Embedded newform 322.2.o.b.5.1

• Label: 322.2.o.b.5.1
• 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			5.1
Character	\(\chi\)	\(=\)	322.5
Dual form			322.2.o.b.129.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.723734 - 0.690079i) q^{2} +(-0.618301 + 3.20805i) q^{3} +(0.0475819 - 0.998867i) q^{4} +(2.71652 + 2.13629i) q^{5} +(1.76632 + 2.74845i) q^{6} +(2.21316 - 1.44980i) q^{7} +(-0.654861 - 0.755750i) q^{8} +(-7.12418 - 2.85209i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 160 q + 8 q^{2} - 6 q^{3} + 8 q^{4} + 11 q^{7} - 16 q^{8} + 12 q^{9}+O(q^{10}) \)

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
							\(3\)	−0.618301	+	3.20805i	−0.356976	+	1.85217i	0.147935	+	0.988997i	\(0.452737\pi\)
−0.504911	+	0.863171i	\(0.668475\pi\)
\(5\)	2.71652	+	2.13629i	1.21486	+	0.955379i	0.999775	−	0.0212155i	\(-0.00675360\pi\)
							0.215088	+	0.976595i	\(0.430996\pi\)
\(7\)	2.21316	−	1.44980i	0.836496	−	0.547973i
							\(11\)	−0.0158005	+	0.0165711i	−0.00476403	+	0.00499637i	−0.726112	−	0.687577i	\(-0.758675\pi\)
0.721348	+	0.692573i	\(0.243523\pi\)
\(13\)	0.552719	+	0.252418i	0.153297	+	0.0700082i	0.490586	−	0.871393i	\(-0.336783\pi\)
							−0.337289	+	0.941401i	\(0.609510\pi\)
\(17\)	2.06537	+	1.06477i	0.500926	+	0.258245i	0.690121	−	0.723694i	\(-0.257557\pi\)
							−0.189195	+	0.981940i	\(0.560588\pi\)
\(19\)	−5.48612	+	2.82829i	−1.25860	+	0.648855i	−0.952903	−	0.303274i	\(-0.901920\pi\)
							−0.305699	+	0.952128i	\(0.598890\pi\)
\(23\)	−2.69976	−	3.96375i	−0.562939	−	0.826499i
							\(29\)	5.60869	−	3.60448i	1.04151	−	0.669336i	0.0961481	−	0.995367i	\(-0.469348\pi\)
0.945359	+	0.326031i	\(0.105711\pi\)
\(31\)	−6.44260	−	2.22981i	−1.15713	−	0.400485i	−0.319958	−	0.947432i	\(-0.603669\pi\)
							−0.837167	+	0.546947i	\(0.815790\pi\)
\(37\)	1.05681	−	2.63978i	0.173738	−	0.433977i	−0.815784	−	0.578356i	\(-0.803694\pi\)
							0.989522	+	0.144380i	\(0.0461186\pi\)
\(41\)	6.47555	+	0.931044i	1.01131	+	0.145405i	0.627999	−	0.778214i	\(-0.283874\pi\)
							0.383312	+	0.923619i	\(0.374783\pi\)
\(43\)	5.98977	+	5.19016i	0.913431	+	0.791493i	0.978470	−	0.206390i	\(-0.0661716\pi\)
							−0.0650385	+	0.997883i	\(0.520717\pi\)
\(47\)	−4.36144	−	2.51808i	−0.636181	−	0.367299i	0.146961	−	0.989142i	\(-0.453051\pi\)
							−0.783142	+	0.621843i	\(0.786384\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	322.2.o.b.5.1	✓	160
7.3	odd	6	inner	322.2.o.b.143.1	yes	160
23.14	odd	22	inner	322.2.o.b.313.1	yes	160
161.129	even	66	inner	322.2.o.b.129.1	yes	160

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
322.2.o.b.5.1	✓	160	1.1	even	1	trivial
322.2.o.b.129.1	yes	160	161.129	even	66	inner
322.2.o.b.143.1	yes	160	7.3	odd	6	inner
322.2.o.b.313.1	yes	160	23.14	odd	22	inner