Embedded newform 950.2.r.a.691.8

• Label: 950.2.r.a.691.8
• Level: $950$
• Weight: $2$
• Character: 950.691
• Analytic conductor: $7.586$
• Analytic rank: $0$
• Dimension: $200$
• 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			691.8
Character	\(\chi\)	\(=\)	950.691
Dual form			950.2.r.a.11.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.104528 - 0.994522i) q^{2} +(-0.955473 + 1.06116i) q^{3} +(-0.978148 - 0.207912i) q^{4} +(0.663784 + 2.13527i) q^{5} +(0.955473 + 1.06116i) q^{6} -1.22437 q^{7} +(-0.309017 + 0.951057i) q^{8} +(0.100453 + 0.955746i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 200 q - 25 q^{2} + 25 q^{4} + q^{5} - 36 q^{7} + 50 q^{8} + 37 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.104528	−	0.994522i	0.0739128	−	0.703233i
							\(3\)	−0.955473	+	1.06116i	−0.551643	+	0.612661i	−0.952893	−	0.303307i	\(-0.901909\pi\)
0.401250	+	0.915968i	\(0.368576\pi\)
\(5\)	0.663784	+	2.13527i	0.296853	+	0.954923i
							\(7\)	−1.22437			−0.462770			−0.231385	−	0.972862i	\(-0.574326\pi\)
−0.231385	+	0.972862i	\(0.574326\pi\)
\(11\)	−2.40985	−	1.75086i	−0.726598	−	0.527904i	0.161888	−	0.986809i	\(-0.448242\pi\)
							−0.888485	+	0.458905i	\(0.848242\pi\)
\(13\)	0.313204	+	2.97993i	0.0868671	+	0.826485i	0.948035	+	0.318165i	\(0.103067\pi\)
							−0.861168	+	0.508320i	\(0.830267\pi\)
\(17\)	0.501156	−	0.106524i	0.121548	−	0.0258359i	−0.146736	−	0.989176i	\(-0.546877\pi\)
							0.268284	+	0.963340i	\(0.413543\pi\)
\(19\)	0.00483604	−	4.35890i	0.00110946	−	0.999999i
							\(23\)	−0.282530	+	0.125791i	−0.0589116	+	0.0262291i	−0.435981	−	0.899956i	\(-0.643599\pi\)
0.377069	+	0.926185i	\(0.376932\pi\)
\(29\)	−7.28487	−	1.54845i	−1.35277	−	0.287539i	−0.526192	−	0.850366i	\(-0.676381\pi\)
							−0.826575	+	0.562827i	\(0.809714\pi\)
\(31\)	−0.0174028	+	0.0535604i	−0.00312564	+	0.00961973i	−0.952607	−	0.304203i	\(-0.901610\pi\)
							0.949482	+	0.313823i	\(0.101610\pi\)
\(37\)	1.46749	−	1.06620i	0.241254	−	0.175281i	−0.460588	−	0.887614i	\(-0.652361\pi\)
							0.701842	+	0.712333i	\(0.252361\pi\)
\(41\)	−2.49741	−	1.11192i	−0.390031	−	0.173653i	0.202346	−	0.979314i	\(-0.435143\pi\)
							−0.592376	+	0.805661i	\(0.701810\pi\)
\(43\)	−0.211978	−	0.367157i	−0.0323263	−	0.0559909i	0.849410	−	0.527734i	\(-0.176958\pi\)
							−0.881736	+	0.471743i	\(0.843625\pi\)
\(47\)	−4.74042	−	1.00761i	−0.691462	−	0.146975i	−0.151239	−	0.988497i	\(-0.548326\pi\)
							−0.540222	+	0.841522i	\(0.681660\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	950.2.r.a.691.8	yes	200
19.11	even	3	inner	950.2.r.a.391.18	yes	200
25.11	even	5	inner	950.2.r.a.311.18	yes	200
475.11	even	15	inner	950.2.r.a.11.8	✓	200

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
950.2.r.a.11.8	✓	200	475.11	even	15	inner
950.2.r.a.311.18	yes	200	25.11	even	5	inner
950.2.r.a.391.18	yes	200	19.11	even	3	inner
950.2.r.a.691.8	yes	200	1.1	even	1	trivial