Embedded newform 350.2.r.a.27.19

• Label: 350.2.r.a.27.19
• Level: $350$
• Weight: $2$
• Character: 350.27
• Analytic conductor: $2.795$
• Analytic rank: $0$
• Dimension: $160$
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 350 = 2 \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	350.r (of order \(20\), degree \(8\), minimal)

Newform invariants

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

Embedding invariants

Embedding label			27.19
Character	\(\chi\)	\(=\)	350.27
Dual form			350.2.r.a.13.19

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.987688 + 0.156434i) q^{2} +(1.66180 - 0.846728i) q^{3} +(0.951057 + 0.309017i) q^{4} +(0.499152 + 2.17964i) q^{5} +(1.77380 - 0.576341i) q^{6} +(-2.54075 - 0.737955i) q^{7} +(0.891007 + 0.453990i) q^{8} +(0.281267 - 0.387131i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 160 q + 8 q^{7}+O(q^{10}) \)

Character values

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

\(n\)	\(101\)	\(127\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{1}{20}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.987688	+	0.156434i	0.698401	+	0.110616i
							\(3\)	1.66180	−	0.846728i	0.959439	−	0.488859i	0.0971473	−	0.995270i	\(-0.469028\pi\)
0.862292	+	0.506411i	\(0.169028\pi\)
\(5\)	0.499152	+	2.17964i	0.223228	+	0.974766i
							\(7\)	−2.54075	−	0.737955i	−0.960314	−	0.278921i
\(11\)	4.96002	−	3.60366i	1.49550	−	1.08655i								0.523371	−	0.852105i	\(-0.324674\pi\)
							0.972130	−	0.234441i	\(-0.0753258\pi\)
\(13\)	0.692430	+	4.37183i	0.192045	+	1.21253i	0.875749	+	0.482767i	\(0.160368\pi\)
							−0.683703	+	0.729760i	\(0.739632\pi\)
\(17\)	−1.52004	−	0.774499i	−0.368664	−	0.187844i	0.259837	−	0.965652i	\(-0.416331\pi\)
							−0.628501	+	0.777809i	\(0.716331\pi\)
\(19\)	−1.94881	−	5.99783i	−0.447089	−	1.37600i	−0.880177	−	0.474646i	\(-0.842576\pi\)
							0.433088	−	0.901351i	\(-0.357424\pi\)
\(23\)	0.284423	−	1.79578i	0.0593063	−	0.374445i	−0.940128	−	0.340821i	\(-0.889295\pi\)
							0.999435	−	0.0336244i	\(-0.0107050\pi\)
\(29\)	−8.17286	−	2.65552i	−1.51766	−	0.493118i	−0.572551	−	0.819869i	\(-0.694046\pi\)
							−0.945111	+	0.326751i	\(0.894046\pi\)
\(31\)	−4.43176	+	1.43997i	−0.795968	+	0.258626i	−0.678644	−	0.734468i	\(-0.737432\pi\)
							−0.117325	+	0.993094i	\(0.537432\pi\)
\(37\)	−0.563587	−	3.55835i	−0.0926531	−	0.584989i	−0.989712	−	0.143077i	\(-0.954300\pi\)
							0.897058	−	0.441912i	\(-0.145700\pi\)
\(41\)	0.248189	−	0.341603i	0.0387606	−	0.0533494i	−0.789197	−	0.614140i	\(-0.789503\pi\)
							0.827957	+	0.560791i	\(0.189503\pi\)
\(43\)	0.714628	−	0.714628i	0.108980	−	0.108980i	−0.650514	−	0.759494i	\(-0.725447\pi\)
							0.759494	+	0.650514i	\(0.225447\pi\)
\(47\)	0.635825	+	1.24788i	0.0927446	+	0.182022i	0.932719	−	0.360605i	\(-0.117430\pi\)
							−0.839974	+	0.542627i	\(0.817430\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	350.2.r.a.27.19	yes	160
7.6	odd	2	inner	350.2.r.a.27.12	yes	160
25.13	odd	20	inner	350.2.r.a.13.12	✓	160
175.13	even	20	inner	350.2.r.a.13.19	yes	160

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
350.2.r.a.13.12	✓	160	25.13	odd	20	inner
350.2.r.a.13.19	yes	160	175.13	even	20	inner
350.2.r.a.27.12	yes	160	7.6	odd	2	inner
350.2.r.a.27.19	yes	160	1.1	even	1	trivial