Embedded newform 221.2.z.a.216.9

• Label: 221.2.z.a.216.9
• 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			216.9
Character	\(\chi\)	\(=\)	221.216
Dual form			221.2.z.a.44.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.139805 + 0.337519i) q^{2} +(-2.15216 + 1.43803i) q^{3} +(1.31984 + 1.31984i) q^{4} +(1.49666 + 2.23991i) q^{5} +(-0.184479 - 0.927438i) q^{6} +(0.0376968 - 0.0564173i) q^{7} +(-1.30503 + 0.540561i) q^{8} +(1.41582 - 3.41808i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 152 q - 8 q^{2} - 16 q^{3} - 8 q^{5} - 8 q^{6} - 8 q^{7} - 24 q^{8} - 16 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.139805	+	0.337519i	−0.0988570	+	0.238662i	−0.965569	−	0.260147i	\(-0.916229\pi\)
							0.866712	+	0.498809i	\(0.166229\pi\)
\(3\)	−2.15216	+	1.43803i	−1.24255	+	0.830245i	−0.990506	−	0.137471i	\(-0.956103\pi\)
							−0.252044	+	0.967716i	\(0.581103\pi\)
\(5\)	1.49666	+	2.23991i	0.669328	+	1.00172i	0.998351	+	0.0573981i	\(0.0182804\pi\)
							−0.329024	+	0.944322i	\(0.606720\pi\)
\(7\)	0.0376968	−	0.0564173i	0.0142481	−	0.0213237i	−0.824277	−	0.566186i	\(-0.808418\pi\)
							0.838525	+	0.544863i	\(0.183418\pi\)
\(11\)	−0.923779	−	4.64415i	−0.278530	−	1.40026i	−0.826109	−	0.563510i	\(-0.809451\pi\)
							0.547580	−	0.836754i	\(-0.315549\pi\)
\(13\)	−1.98576	+	3.00944i	−0.550751	+	0.834669i
							\(17\)	1.41033	+	3.87440i	0.342055	+	0.939680i
\(19\)	−0.167004	+	0.403182i	−0.0383133	+	0.0924964i								−0.941878	−	0.335955i	\(-0.890941\pi\)
							0.903565	+	0.428452i	\(0.140941\pi\)
\(23\)	0.607653	−	0.909418i	0.126705	−	0.189627i	−0.762694	−	0.646760i	\(-0.776124\pi\)
							0.889398	+	0.457133i	\(0.151124\pi\)
\(29\)	1.45279	−	7.30369i	0.269777	−	1.35626i	−0.573686	−	0.819075i	\(-0.694487\pi\)
							0.843463	−	0.537187i	\(-0.180513\pi\)
\(31\)	−0.417831	+	2.10058i	−0.0750446	+	0.377275i	−0.999996	−	0.00281056i	\(-0.999105\pi\)
							0.924951	+	0.380085i	\(0.124105\pi\)
\(37\)	−0.279777	+	1.40653i	−0.0459950	+	0.231233i	−0.996944	−	0.0781138i	\(-0.975110\pi\)
							0.950949	+	0.309346i	\(0.100110\pi\)
\(41\)	−2.06214	−	1.37788i	−0.322053	−	0.215189i	0.384026	−	0.923322i	\(-0.374537\pi\)
							−0.706078	+	0.708134i	\(0.749537\pi\)
\(43\)	7.36036	+	3.04876i	1.12244	+	0.464932i	0.865206	−	0.501417i	\(-0.167188\pi\)
							0.257238	+	0.966348i	\(0.417188\pi\)
\(47\)	10.7304			1.56519			0.782594	−	0.622533i	\(-0.213896\pi\)
							0.782594	+	0.622533i	\(0.213896\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	221.2.z.a.216.9	yes	152
13.5	odd	4		221.2.ba.a.148.11	yes	152
17.10	odd	16		221.2.ba.a.112.11	yes	152
221.44	even	16	inner	221.2.z.a.44.9	✓	152

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
221.2.z.a.44.9	✓	152	221.44	even	16	inner
221.2.z.a.216.9	yes	152	1.1	even	1	trivial
221.2.ba.a.112.11	yes	152	17.10	odd	16
221.2.ba.a.148.11	yes	152	13.5	odd	4