Embedded newform 350.2.x.a.3.12

• Label: 350.2.x.a.3.12
• 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			3.12
Character	\(\chi\)	\(=\)	350.3
Dual form			350.2.x.a.117.12

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.544639 - 0.838671i) q^{2} +(-1.02776 + 2.67740i) q^{3} +(-0.406737 - 0.913545i) q^{4} +(-1.46986 - 1.68509i) q^{5} +(1.68570 + 2.32016i) q^{6} +(0.0482142 - 2.64531i) q^{7} +(-0.987688 - 0.156434i) q^{8} +(-3.88274 - 3.49603i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 320 q + 12 q^{5} - 8 q^{7}+O(q^{10}) \)

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
							\(3\)	−1.02776	+	2.67740i	−0.593375	+	1.54580i	0.226954	+	0.973905i	\(0.427123\pi\)
−0.820330	+	0.571891i	\(0.806210\pi\)
\(5\)	−1.46986	−	1.68509i	−0.657340	−	0.753594i
							\(7\)	0.0482142	−	2.64531i	0.0182233	−	0.999834i
\(11\)	−3.05692	−	3.39505i	−0.921696	−	1.02365i								−0.999644	−	0.0266892i	\(-0.991504\pi\)
							0.0779482	−	0.996957i	\(-0.475163\pi\)
\(13\)	3.02659	+	1.54213i	0.839426	+	0.427709i	0.820180	−	0.572106i	\(-0.193873\pi\)
							0.0192459	+	0.999815i	\(0.493873\pi\)
\(17\)	4.58456	−	5.66145i	1.11192	−	1.37310i	0.191813	−	0.981431i	\(-0.438563\pi\)
							0.920105	−	0.391673i	\(-0.128104\pi\)
\(19\)	1.74637	+	0.777536i	0.400646	+	0.178379i	0.597160	−	0.802122i	\(-0.296296\pi\)
							−0.196515	+	0.980501i	\(0.562962\pi\)
\(23\)	−4.68681	−	3.04365i	−0.977268	−	0.634645i	−0.0459853	−	0.998942i	\(-0.514643\pi\)
							−0.931283	+	0.364297i	\(0.881309\pi\)
\(29\)	−0.582338	+	0.801519i	−0.108137	+	0.148838i	−0.859656	−	0.510874i	\(-0.829322\pi\)
							0.751518	+	0.659712i	\(0.229322\pi\)
\(31\)	−8.88374	+	0.933719i	−1.59557	+	0.167701i	−0.860196	−	0.509964i	\(-0.829659\pi\)
							−0.735371	+	0.677665i	\(0.762992\pi\)
\(37\)	−0.543888	−	10.3780i	−0.0894147	−	1.70613i	−0.563877	−	0.825859i	\(-0.690691\pi\)
							0.474462	−	0.880276i	\(-0.342642\pi\)
\(41\)	−0.915329	+	0.297408i	−0.142950	+	0.0464474i	−0.379618	−	0.925143i	\(-0.623945\pi\)
							0.236668	+	0.971591i	\(0.423945\pi\)
\(43\)	1.12643	+	1.12643i	0.171779	+	0.171779i	0.787760	−	0.615982i	\(-0.211241\pi\)
							−0.615982	+	0.787760i	\(0.711241\pi\)
\(47\)	−2.35951	+	1.91069i	−0.344170	+	0.278703i	−0.785783	−	0.618503i	\(-0.787740\pi\)
							0.441613	+	0.897205i	\(0.354406\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	350.2.x.a.3.12	✓	320
7.5	odd	6	inner	350.2.x.a.103.9	yes	320
25.17	odd	20	inner	350.2.x.a.17.9	yes	320
175.117	even	60	inner	350.2.x.a.117.12	yes	320

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
350.2.x.a.3.12	✓	320	1.1	even	1	trivial
350.2.x.a.17.9	yes	320	25.17	odd	20	inner
350.2.x.a.103.9	yes	320	7.5	odd	6	inner
350.2.x.a.117.12	yes	320	175.117	even	60	inner