Embedded newform 19.4.e.a.6.3

• Label: 19.4.e.a.6.3
• 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			6.3
Character	\(\chi\)	\(=\)	19.6
Dual form			19.4.e.a.16.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.38101 - 1.15881i) q^{2} +(2.58415 + 0.940555i) q^{3} +(-0.824823 + 4.67780i) q^{4} +(-3.13553 - 17.7825i) q^{5} +(4.65867 - 1.69562i) q^{6} +(-14.1277 + 24.4699i) q^{7} +(11.4927 + 19.9060i) q^{8} +(-14.8900 - 12.4942i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 6 q^{2} - 3 q^{3} - 24 q^{4} - 6 q^{5} + 42 q^{6} + 3 q^{7} - 75 q^{8} - 51 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.38101	−	1.15881i	0.488262	−	0.409701i	−0.365141	−	0.930952i	\(-0.618979\pi\)
							0.853403	+	0.521252i	\(0.174535\pi\)
\(3\)	2.58415	+	0.940555i	0.497321	+	0.181010i	0.578489	−	0.815691i	\(-0.303643\pi\)
							−0.0811679	+	0.996700i	\(0.525865\pi\)
\(5\)	−3.13553	−	17.7825i	−0.280451	−	1.59051i	−0.721098	−	0.692833i	\(-0.756362\pi\)
							0.440647	−	0.897680i	\(-0.354749\pi\)
\(7\)	−14.1277	+	24.4699i	−0.762826	+	1.32125i	0.178563	+	0.983928i	\(0.442855\pi\)
							−0.941389	+	0.337324i	\(0.890478\pi\)
\(11\)	−1.89653	−	3.28489i	−0.0519841	−	0.0900391i	0.838862	−	0.544344i	\(-0.183221\pi\)
							−0.890847	+	0.454304i	\(0.849888\pi\)
\(13\)	44.0524	−	16.0338i	0.939841	−	0.342074i	0.173738	−	0.984792i	\(-0.444415\pi\)
							0.766103	+	0.642718i	\(0.222193\pi\)
\(17\)	14.5268	−	12.1895i	0.207252	−	0.173905i	−0.533253	−	0.845956i	\(-0.679031\pi\)
							0.740505	+	0.672051i	\(0.234586\pi\)
\(19\)	75.4905	−	34.0614i	0.911511	−	0.411275i
							\(23\)	−2.85514	+	16.1923i	−0.0258843	+	0.146797i	−0.995011	−	0.0997667i	\(-0.968190\pi\)
0.969127	+	0.246564i	\(0.0793015\pi\)
\(29\)	108.300	+	90.8743i	0.693475	+	0.581894i	0.919909	−	0.392132i	\(-0.128262\pi\)
							−0.226434	+	0.974026i	\(0.572707\pi\)
\(31\)	−89.1238	+	154.367i	−0.516358	+	0.894359i	0.483461	+	0.875366i	\(0.339379\pi\)
							−0.999820	+	0.0189932i	\(0.993954\pi\)
\(37\)	−29.5834			−0.131446			−0.0657228	−	0.997838i	\(-0.520935\pi\)
							−0.0657228	+	0.997838i	\(0.520935\pi\)
\(41\)	−328.747	−	119.654i	−1.25224	−	0.455777i	−0.371080	−	0.928601i	\(-0.621013\pi\)
							−0.881157	+	0.472824i	\(0.843235\pi\)
\(43\)	−13.5956	−	77.1044i	−0.0482165	−	0.273449i	0.951162	−	0.308691i	\(-0.0998909\pi\)
							−0.999379	+	0.0352419i	\(0.988780\pi\)
\(47\)	158.409	+	132.921i	0.491624	+	0.412521i	0.854608	−	0.519274i	\(-0.173798\pi\)
							−0.362984	+	0.931795i	\(0.618242\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	19.4.e.a.6.3	✓	24
3.2	odd	2		171.4.u.b.82.2		24
19.4	even	9		361.4.a.n.1.7		12
19.15	odd	18		361.4.a.m.1.6		12
19.16	even	9	inner	19.4.e.a.16.3	yes	24
57.35	odd	18		171.4.u.b.73.2		24

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
19.4.e.a.6.3	✓	24	1.1	even	1	trivial
19.4.e.a.16.3	yes	24	19.16	even	9	inner
171.4.u.b.73.2		24	57.35	odd	18
171.4.u.b.82.2		24	3.2	odd	2
361.4.a.m.1.6		12	19.15	odd	18
361.4.a.n.1.7		12	19.4	even	9