Embedded newform 20.8.e.b.3.13

• Label: 20.8.e.b.3.13
• 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			3.13
Character	\(\chi\)	\(=\)	20.3
Dual form			20.8.e.b.7.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(6.52813 - 9.24032i) q^{2} +(41.8246 - 41.8246i) q^{3} +(-42.7671 - 120.644i) q^{4} +(-251.552 - 121.846i) q^{5} +(-113.436 - 659.509i) q^{6} +(1013.70 + 1013.70i) q^{7} +(-1393.98 - 392.398i) q^{8} -1311.59i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 36 q + 14 q^{2} - 560 q^{5} + 104 q^{6} - 3052 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	6.52813	−	9.24032i	0.577010	−	0.816737i
							\(3\)	41.8246	−	41.8246i	0.894350	−	0.894350i	−0.100579	−	0.994929i	\(-0.532070\pi\)
0.994929	+	0.100579i	\(0.0320697\pi\)
\(5\)	−251.552	−	121.846i	−0.899980	−	0.435931i
							\(7\)	1013.70	+	1013.70i	1.11703	+	1.11703i	0.992175	+	0.124854i	\(0.0398463\pi\)
0.124854	+	0.992175i	\(0.460154\pi\)
\(11\)		−	4661.02i		−	1.05586i	−0.849288	−	0.527930i	\(-0.822968\pi\)
							0.849288	−	0.527930i	\(-0.177032\pi\)
\(13\)	1233.90	+	1233.90i	0.155767	+	0.155767i	0.780688	−	0.624921i	\(-0.214869\pi\)
							−0.624921	+	0.780688i	\(0.714869\pi\)
\(17\)	15641.6	−	15641.6i	0.772166	−	0.772166i	−0.206319	−	0.978485i	\(-0.566148\pi\)
							0.978485	+	0.206319i	\(0.0661484\pi\)
\(19\)	38225.9			1.27856			0.639278	−	0.768975i	\(-0.279233\pi\)
							0.639278	+	0.768975i	\(0.279233\pi\)
\(23\)	−33492.5	+	33492.5i	−0.573985	+	0.573985i	−0.933240	−	0.359255i	\(-0.883031\pi\)
							0.359255	+	0.933240i	\(0.383031\pi\)
\(29\)			8843.72i			0.0673352i	0.999433	+	0.0336676i	\(0.0107188\pi\)
							−0.999433	+	0.0336676i	\(0.989281\pi\)
\(31\)			35209.7i			0.212274i	0.994352	+	0.106137i	\(0.0338482\pi\)
							−0.994352	+	0.106137i	\(0.966152\pi\)
\(37\)	76255.9	−	76255.9i	0.247495	−	0.247495i	−0.572447	−	0.819942i	\(-0.694006\pi\)
							0.819942	+	0.572447i	\(0.194006\pi\)
\(41\)	−734761.			−1.66495			−0.832477	−	0.554059i	\(-0.813078\pi\)
							−0.832477	+	0.554059i	\(0.813078\pi\)
\(43\)	−223185.	+	223185.i	−0.428080	+	0.428080i	−0.887974	−	0.459894i	\(-0.847888\pi\)
							0.459894	+	0.887974i	\(0.347888\pi\)
\(47\)	663328.	+	663328.i	0.931935	+	0.931935i	0.997827	−	0.0658915i	\(-0.0209891\pi\)
							−0.0658915	+	0.997827i	\(0.520989\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	20.8.e.b.3.13	✓	36
4.3	odd	2	inner	20.8.e.b.3.14	yes	36
5.2	odd	4	inner	20.8.e.b.7.14	yes	36
5.3	odd	4		100.8.e.e.7.5		36
5.4	even	2		100.8.e.e.43.6		36
20.3	even	4		100.8.e.e.7.6		36
20.7	even	4	inner	20.8.e.b.7.13	yes	36
20.19	odd	2		100.8.e.e.43.5		36

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
20.8.e.b.3.13	✓	36	1.1	even	1	trivial
20.8.e.b.3.14	yes	36	4.3	odd	2	inner
20.8.e.b.7.13	yes	36	20.7	even	4	inner
20.8.e.b.7.14	yes	36	5.2	odd	4	inner
100.8.e.e.7.5		36	5.3	odd	4
100.8.e.e.7.6		36	20.3	even	4
100.8.e.e.43.5		36	20.19	odd	2
100.8.e.e.43.6		36	5.4	even	2