Embedded newform 950.2.r.a.311.11

• Label: 950.2.r.a.311.11
• Level: $950$
• Weight: $2$
• Character: 950.311
• 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			311.11
Character	\(\chi\)	\(=\)	950.311
Dual form			950.2.r.a.391.11

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.913545 - 0.406737i) q^{2} +(-0.567647 + 0.120657i) q^{3} +(0.669131 + 0.743145i) q^{4} +(-2.10860 + 0.744192i) q^{5} +(0.567647 + 0.120657i) q^{6} -3.61865 q^{7} +(-0.309017 - 0.951057i) q^{8} +(-2.43297 + 1.08323i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 200 q - 25 q^{2} + 25 q^{4} + q^{5} - 36 q^{7} + 50 q^{8} + 37 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(77\)	\(401\)
\(\chi(n)\)	\(e\left(\frac{4}{5}\right)\)	\(e\left(\frac{1}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.913545	−	0.406737i	−0.645974	−	0.287606i
							\(3\)	−0.567647	+	0.120657i	−0.327731	+	0.0696614i	−0.368839	−	0.929493i	\(-0.620245\pi\)
0.0411080	+	0.999155i	\(0.486911\pi\)
\(5\)	−2.10860	+	0.744192i	−0.942993	+	0.332813i
							\(7\)	−3.61865			−1.36772			−0.683860	−	0.729614i	\(-0.739700\pi\)
−0.683860	+	0.729614i	\(0.739700\pi\)
\(11\)	−3.97707	+	2.88951i	−1.19913	+	0.871221i	−0.994199	−	0.107555i	\(-0.965698\pi\)
							−0.204934	+	0.978776i	\(0.565698\pi\)
\(13\)	−2.12573	+	0.946434i	−0.589570	+	0.262494i	−0.679764	−	0.733431i	\(-0.737918\pi\)
							0.0901940	+	0.995924i	\(0.471251\pi\)
\(17\)	3.60274	−	4.00125i	0.873793	−	0.970445i	−0.125974	−	0.992034i	\(-0.540206\pi\)
							0.999767	+	0.0215882i	\(0.00687229\pi\)
\(19\)	3.36025	−	2.77646i	0.770895	−	0.636963i
							\(23\)	−0.00378193	−	0.0359826i	−0.000788586	−	0.00750290i	0.994121	−	0.108279i	\(-0.0345340\pi\)
−0.994909	+	0.100776i	\(0.967867\pi\)
\(29\)	2.83432	+	3.14783i	0.526320	+	0.584537i	0.946419	−	0.322941i	\(-0.104671\pi\)
							−0.420099	+	0.907478i	\(0.638005\pi\)
\(31\)	−0.351668	−	1.08232i	−0.0631615	−	0.194391i	0.914496	−	0.404595i	\(-0.132587\pi\)
							−0.977658	+	0.210203i	\(0.932587\pi\)
\(37\)	2.51022	+	1.82378i	0.412678	+	0.299828i	0.774685	−	0.632347i	\(-0.217908\pi\)
							−0.362007	+	0.932175i	\(0.617908\pi\)
\(41\)	−0.619557	+	5.89469i	−0.0967586	+	0.920597i	0.833210	+	0.552957i	\(0.186501\pi\)
							−0.929968	+	0.367640i	\(0.880166\pi\)
\(43\)	−6.48508	−	11.2325i	−0.988966	−	1.71294i	−0.622780	−	0.782397i	\(-0.713997\pi\)
							−0.366186	−	0.930542i	\(-0.619336\pi\)
\(47\)	4.19189	+	4.65557i	0.611450	+	0.679084i	0.966766	−	0.255662i	\(-0.0822935\pi\)
							−0.355316	+	0.934746i	\(0.615627\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	950.2.r.a.311.11	yes	200
19.11	even	3	inner	950.2.r.a.11.15	✓	200
25.16	even	5	inner	950.2.r.a.691.15	yes	200
475.391	even	15	inner	950.2.r.a.391.11	yes	200

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
950.2.r.a.11.15	✓	200	19.11	even	3	inner
950.2.r.a.311.11	yes	200	1.1	even	1	trivial
950.2.r.a.391.11	yes	200	475.391	even	15	inner
950.2.r.a.691.15	yes	200	25.16	even	5	inner