Embedded newform 350.2.u.a.9.19

• Label: 350.2.u.a.9.19
• Level: $350$
• Weight: $2$
• Character: 350.9
• Analytic conductor: $2.795$
• Analytic rank: $0$
• Dimension: $160$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 350 = 2 \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	350.u (of order \(30\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.79476407074\)
Analytic rank:	\(0\)
Dimension:	\(160\)
Relative dimension:	\(20\) over \(\Q(\zeta_{30})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{30}]$

Embedding invariants

Embedding label			9.19
Character	\(\chi\)	\(=\)	350.9
Dual form			350.2.u.a.39.19

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.406737 + 0.913545i) q^{2} +(2.06743 + 1.86152i) q^{3} +(-0.669131 + 0.743145i) q^{4} +(0.677494 - 2.13096i) q^{5} +(-0.859686 + 2.64584i) q^{6} +(2.16518 - 1.52052i) q^{7} +(-0.951057 - 0.309017i) q^{8} +(0.495416 + 4.71357i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 160 q - 20 q^{4} - 2 q^{5} + 8 q^{6} - 20 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/350\mathbb{Z}\right)^\times\).

\(n\)	\(101\)	\(127\)
\(\chi(n)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{7}{10}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.406737	+	0.913545i	0.287606	+	0.645974i
							\(3\)	2.06743	+	1.86152i	1.19363	+	1.07475i	0.995518	+	0.0945713i	\(0.0301480\pi\)
0.198113	+	0.980179i	\(0.436519\pi\)
\(5\)	0.677494	−	2.13096i	0.302984	−	0.952996i
							\(7\)	2.16518	−	1.52052i	0.818362	−	0.574703i
\(11\)	−0.571247	+	5.43505i	−0.172237	+	1.63873i								0.477544	+	0.878608i	\(0.341527\pi\)
							−0.649781	+	0.760121i	\(0.725140\pi\)
\(13\)	−2.66753	−	3.67153i	−0.739838	−	1.01830i	−0.998628	−	0.0523665i	\(-0.983324\pi\)
							0.258790	−	0.965934i	\(-0.416676\pi\)
\(17\)	−0.0537151	−	0.252709i	−0.0130278	−	0.0612910i	0.971154	−	0.238455i	\(-0.0766410\pi\)
							−0.984181	+	0.177164i	\(0.943308\pi\)
\(19\)	−1.04316	−	1.15855i	−0.239318	−	0.265789i	0.611507	−	0.791239i	\(-0.290564\pi\)
							−0.850824	+	0.525450i	\(0.823897\pi\)
\(23\)	1.62776	+	3.65601i	0.339411	+	0.762330i	0.999934	+	0.0115238i	\(0.00366822\pi\)
							−0.660522	+	0.750806i	\(0.729665\pi\)
\(29\)	−2.92883	−	9.01402i	−0.543871	−	1.67386i	−0.723660	−	0.690157i	\(-0.757542\pi\)
							0.179789	−	0.983705i	\(-0.442458\pi\)
\(31\)	−6.44883	+	1.37074i	−1.15824	+	0.246192i	−0.746677	−	0.665187i	\(-0.768352\pi\)
							−0.411567	+	0.911379i	\(0.635019\pi\)
\(37\)	−0.913259	+	0.0959874i	−0.150139	+	0.0157802i	−0.179300	−	0.983794i	\(-0.557383\pi\)
							0.0291608	+	0.999575i	\(0.490717\pi\)
\(41\)	−2.95511	+	2.14701i	−0.461510	+	0.335307i	−0.794123	−	0.607757i	\(-0.792070\pi\)
							0.332613	+	0.943063i	\(0.392070\pi\)
\(43\)			6.07475i			0.926391i	0.886256	+	0.463195i	\(0.153297\pi\)
							−0.886256	+	0.463195i	\(0.846703\pi\)
\(47\)	1.91752	−	9.02124i	0.279700	−	1.31588i	−0.583950	−	0.811790i	\(-0.698494\pi\)
							0.863649	−	0.504093i	\(-0.168173\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	350.2.u.a.9.19	✓	160
7.4	even	3	inner	350.2.u.a.109.9	yes	160
25.14	even	10	inner	350.2.u.a.289.9	yes	160
175.39	even	30	inner	350.2.u.a.39.19	yes	160

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
350.2.u.a.9.19	✓	160	1.1	even	1	trivial
350.2.u.a.39.19	yes	160	175.39	even	30	inner
350.2.u.a.109.9	yes	160	7.4	even	3	inner
350.2.u.a.289.9	yes	160	25.14	even	10	inner