Embedded newform 350.2.x.a.3.1

• Label: 350.2.x.a.3.1
• 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			3.1
Character	\(\chi\)	\(=\)	350.3
Dual form			350.2.x.a.117.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.544639 + 0.838671i) q^{2} +(-1.15652 + 3.01283i) q^{3} +(-0.406737 - 0.913545i) q^{4} +(-1.77963 - 1.35386i) q^{5} +(-1.89689 - 2.61084i) q^{6} +(-0.00275235 + 2.64575i) q^{7} +(0.987688 + 0.156434i) q^{8} +(-5.51019 - 4.96140i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 320 q + 12 q^{5} - 8 q^{7}+O(q^{10}) \)

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
							\(3\)	−1.15652	+	3.01283i	−0.667716	+	1.73946i	0.00546846	+	0.999985i	\(0.498259\pi\)
−0.673184	+	0.739475i	\(0.735074\pi\)
\(5\)	−1.77963	−	1.35386i	−0.795873	−	0.605464i
							\(7\)	−0.00275235	+	2.64575i	−0.00104029	+	0.999999i
\(11\)	0.696124	+	0.773125i	0.209889	+	0.233106i								0.838893	−	0.544297i	\(-0.183203\pi\)
							−0.629003	+	0.777403i	\(0.716537\pi\)
\(13\)	−3.97493	−	2.02533i	−1.10245	−	0.561725i	−0.194540	−	0.980895i	\(-0.562321\pi\)
							−0.907908	+	0.419169i	\(0.862321\pi\)
\(17\)	1.70623	−	2.10702i	0.413822	−	0.511027i	−0.527030	−	0.849847i	\(-0.676694\pi\)
							0.940851	+	0.338820i	\(0.110028\pi\)
\(19\)	−2.97009	−	1.32237i	−0.681386	−	0.303373i	0.0367005	−	0.999326i	\(-0.488315\pi\)
							−0.718087	+	0.695954i	\(0.754982\pi\)
\(23\)	4.00771	+	2.60264i	0.835665	+	0.542687i	0.890066	−	0.455832i	\(-0.150658\pi\)
							−0.0544007	+	0.998519i	\(0.517325\pi\)
\(29\)	2.96953	−	4.08721i	0.551428	−	0.758976i	−0.438777	−	0.898596i	\(-0.644588\pi\)
							0.990205	+	0.139620i	\(0.0445881\pi\)
\(31\)	−5.23139	+	0.549842i	−0.939586	+	0.0987545i	−0.561924	−	0.827189i	\(-0.689939\pi\)
							−0.377662	+	0.925943i	\(0.623272\pi\)
\(37\)	0.133100	+	2.53970i	0.0218815	+	0.417524i	0.987506	+	0.157582i	\(0.0503699\pi\)
							−0.965624	+	0.259941i	\(0.916297\pi\)
\(41\)	1.71718	−	0.557946i	0.268178	−	0.0871365i	−0.171841	−	0.985125i	\(-0.554972\pi\)
							0.440019	+	0.897988i	\(0.354972\pi\)
\(43\)	−4.92704	−	4.92704i	−0.751367	−	0.751367i	0.223368	−	0.974734i	\(-0.428295\pi\)
							−0.974734	+	0.223368i	\(0.928295\pi\)
\(47\)	−9.14075	+	7.40204i	−1.33332	+	1.07970i	−0.343227	+	0.939252i	\(0.611520\pi\)
							−0.990088	+	0.140445i	\(0.955147\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	350.2.x.a.3.1	✓	320
7.5	odd	6	inner	350.2.x.a.103.20	yes	320
25.17	odd	20	inner	350.2.x.a.17.20	yes	320
175.117	even	60	inner	350.2.x.a.117.1	yes	320

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
350.2.x.a.3.1	✓	320	1.1	even	1	trivial
350.2.x.a.17.20	yes	320	25.17	odd	20	inner
350.2.x.a.103.20	yes	320	7.5	odd	6	inner
350.2.x.a.117.1	yes	320	175.117	even	60	inner