Embedded newform 19.5.f.a.2.2

• Label: 19.5.f.a.2.2
• Level: $19$
• Weight: $5$
• Character: 19.2
• Analytic conductor: $1.964$
• Analytic rank: $0$
• Dimension: $36$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 19 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	19.f (of order \(18\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.96402929859\)
Analytic rank:	\(0\)
Dimension:	\(36\)
Relative dimension:	\(6\) over \(\Q(\zeta_{18})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{18}]$

Embedding invariants

Embedding label			2.2
Character	\(\chi\)	\(=\)	19.2
Dual form			19.5.f.a.10.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-3.73687 - 0.658911i) q^{2} +(0.577174 - 0.687849i) q^{3} +(-1.50506 - 0.547798i) q^{4} +(-39.5293 + 14.3875i) q^{5} +(-2.61005 + 2.19009i) q^{6} +(-34.7004 - 60.1028i) q^{7} +(57.8416 + 33.3949i) q^{8} +(13.9255 + 78.9754i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 36 q - 6 q^{2} - 18 q^{3} - 48 q^{4} - 6 q^{5} - 48 q^{7} - 9 q^{8} + 246 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{1}{18}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−3.73687	−	0.658911i	−0.934217	−	0.164728i	−0.314236	−	0.949345i	\(-0.601748\pi\)
							−0.619981	+	0.784617i	\(0.712860\pi\)
\(3\)	0.577174	−	0.687849i	0.0641304	−	0.0764276i	−0.733026	−	0.680201i	\(-0.761892\pi\)
							0.797156	+	0.603774i	\(0.206337\pi\)
\(5\)	−39.5293	+	14.3875i	−1.58117	+	0.575499i	−0.975458	−	0.220183i	\(-0.929334\pi\)
							−0.605713	+	0.795683i	\(0.707112\pi\)
\(7\)	−34.7004	−	60.1028i	−0.708171	−	1.22659i	−0.965535	−	0.260274i	\(-0.916187\pi\)
							0.257364	−	0.966315i	\(-0.417146\pi\)
\(11\)	51.3459	−	88.9338i	0.424347	−	0.734990i	−0.572012	−	0.820245i	\(-0.693837\pi\)
							0.996359	+	0.0852549i	\(0.0271705\pi\)
\(13\)	−90.5199	−	107.877i	−0.535621	−	0.638328i	0.428579	−	0.903504i	\(-0.359014\pi\)
							−0.964200	+	0.265176i	\(0.914570\pi\)
\(17\)	−34.7983	+	197.351i	−0.120409	+	0.682876i	0.863519	+	0.504315i	\(0.168255\pi\)
							−0.983929	+	0.178560i	\(0.942856\pi\)
\(19\)	−354.501	+	68.1903i	−0.981998	+	0.188893i
							\(23\)	−173.385	−	63.1068i	−0.327759	−	0.119295i	0.172899	−	0.984940i	\(-0.444687\pi\)
−0.500658	+	0.865645i	\(0.666909\pi\)
\(29\)	−1001.26	+	176.550i	−1.19056	+	0.209929i	−0.733616	−	0.679565i	\(-0.762169\pi\)
							−0.456949	+	0.889493i	\(0.651058\pi\)
\(31\)	12.3987	−	7.15842i	0.0129019	−	0.00744893i	−0.493535	−	0.869726i	\(-0.664296\pi\)
							0.506437	+	0.862277i	\(0.330962\pi\)
\(37\)		−	1152.34i		−	0.841735i	−0.907122	−	0.420868i	\(-0.861726\pi\)
							0.907122	−	0.420868i	\(-0.138274\pi\)
\(41\)	−1461.99	+	1742.34i	−0.869717	+	1.03649i	0.129275	+	0.991609i	\(0.458735\pi\)
							−0.998992	+	0.0448799i	\(0.985709\pi\)
\(43\)	1820.92	−	662.761i	0.984815	−	0.358443i	0.201104	−	0.979570i	\(-0.435547\pi\)
							0.783710	+	0.621127i	\(0.213325\pi\)
\(47\)	72.7550	+	412.614i	0.0329357	+	0.186788i	0.996837	−	0.0794717i	\(-0.0253233\pi\)
							−0.963901	+	0.266259i	\(0.914212\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	19.5.f.a.2.2	✓	36
19.10	odd	18	inner	19.5.f.a.10.2	yes	36

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
19.5.f.a.2.2	✓	36	1.1	even	1	trivial
19.5.f.a.10.2	yes	36	19.10	odd	18	inner