−0.901416	−	0.432954i	\(-0.857471\pi\)
							0.485584	−	0.874190i	\(-0.338607\pi\)
\(47\)	−0.328360			−0.0478962			−0.0239481	−	0.999713i	\(-0.507624\pi\)
							−0.0239481	+	0.999713i	\(0.507624\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	927.2.bg.a.77.25	✓	3264
9.2	odd	6		927.2.bm.a.695.25	yes	3264
103.99	odd	102		927.2.bm.a.923.25	yes	3264
927.614	even	102	inner	927.2.bg.a.614.25	yes	3264

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
927.2.bg.a.77.25	✓	3264	1.1	even	1	trivial
927.2.bg.a.614.25	yes	3264	927.614	even	102	inner
927.2.bm.a.695.25	yes	3264	9.2	odd	6
927.2.bm.a.923.25	yes	3264	103.99	odd	102