Embedded newform 20.6.e.b.3.1

• Label: 20.6.e.b.3.1
• Level: $20$
• Weight: $6$
• Character: 20.3
• Analytic conductor: $3.208$
• Analytic rank: $0$
• Dimension: $24$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 20 = 2^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	20.e (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.20767639626\)
Analytic rank:	\(0\)
Dimension:	\(24\)
Relative dimension:	\(12\) over \(\Q(i)\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			3.1
Character	\(\chi\)	\(=\)	20.3
Dual form			20.6.e.b.7.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-5.45971 + 1.48039i) q^{2} +(3.65704 - 3.65704i) q^{3} +(27.6169 - 16.1650i) q^{4} +(-18.9750 + 52.5828i) q^{5} +(-14.5525 + 25.3802i) q^{6} +(162.920 + 162.920i) q^{7} +(-126.850 + 129.140i) q^{8} +216.252i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 10 q^{2} - 80 q^{5} - 184 q^{6} + 500 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−5.45971	+	1.48039i	−0.965150	+	0.261698i
							\(3\)	3.65704	−	3.65704i	0.234599	−	0.234599i	−0.580010	−	0.814609i	\(-0.696951\pi\)
0.814609	+	0.580010i	\(0.196951\pi\)
\(5\)	−18.9750	+	52.5828i	−0.339436	+	0.940629i
							\(7\)	162.920	+	162.920i	1.25669	+	1.25669i	0.952662	+	0.304031i	\(0.0983325\pi\)
0.304031	+	0.952662i	\(0.401667\pi\)
\(11\)		−	539.141i		−	1.34345i	−0.740802	−	0.671724i	\(-0.765554\pi\)
							0.740802	−	0.671724i	\(-0.234446\pi\)
\(13\)	297.964	+	297.964i	0.488995	+	0.488995i	0.907989	−	0.418994i	\(-0.137617\pi\)
							−0.418994	+	0.907989i	\(0.637617\pi\)
\(17\)	−350.257	+	350.257i	−0.293944	+	0.293944i	−0.838636	−	0.544692i	\(-0.816647\pi\)
							0.544692	+	0.838636i	\(0.316647\pi\)
\(19\)	−220.012			−0.139818			−0.0699090	−	0.997553i	\(-0.522271\pi\)
							−0.0699090	+	0.997553i	\(0.522271\pi\)
\(23\)	−314.008	+	314.008i	−0.123772	+	0.123772i	−0.766279	−	0.642508i	\(-0.777894\pi\)
							0.642508	+	0.766279i	\(0.277894\pi\)
\(29\)		−	1576.37i		−	0.348067i	−0.984740	−	0.174034i	\(-0.944320\pi\)
							0.984740	−	0.174034i	\(-0.0556801\pi\)
\(31\)		−	4101.32i		−	0.766513i	−0.923642	−	0.383256i	\(-0.874803\pi\)
							0.923642	−	0.383256i	\(-0.125197\pi\)
\(37\)	3969.71	−	3969.71i	0.476710	−	0.476710i	−0.427368	−	0.904078i	\(-0.640559\pi\)
							0.904078	+	0.427368i	\(0.140559\pi\)
\(41\)	8309.75			0.772020			0.386010	−	0.922495i	\(-0.373853\pi\)
							0.386010	+	0.922495i	\(0.373853\pi\)
\(43\)	8384.40	−	8384.40i	0.691514	−	0.691514i	−0.271051	−	0.962565i	\(-0.587371\pi\)
							0.962565	+	0.271051i	\(0.0873712\pi\)
\(47\)	−3663.28	−	3663.28i	−0.241894	−	0.241894i	0.575739	−	0.817633i	\(-0.304714\pi\)
							−0.817633	+	0.575739i	\(0.804714\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	20.6.e.b.3.1	✓	24
4.3	odd	2	inner	20.6.e.b.3.6	yes	24
5.2	odd	4	inner	20.6.e.b.7.6	yes	24
5.3	odd	4		100.6.e.e.7.7		24
5.4	even	2		100.6.e.e.43.12		24
20.3	even	4		100.6.e.e.7.12		24
20.7	even	4	inner	20.6.e.b.7.1	yes	24
20.19	odd	2		100.6.e.e.43.7		24

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
20.6.e.b.3.1	✓	24	1.1	even	1	trivial
20.6.e.b.3.6	yes	24	4.3	odd	2	inner
20.6.e.b.7.1	yes	24	20.7	even	4	inner
20.6.e.b.7.6	yes	24	5.2	odd	4	inner
100.6.e.e.7.7		24	5.3	odd	4
100.6.e.e.7.12		24	20.3	even	4
100.6.e.e.43.7		24	20.19	odd	2
100.6.e.e.43.12		24	5.4	even	2