Embedded newform 950.2.x.a.159.9

• Label: 950.2.x.a.159.9
• Level: $950$
• Weight: $2$
• Character: 950.159
• Analytic conductor: $7.586$
• Analytic rank: $0$
• Dimension: $400$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 950 = 2 \cdot 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	950.x (of order \(30\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.58578819202\)
Analytic rank:	\(0\)
Dimension:	\(400\)
Relative dimension:	\(50\) over \(\Q(\zeta_{30})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{30}]$

Embedding invariants

Embedding label			159.9
Character	\(\chi\)	\(=\)	950.159
Dual form			950.2.x.a.239.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.994522 - 0.104528i) q^{2} +(-1.07277 - 0.965928i) q^{3} +(0.978148 + 0.207912i) q^{4} +(-1.93765 - 1.11603i) q^{5} +(0.965928 + 1.07277i) q^{6} -2.14291i q^{7} +(-0.951057 - 0.309017i) q^{8} +(-0.0957633 - 0.911127i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 400 q - 50 q^{4} + 2 q^{5} - 50 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(77\)	\(401\)
\(\chi(n)\)	\(e\left(\frac{7}{10}\right)\)	\(e\left(\frac{1}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.994522	−	0.104528i	−0.703233	−	0.0739128i
							\(3\)	−1.07277	−	0.965928i	−0.619365	−	0.557679i	0.298581	−	0.954384i	\(-0.403487\pi\)
−0.917946	+	0.396706i	\(0.870153\pi\)
\(5\)	−1.93765	−	1.11603i	−0.866542	−	0.499104i
							\(7\)		−	2.14291i		−	0.809942i	−0.914330	−	0.404971i	\(-0.867282\pi\)
0.914330	−	0.404971i	\(-0.132718\pi\)
\(11\)	3.93338	+	2.85777i	1.18596	+	0.861650i	0.992831	−	0.119524i	\(-0.0381369\pi\)
							0.193128	+	0.981174i	\(0.438137\pi\)
\(13\)	6.50209	−	0.683398i	1.80336	−	0.189540i	0.857578	−	0.514354i	\(-0.171968\pi\)
							0.945778	+	0.324813i	\(0.105301\pi\)
\(17\)	0.214960	+	1.01131i	0.0521356	+	0.245279i	0.996497	−	0.0836317i	\(-0.0266519\pi\)
							−0.944361	+	0.328910i	\(0.893319\pi\)
\(19\)	4.34821	+	0.305069i	0.997548	+	0.0699876i
							\(23\)	−1.99340	−	4.47726i	−0.415653	−	0.933573i	−0.993114	−	0.117154i	\(-0.962623\pi\)
0.577460	−	0.816419i	\(-0.304044\pi\)
\(29\)	4.09631	+	0.870697i	0.760665	+	0.161684i	0.571884	−	0.820335i	\(-0.306213\pi\)
							0.188782	+	0.982019i	\(0.439546\pi\)
\(31\)	1.51608	−	4.66602i	0.272296	−	0.838042i	−0.717626	−	0.696429i	\(-0.754771\pi\)
							0.989922	−	0.141613i	\(-0.0452288\pi\)
\(37\)	−5.78607	−	7.96385i	−0.951225	−	1.30925i	−0.950981	−	0.309248i	\(-0.899923\pi\)
							−0.000243290	−	1.00000i	\(-0.500077\pi\)
\(41\)	−3.98832	−	1.77571i	−0.622871	−	0.277320i	0.0709361	−	0.997481i	\(-0.477401\pi\)
							−0.693807	+	0.720161i	\(0.744068\pi\)
\(43\)	0.649982	−	0.375267i	0.0991213	−	0.0572277i	−0.449620	−	0.893220i	\(-0.648441\pi\)
							0.548741	+	0.835992i	\(0.315107\pi\)
\(47\)	−0.264904	+	1.24628i	−0.0386402	+	0.181788i	−0.993235	−	0.116118i	\(-0.962955\pi\)
							0.954595	+	0.297906i	\(0.0962882\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	950.2.x.a.159.9	✓	400
19.11	even	3	inner	950.2.x.a.809.34	yes	400
25.14	even	10	inner	950.2.x.a.539.34	yes	400
475.239	even	30	inner	950.2.x.a.239.9	yes	400

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
950.2.x.a.159.9	✓	400	1.1	even	1	trivial
950.2.x.a.239.9	yes	400	475.239	even	30	inner
950.2.x.a.539.34	yes	400	25.14	even	10	inner
950.2.x.a.809.34	yes	400	19.11	even	3	inner