Embedded newform 19.4.e.a.9.2

• Label: 19.4.e.a.9.2
• Level: $19$
• Weight: $4$
• Character: 19.9
• 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			9.2
Character	\(\chi\)	\(=\)	19.9
Dual form			19.4.e.a.17.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.163781 - 0.928850i) q^{2} +(1.56059 - 1.30949i) q^{3} +(6.68160 - 2.43190i) q^{4} +(-3.55727 - 1.29474i) q^{5} +(-1.47192 - 1.23509i) q^{6} +(-11.7083 + 20.2794i) q^{7} +(-7.12592 - 12.3424i) q^{8} +(-3.96782 + 22.5026i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 6 q^{2} - 3 q^{3} - 24 q^{4} - 6 q^{5} + 42 q^{6} + 3 q^{7} - 75 q^{8} - 51 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{4}{9}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.163781	−	0.928850i	−0.0579055	−	0.328398i	0.942070	−	0.335417i	\(-0.108877\pi\)
							−0.999975	+	0.00701838i	\(0.997766\pi\)
\(3\)	1.56059	−	1.30949i	0.300336	−	0.252012i	−0.480148	−	0.877187i	\(-0.659417\pi\)
							0.780484	+	0.625176i	\(0.214973\pi\)
\(5\)	−3.55727	−	1.29474i	−0.318172	−	0.115805i	0.177997	−	0.984031i	\(-0.443038\pi\)
							−0.496169	+	0.868226i	\(0.665260\pi\)
\(7\)	−11.7083	+	20.2794i	−0.632188	+	1.09498i	0.354915	+	0.934899i	\(0.384510\pi\)
							−0.987103	+	0.160084i	\(0.948824\pi\)
\(11\)	−8.50109	−	14.7243i	−0.233016	−	0.403595i	0.725678	−	0.688034i	\(-0.241526\pi\)
							−0.958694	+	0.284439i	\(0.908193\pi\)
\(13\)	3.66005	+	3.07115i	0.0780859	+	0.0655218i	0.680995	−	0.732288i	\(-0.261547\pi\)
							−0.602909	+	0.797810i	\(0.705992\pi\)
\(17\)	−9.73487	−	55.2092i	−0.138886	−	0.787659i	−0.972075	−	0.234671i	\(-0.924599\pi\)
							0.833189	−	0.552988i	\(-0.186512\pi\)
\(19\)	80.5692	+	19.1731i	0.972833	+	0.231506i
							\(23\)	−83.8179	+	30.5072i	−0.759880	+	0.276574i	−0.692757	−	0.721171i	\(-0.743604\pi\)
−0.0671227	+	0.997745i	\(0.521382\pi\)
\(29\)	29.8807	−	169.462i	0.191335	−	1.08511i	−0.726208	−	0.687475i	\(-0.758719\pi\)
							0.917543	−	0.397637i	\(-0.130170\pi\)
\(31\)	−48.1783	+	83.4472i	−0.279131	+	0.483470i	−0.971169	−	0.238392i	\(-0.923380\pi\)
							0.692038	+	0.721861i	\(0.256713\pi\)
\(37\)	339.998			1.51068			0.755342	−	0.655331i	\(-0.227471\pi\)
							0.755342	+	0.655331i	\(0.227471\pi\)
\(41\)	339.942	−	285.246i	1.29488	−	1.08653i	0.303875	−	0.952712i	\(-0.401719\pi\)
							0.991005	−	0.133822i	\(-0.0427250\pi\)
\(43\)	−253.541	−	92.2813i	−0.899177	−	0.327274i	−0.149254	−	0.988799i	\(-0.547687\pi\)
							−0.749923	+	0.661525i	\(0.769909\pi\)
\(47\)	−84.2516	+	477.815i	−0.261476	+	1.48290i	0.517410	+	0.855737i	\(0.326896\pi\)
							−0.778886	+	0.627165i	\(0.784215\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	19.4.e.a.9.2	✓	24
3.2	odd	2		171.4.u.b.28.3		24
19.6	even	9		361.4.a.n.1.5		12
19.13	odd	18		361.4.a.m.1.8		12
19.17	even	9	inner	19.4.e.a.17.2	yes	24
57.17	odd	18		171.4.u.b.55.3		24

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
19.4.e.a.9.2	✓	24	1.1	even	1	trivial
19.4.e.a.17.2	yes	24	19.17	even	9	inner
171.4.u.b.28.3		24	3.2	odd	2
171.4.u.b.55.3		24	57.17	odd	18
361.4.a.m.1.8		12	19.13	odd	18
361.4.a.n.1.5		12	19.6	even	9