Embedded newform 13.4.c.b.3.1

• Label: 13.4.c.b.3.1
• Level: $13$
• Weight: $4$
• Character: 13.3
• 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			3.1
Root			\(-0.780776 + 1.35234i\) of defining polynomial
Character	\(\chi\)	\(=\)	13.3
Dual form			13.4.c.b.9.1

$q$-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 \(\left(\mathbb{Z}/13\mathbb{Z}\right)^\times\).

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{1}{3}\right)\)

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	0.902174	−	0.431373i	\(-0.141971\pi\)
							−0.824667	+	0.565619i	\(0.808637\pi\)
\(3\)	1.84233	+	3.19101i	0.354556	+	0.614110i	0.987042	−	0.160462i	\(-0.0512985\pi\)
							−0.632486	+	0.774572i	\(0.717965\pi\)
\(5\)	−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	−0.930052	−	0.367428i	\(-0.880239\pi\)
							0.783228	+	0.621735i	\(0.213572\pi\)
\(11\)	11.2116	+	19.4191i	0.307313	+	0.532281i	0.977774	−	0.209664i	\(-0.0672369\pi\)
							−0.670461	+	0.741945i	\(0.733904\pi\)
\(13\)	21.9730	+	41.4027i	0.468786	+	0.883312i
							\(17\)	−33.9924	+	58.8766i	−0.484963	+	0.839981i	−0.999851	−	0.0172769i	\(-0.994500\pi\)
0.514888	+	0.857258i	\(0.327834\pi\)
\(19\)	40.4039	−	69.9816i	0.487857	−	0.844993i	−0.512045	−	0.858958i	\(-0.671112\pi\)
							0.999902	+	0.0139650i	\(0.00444535\pi\)
\(23\)	−70.2656	−	121.704i	−0.637017	−	1.10335i	−0.986084	−	0.166248i	\(-0.946835\pi\)
							0.349067	−	0.937098i	\(-0.386499\pi\)
\(29\)	53.3466	+	92.3990i	0.341594	+	0.591657i	0.984729	−	0.174095i	\(-0.0557001\pi\)
							−0.643135	+	0.765753i	\(0.722367\pi\)
\(31\)	−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	0.870753	−	0.491720i	\(-0.163632\pi\)
							−0.861219	+	0.508234i	\(0.830298\pi\)
\(41\)	−113.884	−	197.254i	−0.433799	−	0.751362i	0.563398	−	0.826186i	\(-0.309494\pi\)
							−0.997197	+	0.0748237i	\(0.976161\pi\)
\(43\)	−13.7647	+	23.8411i	−0.0488162	+	0.0845521i	−0.889401	−	0.457128i	\(-0.848878\pi\)
							0.840585	+	0.541680i	\(0.182212\pi\)
\(47\)	318.617			0.988832			0.494416	−	0.869225i	\(-0.335382\pi\)
							0.494416	+	0.869225i	\(0.335382\pi\)