Embedded newform 19.9.f.a.2.8

• Label: 19.9.f.a.2.8
• Level: $19$
• Weight: $9$
• Character: 19.2
• Analytic conductor: $7.740$
• Analytic rank: $0$
• Dimension: $72$
• CM: no
• 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			2.8
Character	\(\chi\)	\(=\)	19.2
Dual form			19.9.f.a.10.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(7.58647 + 1.33770i) q^{2} +(27.9671 - 33.3299i) q^{3} +(-184.796 - 67.2603i) q^{4} +(295.009 - 107.375i) q^{5} +(256.757 - 215.445i) q^{6} +(-1865.56 - 3231.24i) q^{7} +(-3019.86 - 1743.52i) q^{8} +(810.583 + 4597.05i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 72 q - 6 q^{2} + 162 q^{3} - 720 q^{4} - 6 q^{5} - 720 q^{6} + 3282 q^{7} - 9 q^{8} + 8562 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	7.58647	+	1.33770i	0.474155	+	0.0836062i	0.405619	−	0.914042i	\(-0.367056\pi\)
							0.0685359	+	0.997649i	\(0.478167\pi\)
\(3\)	27.9671	−	33.3299i	0.345273	−	0.411480i	−0.565263	−	0.824911i	\(-0.691225\pi\)
							0.910535	+	0.413431i	\(0.135670\pi\)
\(5\)	295.009	−	107.375i	0.472015	−	0.171799i	−0.0950502	−	0.995472i	\(-0.530301\pi\)
							0.567065	+	0.823673i	\(0.308079\pi\)
\(7\)	−1865.56	−	3231.24i	−0.776991	−	1.34579i	−0.933669	−	0.358138i	\(-0.883412\pi\)
							0.156678	−	0.987650i	\(-0.449922\pi\)
\(11\)	9165.18	−	15874.6i	0.625994	−	1.08425i	−0.362354	−	0.932041i	\(-0.618027\pi\)
							0.988348	−	0.152213i	\(-0.0486399\pi\)
\(13\)	−19210.9	−	22894.7i	−0.672629	−	0.801608i	0.316511	−	0.948589i	\(-0.397489\pi\)
							−0.989139	+	0.146981i	\(0.953044\pi\)
\(17\)	11403.6	−	64673.1i	0.136536	−	0.774333i	−0.837242	−	0.546832i	\(-0.815834\pi\)
							0.973778	−	0.227501i	\(-0.0730554\pi\)
\(19\)	−14163.0	+	129549.i	−0.108678	+	0.994077i
							\(23\)	493986.	+	179796.i	1.76524	+	0.642494i	0.999999	−	0.00125164i	\(-0.000398411\pi\)
0.765239	+	0.643746i	\(0.222621\pi\)
\(29\)	229052.	−	40388.0i	0.323849	−	0.0571032i	−0.00936057	−	0.999956i	\(-0.502980\pi\)
							0.333209	+	0.942853i	\(0.391868\pi\)
\(31\)	−838142.	+	483902.i	−0.907551	+	0.523975i	−0.879642	−	0.475636i	\(-0.842218\pi\)
							−0.0279087	+	0.999610i	\(0.508885\pi\)
\(37\)		−	3.00035e6i		−	1.60090i	−0.599398	−	0.800451i	\(-0.704593\pi\)
							0.599398	−	0.800451i	\(-0.295407\pi\)
\(41\)	1.92017e6	−	2.28837e6i	0.679524	−	0.809825i	−0.310522	−	0.950566i	\(-0.600504\pi\)
							0.990046	+	0.140741i	\(0.0449484\pi\)
\(43\)	2.97017e6	−	1.08105e6i	0.868775	−	0.316208i	0.131104	−	0.991369i	\(-0.458148\pi\)
							0.737671	+	0.675160i	\(0.235926\pi\)
\(47\)	−768416.	−	4.35790e6i	−0.157472	−	0.893071i	−0.956490	−	0.291764i	\(-0.905758\pi\)
							0.799018	−	0.601307i	\(-0.205353\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	19.9.f.a.2.8	✓	72
19.10	odd	18	inner	19.9.f.a.10.8	yes	72

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
19.9.f.a.2.8	✓	72	1.1	even	1	trivial
19.9.f.a.10.8	yes	72	19.10	odd	18	inner