Embedded newform 221.2.ba.a.148.18

• Label: 221.2.ba.a.148.18
• Level: $221$
• Weight: $2$
• Character: 221.148
• 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			148.18
Character	\(\chi\)	\(=\)	221.148
Dual form			221.2.ba.a.112.18

$q$-expansion

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

Character values

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

\(n\)	\(105\)	\(171\)
\(\chi(n)\)	\(e\left(\frac{13}{16}\right)\)	\(e\left(\frac{3}{4}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	2.03115	+	0.841331i	1.43624	+	0.594911i	0.958884	−	0.283799i	\(-0.0915948\pi\)
							0.477357	+	0.878709i	\(0.341595\pi\)
\(3\)	0.600920	−	0.401522i	0.346941	−	0.231819i	−0.369874	−	0.929082i	\(-0.620599\pi\)
							0.716815	+	0.697263i	\(0.245599\pi\)
\(5\)	0.374154	−	0.250002i	0.167327	−	0.111804i	−0.469087	−	0.883152i	\(-0.655417\pi\)
							0.636414	+	0.771348i	\(0.280417\pi\)
\(7\)	−0.830760	−	0.555096i	−0.313998	−	0.209807i	0.388577	−	0.921416i	\(-0.372967\pi\)
							−0.702575	+	0.711609i	\(0.747967\pi\)
\(11\)	−3.21497	+	0.639498i	−0.969351	+	0.192816i	−0.654280	−	0.756252i	\(-0.727028\pi\)
							−0.315071	+	0.949068i	\(0.602028\pi\)
\(13\)	2.09350	−	2.93552i	0.580631	−	0.814167i
							\(17\)	−3.55089	−	2.09551i	−0.861218	−	0.508236i
\(19\)	4.38866	+	1.81784i	1.00683	+	0.417041i								0.824296	−	0.566159i	\(-0.191571\pi\)
							0.182531	+	0.983200i	\(0.441571\pi\)
\(23\)	−2.65190	+	3.96885i	−0.552959	+	0.827562i	−0.997678	−	0.0681082i	\(-0.978304\pi\)
							0.444719	+	0.895670i	\(0.353304\pi\)
\(29\)	0.910635	−	4.57807i	0.169101	−	0.850127i	−0.799339	−	0.600880i	\(-0.794817\pi\)
							0.968440	−	0.249247i	\(-0.0801831\pi\)
\(31\)	−4.89149	−	0.972979i	−0.878538	−	0.174752i	−0.264840	−	0.964292i	\(-0.585319\pi\)
							−0.613699	+	0.789540i	\(0.710319\pi\)
\(37\)	−0.238007	−	0.0473425i	−0.0391281	−	0.00778306i	0.175487	−	0.984482i	\(-0.443850\pi\)
							−0.214616	+	0.976699i	\(0.568850\pi\)
\(41\)	4.19000	−	6.27078i	0.654368	−	0.979331i	−0.344805	−	0.938674i	\(-0.612055\pi\)
							0.999173	−	0.0406564i	\(-0.0129449\pi\)
\(43\)	7.38373	+	3.05844i	1.12601	+	0.466408i	0.866422	−	0.499312i	\(-0.166414\pi\)
							0.259586	+	0.965720i	\(0.416414\pi\)
\(47\)		−	0.215085i		−	0.0313734i	−0.999877	−	0.0156867i	\(-0.995007\pi\)
							0.999877	−	0.0156867i	\(-0.00499344\pi\)

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
221.2.z.a.44.2	✓	152	17.10	odd	16
221.2.z.a.216.2	yes	152	13.8	odd	4
221.2.ba.a.112.18	yes	152	221.112	even	16	inner
221.2.ba.a.148.18	yes	152	1.1	even	1	trivial