Embedded newform 13.4.c.b.9.2

• Label: 13.4.c.b.9.2
• 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			9.2
Root			\(1.28078 + 2.21837i\) of defining polynomial
Character	\(\chi\)	\(=\)	13.9
Dual form			13.4.c.b.3.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(2.28078 - 3.95042i) q^{2} +(-4.34233 + 7.52113i) q^{3} +(-6.40388 - 11.0918i) q^{4} +2.80776 q^{5} +(19.8078 + 34.3081i) q^{6} +(-4.78078 - 8.28055i) q^{7} -21.9309 q^{8} +(-24.2116 - 41.9358i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 5 q^{2} - 5 q^{3} - 5 q^{4} - 30 q^{5} + 38 q^{6} - 15 q^{7} - 30 q^{8} - 35 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	2.28078	−	3.95042i	0.806376	−	1.39668i	−0.108982	−	0.994044i	\(-0.534759\pi\)
							0.915358	−	0.402641i	\(-0.131908\pi\)
\(3\)	−4.34233	+	7.52113i	−0.835682	+	1.44744i	0.0577926	+	0.998329i	\(0.481594\pi\)
							−0.893474	+	0.449114i	\(0.851740\pi\)
\(5\)	2.80776			0.251134			0.125567	−	0.992085i	\(-0.459925\pi\)
							0.125567	+	0.992085i	\(0.459925\pi\)
\(7\)	−4.78078	−	8.28055i	−0.258138	−	0.447108i	0.707605	−	0.706608i	\(-0.249775\pi\)
							−0.965743	+	0.259500i	\(0.916442\pi\)
\(11\)	−19.7116	+	34.1416i	−0.540299	+	0.935825i	0.458588	+	0.888649i	\(0.348355\pi\)
							−0.998887	+	0.0471757i	\(0.984978\pi\)
\(13\)	40.5270	−	23.5492i	0.864628	−	0.502413i
							\(17\)	−1.00758	−	1.74518i	−0.0143749	−	0.0248981i	0.858748	−	0.512397i	\(-0.171242\pi\)
−0.873123	+	0.487499i	\(0.837909\pi\)
\(19\)	30.0961	+	52.1280i	0.363396	+	0.629420i	0.988517	−	0.151107i	\(-0.0482839\pi\)
							−0.625121	+	0.780528i	\(0.714951\pi\)
\(23\)	−2.23438	+	3.87006i	−0.0202565	+	0.0350853i	−0.875976	−	0.482355i	\(-0.839782\pi\)
							0.855719	+	0.517440i	\(0.173115\pi\)
\(29\)	−70.3466	+	121.844i	−0.450449	+	0.780201i	−0.998414	−	0.0563003i	\(-0.982070\pi\)
							0.547964	+	0.836502i	\(0.315403\pi\)
\(31\)	136.155			0.788845			0.394423	−	0.918929i	\(-0.370945\pi\)
							0.394423	+	0.918929i	\(0.370945\pi\)
\(37\)	92.8542	−	160.828i	0.412571	−	0.714594i	−0.582599	−	0.812760i	\(-0.697964\pi\)
							0.995170	+	0.0981657i	\(0.0312975\pi\)
\(41\)	−155.116	+	268.668i	−0.590853	+	1.02339i	0.403265	+	0.915083i	\(0.367875\pi\)
							−0.994118	+	0.108304i	\(0.965458\pi\)
\(43\)	−213.735	−	370.200i	−0.758008	−	1.31291i	−0.943865	−	0.330331i	\(-0.892840\pi\)
							0.185857	−	0.982577i	\(-0.440494\pi\)
\(47\)	−258.617			−0.802622			−0.401311	−	0.915942i	\(-0.631445\pi\)
							−0.401311	+	0.915942i	\(0.631445\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	13.4.c.b.9.2	yes	4
3.2	odd	2		117.4.g.d.100.1		4
4.3	odd	2		208.4.i.e.113.2		4
13.2	odd	12		169.4.e.g.23.4		8
13.3	even	3	inner	13.4.c.b.3.2	✓	4
13.4	even	6		169.4.a.j.1.2		2
13.5	odd	4		169.4.e.g.147.1		8
13.6	odd	12		169.4.b.e.168.4		4
13.7	odd	12		169.4.b.e.168.1		4
13.8	odd	4		169.4.e.g.147.4		8
13.9	even	3		169.4.a.f.1.1		2
13.10	even	6		169.4.c.f.146.1		4
13.11	odd	12		169.4.e.g.23.1		8
13.12	even	2		169.4.c.f.22.1		4
39.17	odd	6		1521.4.a.l.1.1		2
39.29	odd	6		117.4.g.d.55.1		4
39.35	odd	6		1521.4.a.t.1.2		2
52.3	odd	6		208.4.i.e.81.2		4

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
13.4.c.b.3.2	✓	4	13.3	even	3	inner
13.4.c.b.9.2	yes	4	1.1	even	1	trivial
117.4.g.d.55.1		4	39.29	odd	6
117.4.g.d.100.1		4	3.2	odd	2
169.4.a.f.1.1		2	13.9	even	3
169.4.a.j.1.2		2	13.4	even	6
169.4.b.e.168.1		4	13.7	odd	12
169.4.b.e.168.4		4	13.6	odd	12
169.4.c.f.22.1		4	13.12	even	2
169.4.c.f.146.1		4	13.10	even	6
169.4.e.g.23.1		8	13.11	odd	12
169.4.e.g.23.4		8	13.2	odd	12
169.4.e.g.147.1		8	13.5	odd	4
169.4.e.g.147.4		8	13.8	odd	4
208.4.i.e.81.2		4	52.3	odd	6
208.4.i.e.113.2		4	4.3	odd	2
1521.4.a.l.1.1		2	39.17	odd	6
1521.4.a.t.1.2		2	39.35	odd	6