Embedded newform 164.2.o.b.63.12

• Label: 164.2.o.b.63.12
• 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			63.12
Character	\(\chi\)	\(=\)	164.63
Dual form			164.2.o.b.151.12

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.127129 + 1.40849i) q^{2} +(-0.874743 - 2.11182i) q^{3} +(-1.96768 + 0.358118i) q^{4} +(0.653439 - 0.103495i) q^{5} +(2.86326 - 1.50054i) q^{6} +(2.46815 - 2.10800i) q^{7} +(-0.754554 - 2.72592i) q^{8} +(-1.57328 + 1.57328i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 288 q - 12 q^{2} - 20 q^{4} - 32 q^{5} - 28 q^{6} - 24 q^{8} - 40 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.127129	+	1.40849i	0.0898935	+	0.995951i
							\(3\)	−0.874743	−	2.11182i	−0.505033	−	1.21926i	−0.946711	−	0.322084i	\(-0.895617\pi\)
0.441678	−	0.897174i	\(-0.354383\pi\)
\(5\)	0.653439	−	0.103495i	0.292227	−	0.0462842i	−0.00859975	−	0.999963i	\(-0.502737\pi\)
							0.300827	+	0.953679i	\(0.402737\pi\)
\(7\)	2.46815	−	2.10800i	0.932871	−	0.796747i	−0.0464789	−	0.998919i	\(-0.514800\pi\)
							0.979350	+	0.202172i	\(0.0648000\pi\)
\(11\)	1.81494	−	2.96172i	0.547226	−	0.892992i	−0.452773	−	0.891626i	\(-0.649565\pi\)
							0.999999	−	0.00136601i	\(-0.000434815\pi\)
\(13\)	0.0995435	+	1.26482i	0.0276084	+	0.350798i	0.995170	+	0.0981639i	\(0.0312969\pi\)
							−0.967562	+	0.252634i	\(0.918703\pi\)
\(17\)	3.29864	−	0.791934i	0.800038	−	0.192072i	0.187243	−	0.982314i	\(-0.440045\pi\)
							0.612795	+	0.790242i	\(0.290045\pi\)
\(19\)	−4.87618	−	0.383763i	−1.11867	−	0.0880413i	−0.494389	−	0.869240i	\(-0.664608\pi\)
							−0.624282	+	0.781199i	\(0.714608\pi\)
\(23\)	−2.21621	+	6.82078i	−0.462111	+	1.42223i	0.400469	+	0.916310i	\(0.368847\pi\)
							−0.862580	+	0.505921i	\(0.831153\pi\)
\(29\)	5.72758	+	1.37507i	1.06358	+	0.255344i	0.727232	−	0.686392i	\(-0.240806\pi\)
							0.336353	+	0.941736i	\(0.390806\pi\)
\(31\)	0.809308	−	0.587997i	0.145356	−	0.105607i	−0.512731	−	0.858550i	\(-0.671366\pi\)
							0.658087	+	0.752942i	\(0.271366\pi\)
\(37\)	−4.78445	−	3.47611i	−0.786558	−	0.571468i	0.120382	−	0.992728i	\(-0.461588\pi\)
							−0.906940	+	0.421260i	\(0.861588\pi\)
\(41\)	6.40152	+	0.143129i	0.999750	+	0.0223529i
							\(43\)	4.06332	+	7.97472i	0.619651	+	1.21613i	0.961092	+	0.276230i	\(0.0890852\pi\)
−0.341441	+	0.939903i	\(0.610915\pi\)
\(47\)	5.28138	+	4.51072i	0.770368	+	0.657957i	0.945145	−	0.326652i	\(-0.105920\pi\)
							−0.174777	+	0.984608i	\(0.555920\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	164.2.o.b.63.12	yes	288
4.3	odd	2	inner	164.2.o.b.63.6	✓	288
41.28	odd	40	inner	164.2.o.b.151.6	yes	288
164.151	even	40	inner	164.2.o.b.151.12	yes	288

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
164.2.o.b.63.6	✓	288	4.3	odd	2	inner
164.2.o.b.63.12	yes	288	1.1	even	1	trivial
164.2.o.b.151.6	yes	288	41.28	odd	40	inner
164.2.o.b.151.12	yes	288	164.151	even	40	inner