Embedded newform 370.2.v.a.169.1

• Label: 370.2.v.a.169.1
• Level: $370$
• Weight: $2$
• Character: 370.169
• Analytic conductor: $2.954$
• Analytic rank: $0$
• Dimension: $60$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 370 = 2 \cdot 5 \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	370.v (of order \(18\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.95446487479\)
Analytic rank:	\(0\)
Dimension:	\(60\)
Relative dimension:	\(10\) over \(\Q(\zeta_{18})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{18}]$

Embedding invariants

Embedding label			169.1
Character	\(\chi\)	\(=\)	370.169
Dual form			370.2.v.a.289.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.939693 - 0.342020i) q^{2} +(-1.01326 - 2.78391i) q^{3} +(0.766044 + 0.642788i) q^{4} +(-2.07464 - 0.834194i) q^{5} +2.96257i q^{6} +(1.02406 + 0.180569i) q^{7} +(-0.500000 - 0.866025i) q^{8} +(-4.42531 + 3.71328i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 60 q - 3 q^{5} - 30 q^{8} - 6 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/370\mathbb{Z}\right)^\times\).

\(n\)	\(261\)	\(297\)
\(\chi(n)\)	\(e\left(\frac{11}{18}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.939693	−	0.342020i	−0.664463	−	0.241845i
							\(3\)	−1.01326	−	2.78391i	−0.585006	−	1.60729i	−0.779511	−	0.626388i	\(-0.784532\pi\)
0.194505	−	0.980901i	\(-0.437690\pi\)
\(5\)	−2.07464	−	0.834194i	−0.927806	−	0.373063i
							\(7\)	1.02406	+	0.180569i	0.387058	+	0.0682488i	0.363791	−	0.931480i	\(-0.381482\pi\)
0.0232669	+	0.999729i	\(0.492593\pi\)
\(11\)	−1.08964	−	1.88731i	−0.328539	−	0.569046i	0.653683	−	0.756768i	\(-0.273223\pi\)
							−0.982222	+	0.187722i	\(0.939890\pi\)
\(13\)	−3.72808	−	3.12823i	−1.03398	−	0.867615i	−0.0426633	−	0.999090i	\(-0.513584\pi\)
							−0.991319	+	0.131475i	\(0.958029\pi\)
\(17\)	−0.949228	+	0.796496i	−0.230221	+	0.193179i	−0.750600	−	0.660757i	\(-0.770235\pi\)
							0.520378	+	0.853936i	\(0.325791\pi\)
\(19\)	1.59512	+	4.38255i	0.365945	+	1.00543i	0.976888	+	0.213752i	\(0.0685684\pi\)
							−0.610942	+	0.791675i	\(0.709209\pi\)
\(23\)	−2.09433	+	3.62748i	−0.436697	+	0.756382i	−0.997432	−	0.0716133i	\(-0.977185\pi\)
							0.560735	+	0.827995i	\(0.310519\pi\)
\(29\)	6.14779	−	3.54943i	1.14162	−	0.659112i	0.194785	−	0.980846i	\(-0.437599\pi\)
							0.946830	+	0.321734i	\(0.104266\pi\)
\(31\)			3.83594i			0.688955i	0.938795	+	0.344477i	\(0.111944\pi\)
							−0.938795	+	0.344477i	\(0.888056\pi\)
\(37\)	5.16263	+	3.21671i	0.848731	+	0.528824i
							\(41\)	−5.14883	−	4.32038i	−0.804113	−	0.674731i	0.145082	−	0.989420i	\(-0.453655\pi\)
−0.949195	+	0.314689i	\(0.898100\pi\)
\(43\)	2.74566			0.418710			0.209355	−	0.977840i	\(-0.432864\pi\)
							0.209355	+	0.977840i	\(0.432864\pi\)
\(47\)	−10.9104	−	6.29914i	−1.59145	−	0.918824i	−0.993059	−	0.117621i	\(-0.962473\pi\)
							−0.598392	−	0.801204i	\(-0.704193\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	370.2.v.a.169.1	✓	60
5.4	even	2		370.2.v.b.169.10	yes	60
37.30	even	18		370.2.v.b.289.10	yes	60
185.104	even	18	inner	370.2.v.a.289.1	yes	60

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
370.2.v.a.169.1	✓	60	1.1	even	1	trivial
370.2.v.a.289.1	yes	60	185.104	even	18	inner
370.2.v.b.169.10	yes	60	5.4	even	2
370.2.v.b.289.10	yes	60	37.30	even	18