Embedded newform 221.2.ba.a.5.11

• Label: 221.2.ba.a.5.11
• Level: $221$
• Weight: $2$
• Character: 221.5
• Analytic conductor: $1.765$
• Analytic rank: $0$
• Dimension: $152$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 221 = 13 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	221.ba (of order \(16\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.76469388467\)
Analytic rank:	\(0\)
Dimension:	\(152\)
Relative dimension:	\(19\) over \(\Q(\zeta_{16})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{16}]$

Embedding invariants

Embedding label			5.11
Character	\(\chi\)	\(=\)	221.5
Dual form			221.2.ba.a.177.11

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.278442 + 0.115335i) q^{2} +(0.214819 + 0.321499i) q^{3} +(-1.34999 - 1.34999i) q^{4} +(-1.18264 - 1.76994i) q^{5} +(0.0227347 + 0.114295i) q^{6} +(-0.276731 + 0.414157i) q^{7} +(-0.450862 - 1.08848i) q^{8} +(1.09084 - 2.63351i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 152 q - 8 q^{2} - 16 q^{3} - 8 q^{5} - 8 q^{6} - 8 q^{7} + 8 q^{8} - 16 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/221\mathbb{Z}\right)^\times\).

\(n\)	\(105\)	\(171\)
\(\chi(n)\)	\(e\left(\frac{5}{16}\right)\)	\(e\left(\frac{3}{4}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.278442	+	0.115335i	0.196889	+	0.0815539i	0.478949	−	0.877843i	\(-0.341018\pi\)
							−0.282060	+	0.959397i	\(0.591018\pi\)
\(3\)	0.214819	+	0.321499i	0.124026	+	0.185617i	0.888287	−	0.459290i	\(-0.151896\pi\)
							−0.764261	+	0.644907i	\(0.776896\pi\)
\(5\)	−1.18264	−	1.76994i	−0.528891	−	0.791541i	0.466792	−	0.884367i	\(-0.345410\pi\)
							−0.995682	+	0.0928264i	\(0.970410\pi\)
\(7\)	−0.276731	+	0.414157i	−0.104594	+	0.156537i	−0.880075	−	0.474835i	\(-0.842508\pi\)
							0.775480	+	0.631372i	\(0.217508\pi\)
\(11\)	−0.301772	−	1.51711i	−0.0909877	−	0.457426i	−0.999239	−	0.0390012i	\(-0.987582\pi\)
							0.908251	−	0.418425i	\(-0.137418\pi\)
\(13\)	1.89953	−	3.06460i	0.526836	−	0.849967i
							\(17\)	−4.11286	+	0.290436i	−0.997516	+	0.0704411i
\(19\)	0.194078	+	0.0803899i	0.0445246	+	0.0184427i								0.404835	−	0.914390i	\(-0.367329\pi\)
							−0.360310	+	0.932833i	\(0.617329\pi\)
\(23\)	3.53862	+	2.36443i	0.737853	+	0.493017i	0.866814	−	0.498632i	\(-0.166164\pi\)
							−0.128961	+	0.991650i	\(0.541164\pi\)
\(29\)	3.73111	+	0.742164i	0.692850	+	0.137816i	0.528940	−	0.848659i	\(-0.322590\pi\)
							0.163909	+	0.986475i	\(0.447590\pi\)
\(31\)	−1.29279	+	6.49929i	−0.232192	+	1.16731i	0.672123	+	0.740440i	\(0.265383\pi\)
							−0.904315	+	0.426867i	\(0.859617\pi\)
\(37\)	0.237191	−	1.19244i	0.0389939	−	0.196035i	−0.956378	−	0.292131i	\(-0.905636\pi\)
							0.995372	+	0.0960959i	\(0.0306355\pi\)
\(41\)	−1.98354	−	1.32536i	−0.309777	−	0.206986i	0.390957	−	0.920409i	\(-0.372144\pi\)
							−0.700734	+	0.713423i	\(0.747144\pi\)
\(43\)	2.78022	+	1.15161i	0.423980	+	0.175618i	0.584463	−	0.811420i	\(-0.301305\pi\)
							−0.160483	+	0.987039i	\(0.551305\pi\)
\(47\)		−	10.2887i		−	1.50077i	−0.661003	−	0.750383i	\(-0.729869\pi\)
							0.661003	−	0.750383i	\(-0.270131\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	221.2.ba.a.5.11	yes	152
13.8	odd	4		221.2.z.a.73.9	✓	152
17.7	odd	16		221.2.z.a.109.9	yes	152
221.177	even	16	inner	221.2.ba.a.177.11	yes	152

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
221.2.z.a.73.9	✓	152	13.8	odd	4
221.2.z.a.109.9	yes	152	17.7	odd	16
221.2.ba.a.5.11	yes	152	1.1	even	1	trivial
221.2.ba.a.177.11	yes	152	221.177	even	16	inner