Embedded newform 50.6.e.a.9.3

• Label: 50.6.e.a.9.3
• Level: $50$
• Weight: $6$
• Character: 50.9
• 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			9.3
Character	\(\chi\)	\(=\)	50.9
Dual form			50.6.e.a.39.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.35114 + 3.23607i) q^{2} +(-2.02410 + 0.657668i) q^{3} +(-4.94427 - 15.2169i) q^{4} +(24.5492 - 50.2229i) q^{5} +(2.63067 - 8.09638i) q^{6} +244.793i q^{7} +(60.8676 + 19.7771i) q^{8} +(-192.927 + 140.169i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q + 192 q^{4} + 180 q^{5} + 72 q^{6} + 686 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(27\)
\(\chi(n)\)	\(e\left(\frac{7}{10}\right)\)

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
							\(3\)	−2.02410	+	0.657668i	−0.129846	+	0.0421894i	−0.373219	−	0.927743i	\(-0.621746\pi\)
0.243373	+	0.969933i	\(0.421746\pi\)
\(5\)	24.5492	−	50.2229i	0.439149	−	0.898414i
							\(7\)			244.793i			1.88823i	0.329617	+	0.944115i	\(0.393080\pi\)
−0.329617	+	0.944115i	\(0.606920\pi\)
\(11\)	−411.859	−	299.233i	−1.02628	−	0.745638i	−0.0587210	−	0.998274i	\(-0.518702\pi\)
							−0.967561	+	0.252637i	\(0.918702\pi\)
\(13\)	−197.656	−	272.050i	−0.324378	−	0.446468i	0.615420	−	0.788200i	\(-0.288987\pi\)
							−0.939797	+	0.341732i	\(0.888987\pi\)
\(17\)	−623.689	−	202.649i	−0.523415	−	0.170068i	0.0353797	−	0.999374i	\(-0.488736\pi\)
							−0.558795	+	0.829306i	\(0.688736\pi\)
\(19\)	−721.526	+	2220.63i	−0.458530	+	1.41121i	0.408410	+	0.912799i	\(0.366083\pi\)
							−0.866940	+	0.498412i	\(0.833917\pi\)
\(23\)	−1728.42	+	2378.97i	−0.681287	+	0.937711i	−0.999948	−	0.0101573i	\(-0.996767\pi\)
							0.318661	+	0.947869i	\(0.396767\pi\)
\(29\)	−105.865	−	325.818i	−0.0233752	−	0.0719416i	0.938688	−	0.344767i	\(-0.112042\pi\)
							−0.962064	+	0.272825i	\(0.912042\pi\)
\(31\)	−1370.46	+	4217.83i	−0.256130	+	0.788288i	0.737475	+	0.675375i	\(0.236018\pi\)
							−0.993605	+	0.112913i	\(0.963982\pi\)
\(37\)	5817.83	+	8007.55i	0.698645	+	0.961602i	0.999967	+	0.00807994i	\(0.00257195\pi\)
							−0.301322	+	0.953522i	\(0.597428\pi\)
\(41\)	1700.74	−	1235.66i	0.158008	−	0.114800i	−0.505972	−	0.862550i	\(-0.668866\pi\)
							0.663980	+	0.747750i	\(0.268866\pi\)
\(43\)		−	21951.2i		−	1.81045i	−0.424928	−	0.905227i	\(-0.639701\pi\)
							0.424928	−	0.905227i	\(-0.360299\pi\)
\(47\)	14829.0	−	4818.22i	0.979188	−	0.318158i	0.224669	−	0.974435i	\(-0.427870\pi\)
							0.754520	+	0.656278i	\(0.227870\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	50.6.e.a.9.3	✓	48
25.14	even	10	inner	50.6.e.a.39.3	yes	48

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
50.6.e.a.9.3	✓	48	1.1	even	1	trivial
50.6.e.a.39.3	yes	48	25.14	even	10	inner