Embedded newform 19.4.e.a.6.2

• Label: 19.4.e.a.6.2
• 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			6.2
Character	\(\chi\)	\(=\)	19.6
Dual form			19.4.e.a.16.2

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

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

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)

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

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
19.4.e.a.6.2	✓	24	1.1	even	1	trivial
19.4.e.a.16.2	yes	24	19.16	even	9	inner
171.4.u.b.73.3		24	57.35	odd	18
171.4.u.b.82.3		24	3.2	odd	2
361.4.a.m.1.9		12	19.15	odd	18
361.4.a.n.1.4		12	19.4	even	9