Embedded newform 20.10.e.b.3.13

• Label: 20.10.e.b.3.13
• 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			3.13
Character	\(\chi\)	\(=\)	20.3
Dual form			20.10.e.b.7.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(3.74232 - 22.3158i) q^{2} +(-68.1019 + 68.1019i) q^{3} +(-483.990 - 167.026i) q^{4} +(-497.509 - 1305.99i) q^{5} +(1264.89 + 1774.61i) q^{6} +(2745.50 + 2745.50i) q^{7} +(-5538.56 + 10175.6i) q^{8} +10407.3i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q + 30 q^{2} - 1440 q^{5} + 6152 q^{6} - 17100 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	3.74232	−	22.3158i	0.165389	−	0.986228i
							\(3\)	−68.1019	+	68.1019i	−0.485415	+	0.485415i	−0.906856	−	0.421441i	\(-0.861525\pi\)
0.421441	+	0.906856i	\(0.361525\pi\)
\(5\)	−497.509	−	1305.99i	−0.355988	−	0.934490i
							\(7\)	2745.50	+	2745.50i	0.432195	+	0.432195i	0.889374	−	0.457179i	\(-0.151140\pi\)
−0.457179	+	0.889374i	\(0.651140\pi\)
\(11\)			42021.9i			0.865383i	0.901542	+	0.432691i	\(0.142436\pi\)
							−0.901542	+	0.432691i	\(0.857564\pi\)
\(13\)	48297.7	+	48297.7i	0.469010	+	0.469010i	0.901594	−	0.432584i	\(-0.142398\pi\)
							−0.432584	+	0.901594i	\(0.642398\pi\)
\(17\)	26995.1	−	26995.1i	0.0783908	−	0.0783908i	−0.666824	−	0.745215i	\(-0.732347\pi\)
							0.745215	+	0.666824i	\(0.232347\pi\)
\(19\)	−357538.			−0.629406			−0.314703	−	0.949190i	\(-0.601905\pi\)
							−0.314703	+	0.949190i	\(0.601905\pi\)
\(23\)	−798922.	+	798922.i	−0.595291	+	0.595291i	−0.939056	−	0.343765i	\(-0.888298\pi\)
							0.343765	+	0.939056i	\(0.388298\pi\)
\(29\)			5.48824e6i			1.44093i	0.693493	+	0.720464i	\(0.256071\pi\)
							−0.693493	+	0.720464i	\(0.743929\pi\)
\(31\)		−	8.10192e6i		−	1.57565i	−0.615898	−	0.787826i	\(-0.711207\pi\)
							0.615898	−	0.787826i	\(-0.288793\pi\)
\(37\)	−1.32589e7	+	1.32589e7i	−1.16305	+	1.16305i	−0.179250	+	0.983804i	\(0.557367\pi\)
							−0.983804	+	0.179250i	\(0.942633\pi\)
\(41\)	7.49196e6			0.414064			0.207032	−	0.978334i	\(-0.433620\pi\)
							0.207032	+	0.978334i	\(0.433620\pi\)
\(43\)	−2.51880e7	+	2.51880e7i	−1.12353	+	1.12353i	−0.132327	+	0.991206i	\(0.542245\pi\)
							−0.991206	+	0.132327i	\(0.957755\pi\)
\(47\)	1.82438e7	+	1.82438e7i	0.545349	+	0.545349i	0.925092	−	0.379743i	\(-0.123987\pi\)
							−0.379743	+	0.925092i	\(0.623987\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	20.10.e.b.3.13	✓	48
4.3	odd	2	inner	20.10.e.b.3.24	yes	48
5.2	odd	4	inner	20.10.e.b.7.24	yes	48
5.3	odd	4		100.10.e.e.7.1		48
5.4	even	2		100.10.e.e.43.12		48
20.3	even	4		100.10.e.e.7.12		48
20.7	even	4	inner	20.10.e.b.7.13	yes	48
20.19	odd	2		100.10.e.e.43.1		48

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
20.10.e.b.3.13	✓	48	1.1	even	1	trivial
20.10.e.b.3.24	yes	48	4.3	odd	2	inner
20.10.e.b.7.13	yes	48	20.7	even	4	inner
20.10.e.b.7.24	yes	48	5.2	odd	4	inner
100.10.e.e.7.1		48	5.3	odd	4
100.10.e.e.7.12		48	20.3	even	4
100.10.e.e.43.1		48	20.19	odd	2
100.10.e.e.43.12		48	5.4	even	2