Embedded newform 13.4.e.a.4.1

• Label: 13.4.e.a.4.1
• Level: $13$
• Weight: $4$
• Character: 13.4
• Analytic conductor: $0.767$
• Analytic rank: $0$
• Dimension: $2$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 13 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	13.e (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(0.767024830075\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\zeta_{6})\)
Defining polynomial:	\( x^{2} - x + 1 \)
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			4.1
Root			\(0.500000 + 0.866025i\) of defining polynomial
Character	\(\chi\)	\(=\)	13.4
Dual form			13.4.e.a.10.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-3.00000 - 1.73205i) q^{2} +(3.50000 - 6.06218i) q^{3} +(2.00000 + 3.46410i) q^{4} +13.8564i q^{5} +(-21.0000 + 12.1244i) q^{6} +(19.5000 - 11.2583i) q^{7} +13.8564i q^{8} +(-11.0000 - 19.0526i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 6 q^{2} + 7 q^{3} + 4 q^{4} - 42 q^{6} + 39 q^{7} - 22 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{1}{6}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−3.00000	−	1.73205i	−1.06066	−	0.612372i	−0.135045	−	0.990839i	\(-0.543118\pi\)
							−0.925615	+	0.378467i	\(0.876451\pi\)
\(3\)	3.50000	−	6.06218i	0.673575	−	1.16667i	−0.303308	−	0.952893i	\(-0.598091\pi\)
							0.976883	−	0.213774i	\(-0.0685756\pi\)
\(5\)			13.8564i			1.23935i	0.784857	+	0.619677i	\(0.212737\pi\)
							−0.784857	+	0.619677i	\(0.787263\pi\)
\(7\)	19.5000	−	11.2583i	1.05290	−	0.607893i	0.129441	−	0.991587i	\(-0.458682\pi\)
							0.923460	+	0.383694i	\(0.125348\pi\)
\(11\)	−19.5000	−	11.2583i	−0.534497	−	0.308592i	0.208349	−	0.978055i	\(-0.433191\pi\)
							−0.742846	+	0.669462i	\(0.766525\pi\)
\(13\)	−13.0000	+	45.0333i	−0.277350	+	0.960769i
							\(17\)	−13.5000	−	23.3827i	−0.192602	−	0.333596i	0.753510	−	0.657437i	\(-0.228359\pi\)
−0.946112	+	0.323840i	\(0.895026\pi\)
\(19\)	−76.5000	+	44.1673i	−0.923700	+	0.533299i	−0.884814	−	0.465945i	\(-0.845714\pi\)
							−0.0388865	+	0.999244i	\(0.512381\pi\)
\(23\)	−28.5000	+	49.3634i	−0.258377	+	0.447521i	−0.965807	−	0.259261i	\(-0.916521\pi\)
							0.707431	+	0.706783i	\(0.249854\pi\)
\(29\)	34.5000	−	59.7558i	0.220913	−	0.382633i	−0.734172	−	0.678963i	\(-0.762430\pi\)
							0.955086	+	0.296330i	\(0.0957628\pi\)
\(31\)		−	72.7461i		−	0.421471i	−0.977543	−	0.210735i	\(-0.932414\pi\)
							0.977543	−	0.210735i	\(-0.0675858\pi\)
\(37\)	−34.5000	−	19.9186i	−0.153291	−	0.0885026i	0.421393	−	0.906878i	\(-0.361541\pi\)
							−0.574683	+	0.818376i	\(0.694875\pi\)
\(41\)	−340.500	−	196.588i	−1.29700	−	0.748826i	−0.317118	−	0.948386i	\(-0.602715\pi\)
							−0.979886	+	0.199560i	\(0.936049\pi\)
\(43\)	42.5000	+	73.6122i	0.150725	+	0.261064i	0.931494	−	0.363756i	\(-0.118506\pi\)
							−0.780769	+	0.624820i	\(0.785172\pi\)
\(47\)		−	342.946i		−	1.06434i	−0.846639	−	0.532168i	\(-0.821377\pi\)
							0.846639	−	0.532168i	\(-0.178623\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	13.4.e.a.4.1	✓	2
3.2	odd	2		117.4.q.c.82.1		2
4.3	odd	2		208.4.w.a.17.1		2
13.2	odd	12		169.4.c.i.146.1		4
13.3	even	3		169.4.e.b.23.1		2
13.4	even	6		169.4.b.b.168.1		2
13.5	odd	4		169.4.c.i.22.1		4
13.6	odd	12		169.4.a.h.1.2		2
13.7	odd	12		169.4.a.h.1.1		2
13.8	odd	4		169.4.c.i.22.2		4
13.9	even	3		169.4.b.b.168.2		2
13.10	even	6	inner	13.4.e.a.10.1	yes	2
13.11	odd	12		169.4.c.i.146.2		4
13.12	even	2		169.4.e.b.147.1		2
39.20	even	12		1521.4.a.q.1.2		2
39.23	odd	6		117.4.q.c.10.1		2
39.32	even	12		1521.4.a.q.1.1		2
52.23	odd	6		208.4.w.a.49.1		2

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
13.4.e.a.4.1	✓	2	1.1	even	1	trivial
13.4.e.a.10.1	yes	2	13.10	even	6	inner
117.4.q.c.10.1		2	39.23	odd	6
117.4.q.c.82.1		2	3.2	odd	2
169.4.a.h.1.1		2	13.7	odd	12
169.4.a.h.1.2		2	13.6	odd	12
169.4.b.b.168.1		2	13.4	even	6
169.4.b.b.168.2		2	13.9	even	3
169.4.c.i.22.1		4	13.5	odd	4
169.4.c.i.22.2		4	13.8	odd	4
169.4.c.i.146.1		4	13.2	odd	12
169.4.c.i.146.2		4	13.11	odd	12
169.4.e.b.23.1		2	13.3	even	3
169.4.e.b.147.1		2	13.12	even	2
208.4.w.a.17.1		2	4.3	odd	2
208.4.w.a.49.1		2	52.23	odd	6
1521.4.a.q.1.1		2	39.32	even	12
1521.4.a.q.1.2		2	39.20	even	12