Embedded newform 350.2.x.a.3.16

• Label: 350.2.x.a.3.16
• 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			3.16
Character	\(\chi\)	\(=\)	350.3
Dual form			350.2.x.a.117.16

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.544639 - 0.838671i) q^{2} +(-0.0130481 + 0.0339916i) q^{3} +(-0.406737 - 0.913545i) q^{4} +(0.859199 + 2.06441i) q^{5} +(0.0214012 + 0.0294562i) q^{6} +(-1.30429 + 2.30192i) q^{7} +(-0.987688 - 0.156434i) q^{8} +(2.22845 + 2.00650i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 320 q + 12 q^{5} - 8 q^{7}+O(q^{10}) \)

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
							\(3\)	−0.0130481	+	0.0339916i	−0.00753335	+	0.0196251i	−0.937296	−	0.348536i	\(-0.886679\pi\)
0.929762	+	0.368161i	\(0.120012\pi\)
\(5\)	0.859199	+	2.06441i	0.384245	+	0.923231i
							\(7\)	−1.30429	+	2.30192i	−0.492977	+	0.870042i
\(11\)	1.07487	+	1.19377i	0.324086	+	0.359934i								0.883067	−	0.469246i	\(-0.155474\pi\)
							−0.558981	+	0.829180i	\(0.688808\pi\)
\(13\)	2.59604	+	1.32275i	0.720013	+	0.366865i	0.775289	−	0.631606i	\(-0.217604\pi\)
							−0.0552765	+	0.998471i	\(0.517604\pi\)
\(17\)	4.38346	−	5.41312i	1.06314	−	1.31287i	0.115584	−	0.993298i	\(-0.463126\pi\)
							0.947560	−	0.319577i	\(-0.103541\pi\)
\(19\)	−2.73343	−	1.21700i	−0.627092	−	0.279199i	0.0684838	−	0.997652i	\(-0.478184\pi\)
							−0.695575	+	0.718453i	\(0.744851\pi\)
\(23\)	−5.35683	−	3.47877i	−1.11698	−	0.725373i	−0.152176	−	0.988353i	\(-0.548628\pi\)
							−0.964801	+	0.262980i	\(0.915295\pi\)
\(29\)	−4.72535	+	6.50389i	−0.877476	+	1.20774i	0.0996372	+	0.995024i	\(0.468232\pi\)
							−0.977113	+	0.212719i	\(0.931768\pi\)
\(31\)	8.08182	−	0.849434i	1.45154	−	0.152563i	0.654303	−	0.756232i	\(-0.272962\pi\)
							0.797235	+	0.603669i	\(0.206295\pi\)
\(37\)	−0.451619	−	8.61740i	−0.0742456	−	1.41669i	−0.741046	−	0.671455i	\(-0.765670\pi\)
							0.666800	−	0.745237i	\(-0.267663\pi\)
\(41\)	9.04237	−	2.93804i	1.41218	−	0.458846i	0.499072	−	0.866560i	\(-0.333674\pi\)
							0.913109	+	0.407715i	\(0.133674\pi\)
\(43\)	2.85780	+	2.85780i	0.435810	+	0.435810i	0.890599	−	0.454789i	\(-0.150285\pi\)
							−0.454789	+	0.890599i	\(0.650285\pi\)
\(47\)	−6.65883	+	5.39221i	−0.971290	+	0.786535i	−0.976954	−	0.213449i	\(-0.931530\pi\)
							0.00566433	+	0.999984i	\(0.498197\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	350.2.x.a.3.16	✓	320
7.5	odd	6	inner	350.2.x.a.103.5	yes	320
25.17	odd	20	inner	350.2.x.a.17.5	yes	320
175.117	even	60	inner	350.2.x.a.117.16	yes	320

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
350.2.x.a.3.16	✓	320	1.1	even	1	trivial
350.2.x.a.17.5	yes	320	25.17	odd	20	inner
350.2.x.a.103.5	yes	320	7.5	odd	6	inner
350.2.x.a.117.16	yes	320	175.117	even	60	inner