Embedded newform 950.2.r.a.11.16

• Label: 950.2.r.a.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.a.691.16

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.104528 + 0.994522i) q^{2} +(0.831705 + 0.923702i) q^{3} +(-0.978148 + 0.207912i) q^{4} +(2.14903 + 0.617800i) q^{5} +(-0.831705 + 0.923702i) q^{6} +0.754763 q^{7} +(-0.309017 - 0.951057i) q^{8} +(0.152093 - 1.44707i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 200 q - 25 q^{2} + 25 q^{4} + q^{5} - 36 q^{7} + 50 q^{8} + 37 q^{9}+O(q^{10}) \)

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
							\(3\)	0.831705	+	0.923702i	0.480185	+	0.533300i	0.933751	−	0.357923i	\(-0.116515\pi\)
−0.453566	+	0.891223i	\(0.649848\pi\)
\(5\)	2.14903	+	0.617800i	0.961075	+	0.276288i
							\(7\)	0.754763			0.285274			0.142637	−	0.989775i	\(-0.454442\pi\)
0.142637	+	0.989775i	\(0.454442\pi\)
\(11\)	3.79899	−	2.76013i	1.14544	−	0.832210i	0.157571	−	0.987508i	\(-0.449634\pi\)
							0.987868	+	0.155298i	\(0.0496337\pi\)
\(13\)	0.453130	−	4.31125i	0.125676	−	1.19572i	−0.731916	−	0.681395i	\(-0.761374\pi\)
							0.857592	−	0.514330i	\(-0.171960\pi\)
\(17\)	3.11878	+	0.662918i	0.756416	+	0.160781i	0.569949	−	0.821680i	\(-0.306963\pi\)
							0.186467	+	0.982461i	\(0.440296\pi\)
\(19\)	2.86492	+	3.28515i	0.657258	+	0.753666i
							\(23\)	−3.14975	−	1.40236i	−0.656768	−	0.292412i	0.0511662	−	0.998690i	\(-0.483706\pi\)
−0.707935	+	0.706278i	\(0.750373\pi\)
\(29\)	−4.15190	+	0.882514i	−0.770989	+	0.163879i	−0.576579	−	0.817042i	\(-0.695613\pi\)
							−0.194410	+	0.980920i	\(0.562279\pi\)
\(31\)	−1.07682	−	3.31410i	−0.193402	−	0.595230i	−0.999992	−	0.00411814i	\(-0.998689\pi\)
							0.806590	−	0.591112i	\(-0.201311\pi\)
\(37\)	−2.81941	−	2.04842i	−0.463508	−	0.336759i	0.331398	−	0.943491i	\(-0.392480\pi\)
							−0.794906	+	0.606733i	\(0.792480\pi\)
\(41\)	−9.60739	+	4.27748i	−1.50042	+	0.668031i	−0.982307	−	0.187280i	\(-0.940033\pi\)
							−0.518115	+	0.855311i	\(0.673366\pi\)
\(43\)	−5.07230	+	8.78548i	−0.773519	+	1.33977i	0.162105	+	0.986774i	\(0.448172\pi\)
							−0.935623	+	0.353000i	\(0.885162\pi\)
\(47\)	2.05215	−	0.436198i	0.299337	−	0.0636261i	−0.0557948	−	0.998442i	\(-0.517769\pi\)
							0.355132	+	0.934816i	\(0.384436\pi\)

(See \(a_n\) instead)

Twists

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

    

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