Embedded newform 19.7.f.a.3.1

• Label: 19.7.f.a.3.1
• 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			3.1
Character	\(\chi\)	\(=\)	19.3
Dual form			19.7.f.a.13.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-8.55141 + 10.1912i) q^{2} +(-4.72892 - 12.9926i) q^{3} +(-19.6199 - 111.270i) q^{4} +(12.1561 - 68.9408i) q^{5} +(172.849 + 62.9118i) q^{6} +(144.225 + 249.805i) q^{7} +(564.387 + 325.849i) q^{8} +(412.002 - 345.710i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 54 q - 6 q^{2} - 36 q^{3} + 192 q^{4} - 6 q^{5} + 720 q^{6} - 219 q^{7} - 9 q^{8} - 2076 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−8.55141	+	10.1912i	−1.06893	+	1.27390i	−0.108879	+	0.994055i	\(0.534726\pi\)
							−0.960046	+	0.279841i	\(0.909718\pi\)
\(3\)	−4.72892	−	12.9926i	−0.175145	−	0.481207i	0.820795	−	0.571222i	\(-0.193531\pi\)
							−0.995940	+	0.0900152i	\(0.971308\pi\)
\(5\)	12.1561	−	68.9408i	0.0972490	−	0.551527i	−0.896786	−	0.442465i	\(-0.854104\pi\)
							0.994035	−	0.109062i	\(-0.0347847\pi\)
\(7\)	144.225	+	249.805i	0.420482	+	0.728295i	0.995987	−	0.0895028i	\(-0.0285278\pi\)
							−0.575505	+	0.817798i	\(0.695194\pi\)
\(11\)	667.655	−	1156.41i	0.501619	−	0.868830i	−0.498379	−	0.866959i	\(-0.666071\pi\)
							0.999998	−	0.00187087i	\(-0.000595516\pi\)
\(13\)	442.343	−	1215.33i	0.201340	−	0.553176i	−0.797396	−	0.603457i	\(-0.793789\pi\)
							0.998735	+	0.0502810i	\(0.0160117\pi\)
\(17\)	−4735.67	−	3973.70i	−0.963906	−	0.808813i	0.0176781	−	0.999844i	\(-0.494373\pi\)
							−0.981584	+	0.191031i	\(0.938817\pi\)
\(19\)	5837.00	−	3602.13i	0.850999	−	0.525168i
							\(23\)	607.416	+	3444.83i	0.0499233	+	0.283129i	0.999541	−	0.0302802i	\(-0.00963997\pi\)
−0.949618	+	0.313409i	\(0.898529\pi\)
\(29\)	−23758.7	−	28314.5i	−0.974156	−	1.16095i	−0.986949	−	0.161035i	\(-0.948517\pi\)
							0.0127931	−	0.999918i	\(-0.495928\pi\)
\(31\)	−8975.31	+	5181.90i	−0.301276	+	0.173942i	−0.643016	−	0.765853i	\(-0.722317\pi\)
							0.341740	+	0.939795i	\(0.388984\pi\)
\(37\)			80004.9i			1.57947i	0.613448	+	0.789735i	\(0.289782\pi\)
							−0.613448	+	0.789735i	\(0.710218\pi\)
\(41\)	22210.3	+	61022.4i	0.322258	+	0.885396i	0.990008	+	0.141011i	\(0.0450352\pi\)
							−0.667750	+	0.744385i	\(0.732743\pi\)
\(43\)	24169.4	−	137072.i	0.303991	−	1.72402i	−0.324226	−	0.945980i	\(-0.605104\pi\)
							0.628217	−	0.778038i	\(-0.283785\pi\)
\(47\)	−7953.34	+	6673.65i	−0.0766048	+	0.0642791i	−0.680286	−	0.732947i	\(-0.738144\pi\)
							0.603681	+	0.797226i	\(0.293700\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	19.7.f.a.3.1	✓	54
19.13	odd	18	inner	19.7.f.a.13.1	yes	54

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
19.7.f.a.3.1	✓	54	1.1	even	1	trivial
19.7.f.a.13.1	yes	54	19.13	odd	18	inner