Embedded newform 950.2.r.a.11.19

• Label: 950.2.r.a.11.19
• 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			11.19
Character	\(\chi\)	\(=\)	950.11
Dual form			950.2.r.a.691.19

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.104528 + 0.994522i) q^{2} +(1.14496 + 1.27160i) q^{3} +(-0.978148 + 0.207912i) q^{4} +(1.79074 - 1.33913i) q^{5} +(-1.14496 + 1.27160i) q^{6} +1.05033 q^{7} +(-0.309017 - 0.951057i) q^{8} +(0.00753716 - 0.0717113i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 200 q - 25 q^{2} + 25 q^{4} + q^{5} - 36 q^{7} + 50 q^{8} + 37 q^{9}+O(q^{10}) \)

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
							\(3\)	1.14496	+	1.27160i	0.661040	+	0.734160i	0.976675	−	0.214721i	\(-0.0688842\pi\)
−0.315635	+	0.948881i	\(0.602218\pi\)
\(5\)	1.79074	−	1.33913i	0.800842	−	0.598875i
							\(7\)	1.05033			0.396988			0.198494	−	0.980102i	\(-0.436395\pi\)
0.198494	+	0.980102i	\(0.436395\pi\)
\(11\)	0.489273	−	0.355478i	0.147521	−	0.107181i	−0.511576	−	0.859238i	\(-0.670938\pi\)
							0.659098	+	0.752057i	\(0.270938\pi\)
\(13\)	−0.540749	+	5.14489i	−0.149977	+	1.42693i	0.617860	+	0.786288i	\(0.288000\pi\)
							−0.767836	+	0.640646i	\(0.778667\pi\)
\(17\)	−1.19231	−	0.253434i	−0.289179	−	0.0614668i	0.0610383	−	0.998135i	\(-0.480559\pi\)
							−0.350217	+	0.936669i	\(0.613892\pi\)
\(19\)	−2.53625	+	3.54506i	−0.581856	+	0.813292i
							\(23\)	3.85509	+	1.71640i	0.803842	+	0.357894i	0.767178	−	0.641434i	\(-0.221660\pi\)
0.0366641	+	0.999328i	\(0.488327\pi\)
\(29\)	4.89733	−	1.04096i	0.909412	−	0.193301i	0.270625	−	0.962685i	\(-0.412770\pi\)
							0.638787	+	0.769383i	\(0.279436\pi\)
\(31\)	1.92376	+	5.92071i	0.345517	+	1.06339i	0.961306	+	0.275481i	\(0.0888371\pi\)
							−0.615790	+	0.787911i	\(0.711163\pi\)
\(37\)	7.82600	+	5.68592i	1.28659	+	0.934760i	0.999731	−	0.0232137i	\(-0.00738982\pi\)
							0.286856	+	0.957974i	\(0.407390\pi\)
\(41\)	7.52187	−	3.34895i	1.17472	−	0.523019i	0.275835	−	0.961205i	\(-0.411046\pi\)
							0.898884	+	0.438186i	\(0.144379\pi\)
\(43\)	−1.53598	+	2.66040i	−0.234235	+	0.405707i	−0.959050	−	0.283237i	\(-0.908592\pi\)
							0.724815	+	0.688943i	\(0.241925\pi\)
\(47\)	2.26870	−	0.482228i	0.330924	−	0.0703401i	−0.0394533	−	0.999221i	\(-0.512562\pi\)
							0.370378	+	0.928881i	\(0.379228\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	950.2.r.a.11.19	✓	200
19.7	even	3	inner	950.2.r.a.311.7	yes	200
25.16	even	5	inner	950.2.r.a.391.7	yes	200
475.216	even	15	inner	950.2.r.a.691.19	yes	200

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
950.2.r.a.11.19	✓	200	1.1	even	1	trivial
950.2.r.a.311.7	yes	200	19.7	even	3	inner
950.2.r.a.391.7	yes	200	25.16	even	5	inner
950.2.r.a.691.19	yes	200	475.216	even	15	inner