Embedded newform 221.2.z.a.216.17

• Label: 221.2.z.a.216.17
• 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			216.17
Character	\(\chi\)	\(=\)	221.216
Dual form			221.2.z.a.44.17

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.813483 - 1.96392i) q^{2} +(1.03290 - 0.690159i) q^{3} +(-1.78102 - 1.78102i) q^{4} +(-1.32730 - 1.98644i) q^{5} +(-0.515175 - 2.58996i) q^{6} +(0.0574971 - 0.0860505i) q^{7} +(-1.01878 + 0.421991i) q^{8} +(-0.557496 + 1.34591i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 152 q - 8 q^{2} - 16 q^{3} - 8 q^{5} - 8 q^{6} - 8 q^{7} - 24 q^{8} - 16 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.813483	−	1.96392i	0.575220	−	1.38870i	−0.321840	−	0.946794i	\(-0.604301\pi\)
							0.897060	−	0.441909i	\(-0.145699\pi\)
\(3\)	1.03290	−	0.690159i	0.596343	−	0.398464i	−0.220447	−	0.975399i	\(-0.570752\pi\)
							0.816790	+	0.576935i	\(0.195752\pi\)
\(5\)	−1.32730	−	1.98644i	−0.593585	−	0.888363i	0.406091	−	0.913833i	\(-0.366892\pi\)
							−0.999676	+	0.0254700i	\(0.991892\pi\)
\(7\)	0.0574971	−	0.0860505i	0.0217319	−	0.0325240i	−0.820445	−	0.571726i	\(-0.806274\pi\)
							0.842176	+	0.539202i	\(0.181274\pi\)
\(11\)	1.10080	+	5.53411i	0.331904	+	1.66860i	0.681594	+	0.731730i	\(0.261287\pi\)
							−0.349690	+	0.936866i	\(0.613713\pi\)
\(13\)	2.79136	−	2.28217i	0.774184	−	0.632960i
							\(17\)	−3.65806	+	1.90226i	−0.887210	+	0.461366i
\(19\)	−0.203539	+	0.491387i	−0.0466951	+	0.112732i								−0.945506	−	0.325605i	\(-0.894432\pi\)
							0.898811	+	0.438337i	\(0.144432\pi\)
\(23\)	5.10103	−	7.63423i	1.06364	−	1.59185i	0.291302	−	0.956631i	\(-0.405911\pi\)
							0.772335	−	0.635215i	\(-0.219089\pi\)
\(29\)	0.945530	−	4.75350i	0.175581	−	0.882703i	−0.788080	−	0.615573i	\(-0.788925\pi\)
							0.963661	−	0.267130i	\(-0.0860753\pi\)
\(31\)	−2.08701	+	10.4921i	−0.374837	+	1.88443i	0.0849173	+	0.996388i	\(0.472937\pi\)
							−0.459755	+	0.888046i	\(0.652063\pi\)
\(37\)	−1.04169	+	5.23694i	−0.171253	+	0.860948i	0.795640	+	0.605769i	\(0.207134\pi\)
							−0.966894	+	0.255179i	\(0.917866\pi\)
\(41\)	−1.06484	−	0.711502i	−0.166300	−	0.111118i	0.469635	−	0.882860i	\(-0.344385\pi\)
							−0.635935	+	0.771743i	\(0.719385\pi\)
\(43\)	2.69865	+	1.11782i	0.411540	+	0.170466i	0.578841	−	0.815440i	\(-0.303505\pi\)
							−0.167301	+	0.985906i	\(0.553505\pi\)
\(47\)	−2.00950			−0.293116			−0.146558	−	0.989202i	\(-0.546819\pi\)
							−0.146558	+	0.989202i	\(0.546819\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	221.2.z.a.216.17	yes	152
13.5	odd	4		221.2.ba.a.148.3	yes	152
17.10	odd	16		221.2.ba.a.112.3	yes	152
221.44	even	16	inner	221.2.z.a.44.17	✓	152

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
221.2.z.a.44.17	✓	152	221.44	even	16	inner
221.2.z.a.216.17	yes	152	1.1	even	1	trivial
221.2.ba.a.112.3	yes	152	17.10	odd	16
221.2.ba.a.148.3	yes	152	13.5	odd	4