Embedded newform 322.2.o.b.227.2

• Label: 322.2.o.b.227.2
• Level: $322$
• Weight: $2$
• Character: 322.227
• Analytic conductor: $2.571$
• Analytic rank: $0$
• Dimension: $160$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 322 = 2 \cdot 7 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	322.o (of order \(66\), degree \(20\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.57118294509\)
Analytic rank:	\(0\)
Dimension:	\(160\)
Relative dimension:	\(8\) over \(\Q(\zeta_{66})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{66}]$

Embedding invariants

Embedding label			227.2
Character	\(\chi\)	\(=\)	322.227
Dual form			322.2.o.b.61.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.786053 + 0.618159i) q^{2} +(-1.74103 + 1.23978i) q^{3} +(0.235759 - 0.971812i) q^{4} +(0.165201 + 0.0318398i) q^{5} +(0.602159 - 2.05076i) q^{6} +(2.29552 + 1.31552i) q^{7} +(0.415415 + 0.909632i) q^{8} +(0.512917 - 1.48198i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 160 q + 8 q^{2} - 6 q^{3} + 8 q^{4} + 11 q^{7} - 16 q^{8} + 12 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/322\mathbb{Z}\right)^\times\).

\(n\)	\(185\)	\(281\)
\(\chi(n)\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{5}{22}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.786053	+	0.618159i	−0.555823	+	0.437104i
							\(3\)	−1.74103	+	1.23978i	−1.00518	+	0.715787i	−0.959076	−	0.283149i	\(-0.908621\pi\)
−0.0461066	+	0.998937i	\(0.514681\pi\)
\(5\)	0.165201	+	0.0318398i	0.0738799	+	0.0142392i	0.226057	−	0.974114i	\(-0.427416\pi\)
							−0.152177	+	0.988353i	\(0.548628\pi\)
\(7\)	2.29552	+	1.31552i	0.867624	+	0.497221i
							\(11\)	−3.26083	+	4.14649i	−0.983179	+	1.25021i	−0.0150170	+	0.999887i	\(0.504780\pi\)
−0.968162	+	0.250326i	\(0.919462\pi\)
\(13\)	0.529163	−	0.823394i	0.146763	−	0.228368i	−0.760087	−	0.649822i	\(-0.774844\pi\)
							0.906850	+	0.421453i	\(0.138480\pi\)
\(17\)	0.0531311	−	0.0506604i	0.0128862	−	0.0122869i	−0.683609	−	0.729849i	\(-0.739590\pi\)
							0.696495	+	0.717562i	\(0.254742\pi\)
\(19\)	−1.09237	−	1.04158i	−0.250608	−	0.238954i	0.554353	−	0.832281i	\(-0.312966\pi\)
							−0.804961	+	0.593327i	\(0.797814\pi\)
\(23\)	−4.16605	+	2.37572i	−0.868681	+	0.495373i
							\(29\)	−6.66227	−	1.95622i	−1.23715	−	0.363261i	−0.403207	−	0.915109i	\(-0.632105\pi\)
−0.833945	+	0.551848i	\(0.813923\pi\)
\(31\)	−0.727648	+	7.62027i	−0.130689	+	1.36864i	0.657614	+	0.753355i	\(0.271566\pi\)
							−0.788303	+	0.615287i	\(0.789040\pi\)
\(37\)	1.21221	+	0.419551i	0.199287	+	0.0689738i	0.424888	−	0.905246i	\(-0.360313\pi\)
							−0.225601	+	0.974220i	\(0.572435\pi\)
\(41\)	7.09002	+	6.14354i	1.10728	+	0.959460i	0.999398	−	0.0347044i	\(-0.0110490\pi\)
							0.107878	+	0.994164i	\(0.465594\pi\)
\(43\)	−1.49860	−	0.684388i	−0.228534	−	0.104368i	0.297857	−	0.954610i	\(-0.403728\pi\)
							−0.526391	+	0.850242i	\(0.676455\pi\)
\(47\)	4.26026	−	2.45966i	0.621423	−	0.358779i	−0.156000	−	0.987757i	\(-0.549860\pi\)
							0.777423	+	0.628979i	\(0.216527\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	322.2.o.b.227.2	yes	160
7.5	odd	6	inner	322.2.o.b.89.7	yes	160
23.15	odd	22	inner	322.2.o.b.199.7	yes	160
161.61	even	66	inner	322.2.o.b.61.2	✓	160

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
322.2.o.b.61.2	✓	160	161.61	even	66	inner
322.2.o.b.89.7	yes	160	7.5	odd	6	inner
322.2.o.b.199.7	yes	160	23.15	odd	22	inner
322.2.o.b.227.2	yes	160	1.1	even	1	trivial