Embedded newform 322.2.m.a.39.2

• Label: 322.2.m.a.39.2
• Level: $322$
• Weight: $2$
• Character: 322.39
• 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.m (of order \(33\), degree \(20\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.57118294509\)
Analytic rank:	\(0\)
Dimension:	\(160\)
Relative dimension:	\(8\) over \(\Q(\zeta_{33})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{33}]$

Embedding invariants

Embedding label			39.2
Character	\(\chi\)	\(=\)	322.39
Dual form			322.2.m.a.289.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.981929 - 0.189251i) q^{2} +(-0.121146 - 2.54317i) q^{3} +(0.928368 + 0.371662i) q^{4} +(1.79650 - 2.52283i) q^{5} +(-0.362342 + 2.52014i) q^{6} +(-0.443953 - 2.60824i) q^{7} +(-0.841254 - 0.540641i) q^{8} +(-3.46663 + 0.331023i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 160 q - 8 q^{2} + 2 q^{3} + 8 q^{4} - 2 q^{5} + 4 q^{6} + 15 q^{7} + 16 q^{8} + 28 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(185\)	\(281\)
\(\chi(n)\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{4}{11}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.981929	−	0.189251i	−0.694328	−	0.133821i
							\(3\)	−0.121146	−	2.54317i	−0.0699438	−	1.46830i	−0.712289	−	0.701886i	\(-0.752342\pi\)
0.642346	−	0.766415i	\(-0.277961\pi\)
\(5\)	1.79650	−	2.52283i	0.803419	−	1.12824i	−0.185913	−	0.982566i	\(-0.559524\pi\)
							0.989332	−	0.145678i	\(-0.0465365\pi\)
\(7\)	−0.443953	−	2.60824i	−0.167799	−	0.985821i
							\(11\)	0.875774	−	0.168792i	0.264056	−	0.0508926i	−0.0555048	−	0.998458i	\(-0.517677\pi\)
0.319561	+	0.947566i	\(0.396465\pi\)
\(13\)	3.31200	+	0.972490i	0.918582	+	0.269720i	0.706649	−	0.707564i	\(-0.250206\pi\)
							0.211933	+	0.977284i	\(0.432024\pi\)
\(17\)	4.73327	+	3.72229i	1.14799	+	0.902787i	0.996326	−	0.0856455i	\(-0.0272952\pi\)
							0.151661	+	0.988433i	\(0.451538\pi\)
\(19\)	−3.65295	+	2.87271i	−0.838045	+	0.659046i	−0.942028	−	0.335533i	\(-0.891084\pi\)
							0.103984	+	0.994579i	\(0.466841\pi\)
\(23\)	−0.523312	+	4.76719i	−0.109118	+	0.994029i
							\(29\)	1.29353	−	8.99667i	0.240202	−	1.67064i	−0.410925	−	0.911669i	\(-0.634794\pi\)
0.651126	−	0.758969i	\(-0.274297\pi\)
\(31\)	−8.06023	+	4.15533i	−1.44766	+	0.746320i	−0.989768	−	0.142688i	\(-0.954425\pi\)
							−0.457891	+	0.889008i	\(0.651395\pi\)
\(37\)	−2.38713	+	0.227943i	−0.392442	+	0.0374737i	−0.289412	−	0.957205i	\(-0.593460\pi\)
							−0.103030	+	0.994678i	\(0.532854\pi\)
\(41\)	3.41574	+	7.47944i	0.533450	+	1.16809i	0.964092	+	0.265569i	\(0.0855599\pi\)
							−0.430642	+	0.902523i	\(0.641713\pi\)
\(43\)	4.11741	−	2.64610i	0.627899	−	0.403526i	−0.187632	−	0.982239i	\(-0.560081\pi\)
							0.815531	+	0.578713i	\(0.196445\pi\)
\(47\)	3.32135	+	5.75275i	0.484469	+	0.839124i	0.999841	−	0.0178421i	\(-0.00567961\pi\)
							−0.515372	+	0.856967i	\(0.672346\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	322.2.m.a.39.2	✓	160
7.2	even	3	inner	322.2.m.a.177.7	yes	160
23.13	even	11	inner	322.2.m.a.151.7	yes	160
161.128	even	33	inner	322.2.m.a.289.2	yes	160

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
322.2.m.a.39.2	✓	160	1.1	even	1	trivial
322.2.m.a.151.7	yes	160	23.13	even	11	inner
322.2.m.a.177.7	yes	160	7.2	even	3	inner
322.2.m.a.289.2	yes	160	161.128	even	33	inner