Embedded newform 19.7.f.a.3.9

• Label: 19.7.f.a.3.9
• 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			3.9
Character	\(\chi\)	\(=\)	19.3
Dual form			19.7.f.a.13.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(8.86541 - 10.5654i) q^{2} +(10.1562 + 27.9038i) q^{3} +(-21.9184 - 124.305i) q^{4} +(30.9255 - 175.387i) q^{5} +(384.853 + 140.075i) q^{6} +(131.445 + 227.670i) q^{7} +(-743.213 - 429.094i) q^{8} +(-117.030 + 98.1996i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 54 q - 6 q^{2} - 36 q^{3} + 192 q^{4} - 6 q^{5} + 720 q^{6} - 219 q^{7} - 9 q^{8} - 2076 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	8.86541	−	10.5654i	1.10818	−	1.32067i	0.165785	−	0.986162i	\(-0.446984\pi\)
							0.942392	−	0.334512i	\(-0.108571\pi\)
\(3\)	10.1562	+	27.9038i	0.376154	+	1.03348i	0.972937	+	0.231072i	\(0.0742233\pi\)
							−0.596782	+	0.802403i	\(0.703554\pi\)
\(5\)	30.9255	−	175.387i	0.247404	−	1.40310i	−0.567440	−	0.823415i	\(-0.692066\pi\)
							0.814843	−	0.579681i	\(-0.196823\pi\)
\(7\)	131.445	+	227.670i	0.383222	+	0.663761i	0.991521	−	0.129948i	\(-0.0414810\pi\)
							−0.608299	+	0.793708i	\(0.708148\pi\)
\(11\)	−715.771	+	1239.75i	−0.537769	+	0.931444i	0.461254	+	0.887268i	\(0.347400\pi\)
							−0.999024	+	0.0441760i	\(0.985934\pi\)
\(13\)	−1247.94	+	3428.68i	−0.568019	+	1.56062i	0.239574	+	0.970878i	\(0.422992\pi\)
							−0.807593	+	0.589740i	\(0.799230\pi\)
\(17\)	−2322.87	−	1949.12i	−0.472801	−	0.396727i	0.375014	−	0.927019i	\(-0.377638\pi\)
							−0.847815	+	0.530292i	\(0.822082\pi\)
\(19\)	5432.97	+	4186.73i	0.792094	+	0.610399i
							\(23\)	−2925.86	−	16593.4i	−0.240475	−	1.36380i	−0.830771	−	0.556614i	\(-0.812100\pi\)
0.590296	−	0.807187i	\(-0.299011\pi\)
\(29\)	−20087.9	−	23939.8i	−0.823644	−	0.981581i	0.176352	−	0.984327i	\(-0.443570\pi\)
							−0.999996	+	0.00274631i	\(0.999126\pi\)
\(31\)	−18227.4	+	10523.6i	−0.611842	+	0.353247i	−0.773686	−	0.633569i	\(-0.781589\pi\)
							0.161844	+	0.986816i	\(0.448256\pi\)
\(37\)			56068.7i			1.10692i	0.832876	+	0.553459i	\(0.186693\pi\)
							−0.832876	+	0.553459i	\(0.813307\pi\)
\(41\)	−8613.60	−	23665.7i	−0.124978	−	0.343374i	0.861387	−	0.507950i	\(-0.169597\pi\)
							−0.986364	+	0.164576i	\(0.947374\pi\)
\(43\)	3667.17	−	20797.5i	0.0461238	−	0.261581i	−0.953022	−	0.302901i	\(-0.902045\pi\)
							0.999146	+	0.0413194i	\(0.0131561\pi\)
\(47\)	51565.9	−	43269.0i	0.496672	−	0.416757i	−0.359738	−	0.933053i	\(-0.617134\pi\)
							0.856410	+	0.516296i	\(0.172690\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	19.7.f.a.3.9	✓	54
19.13	odd	18	inner	19.7.f.a.13.9	yes	54

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
19.7.f.a.3.9	✓	54	1.1	even	1	trivial
19.7.f.a.13.9	yes	54	19.13	odd	18	inner