Embedded newform 50.6.e.a.39.10

• Label: 50.6.e.a.39.10
• Level: $50$
• Weight: $6$
• Character: 50.39
• Analytic conductor: $8.019$
• Analytic rank: $0$
• Dimension: $48$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 50 = 2 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	50.e (of order \(10\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(8.01919099065\)
Analytic rank:	\(0\)
Dimension:	\(48\)
Relative dimension:	\(12\) over \(\Q(\zeta_{10})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{10}]$

Embedding invariants

Embedding label			39.10
Character	\(\chi\)	\(=\)	50.39
Dual form			50.6.e.a.9.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(2.35114 + 3.23607i) q^{2} +(-1.61240 - 0.523901i) q^{3} +(-4.94427 + 15.2169i) q^{4} +(-47.4044 + 29.6281i) q^{5} +(-2.09560 - 6.44960i) q^{6} -48.2855i q^{7} +(-60.8676 + 19.7771i) q^{8} +(-194.266 - 141.142i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q + 192 q^{4} + 180 q^{5} + 72 q^{6} + 686 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/50\mathbb{Z}\right)^\times\).

\(n\)	\(27\)
\(\chi(n)\)	\(e\left(\frac{3}{10}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	2.35114	+	3.23607i	0.415627	+	0.572061i
							\(3\)	−1.61240	−	0.523901i	−0.103436	−	0.0336082i	0.256842	−	0.966453i	\(-0.417318\pi\)
−0.360278	+	0.932845i	\(0.617318\pi\)
\(5\)	−47.4044	+	29.6281i	−0.847995	+	0.530004i
							\(7\)		−	48.2855i		−	0.372454i	−0.982507	−	0.186227i	\(-0.940374\pi\)
0.982507	−	0.186227i	\(-0.0596259\pi\)
\(11\)	−272.832	+	198.224i	−0.679851	+	0.493941i	−0.873308	−	0.487168i	\(-0.838030\pi\)
							0.193457	+	0.981109i	\(0.438030\pi\)
\(13\)	−83.4013	+	114.792i	−0.136872	+	0.188388i	−0.871951	−	0.489594i	\(-0.837145\pi\)
							0.735079	+	0.677982i	\(0.237145\pi\)
\(17\)	−1113.26	+	361.719i	−0.934271	+	0.303563i	−0.736308	−	0.676647i	\(-0.763433\pi\)
							−0.197963	+	0.980210i	\(0.563433\pi\)
\(19\)	−199.008	−	612.483i	−0.126470	−	0.389233i	0.867696	−	0.497094i	\(-0.165600\pi\)
							−0.994166	+	0.107861i	\(0.965600\pi\)
\(23\)	1150.01	+	1582.85i	0.453296	+	0.623908i	0.973102	−	0.230376i	\(-0.0739957\pi\)
							−0.519806	+	0.854284i	\(0.673996\pi\)
\(29\)	−885.345	+	2724.81i	−0.195487	+	0.601647i	0.804484	+	0.593975i	\(0.202442\pi\)
							−0.999971	+	0.00767204i	\(0.997558\pi\)
\(31\)	2483.38	+	7643.05i	0.464129	+	1.42844i	0.860075	+	0.510168i	\(0.170417\pi\)
							−0.395946	+	0.918274i	\(0.629583\pi\)
\(37\)	−1591.42	+	2190.40i	−0.191108	+	0.263038i	−0.893809	−	0.448447i	\(-0.851977\pi\)
							0.702701	+	0.711485i	\(0.251977\pi\)
\(41\)	12188.9	+	8855.72i	1.13241	+	0.822743i	0.986044	−	0.166487i	\(-0.0532425\pi\)
							0.146365	+	0.989231i	\(0.453243\pi\)
\(43\)		−	2669.38i		−	0.220160i	−0.993923	−	0.110080i	\(-0.964889\pi\)
							0.993923	−	0.110080i	\(-0.0351108\pi\)
\(47\)	−9632.03	−	3129.64i	−0.636024	−	0.206657i	−0.0267821	−	0.999641i	\(-0.508526\pi\)
							−0.609242	+	0.792985i	\(0.708526\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	50.6.e.a.39.10	yes	48
25.9	even	10	inner	50.6.e.a.9.10	✓	48

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
50.6.e.a.9.10	✓	48	25.9	even	10	inner
50.6.e.a.39.10	yes	48	1.1	even	1	trivial