Embedded newform 350.2.u.a.9.14

• Label: 350.2.u.a.9.14
• 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			9.14
Character	\(\chi\)	\(=\)	350.9
Dual form			350.2.u.a.39.14

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.406737 + 0.913545i) q^{2} +(-0.383437 - 0.345248i) q^{3} +(-0.669131 + 0.743145i) q^{4} +(2.15736 - 0.588036i) q^{5} +(0.159442 - 0.490712i) q^{6} +(0.792205 - 2.52436i) q^{7} +(-0.951057 - 0.309017i) q^{8} +(-0.285758 - 2.71880i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 160 q - 20 q^{4} - 2 q^{5} + 8 q^{6} - 20 q^{9}+O(q^{10}) \)

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
							\(3\)	−0.383437	−	0.345248i	−0.221378	−	0.199329i	0.550978	−	0.834520i	\(-0.314255\pi\)
−0.772355	+	0.635191i	\(0.780922\pi\)
\(5\)	2.15736	−	0.588036i	0.964802	−	0.262978i
							\(7\)	0.792205	−	2.52436i	0.299426	−	0.954120i
\(11\)	0.0983024	−	0.935285i	0.0296393	−	0.281999i								−0.969656	−	0.244472i	\(-0.921385\pi\)
							0.999296	−	0.0375269i	\(-0.0119480\pi\)
\(13\)	3.32718	+	4.57947i	0.922794	+	1.27012i	0.962605	+	0.270910i	\(0.0873245\pi\)
							−0.0398109	+	0.999207i	\(0.512676\pi\)
\(17\)	−1.09803	−	5.16581i	−0.266311	−	1.25289i	−0.884379	−	0.466769i	\(-0.845418\pi\)
							0.618069	−	0.786124i	\(-0.287915\pi\)
\(19\)	0.322527	+	0.358203i	0.0739928	+	0.0821773i	0.779004	−	0.627018i	\(-0.215725\pi\)
							−0.705012	+	0.709196i	\(0.749058\pi\)
\(23\)	1.83352	+	4.11815i	0.382315	+	0.858694i	0.997524	+	0.0703298i	\(0.0224052\pi\)
							−0.615208	+	0.788364i	\(0.710928\pi\)
\(29\)	2.72137	+	8.37551i	0.505345	+	1.55529i	0.800189	+	0.599748i	\(0.204732\pi\)
							−0.294844	+	0.955546i	\(0.595268\pi\)
\(31\)	−7.73931	+	1.64504i	−1.39002	+	0.295458i	−0.841310	−	0.540554i	\(-0.818215\pi\)
							−0.548712	+	0.836012i	\(0.684881\pi\)
\(37\)	1.81896	−	0.191181i	0.299036	−	0.0314299i	0.0461776	−	0.998933i	\(-0.485296\pi\)
							0.252858	+	0.967503i	\(0.418629\pi\)
\(41\)	3.66686	−	2.66413i	0.572667	−	0.416067i	−0.263406	−	0.964685i	\(-0.584846\pi\)
							0.836073	+	0.548618i	\(0.184846\pi\)
\(43\)		−	1.60652i		−	0.244992i	−0.992469	−	0.122496i	\(-0.960910\pi\)
							0.992469	−	0.122496i	\(-0.0390899\pi\)
\(47\)	0.268435	−	1.26289i	0.0391553	−	0.184211i	−0.954224	−	0.299093i	\(-0.903316\pi\)
							0.993379	+	0.114882i	\(0.0366491\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	350.2.u.a.9.14	✓	160
7.4	even	3	inner	350.2.u.a.109.4	yes	160
25.14	even	10	inner	350.2.u.a.289.4	yes	160
175.39	even	30	inner	350.2.u.a.39.14	yes	160

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
350.2.u.a.9.14	✓	160	1.1	even	1	trivial
350.2.u.a.39.14	yes	160	175.39	even	30	inner
350.2.u.a.109.4	yes	160	7.4	even	3	inner
350.2.u.a.289.4	yes	160	25.14	even	10	inner