Embedded newform 20.8.e.b.3.6

• Label: 20.8.e.b.3.6
• 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			3.6
Character	\(\chi\)	\(=\)	20.3
Dual form			20.8.e.b.7.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-6.36342 + 9.35451i) q^{2} +(-56.5129 + 56.5129i) q^{3} +(-47.0138 - 119.053i) q^{4} +(-240.823 + 141.877i) q^{5} +(-169.035 - 888.266i) q^{6} +(764.897 + 764.897i) q^{7} +(1412.85 + 317.795i) q^{8} -4200.41i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 36 q + 14 q^{2} - 560 q^{5} + 104 q^{6} - 3052 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−6.36342	+	9.35451i	−0.562452	+	0.826830i
							\(3\)	−56.5129	+	56.5129i	−1.20843	+	1.20843i	−0.236901	+	0.971534i	\(0.576132\pi\)
−0.971534	+	0.236901i	\(0.923868\pi\)
\(5\)	−240.823	+	141.877i	−0.861596	+	0.507595i
							\(7\)	764.897	+	764.897i	0.842868	+	0.842868i	0.989231	−	0.146363i	\(-0.0467567\pi\)
−0.146363	+	0.989231i	\(0.546757\pi\)
\(11\)			1507.09i			0.341400i	0.985323	+	0.170700i	\(0.0546029\pi\)
							−0.985323	+	0.170700i	\(0.945397\pi\)
\(13\)	−5391.78	−	5391.78i	−0.680660	−	0.680660i	0.279489	−	0.960149i	\(-0.409835\pi\)
							−0.960149	+	0.279489i	\(0.909835\pi\)
\(17\)	2875.49	−	2875.49i	0.141952	−	0.141952i	−0.632560	−	0.774512i	\(-0.717996\pi\)
							0.774512	+	0.632560i	\(0.217996\pi\)
\(19\)	−1541.88			−0.0515719			−0.0257859	−	0.999667i	\(-0.508209\pi\)
							−0.0257859	+	0.999667i	\(0.508209\pi\)
\(23\)	−49300.6	+	49300.6i	−0.844898	+	0.844898i	−0.989491	−	0.144593i	\(-0.953813\pi\)
							0.144593	+	0.989491i	\(0.453813\pi\)
\(29\)			84003.4i			0.639594i	0.947486	+	0.319797i	\(0.103615\pi\)
							−0.947486	+	0.319797i	\(0.896385\pi\)
\(31\)		−	303876.i		−	1.83202i	−0.401155	−	0.916010i	\(-0.631391\pi\)
							0.401155	−	0.916010i	\(-0.368609\pi\)
\(37\)	174112.	−	174112.i	0.565098	−	0.565098i	−0.365653	−	0.930751i	\(-0.619154\pi\)
							0.930751	+	0.365653i	\(0.119154\pi\)
\(41\)	114783.			0.260097			0.130049	−	0.991508i	\(-0.458487\pi\)
							0.130049	+	0.991508i	\(0.458487\pi\)
\(43\)	−585971.	+	585971.i	−1.12392	+	1.12392i	−0.132776	+	0.991146i	\(0.542389\pi\)
							−0.991146	+	0.132776i	\(0.957611\pi\)
\(47\)	161312.	+	161312.i	0.226634	+	0.226634i	0.811285	−	0.584651i	\(-0.198769\pi\)
							−0.584651	+	0.811285i	\(0.698769\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	20.8.e.b.3.6	yes	36
4.3	odd	2	inner	20.8.e.b.3.4	✓	36
5.2	odd	4	inner	20.8.e.b.7.4	yes	36
5.3	odd	4		100.8.e.e.7.15		36
5.4	even	2		100.8.e.e.43.13		36
20.3	even	4		100.8.e.e.7.13		36
20.7	even	4	inner	20.8.e.b.7.6	yes	36
20.19	odd	2		100.8.e.e.43.15		36

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
20.8.e.b.3.4	✓	36	4.3	odd	2	inner
20.8.e.b.3.6	yes	36	1.1	even	1	trivial
20.8.e.b.7.4	yes	36	5.2	odd	4	inner
20.8.e.b.7.6	yes	36	20.7	even	4	inner
100.8.e.e.7.13		36	20.3	even	4
100.8.e.e.7.15		36	5.3	odd	4
100.8.e.e.43.13		36	5.4	even	2
100.8.e.e.43.15		36	20.19	odd	2