Embedded newform 164.2.i.b.55.17

• Label: 164.2.i.b.55.17
• Level: $164$
• Weight: $2$
• Character: 164.55
• Analytic conductor: $1.310$
• Analytic rank: $0$
• Dimension: $72$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 164 = 2^{2} \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	164.i (of order \(8\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.30954659315\)
Analytic rank:	\(0\)
Dimension:	\(72\)
Relative dimension:	\(18\) over \(\Q(\zeta_{8})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{8}]$

Embedding invariants

Embedding label			55.17
Character	\(\chi\)	\(=\)	164.55
Dual form			164.2.i.b.3.17

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.27972 - 0.601931i) q^{2} +(0.337920 - 0.139971i) q^{3} +(1.27536 - 1.54060i) q^{4} +(1.59866 - 1.59866i) q^{5} +(0.348189 - 0.382528i) q^{6} +(-3.95387 + 1.63775i) q^{7} +(0.704761 - 2.73922i) q^{8} +(-2.02672 + 2.02672i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 72 q - 8 q^{2} - 8 q^{5} + 8 q^{6} + 4 q^{8}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/164\mathbb{Z}\right)^\times\).

\(n\)	\(83\)	\(129\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{5}{8}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.27972	−	0.601931i	0.904898	−	0.425630i
							\(3\)	0.337920	−	0.139971i	0.195098	−	0.0808123i	−0.282995	−	0.959121i	\(-0.591328\pi\)
0.478093	+	0.878309i	\(0.341328\pi\)
\(5\)	1.59866	−	1.59866i	0.714944	−	0.714944i	−0.252621	−	0.967565i	\(-0.581293\pi\)
							0.967565	+	0.252621i	\(0.0812927\pi\)
\(7\)	−3.95387	+	1.63775i	−1.49442	+	0.619010i	−0.972274	−	0.233845i	\(-0.924869\pi\)
							−0.522148	+	0.852855i	\(0.674869\pi\)
\(11\)	0.482129	+	1.16396i	0.145367	+	0.350948i	0.979746	−	0.200244i	\(-0.0641734\pi\)
							−0.834379	+	0.551191i	\(0.814173\pi\)
\(13\)	1.71388	+	4.13766i	0.475344	+	1.14758i	0.961770	+	0.273860i	\(0.0883004\pi\)
							−0.486426	+	0.873722i	\(0.661700\pi\)
\(17\)	1.72482	−	4.16409i	0.418331	−	1.00994i	−0.564500	−	0.825433i	\(-0.690931\pi\)
							0.982831	−	0.184508i	\(-0.0590691\pi\)
\(19\)	1.71167	+	0.708996i	0.392684	+	0.162655i	0.570284	−	0.821448i	\(-0.306833\pi\)
							−0.177600	+	0.984103i	\(0.556833\pi\)
\(23\)	0.562036			0.117193			0.0585963	−	0.998282i	\(-0.481338\pi\)
							0.0585963	+	0.998282i	\(0.481338\pi\)
\(29\)	−0.0734894	−	0.177419i	−0.0136466	−	0.0329459i	0.916909	−	0.399096i	\(-0.130676\pi\)
							−0.930556	+	0.366150i	\(0.880676\pi\)
\(31\)	−8.53976			−1.53379			−0.766893	−	0.641775i	\(-0.778198\pi\)
							−0.766893	+	0.641775i	\(0.778198\pi\)
\(37\)	−3.69809			−0.607962			−0.303981	−	0.952678i	\(-0.598316\pi\)
							−0.303981	+	0.952678i	\(0.598316\pi\)
\(41\)	−6.23692	−	1.44943i	−0.974043	−	0.226364i
							\(43\)	8.35977	−	8.35977i	1.27485	−	1.27485i	0.331343	−	0.943510i	\(-0.392498\pi\)
0.943510	−	0.331343i	\(-0.107502\pi\)
\(47\)	8.03647	+	3.32881i	1.17224	+	0.485558i	0.881932	−	0.471376i	\(-0.156243\pi\)
							0.290307	+	0.956934i	\(0.406243\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	164.2.i.b.55.17	yes	72
4.3	odd	2	inner	164.2.i.b.55.8	yes	72
41.3	odd	8	inner	164.2.i.b.3.8	✓	72
164.3	even	8	inner	164.2.i.b.3.17	yes	72

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
164.2.i.b.3.8	✓	72	41.3	odd	8	inner
164.2.i.b.3.17	yes	72	164.3	even	8	inner
164.2.i.b.55.8	yes	72	4.3	odd	2	inner
164.2.i.b.55.17	yes	72	1.1	even	1	trivial