Embedded newform 164.2.o.b.71.3

• Label: 164.2.o.b.71.3
• Level: $164$
• Weight: $2$
• Character: 164.71
• 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			71.3
Character	\(\chi\)	\(=\)	164.71
Dual form			164.2.o.b.67.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.21706 + 0.720261i) q^{2} +(-2.70129 - 1.11891i) q^{3} +(0.962447 - 1.75320i) q^{4} +(0.127570 - 0.250370i) q^{5} +(4.09353 - 0.583858i) q^{6} +(-1.61669 + 0.388132i) q^{7} +(0.0914077 + 2.82695i) q^{8} +(3.92368 + 3.92368i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 288 q - 12 q^{2} - 20 q^{4} - 32 q^{5} - 28 q^{6} - 24 q^{8} - 40 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(83\)	\(129\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{23}{40}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.21706	+	0.720261i	−0.860588	+	0.509302i
							\(3\)	−2.70129	−	1.11891i	−1.55959	−	0.646003i	−0.574572	−	0.818454i	\(-0.694831\pi\)
−0.985018	+	0.172451i	\(0.944831\pi\)
\(5\)	0.127570	−	0.250370i	0.0570510	−	0.111969i	−0.860720	−	0.509078i	\(-0.829987\pi\)
							0.917771	+	0.397109i	\(0.129987\pi\)
\(7\)	−1.61669	+	0.388132i	−0.611050	+	0.146700i	−0.527148	−	0.849774i	\(-0.676738\pi\)
							−0.0839027	+	0.996474i	\(0.526738\pi\)
\(11\)	3.24090	+	2.76799i	0.977167	+	0.834579i	0.986230	−	0.165380i	\(-0.0528850\pi\)
							−0.00906324	+	0.999959i	\(0.502885\pi\)
\(13\)	−5.63952	+	3.45590i	−1.56412	+	0.958495i	−0.573910	+	0.818919i	\(0.694574\pi\)
							−0.990211	+	0.139576i	\(0.955426\pi\)
\(17\)	3.49754	−	0.275262i	0.848278	−	0.0667609i	0.353135	−	0.935572i	\(-0.385116\pi\)
							0.495143	+	0.868812i	\(0.335116\pi\)
\(19\)	−2.44503	+	3.98992i	−0.560927	+	0.915350i	0.438931	+	0.898521i	\(0.355357\pi\)
							−0.999858	+	0.0168296i	\(0.994643\pi\)
\(23\)	2.69833	+	1.96045i	0.562640	+	0.408782i	0.832424	−	0.554139i	\(-0.186952\pi\)
							−0.269784	+	0.962921i	\(0.586952\pi\)
\(29\)	−3.01252	−	0.237090i	−0.559411	−	0.0440266i	−0.204398	−	0.978888i	\(-0.565524\pi\)
							−0.355013	+	0.934861i	\(0.615524\pi\)
\(31\)	0.632612	−	1.94698i	0.113620	−	0.349688i	−0.878036	−	0.478594i	\(-0.841147\pi\)
							0.991657	+	0.128906i	\(0.0411466\pi\)
\(37\)	−1.81658	−	5.59087i	−0.298645	−	0.919134i	−0.981973	−	0.189022i	\(-0.939468\pi\)
							0.683328	−	0.730111i	\(-0.260532\pi\)
\(41\)	−5.59237	+	3.11856i	−0.873381	+	0.487037i
							\(43\)	0.404378	+	2.55315i	0.0616671	+	0.389351i	0.999146	+	0.0413098i	\(0.0131531\pi\)
−0.937479	+	0.348041i	\(0.886847\pi\)
\(47\)	−0.0461094	−	0.0110699i	−0.00672575	−	0.00161471i	0.230081	−	0.973171i	\(-0.426101\pi\)
							−0.236807	+	0.971557i	\(0.576101\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	164.2.o.b.71.3	yes	288
4.3	odd	2	inner	164.2.o.b.71.1	yes	288
41.26	odd	40	inner	164.2.o.b.67.1	✓	288
164.67	even	40	inner	164.2.o.b.67.3	yes	288

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
164.2.o.b.67.1	✓	288	41.26	odd	40	inner
164.2.o.b.67.3	yes	288	164.67	even	40	inner
164.2.o.b.71.1	yes	288	4.3	odd	2	inner
164.2.o.b.71.3	yes	288	1.1	even	1	trivial