Embedded newform 19.4.e.a.16.4

• Label: 19.4.e.a.16.4
• Level: $19$
• Weight: $4$
• Character: 19.16
• 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			16.4
Character	\(\chi\)	\(=\)	19.16
Dual form			19.4.e.a.6.4

$q$-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 \(\left(\mathbb{Z}/19\mathbb{Z}\right)^\times\).

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

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

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
19.4.e.a.6.4	✓	24	19.6	even	9	inner
19.4.e.a.16.4	yes	24	1.1	even	1	trivial
171.4.u.b.73.1		24	3.2	odd	2
171.4.u.b.82.1		24	57.44	odd	18
361.4.a.m.1.3		12	19.14	odd	18
361.4.a.n.1.10		12	19.5	even	9