Embedded newform 20.10.e.b.3.5

• Label: 20.10.e.b.3.5
• 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			3.5
Character	\(\chi\)	\(=\)	20.3
Dual form			20.10.e.b.7.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-19.1048 - 12.1247i) q^{2} +(-48.3412 + 48.3412i) q^{3} +(217.985 + 463.278i) q^{4} +(-1393.61 - 104.823i) q^{5} +(1509.67 - 337.427i) q^{6} +(5614.42 + 5614.42i) q^{7} +(1452.53 - 11493.8i) q^{8} +15009.3i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q + 30 q^{2} - 1440 q^{5} + 6152 q^{6} - 17100 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−19.1048	−	12.1247i	−0.844320	−	0.535839i
							\(3\)	−48.3412	+	48.3412i	−0.344566	+	0.344566i	−0.858081	−	0.513515i	\(-0.828343\pi\)
0.513515	+	0.858081i	\(0.328343\pi\)
\(5\)	−1393.61	−	104.823i	−0.997183	−	0.0750055i
							\(7\)	5614.42	+	5614.42i	0.883819	+	0.883819i	0.993920	−	0.110101i	\(-0.0351175\pi\)
−0.110101	+	0.993920i	\(0.535117\pi\)
\(11\)		−	59408.0i		−	1.22343i	−0.791080	−	0.611713i	\(-0.790481\pi\)
							0.791080	−	0.611713i	\(-0.209519\pi\)
\(13\)	−124255.	−	124255.i	−1.20661	−	1.20661i	−0.972117	−	0.234496i	\(-0.924656\pi\)
							−0.234496	−	0.972117i	\(-0.575344\pi\)
\(17\)	211436.	−	211436.i	0.613988	−	0.613988i	−0.329995	−	0.943983i	\(-0.607047\pi\)
							0.943983	+	0.329995i	\(0.107047\pi\)
\(19\)	−238194.			−0.419314			−0.209657	−	0.977775i	\(-0.567235\pi\)
							−0.209657	+	0.977775i	\(0.567235\pi\)
\(23\)	1.77559e6	−	1.77559e6i	1.32302	−	1.32302i	0.411701	−	0.911319i	\(-0.364935\pi\)
							0.911319	−	0.411701i	\(-0.135065\pi\)
\(29\)			2.63368e6i			0.691468i	0.938333	+	0.345734i	\(0.112370\pi\)
							−0.938333	+	0.345734i	\(0.887630\pi\)
\(31\)		−	2.40344e6i		−	0.467419i	−0.972307	−	0.233709i	\(-0.924914\pi\)
							0.972307	−	0.233709i	\(-0.0750864\pi\)
\(37\)	2.79682e6	−	2.79682e6i	0.245334	−	0.245334i	−0.573719	−	0.819052i	\(-0.694500\pi\)
							0.819052	+	0.573719i	\(0.194500\pi\)
\(41\)	−4.05163e6			−0.223925			−0.111962	−	0.993712i	\(-0.535714\pi\)
							−0.111962	+	0.993712i	\(0.535714\pi\)
\(43\)	1.38414e7	−	1.38414e7i	0.617408	−	0.617408i	−0.327458	−	0.944866i	\(-0.606192\pi\)
							0.944866	+	0.327458i	\(0.106192\pi\)
\(47\)	−6.81666e6	−	6.81666e6i	−0.203766	−	0.203766i	0.597845	−	0.801611i	\(-0.296024\pi\)
							−0.801611	+	0.597845i	\(0.796024\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	20.10.e.b.3.5	✓	48
4.3	odd	2	inner	20.10.e.b.3.16	yes	48
5.2	odd	4	inner	20.10.e.b.7.16	yes	48
5.3	odd	4		100.10.e.e.7.9		48
5.4	even	2		100.10.e.e.43.20		48
20.3	even	4		100.10.e.e.7.20		48
20.7	even	4	inner	20.10.e.b.7.5	yes	48
20.19	odd	2		100.10.e.e.43.9		48

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
20.10.e.b.3.5	✓	48	1.1	even	1	trivial
20.10.e.b.3.16	yes	48	4.3	odd	2	inner
20.10.e.b.7.5	yes	48	20.7	even	4	inner
20.10.e.b.7.16	yes	48	5.2	odd	4	inner
100.10.e.e.7.9		48	5.3	odd	4
100.10.e.e.7.20		48	20.3	even	4
100.10.e.e.43.9		48	20.19	odd	2
100.10.e.e.43.20		48	5.4	even	2