Embedded newform 19.4.e.a.6.4

• Label: 19.4.e.a.6.4
• Level: $19$
• Weight: $4$
• Character: 19.6
• Analytic conductor: $1.121$
• Analytic rank: $0$
• Dimension: $24$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 19 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	19.e (of order \(9\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.12103629011\)
Analytic rank:	\(0\)
Dimension:	\(24\)
Relative dimension:	\(4\) over \(\Q(\zeta_{9})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{9}]$

Embedding invariants

Embedding label			6.4
Character	\(\chi\)	\(=\)	19.6
Dual form			19.4.e.a.16.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(2.98693 - 2.50633i) q^{2} +(-3.72413 - 1.35547i) q^{3} +(1.25086 - 7.09397i) q^{4} +(2.58856 + 14.6804i) q^{5} +(-14.5210 + 5.28519i) q^{6} +(5.35146 - 9.26900i) q^{7} +(1.55302 + 2.68992i) q^{8} +(-8.65137 - 7.25936i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 6 q^{2} - 3 q^{3} - 24 q^{4} - 6 q^{5} + 42 q^{6} + 3 q^{7} - 75 q^{8} - 51 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/19\mathbb{Z}\right)^\times\).

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{7}{9}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	2.98693	−	2.50633i	1.05604	−	0.886121i	0.0623221	−	0.998056i	\(-0.480149\pi\)
							0.993715	+	0.111935i	\(0.0357049\pi\)
\(3\)	−3.72413	−	1.35547i	−0.716709	−	0.260861i	−0.0421809	−	0.999110i	\(-0.513431\pi\)
							−0.674528	+	0.738249i	\(0.735653\pi\)
\(5\)	2.58856	+	14.6804i	0.231527	+	1.31306i	0.849805	+	0.527097i	\(0.176720\pi\)
							−0.618277	+	0.785960i	\(0.712169\pi\)
\(7\)	5.35146	−	9.26900i	0.288952	−	0.500479i	−0.684608	−	0.728911i	\(-0.740027\pi\)
							0.973560	+	0.228432i	\(0.0733600\pi\)
\(11\)	−14.6256	−	25.3323i	−0.400890	−	0.694361i	0.592944	−	0.805244i	\(-0.297966\pi\)
							−0.993834	+	0.110883i	\(0.964632\pi\)
\(13\)	−76.4727	+	27.8338i	−1.63152	+	0.593823i	−0.985526	−	0.169522i	\(-0.945778\pi\)
							−0.645990	+	0.763346i	\(0.723555\pi\)
\(17\)	61.3408	−	51.4710i	0.875137	−	0.734327i	−0.0900365	−	0.995938i	\(-0.528698\pi\)
							0.965173	+	0.261612i	\(0.0842539\pi\)
\(19\)	81.6408	+	13.9203i	0.985773	+	0.168081i
							\(23\)	11.6670	−	66.1669i	0.105771	−	0.599858i	−0.885138	−	0.465328i	\(-0.845936\pi\)
0.990909	−	0.134530i	\(-0.0429526\pi\)
\(29\)	1.78142	+	1.49479i	0.0114070	+	0.00957158i	0.648473	−	0.761237i	\(-0.275408\pi\)
							−0.637066	+	0.770809i	\(0.719852\pi\)
\(31\)	−68.9182	+	119.370i	−0.399292	+	0.691595i	−0.993639	−	0.112614i	\(-0.964078\pi\)
							0.594346	+	0.804209i	\(0.297411\pi\)
\(37\)	−305.379			−1.35687			−0.678433	−	0.734662i	\(-0.737341\pi\)
							−0.678433	+	0.734662i	\(0.737341\pi\)
\(41\)	213.997	+	77.8884i	0.815138	+	0.296686i	0.715744	−	0.698362i	\(-0.246088\pi\)
							0.0993934	+	0.995048i	\(0.468310\pi\)
\(43\)	−29.9719	−	169.979i	−0.106295	−	0.602828i	−0.990695	−	0.136099i	\(-0.956544\pi\)
							0.884400	−	0.466729i	\(-0.154568\pi\)
\(47\)	−324.048	−	271.908i	−1.00569	−	0.843870i	−0.0179234	−	0.999839i	\(-0.505706\pi\)
							−0.987762	+	0.155969i	\(0.950150\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	19.4.e.a.6.4	✓	24
3.2	odd	2		171.4.u.b.82.1		24
19.4	even	9		361.4.a.n.1.10		12
19.15	odd	18		361.4.a.m.1.3		12
19.16	even	9	inner	19.4.e.a.16.4	yes	24
57.35	odd	18		171.4.u.b.73.1		24

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
19.4.e.a.6.4	✓	24	1.1	even	1	trivial
19.4.e.a.16.4	yes	24	19.16	even	9	inner
171.4.u.b.73.1		24	57.35	odd	18
171.4.u.b.82.1		24	3.2	odd	2
361.4.a.m.1.3		12	19.15	odd	18
361.4.a.n.1.10		12	19.4	even	9