Embedded newform 20.8.e.b.3.17

• Label: 20.8.e.b.3.17
• 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			3.17
Character	\(\chi\)	\(=\)	20.3
Dual form			20.8.e.b.7.17

$q$-expansion

\(f(q)\)	\(=\)	\(q+(10.6961 - 3.68683i) q^{2} +(32.6647 - 32.6647i) q^{3} +(100.815 - 78.8696i) q^{4} +(145.214 + 238.826i) q^{5} +(228.957 - 469.815i) q^{6} +(-669.751 - 669.751i) q^{7} +(787.548 - 1215.29i) q^{8} +53.0351i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 36 q + 14 q^{2} - 560 q^{5} + 104 q^{6} - 3052 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	10.6961	−	3.68683i	0.945414	−	0.325873i
							\(3\)	32.6647	−	32.6647i	0.698480	−	0.698480i	−0.265602	−	0.964083i	\(-0.585571\pi\)
0.964083	+	0.265602i	\(0.0855708\pi\)
\(5\)	145.214	+	238.826i	0.519534	+	0.854450i
							\(7\)	−669.751	−	669.751i	−0.738024	−	0.738024i	0.234171	−	0.972195i	\(-0.424762\pi\)
−0.972195	+	0.234171i	\(0.924762\pi\)
\(11\)		−	705.674i		−	0.159856i	−0.996801	−	0.0799281i	\(-0.974531\pi\)
							0.996801	−	0.0799281i	\(-0.0254691\pi\)
\(13\)	1970.86	+	1970.86i	0.248802	+	0.248802i	0.820479	−	0.571677i	\(-0.193707\pi\)
							−0.571677	+	0.820479i	\(0.693707\pi\)
\(17\)	−19602.6	+	19602.6i	−0.967706	+	0.967706i	−0.999495	−	0.0317887i	\(-0.989880\pi\)
							0.0317887	+	0.999495i	\(0.489880\pi\)
\(19\)	23301.4			0.779372			0.389686	−	0.920948i	\(-0.372584\pi\)
							0.389686	+	0.920948i	\(0.372584\pi\)
\(23\)	−73788.6	+	73788.6i	−1.26457	+	1.26457i	−0.315711	+	0.948855i	\(0.602243\pi\)
							−0.948855	+	0.315711i	\(0.897757\pi\)
\(29\)			43042.9i			0.327725i	0.986483	+	0.163862i	\(0.0523953\pi\)
							−0.986483	+	0.163862i	\(0.947605\pi\)
\(31\)		−	147533.i		−	0.889452i	−0.895667	−	0.444726i	\(-0.853301\pi\)
							0.895667	−	0.444726i	\(-0.146699\pi\)
\(37\)	406255.	−	406255.i	1.31854	−	1.31854i	0.403607	−	0.914932i	\(-0.367756\pi\)
							0.914932	−	0.403607i	\(-0.132244\pi\)
\(41\)	179416.			0.406552			0.203276	−	0.979121i	\(-0.434841\pi\)
							0.203276	+	0.979121i	\(0.434841\pi\)
\(43\)	−317814.	+	317814.i	−0.609584	+	0.609584i	−0.942837	−	0.333253i	\(-0.891854\pi\)
							0.333253	+	0.942837i	\(0.391854\pi\)
\(47\)	−886147.	−	886147.i	−1.24498	−	1.24498i	−0.957908	−	0.287074i	\(-0.907317\pi\)
							−0.287074	−	0.957908i	\(-0.592683\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	20.8.e.b.3.17	yes	36
4.3	odd	2	inner	20.8.e.b.3.11	✓	36
5.2	odd	4	inner	20.8.e.b.7.11	yes	36
5.3	odd	4		100.8.e.e.7.8		36
5.4	even	2		100.8.e.e.43.2		36
20.3	even	4		100.8.e.e.7.2		36
20.7	even	4	inner	20.8.e.b.7.17	yes	36
20.19	odd	2		100.8.e.e.43.8		36

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
20.8.e.b.3.11	✓	36	4.3	odd	2	inner
20.8.e.b.3.17	yes	36	1.1	even	1	trivial
20.8.e.b.7.11	yes	36	5.2	odd	4	inner
20.8.e.b.7.17	yes	36	20.7	even	4	inner
100.8.e.e.7.2		36	20.3	even	4
100.8.e.e.7.8		36	5.3	odd	4
100.8.e.e.43.2		36	5.4	even	2
100.8.e.e.43.8		36	20.19	odd	2