Embedded newform 322.2.m.a.39.1

• Label: 322.2.m.a.39.1
• 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			39.1
Character	\(\chi\)	\(=\)	322.39
Dual form			322.2.m.a.289.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.981929 - 0.189251i) q^{2} +(-0.150806 - 3.16580i) q^{3} +(0.928368 + 0.371662i) q^{4} +(-0.753896 + 1.05870i) q^{5} +(-0.451052 + 3.13713i) q^{6} +(-0.989718 + 2.45366i) q^{7} +(-0.841254 - 0.540641i) q^{8} +(-7.01315 + 0.669675i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 160 q - 8 q^{2} + 2 q^{3} + 8 q^{4} - 2 q^{5} + 4 q^{6} + 15 q^{7} + 16 q^{8} + 28 q^{9}+O(q^{10}) \)

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
							\(3\)	−0.150806	−	3.16580i	−0.0870678	−	1.82778i	−0.446701	−	0.894684i	\(-0.647401\pi\)
0.359633	−	0.933094i	\(-0.382902\pi\)
\(5\)	−0.753896	+	1.05870i	−0.337153	+	0.473464i	−0.947994	−	0.318289i	\(-0.896892\pi\)
							0.610841	+	0.791753i	\(0.290831\pi\)
\(7\)	−0.989718	+	2.45366i	−0.374078	+	0.927397i
							\(11\)	−1.07021	+	0.206266i	−0.322680	+	0.0621914i	−0.348019	−	0.937487i	\(-0.613146\pi\)
0.0253392	+	0.999679i	\(0.491933\pi\)
\(13\)	−5.33434	−	1.56630i	−1.47948	−	0.434414i	−0.560312	−	0.828282i	\(-0.689319\pi\)
							−0.919167	+	0.393867i	\(0.871137\pi\)
\(17\)	−2.35366	−	1.85093i	−0.570845	−	0.448918i	0.290545	−	0.956861i	\(-0.406163\pi\)
							−0.861390	+	0.507944i	\(0.830406\pi\)
\(19\)	−4.33404	+	3.40833i	−0.994297	+	0.781924i	−0.975969	−	0.217908i	\(-0.930077\pi\)
							−0.0183281	+	0.999832i	\(0.505834\pi\)
\(23\)	4.47278	−	1.73040i	0.932638	−	0.360813i
							\(29\)	0.731931	−	5.09069i	0.135916	−	0.945318i	−0.801720	−	0.597700i	\(-0.796082\pi\)
0.937636	−	0.347618i	\(-0.113009\pi\)
\(31\)	−0.404606	+	0.208589i	−0.0726693	+	0.0374637i	−0.494178	−	0.869361i	\(-0.664531\pi\)
							0.421509	+	0.906824i	\(0.361501\pi\)
\(37\)	0.425463	−	0.0406268i	0.0699456	−	0.00667900i	−0.0600245	−	0.998197i	\(-0.519118\pi\)
							0.129970	+	0.991518i	\(0.458512\pi\)
\(41\)	−0.600330	−	1.31454i	−0.0937557	−	0.205296i	0.856944	−	0.515410i	\(-0.172360\pi\)
							−0.950700	+	0.310113i	\(0.899633\pi\)
\(43\)	0.0241117	−	0.0154956i	0.00367700	−	0.00236306i	−0.538801	−	0.842433i	\(-0.681123\pi\)
							0.542478	+	0.840070i	\(0.317486\pi\)
\(47\)	−4.80350	−	8.31991i	−0.700663	−	1.21358i	−0.968234	−	0.250045i	\(-0.919554\pi\)
							0.267572	−	0.963538i	\(-0.413779\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	322.2.m.a.39.1	✓	160
7.2	even	3	inner	322.2.m.a.177.8	yes	160
23.13	even	11	inner	322.2.m.a.151.8	yes	160
161.128	even	33	inner	322.2.m.a.289.1	yes	160

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
322.2.m.a.39.1	✓	160	1.1	even	1	trivial
322.2.m.a.151.8	yes	160	23.13	even	11	inner
322.2.m.a.177.8	yes	160	7.2	even	3	inner
322.2.m.a.289.1	yes	160	161.128	even	33	inner