Embedded newform 50.6.e.a.9.4

• Label: 50.6.e.a.9.4
• 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			9.4
Character	\(\chi\)	\(=\)	50.9
Dual form			50.6.e.a.39.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.35114 + 3.23607i) q^{2} +(-0.401275 + 0.130382i) q^{3} +(-4.94427 - 15.2169i) q^{4} +(-3.09691 + 55.8159i) q^{5} +(0.521529 - 1.60510i) q^{6} -69.1609i q^{7} +(60.8676 + 19.7771i) q^{8} +(-196.447 + 142.727i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q + 192 q^{4} + 180 q^{5} + 72 q^{6} + 686 q^{9}+O(q^{10}) \)

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
							\(3\)	−0.401275	+	0.130382i	−0.0257418	+	0.00836402i	−0.321860	−	0.946787i	\(-0.604308\pi\)
0.296118	+	0.955151i	\(0.404308\pi\)
\(5\)	−3.09691	+	55.8159i	−0.0553992	+	0.998464i
							\(7\)		−	69.1609i		−	0.533477i	−0.963769	−	0.266738i	\(-0.914054\pi\)
0.963769	−	0.266738i	\(-0.0859460\pi\)
\(11\)	−243.694	−	177.054i	−0.607243	−	0.441188i	0.241199	−	0.970476i	\(-0.422459\pi\)
							−0.848442	+	0.529288i	\(0.822459\pi\)
\(13\)	−166.770	−	229.539i	−0.273690	−	0.376703i	0.649941	−	0.759985i	\(-0.274794\pi\)
							−0.923631	+	0.383282i	\(0.874794\pi\)
\(17\)	−2031.30	−	660.010i	−1.70472	−	0.553896i	−0.715277	−	0.698841i	\(-0.753700\pi\)
							−0.989440	+	0.144945i	\(0.953700\pi\)
\(19\)	935.805	−	2880.11i	0.594704	−	1.83031i	0.0385128	−	0.999258i	\(-0.487738\pi\)
							0.556192	−	0.831054i	\(-0.312262\pi\)
\(23\)	−1812.48	+	2494.66i	−0.714420	+	0.983315i	0.285271	+	0.958447i	\(0.407917\pi\)
							−0.999691	+	0.0248679i	\(0.992083\pi\)
\(29\)	2499.24	+	7691.88i	0.551841	+	1.69839i	0.704144	+	0.710058i	\(0.251331\pi\)
							−0.152303	+	0.988334i	\(0.548669\pi\)
\(31\)	−861.952	+	2652.82i	−0.161094	+	0.495796i	−0.998727	−	0.0504369i	\(-0.983939\pi\)
							0.837633	+	0.546233i	\(0.183939\pi\)
\(37\)	−1827.10	−	2514.79i	−0.219411	−	0.301993i	0.685096	−	0.728453i	\(-0.259760\pi\)
							−0.904506	+	0.426460i	\(0.859760\pi\)
\(41\)	4844.56	−	3519.78i	0.450085	−	0.327006i	−0.339544	−	0.940590i	\(-0.610273\pi\)
							0.789629	+	0.613584i	\(0.210273\pi\)
\(43\)			16482.4i			1.35941i	0.733487	+	0.679703i	\(0.237891\pi\)
							−0.733487	+	0.679703i	\(0.762109\pi\)
\(47\)	−10143.4	+	3295.79i	−0.669790	+	0.217628i	−0.624120	−	0.781329i	\(-0.714542\pi\)
							−0.0456700	+	0.998957i	\(0.514542\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	50.6.e.a.9.4	✓	48
25.14	even	10	inner	50.6.e.a.39.4	yes	48

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
50.6.e.a.9.4	✓	48	1.1	even	1	trivial
50.6.e.a.39.4	yes	48	25.14	even	10	inner