Embedded newform 50.6.e.a.19.8

• Label: 50.6.e.a.19.8
• 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			19.8
Character	\(\chi\)	\(=\)	50.19
Dual form			50.6.e.a.29.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(3.80423 - 1.23607i) q^{2} +(-8.70645 - 11.9834i) q^{3} +(12.9443 - 9.40456i) q^{4} +(45.9076 + 31.8981i) q^{5} +(-47.9336 - 34.8258i) q^{6} -137.000i q^{7} +(37.6183 - 51.7771i) q^{8} +(7.29147 - 22.4408i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q + 192 q^{4} + 180 q^{5} + 72 q^{6} + 686 q^{9}+O(q^{10}) \)

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
							\(3\)	−8.70645	−	11.9834i	−0.558519	−	0.768736i	0.432618	−	0.901577i	\(-0.357590\pi\)
−0.991137	+	0.132842i	\(0.957590\pi\)
\(5\)	45.9076	+	31.8981i	0.821221	+	0.570611i
							\(7\)		−	137.000i		−	1.05675i	−0.849010	−	0.528377i	\(-0.822801\pi\)
0.849010	−	0.528377i	\(-0.177199\pi\)
\(11\)	−100.652	−	309.775i	−0.250808	−	0.771908i	−0.994627	−	0.103526i	\(-0.966987\pi\)
							0.743819	−	0.668381i	\(-0.233013\pi\)
\(13\)	−943.119	−	306.438i	−1.54778	−	0.502903i	−0.594268	−	0.804267i	\(-0.702558\pi\)
							−0.953509	+	0.301364i	\(0.902558\pi\)
\(17\)	−80.3345	+	110.571i	−0.0674186	+	0.0927938i	−0.841395	−	0.540421i	\(-0.818265\pi\)
							0.773976	+	0.633215i	\(0.218265\pi\)
\(19\)	1827.95	+	1328.08i	1.16166	+	0.843996i	0.989987	−	0.141158i	\(-0.0450826\pi\)
							0.171674	+	0.985154i	\(0.445083\pi\)
\(23\)	4525.98	−	1470.58i	1.78399	−	0.579654i	0.784797	−	0.619753i	\(-0.212767\pi\)
							0.999196	+	0.0400981i	\(0.0127671\pi\)
\(29\)	2594.62	−	1885.10i	0.572900	−	0.416236i	−0.263258	−	0.964726i	\(-0.584797\pi\)
							0.836157	+	0.548490i	\(0.184797\pi\)
\(31\)	3751.44	+	2725.58i	0.701121	+	0.509395i	0.880297	−	0.474423i	\(-0.157343\pi\)
							−0.179176	+	0.983817i	\(0.557343\pi\)
\(37\)	−6518.38	−	2117.95i	−0.782772	−	0.254338i	−0.109749	−	0.993959i	\(-0.535005\pi\)
							−0.673023	+	0.739621i	\(0.735005\pi\)
\(41\)	−3935.23	+	12111.4i	−0.365604	+	1.12521i	0.583998	+	0.811755i	\(0.301487\pi\)
							−0.949602	+	0.313458i	\(0.898513\pi\)
\(43\)			10691.0i			0.881751i	0.897568	+	0.440875i	\(0.145332\pi\)
							−0.897568	+	0.440875i	\(0.854668\pi\)
\(47\)	−4637.28	−	6382.67i	−0.306210	−	0.421461i	0.627985	−	0.778226i	\(-0.283880\pi\)
							−0.934194	+	0.356764i	\(0.883880\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	50.6.e.a.19.8	✓	48
25.4	even	10	inner	50.6.e.a.29.8	yes	48

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
50.6.e.a.19.8	✓	48	1.1	even	1	trivial
50.6.e.a.29.8	yes	48	25.4	even	10	inner