Embedded newform 221.2.ba.a.5.9

• Label: 221.2.ba.a.5.9
• Level: $221$
• Weight: $2$
• Character: 221.5
• Analytic conductor: $1.765$
• Analytic rank: $0$
• Dimension: $152$
• CM: no
• 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			5.9
Character	\(\chi\)	\(=\)	221.5
Dual form			221.2.ba.a.177.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.273153 - 0.113144i) q^{2} +(1.37505 + 2.05790i) q^{3} +(-1.35240 - 1.35240i) q^{4} +(1.39301 + 2.08478i) q^{5} +(-0.142759 - 0.717699i) q^{6} +(1.43172 - 2.14272i) q^{7} +(0.442684 + 1.06873i) q^{8} +(-1.19616 + 2.88778i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 152 q - 8 q^{2} - 16 q^{3} - 8 q^{5} - 8 q^{6} - 8 q^{7} + 8 q^{8} - 16 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.273153	−	0.113144i	−0.193148	−	0.0800046i	0.284013	−	0.958820i	\(-0.408334\pi\)
							−0.477161	+	0.878816i	\(0.658334\pi\)
\(3\)	1.37505	+	2.05790i	0.793883	+	1.18813i	0.978688	+	0.205355i	\(0.0658348\pi\)
							−0.184805	+	0.982775i	\(0.559165\pi\)
\(5\)	1.39301	+	2.08478i	0.622971	+	0.932343i	0.999981	+	0.00613334i	\(0.00195232\pi\)
							−0.377010	+	0.926209i	\(0.623048\pi\)
\(7\)	1.43172	−	2.14272i	0.541140	−	0.809873i	−0.455631	−	0.890169i	\(-0.650586\pi\)
							0.996771	+	0.0802955i	\(0.0255864\pi\)
\(11\)	0.960885	+	4.83070i	0.289718	+	1.45651i	0.801810	+	0.597579i	\(0.203871\pi\)
							−0.512092	+	0.858931i	\(0.671129\pi\)
\(13\)	−1.03718	−	3.45315i	−0.287663	−	0.957732i
							\(17\)	0.647015	+	4.07202i	0.156924	+	0.987611i
\(19\)	−1.22477	−	0.507317i	−0.280982	−	0.116387i								0.237742	−	0.971328i	\(-0.423593\pi\)
							−0.518724	+	0.854942i	\(0.673593\pi\)
\(23\)	−6.24179	−	4.17063i	−1.30150	−	0.869636i	−0.304931	−	0.952374i	\(-0.598633\pi\)
							−0.996571	+	0.0827383i	\(0.973633\pi\)
\(29\)	3.88702	+	0.773176i	0.721801	+	0.143575i	0.542303	−	0.840183i	\(-0.317553\pi\)
							0.179498	+	0.983758i	\(0.442553\pi\)
\(31\)	0.989602	−	4.97507i	0.177738	−	0.893548i	−0.784248	−	0.620448i	\(-0.786951\pi\)
							0.961985	−	0.273101i	\(-0.0880492\pi\)
\(37\)	1.40213	−	7.04899i	0.230509	−	1.15885i	−0.676079	−	0.736829i	\(-0.736322\pi\)
							0.906588	−	0.422017i	\(-0.138678\pi\)
\(41\)	−6.93478	−	4.63367i	−1.08303	−	0.723658i	−0.119926	−	0.992783i	\(-0.538266\pi\)
							−0.963105	+	0.269125i	\(0.913266\pi\)
\(43\)	6.27737	+	2.60017i	0.957289	+	0.396522i	0.805966	−	0.591962i	\(-0.201646\pi\)
							0.151324	+	0.988484i	\(0.451646\pi\)
\(47\)		−	5.63007i		−	0.821230i	−0.911809	−	0.410615i	\(-0.865314\pi\)
							0.911809	−	0.410615i	\(-0.134686\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	221.2.ba.a.5.9	yes	152
13.8	odd	4		221.2.z.a.73.11	✓	152
17.7	odd	16		221.2.z.a.109.11	yes	152
221.177	even	16	inner	221.2.ba.a.177.9	yes	152

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
221.2.z.a.73.11	✓	152	13.8	odd	4
221.2.z.a.109.11	yes	152	17.7	odd	16
221.2.ba.a.5.9	yes	152	1.1	even	1	trivial
221.2.ba.a.177.9	yes	152	221.177	even	16	inner