Embedded newform 322.2.o.b.17.1

• Label: 322.2.o.b.17.1
• Level: $322$
• Weight: $2$
• Character: 322.17
• 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			17.1
Character	\(\chi\)	\(=\)	322.17
Dual form			322.2.o.b.19.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.995472 + 0.0950560i) q^{2} +(-2.26151 - 2.37180i) q^{3} +(0.981929 - 0.189251i) q^{4} +(2.68625 + 1.38486i) q^{5} +(2.47672 + 2.14609i) q^{6} +(2.03190 - 1.69451i) q^{7} +(-0.959493 + 0.281733i) q^{8} +(-0.368277 + 7.73108i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 160 q + 8 q^{2} - 6 q^{3} + 8 q^{4} + 11 q^{7} - 16 q^{8} + 12 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(185\)	\(281\)
\(\chi(n)\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{7}{22}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.995472	+	0.0950560i	−0.703905	+	0.0672148i
							\(3\)	−2.26151	−	2.37180i	−1.30568	−	1.36936i	−0.887371	−	0.461055i	\(-0.847471\pi\)
−0.418310	−	0.908304i	\(-0.637377\pi\)
\(5\)	2.68625	+	1.38486i	1.20133	+	0.619328i	0.938421	−	0.345493i	\(-0.112288\pi\)
							0.262908	+	0.964821i	\(0.415319\pi\)
\(7\)	2.03190	−	1.69451i	0.767988	−	0.640465i
							\(11\)	−0.443364	+	4.64312i	−0.133679	+	1.39995i	0.640972	+	0.767564i	\(0.278531\pi\)
−0.774652	+	0.632388i	\(0.782075\pi\)
\(13\)	4.52750	−	0.650956i	1.25570	−	0.180543i	0.517830	−	0.855483i	\(-0.326740\pi\)
							0.737872	+	0.674941i	\(0.235831\pi\)
\(17\)	0.845334	+	2.44243i	0.205024	+	0.592377i	0.999896	−	0.0143993i	\(-0.00458358\pi\)
							−0.794873	+	0.606776i	\(0.792462\pi\)
\(19\)	0.596147	−	1.72245i	0.136766	−	0.395158i	−0.855732	−	0.517418i	\(-0.826893\pi\)
							0.992498	+	0.122260i	\(0.0390143\pi\)
\(23\)	2.37257	−	4.16784i	0.494716	−	0.869055i
							\(29\)	2.97125	−	3.42900i	0.551747	−	0.636750i	−0.409542	−	0.912291i	\(-0.634312\pi\)
0.961289	+	0.275541i	\(0.0888570\pi\)
\(31\)	0.446671	+	0.108361i	0.0802244	+	0.0194622i	0.275670	−	0.961252i	\(-0.411100\pi\)
							−0.195446	+	0.980715i	\(0.562615\pi\)
\(37\)	−8.19202	−	0.390234i	−1.34676	−	0.0641541i	−0.638236	−	0.769841i	\(-0.720335\pi\)
							−0.708524	+	0.705687i	\(0.750638\pi\)
\(41\)	−0.0349256	−	0.0543452i	−0.00545446	−	0.00848730i	0.838516	−	0.544878i	\(-0.183424\pi\)
							−0.843970	+	0.536390i	\(0.819788\pi\)
\(43\)	−0.939843	+	3.20081i	−0.143325	+	0.488119i	−0.999596	−	0.0284065i	\(-0.990957\pi\)
							0.856272	+	0.516526i	\(0.172775\pi\)
\(47\)	−4.32588	+	2.49755i	−0.630994	+	0.364305i	−0.781137	−	0.624360i	\(-0.785360\pi\)
							0.150143	+	0.988664i	\(0.452027\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	322.2.o.b.17.1	✓	160
7.5	odd	6	inner	322.2.o.b.201.1	yes	160
23.19	odd	22	inner	322.2.o.b.157.1	yes	160
161.19	even	66	inner	322.2.o.b.19.1	yes	160

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
322.2.o.b.17.1	✓	160	1.1	even	1	trivial
322.2.o.b.19.1	yes	160	161.19	even	66	inner
322.2.o.b.157.1	yes	160	23.19	odd	22	inner
322.2.o.b.201.1	yes	160	7.5	odd	6	inner