Embedded newform 20.6.e.b.7.1

• Label: 20.6.e.b.7.1
• Level: $20$
• Weight: $6$
• Character: 20.7
• 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			7.1
Character	\(\chi\)	\(=\)	20.7
Dual form			20.6.e.b.3.1

$q$-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 \(\left(\mathbb{Z}/20\mathbb{Z}\right)^\times\).

\(n\)	\(11\)	\(17\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{1}{4}\right)\)

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	0.814609	−	0.580010i	\(-0.196951\pi\)
−0.580010	+	0.814609i	\(0.696951\pi\)
\(5\)	−18.9750	−	52.5828i	−0.339436	−	0.940629i
							\(7\)	162.920	−	162.920i	1.25669	−	1.25669i	0.304031	−	0.952662i	\(-0.401667\pi\)
0.952662	−	0.304031i	\(-0.0983325\pi\)
\(11\)			539.141i			1.34345i	0.740802	+	0.671724i	\(0.234446\pi\)
							−0.740802	+	0.671724i	\(0.765554\pi\)
\(13\)	297.964	−	297.964i	0.488995	−	0.488995i	−0.418994	−	0.907989i	\(-0.637617\pi\)
							0.907989	+	0.418994i	\(0.137617\pi\)
\(17\)	−350.257	−	350.257i	−0.293944	−	0.293944i	0.544692	−	0.838636i	\(-0.316647\pi\)
							−0.838636	+	0.544692i	\(0.816647\pi\)
\(19\)	−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	0.642508	−	0.766279i	\(-0.277894\pi\)
							−0.766279	+	0.642508i	\(0.777894\pi\)
\(29\)			1576.37i			0.348067i	0.984740	+	0.174034i	\(0.0556801\pi\)
							−0.984740	+	0.174034i	\(0.944320\pi\)
\(31\)			4101.32i			0.766513i	0.923642	+	0.383256i	\(0.125197\pi\)
							−0.923642	+	0.383256i	\(0.874803\pi\)
\(37\)	3969.71	+	3969.71i	0.476710	+	0.476710i	0.904078	−	0.427368i	\(-0.140559\pi\)
							−0.427368	+	0.904078i	\(0.640559\pi\)
\(41\)	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	0.962565	−	0.271051i	\(-0.0873712\pi\)
							−0.271051	+	0.962565i	\(0.587371\pi\)
\(47\)	−3663.28	+	3663.28i	−0.241894	+	0.241894i	−0.817633	−	0.575739i	\(-0.804714\pi\)
							0.575739	+	0.817633i	\(0.304714\pi\)

(See \(a_n\) instead)

Twists

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

    

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