Embedded newform 20.6.e.b.7.7

• Label: 20.6.e.b.7.7
• 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			7.7
Character	\(\chi\)	\(=\)	20.7
Dual form			20.6.e.b.3.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.216011 + 5.65273i) q^{2} +(-10.7585 - 10.7585i) q^{3} +(-31.9067 + 2.44211i) q^{4} +(-55.8903 - 1.12733i) q^{5} +(58.4911 - 63.1390i) q^{6} +(-26.5759 + 26.5759i) q^{7} +(-20.6968 - 179.832i) q^{8} -11.5083i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 10 q^{2} - 80 q^{5} - 184 q^{6} + 500 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.216011	+	5.65273i	0.0381858	+	0.999271i
							\(3\)	−10.7585	−	10.7585i	−0.690160	−	0.690160i	0.272107	−	0.962267i	\(-0.412279\pi\)
−0.962267	+	0.272107i	\(0.912279\pi\)
\(5\)	−55.8903	−	1.12733i	−0.999797	−	0.0201663i
							\(7\)	−26.5759	+	26.5759i	−0.204995	+	0.204995i	−0.802136	−	0.597141i	\(-0.796303\pi\)
0.597141	+	0.802136i	\(0.296303\pi\)
\(11\)			430.225i			1.07205i	0.844203	+	0.536024i	\(0.180074\pi\)
							−0.844203	+	0.536024i	\(0.819926\pi\)
\(13\)	−557.732	+	557.732i	−0.915307	+	0.915307i	−0.996683	−	0.0813761i	\(-0.974069\pi\)
							0.0813761	+	0.996683i	\(0.474069\pi\)
\(17\)	−780.679	−	780.679i	−0.655164	−	0.655164i	0.299068	−	0.954232i	\(-0.403324\pi\)
							−0.954232	+	0.299068i	\(0.903324\pi\)
\(19\)	1417.86			0.901049			0.450525	−	0.892764i	\(-0.351237\pi\)
							0.450525	+	0.892764i	\(0.351237\pi\)
\(23\)	−3110.17	−	3110.17i	−1.22593	−	1.22593i	−0.965491	−	0.260436i	\(-0.916134\pi\)
							−0.260436	−	0.965491i	\(-0.583866\pi\)
\(29\)			1876.98i			0.414443i	0.978294	+	0.207222i	\(0.0664422\pi\)
							−0.978294	+	0.207222i	\(0.933558\pi\)
\(31\)		−	5923.06i		−	1.10698i	−0.832854	−	0.553492i	\(-0.813295\pi\)
							0.832854	−	0.553492i	\(-0.186705\pi\)
\(37\)	−517.920	−	517.920i	−0.0621954	−	0.0621954i	0.675325	−	0.737520i	\(-0.264003\pi\)
							−0.737520	+	0.675325i	\(0.764003\pi\)
\(41\)	−8378.40			−0.778397			−0.389199	−	0.921154i	\(-0.627248\pi\)
							−0.389199	+	0.921154i	\(0.627248\pi\)
\(43\)	2301.18	+	2301.18i	0.189793	+	0.189793i	0.795607	−	0.605814i	\(-0.207152\pi\)
							−0.605814	+	0.795607i	\(0.707152\pi\)
\(47\)	−17060.7	+	17060.7i	−1.12655	+	1.12655i	−0.135821	+	0.990733i	\(0.543367\pi\)
							−0.990733	+	0.135821i	\(0.956633\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	20.6.e.b.7.7	yes	24
4.3	odd	2	inner	20.6.e.b.7.12	yes	24
5.2	odd	4		100.6.e.e.43.1		24
5.3	odd	4	inner	20.6.e.b.3.12	yes	24
5.4	even	2		100.6.e.e.7.6		24
20.3	even	4	inner	20.6.e.b.3.7	✓	24
20.7	even	4		100.6.e.e.43.6		24
20.19	odd	2		100.6.e.e.7.1		24

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
20.6.e.b.3.7	✓	24	20.3	even	4	inner
20.6.e.b.3.12	yes	24	5.3	odd	4	inner
20.6.e.b.7.7	yes	24	1.1	even	1	trivial
20.6.e.b.7.12	yes	24	4.3	odd	2	inner
100.6.e.e.7.1		24	20.19	odd	2
100.6.e.e.7.6		24	5.4	even	2
100.6.e.e.43.1		24	5.2	odd	4
100.6.e.e.43.6		24	20.7	even	4