Embedded newform 950.2.r.b.11.5

• Label: 950.2.r.b.11.5
• 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			11.5
Character	\(\chi\)	\(=\)	950.11
Dual form			950.2.r.b.691.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.104528 - 0.994522i) q^{2} +(-1.54095 - 1.71140i) q^{3} +(-0.978148 + 0.207912i) q^{4} +(1.52065 - 1.63939i) q^{5} +(-1.54095 + 1.71140i) q^{6} +0.0891276 q^{7} +(0.309017 + 0.951057i) q^{8} +(-0.240774 + 2.29081i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 200 q + 25 q^{2} + 25 q^{4} + q^{5} + 28 q^{7} - 50 q^{8} + 13 q^{9}+O(q^{10}) \)

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
							\(3\)	−1.54095	−	1.71140i	−0.889670	−	0.988078i	0.110313	−	0.993897i	\(-0.464815\pi\)
−0.999983	+	0.00581866i	\(0.998148\pi\)
\(5\)	1.52065	−	1.63939i	0.680057	−	0.733159i
							\(7\)	0.0891276			0.0336871			0.0168435	−	0.999858i	\(-0.494638\pi\)
0.0168435	+	0.999858i	\(0.494638\pi\)
\(11\)	2.28335	−	1.65895i	0.688455	−	0.500192i	−0.187697	−	0.982227i	\(-0.560102\pi\)
							0.876152	+	0.482035i	\(0.160102\pi\)
\(13\)	0.165881	−	1.57826i	0.0460072	−	0.437729i	−0.947137	−	0.320831i	\(-0.896038\pi\)
							0.993144	−	0.116899i	\(-0.0372953\pi\)
\(17\)	−4.98113	−	1.05877i	−1.20810	−	0.256790i	−0.440520	−	0.897743i	\(-0.645206\pi\)
							−0.767580	+	0.640953i	\(0.778539\pi\)
\(19\)	0.358994	−	4.34409i	0.0823589	−	0.996603i
							\(23\)	−3.31669	−	1.47669i	−0.691578	−	0.307910i	0.0306848	−	0.999529i	\(-0.490231\pi\)
−0.722263	+	0.691619i	\(0.756898\pi\)
\(29\)	1.15348	−	0.245180i	0.214196	−	0.0455289i	−0.0995634	−	0.995031i	\(-0.531745\pi\)
							0.313760	+	0.949502i	\(0.398411\pi\)
\(31\)	0.168923	+	0.519893i	0.0303395	+	0.0933755i	0.965080	−	0.261957i	\(-0.0843678\pi\)
							−0.934740	+	0.355332i	\(0.884368\pi\)
\(37\)	3.51822	+	2.55614i	0.578392	+	0.420226i	0.838144	−	0.545449i	\(-0.183641\pi\)
							−0.259752	+	0.965675i	\(0.583641\pi\)
\(41\)	8.87471	−	3.95128i	1.38600	−	0.617086i	0.427978	−	0.903789i	\(-0.359226\pi\)
							0.958019	+	0.286703i	\(0.0925594\pi\)
\(43\)	−4.70498	+	8.14926i	−0.717502	+	1.24275i	0.244485	+	0.969653i	\(0.421381\pi\)
							−0.961987	+	0.273097i	\(0.911952\pi\)
\(47\)	−7.89435	+	1.67800i	−1.15151	+	0.244761i	−0.743833	−	0.668366i	\(-0.766994\pi\)
							−0.407676	+	0.913127i	\(0.633661\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	950.2.r.b.11.5	✓	200
19.7	even	3	inner	950.2.r.b.311.21	yes	200
25.16	even	5	inner	950.2.r.b.391.21	yes	200
475.216	even	15	inner	950.2.r.b.691.5	yes	200

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
950.2.r.b.11.5	✓	200	1.1	even	1	trivial
950.2.r.b.311.21	yes	200	19.7	even	3	inner
950.2.r.b.391.21	yes	200	25.16	even	5	inner
950.2.r.b.691.5	yes	200	475.216	even	15	inner