Embedded newform 927.2.bg.a.77.38

• Label: 927.2.bg.a.77.38
• 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			77.38
Character	\(\chi\)	\(=\)	927.77
Dual form			927.2.bg.a.614.38

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.727321 + 0.307848i) q^{2} +(-1.71392 - 0.249946i) q^{3} +(-0.958043 + 0.988015i) q^{4} +(1.54964 - 1.09697i) q^{5} +(1.32352 - 0.345837i) q^{6} +(-4.11060 + 1.59245i) q^{7} +(0.963255 - 2.48645i) q^{8} +(2.87505 + 0.856777i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3264 q - 48 q^{2} - 37 q^{3} - 118 q^{4} - 48 q^{5} - 37 q^{6} - 17 q^{7} - 23 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.727321	+	0.307848i	−0.514293	+	0.217682i	−0.631428	−	0.775434i	\(-0.717531\pi\)
							0.117135	+	0.993116i	\(0.462629\pi\)
\(3\)	−1.71392	−	0.249946i	−0.989533	−	0.144307i
							\(5\)	1.54964	−	1.09697i	0.693021	−	0.490579i	−0.176611	−	0.984281i	\(-0.556513\pi\)
0.869631	+	0.493702i	\(0.164356\pi\)
\(7\)	−4.11060	+	1.59245i	−1.55366	+	0.601891i	−0.976648	−	0.214844i	\(-0.931076\pi\)
							−0.577012	+	0.816735i	\(0.695782\pi\)
\(11\)	−1.15771	+	1.53305i	−0.349062	+	0.462232i	−0.938348	−	0.345692i	\(-0.887644\pi\)
							0.589287	+	0.807924i	\(0.299409\pi\)
\(13\)	0.858142	−	5.52824i	0.238006	−	1.53326i	−0.504496	−	0.863414i	\(-0.668322\pi\)
							0.742502	−	0.669844i	\(-0.233639\pi\)
\(17\)	1.77203	+	0.624441i	0.429781	+	0.151449i	0.538504	−	0.842623i	\(-0.318990\pi\)
							−0.108722	+	0.994072i	\(0.534676\pi\)
\(19\)	−5.68943	+	0.350912i	−1.30524	+	0.0805047i	−0.699200	−	0.714926i	\(-0.746460\pi\)
							−0.606044	+	0.795431i	\(0.707245\pi\)
\(23\)	−0.149946	−	1.21093i	−0.0312659	−	0.252497i	−0.999991	−	0.00435849i	\(-0.998613\pi\)
							0.968725	−	0.248138i	\(-0.0798187\pi\)
\(29\)	1.52040	−	3.30469i	0.282331	−	0.613666i	−0.713915	−	0.700232i	\(-0.753080\pi\)
							0.996246	+	0.0865664i	\(0.0275895\pi\)
\(31\)	3.53392	−	0.108879i	0.634710	−	0.0195552i	0.288153	−	0.957584i	\(-0.406959\pi\)
							0.346557	+	0.938029i	\(0.387351\pi\)
\(37\)	4.70495	+	5.16109i	0.773489	+	0.848477i	0.991853	−	0.127390i	\(-0.0406600\pi\)
							−0.218363	+	0.975868i	\(0.570072\pi\)
\(41\)	−1.76622	−	0.163664i	−0.275837	−	0.0255601i	−0.0465397	−	0.998916i	\(-0.514819\pi\)
							−0.229298	+	0.973356i	\(0.573643\pi\)
\(43\)	0.714971	+	2.24747i	0.109032	+	0.342736i	0.991923	−	0.126843i	\(-0.0404844\pi\)
							−0.882891	+	0.469578i	\(0.844406\pi\)
\(47\)	7.82023			1.14070			0.570349	−	0.821402i	\(-0.306808\pi\)
							0.570349	+	0.821402i	\(0.306808\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	927.2.bg.a.77.38	✓	3264
9.2	odd	6		927.2.bm.a.695.38	yes	3264
103.99	odd	102		927.2.bm.a.923.38	yes	3264
927.614	even	102	inner	927.2.bg.a.614.38	yes	3264

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
927.2.bg.a.77.38	✓	3264	1.1	even	1	trivial
927.2.bg.a.614.38	yes	3264	927.614	even	102	inner
927.2.bm.a.695.38	yes	3264	9.2	odd	6
927.2.bm.a.923.38	yes	3264	103.99	odd	102