Embedded newform 20.10.e.b.3.4

• Label: 20.10.e.b.3.4
• Level: $20$
• Weight: $10$
• Character: 20.3
• Analytic conductor: $10.301$
• Analytic rank: $0$
• Dimension: $48$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 20 = 2^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	20.e (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(10.3007167233\)
Analytic rank:	\(0\)
Dimension:	\(48\)
Relative dimension:	\(24\) over \(\Q(i)\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			3.4
Character	\(\chi\)	\(=\)	20.3
Dual form			20.10.e.b.7.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-20.8440 + 8.80488i) q^{2} +(-99.4230 + 99.4230i) q^{3} +(356.948 - 367.059i) q^{4} +(-1395.00 - 84.3181i) q^{5} +(1196.97 - 2947.78i) q^{6} +(-5882.34 - 5882.34i) q^{7} +(-4208.33 + 10793.9i) q^{8} -86.8475i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q + 30 q^{2} - 1440 q^{5} + 6152 q^{6} - 17100 q^{8}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/20\mathbb{Z}\right)^\times\).

\(n\)	\(11\)	\(17\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{3}{4}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−20.8440	+	8.80488i	−0.921185	+	0.389124i
							\(3\)	−99.4230	+	99.4230i	−0.708665	+	0.708665i	−0.966254	−	0.257589i	\(-0.917072\pi\)
0.257589	+	0.966254i	\(0.417072\pi\)
\(5\)	−1395.00	−	84.3181i	−0.998178	−	0.0603331i
							\(7\)	−5882.34	−	5882.34i	−0.925995	−	0.925995i	0.0714491	−	0.997444i	\(-0.477238\pi\)
−0.997444	+	0.0714491i	\(0.977238\pi\)
\(11\)		−	32303.3i		−	0.665242i	−0.943061	−	0.332621i	\(-0.892067\pi\)
							0.943061	−	0.332621i	\(-0.107933\pi\)
\(13\)	117532.	+	117532.i	1.14133	+	1.14133i	0.988207	+	0.153123i	\(0.0489331\pi\)
							0.153123	+	0.988207i	\(0.451067\pi\)
\(17\)	−91183.0	+	91183.0i	−0.264785	+	0.264785i	−0.826995	−	0.562210i	\(-0.809951\pi\)
							0.562210	+	0.826995i	\(0.309951\pi\)
\(19\)	611704.			1.07684			0.538418	−	0.842678i	\(-0.319022\pi\)
							0.538418	+	0.842678i	\(0.319022\pi\)
\(23\)	−678729.	+	678729.i	−0.505733	+	0.505733i	−0.913214	−	0.407481i	\(-0.866407\pi\)
							0.407481	+	0.913214i	\(0.366407\pi\)
\(29\)		−	2.06684e6i		−	0.542645i	−0.962488	−	0.271323i	\(-0.912539\pi\)
							0.962488	−	0.271323i	\(-0.0874611\pi\)
\(31\)		−	6.14607e6i		−	1.19528i	−0.801765	−	0.597640i	\(-0.796105\pi\)
							0.801765	−	0.597640i	\(-0.203895\pi\)
\(37\)	620760.	−	620760.i	0.0544522	−	0.0544522i	−0.679356	−	0.733809i	\(-0.737741\pi\)
							0.733809	+	0.679356i	\(0.237741\pi\)
\(41\)	2.64523e7			1.46196			0.730982	−	0.682397i	\(-0.239063\pi\)
							0.730982	+	0.682397i	\(0.239063\pi\)
\(43\)	2.48600e7	−	2.48600e7i	1.10890	−	1.10890i	0.115609	−	0.993295i	\(-0.463118\pi\)
							0.993295	−	0.115609i	\(-0.0368818\pi\)
\(47\)	2.21054e7	+	2.21054e7i	0.660781	+	0.660781i	0.955564	−	0.294783i	\(-0.0952474\pi\)
							−0.294783	+	0.955564i	\(0.595247\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	20.10.e.b.3.4	✓	48
4.3	odd	2	inner	20.10.e.b.3.8	yes	48
5.2	odd	4	inner	20.10.e.b.7.8	yes	48
5.3	odd	4		100.10.e.e.7.17		48
5.4	even	2		100.10.e.e.43.21		48
20.3	even	4		100.10.e.e.7.21		48
20.7	even	4	inner	20.10.e.b.7.4	yes	48
20.19	odd	2		100.10.e.e.43.17		48

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
20.10.e.b.3.4	✓	48	1.1	even	1	trivial
20.10.e.b.3.8	yes	48	4.3	odd	2	inner
20.10.e.b.7.4	yes	48	20.7	even	4	inner
20.10.e.b.7.8	yes	48	5.2	odd	4	inner
100.10.e.e.7.17		48	5.3	odd	4
100.10.e.e.7.21		48	20.3	even	4
100.10.e.e.43.17		48	20.19	odd	2
100.10.e.e.43.21		48	5.4	even	2