Embedded newform 19.4.e.a.4.3

• Label: 19.4.e.a.4.3
• Level: $19$
• Weight: $4$
• Character: 19.4
• 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			4.3
Character	\(\chi\)	\(=\)	19.4
Dual form			19.4.e.a.5.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.579114 + 0.210780i) q^{2} +(1.63522 + 9.27381i) q^{3} +(-5.83741 - 4.89817i) q^{4} +(9.88078 - 8.29096i) q^{5} +(-1.00776 + 5.71527i) q^{6} +(6.05695 - 10.4909i) q^{7} +(-4.81321 - 8.33672i) q^{8} +(-57.9579 + 21.0950i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 6 q^{2} - 3 q^{3} - 24 q^{4} - 6 q^{5} + 42 q^{6} + 3 q^{7} - 75 q^{8} - 51 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{1}{9}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.579114	+	0.210780i	0.204748	+	0.0745221i	0.442358	−	0.896838i	\(-0.354142\pi\)
							−0.237611	+	0.971361i	\(0.576364\pi\)
\(3\)	1.63522	+	9.27381i	0.314699	+	1.78475i	0.573903	+	0.818923i	\(0.305429\pi\)
							−0.259204	+	0.965823i	\(0.583460\pi\)
\(5\)	9.88078	−	8.29096i	0.883764	−	0.741566i	−0.0831853	−	0.996534i	\(-0.526509\pi\)
							0.966950	+	0.254968i	\(0.0820649\pi\)
\(7\)	6.05695	−	10.4909i	0.327045	−	0.566458i	−0.654879	−	0.755733i	\(-0.727281\pi\)
							0.981924	+	0.189276i	\(0.0606140\pi\)
\(11\)	1.51059	+	2.61642i	0.0414055	+	0.0717164i	0.885985	−	0.463713i	\(-0.153483\pi\)
							−0.844580	+	0.535429i	\(0.820150\pi\)
\(13\)	−2.02601	+	11.4901i	−0.0432242	+	0.245136i	−0.998763	−	0.0497298i	\(-0.984164\pi\)
							0.955539	+	0.294866i	\(0.0952751\pi\)
\(17\)	−104.502	−	38.0357i	−1.49091	−	0.542647i	−0.537222	−	0.843441i	\(-0.680526\pi\)
							−0.953689	+	0.300794i	\(0.902748\pi\)
\(19\)	82.6622	+	5.09509i	0.998106	+	0.0615208i
							\(23\)	−0.581304	−	0.487772i	−0.00527001	−	0.00442206i	0.640149	−	0.768251i	\(-0.278873\pi\)
−0.645419	+	0.763829i	\(0.723317\pi\)
\(29\)	−96.6410	+	35.1745i	−0.618820	+	0.225232i	−0.632358	−	0.774676i	\(-0.717913\pi\)
							0.0135379	+	0.999908i	\(0.495691\pi\)
\(31\)	−59.9698	+	103.871i	−0.347448	+	0.601798i	−0.985795	−	0.167951i	\(-0.946285\pi\)
							0.638347	+	0.769748i	\(0.279618\pi\)
\(37\)	336.206			1.49384			0.746919	−	0.664915i	\(-0.231532\pi\)
							0.746919	+	0.664915i	\(0.231532\pi\)
\(41\)	−17.2295	−	97.7133i	−0.0656291	−	0.372201i	−0.999879	−	0.0155811i	\(-0.995040\pi\)
							0.934249	−	0.356620i	\(-0.116071\pi\)
\(43\)	88.1148	−	73.9371i	0.312497	−	0.262216i	−0.473026	−	0.881048i	\(-0.656838\pi\)
							0.785523	+	0.618832i	\(0.212394\pi\)
\(47\)	298.069	−	108.488i	0.925059	−	0.336694i	0.164810	−	0.986325i	\(-0.447299\pi\)
							0.760249	+	0.649631i	\(0.225077\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	19.4.e.a.4.3	✓	24
3.2	odd	2		171.4.u.b.118.2		24
19.5	even	9	inner	19.4.e.a.5.3	yes	24
19.9	even	9		361.4.a.n.1.6		12
19.10	odd	18		361.4.a.m.1.7		12
57.5	odd	18		171.4.u.b.100.2		24

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
19.4.e.a.4.3	✓	24	1.1	even	1	trivial
19.4.e.a.5.3	yes	24	19.5	even	9	inner
171.4.u.b.100.2		24	57.5	odd	18
171.4.u.b.118.2		24	3.2	odd	2
361.4.a.m.1.7		12	19.10	odd	18
361.4.a.n.1.6		12	19.9	even	9