Embedded newform 19.5.f.a.3.2

• Label: 19.5.f.a.3.2
• Level: $19$
• Weight: $5$
• Character: 19.3
• 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			3.2
Character	\(\chi\)	\(=\)	19.3
Dual form			19.5.f.a.13.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-3.41023 + 4.06416i) q^{2} +(-4.67230 - 12.8370i) q^{3} +(-2.10931 - 11.9625i) q^{4} +(3.16499 - 17.9496i) q^{5} +(68.1054 + 24.7883i) q^{6} +(-22.5528 - 39.0626i) q^{7} +(-17.7027 - 10.2207i) q^{8} +(-80.9096 + 67.8913i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 36 q - 6 q^{2} - 18 q^{3} - 48 q^{4} - 6 q^{5} - 48 q^{7} - 9 q^{8} + 246 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{13}{18}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−3.41023	+	4.06416i	−0.852558	+	1.01604i	0.147080	+	0.989125i	\(0.453013\pi\)
							−0.999638	+	0.0269146i	\(0.991432\pi\)
\(3\)	−4.67230	−	12.8370i	−0.519145	−	1.42634i	−0.871465	−	0.490458i	\(-0.836829\pi\)
							0.352320	−	0.935880i	\(-0.385393\pi\)
\(5\)	3.16499	−	17.9496i	0.126600	−	0.717982i	−0.853745	−	0.520691i	\(-0.825674\pi\)
							0.980345	−	0.197291i	\(-0.0632145\pi\)
\(7\)	−22.5528	−	39.0626i	−0.460262	−	0.797197i	0.538712	−	0.842490i	\(-0.318911\pi\)
							−0.998974	+	0.0452933i	\(0.985578\pi\)
\(11\)	−70.6688	+	122.402i	−0.584040	+	1.01159i	0.410955	+	0.911656i	\(0.365195\pi\)
							−0.994994	+	0.0999305i	\(0.968138\pi\)
\(13\)	78.7139	−	216.265i	0.465763	−	1.27967i	−0.455328	−	0.890324i	\(-0.650478\pi\)
							0.921090	−	0.389349i	\(-0.127300\pi\)
\(17\)	382.445	+	320.910i	1.32334	+	1.11041i	0.985585	+	0.169181i	\(0.0541122\pi\)
							0.337756	+	0.941234i	\(0.390332\pi\)
\(19\)	−237.683	−	271.712i	−0.658403	−	0.752666i
							\(23\)	−52.8607	−	299.788i	−0.0999256	−	0.566706i	−0.993126	−	0.117048i	\(-0.962657\pi\)
0.893201	−	0.449658i	\(-0.148454\pi\)
\(29\)	−487.707	−	581.227i	−0.579913	−	0.691114i	0.393721	−	0.919230i	\(-0.371188\pi\)
							−0.973634	+	0.228116i	\(0.926743\pi\)
\(31\)	662.689	−	382.603i	0.689582	−	0.398131i	−0.113873	−	0.993495i	\(-0.536326\pi\)
							0.803456	+	0.595365i	\(0.202992\pi\)
\(37\)		−	660.070i		−	0.482155i	−0.970506	−	0.241078i	\(-0.922499\pi\)
							0.970506	−	0.241078i	\(-0.0775008\pi\)
\(41\)	74.5789	+	204.904i	0.0443658	+	0.121894i	0.959897	−	0.280353i	\(-0.0904515\pi\)
							−0.915531	+	0.402247i	\(0.868229\pi\)
\(43\)	−197.042	+	1117.48i	−0.106567	+	0.604371i	0.884016	+	0.467457i	\(0.154830\pi\)
							−0.990583	+	0.136915i	\(0.956281\pi\)
\(47\)	1568.45	−	1316.08i	0.710026	−	0.595782i	−0.214581	−	0.976706i	\(-0.568838\pi\)
							0.924606	+	0.380924i	\(0.124394\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	19.5.f.a.3.2	✓	36
19.13	odd	18	inner	19.5.f.a.13.2	yes	36

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
19.5.f.a.3.2	✓	36	1.1	even	1	trivial
19.5.f.a.13.2	yes	36	19.13	odd	18	inner