Embedded newform 164.2.o.b.11.4

• Label: 164.2.o.b.11.4
• Level: $164$
• Weight: $2$
• Character: 164.11
• 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			11.4
Character	\(\chi\)	\(=\)	164.11
Dual form			164.2.o.b.15.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.16191 - 0.806201i) q^{2} +(0.107548 - 0.259643i) q^{3} +(0.700081 + 1.87347i) q^{4} +(0.160579 - 0.315153i) q^{5} +(-0.334286 + 0.214978i) q^{6} +(-0.730639 - 3.04333i) q^{7} +(0.696960 - 2.74121i) q^{8} +(2.06547 + 2.06547i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 288 q - 12 q^{2} - 20 q^{4} - 32 q^{5} - 28 q^{6} - 24 q^{8} - 40 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(83\)	\(129\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{3}{40}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.16191	−	0.806201i	−0.821596	−	0.570070i
							\(3\)	0.107548	−	0.259643i	0.0620928	−	0.149905i	−0.889788	−	0.456375i	\(-0.849148\pi\)
0.951881	+	0.306469i	\(0.0991477\pi\)
\(5\)	0.160579	−	0.315153i	0.0718129	−	0.140941i	−0.852309	−	0.523039i	\(-0.824798\pi\)
							0.924122	+	0.382098i	\(0.124798\pi\)
\(7\)	−0.730639	−	3.04333i	−0.276156	−	1.15027i	−0.922120	−	0.386904i	\(-0.873545\pi\)
							0.645965	−	0.763367i	\(-0.276455\pi\)
\(11\)	2.95352	−	3.45813i	0.890521	−	1.04267i	−0.108300	−	0.994118i	\(-0.534541\pi\)
							0.998821	−	0.0485474i	\(-0.0154592\pi\)
\(13\)	0.756418	+	1.23436i	0.209793	+	0.342350i	0.940177	−	0.340686i	\(-0.110659\pi\)
							−0.730385	+	0.683036i	\(0.760659\pi\)
\(17\)	−0.587614	−	7.46634i	−0.142517	−	1.81085i	−0.488447	−	0.872593i	\(-0.662436\pi\)
							0.345930	−	0.938260i	\(-0.387564\pi\)
\(19\)	−3.57340	−	2.18978i	−0.819794	−	0.502370i	0.0482633	−	0.998835i	\(-0.484631\pi\)
							−0.868057	+	0.496464i	\(0.834631\pi\)
\(23\)	1.40629	+	1.02173i	0.293233	+	0.213046i	0.724669	−	0.689098i	\(-0.241993\pi\)
							−0.431436	+	0.902144i	\(0.641993\pi\)
\(29\)	−0.426203	+	5.41542i	−0.0791439	+	1.00562i	0.821257	+	0.570558i	\(0.193273\pi\)
							−0.900401	+	0.435060i	\(0.856727\pi\)
\(31\)	−2.29687	+	7.06903i	−0.412530	+	1.26964i	0.501912	+	0.864919i	\(0.332630\pi\)
							−0.914442	+	0.404717i	\(0.867370\pi\)
\(37\)	−0.0407975	−	0.125562i	−0.00670707	−	0.0206423i	0.947647	−	0.319320i	\(-0.103454\pi\)
							−0.954354	+	0.298678i	\(0.903454\pi\)
\(41\)	4.51550	−	4.53985i	0.705203	−	0.709005i
							\(43\)	1.66707	+	10.5255i	0.254226	+	1.60512i	0.702811	+	0.711377i	\(0.251928\pi\)
−0.448585	+	0.893740i	\(0.648072\pi\)
\(47\)	−0.405424	+	1.68871i	−0.0591371	+	0.246324i	−0.994107	−	0.108403i	\(-0.965426\pi\)
							0.934970	+	0.354727i	\(0.115426\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	164.2.o.b.11.4	✓	288
4.3	odd	2	inner	164.2.o.b.11.6	yes	288
41.15	odd	40	inner	164.2.o.b.15.6	yes	288
164.15	even	40	inner	164.2.o.b.15.4	yes	288

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
164.2.o.b.11.4	✓	288	1.1	even	1	trivial
164.2.o.b.11.6	yes	288	4.3	odd	2	inner
164.2.o.b.15.4	yes	288	164.15	even	40	inner
164.2.o.b.15.6	yes	288	41.15	odd	40	inner