Embedded newform 221.2.z.a.44.2

• Label: 221.2.z.a.44.2
• 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			44.2
Character	\(\chi\)	\(=\)	221.44
Dual form			221.2.z.a.216.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.841331 - 2.03115i) q^{2} +(0.600920 + 0.401522i) q^{3} +(-2.00353 + 2.00353i) q^{4} +(0.250002 - 0.374154i) q^{5} +(0.309980 - 1.55837i) q^{6} +(-0.555096 - 0.830760i) q^{7} +(1.69279 + 0.701176i) q^{8} +(-0.948165 - 2.28907i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 152 q - 8 q^{2} - 16 q^{3} - 8 q^{5} - 8 q^{6} - 8 q^{7} - 24 q^{8} - 16 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.841331	−	2.03115i	−0.594911	−	1.43624i	−0.878709	−	0.477357i	\(-0.841595\pi\)
							0.283799	−	0.958884i	\(-0.408405\pi\)
\(3\)	0.600920	+	0.401522i	0.346941	+	0.231819i	0.716815	−	0.697263i	\(-0.245599\pi\)
							−0.369874	+	0.929082i	\(0.620599\pi\)
\(5\)	0.250002	−	0.374154i	0.111804	−	0.167327i	−0.771348	−	0.636414i	\(-0.780417\pi\)
							0.883152	+	0.469087i	\(0.155417\pi\)
\(7\)	−0.555096	−	0.830760i	−0.209807	−	0.313998i	0.711609	−	0.702575i	\(-0.247967\pi\)
							−0.921416	+	0.388577i	\(0.872967\pi\)
\(11\)	0.639498	−	3.21497i	0.192816	−	0.969351i	−0.756252	−	0.654280i	\(-0.772972\pi\)
							0.949068	−	0.315071i	\(-0.102028\pi\)
\(13\)	−2.93552	−	2.09350i	−0.814167	−	0.580631i
							\(17\)	3.55089	−	2.09551i	0.861218	−	0.508236i
\(19\)	−1.81784	−	4.38866i	−0.417041	−	1.00683i								−0.983200	−	0.182531i	\(-0.941571\pi\)
							0.566159	−	0.824296i	\(-0.308429\pi\)
\(23\)	2.65190	+	3.96885i	0.552959	+	0.827562i	0.997678	−	0.0681082i	\(-0.0216963\pi\)
							−0.444719	+	0.895670i	\(0.646696\pi\)
\(29\)	0.910635	+	4.57807i	0.169101	+	0.850127i	0.968440	+	0.249247i	\(0.0801831\pi\)
							−0.799339	+	0.600880i	\(0.794817\pi\)
\(31\)	0.972979	+	4.89149i	0.174752	+	0.878538i	0.964292	+	0.264840i	\(0.0853190\pi\)
							−0.789540	+	0.613699i	\(0.789681\pi\)
\(37\)	−0.0473425	−	0.238007i	−0.00778306	−	0.0391281i	0.976699	−	0.214616i	\(-0.0688498\pi\)
							−0.984482	+	0.175487i	\(0.943850\pi\)
\(41\)	6.27078	−	4.19000i	0.979331	−	0.654368i	0.0406564	−	0.999173i	\(-0.487055\pi\)
							0.938674	+	0.344805i	\(0.112055\pi\)
\(43\)	−7.38373	+	3.05844i	−1.12601	+	0.466408i	−0.866422	−	0.499312i	\(-0.833586\pi\)
							−0.259586	+	0.965720i	\(0.583586\pi\)
\(47\)	−0.215085			−0.0313734			−0.0156867	−	0.999877i	\(-0.504993\pi\)
							−0.0156867	+	0.999877i	\(0.504993\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	221.2.z.a.44.2	✓	152
13.8	odd	4		221.2.ba.a.112.18	yes	152
17.12	odd	16		221.2.ba.a.148.18	yes	152
221.216	even	16	inner	221.2.z.a.216.2	yes	152

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
221.2.z.a.44.2	✓	152	1.1	even	1	trivial
221.2.z.a.216.2	yes	152	221.216	even	16	inner
221.2.ba.a.112.18	yes	152	13.8	odd	4
221.2.ba.a.148.18	yes	152	17.12	odd	16