Embedded newform 50.6.e.a.19.5

• Label: 50.6.e.a.19.5
• 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			19.5
Character	\(\chi\)	\(=\)	50.19
Dual form			50.6.e.a.29.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-3.80423 + 1.23607i) q^{2} +(9.57478 + 13.1786i) q^{3} +(12.9443 - 9.40456i) q^{4} +(-0.421737 - 55.9001i) q^{5} +(-52.7142 - 38.2991i) q^{6} -120.466i q^{7} +(-37.6183 + 51.7771i) q^{8} +(-6.90673 + 21.2567i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q + 192 q^{4} + 180 q^{5} + 72 q^{6} + 686 q^{9}+O(q^{10}) \)

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
							\(3\)	9.57478	+	13.1786i	0.614222	+	0.845405i	0.996916	−	0.0784706i	\(-0.0250037\pi\)
−0.382694	+	0.923875i	\(0.625004\pi\)
\(5\)	−0.421737	−	55.9001i	−0.00754426	−	0.999972i
							\(7\)		−	120.466i		−	0.929221i	−0.885515	−	0.464611i	\(-0.846194\pi\)
0.885515	−	0.464611i	\(-0.153806\pi\)
\(11\)	−31.8126	−	97.9092i	−0.0792717	−	0.243973i	0.903565	−	0.428451i	\(-0.140941\pi\)
							−0.982837	+	0.184478i	\(0.940941\pi\)
\(13\)	255.831	+	83.1244i	0.419850	+	0.136418i	0.511320	−	0.859390i	\(-0.329157\pi\)
							−0.0914702	+	0.995808i	\(0.529157\pi\)
\(17\)	1171.73	−	1612.75i	0.983347	−	1.35346i	0.0483396	−	0.998831i	\(-0.484607\pi\)
							0.935007	−	0.354629i	\(-0.115393\pi\)
\(19\)	666.349	+	484.131i	0.423465	+	0.307665i	0.779031	−	0.626986i	\(-0.215712\pi\)
							−0.355565	+	0.934651i	\(0.615712\pi\)
\(23\)	1370.08	−	445.166i	0.540041	−	0.175470i	−0.0262803	−	0.999655i	\(-0.508366\pi\)
							0.566321	+	0.824185i	\(0.308366\pi\)
\(29\)	−5446.90	+	3957.41i	−1.20269	+	0.873807i	−0.994547	−	0.104290i	\(-0.966743\pi\)
							−0.208146	+	0.978098i	\(0.566743\pi\)
\(31\)	4586.01	+	3331.93i	0.857098	+	0.622718i	0.927094	−	0.374829i	\(-0.122299\pi\)
							−0.0699960	+	0.997547i	\(0.522299\pi\)
\(37\)	−10575.6	−	3436.21i	−1.26999	−	0.412644i	−0.404945	−	0.914341i	\(-0.632709\pi\)
							−0.865044	+	0.501696i	\(0.832709\pi\)
\(41\)	2553.57	−	7859.08i	0.237240	−	0.730150i	−0.759576	−	0.650418i	\(-0.774594\pi\)
							0.996816	−	0.0797317i	\(-0.0254064\pi\)
\(43\)		−	352.908i		−	0.0291065i	−0.999894	−	0.0145533i	\(-0.995367\pi\)
							0.999894	−	0.0145533i	\(-0.00463261\pi\)
\(47\)	−9385.67	−	12918.3i	−0.619756	−	0.853021i	0.377579	−	0.925977i	\(-0.376757\pi\)
							−0.997335	+	0.0729564i	\(0.976757\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	50.6.e.a.19.5	✓	48
25.4	even	10	inner	50.6.e.a.29.5	yes	48

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
50.6.e.a.19.5	✓	48	1.1	even	1	trivial
50.6.e.a.29.5	yes	48	25.4	even	10	inner