Embedded newform 350.2.x.a.3.2

• Label: 350.2.x.a.3.2
• 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			3.2
Character	\(\chi\)	\(=\)	350.3
Dual form			350.2.x.a.117.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.544639 + 0.838671i) q^{2} +(-0.817783 + 2.13040i) q^{3} +(-0.406737 - 0.913545i) q^{4} +(2.21313 - 0.319493i) q^{5} +(-1.34131 - 1.84615i) q^{6} +(2.63815 - 0.200418i) q^{7} +(0.987688 + 0.156434i) q^{8} +(-1.64039 - 1.47701i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 320 q + 12 q^{5} - 8 q^{7}+O(q^{10}) \)

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
							\(3\)	−0.817783	+	2.13040i	−0.472147	+	1.22999i	0.466320	+	0.884616i	\(0.345580\pi\)
−0.938467	+	0.345369i	\(0.887754\pi\)
\(5\)	2.21313	−	0.319493i	0.989740	−	0.142882i
							\(7\)	2.63815	−	0.200418i	0.997127	−	0.0757510i
\(11\)	1.08090	+	1.20046i	0.325902	+	0.361951i								0.883723	−	0.468010i	\(-0.155029\pi\)
							−0.557821	+	0.829962i	\(0.688362\pi\)
\(13\)	5.14459	+	2.62130i	1.42685	+	0.727018i	0.985398	−	0.170266i	\(-0.0544626\pi\)
							0.441455	+	0.897284i	\(0.354463\pi\)
\(17\)	1.96313	−	2.42426i	0.476128	−	0.587969i	−0.481331	−	0.876539i	\(-0.659846\pi\)
							0.957459	+	0.288570i	\(0.0931797\pi\)
\(19\)	−5.54316	−	2.46797i	−1.27169	−	0.566192i	−0.343797	−	0.939044i	\(-0.611713\pi\)
							−0.927891	+	0.372852i	\(0.878380\pi\)
\(23\)	−5.31942	−	3.45447i	−1.10918	−	0.720307i	−0.146028	−	0.989280i	\(-0.546649\pi\)
							−0.963148	+	0.268973i	\(0.913316\pi\)
\(29\)	−3.32918	+	4.58222i	−0.618212	+	0.850896i	−0.997221	−	0.0744966i	\(-0.976265\pi\)
							0.379009	+	0.925393i	\(0.376265\pi\)
\(31\)	−3.23124	+	0.339617i	−0.580348	+	0.0609970i	−0.390154	−	0.920750i	\(-0.627578\pi\)
							−0.190194	+	0.981747i	\(0.560912\pi\)
\(37\)	0.230706	+	4.40213i	0.0379278	+	0.723705i	0.949729	+	0.313073i	\(0.101359\pi\)
							−0.911801	+	0.410632i	\(0.865308\pi\)
\(41\)	2.21303	−	0.719058i	0.345618	−	0.112298i	−0.131064	−	0.991374i	\(-0.541839\pi\)
							0.476682	+	0.879076i	\(0.341839\pi\)
\(43\)	2.56812	+	2.56812i	0.391635	+	0.391635i	0.875270	−	0.483635i	\(-0.160684\pi\)
							−0.483635	+	0.875270i	\(0.660684\pi\)
\(47\)	5.91088	−	4.78654i	0.862191	−	0.698188i	−0.0925240	−	0.995710i	\(-0.529493\pi\)
							0.954715	+	0.297522i	\(0.0961602\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	350.2.x.a.3.2	✓	320
7.5	odd	6	inner	350.2.x.a.103.19	yes	320
25.17	odd	20	inner	350.2.x.a.17.19	yes	320
175.117	even	60	inner	350.2.x.a.117.2	yes	320

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
350.2.x.a.3.2	✓	320	1.1	even	1	trivial
350.2.x.a.17.19	yes	320	25.17	odd	20	inner
350.2.x.a.103.19	yes	320	7.5	odd	6	inner
350.2.x.a.117.2	yes	320	175.117	even	60	inner