Embedded newform 221.2.z.a.216.11

• Label: 221.2.z.a.216.11
• Level: $221$
• Weight: $2$
• Character: 221.216
• 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.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			216.11
Character	\(\chi\)	\(=\)	221.216
Dual form			221.2.z.a.44.11

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.271988 - 0.656637i) q^{2} +(-0.671616 + 0.448759i) q^{3} +(1.05702 + 1.05702i) q^{4} +(0.388698 + 0.581728i) q^{5} +(0.112001 + 0.563065i) q^{6} +(-1.33635 + 1.99999i) q^{7} +(2.29485 - 0.950557i) q^{8} +(-0.898367 + 2.16885i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 152 q - 8 q^{2} - 16 q^{3} - 8 q^{5} - 8 q^{6} - 8 q^{7} - 24 q^{8} - 16 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(105\)	\(171\)
\(\chi(n)\)	\(e\left(\frac{13}{16}\right)\)	\(e\left(\frac{1}{4}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.271988	−	0.656637i	0.192325	−	0.464313i	−0.798073	−	0.602561i	\(-0.794147\pi\)
							0.990398	+	0.138248i	\(0.0441471\pi\)
\(3\)	−0.671616	+	0.448759i	−0.387758	+	0.259091i	−0.734134	−	0.679004i	\(-0.762412\pi\)
							0.346377	+	0.938095i	\(0.387412\pi\)
\(5\)	0.388698	+	0.581728i	0.173831	+	0.260157i	0.908147	−	0.418651i	\(-0.137497\pi\)
							−0.734316	+	0.678808i	\(0.762497\pi\)
\(7\)	−1.33635	+	1.99999i	−0.505093	+	0.755924i	−0.993145	−	0.116890i	\(-0.962708\pi\)
							0.488052	+	0.872814i	\(0.337708\pi\)
\(11\)	0.352170	+	1.77048i	0.106183	+	0.533819i	0.996860	+	0.0791824i	\(0.0252310\pi\)
							−0.890677	+	0.454637i	\(0.849769\pi\)
\(13\)	0.196068	−	3.60022i	0.0543795	−	0.998520i
							\(17\)	3.75802	+	1.69625i	0.911454	+	0.411402i
\(19\)	1.71898	−	4.14998i	0.394361	−	0.952071i								−0.594617	−	0.804009i	\(-0.702696\pi\)
							0.988978	−	0.148062i	\(-0.0473036\pi\)
\(23\)	−1.62153	+	2.42679i	−0.338112	+	0.506021i	−0.961096	−	0.276214i	\(-0.910920\pi\)
							0.622984	+	0.782235i	\(0.285920\pi\)
\(29\)	−0.192481	+	0.967666i	−0.0357428	+	0.179691i	−0.994533	−	0.104424i	\(-0.966700\pi\)
							0.958790	+	0.284116i	\(0.0916999\pi\)
\(31\)	0.0128357	−	0.0645293i	0.00230535	−	0.0115898i	−0.979615	−	0.200882i	\(-0.935619\pi\)
							0.981921	+	0.189292i	\(0.0606193\pi\)
\(37\)	1.52114	−	7.64730i	0.250074	−	1.25721i	−0.627822	−	0.778357i	\(-0.716053\pi\)
							0.877896	−	0.478852i	\(-0.158947\pi\)
\(41\)	−3.81696	−	2.55041i	−0.596109	−	0.398307i	0.220594	−	0.975366i	\(-0.429200\pi\)
							−0.816703	+	0.577058i	\(0.804200\pi\)
\(43\)	−4.48766	−	1.85885i	−0.684361	−	0.283472i	0.0132876	−	0.999912i	\(-0.495770\pi\)
							−0.697649	+	0.716440i	\(0.745770\pi\)
\(47\)	−6.47680			−0.944738			−0.472369	−	0.881401i	\(-0.656601\pi\)
							−0.472369	+	0.881401i	\(0.656601\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	221.2.z.a.216.11	yes	152
13.5	odd	4		221.2.ba.a.148.9	yes	152
17.10	odd	16		221.2.ba.a.112.9	yes	152
221.44	even	16	inner	221.2.z.a.44.11	✓	152

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
221.2.z.a.44.11	✓	152	221.44	even	16	inner
221.2.z.a.216.11	yes	152	1.1	even	1	trivial
221.2.ba.a.112.9	yes	152	17.10	odd	16
221.2.ba.a.148.9	yes	152	13.5	odd	4