Embedded newform 322.2.o.b.17.4

• Label: 322.2.o.b.17.4
• 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			17.4
Character	\(\chi\)	\(=\)	322.17
Dual form			322.2.o.b.19.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.995472 + 0.0950560i) q^{2} +(-0.855603 - 0.897331i) q^{3} +(0.981929 - 0.189251i) q^{4} +(-2.48995 - 1.28366i) q^{5} +(0.937026 + 0.811937i) q^{6} +(-1.18187 + 2.36710i) q^{7} +(-0.959493 + 0.281733i) q^{8} +(0.0696000 - 1.46108i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 160 q + 8 q^{2} - 6 q^{3} + 8 q^{4} + 11 q^{7} - 16 q^{8} + 12 q^{9}+O(q^{10}) \)

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
							\(3\)	−0.855603	−	0.897331i	−0.493983	−	0.518074i	0.428798	−	0.903401i	\(-0.358937\pi\)
−0.922780	+	0.385326i	\(0.874089\pi\)
\(5\)	−2.48995	−	1.28366i	−1.11354	−	0.574070i	−0.199583	−	0.979881i	\(-0.563959\pi\)
							−0.913957	+	0.405811i	\(0.866989\pi\)
\(7\)	−1.18187	+	2.36710i	−0.446705	+	0.894681i
							\(11\)	−0.550496	+	5.76505i	−0.165981	+	1.73823i	0.402401	+	0.915463i	\(0.368176\pi\)
−0.568382	+	0.822765i	\(0.692430\pi\)
\(13\)	3.86775	−	0.556099i	1.07272	−	0.154234i	0.416751	−	0.909021i	\(-0.363169\pi\)
							0.655970	+	0.754787i	\(0.272260\pi\)
\(17\)	1.36434	+	3.94200i	0.330901	+	0.956076i	0.980623	+	0.195904i	\(0.0627641\pi\)
							−0.649722	+	0.760172i	\(0.725115\pi\)
\(19\)	1.17699	−	3.40068i	0.270019	−	0.780170i	−0.725723	−	0.687987i	\(-0.758495\pi\)
							0.995742	−	0.0921824i	\(-0.0293843\pi\)
\(23\)	3.73301	+	3.01075i	0.778387	+	0.627785i
							\(29\)	−5.50614	+	6.35443i	−1.02247	+	1.17999i	−0.0389357	+	0.999242i	\(0.512397\pi\)
−0.983530	+	0.180746i	\(0.942149\pi\)
\(31\)	7.23514	+	1.75523i	1.29947	+	0.315248i	0.825054	−	0.565053i	\(-0.191144\pi\)
							0.474415	+	0.880301i	\(0.342660\pi\)
\(37\)	−2.02002	−	0.0962254i	−0.332089	−	0.0158194i	−0.119125	−	0.992879i	\(-0.538009\pi\)
							−0.212965	+	0.977060i	\(0.568312\pi\)
\(41\)	0.229602	+	0.357267i	0.0358578	+	0.0557958i	0.858720	−	0.512446i	\(-0.171260\pi\)
							−0.822862	+	0.568241i	\(0.807624\pi\)
\(43\)	−3.34821	+	11.4030i	−0.510597	+	1.73893i	0.150476	+	0.988614i	\(0.451919\pi\)
							−0.661074	+	0.750321i	\(0.729899\pi\)
\(47\)	−3.62891	+	2.09515i	−0.529330	+	0.305609i	−0.740744	−	0.671788i	\(-0.765527\pi\)
							0.211413	+	0.977397i	\(0.432193\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	322.2.o.b.17.4	✓	160
7.5	odd	6	inner	322.2.o.b.201.4	yes	160
23.19	odd	22	inner	322.2.o.b.157.4	yes	160
161.19	even	66	inner	322.2.o.b.19.4	yes	160

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
322.2.o.b.17.4	✓	160	1.1	even	1	trivial
322.2.o.b.19.4	yes	160	161.19	even	66	inner
322.2.o.b.157.4	yes	160	23.19	odd	22	inner
322.2.o.b.201.4	yes	160	7.5	odd	6	inner