Embedded newform 322.2.o.b.5.3

• Label: 322.2.o.b.5.3
• Level: $322$
• Weight: $2$
• Character: 322.5
• 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			5.3
Character	\(\chi\)	\(=\)	322.5
Dual form			322.2.o.b.129.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.723734 - 0.690079i) q^{2} +(-0.219533 + 1.13904i) q^{3} +(0.0475819 - 0.998867i) q^{4} +(-0.936223 - 0.736254i) q^{5} +(0.627147 + 0.975860i) q^{6} +(0.155114 - 2.64120i) q^{7} +(-0.654861 - 0.755750i) q^{8} +(1.53588 + 0.614873i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 160 q + 8 q^{2} - 6 q^{3} + 8 q^{4} + 11 q^{7} - 16 q^{8} + 12 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.723734	−	0.690079i	0.511757	−	0.487960i
							\(3\)	−0.219533	+	1.13904i	−0.126747	+	0.657627i	0.861864	+	0.507139i	\(0.169297\pi\)
−0.988612	+	0.150489i	\(0.951915\pi\)
\(5\)	−0.936223	−	0.736254i	−0.418692	−	0.329263i	0.386487	−	0.922295i	\(-0.373688\pi\)
							−0.805179	+	0.593032i	\(0.797931\pi\)
\(7\)	0.155114	−	2.64120i	0.0586274	−	0.998280i
							\(11\)	3.65556	−	3.83384i	1.10219	−	1.15595i	0.115509	−	0.993306i	\(-0.463150\pi\)
0.986685	−	0.162641i	\(-0.0520013\pi\)
\(13\)	0.859065	+	0.392322i	0.238262	+	0.108811i	0.530968	−	0.847392i	\(-0.321828\pi\)
							−0.292706	+	0.956202i	\(0.594556\pi\)
\(17\)	0.487928	+	0.251544i	0.118340	+	0.0610084i	0.516380	−	0.856360i	\(-0.327279\pi\)
							−0.398040	+	0.917368i	\(0.630310\pi\)
\(19\)	0.104897	−	0.0540784i	0.0240651	−	0.0124064i	−0.446152	−	0.894957i	\(-0.647206\pi\)
							0.470217	+	0.882551i	\(0.344176\pi\)
\(23\)	2.98226	+	3.75581i	0.621844	+	0.783141i
							\(29\)	−4.72275	+	3.03513i	−0.876993	+	0.563609i	−0.899884	−	0.436129i	\(-0.856349\pi\)
0.0228911	+	0.999738i	\(0.492713\pi\)
\(31\)	−0.780734	−	0.270215i	−0.140224	−	0.0485320i	0.256055	−	0.966662i	\(-0.417577\pi\)
							−0.396279	+	0.918130i	\(0.629699\pi\)
\(37\)	0.634438	−	1.58475i	0.104301	−	0.260531i	−0.867033	−	0.498251i	\(-0.833976\pi\)
							0.971334	+	0.237720i	\(0.0764000\pi\)
\(41\)	−10.5919	−	1.52288i	−1.65417	−	0.237834i	−0.748905	−	0.662677i	\(-0.769420\pi\)
							−0.905267	+	0.424843i	\(0.860329\pi\)
\(43\)	6.81134	+	5.90206i	1.03872	+	0.900056i	0.995088	−	0.0989919i	\(-0.0315618\pi\)
							0.0436315	+	0.999048i	\(0.486107\pi\)
\(47\)	3.80395	+	2.19621i	0.554863	+	0.320350i	0.751081	−	0.660210i	\(-0.229533\pi\)
							−0.196218	+	0.980560i	\(0.562866\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	322.2.o.b.5.3	✓	160
7.3	odd	6	inner	322.2.o.b.143.3	yes	160
23.14	odd	22	inner	322.2.o.b.313.3	yes	160
161.129	even	66	inner	322.2.o.b.129.3	yes	160

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
322.2.o.b.5.3	✓	160	1.1	even	1	trivial
322.2.o.b.129.3	yes	160	161.129	even	66	inner
322.2.o.b.143.3	yes	160	7.3	odd	6	inner
322.2.o.b.313.3	yes	160	23.14	odd	22	inner