Embedded newform 19.9.f.a.2.9

• Label: 19.9.f.a.2.9
• Level: $19$
• Weight: $9$
• Character: 19.2
• Analytic conductor: $7.740$
• Analytic rank: $0$
• Dimension: $72$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 19 \)
Weight:	\( k \)	\(=\)	\( 9 \)
Character orbit:	\([\chi]\)	\(=\)	19.f (of order \(18\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.74019359116\)
Analytic rank:	\(0\)
Dimension:	\(72\)
Relative dimension:	\(12\) over \(\Q(\zeta_{18})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{18}]$

Embedding invariants

Embedding label			2.9
Character	\(\chi\)	\(=\)	19.2
Dual form			19.9.f.a.10.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(14.3082 + 2.52292i) q^{2} +(-93.7652 + 111.745i) q^{3} +(-42.2029 - 15.3606i) q^{4} +(404.530 - 147.237i) q^{5} +(-1623.53 + 1362.30i) q^{6} +(-1320.16 - 2286.59i) q^{7} +(-3786.18 - 2185.95i) q^{8} +(-2555.73 - 14494.2i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 72 q - 6 q^{2} + 162 q^{3} - 720 q^{4} - 6 q^{5} - 720 q^{6} + 3282 q^{7} - 9 q^{8} + 8562 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	14.3082	+	2.52292i	0.894260	+	0.157682i	0.601848	−	0.798610i	\(-0.294431\pi\)
							0.292412	+	0.956293i	\(0.405542\pi\)
\(3\)	−93.7652	+	111.745i	−1.15759	+	1.37957i	−0.245597	+	0.969372i	\(0.578984\pi\)
							−0.911998	+	0.410195i	\(0.865460\pi\)
\(5\)	404.530	−	147.237i	0.647248	−	0.235579i	0.00252704	−	0.999997i	\(-0.499196\pi\)
							0.644721	+	0.764418i	\(0.276973\pi\)
\(7\)	−1320.16	−	2286.59i	−0.549839	−	0.952349i	−0.998285	−	0.0585391i	\(-0.981356\pi\)
							0.448446	−	0.893810i	\(-0.351978\pi\)
\(11\)	−14584.5	+	25261.1i	−0.996141	+	1.72537i	−0.422065	+	0.906566i	\(0.638695\pi\)
							−0.574076	+	0.818802i	\(0.694639\pi\)
\(13\)	9788.86	+	11665.9i	0.342735	+	0.408456i	0.909687	−	0.415295i	\(-0.136322\pi\)
							−0.566951	+	0.823751i	\(0.691877\pi\)
\(17\)	6335.02	−	35927.7i	0.0758494	−	0.430163i	−0.923109	−	0.384538i	\(-0.874361\pi\)
							0.998959	−	0.0456255i	\(-0.0145281\pi\)
\(19\)	−97792.2	+	86140.9i	−0.750395	+	0.660990i
							\(23\)	−257997.	−	93903.3i	−0.921943	−	0.335560i	−0.162931	−	0.986637i	\(-0.552095\pi\)
−0.759011	+	0.651078i	\(0.774317\pi\)
\(29\)	247449.	−	43631.9i	0.349859	−	0.0616896i	0.00404290	−	0.999992i	\(-0.498713\pi\)
							0.345816	+	0.938302i	\(0.387602\pi\)
\(31\)	359622.	−	207628.i	0.389403	−	0.224822i	−0.292499	−	0.956266i	\(-0.594487\pi\)
							0.681901	+	0.731444i	\(0.261153\pi\)
\(37\)			2.34782e6i			1.25273i	0.779529	+	0.626367i	\(0.215459\pi\)
							−0.779529	+	0.626367i	\(0.784541\pi\)
\(41\)	−482516.	+	575040.i	−0.170756	+	0.203499i	−0.844635	−	0.535343i	\(-0.820183\pi\)
							0.673879	+	0.738842i	\(0.264627\pi\)
\(43\)	137126.	−	49909.9i	0.0401095	−	0.0145987i	−0.321887	−	0.946778i	\(-0.604317\pi\)
							0.361997	+	0.932179i	\(0.382095\pi\)
\(47\)	−175535.	−	995506.i	−0.0359725	−	0.204010i	0.961524	−	0.274719i	\(-0.0885850\pi\)
							−0.997497	+	0.0707089i	\(0.977474\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	19.9.f.a.2.9	✓	72
19.10	odd	18	inner	19.9.f.a.10.9	yes	72

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
19.9.f.a.2.9	✓	72	1.1	even	1	trivial
19.9.f.a.10.9	yes	72	19.10	odd	18	inner