Embedded newform 221.2.ba.a.5.18

• Label: 221.2.ba.a.5.18
• Level: $221$
• Weight: $2$
• Character: 221.5
• 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.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			5.18
Character	\(\chi\)	\(=\)	221.5
Dual form			221.2.ba.a.177.18

$q$-expansion

\(f(q)\)	\(=\)	\(q+(2.09964 + 0.869700i) q^{2} +(-0.864818 - 1.29429i) q^{3} +(2.23790 + 2.23790i) q^{4} +(0.867285 + 1.29798i) q^{5} +(-0.690162 - 3.46968i) q^{6} +(0.864770 - 1.29422i) q^{7} +(1.01309 + 2.44582i) q^{8} +(0.220771 - 0.532988i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 152 q - 8 q^{2} - 16 q^{3} - 8 q^{5} - 8 q^{6} - 8 q^{7} + 8 q^{8} - 16 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/221\mathbb{Z}\right)^\times\).

\(n\)	\(105\)	\(171\)
\(\chi(n)\)	\(e\left(\frac{5}{16}\right)\)	\(e\left(\frac{3}{4}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	2.09964	+	0.869700i	1.48467	+	0.614971i	0.970150	−	0.242507i	\(-0.0779696\pi\)
							0.514521	+	0.857478i	\(0.327970\pi\)
\(3\)	−0.864818	−	1.29429i	−0.499303	−	0.747259i	0.493142	−	0.869949i	\(-0.335848\pi\)
							−0.992445	+	0.122689i	\(0.960848\pi\)
\(5\)	0.867285	+	1.29798i	0.387862	+	0.580476i	0.973099	−	0.230386i	\(-0.0739989\pi\)
							−0.585238	+	0.810862i	\(0.698999\pi\)
\(7\)	0.864770	−	1.29422i	0.326853	−	0.489169i	−0.631256	−	0.775574i	\(-0.717460\pi\)
							0.958109	+	0.286405i	\(0.0924603\pi\)
\(11\)	1.12148	+	5.63807i	0.338139	+	1.69994i	0.658446	+	0.752628i	\(0.271214\pi\)
							−0.320307	+	0.947314i	\(0.603786\pi\)
\(13\)	−2.75680	−	2.32380i	−0.764600	−	0.644505i
							\(17\)	−3.48774	+	2.19902i	−0.845900	+	0.533341i
\(19\)	−4.09276	−	1.69528i	−0.938944	−	0.388923i								−0.139879	−	0.990169i	\(-0.544671\pi\)
							−0.799065	+	0.601245i	\(0.794671\pi\)
\(23\)	3.37928	+	2.25796i	0.704628	+	0.470818i	0.855545	−	0.517729i	\(-0.173222\pi\)
							−0.150916	+	0.988547i	\(0.548222\pi\)
\(29\)	−5.78029	−	1.14977i	−1.07337	−	0.213507i	−0.373396	−	0.927672i	\(-0.621807\pi\)
							−0.699977	+	0.714165i	\(0.746807\pi\)
\(31\)	−0.0898618	+	0.451766i	−0.0161397	+	0.0811395i	−0.988016	−	0.154354i	\(-0.950671\pi\)
							0.971876	+	0.235493i	\(0.0756705\pi\)
\(37\)	0.253362	−	1.27373i	0.0416524	−	0.209401i	−0.954355	−	0.298676i	\(-0.903455\pi\)
							0.996007	+	0.0892752i	\(0.0284550\pi\)
\(41\)	−0.718783	−	0.480275i	−0.112255	−	0.0750064i	0.498175	−	0.867076i	\(-0.334004\pi\)
							−0.610430	+	0.792070i	\(0.709004\pi\)
\(43\)	−0.398771	−	0.165176i	−0.0608120	−	0.0251892i	0.352070	−	0.935974i	\(-0.385478\pi\)
							−0.412882	+	0.910784i	\(0.635478\pi\)
\(47\)			4.53080i			0.660886i	0.943826	+	0.330443i	\(0.107198\pi\)
							−0.943826	+	0.330443i	\(0.892802\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	221.2.ba.a.5.18	yes	152
13.8	odd	4		221.2.z.a.73.2	✓	152
17.7	odd	16		221.2.z.a.109.2	yes	152
221.177	even	16	inner	221.2.ba.a.177.18	yes	152

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
221.2.z.a.73.2	✓	152	13.8	odd	4
221.2.z.a.109.2	yes	152	17.7	odd	16
221.2.ba.a.5.18	yes	152	1.1	even	1	trivial
221.2.ba.a.177.18	yes	152	221.177	even	16	inner