Embedded newform 350.2.x.a.3.10

• Label: 350.2.x.a.3.10
• 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			3.10
Character	\(\chi\)	\(=\)	350.3
Dual form			350.2.x.a.117.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.544639 + 0.838671i) q^{2} +(1.00842 - 2.62703i) q^{3} +(-0.406737 - 0.913545i) q^{4} +(-0.333527 - 2.21105i) q^{5} +(1.65399 + 2.27652i) q^{6} +(2.63222 + 0.267259i) q^{7} +(0.987688 + 0.156434i) q^{8} +(-3.65494 - 3.29092i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 320 q + 12 q^{5} - 8 q^{7}+O(q^{10}) \)

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
							\(3\)	1.00842	−	2.62703i	0.582213	−	1.51672i	−0.253379	−	0.967367i	\(-0.581542\pi\)
0.835592	−	0.549350i	\(-0.185125\pi\)
\(5\)	−0.333527	−	2.21105i	−0.149158	−	0.988813i
							\(7\)	2.63222	+	0.267259i	0.994885	+	0.101015i
\(11\)	3.33150	+	3.70000i	1.00448	+	1.11559i								0.993289	+	0.115656i	\(0.0368969\pi\)
							0.0111952	+	0.999937i	\(0.496436\pi\)
\(13\)	−3.79623	−	1.93427i	−1.05288	−	0.536471i	−0.160167	−	0.987090i	\(-0.551203\pi\)
							−0.892716	+	0.450619i	\(0.851203\pi\)
\(17\)	2.19117	−	2.70587i	0.531437	−	0.656270i	−0.438755	−	0.898606i	\(-0.644581\pi\)
							0.970192	+	0.242337i	\(0.0779139\pi\)
\(19\)	−2.37160	−	1.05591i	−0.544083	−	0.242241i	0.116243	−	0.993221i	\(-0.462915\pi\)
							−0.660325	+	0.750980i	\(0.729582\pi\)
\(23\)	−0.894201	−	0.580701i	−0.186454	−	0.121085i	0.448037	−	0.894015i	\(-0.352123\pi\)
							−0.634491	+	0.772931i	\(0.718790\pi\)
\(29\)	0.289574	−	0.398564i	0.0537725	−	0.0740115i	−0.781284	−	0.624176i	\(-0.785435\pi\)
							0.835056	+	0.550164i	\(0.185435\pi\)
\(31\)	−3.43310	+	0.360833i	−0.616602	+	0.0648075i	−0.407680	−	0.913125i	\(-0.633662\pi\)
							−0.208922	+	0.977932i	\(0.566996\pi\)
\(37\)	0.427996	+	8.16666i	0.0703622	+	1.34259i	0.774521	+	0.632548i	\(0.217991\pi\)
							−0.704159	+	0.710042i	\(0.748676\pi\)
\(41\)	−6.47290	+	2.10317i	−1.01090	+	0.328460i	−0.767212	−	0.641394i	\(-0.778357\pi\)
							−0.243686	+	0.969854i	\(0.578357\pi\)
\(43\)	3.91580	+	3.91580i	0.597154	+	0.597154i	0.939554	−	0.342400i	\(-0.111240\pi\)
							−0.342400	+	0.939554i	\(0.611240\pi\)
\(47\)	8.38541	−	6.79037i	1.22314	−	0.990477i	0.223261	−	0.974759i	\(-0.428330\pi\)
							0.999876	−	0.0157188i	\(-0.00500365\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	350.2.x.a.3.10	✓	320
7.5	odd	6	inner	350.2.x.a.103.11	yes	320
25.17	odd	20	inner	350.2.x.a.17.11	yes	320
175.117	even	60	inner	350.2.x.a.117.10	yes	320

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
350.2.x.a.3.10	✓	320	1.1	even	1	trivial
350.2.x.a.17.11	yes	320	25.17	odd	20	inner
350.2.x.a.103.11	yes	320	7.5	odd	6	inner
350.2.x.a.117.10	yes	320	175.117	even	60	inner