Embedded newform 164.2.o.b.135.5

• Label: 164.2.o.b.135.5
• Level: $164$
• Weight: $2$
• Character: 164.135
• 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			135.5
Character	\(\chi\)	\(=\)	164.135
Dual form			164.2.o.b.147.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.04799 + 0.949588i) q^{2} +(-0.956948 + 2.31028i) q^{3} +(0.196566 - 1.99032i) q^{4} +(-0.300195 + 0.152957i) q^{5} +(-1.19094 - 3.32985i) q^{6} +(-3.46827 + 2.12536i) q^{7} +(1.68398 + 2.27249i) q^{8} +(-2.30031 - 2.30031i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 288 q - 12 q^{2} - 20 q^{4} - 32 q^{5} - 28 q^{6} - 24 q^{8} - 40 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(83\)	\(129\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{27}{40}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.04799	+	0.949588i	−0.741041	+	0.671460i
							\(3\)	−0.956948	+	2.31028i	−0.552494	+	1.33384i	0.363105	+	0.931748i	\(0.381717\pi\)
−0.915600	+	0.402091i	\(0.868283\pi\)
\(5\)	−0.300195	+	0.152957i	−0.134251	+	0.0684044i	−0.519826	−	0.854272i	\(-0.674003\pi\)
							0.385575	+	0.922677i	\(0.374003\pi\)
\(7\)	−3.46827	+	2.12536i	−1.31088	+	0.803310i	−0.989624	−	0.143681i	\(-0.954106\pi\)
							−0.321259	+	0.946991i	\(0.604106\pi\)
\(11\)	0.267701	−	3.40147i	0.0807150	−	1.02558i	−0.814563	−	0.580074i	\(-0.803023\pi\)
							0.895278	−	0.445507i	\(-0.146977\pi\)
\(13\)	−2.06311	+	0.495308i	−0.572202	+	0.137374i	−0.509213	−	0.860641i	\(-0.670063\pi\)
							−0.0629895	+	0.998014i	\(0.520063\pi\)
\(17\)	−0.262937	−	0.307860i	−0.0637717	−	0.0746670i	0.727603	−	0.685998i	\(-0.240634\pi\)
							−0.791375	+	0.611331i	\(0.790634\pi\)
\(19\)	0.0388910	−	0.161993i	0.00892221	−	0.0371637i	−0.967731	−	0.251985i	\(-0.918917\pi\)
							0.976653	+	0.214821i	\(0.0689168\pi\)
\(23\)	2.58078	−	1.87505i	0.538130	−	0.390974i	−0.285260	−	0.958450i	\(-0.592080\pi\)
							0.823390	+	0.567476i	\(0.192080\pi\)
\(29\)	−5.71265	+	6.68865i	−1.06081	+	1.24205i	−0.0916070	+	0.995795i	\(0.529200\pi\)
							−0.969205	+	0.246256i	\(0.920800\pi\)
\(31\)	3.33979	+	10.2788i	0.599844	+	1.84613i	0.528964	+	0.848644i	\(0.322580\pi\)
							0.0708791	+	0.997485i	\(0.477420\pi\)
\(37\)	−1.96477	+	6.04695i	−0.323007	+	0.994112i	0.649326	+	0.760510i	\(0.275051\pi\)
							−0.972332	+	0.233602i	\(0.924949\pi\)
\(41\)	3.05900	−	5.62517i	0.477736	−	0.878504i
							\(43\)	−6.05122	−	0.958419i	−0.922802	−	0.146157i	−0.323085	−	0.946370i	\(-0.604720\pi\)
−0.599717	+	0.800212i	\(0.704720\pi\)
\(47\)	5.56578	+	3.41071i	0.811852	+	0.497504i	0.865412	−	0.501061i	\(-0.167057\pi\)
							−0.0535597	+	0.998565i	\(0.517057\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	164.2.o.b.135.5	✓	288
4.3	odd	2	inner	164.2.o.b.135.13	yes	288
41.24	odd	40	inner	164.2.o.b.147.13	yes	288
164.147	even	40	inner	164.2.o.b.147.5	yes	288

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
164.2.o.b.135.5	✓	288	1.1	even	1	trivial
164.2.o.b.135.13	yes	288	4.3	odd	2	inner
164.2.o.b.147.5	yes	288	164.147	even	40	inner
164.2.o.b.147.13	yes	288	41.24	odd	40	inner