Embedded newform 221.2.z.a.216.7

• Label: 221.2.z.a.216.7
• 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			216.7
Character	\(\chi\)	\(=\)	221.216
Dual form			221.2.z.a.44.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.306289 + 0.739446i) q^{2} +(-0.471645 + 0.315143i) q^{3} +(0.961245 + 0.961245i) q^{4} +(-1.97622 - 2.95762i) q^{5} +(-0.0885718 - 0.445281i) q^{6} +(-2.57309 + 3.85090i) q^{7} +(-2.48410 + 1.02895i) q^{8} +(-1.02492 + 2.47437i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 152 q - 8 q^{2} - 16 q^{3} - 8 q^{5} - 8 q^{6} - 8 q^{7} - 24 q^{8} - 16 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.306289	+	0.739446i	−0.216579	+	0.522868i	−0.994408	−	0.105608i	\(-0.966321\pi\)
							0.777829	+	0.628476i	\(0.216321\pi\)
\(3\)	−0.471645	+	0.315143i	−0.272304	+	0.181948i	−0.684223	−	0.729273i	\(-0.739859\pi\)
							0.411919	+	0.911220i	\(0.364859\pi\)
\(5\)	−1.97622	−	2.95762i	−0.883793	−	1.32269i	−0.945846	−	0.324616i	\(-0.894765\pi\)
							0.0620533	−	0.998073i	\(-0.480235\pi\)
\(7\)	−2.57309	+	3.85090i	−0.972535	+	1.45550i	−0.0841226	+	0.996455i	\(0.526809\pi\)
							−0.888412	+	0.459046i	\(0.848191\pi\)
\(11\)	−0.0669719	−	0.336691i	−0.0201928	−	0.101516i	0.969375	−	0.245587i	\(-0.0789807\pi\)
							−0.989567	+	0.144071i	\(0.953981\pi\)
\(13\)	2.43240	+	2.66147i	0.674626	+	0.738159i
							\(17\)	−2.96019	+	2.87007i	−0.717951	+	0.696094i
\(19\)	0.877004	−	2.11728i	0.201199	−	0.485736i								−0.790786	−	0.612092i	\(-0.790328\pi\)
							0.991985	+	0.126356i	\(0.0403281\pi\)
\(23\)	2.35817	−	3.52926i	0.491713	−	0.735901i	−0.499767	−	0.866160i	\(-0.666581\pi\)
							0.991480	+	0.130259i	\(0.0415810\pi\)
\(29\)	−0.896391	+	4.50646i	−0.166456	+	0.836829i	0.803829	+	0.594861i	\(0.202793\pi\)
							−0.970284	+	0.241968i	\(0.922207\pi\)
\(31\)	1.86788	−	9.39049i	0.335482	−	1.68658i	−0.333059	−	0.942906i	\(-0.608081\pi\)
							0.668541	−	0.743676i	\(-0.266919\pi\)
\(37\)	−0.867840	+	4.36293i	−0.142672	+	0.717261i	0.841530	+	0.540211i	\(0.181655\pi\)
							−0.984202	+	0.177050i	\(0.943345\pi\)
\(41\)	6.46108	+	4.31715i	1.00905	+	0.674226i	0.946128	−	0.323792i	\(-0.104958\pi\)
							0.0629226	+	0.998018i	\(0.479958\pi\)
\(43\)	10.0247	+	4.15236i	1.52875	+	0.633229i	0.979321	−	0.202312i	\(-0.0648456\pi\)
							0.549428	+	0.835541i	\(0.314846\pi\)
\(47\)	−4.43180			−0.646444			−0.323222	−	0.946323i	\(-0.604766\pi\)
							−0.323222	+	0.946323i	\(0.604766\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	221.2.z.a.216.7	yes	152
13.5	odd	4		221.2.ba.a.148.13	yes	152
17.10	odd	16		221.2.ba.a.112.13	yes	152
221.44	even	16	inner	221.2.z.a.44.7	✓	152

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
221.2.z.a.44.7	✓	152	221.44	even	16	inner
221.2.z.a.216.7	yes	152	1.1	even	1	trivial
221.2.ba.a.112.13	yes	152	17.10	odd	16
221.2.ba.a.148.13	yes	152	13.5	odd	4