Embedded newform 322.2.m.a.39.4

• Label: 322.2.m.a.39.4
• 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			39.4
Character	\(\chi\)	\(=\)	322.39
Dual form			322.2.m.a.289.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.981929 - 0.189251i) q^{2} +(-0.0525591 - 1.10335i) q^{3} +(0.928368 + 0.371662i) q^{4} +(-1.42783 + 2.00510i) q^{5} +(-0.157201 + 1.09336i) q^{6} +(-2.61494 - 0.402583i) q^{7} +(-0.841254 - 0.540641i) q^{8} +(1.77179 - 0.169186i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 160 q - 8 q^{2} + 2 q^{3} + 8 q^{4} - 2 q^{5} + 4 q^{6} + 15 q^{7} + 16 q^{8} + 28 q^{9}+O(q^{10}) \)

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
							\(3\)	−0.0525591	−	1.10335i	−0.0303450	−	0.637020i	−0.961897	−	0.273412i	\(-0.911848\pi\)
0.931552	−	0.363608i	\(-0.118455\pi\)
\(5\)	−1.42783	+	2.00510i	−0.638544	+	0.896710i	−0.999354	−	0.0359297i	\(-0.988561\pi\)
							0.360810	+	0.932639i	\(0.382500\pi\)
\(7\)	−2.61494	−	0.402583i	−0.988356	−	0.152162i
							\(11\)	5.05954	−	0.975146i	1.52551	−	0.294017i	0.643571	−	0.765386i	\(-0.277452\pi\)
0.881936	+	0.471369i	\(0.156240\pi\)
\(13\)	5.43469	+	1.59577i	1.50731	+	0.442586i	0.928019	−	0.372534i	\(-0.121511\pi\)
							0.579292	+	0.815120i	\(0.303329\pi\)
\(17\)	−5.02790	−	3.95398i	−1.21944	−	0.958981i	−0.219582	−	0.975594i	\(-0.570469\pi\)
							−0.999862	+	0.0166130i	\(0.994712\pi\)
\(19\)	4.87198	−	3.83136i	1.11771	−	0.878975i	0.124146	−	0.992264i	\(-0.460381\pi\)
							0.993562	+	0.113289i	\(0.0361385\pi\)
\(23\)	3.00044	+	3.74130i	0.625635	+	0.780116i
							\(29\)	−0.427612	+	2.97411i	−0.0794056	+	0.552278i	0.910820	+	0.412804i	\(0.135450\pi\)
−0.990226	+	0.139474i	\(0.955459\pi\)
\(31\)	1.93895	−	0.999597i	0.348245	−	0.179533i	−0.275225	−	0.961380i	\(-0.588752\pi\)
							0.623471	+	0.781847i	\(0.285722\pi\)
\(37\)	1.18666	−	0.113313i	0.195086	−	0.0186285i	0.00294454	−	0.999996i	\(-0.499063\pi\)
							0.192142	+	0.981367i	\(0.438457\pi\)
\(41\)	0.103629	+	0.226916i	0.0161842	+	0.0354384i	0.917551	−	0.397617i	\(-0.130163\pi\)
							−0.901367	+	0.433055i	\(0.857435\pi\)
\(43\)	5.27665	−	3.39110i	0.804682	−	0.517138i	−0.0724585	−	0.997371i	\(-0.523084\pi\)
							0.877141	+	0.480234i	\(0.159448\pi\)
\(47\)	−0.168216	−	0.291359i	−0.0245369	−	0.0424991i	0.853496	−	0.521099i	\(-0.174478\pi\)
							−0.878033	+	0.478600i	\(0.841144\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	322.2.m.a.39.4	✓	160
7.2	even	3	inner	322.2.m.a.177.5	yes	160
23.13	even	11	inner	322.2.m.a.151.5	yes	160
161.128	even	33	inner	322.2.m.a.289.4	yes	160

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
322.2.m.a.39.4	✓	160	1.1	even	1	trivial
322.2.m.a.151.5	yes	160	23.13	even	11	inner
322.2.m.a.177.5	yes	160	7.2	even	3	inner
322.2.m.a.289.4	yes	160	161.128	even	33	inner