Embedded newform 322.2.o.b.5.6

• Label: 322.2.o.b.5.6
• 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			5.6
Character	\(\chi\)	\(=\)	322.5
Dual form			322.2.o.b.129.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.723734 - 0.690079i) q^{2} +(0.340605 - 1.76722i) q^{3} +(0.0475819 - 0.998867i) q^{4} +(-0.793626 - 0.624115i) q^{5} +(-0.973017 - 1.51404i) q^{6} +(-0.911188 - 2.48390i) q^{7} +(-0.654861 - 0.755750i) q^{8} +(-0.221966 - 0.0888616i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 160 q + 8 q^{2} - 6 q^{3} + 8 q^{4} + 11 q^{7} - 16 q^{8} + 12 q^{9}+O(q^{10}) \)

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
							\(3\)	0.340605	−	1.76722i	0.196648	−	1.02031i	−0.740680	−	0.671858i	\(-0.765496\pi\)
0.937328	−	0.348449i	\(-0.113292\pi\)
\(5\)	−0.793626	−	0.624115i	−0.354920	−	0.279113i	0.424740	−	0.905316i	\(-0.360366\pi\)
							−0.779660	+	0.626203i	\(0.784608\pi\)
\(7\)	−0.911188	−	2.48390i	−0.344397	−	0.938824i
							\(11\)	−2.73144	+	2.86465i	−0.823561	+	0.863726i	−0.992543	−	0.121893i	\(-0.961103\pi\)
0.168982	+	0.985619i	\(0.445952\pi\)
\(13\)	−0.912364	−	0.416663i	−0.253044	−	0.115561i	0.284855	−	0.958571i	\(-0.408055\pi\)
							−0.537899	+	0.843009i	\(0.680782\pi\)
\(17\)	4.75133	+	2.44948i	1.15237	+	0.594086i	0.925092	−	0.379744i	\(-0.123988\pi\)
							0.227275	+	0.973831i	\(0.427018\pi\)
\(19\)	6.13119	−	3.16085i	1.40659	−	0.725148i	0.423096	−	0.906085i	\(-0.360943\pi\)
							0.983495	+	0.180937i	\(0.0579129\pi\)
\(23\)	−4.39477	+	1.91989i	−0.916373	+	0.400325i
							\(29\)	7.16293	−	4.60334i	1.33012	−	0.854818i	0.333981	−	0.942580i	\(-0.391608\pi\)
0.996141	+	0.0877619i	\(0.0279715\pi\)
\(31\)	−4.32922	−	1.49836i	−0.777551	−	0.269113i	−0.0906684	−	0.995881i	\(-0.528900\pi\)
							−0.686883	+	0.726768i	\(0.741022\pi\)
\(37\)	−3.14976	+	7.86771i	−0.517817	+	1.29344i	0.407138	+	0.913367i	\(0.366527\pi\)
							−0.924955	+	0.380078i	\(0.875897\pi\)
\(41\)	10.6413	+	1.52998i	1.66188	+	0.238943i	0.908273	−	0.418378i	\(-0.137401\pi\)
							0.753611	+	0.657321i	\(0.228310\pi\)
\(43\)	0.748530	+	0.648605i	0.114150	+	0.0989113i	0.710065	−	0.704136i	\(-0.248665\pi\)
							−0.595916	+	0.803047i	\(0.703211\pi\)
\(47\)	−3.86559	−	2.23180i	−0.563854	−	0.325541i	0.190837	−	0.981622i	\(-0.438880\pi\)
							−0.754691	+	0.656081i	\(0.772213\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	322.2.o.b.5.6	✓	160
7.3	odd	6	inner	322.2.o.b.143.6	yes	160
23.14	odd	22	inner	322.2.o.b.313.6	yes	160
161.129	even	66	inner	322.2.o.b.129.6	yes	160

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
322.2.o.b.5.6	✓	160	1.1	even	1	trivial
322.2.o.b.129.6	yes	160	161.129	even	66	inner
322.2.o.b.143.6	yes	160	7.3	odd	6	inner
322.2.o.b.313.6	yes	160	23.14	odd	22	inner