Embedded newform 20.4.e.b.3.1

• Label: 20.4.e.b.3.1
• 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			3.1
Root			\(-1.13579 - 1.64620i\) of defining polynomial
Character	\(\chi\)	\(=\)	20.3
Dual form			20.4.e.b.7.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.78199 + 0.510409i) q^{2} +(4.02923 - 4.02923i) q^{3} +(7.47897 - 2.83991i) q^{4} +(10.9349 + 2.32970i) q^{5} +(-9.15273 + 13.2658i) q^{6} +(-14.4440 - 14.4440i) q^{7} +(-19.3569 + 11.7179i) q^{8} -5.46937i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 6 q^{2} + 8 q^{6} - 12 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.78199	+	0.510409i	−0.983583	+	0.180457i
							\(3\)	4.02923	−	4.02923i	0.775425	−	0.775425i	−0.203624	−	0.979049i	\(-0.565272\pi\)
0.979049	+	0.203624i	\(0.0652719\pi\)
\(5\)	10.9349	+	2.32970i	0.978049	+	0.208375i
							\(7\)	−14.4440	−	14.4440i	−0.779902	−	0.779902i	0.199912	−	0.979814i	\(-0.435934\pi\)
−0.979814	+	0.199912i	\(0.935934\pi\)
\(11\)			47.0607i			1.28994i	0.764209	+	0.644969i	\(0.223130\pi\)
							−0.764209	+	0.644969i	\(0.776870\pi\)
\(13\)	−8.79525	−	8.79525i	−0.187643	−	0.187643i	0.607033	−	0.794676i	\(-0.292360\pi\)
							−0.794676	+	0.607033i	\(0.792360\pi\)
\(17\)	−26.4898	+	26.4898i	−0.377925	+	0.377925i	−0.870353	−	0.492428i	\(-0.836109\pi\)
							0.492428	+	0.870353i	\(0.336109\pi\)
\(19\)	−49.8054			−0.601376			−0.300688	−	0.953723i	\(-0.597216\pi\)
							−0.300688	+	0.953723i	\(0.597216\pi\)
\(23\)	−41.2762	+	41.2762i	−0.374204	+	0.374204i	−0.869006	−	0.494802i	\(-0.835241\pi\)
							0.494802	+	0.869006i	\(0.335241\pi\)
\(29\)		−	247.406i		−	1.58421i	−0.610382	−	0.792107i	\(-0.708984\pi\)
							0.610382	−	0.792107i	\(-0.291016\pi\)
\(31\)		−	62.3240i		−	0.361088i	−0.983567	−	0.180544i	\(-0.942214\pi\)
							0.983567	−	0.180544i	\(-0.0577858\pi\)
\(37\)	−73.2182	+	73.2182i	−0.325324	+	0.325324i	−0.850805	−	0.525481i	\(-0.823885\pi\)
							0.525481	+	0.850805i	\(0.323885\pi\)
\(41\)	118.624			0.451854			0.225927	−	0.974144i	\(-0.427459\pi\)
							0.225927	+	0.974144i	\(0.427459\pi\)
\(43\)	245.335	−	245.335i	0.870076	−	0.870076i	−0.122404	−	0.992480i	\(-0.539060\pi\)
							0.992480	+	0.122404i	\(0.0390605\pi\)
\(47\)	125.525	+	125.525i	0.389567	+	0.389567i	0.874533	−	0.484966i	\(-0.161168\pi\)
							−0.484966	+	0.874533i	\(0.661168\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	20.4.e.b.3.1	✓	12
3.2	odd	2		180.4.k.e.163.6		12
4.3	odd	2	inner	20.4.e.b.3.4	yes	12
5.2	odd	4	inner	20.4.e.b.7.4	yes	12
5.3	odd	4		100.4.e.e.7.3		12
5.4	even	2		100.4.e.e.43.6		12
8.3	odd	2		320.4.n.k.63.5		12
8.5	even	2		320.4.n.k.63.2		12
12.11	even	2		180.4.k.e.163.3		12
15.2	even	4		180.4.k.e.127.3		12
20.3	even	4		100.4.e.e.7.6		12
20.7	even	4	inner	20.4.e.b.7.1	yes	12
20.19	odd	2		100.4.e.e.43.3		12
40.27	even	4		320.4.n.k.127.2		12
40.37	odd	4		320.4.n.k.127.5		12
60.47	odd	4		180.4.k.e.127.6		12

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
20.4.e.b.3.1	✓	12	1.1	even	1	trivial
20.4.e.b.3.4	yes	12	4.3	odd	2	inner
20.4.e.b.7.1	yes	12	20.7	even	4	inner
20.4.e.b.7.4	yes	12	5.2	odd	4	inner
100.4.e.e.7.3		12	5.3	odd	4
100.4.e.e.7.6		12	20.3	even	4
100.4.e.e.43.3		12	20.19	odd	2
100.4.e.e.43.6		12	5.4	even	2
180.4.k.e.127.3		12	15.2	even	4
180.4.k.e.127.6		12	60.47	odd	4
180.4.k.e.163.3		12	12.11	even	2
180.4.k.e.163.6		12	3.2	odd	2
320.4.n.k.63.2		12	8.5	even	2
320.4.n.k.63.5		12	8.3	odd	2
320.4.n.k.127.2		12	40.27	even	4
320.4.n.k.127.5		12	40.37	odd	4