Embedded newform 950.2.r.b.11.8

• Label: 950.2.r.b.11.8
• 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			11.8
Character	\(\chi\)	\(=\)	950.11
Dual form			950.2.r.b.691.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.104528 - 0.994522i) q^{2} +(-1.00310 - 1.11405i) q^{3} +(-0.978148 + 0.207912i) q^{4} +(2.10151 + 0.763969i) q^{5} +(-1.00310 + 1.11405i) q^{6} -3.65812 q^{7} +(0.309017 + 0.951057i) q^{8} +(0.0786767 - 0.748558i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 200 q + 25 q^{2} + 25 q^{4} + q^{5} + 28 q^{7} - 50 q^{8} + 13 q^{9}+O(q^{10}) \)

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
							\(3\)	−1.00310	−	1.11405i	−0.579139	−	0.643199i	0.380384	−	0.924828i	\(-0.375792\pi\)
−0.959523	+	0.281630i	\(0.909125\pi\)
\(5\)	2.10151	+	0.763969i	0.939825	+	0.341658i
							\(7\)	−3.65812			−1.38264			−0.691320	−	0.722549i	\(-0.742970\pi\)
−0.691320	+	0.722549i	\(0.742970\pi\)
\(11\)	−0.0737076	+	0.0535517i	−0.0222237	+	0.0161464i	−0.598842	−	0.800867i	\(-0.704372\pi\)
							0.576618	+	0.817014i	\(0.304372\pi\)
\(13\)	0.453249	−	4.31238i	0.125709	−	1.19604i	−0.731782	−	0.681539i	\(-0.761311\pi\)
							0.857490	−	0.514500i	\(-0.172022\pi\)
\(17\)	2.65247	+	0.563800i	0.643319	+	0.136742i	0.518008	−	0.855376i	\(-0.326674\pi\)
							0.125311	+	0.992117i	\(0.460007\pi\)
\(19\)	−3.99083	+	1.75309i	−0.915558	+	0.402186i
							\(23\)	−6.40407	−	2.85127i	−1.33534	−	0.594532i	−0.390060	−	0.920790i	\(-0.627546\pi\)
−0.945281	+	0.326258i	\(0.894212\pi\)
\(29\)	−2.94825	+	0.626670i	−0.547476	+	0.116370i	−0.473342	−	0.880879i	\(-0.656953\pi\)
							−0.0741340	+	0.997248i	\(0.523619\pi\)
\(31\)	0.909540	+	2.79928i	0.163358	+	0.502765i	0.998912	−	0.0466444i	\(-0.0148528\pi\)
							−0.835553	+	0.549409i	\(0.814853\pi\)
\(37\)	−2.48733	−	1.80715i	−0.408915	−	0.297094i	0.364247	−	0.931302i	\(-0.381326\pi\)
							−0.773162	+	0.634208i	\(0.781326\pi\)
\(41\)	−8.48822	+	3.77920i	−1.32564	+	0.590212i	−0.942724	−	0.333573i	\(-0.891746\pi\)
							−0.382913	+	0.923784i	\(0.625079\pi\)
\(43\)	0.163007	−	0.282336i	0.0248583	−	0.0430558i	−0.853329	−	0.521373i	\(-0.825420\pi\)
							0.878187	+	0.478318i	\(0.158753\pi\)
\(47\)	−0.0345631	+	0.00734662i	−0.00504155	+	0.00107161i	−0.210432	−	0.977609i	\(-0.567487\pi\)
							0.205390	+	0.978680i	\(0.434154\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	950.2.r.b.11.8	✓	200
19.7	even	3	inner	950.2.r.b.311.18	yes	200
25.16	even	5	inner	950.2.r.b.391.18	yes	200
475.216	even	15	inner	950.2.r.b.691.8	yes	200

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
950.2.r.b.11.8	✓	200	1.1	even	1	trivial
950.2.r.b.311.18	yes	200	19.7	even	3	inner
950.2.r.b.391.18	yes	200	25.16	even	5	inner
950.2.r.b.691.8	yes	200	475.216	even	15	inner