Embedded newform 20.4.e.b.3.6

• Label: 20.4.e.b.3.6
• Level: $20$
• Weight: $4$
• Character: 20.3
• Analytic conductor: $1.180$
• Analytic rank: $0$
• Dimension: $12$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 20 = 2^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	20.e (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.18003820011\)
Analytic rank:	\(0\)
Dimension:	\(12\)
Relative dimension:	\(6\) over \(\Q(i)\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Defining polynomial:	\( x^{12} - 7x^{10} + 44x^{8} - 156x^{6} + 704x^{4} - 1792x^{2} + 4096 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index:	\( 2^{14} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			3.6
Root			\(1.76129 + 0.947553i\) of defining polynomial
Character	\(\chi\)	\(=\)	20.3
Dual form			20.4.e.b.7.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(2.70884 + 0.813737i) q^{2} +(-2.61822 + 2.61822i) q^{3} +(6.67566 + 4.40857i) q^{4} +(-0.435501 - 11.1719i) q^{5} +(-9.22289 + 4.96181i) q^{6} +(-17.7783 - 17.7783i) q^{7} +(14.4959 + 17.3744i) q^{8} +13.2899i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 6 q^{2} + 8 q^{6} - 12 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	2.70884	+	0.813737i	0.957721	+	0.287699i
							\(3\)	−2.61822	+	2.61822i	−0.503877	+	0.503877i	−0.912640	−	0.408764i	\(-0.865960\pi\)
0.408764	+	0.912640i	\(0.365960\pi\)
\(5\)	−0.435501	−	11.1719i	−0.0389524	−	0.999241i
							\(7\)	−17.7783	−	17.7783i	−0.959936	−	0.959936i	0.0392914	−	0.999228i	\(-0.487490\pi\)
−0.999228	+	0.0392914i	\(0.987490\pi\)
\(11\)			7.37590i			0.202174i	0.994878	+	0.101087i	\(0.0322321\pi\)
							−0.994878	+	0.101087i	\(0.967768\pi\)
\(13\)	−2.68249	−	2.68249i	−0.0572300	−	0.0572300i	0.677913	−	0.735143i	\(-0.262885\pi\)
							−0.735143	+	0.677913i	\(0.762885\pi\)
\(17\)	−20.2367	+	20.2367i	−0.288713	+	0.288713i	−0.836571	−	0.547858i	\(-0.815443\pi\)
							0.547858	+	0.836571i	\(0.315443\pi\)
\(19\)	135.808			1.63981			0.819906	−	0.572498i	\(-0.194025\pi\)
							0.819906	+	0.572498i	\(0.194025\pi\)
\(23\)	−71.0426	+	71.0426i	−0.644061	+	0.644061i	−0.951551	−	0.307490i	\(-0.900511\pi\)
							0.307490	+	0.951551i	\(0.400511\pi\)
\(29\)			34.2890i			0.219562i	0.993956	+	0.109781i	\(0.0350150\pi\)
							−0.993956	+	0.109781i	\(0.964985\pi\)
\(31\)		−	187.974i		−	1.08907i	−0.838739	−	0.544533i	\(-0.816707\pi\)
							0.838739	−	0.544533i	\(-0.183293\pi\)
\(37\)	250.679	−	250.679i	1.11382	−	1.11382i	0.121190	−	0.992629i	\(-0.461329\pi\)
							0.992629	−	0.121190i	\(-0.0386712\pi\)
\(41\)	−211.105			−0.804122			−0.402061	−	0.915613i	\(-0.631706\pi\)
							−0.402061	+	0.915613i	\(0.631706\pi\)
\(43\)	46.7326	−	46.7326i	0.165736	−	0.165736i	−0.619366	−	0.785102i	\(-0.712610\pi\)
							0.785102	+	0.619366i	\(0.212610\pi\)
\(47\)	189.707	+	189.707i	0.588757	+	0.588757i	0.937295	−	0.348538i	\(-0.113322\pi\)
							−0.348538	+	0.937295i	\(0.613322\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	20.4.e.b.3.6	yes	12
3.2	odd	2		180.4.k.e.163.1		12
4.3	odd	2	inner	20.4.e.b.3.3	✓	12
5.2	odd	4	inner	20.4.e.b.7.3	yes	12
5.3	odd	4		100.4.e.e.7.4		12
5.4	even	2		100.4.e.e.43.1		12
8.3	odd	2		320.4.n.k.63.3		12
8.5	even	2		320.4.n.k.63.4		12
12.11	even	2		180.4.k.e.163.4		12
15.2	even	4		180.4.k.e.127.4		12
20.3	even	4		100.4.e.e.7.1		12
20.7	even	4	inner	20.4.e.b.7.6	yes	12
20.19	odd	2		100.4.e.e.43.4		12
40.27	even	4		320.4.n.k.127.4		12
40.37	odd	4		320.4.n.k.127.3		12
60.47	odd	4		180.4.k.e.127.1		12

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
20.4.e.b.3.3	✓	12	4.3	odd	2	inner
20.4.e.b.3.6	yes	12	1.1	even	1	trivial
20.4.e.b.7.3	yes	12	5.2	odd	4	inner
20.4.e.b.7.6	yes	12	20.7	even	4	inner
100.4.e.e.7.1		12	20.3	even	4
100.4.e.e.7.4		12	5.3	odd	4
100.4.e.e.43.1		12	5.4	even	2
100.4.e.e.43.4		12	20.19	odd	2
180.4.k.e.127.1		12	60.47	odd	4
180.4.k.e.127.4		12	15.2	even	4
180.4.k.e.163.1		12	3.2	odd	2
180.4.k.e.163.4		12	12.11	even	2
320.4.n.k.63.3		12	8.3	odd	2
320.4.n.k.63.4		12	8.5	even	2
320.4.n.k.127.3		12	40.37	odd	4
320.4.n.k.127.4		12	40.27	even	4