Embedded newform 950.2.r.a.11.20

• Label: 950.2.r.a.11.20
• 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			11.20
Character	\(\chi\)	\(=\)	950.11
Dual form			950.2.r.a.691.20

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.104528 + 0.994522i) q^{2} +(1.32430 + 1.47079i) q^{3} +(-0.978148 + 0.207912i) q^{4} +(0.202278 - 2.22690i) q^{5} +(-1.32430 + 1.47079i) q^{6} -4.37854 q^{7} +(-0.309017 - 0.951057i) q^{8} +(-0.0958531 + 0.911982i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 200 q - 25 q^{2} + 25 q^{4} + q^{5} - 36 q^{7} + 50 q^{8} + 37 q^{9}+O(q^{10}) \)

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
							\(3\)	1.32430	+	1.47079i	0.764588	+	0.849161i	0.992208	−	0.124589i	\(-0.0397613\pi\)
−0.227621	+	0.973750i	\(0.573095\pi\)
\(5\)	0.202278	−	2.22690i	0.0904615	−	0.995900i
							\(7\)	−4.37854			−1.65493			−0.827466	−	0.561515i	\(-0.810219\pi\)
−0.827466	+	0.561515i	\(0.810219\pi\)
\(11\)	4.04734	−	2.94057i	1.22032	−	0.886614i	0.224193	−	0.974545i	\(-0.428025\pi\)
							0.996127	+	0.0879304i	\(0.0280253\pi\)
\(13\)	0.464803	−	4.42231i	0.128913	−	1.22653i	−0.718475	−	0.695553i	\(-0.755160\pi\)
							0.847388	−	0.530974i	\(-0.178174\pi\)
\(17\)	−5.81164	−	1.23530i	−1.40953	−	0.299605i	−0.560590	−	0.828093i	\(-0.689426\pi\)
							−0.848940	+	0.528489i	\(0.822759\pi\)
\(19\)	3.66980	−	2.35215i	0.841909	−	0.539620i
							\(23\)	5.39421	+	2.40166i	1.12477	+	0.500780i	0.882915	−	0.469533i	\(-0.155578\pi\)
0.241855	+	0.970312i	\(0.422244\pi\)
\(29\)	−4.96795	+	1.05597i	−0.922526	+	0.196089i	−0.644606	−	0.764515i	\(-0.722978\pi\)
							−0.277920	+	0.960604i	\(0.589645\pi\)
\(31\)	0.550448	+	1.69411i	0.0988635	+	0.304270i	0.988241	−	0.152903i	\(-0.0488620\pi\)
							−0.889378	+	0.457173i	\(0.848862\pi\)
\(37\)	−1.37509	−	0.999063i	−0.226064	−	0.164245i	0.468988	−	0.883204i	\(-0.344619\pi\)
							−0.695052	+	0.718959i	\(0.744619\pi\)
\(41\)	5.53600	−	2.46479i	0.864578	−	0.384935i	0.0739721	−	0.997260i	\(-0.476432\pi\)
							0.790606	+	0.612325i	\(0.209766\pi\)
\(43\)	1.85684	−	3.21614i	0.283166	−	0.490457i	−0.688997	−	0.724764i	\(-0.741949\pi\)
							0.972163	+	0.234307i	\(0.0752820\pi\)
\(47\)	−3.83456	+	0.815061i	−0.559328	+	0.118889i	−0.478897	−	0.877871i	\(-0.658963\pi\)
							−0.0804313	+	0.996760i	\(0.525630\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	950.2.r.a.11.20	✓	200
19.7	even	3	inner	950.2.r.a.311.6	yes	200
25.16	even	5	inner	950.2.r.a.391.6	yes	200
475.216	even	15	inner	950.2.r.a.691.20	yes	200

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
950.2.r.a.11.20	✓	200	1.1	even	1	trivial
950.2.r.a.311.6	yes	200	19.7	even	3	inner
950.2.r.a.391.6	yes	200	25.16	even	5	inner
950.2.r.a.691.20	yes	200	475.216	even	15	inner