Embedded newform 164.2.o.b.11.11

• Label: 164.2.o.b.11.11
• Level: $164$
• Weight: $2$
• Character: 164.11
• Analytic conductor: $1.310$
• Analytic rank: $0$
• Dimension: $288$
• 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			11.11
Character	\(\chi\)	\(=\)	164.11
Dual form			164.2.o.b.15.11

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.0797897 + 1.41196i) q^{2} +(-0.992268 + 2.39555i) q^{3} +(-1.98727 + 0.225320i) q^{4} +(0.0861519 - 0.169083i) q^{5} +(-3.46159 - 1.20990i) q^{6} +(0.388327 + 1.61750i) q^{7} +(-0.476706 - 2.78797i) q^{8} +(-2.63273 - 2.63273i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 288 q - 12 q^{2} - 20 q^{4} - 32 q^{5} - 28 q^{6} - 24 q^{8} - 40 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.0797897	+	1.41196i	0.0564198	+	0.998407i
							\(3\)	−0.992268	+	2.39555i	−0.572886	+	1.38307i	0.326201	+	0.945301i	\(0.394231\pi\)
−0.899087	+	0.437770i	\(0.855769\pi\)
\(5\)	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	\(0.0267913\pi\)
							−0.849686	+	0.527289i	\(0.823209\pi\)
\(11\)	0.130843	−	0.153197i	0.0394506	−	0.0461908i	−0.740330	−	0.672244i	\(-0.765331\pi\)
							0.779780	+	0.626053i	\(0.215331\pi\)
\(13\)	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	−0.00140827	−	0.999999i	\(-0.500448\pi\)
							−0.891645	+	0.452735i	\(0.850448\pi\)
\(23\)	4.61318	+	3.35167i	0.961915	+	0.698872i	0.953595	−	0.301093i	\(-0.0973515\pi\)
							0.00832035	+	0.999965i	\(0.497352\pi\)
\(29\)	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	\(0.462700\pi\)
							−0.678339	+	0.734749i	\(0.737300\pi\)
\(37\)	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	\(-0.985840\pi\)
0.936374	−	0.351004i	\(-0.114160\pi\)
\(47\)	2.73427	−	11.3891i	0.398835	−	1.66127i	−0.306105	−	0.951998i	\(-0.599026\pi\)
							0.704939	−	0.709268i	\(-0.250974\pi\)

(See \(a_n\) instead)

Twists

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

    

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