Embedded newform 164.2.o.b.11.8

• Label: 164.2.o.b.11.8
• Level: $164$
• Weight: $2$
• Character: 164.11
• Analytic conductor: $1.310$
• Analytic rank: $0$
• Dimension: $288$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 164 = 2^{2} \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	164.o (of order \(40\), degree \(16\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.30954659315\)
Analytic rank:	\(0\)
Dimension:	\(288\)
Relative dimension:	\(18\) over \(\Q(\zeta_{40})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{40}]$

Embedding invariants

Embedding label			11.8
Character	\(\chi\)	\(=\)	164.11
Dual form			164.2.o.b.15.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.360435 - 1.36751i) q^{2} +(0.992268 - 2.39555i) q^{3} +(-1.74017 + 0.985799i) q^{4} +(0.0861519 - 0.169083i) q^{5} +(-3.63359 - 0.493498i) q^{6} +(-0.388327 - 1.61750i) q^{7} +(1.97531 + 2.02439i) q^{8} +(-2.63273 - 2.63273i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 288 q - 12 q^{2} - 20 q^{4} - 32 q^{5} - 28 q^{6} - 24 q^{8} - 40 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(83\)	\(129\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{3}{40}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.360435	−	1.36751i	−0.254866	−	0.966976i
							\(3\)	0.992268	−	2.39555i	0.572886	−	1.38307i	−0.326201	−	0.945301i	\(-0.605769\pi\)
0.899087	−	0.437770i	\(-0.144231\pi\)
\(5\)	0.0861519	−	0.169083i	0.0385283	−	0.0756161i	−0.870940	−	0.491390i	\(-0.836489\pi\)
							0.909468	+	0.415774i	\(0.136489\pi\)
\(7\)	−0.388327	−	1.61750i	−0.146774	−	0.611357i	−0.996460	−	0.0840681i	\(-0.973209\pi\)
							0.849686	−	0.527289i	\(-0.176791\pi\)
\(11\)	−0.130843	+	0.153197i	−0.0394506	+	0.0461908i	−0.779780	−	0.626053i	\(-0.784669\pi\)
							0.740330	+	0.672244i	\(0.234669\pi\)
\(13\)	0.0548598	+	0.0895230i	0.0152154	+	0.0248292i	0.860157	−	0.510029i	\(-0.170365\pi\)
							−0.844942	+	0.534858i	\(0.820365\pi\)
\(17\)	0.293344	+	3.72728i	0.0711463	+	0.903999i	0.924134	+	0.382068i	\(0.124788\pi\)
							−0.852988	+	0.521931i	\(0.825212\pi\)
\(19\)	3.89273	+	2.38547i	0.893053	+	0.547264i	0.891645	−	0.452735i	\(-0.149552\pi\)
							0.00140827	+	0.999999i	\(0.499552\pi\)
\(23\)	−4.61318	−	3.35167i	−0.961915	−	0.698872i	−0.00832035	−	0.999965i	\(-0.502648\pi\)
							−0.953595	+	0.301093i	\(0.902648\pi\)
\(29\)	0.459163	−	5.83421i	0.0852643	−	1.08339i	−0.794275	−	0.607559i	\(-0.792149\pi\)
							0.879539	−	0.475827i	\(-0.157851\pi\)
\(31\)	3.12589	−	9.62049i	0.561426	−	1.72789i	−0.116914	−	0.993142i	\(-0.537300\pi\)
							0.678339	−	0.734749i	\(-0.262700\pi\)
\(37\)	2.93725	+	9.03994i	0.482881	+	1.48616i	0.835026	+	0.550211i	\(0.185453\pi\)
							−0.352145	+	0.935946i	\(0.614547\pi\)
\(41\)	3.90171	−	5.07707i	0.609345	−	0.792905i
							\(43\)	0.410737	+	2.59329i	0.0626368	+	0.395473i	0.999011	+	0.0444693i	\(0.0141597\pi\)
−0.936374	+	0.351004i	\(0.885840\pi\)
\(47\)	−2.73427	+	11.3891i	−0.398835	+	1.66127i	0.306105	+	0.951998i	\(0.400974\pi\)
							−0.704939	+	0.709268i	\(0.749026\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	164.2.o.b.11.8	✓	288
4.3	odd	2	inner	164.2.o.b.11.11	yes	288
41.15	odd	40	inner	164.2.o.b.15.11	yes	288
164.15	even	40	inner	164.2.o.b.15.8	yes	288

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
164.2.o.b.11.8	✓	288	1.1	even	1	trivial
164.2.o.b.11.11	yes	288	4.3	odd	2	inner
164.2.o.b.15.8	yes	288	164.15	even	40	inner
164.2.o.b.15.11	yes	288	41.15	odd	40	inner