Embedded newform 19.7.f.a.3.4

• Label: 19.7.f.a.3.4
• Level: $19$
• Weight: $7$
• Character: 19.3
• Analytic conductor: $4.371$
• Analytic rank: $0$
• Dimension: $54$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 19 \)
Weight:	\( k \)	\(=\)	\( 7 \)
Character orbit:	\([\chi]\)	\(=\)	19.f (of order \(18\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.37102758878\)
Analytic rank:	\(0\)
Dimension:	\(54\)
Relative dimension:	\(9\) over \(\Q(\zeta_{18})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{18}]$

Embedding invariants

Embedding label			3.4
Character	\(\chi\)	\(=\)	19.3
Dual form			19.7.f.a.13.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.65924 + 3.16916i) q^{2} +(-16.9615 - 46.6013i) q^{3} +(8.14146 + 46.1725i) q^{4} +(-28.7281 + 162.925i) q^{5} +(192.792 + 70.1705i) q^{6} +(152.866 + 264.772i) q^{7} +(-397.277 - 229.368i) q^{8} +(-1325.54 + 1112.26i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 54 q - 6 q^{2} - 36 q^{3} + 192 q^{4} - 6 q^{5} + 720 q^{6} - 219 q^{7} - 9 q^{8} - 2076 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.65924	+	3.16916i	−0.332406	+	0.396146i	−0.906197	−	0.422856i	\(-0.861028\pi\)
							0.573791	+	0.819002i	\(0.305472\pi\)
\(3\)	−16.9615	−	46.6013i	−0.628203	−	1.72597i	−0.685957	−	0.727642i	\(-0.740616\pi\)
							0.0577544	−	0.998331i	\(-0.481606\pi\)
\(5\)	−28.7281	+	162.925i	−0.229825	+	1.30340i	0.623418	+	0.781889i	\(0.285744\pi\)
							−0.853243	+	0.521514i	\(0.825367\pi\)
\(7\)	152.866	+	264.772i	0.445674	+	0.771930i	0.998099	−	0.0616329i	\(-0.0196308\pi\)
							−0.552425	+	0.833563i	\(0.686297\pi\)
\(11\)	365.927	−	633.805i	0.274927	−	0.476187i	−0.695190	−	0.718826i	\(-0.744680\pi\)
							0.970117	+	0.242639i	\(0.0780131\pi\)
\(13\)	−1022.45	+	2809.15i	−0.465383	+	1.27863i	0.456002	+	0.889979i	\(0.349281\pi\)
							−0.921385	+	0.388651i	\(0.872941\pi\)
\(17\)	−272.979	−	229.057i	−0.0555626	−	0.0466226i	0.614583	−	0.788852i	\(-0.289324\pi\)
							−0.670146	+	0.742229i	\(0.733769\pi\)
\(19\)	−6120.16	−	3096.70i	−0.892281	−	0.451480i
							\(23\)	858.998	+	4871.62i	0.0706006	+	0.400396i	0.999545	+	0.0301758i	\(0.00960671\pi\)
−0.928944	+	0.370220i	\(0.879282\pi\)
\(29\)	−3477.35	−	4144.14i	−0.142579	−	0.169918i	0.690029	−	0.723781i	\(-0.257598\pi\)
							−0.832608	+	0.553863i	\(0.813153\pi\)
\(31\)	19415.1	−	11209.3i	0.651710	−	0.376265i	−0.137401	−	0.990516i	\(-0.543875\pi\)
							0.789111	+	0.614250i	\(0.210542\pi\)
\(37\)			88423.0i			1.74566i	0.488023	+	0.872831i	\(0.337718\pi\)
							−0.488023	+	0.872831i	\(0.662282\pi\)
\(41\)	−19227.1	−	52826.0i	−0.278973	−	0.766472i	−0.997480	−	0.0709521i	\(-0.977396\pi\)
							0.718507	−	0.695520i	\(-0.244826\pi\)
\(43\)	−17705.6	+	100413.i	−0.222692	+	1.26295i	0.644357	+	0.764725i	\(0.277125\pi\)
							−0.867049	+	0.498224i	\(0.833986\pi\)
\(47\)	37736.3	−	31664.5i	0.363467	−	0.304985i	−0.442704	−	0.896668i	\(-0.645981\pi\)
							0.806171	+	0.591683i	\(0.201536\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	19.7.f.a.3.4	✓	54
19.13	odd	18	inner	19.7.f.a.13.4	yes	54

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
19.7.f.a.3.4	✓	54	1.1	even	1	trivial
19.7.f.a.13.4	yes	54	19.13	odd	18	inner