Embedded newform 164.2.o.b.71.14

• Label: 164.2.o.b.71.14
• 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			71.14
Character	\(\chi\)	\(=\)	164.71
Dual form			164.2.o.b.67.14

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.788208 - 1.17419i) q^{2} +(-0.897189 - 0.371628i) q^{3} +(-0.757456 - 1.85102i) q^{4} +(0.598264 - 1.17416i) q^{5} +(-1.14353 + 0.760552i) q^{6} +(0.523162 - 0.125600i) q^{7} +(-2.77048 - 0.569587i) q^{8} +(-1.45448 - 1.45448i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 288 q - 12 q^{2} - 20 q^{4} - 32 q^{5} - 28 q^{6} - 24 q^{8} - 40 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.788208	−	1.17419i	0.557347	−	0.830279i
							\(3\)	−0.897189	−	0.371628i	−0.517992	−	0.214559i	0.108343	−	0.994114i	\(-0.465446\pi\)
−0.626335	+	0.779554i	\(0.715446\pi\)
\(5\)	0.598264	−	1.17416i	0.267552	−	0.525100i	−0.717670	−	0.696383i	\(-0.754791\pi\)
							0.985222	+	0.171283i	\(0.0547914\pi\)
\(7\)	0.523162	−	0.125600i	0.197737	−	0.0474724i	−0.133367	−	0.991067i	\(-0.542579\pi\)
							0.331104	+	0.943594i	\(0.392579\pi\)
\(11\)	1.62527	+	1.38812i	0.490039	+	0.418533i	0.859829	−	0.510583i	\(-0.170570\pi\)
							−0.369790	+	0.929115i	\(0.620570\pi\)
\(13\)	0.521560	−	0.319612i	0.144655	−	0.0886446i	−0.448288	−	0.893889i	\(-0.647966\pi\)
							0.592943	+	0.805245i	\(0.297966\pi\)
\(17\)	2.03503	−	0.160160i	0.493567	−	0.0388445i	0.170766	−	0.985312i	\(-0.445376\pi\)
							0.322800	+	0.946467i	\(0.395376\pi\)
\(19\)	1.57987	−	2.57812i	0.362448	−	0.591461i	−0.618488	−	0.785794i	\(-0.712254\pi\)
							0.980936	+	0.194333i	\(0.0622544\pi\)
\(23\)	3.79473	+	2.75703i	0.791255	+	0.574880i	0.908336	−	0.418242i	\(-0.137354\pi\)
							−0.117081	+	0.993122i	\(0.537354\pi\)
\(29\)	3.49845	+	0.275334i	0.649645	+	0.0511282i	0.399000	−	0.916951i	\(-0.369357\pi\)
							0.250645	+	0.968079i	\(0.419357\pi\)
\(31\)	−1.31399	+	4.04406i	−0.236000	+	0.726335i	0.760987	+	0.648767i	\(0.224715\pi\)
							−0.996987	+	0.0775671i	\(0.975285\pi\)
\(37\)	−1.75846	−	5.41198i	−0.289089	−	0.889724i	−0.985143	−	0.171735i	\(-0.945063\pi\)
							0.696054	−	0.717989i	\(-0.254937\pi\)
\(41\)	2.95463	+	5.68068i	0.461436	+	0.887173i
							\(43\)	−0.859744	−	5.42821i	−0.131110	−	0.827794i	−0.962337	−	0.271861i	\(-0.912361\pi\)
0.831227	−	0.555933i	\(-0.187639\pi\)
\(47\)	−2.27013	−	0.545011i	−0.331133	−	0.0794980i	0.0644699	−	0.997920i	\(-0.479464\pi\)
							−0.395603	+	0.918422i	\(0.629464\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	164.2.o.b.71.14	yes	288
4.3	odd	2	inner	164.2.o.b.71.16	yes	288
41.26	odd	40	inner	164.2.o.b.67.16	yes	288
164.67	even	40	inner	164.2.o.b.67.14	✓	288

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
164.2.o.b.67.14	✓	288	164.67	even	40	inner
164.2.o.b.67.16	yes	288	41.26	odd	40	inner
164.2.o.b.71.14	yes	288	1.1	even	1	trivial
164.2.o.b.71.16	yes	288	4.3	odd	2	inner