Embedded newform 164.2.o.b.63.13

• Label: 164.2.o.b.63.13
• Level: $164$
• Weight: $2$
• Character: 164.63
• 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			63.13
Character	\(\chi\)	\(=\)	164.63
Dual form			164.2.o.b.151.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.607797 - 1.27694i) q^{2} +(-0.536253 - 1.29463i) q^{3} +(-1.26117 - 1.55224i) q^{4} +(-2.42661 + 0.384338i) q^{5} +(-1.97910 - 0.102107i) q^{6} +(1.23965 - 1.05876i) q^{7} +(-2.74866 + 0.666988i) q^{8} +(0.732822 - 0.732822i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 288 q - 12 q^{2} - 20 q^{4} - 32 q^{5} - 28 q^{6} - 24 q^{8} - 40 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(83\)	\(129\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{29}{40}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.607797	−	1.27694i	0.429777	−	0.902935i
							\(3\)	−0.536253	−	1.29463i	−0.309606	−	0.747455i	−0.999718	−	0.0237521i	\(-0.992439\pi\)
0.690112	−	0.723703i	\(-0.257561\pi\)
\(5\)	−2.42661	+	0.384338i	−1.08521	+	0.171881i	−0.673320	−	0.739351i	\(-0.735133\pi\)
							−0.411894	+	0.911232i	\(0.635133\pi\)
\(7\)	1.23965	−	1.05876i	0.468543	−	0.400174i	−0.383586	−	0.923505i	\(-0.625311\pi\)
							0.852129	+	0.523331i	\(0.175311\pi\)
\(11\)	−1.25313	+	2.04493i	−0.377834	+	0.616570i	−0.983874	−	0.178865i	\(-0.942757\pi\)
							0.606039	+	0.795435i	\(0.292757\pi\)
\(13\)	−0.384245	−	4.88229i	−0.106570	−	1.35410i	−0.785026	−	0.619463i	\(-0.787350\pi\)
							0.678456	−	0.734641i	\(-0.262650\pi\)
\(17\)	3.19707	−	0.767548i	0.775403	−	0.186158i	0.173615	−	0.984814i	\(-0.444455\pi\)
							0.601788	+	0.798656i	\(0.294455\pi\)
\(19\)	5.60081	+	0.440793i	1.28491	+	0.101125i	0.702451	−	0.711732i	\(-0.252089\pi\)
							0.582463	+	0.812857i	\(0.302089\pi\)
\(23\)	0.304199	−	0.936228i	0.0634299	−	0.195217i	−0.914319	−	0.404994i	\(-0.867274\pi\)
							0.977749	+	0.209777i	\(0.0672737\pi\)
\(29\)	3.88623	+	0.933001i	0.721655	+	0.173254i	0.577605	−	0.816316i	\(-0.303987\pi\)
							0.144050	+	0.989570i	\(0.453987\pi\)
\(31\)	5.92894	−	4.30763i	1.06487	−	0.773673i	0.0898859	−	0.995952i	\(-0.471350\pi\)
							0.974983	+	0.222280i	\(0.0713498\pi\)
\(37\)	−7.90526	−	5.74351i	−1.29962	−	0.944227i	−0.299666	−	0.954044i	\(-0.596875\pi\)
							−0.999952	+	0.00981690i	\(0.996875\pi\)
\(41\)	1.79022	+	6.14777i	0.279585	+	0.960121i
							\(43\)	1.81302	+	3.55824i	0.276482	+	0.542627i	0.986935	−	0.161118i	\(-0.0515099\pi\)
−0.710453	+	0.703745i	\(0.751510\pi\)
\(47\)	4.14925	+	3.54380i	0.605231	+	0.516916i	0.898419	−	0.439139i	\(-0.144716\pi\)
							−0.293189	+	0.956055i	\(0.594716\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	164.2.o.b.63.13	✓	288
4.3	odd	2	inner	164.2.o.b.63.17	yes	288
41.28	odd	40	inner	164.2.o.b.151.17	yes	288
164.151	even	40	inner	164.2.o.b.151.13	yes	288

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
164.2.o.b.63.13	✓	288	1.1	even	1	trivial
164.2.o.b.63.17	yes	288	4.3	odd	2	inner
164.2.o.b.151.13	yes	288	164.151	even	40	inner
164.2.o.b.151.17	yes	288	41.28	odd	40	inner