Embedded newform 19.9.f.a.3.1

• Label: 19.9.f.a.3.1
• Level: $19$
• Weight: $9$
• Character: 19.3
• 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			3.1
Character	\(\chi\)	\(=\)	19.3
Dual form			19.9.f.a.13.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-19.5948 + 23.3522i) q^{2} +(-44.0282 - 120.966i) q^{3} +(-116.914 - 663.050i) q^{4} +(-107.009 + 606.875i) q^{5} +(3687.55 + 1342.16i) q^{6} +(-1489.58 - 2580.03i) q^{7} +(11016.2 + 6360.18i) q^{8} +(-7668.38 + 6434.54i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 72 q - 6 q^{2} + 162 q^{3} - 720 q^{4} - 6 q^{5} - 720 q^{6} + 3282 q^{7} - 9 q^{8} + 8562 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−19.5948	+	23.3522i	−1.22467	+	1.45951i	−0.379353	+	0.925252i	\(0.623853\pi\)
							−0.845322	+	0.534258i	\(0.820591\pi\)
\(3\)	−44.0282	−	120.966i	−0.543558	−	1.49341i	−0.842263	−	0.539067i	\(-0.818777\pi\)
							0.298705	−	0.954346i	\(-0.403445\pi\)
\(5\)	−107.009	+	606.875i	−0.171214	+	0.971001i	0.771210	+	0.636580i	\(0.219652\pi\)
							−0.942424	+	0.334420i	\(0.891459\pi\)
\(7\)	−1489.58	−	2580.03i	−0.620399	−	1.07456i	−0.989411	−	0.145138i	\(-0.953637\pi\)
							0.369012	−	0.929425i	\(-0.379696\pi\)
\(11\)	833.661	−	1443.94i	0.0569402	−	0.0986233i	−0.836150	−	0.548500i	\(-0.815199\pi\)
							0.893090	+	0.449877i	\(0.148532\pi\)
\(13\)	−8401.98	+	23084.3i	−0.294177	+	0.808244i	0.701267	+	0.712898i	\(0.252618\pi\)
							−0.995444	+	0.0953457i	\(0.969604\pi\)
\(17\)	17302.3	+	14518.3i	0.207161	+	0.173829i	0.740465	−	0.672095i	\(-0.234605\pi\)
							−0.533304	+	0.845924i	\(0.679050\pi\)
\(19\)	112007.	+	66618.3i	0.859470	+	0.511186i
							\(23\)	66596.1	+	377685.i	0.237978	+	1.34964i	0.836250	+	0.548349i	\(0.184743\pi\)
−0.598271	+	0.801294i	\(0.704145\pi\)
\(29\)	−112496.	−	134068.i	−0.159054	−	0.189553i	0.680632	−	0.732626i	\(-0.261705\pi\)
							−0.839686	+	0.543072i	\(0.817261\pi\)
\(31\)	734649.	−	424150.i	0.795487	−	0.459275i	−0.0464033	−	0.998923i	\(-0.514776\pi\)
							0.841891	+	0.539648i	\(0.181443\pi\)
\(37\)		−	2.30725e6i		−	1.23108i	−0.788104	−	0.615542i	\(-0.788937\pi\)
							0.788104	−	0.615542i	\(-0.211063\pi\)
\(41\)	1.11706e6	+	3.06909e6i	0.395312	+	1.08611i	0.964541	+	0.263931i	\(0.0850192\pi\)
							−0.569230	+	0.822178i	\(0.692759\pi\)
\(43\)	−487672.	+	2.76573e6i	−0.142644	+	0.808975i	0.826584	+	0.562813i	\(0.190281\pi\)
							−0.969229	+	0.246163i	\(0.920830\pi\)
\(47\)	−1.16357e6	+	976354.i	−0.238453	+	0.200086i	−0.754181	−	0.656667i	\(-0.771966\pi\)
							0.515728	+	0.856752i	\(0.327521\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	19.9.f.a.3.1	✓	72
19.13	odd	18	inner	19.9.f.a.13.1	yes	72

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
19.9.f.a.3.1	✓	72	1.1	even	1	trivial
19.9.f.a.13.1	yes	72	19.13	odd	18	inner