Embedded newform 950.2.bd.a.27.9

• Label: 950.2.bd.a.27.9
• Level: $950$
• Weight: $2$
• Character: 950.27
• Analytic conductor: $7.586$
• Analytic rank: $0$
• Dimension: $800$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 950 = 2 \cdot 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	950.bd (of order \(60\), degree \(16\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.58578819202\)
Analytic rank:	\(0\)
Dimension:	\(800\)
Relative dimension:	\(50\) over \(\Q(\zeta_{60})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{60}]$

Embedding invariants

Embedding label			27.9
Character	\(\chi\)	\(=\)	950.27
Dual form			950.2.bd.a.563.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.777146 - 0.629320i) q^{2} +(-0.0710372 + 1.35547i) q^{3} +(0.207912 + 0.978148i) q^{4} +(-1.70399 + 1.44791i) q^{5} +(0.908231 - 1.00869i) q^{6} +(-0.514487 - 0.514487i) q^{7} +(0.453990 - 0.891007i) q^{8} +(1.15131 + 0.121008i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 800 q - 4 q^{5} - 8 q^{7}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/950\mathbb{Z}\right)^\times\).

\(n\)	\(77\)	\(401\)
\(\chi(n)\)	\(e\left(\frac{1}{20}\right)\)	\(e\left(\frac{1}{6}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.777146	−	0.629320i	−0.549525	−	0.444997i
							\(3\)	−0.0710372	+	1.35547i	−0.0410133	+	0.782581i	0.898271	+	0.439442i	\(0.144824\pi\)
−0.939284	+	0.343139i	\(0.888510\pi\)
\(5\)	−1.70399	+	1.44791i	−0.762045	+	0.647524i
							\(7\)	−0.514487	−	0.514487i	−0.194458	−	0.194458i	0.603161	−	0.797619i	\(-0.293907\pi\)
−0.797619	+	0.603161i	\(0.793907\pi\)
\(11\)	−1.19005	+	0.864625i	−0.358815	+	0.260694i	−0.752558	−	0.658526i	\(-0.771180\pi\)
							0.393743	+	0.919221i	\(0.371180\pi\)
\(13\)	−2.77697	+	2.24875i	−0.770194	+	0.623691i	−0.931720	−	0.363178i	\(-0.881692\pi\)
							0.161526	+	0.986868i	\(0.448358\pi\)
\(17\)	0.199034	+	0.306486i	0.0482729	+	0.0743337i	0.861983	−	0.506937i	\(-0.169222\pi\)
							−0.813710	+	0.581271i	\(0.802556\pi\)
\(19\)	−4.34379	+	0.362573i	−0.996535	+	0.0831799i
							\(23\)	0.356702	−	0.136925i	0.0743776	−	0.0285509i	−0.320895	−	0.947115i	\(-0.603984\pi\)
0.395272	+	0.918564i	\(0.370650\pi\)
\(29\)	0.306073	−	0.0650579i	0.0568364	−	0.0120809i	−0.179406	−	0.983775i	\(-0.557417\pi\)
							0.236242	+	0.971694i	\(0.424084\pi\)
\(31\)	7.10237	−	2.30770i	1.27562	−	0.414475i	0.408587	−	0.912719i	\(-0.366022\pi\)
							0.867037	+	0.498244i	\(0.166022\pi\)
\(37\)	−7.64390	+	1.21067i	−1.25665	+	0.199034i	−0.749023	−	0.662544i	\(-0.769477\pi\)
							−0.507626	+	0.861578i	\(0.669477\pi\)
\(41\)	−4.61360	−	10.3623i	−0.720523	−	1.61832i	−0.784319	−	0.620357i	\(-0.786988\pi\)
							0.0637967	−	0.997963i	\(-0.479679\pi\)
\(43\)	1.39685	+	5.21311i	0.213018	+	0.794992i	0.986855	+	0.161608i	\(0.0516680\pi\)
							−0.773837	+	0.633384i	\(0.781665\pi\)
\(47\)	−10.6776	−	6.93412i	−1.55749	−	1.01144i	−0.980924	−	0.194393i	\(-0.937726\pi\)
							−0.576565	−	0.817052i	\(-0.695607\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	950.2.bd.a.27.9	✓	800
19.12	odd	6	inner	950.2.bd.a.677.17	yes	800
25.13	odd	20	inner	950.2.bd.a.863.17	yes	800
475.88	even	60	inner	950.2.bd.a.563.9	yes	800

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
950.2.bd.a.27.9	✓	800	1.1	even	1	trivial
950.2.bd.a.563.9	yes	800	475.88	even	60	inner
950.2.bd.a.677.17	yes	800	19.12	odd	6	inner
950.2.bd.a.863.17	yes	800	25.13	odd	20	inner