Embedded newform 221.2.ba.a.148.6

• Label: 221.2.ba.a.148.6
• Level: $221$
• Weight: $2$
• Character: 221.148
• 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.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			148.6
Character	\(\chi\)	\(=\)	221.148
Dual form			221.2.ba.a.112.6

$q$-expansion

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.23037	−	0.509634i	−0.870000	−	0.360366i	−0.0973894	−	0.995246i	\(-0.531049\pi\)
							−0.772610	+	0.634881i	\(0.781049\pi\)
\(3\)	0.464497	−	0.310367i	0.268177	−	0.179190i	−0.414210	−	0.910182i	\(-0.635942\pi\)
							0.682387	+	0.730991i	\(0.260942\pi\)
\(5\)	−2.21702	+	1.48137i	−0.991484	+	0.662488i	−0.941764	−	0.336275i	\(-0.890833\pi\)
							−0.0497197	+	0.998763i	\(0.515833\pi\)
\(7\)	1.30911	+	0.874722i	0.494798	+	0.330614i	0.777807	−	0.628504i	\(-0.216332\pi\)
							−0.283008	+	0.959118i	\(0.591332\pi\)
\(11\)	4.45356	−	0.885867i	1.34280	−	0.267099i	0.529185	−	0.848507i	\(-0.322498\pi\)
							0.813613	+	0.581408i	\(0.197498\pi\)
\(13\)	3.60520	−	0.0500371i	0.999904	−	0.0138778i
							\(17\)	−3.85331	+	1.46696i	−0.934566	+	0.355790i
\(19\)	2.35402	+	0.975068i	0.540050	+	0.223696i								0.635998	−	0.771690i	\(-0.280588\pi\)
							−0.0959486	+	0.995386i	\(0.530588\pi\)
\(23\)	0.739094	−	1.10613i	0.154112	−	0.230644i	−0.746378	−	0.665523i	\(-0.768209\pi\)
							0.900489	+	0.434878i	\(0.143209\pi\)
\(29\)	−1.82100	+	9.15477i	−0.338151	+	1.70000i	0.320253	+	0.947332i	\(0.396232\pi\)
							−0.658403	+	0.752665i	\(0.728768\pi\)
\(31\)	3.61366	+	0.718801i	0.649032	+	0.129101i	0.508618	−	0.860992i	\(-0.330157\pi\)
							0.140414	+	0.990093i	\(0.455157\pi\)
\(37\)	1.48757	+	0.295896i	0.244555	+	0.0486450i	0.315846	−	0.948811i	\(-0.397712\pi\)
							−0.0712906	+	0.997456i	\(0.522712\pi\)
\(41\)	−2.58304	+	3.86579i	−0.403402	+	0.603734i	−0.976439	−	0.215796i	\(-0.930765\pi\)
							0.573036	+	0.819530i	\(0.305765\pi\)
\(43\)	−2.77166	−	1.14806i	−0.422674	−	0.175077i	0.161200	−	0.986922i	\(-0.448464\pi\)
							−0.583873	+	0.811845i	\(0.698464\pi\)
\(47\)			6.92669i			1.01036i	0.863014	+	0.505181i	\(0.168574\pi\)
							−0.863014	+	0.505181i	\(0.831426\pi\)

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
221.2.z.a.44.14	✓	152	17.10	odd	16
221.2.z.a.216.14	yes	152	13.8	odd	4
221.2.ba.a.112.6	yes	152	221.112	even	16	inner
221.2.ba.a.148.6	yes	152	1.1	even	1	trivial