Embedded newform 19.9.f.a.10.8

• Label: 19.9.f.a.10.8
• Level: $19$
• Weight: $9$
• Character: 19.10
• 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			10.8
Character	\(\chi\)	\(=\)	19.10
Dual form			19.9.f.a.2.8

$q$-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 \(\left(\mathbb{Z}/19\mathbb{Z}\right)^\times\).

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{17}{18}\right)\)

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

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
19.9.f.a.2.8	✓	72	19.2	odd	18	inner
19.9.f.a.10.8	yes	72	1.1	even	1	trivial