Embedded newform 950.2.r.b.391.17

• Label: 950.2.r.b.391.17
• Level: $950$
• Weight: $2$
• Character: 950.391
• 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			391.17
Character	\(\chi\)	\(=\)	950.391
Dual form			950.2.r.b.311.17

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.913545 - 0.406737i) q^{2} +(1.18362 + 0.251586i) q^{3} +(0.669131 - 0.743145i) q^{4} +(2.07760 + 0.826782i) q^{5} +(1.18362 - 0.251586i) q^{6} +2.11907 q^{7} +(0.309017 - 0.951057i) q^{8} +(-1.40298 - 0.624646i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 200 q + 25 q^{2} + 25 q^{4} + q^{5} + 28 q^{7} - 50 q^{8} + 13 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(77\)	\(401\)
\(\chi(n)\)	\(e\left(\frac{1}{5}\right)\)	\(e\left(\frac{2}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.913545	−	0.406737i	0.645974	−	0.287606i
							\(3\)	1.18362	+	0.251586i	0.683363	+	0.145253i	0.536496	−	0.843903i	\(-0.319748\pi\)
0.146867	+	0.989156i	\(0.453081\pi\)
\(5\)	2.07760	+	0.826782i	0.929132	+	0.369748i
							\(7\)	2.11907			0.800935			0.400467	−	0.916311i	\(-0.368848\pi\)
0.400467	+	0.916311i	\(0.368848\pi\)
\(11\)	4.47327	+	3.25002i	1.34874	+	0.979919i	0.999073	+	0.0430453i	\(0.0137060\pi\)
							0.349669	+	0.936873i	\(0.386294\pi\)
\(13\)	−4.45091	−	1.98167i	−1.23446	−	0.549617i	−0.317371	−	0.948301i	\(-0.602800\pi\)
							−0.917088	+	0.398685i	\(0.869467\pi\)
\(17\)	−1.17253	−	1.30222i	−0.284380	−	0.315836i	0.583982	−	0.811766i	\(-0.301494\pi\)
							−0.868362	+	0.495931i	\(0.834827\pi\)
\(19\)	0.276343	−	4.35013i	0.0633973	−	0.997988i
							\(23\)	−0.522624	+	4.97243i	−0.108975	+	1.03682i	0.794231	+	0.607616i	\(0.207874\pi\)
−0.903206	+	0.429208i	\(0.858793\pi\)
\(29\)	3.56817	−	3.96286i	0.662593	−	0.735884i	−0.314368	−	0.949301i	\(-0.601793\pi\)
							0.976961	+	0.213417i	\(0.0684594\pi\)
\(31\)	−3.19820	+	9.84304i	−0.574413	+	1.76786i	0.0637549	+	0.997966i	\(0.479692\pi\)
							−0.638168	+	0.769897i	\(0.720308\pi\)
\(37\)	−7.01299	+	5.09523i	−1.15293	+	0.837651i	−0.988867	−	0.148799i	\(-0.952459\pi\)
							−0.164060	+	0.986450i	\(0.552459\pi\)
\(41\)	−0.989729	−	9.41664i	−0.154570	−	1.47063i	−0.746900	−	0.664937i	\(-0.768459\pi\)
							0.592330	−	0.805695i	\(-0.298208\pi\)
\(43\)	−0.846313	+	1.46586i	−0.129061	+	0.223541i	−0.923313	−	0.384048i	\(-0.874530\pi\)
							0.794252	+	0.607589i	\(0.207863\pi\)
\(47\)	−1.56714	+	1.74049i	−0.228591	+	0.253876i	−0.846519	−	0.532359i	\(-0.821306\pi\)
							0.617928	+	0.786235i	\(0.287972\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	950.2.r.b.391.17	yes	200
19.7	even	3	inner	950.2.r.b.691.9	yes	200
25.11	even	5	inner	950.2.r.b.11.9	✓	200
475.311	even	15	inner	950.2.r.b.311.17	yes	200

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
950.2.r.b.11.9	✓	200	25.11	even	5	inner
950.2.r.b.311.17	yes	200	475.311	even	15	inner
950.2.r.b.391.17	yes	200	1.1	even	1	trivial
950.2.r.b.691.9	yes	200	19.7	even	3	inner