Embedded newform 950.2.r.b.11.16

• Label: 950.2.r.b.11.16
• Level: $950$
• Weight: $2$
• Character: 950.11
• Analytic conductor: $7.586$
• Analytic rank: $0$
• Dimension: $200$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 950 = 2 \cdot 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	950.r (of order \(15\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.58578819202\)
Analytic rank:	\(0\)
Dimension:	\(200\)
Relative dimension:	\(25\) over \(\Q(\zeta_{15})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{15}]$

Embedding invariants

Embedding label			11.16
Character	\(\chi\)	\(=\)	950.11
Dual form			950.2.r.b.691.16

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.104528 - 0.994522i) q^{2} +(0.459328 + 0.510136i) q^{3} +(-0.978148 + 0.207912i) q^{4} +(-2.23563 - 0.0442828i) q^{5} +(0.459328 - 0.510136i) q^{6} +4.99570 q^{7} +(0.309017 + 0.951057i) q^{8} +(0.264329 - 2.51493i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 200 q + 25 q^{2} + 25 q^{4} + q^{5} + 28 q^{7} - 50 q^{8} + 13 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(77\)	\(401\)
\(\chi(n)\)	\(e\left(\frac{4}{5}\right)\)	\(e\left(\frac{2}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.104528	−	0.994522i	−0.0739128	−	0.703233i
							\(3\)	0.459328	+	0.510136i	0.265193	+	0.294527i	0.861004	−	0.508598i	\(-0.169836\pi\)
−0.595811	+	0.803125i	\(0.703169\pi\)
\(5\)	−2.23563	−	0.0442828i	−0.999804	−	0.0198039i
							\(7\)	4.99570			1.88820			0.944099	−	0.329661i	\(-0.106934\pi\)
0.944099	+	0.329661i	\(0.106934\pi\)
\(11\)	−1.51517	+	1.10083i	−0.456840	+	0.331914i	−0.792290	−	0.610144i	\(-0.791112\pi\)
							0.335450	+	0.942058i	\(0.391112\pi\)
\(13\)	−0.629774	+	5.99190i	−0.174668	+	1.66185i	0.459140	+	0.888364i	\(0.348158\pi\)
							−0.633808	+	0.773491i	\(0.718509\pi\)
\(17\)	−0.234714	−	0.0498900i	−0.0569265	−	0.0121001i	0.179360	−	0.983783i	\(-0.442597\pi\)
							−0.236287	+	0.971683i	\(0.575931\pi\)
\(19\)	4.10575	−	1.46383i	0.941924	−	0.335826i
							\(23\)	1.49321	+	0.664819i	0.311355	+	0.138624i	0.556465	−	0.830871i	\(-0.312157\pi\)
−0.245110	+	0.969495i	\(0.578824\pi\)
\(29\)	−1.21648	+	0.258572i	−0.225895	+	0.0480156i	−0.319469	−	0.947597i	\(-0.603504\pi\)
							0.0935732	+	0.995612i	\(0.470171\pi\)
\(31\)	−1.44168	−	4.43704i	−0.258933	−	0.796915i	−0.993029	−	0.117869i	\(-0.962394\pi\)
							0.734096	−	0.679046i	\(-0.237606\pi\)
\(37\)	9.21730	+	6.69676i	1.51531	+	1.10094i	0.963750	+	0.266808i	\(0.0859690\pi\)
							0.551565	+	0.834132i	\(0.314031\pi\)
\(41\)	0.739673	−	0.329324i	0.115518	−	0.0514317i	−0.348164	−	0.937434i	\(-0.613195\pi\)
							0.463682	+	0.886002i	\(0.346528\pi\)
\(43\)	−0.638878	+	1.10657i	−0.0974280	+	0.168750i	−0.910619	−	0.413246i	\(-0.864395\pi\)
							0.813191	+	0.581996i	\(0.197728\pi\)
\(47\)	−8.50472	+	1.80773i	−1.24054	+	0.263685i	−0.781034	−	0.624489i	\(-0.785307\pi\)
							−0.459507	+	0.888174i	\(0.651974\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	950.2.r.b.11.16	✓	200
19.7	even	3	inner	950.2.r.b.311.10	yes	200
25.16	even	5	inner	950.2.r.b.391.10	yes	200
475.216	even	15	inner	950.2.r.b.691.16	yes	200

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
950.2.r.b.11.16	✓	200	1.1	even	1	trivial
950.2.r.b.311.10	yes	200	19.7	even	3	inner
950.2.r.b.391.10	yes	200	25.16	even	5	inner
950.2.r.b.691.16	yes	200	475.216	even	15	inner