Embedded newform 13.4.c.b.9.1

• Label: 13.4.c.b.9.1
• Level: $13$
• Weight: $4$
• Character: 13.9
• Analytic conductor: $0.767$
• Analytic rank: $0$
• Dimension: $4$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 13 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	13.c (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(0.767024830075\)
Analytic rank:	\(0\)
Dimension:	\(4\)
Relative dimension:	\(2\) over \(\Q(\zeta_{3})\)
Coefficient field:	\(\Q(\sqrt{-3}, \sqrt{17})\)
Defining polynomial:	\( x^{4} - x^{3} + 5x^{2} + 4x + 16 \)
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			9.1
Root			\(-0.780776 - 1.35234i\) of defining polynomial
Character	\(\chi\)	\(=\)	13.9
Dual form			13.4.c.b.3.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.219224 - 0.379706i) q^{2} +(1.84233 - 3.19101i) q^{3} +(3.90388 + 6.76172i) q^{4} -17.8078 q^{5} +(-0.807764 - 1.39909i) q^{6} +(-2.71922 - 4.70983i) q^{7} +6.93087 q^{8} +(6.71165 + 11.6249i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 5 q^{2} - 5 q^{3} - 5 q^{4} - 30 q^{5} + 38 q^{6} - 15 q^{7} - 30 q^{8} - 35 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/13\mathbb{Z}\right)^\times\).

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{2}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.219224	−	0.379706i	0.0775072	−	0.134246i	−0.824667	−	0.565619i	\(-0.808637\pi\)
							0.902174	+	0.431373i	\(0.141971\pi\)
\(3\)	1.84233	−	3.19101i	0.354556	−	0.614110i	−0.632486	−	0.774572i	\(-0.717965\pi\)
							0.987042	+	0.160462i	\(0.0512985\pi\)
\(5\)	−17.8078			−1.59277			−0.796387	−	0.604787i	\(-0.793258\pi\)
							−0.796387	+	0.604787i	\(0.793258\pi\)
\(7\)	−2.71922	−	4.70983i	−0.146824	−	0.254307i	0.783228	−	0.621735i	\(-0.213572\pi\)
							−0.930052	+	0.367428i	\(0.880239\pi\)
\(11\)	11.2116	−	19.4191i	0.307313	−	0.532281i	−0.670461	−	0.741945i	\(-0.733904\pi\)
							0.977774	+	0.209664i	\(0.0672369\pi\)
\(13\)	21.9730	−	41.4027i	0.468786	−	0.883312i
							\(17\)	−33.9924	−	58.8766i	−0.484963	−	0.839981i	0.514888	−	0.857258i	\(-0.327834\pi\)
−0.999851	+	0.0172769i	\(0.994500\pi\)
\(19\)	40.4039	+	69.9816i	0.487857	+	0.844993i	0.999902	−	0.0139650i	\(-0.00444535\pi\)
							−0.512045	+	0.858958i	\(0.671112\pi\)
\(23\)	−70.2656	+	121.704i	−0.637017	+	1.10335i	0.349067	+	0.937098i	\(0.386499\pi\)
							−0.986084	+	0.166248i	\(0.946835\pi\)
\(29\)	53.3466	−	92.3990i	0.341594	−	0.591657i	−0.643135	−	0.765753i	\(-0.722367\pi\)
							0.984729	+	0.174095i	\(0.0557001\pi\)
\(31\)	−276.155			−1.59997			−0.799983	−	0.600023i	\(-0.795158\pi\)
							−0.799983	+	0.600023i	\(0.795158\pi\)
\(37\)	2.14584	−	3.71670i	0.00953442	−	0.0165141i	−0.861219	−	0.508234i	\(-0.830298\pi\)
							0.870753	+	0.491720i	\(0.163632\pi\)
\(41\)	−113.884	+	197.254i	−0.433799	+	0.751362i	−0.997197	−	0.0748237i	\(-0.976161\pi\)
							0.563398	+	0.826186i	\(0.309494\pi\)
\(43\)	−13.7647	−	23.8411i	−0.0488162	−	0.0845521i	0.840585	−	0.541680i	\(-0.182212\pi\)
							−0.889401	+	0.457128i	\(0.848878\pi\)
\(47\)	318.617			0.988832			0.494416	−	0.869225i	\(-0.335382\pi\)
							0.494416	+	0.869225i	\(0.335382\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	13.4.c.b.9.1	yes	4
3.2	odd	2		117.4.g.d.100.2		4
4.3	odd	2		208.4.i.e.113.1		4
13.2	odd	12		169.4.e.g.23.3		8
13.3	even	3	inner	13.4.c.b.3.1	✓	4
13.4	even	6		169.4.a.j.1.1		2
13.5	odd	4		169.4.e.g.147.2		8
13.6	odd	12		169.4.b.e.168.3		4
13.7	odd	12		169.4.b.e.168.2		4
13.8	odd	4		169.4.e.g.147.3		8
13.9	even	3		169.4.a.f.1.2		2
13.10	even	6		169.4.c.f.146.2		4
13.11	odd	12		169.4.e.g.23.2		8
13.12	even	2		169.4.c.f.22.2		4
39.17	odd	6		1521.4.a.l.1.2		2
39.29	odd	6		117.4.g.d.55.2		4
39.35	odd	6		1521.4.a.t.1.1		2
52.3	odd	6		208.4.i.e.81.1		4

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
13.4.c.b.3.1	✓	4	13.3	even	3	inner
13.4.c.b.9.1	yes	4	1.1	even	1	trivial
117.4.g.d.55.2		4	39.29	odd	6
117.4.g.d.100.2		4	3.2	odd	2
169.4.a.f.1.2		2	13.9	even	3
169.4.a.j.1.1		2	13.4	even	6
169.4.b.e.168.2		4	13.7	odd	12
169.4.b.e.168.3		4	13.6	odd	12
169.4.c.f.22.2		4	13.12	even	2
169.4.c.f.146.2		4	13.10	even	6
169.4.e.g.23.2		8	13.11	odd	12
169.4.e.g.23.3		8	13.2	odd	12
169.4.e.g.147.2		8	13.5	odd	4
169.4.e.g.147.3		8	13.8	odd	4
208.4.i.e.81.1		4	52.3	odd	6
208.4.i.e.113.1		4	4.3	odd	2
1521.4.a.l.1.2		2	39.17	odd	6
1521.4.a.t.1.1		2	39.35	odd	6