Embedded newform 164.2.o.b.111.3

• Label: 164.2.o.b.111.3
• Level: $164$
• Weight: $2$
• Character: 164.111
• 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			111.3
Character	\(\chi\)	\(=\)	164.111
Dual form			164.2.o.b.99.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.24842 + 0.664422i) q^{2} +(-0.971394 - 0.402365i) q^{3} +(1.11709 - 1.65895i) q^{4} +(-1.22084 + 0.622051i) q^{5} +(1.48004 - 0.143097i) q^{6} +(-0.0357653 - 0.0583637i) q^{7} +(-0.292346 + 2.81328i) q^{8} +(-1.33961 - 1.33961i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 288 q - 12 q^{2} - 20 q^{4} - 32 q^{5} - 28 q^{6} - 24 q^{8} - 40 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(83\)	\(129\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{7}{40}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.24842	+	0.664422i	−0.882764	+	0.469817i
							\(3\)	−0.971394	−	0.402365i	−0.560835	−	0.232305i	0.0842128	−	0.996448i	\(-0.473162\pi\)
−0.645048	+	0.764142i	\(0.723162\pi\)
\(5\)	−1.22084	+	0.622051i	−0.545978	+	0.278190i	−0.705146	−	0.709062i	\(-0.749119\pi\)
							0.159169	+	0.987251i	\(0.449119\pi\)
\(7\)	−0.0357653	−	0.0583637i	−0.0135180	−	0.0220594i	0.845810	−	0.533485i	\(-0.179118\pi\)
							−0.859328	+	0.511425i	\(0.829118\pi\)
\(11\)	−5.58920	−	0.439880i	−1.68521	−	0.132629i	−0.800673	−	0.599102i	\(-0.795525\pi\)
							−0.884536	+	0.466473i	\(0.845525\pi\)
\(13\)	−1.03765	−	4.32214i	−0.287793	−	1.19875i	−0.909570	−	0.415551i	\(-0.863589\pi\)
							0.621776	−	0.783195i	\(-0.286411\pi\)
\(17\)	1.66925	−	1.42567i	0.404852	−	0.345776i	−0.423587	−	0.905855i	\(-0.639229\pi\)
							0.828439	+	0.560079i	\(0.189229\pi\)
\(19\)	−1.78685	−	0.428985i	−0.409932	−	0.0984159i	0.0232343	−	0.999730i	\(-0.492604\pi\)
							−0.433166	+	0.901314i	\(0.642604\pi\)
\(23\)	−0.143714	+	0.104414i	−0.0299664	+	0.0217718i	−0.602668	−	0.797992i	\(-0.705896\pi\)
							0.572701	+	0.819764i	\(0.305896\pi\)
\(29\)	−4.48448	−	3.83011i	−0.832748	−	0.711234i	0.127059	−	0.991895i	\(-0.459446\pi\)
							−0.959807	+	0.280661i	\(0.909446\pi\)
\(31\)	−0.761616	−	2.34401i	−0.136790	−	0.420997i	0.859074	−	0.511852i	\(-0.171040\pi\)
							−0.995864	+	0.0908543i	\(0.971040\pi\)
\(37\)	−1.58795	+	4.88721i	−0.261057	+	0.803452i	0.731518	+	0.681822i	\(0.238812\pi\)
							−0.992576	+	0.121630i	\(0.961188\pi\)
\(41\)	4.25278	+	4.78684i	0.664173	+	0.747579i
							\(43\)	11.3727	+	1.80126i	1.73432	+	0.274689i	0.942047	−	0.335480i	\(-0.108899\pi\)
0.792271	+	0.610169i	\(0.208899\pi\)
\(47\)	−2.70915	+	4.42093i	−0.395170	+	0.644859i	−0.986910	−	0.161270i	\(-0.948441\pi\)
							0.591741	+	0.806129i	\(0.298441\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	164.2.o.b.111.3	yes	288
4.3	odd	2	inner	164.2.o.b.111.14	yes	288
41.17	odd	40	inner	164.2.o.b.99.14	yes	288
164.99	even	40	inner	164.2.o.b.99.3	✓	288

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
164.2.o.b.99.3	✓	288	164.99	even	40	inner
164.2.o.b.99.14	yes	288	41.17	odd	40	inner
164.2.o.b.111.3	yes	288	1.1	even	1	trivial
164.2.o.b.111.14	yes	288	4.3	odd	2	inner