Embedded newform 221.2.z.a.44.4

• Label: 221.2.z.a.44.4
• Level: $221$
• Weight: $2$
• Character: 221.44
• 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.z (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			44.4
Character	\(\chi\)	\(=\)	221.44
Dual form			221.2.z.a.216.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.728016 - 1.75759i) q^{2} +(-0.779718 - 0.520991i) q^{3} +(-1.14489 + 1.14489i) q^{4} +(-1.76865 + 2.64697i) q^{5} +(-0.348039 + 1.74971i) q^{6} +(2.64641 + 3.96063i) q^{7} +(-0.669433 - 0.277288i) q^{8} +(-0.811522 - 1.95919i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 152 q - 8 q^{2} - 16 q^{3} - 8 q^{5} - 8 q^{6} - 8 q^{7} - 24 q^{8} - 16 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.728016	−	1.75759i	−0.514785	−	1.24280i	−0.941070	−	0.338212i	\(-0.890178\pi\)
							0.426285	−	0.904589i	\(-0.359822\pi\)
\(3\)	−0.779718	−	0.520991i	−0.450170	−	0.300794i	0.309741	−	0.950821i	\(-0.399758\pi\)
							−0.759911	+	0.650027i	\(0.774758\pi\)
\(5\)	−1.76865	+	2.64697i	−0.790964	+	1.18376i	0.188487	+	0.982076i	\(0.439642\pi\)
							−0.979450	+	0.201685i	\(0.935358\pi\)
\(7\)	2.64641	+	3.96063i	1.00025	+	1.49698i	0.862209	+	0.506553i	\(0.169080\pi\)
							0.138040	+	0.990427i	\(0.455920\pi\)
\(11\)	−0.840561	+	4.22578i	−0.253439	+	1.27412i	0.618997	+	0.785393i	\(0.287539\pi\)
							−0.872436	+	0.488729i	\(0.837461\pi\)
\(13\)	−0.196515	+	3.60019i	−0.0545034	+	0.998514i
							\(17\)	−0.283954	−	4.11332i	−0.0688688	−	0.997626i
\(19\)	−0.935706	−	2.25899i	−0.214666	−	0.518249i								0.779464	−	0.626448i	\(-0.215492\pi\)
							−0.994129	+	0.108199i	\(0.965492\pi\)
\(23\)	−0.411512	−	0.615871i	−0.0858061	−	0.128418i	0.786084	−	0.618120i	\(-0.212105\pi\)
							−0.871890	+	0.489702i	\(0.837105\pi\)
\(29\)	1.26782	+	6.37378i	0.235429	+	1.18358i	0.899842	+	0.436216i	\(0.143682\pi\)
							−0.664413	+	0.747366i	\(0.731318\pi\)
\(31\)	−0.451196	−	2.26831i	−0.0810371	−	0.407401i	−0.999917	−	0.0128643i	\(-0.995905\pi\)
							0.918880	−	0.394537i	\(-0.129095\pi\)
\(37\)	1.05065	+	5.28200i	0.172727	+	0.868355i	0.965812	+	0.259244i	\(0.0834734\pi\)
							−0.793085	+	0.609111i	\(0.791527\pi\)
\(41\)	−5.16482	+	3.45102i	−0.806609	+	0.538959i	−0.889143	−	0.457629i	\(-0.848699\pi\)
							0.0825339	+	0.996588i	\(0.473699\pi\)
\(43\)	2.32276	−	0.962120i	0.354218	−	0.146722i	−0.198477	−	0.980106i	\(-0.563599\pi\)
							0.552695	+	0.833384i	\(0.313599\pi\)
\(47\)	8.28997			1.20922			0.604608	−	0.796523i	\(-0.293330\pi\)
							0.604608	+	0.796523i	\(0.293330\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	221.2.z.a.44.4	✓	152
13.8	odd	4		221.2.ba.a.112.16	yes	152
17.12	odd	16		221.2.ba.a.148.16	yes	152
221.216	even	16	inner	221.2.z.a.216.4	yes	152

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
221.2.z.a.44.4	✓	152	1.1	even	1	trivial
221.2.z.a.216.4	yes	152	221.216	even	16	inner
221.2.ba.a.112.16	yes	152	13.8	odd	4
221.2.ba.a.148.16	yes	152	17.12	odd	16