Embedded newform 164.2.o.b.7.1

• Label: 164.2.o.b.7.1
• Level: $164$
• Weight: $2$
• Character: 164.7
• 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			7.1
Character	\(\chi\)	\(=\)	164.7
Dual form			164.2.o.b.47.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.41078 + 0.0984343i) q^{2} +(-1.75033 - 0.725011i) q^{3} +(1.98062 - 0.277739i) q^{4} +(0.186813 + 1.17949i) q^{5} +(2.54071 + 0.850542i) q^{6} +(0.145426 - 1.84782i) q^{7} +(-2.76689 + 0.586791i) q^{8} +(0.416702 + 0.416702i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 288 q - 12 q^{2} - 20 q^{4} - 32 q^{5} - 28 q^{6} - 24 q^{8} - 40 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(83\)	\(129\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{39}{40}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.41078	+	0.0984343i	−0.997575	+	0.0696035i
							\(3\)	−1.75033	−	0.725011i	−1.01055	−	0.418586i	−0.184898	−	0.982758i	\(-0.559195\pi\)
−0.825657	+	0.564172i	\(0.809195\pi\)
\(5\)	0.186813	+	1.17949i	0.0835452	+	0.527483i	0.993596	+	0.112988i	\(0.0360420\pi\)
							−0.910051	+	0.414496i	\(0.863958\pi\)
\(7\)	0.145426	−	1.84782i	0.0549660	−	0.698409i	−0.906273	−	0.422693i	\(-0.861085\pi\)
							0.961239	−	0.275717i	\(-0.0889151\pi\)
\(11\)	−0.146490	−	0.610176i	−0.0441685	−	0.183975i	0.945925	−	0.324386i	\(-0.105158\pi\)
							−0.990093	+	0.140411i	\(0.955158\pi\)
\(13\)	−4.66385	−	3.98331i	−1.29352	−	1.10477i	−0.988306	−	0.152487i	\(-0.951272\pi\)
							−0.305214	−	0.952284i	\(-0.598728\pi\)
\(17\)	−2.93304	−	1.79737i	−0.711368	−	0.435927i	0.119181	−	0.992873i	\(-0.461973\pi\)
							−0.830549	+	0.556946i	\(0.811973\pi\)
\(19\)	−4.02606	−	4.71391i	−0.923642	−	1.08145i	−0.996457	−	0.0841000i	\(-0.973198\pi\)
							0.0728157	−	0.997345i	\(-0.476802\pi\)
\(23\)	−0.173741	+	0.534719i	−0.0362274	+	0.111497i	−0.967535	−	0.252737i	\(-0.918669\pi\)
							0.931307	+	0.364234i	\(0.118669\pi\)
\(29\)	0.0538392	−	0.0329927i	0.00999769	−	0.00612659i	−0.517491	−	0.855689i	\(-0.673134\pi\)
							0.527488	+	0.849562i	\(0.323134\pi\)
\(31\)	2.49334	−	1.81152i	0.447817	−	0.325358i	−0.340916	−	0.940094i	\(-0.610737\pi\)
							0.788733	+	0.614735i	\(0.210737\pi\)
\(37\)	1.77641	+	1.29064i	0.292040	+	0.212179i	0.724152	−	0.689641i	\(-0.242231\pi\)
							−0.432112	+	0.901820i	\(0.642231\pi\)
\(41\)	0.857221	+	6.34548i	0.133875	+	0.990998i
							\(43\)	−6.40071	+	3.26133i	−0.976100	+	0.497348i	−0.867877	−	0.496779i	\(-0.834516\pi\)
−0.108223	+	0.994127i	\(0.534516\pi\)
\(47\)	−0.821840	−	10.4425i	−0.119878	−	1.52319i	−0.703225	−	0.710967i	\(-0.748257\pi\)
							0.583348	−	0.812223i	\(-0.301743\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	164.2.o.b.7.1	✓	288
4.3	odd	2	inner	164.2.o.b.7.14	yes	288
41.6	odd	40	inner	164.2.o.b.47.14	yes	288
164.47	even	40	inner	164.2.o.b.47.1	yes	288

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
164.2.o.b.7.1	✓	288	1.1	even	1	trivial
164.2.o.b.7.14	yes	288	4.3	odd	2	inner
164.2.o.b.47.1	yes	288	164.47	even	40	inner
164.2.o.b.47.14	yes	288	41.6	odd	40	inner