Embedded newform 950.2.r.a.11.17

• Label: 950.2.r.a.11.17
• 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			11.17
Character	\(\chi\)	\(=\)	950.11
Dual form			950.2.r.a.691.17

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.104528 + 0.994522i) q^{2} +(0.856524 + 0.951266i) q^{3} +(-0.978148 + 0.207912i) q^{4} +(-1.58933 - 1.57290i) q^{5} +(-0.856524 + 0.951266i) q^{6} +1.37782 q^{7} +(-0.309017 - 0.951057i) q^{8} +(0.142311 - 1.35400i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 200 q - 25 q^{2} + 25 q^{4} + q^{5} - 36 q^{7} + 50 q^{8} + 37 q^{9}+O(q^{10}) \)

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
							\(3\)	0.856524	+	0.951266i	0.494514	+	0.549214i	0.937804	−	0.347164i	\(-0.112855\pi\)
−0.443290	+	0.896378i	\(0.646189\pi\)
\(5\)	−1.58933	−	1.57290i	−0.710771	−	0.703424i
							\(7\)	1.37782			0.520769			0.260384	−	0.965505i	\(-0.416151\pi\)
0.260384	+	0.965505i	\(0.416151\pi\)
\(11\)	−1.54709	+	1.12403i	−0.466466	+	0.338907i	−0.796062	−	0.605215i	\(-0.793087\pi\)
							0.329596	+	0.944122i	\(0.393087\pi\)
\(13\)	0.223986	−	2.13109i	0.0621226	−	0.591057i	−0.918537	−	0.395335i	\(-0.870628\pi\)
							0.980659	−	0.195722i	\(-0.0627050\pi\)
\(17\)	5.26275	+	1.11863i	1.27640	+	0.271308i	0.795762	−	0.605609i	\(-0.207070\pi\)
							0.480642	+	0.876917i	\(0.340404\pi\)
\(19\)	4.30625	+	0.675463i	0.987920	+	0.154962i
							\(23\)	3.10291	+	1.38150i	0.647001	+	0.288063i	0.703881	−	0.710318i	\(-0.251449\pi\)
−0.0568803	+	0.998381i	\(0.518115\pi\)
\(29\)	5.04315	−	1.07195i	0.936490	−	0.199057i	0.285703	−	0.958318i	\(-0.407773\pi\)
							0.650786	+	0.759261i	\(0.274439\pi\)
\(31\)	1.90338	+	5.85799i	0.341857	+	1.05213i	0.963245	+	0.268624i	\(0.0865689\pi\)
							−0.621389	+	0.783503i	\(0.713431\pi\)
\(37\)	−8.36877	−	6.08027i	−1.37582	−	0.999590i	−0.997257	−	0.0740129i	\(-0.976419\pi\)
							−0.378560	−	0.925577i	\(-0.623581\pi\)
\(41\)	5.91668	−	2.63428i	0.924030	−	0.411405i	0.111129	−	0.993806i	\(-0.464553\pi\)
							0.812902	+	0.582401i	\(0.197887\pi\)
\(43\)	0.828224	−	1.43453i	0.126303	−	0.218763i	−0.795939	−	0.605377i	\(-0.793022\pi\)
							0.922242	+	0.386614i	\(0.126356\pi\)
\(47\)	−2.75322	+	0.585215i	−0.401599	+	0.0853624i	−0.404283	−	0.914634i	\(-0.632479\pi\)
							0.00268426	+	0.999996i	\(0.499146\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	950.2.r.a.11.17	✓	200
19.7	even	3	inner	950.2.r.a.311.9	yes	200
25.16	even	5	inner	950.2.r.a.391.9	yes	200
475.216	even	15	inner	950.2.r.a.691.17	yes	200

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
950.2.r.a.11.17	✓	200	1.1	even	1	trivial
950.2.r.a.311.9	yes	200	19.7	even	3	inner
950.2.r.a.391.9	yes	200	25.16	even	5	inner
950.2.r.a.691.17	yes	200	475.216	even	15	inner