Embedded newform 164.2.o.b.71.1

• Label: 164.2.o.b.71.1
• 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			71.1
Character	\(\chi\)	\(=\)	164.71
Dual form			164.2.o.b.67.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.38006 - 0.308918i) q^{2} +(2.70129 + 1.11891i) q^{3} +(1.80914 + 0.852653i) q^{4} +(0.127570 - 0.250370i) q^{5} +(-3.38229 - 2.37864i) q^{6} +(1.61669 - 0.388132i) q^{7} +(-2.23332 - 1.73559i) q^{8} +(3.92368 + 3.92368i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 288 q - 12 q^{2} - 20 q^{4} - 32 q^{5} - 28 q^{6} - 24 q^{8} - 40 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.38006	−	0.308918i	−0.975851	−	0.218438i
							\(3\)	2.70129	+	1.11891i	1.55959	+	0.646003i	0.985018	−	0.172451i	\(-0.0551686\pi\)
0.574572	+	0.818454i	\(0.305169\pi\)
\(5\)	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	0.0839027	−	0.996474i	\(-0.473262\pi\)
							0.527148	+	0.849774i	\(0.323262\pi\)
\(11\)	−3.24090	−	2.76799i	−0.977167	−	0.834579i	0.00906324	−	0.999959i	\(-0.497115\pi\)
							−0.986230	+	0.165380i	\(0.947115\pi\)
\(13\)	−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	\(-0.644643\pi\)
							0.999858	−	0.0168296i	\(-0.00535727\pi\)
\(23\)	−2.69833	−	1.96045i	−0.562640	−	0.408782i	0.269784	−	0.962921i	\(-0.413048\pi\)
							−0.832424	+	0.554139i	\(0.813048\pi\)
\(29\)	−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	−0.991657	−	0.128906i	\(-0.958853\pi\)
							0.878036	+	0.478594i	\(0.158853\pi\)
\(37\)	−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	\(-0.986847\pi\)
0.937479	−	0.348041i	\(-0.113153\pi\)
\(47\)	0.0461094	+	0.0110699i	0.00672575	+	0.00161471i	0.236807	−	0.971557i	\(-0.423899\pi\)
							−0.230081	+	0.973171i	\(0.573899\pi\)

(See \(a_n\) instead)

Twists

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

    

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