Embedded newform 350.2.r.a.13.10

• Label: 350.2.r.a.13.10
• Level: $350$
• Weight: $2$
• Character: 350.13
• 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.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			13.10
Character	\(\chi\)	\(=\)	350.13
Dual form			350.2.r.a.27.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.987688 + 0.156434i) q^{2} +(2.24936 + 1.14611i) q^{3} +(0.951057 - 0.309017i) q^{4} +(-0.898072 + 2.04780i) q^{5} +(-2.40096 - 0.780120i) q^{6} +(-1.82192 - 1.91850i) q^{7} +(-0.891007 + 0.453990i) q^{8} +(1.98272 + 2.72898i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 160 q + 8 q^{7}+O(q^{10}) \)

Character values

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

\(n\)	\(101\)	\(127\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{19}{20}\right)\)

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
							\(3\)	2.24936	+	1.14611i	1.29867	+	0.661706i	0.960211	−	0.279275i	\(-0.0900940\pi\)
0.338460	+	0.940981i	\(0.390094\pi\)
\(5\)	−0.898072	+	2.04780i	−0.401630	+	0.915802i
							\(7\)	−1.82192	−	1.91850i	−0.688619	−	0.725123i
\(11\)	3.41872	+	2.48385i	1.03078	+	0.748908i								0.968465	−	0.249148i	\(-0.0801506\pi\)
							0.0623183	+	0.998056i	\(0.480151\pi\)
\(13\)	−0.770695	+	4.86597i	−0.213752	+	1.34958i	0.614364	+	0.789023i	\(0.289413\pi\)
							−0.828117	+	0.560556i	\(0.810587\pi\)
\(17\)	4.18702	−	2.13339i	1.01550	−	0.517423i	0.134688	−	0.990888i	\(-0.456997\pi\)
							0.880813	+	0.473465i	\(0.156997\pi\)
\(19\)	−2.18882	+	6.73649i	−0.502150	+	1.54546i	0.303361	+	0.952876i	\(0.401891\pi\)
							−0.805510	+	0.592582i	\(0.798109\pi\)
\(23\)	−0.306485	−	1.93507i	−0.0639066	−	0.403490i	−0.998818	−	0.0486089i	\(-0.984521\pi\)
							0.934911	−	0.354881i	\(-0.115479\pi\)
\(29\)	1.31572	−	0.427503i	0.244323	−	0.0793853i	−0.184296	−	0.982871i	\(-0.559000\pi\)
							0.428619	+	0.903486i	\(0.359000\pi\)
\(31\)	−2.19010	−	0.711605i	−0.393353	−	0.127808i	0.105661	−	0.994402i	\(-0.466304\pi\)
							−0.499014	+	0.866594i	\(0.666304\pi\)
\(37\)	1.59378	−	10.0627i	0.262016	−	1.65430i	−0.408758	−	0.912643i	\(-0.634038\pi\)
							0.670774	−	0.741662i	\(-0.265962\pi\)
\(41\)	3.30195	+	4.54475i	0.515678	+	0.709770i	0.984864	−	0.173329i	\(-0.0554525\pi\)
							−0.469186	+	0.883099i	\(0.655453\pi\)
\(43\)	−2.96957	−	2.96957i	−0.452855	−	0.452855i	0.443446	−	0.896301i	\(-0.353756\pi\)
							−0.896301	+	0.443446i	\(0.853756\pi\)
\(47\)	−0.385551	+	0.756687i	−0.0562385	+	0.110374i	−0.917415	−	0.397932i	\(-0.869728\pi\)
							0.861177	+	0.508306i	\(0.169728\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	350.2.r.a.13.10	yes	160
7.6	odd	2	inner	350.2.r.a.13.1	✓	160
25.2	odd	20	inner	350.2.r.a.27.1	yes	160
175.27	even	20	inner	350.2.r.a.27.10	yes	160

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
350.2.r.a.13.1	✓	160	7.6	odd	2	inner
350.2.r.a.13.10	yes	160	1.1	even	1	trivial
350.2.r.a.27.1	yes	160	25.2	odd	20	inner
350.2.r.a.27.10	yes	160	175.27	even	20	inner