Embedded newform 20.4.e.b.7.4

• Label: 20.4.e.b.7.4
• Level: $20$
• Weight: $4$
• Character: 20.7
• 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			7.4
Root			\(1.13579 + 1.64620i\) of defining polynomial
Character	\(\chi\)	\(=\)	20.7
Dual form			20.4.e.b.3.4

$q$-expansion

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/20\mathbb{Z}\right)^\times\).

\(n\)	\(11\)	\(17\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{1}{4}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.510409	−	2.78199i	−0.180457	−	0.983583i
							\(3\)	−4.02923	−	4.02923i	−0.775425	−	0.775425i	0.203624	−	0.979049i	\(-0.434728\pi\)
−0.979049	+	0.203624i	\(0.934728\pi\)
\(5\)	10.9349	−	2.32970i	0.978049	−	0.208375i
							\(7\)	14.4440	−	14.4440i	0.779902	−	0.779902i	−0.199912	−	0.979814i	\(-0.564066\pi\)
0.979814	+	0.199912i	\(0.0640655\pi\)
\(11\)			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	−0.794676	−	0.607033i	\(-0.792360\pi\)
							0.607033	+	0.794676i	\(0.292360\pi\)
\(17\)	−26.4898	−	26.4898i	−0.377925	−	0.377925i	0.492428	−	0.870353i	\(-0.336109\pi\)
							−0.870353	+	0.492428i	\(0.836109\pi\)
\(19\)	49.8054			0.601376			0.300688	−	0.953723i	\(-0.402784\pi\)
							0.300688	+	0.953723i	\(0.402784\pi\)
\(23\)	41.2762	+	41.2762i	0.374204	+	0.374204i	0.869006	−	0.494802i	\(-0.164759\pi\)
							−0.494802	+	0.869006i	\(0.664759\pi\)
\(29\)			247.406i			1.58421i	0.610382	+	0.792107i	\(0.291016\pi\)
							−0.610382	+	0.792107i	\(0.708984\pi\)
\(31\)		−	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	0.525481	−	0.850805i	\(-0.323885\pi\)
							−0.850805	+	0.525481i	\(0.823885\pi\)
\(41\)	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	\(-0.460940\pi\)
							−0.992480	+	0.122404i	\(0.960940\pi\)
\(47\)	−125.525	+	125.525i	−0.389567	+	0.389567i	−0.874533	−	0.484966i	\(-0.838832\pi\)
							0.484966	+	0.874533i	\(0.338832\pi\)

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
20.4.e.b.3.1	✓	12	5.3	odd	4	inner
20.4.e.b.3.4	yes	12	20.3	even	4	inner
20.4.e.b.7.1	yes	12	4.3	odd	2	inner
20.4.e.b.7.4	yes	12	1.1	even	1	trivial
100.4.e.e.7.3		12	5.4	even	2
100.4.e.e.7.6		12	20.19	odd	2
100.4.e.e.43.3		12	20.7	even	4
100.4.e.e.43.6		12	5.2	odd	4
180.4.k.e.127.3		12	3.2	odd	2
180.4.k.e.127.6		12	12.11	even	2
180.4.k.e.163.3		12	60.23	odd	4
180.4.k.e.163.6		12	15.8	even	4
320.4.n.k.63.2		12	40.13	odd	4
320.4.n.k.63.5		12	40.3	even	4
320.4.n.k.127.2		12	8.3	odd	2
320.4.n.k.127.5		12	8.5	even	2