Embedded newform 950.2.r.a.11.15

• Label: 950.2.r.a.11.15
• 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			11.15
Character	\(\chi\)	\(=\)	950.11
Dual form			950.2.r.a.691.15

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.104528 + 0.994522i) q^{2} +(0.388315 + 0.431268i) q^{3} +(-0.978148 + 0.207912i) q^{4} +(1.69879 + 1.45400i) q^{5} +(-0.388315 + 0.431268i) q^{6} -3.61865 q^{7} +(-0.309017 - 0.951057i) q^{8} +(0.278382 - 2.64863i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 200 q - 25 q^{2} + 25 q^{4} + q^{5} - 36 q^{7} + 50 q^{8} + 37 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(77\)	\(401\)
\(\chi(n)\)	\(e\left(\frac{4}{5}\right)\)	\(e\left(\frac{2}{3}\right)\)

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
							\(3\)	0.388315	+	0.431268i	0.224194	+	0.248993i	0.844739	−	0.535178i	\(-0.179755\pi\)
−0.620545	+	0.784170i	\(0.713089\pi\)
\(5\)	1.69879	+	1.45400i	0.759721	+	0.650249i
							\(7\)	−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	\(0.565698\pi\)
\(13\)	0.243227	−	2.31415i	0.0674590	−	0.641830i	−0.907593	−	0.419852i	\(-0.862082\pi\)
							0.975052	−	0.221978i	\(-0.0712513\pi\)
\(17\)	−5.26655	−	1.11944i	−1.27733	−	0.271504i	−0.481188	−	0.876618i	\(-0.659794\pi\)
							−0.796139	+	0.605114i	\(0.793128\pi\)
\(19\)	−4.35046	−	0.271093i	−0.998064	−	0.0621930i
							\(23\)	0.0330528	+	0.0147161i	0.00689199	+	0.00306851i	0.410180	−	0.912005i	\(-0.365466\pi\)
−0.403288	+	0.915073i	\(0.632133\pi\)
\(29\)	−4.14326	+	0.880677i	−0.769384	+	0.163538i	−0.575849	−	0.817556i	\(-0.695329\pi\)
							−0.193535	+	0.981093i	\(0.561995\pi\)
\(31\)	−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	\(0.617908\pi\)
\(41\)	5.41473	−	2.41079i	0.845639	−	0.376503i	0.0622700	−	0.998059i	\(-0.480166\pi\)
							0.783369	+	0.621556i	\(0.213499\pi\)
\(43\)	−6.48508	+	11.2325i	−0.988966	+	1.71294i	−0.366186	+	0.930542i	\(0.619336\pi\)
							−0.622780	+	0.782397i	\(0.713997\pi\)
\(47\)	−6.12778	+	1.30250i	−0.893829	+	0.189989i	−0.631856	−	0.775086i	\(-0.717707\pi\)
							−0.261973	+	0.965075i	\(0.584373\pi\)

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
950.2.r.a.11.15	✓	200	1.1	even	1	trivial
950.2.r.a.311.11	yes	200	19.7	even	3	inner
950.2.r.a.391.11	yes	200	25.16	even	5	inner
950.2.r.a.691.15	yes	200	475.216	even	15	inner