Embedded newform 950.2.bd.a.27.14

• Label: 950.2.bd.a.27.14
• 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			27.14
Character	\(\chi\)	\(=\)	950.27
Dual form			950.2.bd.a.563.14

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.777146 - 0.629320i) q^{2} +(0.0138662 - 0.264584i) q^{3} +(0.207912 + 0.978148i) q^{4} +(0.889244 + 2.05164i) q^{5} +(-0.177284 + 0.196894i) q^{6} +(2.41122 + 2.41122i) q^{7} +(0.453990 - 0.891007i) q^{8} +(2.91375 + 0.306248i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 800 q - 4 q^{5} - 8 q^{7}+O(q^{10}) \)

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
							\(3\)	0.0138662	−	0.264584i	0.00800568	−	0.152757i	−0.991702	−	0.128561i	\(-0.958964\pi\)
0.999707	−	0.0241961i	\(-0.00770260\pi\)
\(5\)	0.889244	+	2.05164i	0.397682	+	0.917523i
							\(7\)	2.41122	+	2.41122i	0.911356	+	0.911356i	0.996379	−	0.0850232i	\(-0.0270964\pi\)
−0.0850232	+	0.996379i	\(0.527096\pi\)
\(11\)	1.19215	−	0.866149i	0.359447	−	0.261154i	−0.393374	−	0.919378i	\(-0.628692\pi\)
							0.752822	+	0.658225i	\(0.228692\pi\)
\(13\)	0.676922	−	0.548160i	0.187744	−	0.152032i	−0.530840	−	0.847472i	\(-0.678124\pi\)
							0.718585	+	0.695439i	\(0.244790\pi\)
\(17\)	−0.675999	−	1.04095i	−0.163954	−	0.252467i	0.747139	−	0.664668i	\(-0.231427\pi\)
							−0.911093	+	0.412201i	\(0.864760\pi\)
\(19\)	−2.95106	+	3.20800i	−0.677019	+	0.735965i
							\(23\)	3.28353	−	1.26043i	0.684664	−	0.262818i	0.00892644	−	0.999960i	\(-0.497159\pi\)
0.675738	+	0.737142i	\(0.263825\pi\)
\(29\)	−0.400677	+	0.0851666i	−0.0744039	+	0.0158150i	−0.244963	−	0.969532i	\(-0.578776\pi\)
							0.170559	+	0.985347i	\(0.445443\pi\)
\(31\)	−0.434137	+	0.141060i	−0.0779733	+	0.0253351i	−0.347744	−	0.937590i	\(-0.613052\pi\)
							0.269771	+	0.962925i	\(0.413052\pi\)
\(37\)	−7.34397	+	1.16317i	−1.20734	+	0.191224i	−0.727485	−	0.686124i	\(-0.759311\pi\)
							−0.479856	+	0.877348i	\(0.659311\pi\)
\(41\)	3.03485	+	6.81638i	0.473964	+	1.06454i	0.979447	+	0.201703i	\(0.0646477\pi\)
							−0.505483	+	0.862837i	\(0.668686\pi\)
\(43\)	−1.24168	−	4.63403i	−0.189355	−	0.706682i	−0.993656	−	0.112461i	\(-0.964127\pi\)
							0.804301	−	0.594222i	\(-0.202540\pi\)
\(47\)	8.32773	+	5.40809i	1.21472	+	0.788852i	0.982942	−	0.183916i	\(-0.0588774\pi\)
							0.231783	+	0.972768i	\(0.425544\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	950.2.bd.a.27.14	✓	800
19.12	odd	6	inner	950.2.bd.a.677.12	yes	800
25.13	odd	20	inner	950.2.bd.a.863.12	yes	800
475.88	even	60	inner	950.2.bd.a.563.14	yes	800

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
950.2.bd.a.27.14	✓	800	1.1	even	1	trivial
950.2.bd.a.563.14	yes	800	475.88	even	60	inner
950.2.bd.a.677.12	yes	800	19.12	odd	6	inner
950.2.bd.a.863.12	yes	800	25.13	odd	20	inner