Embedded newform 221.2.z.a.44.14

• Label: 221.2.z.a.44.14
• 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			44.14
Character	\(\chi\)	\(=\)	221.44
Dual form			221.2.z.a.216.14

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.509634 + 1.23037i) q^{2} +(0.464497 + 0.310367i) q^{3} +(0.160142 - 0.160142i) q^{4} +(-1.48137 + 2.21702i) q^{5} +(-0.145141 + 0.729674i) q^{6} +(0.874722 + 1.30911i) q^{7} +(2.73938 + 1.13469i) q^{8} +(-1.02862 - 2.48331i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 152 q - 8 q^{2} - 16 q^{3} - 8 q^{5} - 8 q^{6} - 8 q^{7} - 24 q^{8} - 16 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.509634	+	1.23037i	0.360366	+	0.870000i	0.995246	+	0.0973894i	\(0.0310492\pi\)
							−0.634881	+	0.772610i	\(0.718951\pi\)
\(3\)	0.464497	+	0.310367i	0.268177	+	0.179190i	0.682387	−	0.730991i	\(-0.260942\pi\)
							−0.414210	+	0.910182i	\(0.635942\pi\)
\(5\)	−1.48137	+	2.21702i	−0.662488	+	0.991484i	0.336275	+	0.941764i	\(0.390833\pi\)
							−0.998763	+	0.0497197i	\(0.984167\pi\)
\(7\)	0.874722	+	1.30911i	0.330614	+	0.494798i	0.959118	−	0.283008i	\(-0.0913323\pi\)
							−0.628504	+	0.777807i	\(0.716332\pi\)
\(11\)	−0.885867	+	4.45356i	−0.267099	+	1.34280i	0.581408	+	0.813613i	\(0.302502\pi\)
							−0.848507	+	0.529185i	\(0.822498\pi\)
\(13\)	−0.0500371	−	3.60520i	−0.0138778	−	0.999904i
							\(17\)	3.85331	+	1.46696i	0.934566	+	0.355790i
\(19\)	−0.975068	−	2.35402i	−0.223696	−	0.540050i								0.771690	−	0.635998i	\(-0.219412\pi\)
							−0.995386	+	0.0959486i	\(0.969412\pi\)
\(23\)	−0.739094	−	1.10613i	−0.154112	−	0.230644i	0.746378	−	0.665523i	\(-0.231791\pi\)
							−0.900489	+	0.434878i	\(0.856791\pi\)
\(29\)	−1.82100	−	9.15477i	−0.338151	−	1.70000i	−0.658403	−	0.752665i	\(-0.728768\pi\)
							0.320253	−	0.947332i	\(-0.396232\pi\)
\(31\)	−0.718801	−	3.61366i	−0.129101	−	0.649032i	−0.990093	−	0.140414i	\(-0.955157\pi\)
							0.860992	−	0.508618i	\(-0.169843\pi\)
\(37\)	0.295896	+	1.48757i	0.0486450	+	0.244555i	0.997456	−	0.0712906i	\(-0.0227118\pi\)
							−0.948811	+	0.315846i	\(0.897712\pi\)
\(41\)	−3.86579	+	2.58304i	−0.603734	+	0.403402i	−0.819530	−	0.573036i	\(-0.805765\pi\)
							0.215796	+	0.976439i	\(0.430765\pi\)
\(43\)	2.77166	−	1.14806i	0.422674	−	0.175077i	−0.161200	−	0.986922i	\(-0.551536\pi\)
							0.583873	+	0.811845i	\(0.301536\pi\)
\(47\)	6.92669			1.01036			0.505181	−	0.863014i	\(-0.331426\pi\)
							0.505181	+	0.863014i	\(0.331426\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	221.2.z.a.44.14	✓	152
13.8	odd	4		221.2.ba.a.112.6	yes	152
17.12	odd	16		221.2.ba.a.148.6	yes	152
221.216	even	16	inner	221.2.z.a.216.14	yes	152

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
221.2.z.a.44.14	✓	152	1.1	even	1	trivial
221.2.z.a.216.14	yes	152	221.216	even	16	inner
221.2.ba.a.112.6	yes	152	13.8	odd	4
221.2.ba.a.148.6	yes	152	17.12	odd	16