Embedded newform 322.2.o.a.171.8

• Label: 322.2.o.a.171.8
• Level: $322$
• Weight: $2$
• Character: 322.171
• 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			171.8
Character	\(\chi\)	\(=\)	322.171
Dual form			322.2.o.a.145.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.327068 - 0.945001i) q^{2} +(1.37509 + 2.66731i) q^{3} +(-0.786053 - 0.618159i) q^{4} +(-4.34646 + 0.415037i) q^{5} +(2.97036 - 0.427073i) q^{6} +(2.28714 + 1.33003i) q^{7} +(-0.841254 + 0.540641i) q^{8} +(-3.48348 + 4.89187i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 160 q - 8 q^{2} + 6 q^{3} + 8 q^{4} - 11 q^{7} + 16 q^{8} - 32 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(185\)	\(281\)
\(\chi(n)\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{3}{22}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.327068	−	0.945001i	0.231272	−	0.668216i
							\(3\)	1.37509	+	2.66731i	0.793910	+	1.53997i	0.840971	+	0.541079i	\(0.181984\pi\)
−0.0470610	+	0.998892i	\(0.514986\pi\)
\(5\)	−4.34646	+	0.415037i	−1.94380	+	0.185610i	−0.992462	−	0.122552i	\(-0.960892\pi\)
							−0.951334	+	0.308162i	\(0.900286\pi\)
\(7\)	2.28714	+	1.33003i	0.864458	+	0.502704i
							\(11\)	−3.38674	+	1.17216i	−1.02114	+	0.353420i	−0.785792	−	0.618491i	\(-0.787744\pi\)
−0.235348	+	0.971911i	\(0.575623\pi\)
\(13\)	0.516121	+	1.75775i	0.143146	+	0.487512i	0.999587	−	0.0287337i	\(-0.00914748\pi\)
							−0.856441	+	0.516245i	\(0.827329\pi\)
\(17\)	3.05481	+	1.22296i	0.740901	+	0.296612i	0.711241	−	0.702948i	\(-0.248133\pi\)
							0.0296599	+	0.999560i	\(0.490558\pi\)
\(19\)	2.87964	−	1.15283i	0.660634	−	0.264478i	−0.0170238	−	0.999855i	\(-0.505419\pi\)
							0.677658	+	0.735377i	\(0.262995\pi\)
\(23\)	−4.55215	+	1.50930i	−0.949188	+	0.314710i
							\(29\)	−0.135813	−	0.944599i	−0.0252198	−	0.175408i	0.973318	−	0.229459i	\(-0.0736957\pi\)
−0.998538	+	0.0540512i	\(0.982787\pi\)
\(31\)	0.781048	−	0.0372059i	0.140280	−	0.00668238i	0.0226757	−	0.999743i	\(-0.492781\pi\)
							0.117605	+	0.993060i	\(0.462478\pi\)
\(37\)	−1.24014	−	0.883100i	−0.203878	−	0.145181i	0.473558	−	0.880763i	\(-0.342969\pi\)
							−0.677436	+	0.735582i	\(0.736909\pi\)
\(41\)	8.45144	+	3.85964i	1.31989	+	0.602775i	0.945840	−	0.324632i	\(-0.105240\pi\)
							0.374053	+	0.927407i	\(0.377968\pi\)
\(43\)	−2.61143	+	4.06347i	−0.398240	+	0.619674i	−0.981238	−	0.192800i	\(-0.938243\pi\)
							0.582998	+	0.812474i	\(0.301880\pi\)
\(47\)	10.0384	−	5.79570i	1.46426	−	0.845389i	0.465053	−	0.885283i	\(-0.346035\pi\)
							0.999204	+	0.0398932i	\(0.0127018\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	322.2.o.a.171.8	yes	160
7.5	odd	6	inner	322.2.o.a.33.8	✓	160
23.7	odd	22	inner	322.2.o.a.283.8	yes	160
161.145	even	66	inner	322.2.o.a.145.8	yes	160

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
322.2.o.a.33.8	✓	160	7.5	odd	6	inner
322.2.o.a.145.8	yes	160	161.145	even	66	inner
322.2.o.a.171.8	yes	160	1.1	even	1	trivial
322.2.o.a.283.8	yes	160	23.7	odd	22	inner