Embedded newform 20.4.e.b.3.5

• Label: 20.4.e.b.3.5
• 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			3.5
Root			\(1.83244 - 0.801352i\) of defining polynomial
Character	\(\chi\)	\(=\)	20.3
Dual form			20.4.e.b.7.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.03109 + 2.63379i) q^{2} +(5.55970 - 5.55970i) q^{3} +(-5.87372 + 5.43134i) q^{4} +(-10.4994 + 3.84216i) q^{5} +(20.3756 + 8.91056i) q^{6} +(-1.14202 - 1.14202i) q^{7} +(-20.3613 - 9.86997i) q^{8} -34.8205i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 6 q^{2} + 8 q^{6} - 12 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.03109	+	2.63379i	0.364544	+	0.931186i
							\(3\)	5.55970	−	5.55970i	1.06996	−	1.06996i	0.0726035	−	0.997361i	\(-0.476869\pi\)
0.997361	−	0.0726035i	\(-0.0231308\pi\)
\(5\)	−10.4994	+	3.84216i	−0.939097	+	0.343653i
							\(7\)	−1.14202	−	1.14202i	−0.0616631	−	0.0616631i	0.675603	−	0.737266i	\(-0.263883\pi\)
−0.737266	+	0.675603i	\(0.763883\pi\)
\(11\)		−	27.0350i		−	0.741033i	−0.928826	−	0.370516i	\(-0.879181\pi\)
							0.928826	−	0.370516i	\(-0.120819\pi\)
\(13\)	40.4777	+	40.4777i	0.863577	+	0.863577i	0.991752	−	0.128174i	\(-0.0409117\pi\)
							−0.128174	+	0.991752i	\(0.540912\pi\)
\(17\)	−36.2735	+	36.2735i	−0.517507	+	0.517507i	−0.916816	−	0.399309i	\(-0.869250\pi\)
							0.399309	+	0.916816i	\(0.369250\pi\)
\(19\)	56.8829			0.686834			0.343417	−	0.939183i	\(-0.388416\pi\)
							0.343417	+	0.939183i	\(0.388416\pi\)
\(23\)	−54.9839	+	54.9839i	−0.498475	+	0.498475i	−0.910963	−	0.412488i	\(-0.864660\pi\)
							0.412488	+	0.910963i	\(0.364660\pi\)
\(29\)			57.1173i			0.365739i	0.983137	+	0.182869i	\(0.0585385\pi\)
							−0.983137	+	0.182869i	\(0.941461\pi\)
\(31\)		−	190.845i		−	1.10570i	−0.833281	−	0.552850i	\(-0.813540\pi\)
							0.833281	−	0.552850i	\(-0.186460\pi\)
\(37\)	−50.4605	+	50.4605i	−0.224207	+	0.224207i	−0.810267	−	0.586060i	\(-0.800678\pi\)
							0.586060	+	0.810267i	\(0.300678\pi\)
\(41\)	−71.5197			−0.272427			−0.136213	−	0.990680i	\(-0.543493\pi\)
							−0.136213	+	0.990680i	\(0.543493\pi\)
\(43\)	−66.9381	+	66.9381i	−0.237394	+	0.237394i	−0.815770	−	0.578376i	\(-0.803687\pi\)
							0.578376	+	0.815770i	\(0.303687\pi\)
\(47\)	−343.017	−	343.017i	−1.06456	−	1.06456i	−0.997767	−	0.0667913i	\(-0.978724\pi\)
							−0.0667913	−	0.997767i	\(-0.521276\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	20.4.e.b.3.5	yes	12
3.2	odd	2		180.4.k.e.163.2		12
4.3	odd	2	inner	20.4.e.b.3.2	✓	12
5.2	odd	4	inner	20.4.e.b.7.2	yes	12
5.3	odd	4		100.4.e.e.7.5		12
5.4	even	2		100.4.e.e.43.2		12
8.3	odd	2		320.4.n.k.63.6		12
8.5	even	2		320.4.n.k.63.1		12
12.11	even	2		180.4.k.e.163.5		12
15.2	even	4		180.4.k.e.127.5		12
20.3	even	4		100.4.e.e.7.2		12
20.7	even	4	inner	20.4.e.b.7.5	yes	12
20.19	odd	2		100.4.e.e.43.5		12
40.27	even	4		320.4.n.k.127.1		12
40.37	odd	4		320.4.n.k.127.6		12
60.47	odd	4		180.4.k.e.127.2		12

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
20.4.e.b.3.2	✓	12	4.3	odd	2	inner
20.4.e.b.3.5	yes	12	1.1	even	1	trivial
20.4.e.b.7.2	yes	12	5.2	odd	4	inner
20.4.e.b.7.5	yes	12	20.7	even	4	inner
100.4.e.e.7.2		12	20.3	even	4
100.4.e.e.7.5		12	5.3	odd	4
100.4.e.e.43.2		12	5.4	even	2
100.4.e.e.43.5		12	20.19	odd	2
180.4.k.e.127.2		12	60.47	odd	4
180.4.k.e.127.5		12	15.2	even	4
180.4.k.e.163.2		12	3.2	odd	2
180.4.k.e.163.5		12	12.11	even	2
320.4.n.k.63.1		12	8.5	even	2
320.4.n.k.63.6		12	8.3	odd	2
320.4.n.k.127.1		12	40.27	even	4
320.4.n.k.127.6		12	40.37	odd	4