Embedded newform 164.2.o.b.95.17

• Label: 164.2.o.b.95.17
• Level: $164$
• Weight: $2$
• Character: 164.95
• 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			95.17
Character	\(\chi\)	\(=\)	164.95
Dual form			164.2.o.b.19.17

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.34290 - 0.443407i) q^{2} +(-1.70788 - 0.707429i) q^{3} +(1.60678 - 1.19091i) q^{4} +(-0.612191 - 0.0969615i) q^{5} +(-2.60720 - 0.192720i) q^{6} +(2.49533 - 2.92165i) q^{7} +(1.62969 - 2.31173i) q^{8} +(0.295094 + 0.295094i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 288 q - 12 q^{2} - 20 q^{4} - 32 q^{5} - 28 q^{6} - 24 q^{8} - 40 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(83\)	\(129\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{31}{40}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.34290	−	0.443407i	0.949576	−	0.313536i
							\(3\)	−1.70788	−	0.707429i	−0.986048	−	0.408434i	−0.169385	−	0.985550i	\(-0.554178\pi\)
−0.816662	+	0.577116i	\(0.804178\pi\)
\(5\)	−0.612191	−	0.0969615i	−0.273780	−	0.0433625i	0.0180346	−	0.999837i	\(-0.494259\pi\)
							−0.291815	+	0.956475i	\(0.594259\pi\)
\(7\)	2.49533	−	2.92165i	0.943145	−	1.10428i	−0.0512637	−	0.998685i	\(-0.516325\pi\)
							0.994409	−	0.105596i	\(-0.0336751\pi\)
\(11\)	−4.78844	+	2.93436i	−1.44377	+	0.884743i	−0.999935	−	0.0113654i	\(-0.996382\pi\)
							−0.443835	+	0.896109i	\(0.646382\pi\)
\(13\)	6.40559	+	0.504131i	1.77659	+	0.139821i	0.923909	−	0.382611i	\(-0.124975\pi\)
							0.852681	+	0.522432i	\(0.174975\pi\)
\(17\)	−0.473405	+	1.97187i	−0.114818	+	0.478250i	0.885100	+	0.465401i	\(0.154090\pi\)
							−0.999917	+	0.0128486i	\(0.995910\pi\)
\(19\)	0.321114	+	4.08013i	0.0736685	+	0.936047i	0.917060	+	0.398748i	\(0.130555\pi\)
							−0.843392	+	0.537299i	\(0.819445\pi\)
\(23\)	0.249522	+	0.767951i	0.0520290	+	0.160129i	0.973695	−	0.227856i	\(-0.0731714\pi\)
							−0.921666	+	0.387984i	\(0.873171\pi\)
\(29\)	0.979818	+	4.08124i	0.181948	+	0.757866i	0.986470	+	0.163944i	\(0.0524215\pi\)
							−0.804522	+	0.593923i	\(0.797578\pi\)
\(31\)	4.06545	+	2.95373i	0.730177	+	0.530505i	0.889620	−	0.456702i	\(-0.150970\pi\)
							−0.159442	+	0.987207i	\(0.550970\pi\)
\(37\)	−5.44526	+	3.95621i	−0.895195	+	0.650397i	−0.937227	−	0.348719i	\(-0.886617\pi\)
							0.0420324	+	0.999116i	\(0.486617\pi\)
\(41\)	4.75858	−	4.28438i	0.743165	−	0.669108i
							\(43\)	2.29364	−	4.50151i	0.349776	−	0.686474i	−0.647353	−	0.762190i	\(-0.724124\pi\)
0.997129	+	0.0757158i	\(0.0241242\pi\)
\(47\)	−1.09120	−	1.27763i	−0.159168	−	0.186361i	0.675091	−	0.737734i	\(-0.264104\pi\)
							−0.834259	+	0.551373i	\(0.814104\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	164.2.o.b.95.17	yes	288
4.3	odd	2	inner	164.2.o.b.95.13	yes	288
41.19	odd	40	inner	164.2.o.b.19.13	✓	288
164.19	even	40	inner	164.2.o.b.19.17	yes	288

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
164.2.o.b.19.13	✓	288	41.19	odd	40	inner
164.2.o.b.19.17	yes	288	164.19	even	40	inner
164.2.o.b.95.13	yes	288	4.3	odd	2	inner
164.2.o.b.95.17	yes	288	1.1	even	1	trivial