Embedded newform 19.4.e.a.16.2

• Label: 19.4.e.a.16.2
• 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			16.2
Character	\(\chi\)	\(=\)	19.16
Dual form			19.4.e.a.6.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.74757 - 1.46638i) q^{2} +(4.00345 - 1.45714i) q^{3} +(-0.485473 - 2.75325i) q^{4} +(1.27381 - 7.22413i) q^{5} +(-9.13303 - 3.32415i) q^{6} +(13.1019 + 22.6932i) q^{7} +(-12.3141 + 21.3286i) q^{8} +(-6.77881 + 5.68809i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 6 q^{2} - 3 q^{3} - 24 q^{4} - 6 q^{5} + 42 q^{6} + 3 q^{7} - 75 q^{8} - 51 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.74757	−	1.46638i	−0.617858	−	0.518445i	0.279271	−	0.960212i	\(-0.409907\pi\)
							−0.897130	+	0.441767i	\(0.854352\pi\)
\(3\)	4.00345	−	1.45714i	0.770465	−	0.280426i	0.0732742	−	0.997312i	\(-0.476655\pi\)
							0.697191	+	0.716885i	\(0.254433\pi\)
\(5\)	1.27381	−	7.22413i	0.113933	−	0.646145i	−0.873340	−	0.487111i	\(-0.838051\pi\)
							0.987273	−	0.159035i	\(-0.0508381\pi\)
\(7\)	13.1019	+	22.6932i	0.707437	+	1.22532i	0.965805	+	0.259270i	\(0.0834819\pi\)
							−0.258368	+	0.966046i	\(0.583185\pi\)
\(11\)	7.39009	−	12.8000i	0.202563	−	0.350850i	−0.746790	−	0.665059i	\(-0.768406\pi\)
							0.949354	+	0.314210i	\(0.101740\pi\)
\(13\)	22.9964	+	8.37001i	0.490620	+	0.178571i	0.575470	−	0.817823i	\(-0.304819\pi\)
							−0.0848505	+	0.996394i	\(0.527041\pi\)
\(17\)	−25.4317	−	21.3397i	−0.362829	−	0.304450i	0.443088	−	0.896478i	\(-0.353883\pi\)
							−0.805917	+	0.592028i	\(0.798327\pi\)
\(19\)	−18.8175	+	80.6530i	−0.227212	+	0.973845i
							\(23\)	−35.7418	−	202.702i	−0.324029	−	1.83766i	−0.516410	−	0.856341i	\(-0.672732\pi\)
0.192381	−	0.981320i	\(-0.438379\pi\)
\(29\)	−142.667	+	119.712i	−0.913540	+	0.766551i	−0.972789	−	0.231692i	\(-0.925574\pi\)
							0.0592490	+	0.998243i	\(0.481129\pi\)
\(31\)	−70.1272	−	121.464i	−0.406297	−	0.703727i	0.588174	−	0.808734i	\(-0.299847\pi\)
							−0.994471	+	0.105007i	\(0.966514\pi\)
\(37\)	−258.905			−1.15037			−0.575186	−	0.818023i	\(-0.695070\pi\)
							−0.575186	+	0.818023i	\(0.695070\pi\)
\(41\)	−3.41538	+	1.24310i	−0.0130096	+	0.00473509i	−0.348517	−	0.937303i	\(-0.613315\pi\)
							0.335507	+	0.942038i	\(0.391092\pi\)
\(43\)	90.2743	−	511.971i	0.320156	−	1.81569i	−0.221577	−	0.975143i	\(-0.571121\pi\)
							0.541733	−	0.840551i	\(-0.317768\pi\)
\(47\)	−6.20631	+	5.20771i	−0.0192614	+	0.0161622i	−0.652367	−	0.757903i	\(-0.726224\pi\)
							0.633106	+	0.774065i	\(0.281780\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	19.4.e.a.16.2	yes	24
3.2	odd	2		171.4.u.b.73.3		24
19.5	even	9		361.4.a.n.1.4		12
19.6	even	9	inner	19.4.e.a.6.2	✓	24
19.14	odd	18		361.4.a.m.1.9		12
57.44	odd	18		171.4.u.b.82.3		24

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
19.4.e.a.6.2	✓	24	19.6	even	9	inner
19.4.e.a.16.2	yes	24	1.1	even	1	trivial
171.4.u.b.73.3		24	3.2	odd	2
171.4.u.b.82.3		24	57.44	odd	18
361.4.a.m.1.9		12	19.14	odd	18
361.4.a.n.1.4		12	19.5	even	9