Embedded newform 50.6.e.a.19.7

• Label: 50.6.e.a.19.7
• Level: $50$
• Weight: $6$
• Character: 50.19
• 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			19.7
Character	\(\chi\)	\(=\)	50.19
Dual form			50.6.e.a.29.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(3.80423 - 1.23607i) q^{2} +(-16.9585 - 23.3414i) q^{3} +(12.9443 - 9.40456i) q^{4} +(-8.25144 - 55.2894i) q^{5} +(-93.3657 - 67.8341i) q^{6} +138.453i q^{7} +(37.6183 - 51.7771i) q^{8} +(-182.139 + 560.566i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q + 192 q^{4} + 180 q^{5} + 72 q^{6} + 686 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(27\)
\(\chi(n)\)	\(e\left(\frac{9}{10}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	3.80423	−	1.23607i	0.672499	−	0.218508i
							\(3\)	−16.9585	−	23.3414i	−1.08789	−	1.49735i	−0.850527	−	0.525932i	\(-0.823717\pi\)
−0.237364	−	0.971421i	\(-0.576283\pi\)
\(5\)	−8.25144	−	55.2894i	−0.147606	−	0.989046i
							\(7\)			138.453i			1.06797i	0.845495	+	0.533983i	\(0.179306\pi\)
−0.845495	+	0.533983i	\(0.820694\pi\)
\(11\)	−71.4074	−	219.769i	−0.177935	−	0.547628i	0.821820	−	0.569747i	\(-0.192959\pi\)
							−0.999755	+	0.0221190i	\(0.992959\pi\)
\(13\)	−251.878	−	81.8400i	−0.413363	−	0.134310i	0.0949504	−	0.995482i	\(-0.469731\pi\)
							−0.508313	+	0.861172i	\(0.669731\pi\)
\(17\)	323.609	−	445.410i	0.271581	−	0.373799i	−0.651342	−	0.758784i	\(-0.725794\pi\)
							0.922923	+	0.384986i	\(0.125794\pi\)
\(19\)	−1503.87	−	1092.63i	−0.955713	−	0.694366i	−0.00356161	−	0.999994i	\(-0.501134\pi\)
							−0.952151	+	0.305628i	\(0.901134\pi\)
\(23\)	3180.04	−	1033.26i	1.25347	−	0.407277i	0.394306	−	0.918979i	\(-0.370985\pi\)
							0.859163	+	0.511702i	\(0.170985\pi\)
\(29\)	−3656.43	+	2656.55i	−0.807350	+	0.586574i	−0.913061	−	0.407823i	\(-0.866288\pi\)
							0.105711	+	0.994397i	\(0.466288\pi\)
\(31\)	−8606.44	−	6252.94i	−1.60849	−	1.16864i	−0.867984	−	0.496591i	\(-0.834585\pi\)
							−0.740508	−	0.672047i	\(-0.765415\pi\)
\(37\)	12117.1	+	3937.07i	1.45510	+	0.472790i	0.926569	−	0.376124i	\(-0.122743\pi\)
							0.528530	+	0.848915i	\(0.322743\pi\)
\(41\)	2161.21	−	6651.52i	0.200788	−	0.617961i	−0.799072	−	0.601235i	\(-0.794676\pi\)
							0.999860	−	0.0167264i	\(-0.00532442\pi\)
\(43\)		−	14511.7i		−	1.19687i	−0.801170	−	0.598437i	\(-0.795789\pi\)
							0.801170	−	0.598437i	\(-0.204211\pi\)
\(47\)	−9939.24	−	13680.2i	−0.656309	−	0.903332i	0.343043	−	0.939320i	\(-0.388542\pi\)
							−0.999352	+	0.0359878i	\(0.988542\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	50.6.e.a.19.7	✓	48
25.4	even	10	inner	50.6.e.a.29.7	yes	48

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
50.6.e.a.19.7	✓	48	1.1	even	1	trivial
50.6.e.a.29.7	yes	48	25.4	even	10	inner