Embedded newform 20.10.e.b.3.18

• Label: 20.10.e.b.3.18
• 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			3.18
Character	\(\chi\)	\(=\)	20.3
Dual form			20.10.e.b.7.18

$q$-expansion

\(f(q)\)	\(=\)	\(q+(15.5607 - 16.4276i) q^{2} +(139.769 - 139.769i) q^{3} +(-27.7316 - 511.248i) q^{4} +(1375.58 - 246.768i) q^{5} +(-121.170 - 4470.96i) q^{6} +(3052.81 + 3052.81i) q^{7} +(-8830.10 - 7499.80i) q^{8} -19387.7i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q + 30 q^{2} - 1440 q^{5} + 6152 q^{6} - 17100 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	15.5607	−	16.4276i	0.687691	−	0.726004i
							\(3\)	139.769	−	139.769i	0.996242	−	0.996242i	−0.00375131	−	0.999993i	\(-0.501194\pi\)
0.999993	+	0.00375131i	\(0.00119408\pi\)
\(5\)	1375.58	−	246.768i	0.984288	−	0.176573i
							\(7\)	3052.81	+	3052.81i	0.480573	+	0.480573i	0.905315	−	0.424742i	\(-0.139635\pi\)
−0.424742	+	0.905315i	\(0.639635\pi\)
\(11\)			94007.5i			1.93595i	0.251037	+	0.967977i	\(0.419228\pi\)
							−0.251037	+	0.967977i	\(0.580772\pi\)
\(13\)	−57102.7	−	57102.7i	−0.554512	−	0.554512i	0.373228	−	0.927740i	\(-0.378251\pi\)
							−0.927740	+	0.373228i	\(0.878251\pi\)
\(17\)	19143.6	−	19143.6i	0.0555910	−	0.0555910i	−0.678765	−	0.734356i	\(-0.737484\pi\)
							0.734356	+	0.678765i	\(0.237484\pi\)
\(19\)	−674520.			−1.18742			−0.593709	−	0.804680i	\(-0.702337\pi\)
							−0.593709	+	0.804680i	\(0.702337\pi\)
\(23\)	576591.	−	576591.i	0.429628	−	0.429628i	−0.458874	−	0.888502i	\(-0.651747\pi\)
							0.888502	+	0.458874i	\(0.151747\pi\)
\(29\)			4.76535e6i			1.25113i	0.780170	+	0.625567i	\(0.215132\pi\)
							−0.780170	+	0.625567i	\(0.784868\pi\)
\(31\)			4.09142e6i			0.795694i	0.917452	+	0.397847i	\(0.130243\pi\)
							−0.917452	+	0.397847i	\(0.869757\pi\)
\(37\)	6.25292e6	−	6.25292e6i	0.548498	−	0.548498i	−0.377508	−	0.926006i	\(-0.623219\pi\)
							0.926006	+	0.377508i	\(0.123219\pi\)
\(41\)	−5.40848e6			−0.298915			−0.149458	−	0.988768i	\(-0.547753\pi\)
							−0.149458	+	0.988768i	\(0.547753\pi\)
\(43\)	2.13080e7	−	2.13080e7i	0.950464	−	0.950464i	−0.0483658	−	0.998830i	\(-0.515401\pi\)
							0.998830	+	0.0483658i	\(0.0154013\pi\)
\(47\)	−2.28239e7	−	2.28239e7i	−0.682258	−	0.682258i	0.278251	−	0.960509i	\(-0.410245\pi\)
							−0.960509	+	0.278251i	\(0.910245\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	20.10.e.b.3.18	✓	48
4.3	odd	2	inner	20.10.e.b.3.19	yes	48
5.2	odd	4	inner	20.10.e.b.7.19	yes	48
5.3	odd	4		100.10.e.e.7.6		48
5.4	even	2		100.10.e.e.43.7		48
20.3	even	4		100.10.e.e.7.7		48
20.7	even	4	inner	20.10.e.b.7.18	yes	48
20.19	odd	2		100.10.e.e.43.6		48

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
20.10.e.b.3.18	✓	48	1.1	even	1	trivial
20.10.e.b.3.19	yes	48	4.3	odd	2	inner
20.10.e.b.7.18	yes	48	20.7	even	4	inner
20.10.e.b.7.19	yes	48	5.2	odd	4	inner
100.10.e.e.7.6		48	5.3	odd	4
100.10.e.e.7.7		48	20.3	even	4
100.10.e.e.43.6		48	20.19	odd	2
100.10.e.e.43.7		48	5.4	even	2