Embedded newform 20.6.e.b.7.4

• Label: 20.6.e.b.7.4
• 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			7.4
Character	\(\chi\)	\(=\)	20.7
Dual form			20.6.e.b.3.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-3.91187 + 4.08623i) q^{2} +(-6.89992 - 6.89992i) q^{3} +(-1.39448 - 31.9696i) q^{4} +(53.4769 - 16.2857i) q^{5} +(55.1863 - 1.20301i) q^{6} +(57.8425 - 57.8425i) q^{7} +(136.090 + 119.363i) q^{8} -147.782i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 10 q^{2} - 80 q^{5} - 184 q^{6} + 500 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−3.91187	+	4.08623i	−0.691528	+	0.722349i
							\(3\)	−6.89992	−	6.89992i	−0.442630	−	0.442630i	0.450265	−	0.892895i	\(-0.351330\pi\)
−0.892895	+	0.450265i	\(0.851330\pi\)
\(5\)	53.4769	−	16.2857i	0.956624	−	0.291327i
							\(7\)	57.8425	−	57.8425i	0.446171	−	0.446171i	−0.447908	−	0.894080i	\(-0.647831\pi\)
0.894080	+	0.447908i	\(0.147831\pi\)
\(11\)		−	628.542i		−	1.56622i	−0.621884	−	0.783110i	\(-0.713632\pi\)
							0.621884	−	0.783110i	\(-0.286368\pi\)
\(13\)	−402.925	+	402.925i	−0.661250	+	0.661250i	−0.955675	−	0.294425i	\(-0.904872\pi\)
							0.294425	+	0.955675i	\(0.404872\pi\)
\(17\)	366.407	+	366.407i	0.307497	+	0.307497i	0.843938	−	0.536441i	\(-0.180231\pi\)
							−0.536441	+	0.843938i	\(0.680231\pi\)
\(19\)	1240.95			0.788625			0.394312	−	0.918976i	\(-0.370983\pi\)
							0.394312	+	0.918976i	\(0.370983\pi\)
\(23\)	12.5222	+	12.5222i	0.00493582	+	0.00493582i	0.709570	−	0.704635i	\(-0.248889\pi\)
							−0.704635	+	0.709570i	\(0.748889\pi\)
\(29\)		−	3633.44i		−	0.802275i	−0.916018	−	0.401138i	\(-0.868615\pi\)
							0.916018	−	0.401138i	\(-0.131385\pi\)
\(31\)			8717.82i			1.62931i	0.579946	+	0.814655i	\(0.303073\pi\)
							−0.579946	+	0.814655i	\(0.696927\pi\)
\(37\)	6668.22	+	6668.22i	0.800767	+	0.800767i	0.983215	−	0.182449i	\(-0.0584024\pi\)
							−0.182449	+	0.983215i	\(0.558402\pi\)
\(41\)	−9444.76			−0.877468			−0.438734	−	0.898617i	\(-0.644573\pi\)
							−0.438734	+	0.898617i	\(0.644573\pi\)
\(43\)	−7764.81	−	7764.81i	−0.640412	−	0.640412i	0.310245	−	0.950657i	\(-0.399589\pi\)
							−0.950657	+	0.310245i	\(0.899589\pi\)
\(47\)	−1958.63	+	1958.63i	−0.129333	+	0.129333i	−0.768810	−	0.639477i	\(-0.779151\pi\)
							0.639477	+	0.768810i	\(0.279151\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	20.6.e.b.7.4	yes	24
4.3	odd	2	inner	20.6.e.b.7.10	yes	24
5.2	odd	4		100.6.e.e.43.3		24
5.3	odd	4	inner	20.6.e.b.3.10	yes	24
5.4	even	2		100.6.e.e.7.9		24
20.3	even	4	inner	20.6.e.b.3.4	✓	24
20.7	even	4		100.6.e.e.43.9		24
20.19	odd	2		100.6.e.e.7.3		24

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
20.6.e.b.3.4	✓	24	20.3	even	4	inner
20.6.e.b.3.10	yes	24	5.3	odd	4	inner
20.6.e.b.7.4	yes	24	1.1	even	1	trivial
20.6.e.b.7.10	yes	24	4.3	odd	2	inner
100.6.e.e.7.3		24	20.19	odd	2
100.6.e.e.7.9		24	5.4	even	2
100.6.e.e.43.3		24	5.2	odd	4
100.6.e.e.43.9		24	20.7	even	4