Embedded newform 350.2.x.a.3.4

• Label: 350.2.x.a.3.4
• Level: $350$
• Weight: $2$
• Character: 350.3
• Analytic conductor: $2.795$
• Analytic rank: $0$
• Dimension: $320$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 350 = 2 \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	350.x (of order \(60\), degree \(16\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.79476407074\)
Analytic rank:	\(0\)
Dimension:	\(320\)
Relative dimension:	\(20\) over \(\Q(\zeta_{60})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{60}]$

Embedding invariants

Embedding label			3.4
Character	\(\chi\)	\(=\)	350.3
Dual form			350.2.x.a.117.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.544639 + 0.838671i) q^{2} +(-0.316438 + 0.824348i) q^{3} +(-0.406737 - 0.913545i) q^{4} +(0.514287 - 2.17612i) q^{5} +(-0.519012 - 0.714359i) q^{6} +(-2.61096 - 0.427657i) q^{7} +(0.987688 + 0.156434i) q^{8} +(1.65002 + 1.48568i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 320 q + 12 q^{5} - 8 q^{7}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/350\mathbb{Z}\right)^\times\).

\(n\)	\(101\)	\(127\)
\(\chi(n)\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{7}{20}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.544639	+	0.838671i	−0.385118	+	0.593030i
							\(3\)	−0.316438	+	0.824348i	−0.182695	+	0.475937i	−0.994090	−	0.108561i	\(-0.965376\pi\)
0.811394	+	0.584499i	\(0.198709\pi\)
\(5\)	0.514287	−	2.17612i	0.229996	−	0.973192i
							\(7\)	−2.61096	−	0.427657i	−0.986850	−	0.161639i
\(11\)	2.53990	+	2.82085i	0.765809	+	0.850517i								0.992346	−	0.123485i	\(-0.0394071\pi\)
							−0.226537	+	0.974003i	\(0.572740\pi\)
\(13\)	1.44839	+	0.737992i	0.401711	+	0.204682i	0.643161	−	0.765731i	\(-0.277623\pi\)
							−0.241449	+	0.970413i	\(0.577623\pi\)
\(17\)	4.18720	−	5.17076i	1.01554	−	1.25409i	0.0486769	−	0.998815i	\(-0.484500\pi\)
							0.966868	−	0.255278i	\(-0.0821671\pi\)
\(19\)	7.77424	+	3.46131i	1.78353	+	0.794080i	0.980050	+	0.198751i	\(0.0636886\pi\)
							0.803482	+	0.595329i	\(0.202978\pi\)
\(23\)	2.06336	+	1.33996i	0.430241	+	0.279402i	0.741538	−	0.670911i	\(-0.234097\pi\)
							−0.311297	+	0.950313i	\(0.600763\pi\)
\(29\)	−3.41045	+	4.69408i	−0.633304	+	0.871668i	−0.998236	−	0.0593660i	\(-0.981092\pi\)
							0.364932	+	0.931034i	\(0.381092\pi\)
\(31\)	5.27597	−	0.554527i	0.947592	−	0.0995959i	0.381892	−	0.924207i	\(-0.375273\pi\)
							0.565700	+	0.824611i	\(0.308606\pi\)
\(37\)	−0.343285	−	6.55026i	−0.0564356	−	1.07686i	−0.869263	−	0.494351i	\(-0.835406\pi\)
							0.812827	−	0.582505i	\(-0.197927\pi\)
\(41\)	−4.80423	+	1.56099i	−0.750294	+	0.243785i	−0.659108	−	0.752048i	\(-0.729066\pi\)
							−0.0911866	+	0.995834i	\(0.529066\pi\)
\(43\)	−3.22963	−	3.22963i	−0.492514	−	0.492514i	0.416584	−	0.909097i	\(-0.363227\pi\)
							−0.909097	+	0.416584i	\(0.863227\pi\)
\(47\)	−2.53251	+	2.05079i	−0.369405	+	0.299138i	−0.795991	−	0.605309i	\(-0.793050\pi\)
							0.426586	+	0.904447i	\(0.359716\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	350.2.x.a.3.4	✓	320
7.5	odd	6	inner	350.2.x.a.103.17	yes	320
25.17	odd	20	inner	350.2.x.a.17.17	yes	320
175.117	even	60	inner	350.2.x.a.117.4	yes	320

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
350.2.x.a.3.4	✓	320	1.1	even	1	trivial
350.2.x.a.17.17	yes	320	25.17	odd	20	inner
350.2.x.a.103.17	yes	320	7.5	odd	6	inner
350.2.x.a.117.4	yes	320	175.117	even	60	inner