Embedded newform 20.8.e.b.3.18

• Label: 20.8.e.b.3.18
• Level: $20$
• Weight: $8$
• Character: 20.3
• Analytic conductor: $6.248$
• Analytic rank: $0$
• Dimension: $36$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 20 = 2^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	20.e (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(6.24770050968\)
Analytic rank:	\(0\)
Dimension:	\(36\)
Relative dimension:	\(18\) 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			20.8.e.b.7.18

$q$-expansion

\(f(q)\)	\(=\)	\(q+(11.3129 - 0.138397i) q^{2} +(-19.3526 + 19.3526i) q^{3} +(127.962 - 3.13132i) q^{4} +(191.974 - 203.153i) q^{5} +(-216.255 + 221.612i) q^{6} +(939.327 + 939.327i) q^{7} +(1447.18 - 53.1337i) q^{8} +1437.95i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 36 q + 14 q^{2} - 560 q^{5} + 104 q^{6} - 3052 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	11.3129	−	0.138397i	0.999925	−	0.0122326i
							\(3\)	−19.3526	+	19.3526i	−0.413824	+	0.413824i	−0.883068	−	0.469244i	\(-0.844526\pi\)
0.469244	+	0.883068i	\(0.344526\pi\)
\(5\)	191.974	−	203.153i	0.686827	−	0.726821i
							\(7\)	939.327	+	939.327i	1.03508	+	1.03508i	0.999362	+	0.0357180i	\(0.0113718\pi\)
0.0357180	+	0.999362i	\(0.488628\pi\)
\(11\)		−	1453.91i		−	0.329354i	−0.986348	−	0.164677i	\(-0.947342\pi\)
							0.986348	−	0.164677i	\(-0.0526581\pi\)
\(13\)	−8589.52	−	8589.52i	−1.08434	−	1.08434i	−0.996099	−	0.0882460i	\(-0.971874\pi\)
							−0.0882460	−	0.996099i	\(-0.528126\pi\)
\(17\)	11770.4	−	11770.4i	0.581059	−	0.581059i	−0.354136	−	0.935194i	\(-0.615225\pi\)
							0.935194	+	0.354136i	\(0.115225\pi\)
\(19\)	−38962.7			−1.30320			−0.651601	−	0.758562i	\(-0.725902\pi\)
							−0.651601	+	0.758562i	\(0.725902\pi\)
\(23\)	−47213.1	+	47213.1i	−0.809123	+	0.809123i	−0.984501	−	0.175378i	\(-0.943885\pi\)
							0.175378	+	0.984501i	\(0.443885\pi\)
\(29\)			18289.2i			0.139252i	0.997573	+	0.0696260i	\(0.0221806\pi\)
							−0.997573	+	0.0696260i	\(0.977819\pi\)
\(31\)			73973.2i			0.445973i	0.974822	+	0.222986i	\(0.0715805\pi\)
							−0.974822	+	0.222986i	\(0.928419\pi\)
\(37\)	−117886.	+	117886.i	−0.382608	+	0.382608i	−0.872041	−	0.489433i	\(-0.837204\pi\)
							0.489433	+	0.872041i	\(0.337204\pi\)
\(41\)	279248.			0.632771			0.316385	−	0.948631i	\(-0.397531\pi\)
							0.316385	+	0.948631i	\(0.397531\pi\)
\(43\)	104477.	−	104477.i	0.200392	−	0.200392i	−0.599776	−	0.800168i	\(-0.704744\pi\)
							0.800168	+	0.599776i	\(0.204744\pi\)
\(47\)	−667707.	−	667707.i	−0.938087	−	0.938087i	0.0601047	−	0.998192i	\(-0.480857\pi\)
							−0.998192	+	0.0601047i	\(0.980857\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	20.8.e.b.3.18	yes	36
4.3	odd	2	inner	20.8.e.b.3.9	✓	36
5.2	odd	4	inner	20.8.e.b.7.9	yes	36
5.3	odd	4		100.8.e.e.7.10		36
5.4	even	2		100.8.e.e.43.1		36
20.3	even	4		100.8.e.e.7.1		36
20.7	even	4	inner	20.8.e.b.7.18	yes	36
20.19	odd	2		100.8.e.e.43.10		36

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
20.8.e.b.3.9	✓	36	4.3	odd	2	inner
20.8.e.b.3.18	yes	36	1.1	even	1	trivial
20.8.e.b.7.9	yes	36	5.2	odd	4	inner
20.8.e.b.7.18	yes	36	20.7	even	4	inner
100.8.e.e.7.1		36	20.3	even	4
100.8.e.e.7.10		36	5.3	odd	4
100.8.e.e.43.1		36	5.4	even	2
100.8.e.e.43.10		36	20.19	odd	2