Embedded newform 19.7.f.a.2.5

• Label: 19.7.f.a.2.5
• Level: $19$
• Weight: $7$
• Character: 19.2
• 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			2.5
Character	\(\chi\)	\(=\)	19.2
Dual form			19.7.f.a.10.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.612104 - 0.107930i) q^{2} +(29.3902 - 35.0259i) q^{3} +(-59.7773 - 21.7572i) q^{4} +(-60.2750 + 21.9383i) q^{5} +(-21.7702 + 18.2674i) q^{6} +(-70.8586 - 122.731i) q^{7} +(68.6913 + 39.6589i) q^{8} +(-236.440 - 1340.92i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 54 q - 6 q^{2} - 36 q^{3} + 192 q^{4} - 6 q^{5} + 720 q^{6} - 219 q^{7} - 9 q^{8} - 2076 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{1}{18}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.612104	−	0.107930i	−0.0765130	−	0.0134913i	0.135261	−	0.990810i	\(-0.456813\pi\)
							−0.211774	+	0.977319i	\(0.567924\pi\)
\(3\)	29.3902	−	35.0259i	1.08853	−	1.29726i	0.136704	−	0.990612i	\(-0.456349\pi\)
							0.951824	−	0.306645i	\(-0.0992063\pi\)
\(5\)	−60.2750	+	21.9383i	−0.482200	+	0.175507i	−0.571671	−	0.820483i	\(-0.693705\pi\)
							0.0894708	+	0.995989i	\(0.471482\pi\)
\(7\)	−70.8586	−	122.731i	−0.206585	−	0.357815i	0.744052	−	0.668122i	\(-0.232902\pi\)
							−0.950637	+	0.310307i	\(0.899568\pi\)
\(11\)	994.377	−	1722.31i	0.747090	−	1.29400i	−0.202122	−	0.979360i	\(-0.564784\pi\)
							0.949212	−	0.314638i	\(-0.101883\pi\)
\(13\)	1619.46	+	1930.00i	0.737125	+	0.878472i	0.996174	−	0.0873927i	\(-0.0278535\pi\)
							−0.259049	+	0.965864i	\(0.583409\pi\)
\(17\)	−359.351	+	2037.98i	−0.0731428	+	0.414814i	0.926148	+	0.377160i	\(0.123099\pi\)
							−0.999291	+	0.0376533i	\(0.988012\pi\)
\(19\)	1307.84	−	6733.16i	0.190675	−	0.981653i
							\(23\)	10910.3	+	3971.02i	0.896710	+	0.326376i	0.748934	−	0.662645i	\(-0.230566\pi\)
0.147777	+	0.989021i	\(0.452788\pi\)
\(29\)	−8277.72	+	1459.59i	−0.339404	+	0.0598460i	−0.340752	−	0.940153i	\(-0.610682\pi\)
							0.00134871	+	0.999999i	\(0.499571\pi\)
\(31\)	42628.1	−	24611.3i	1.43091	−	0.826133i	0.433715	−	0.901050i	\(-0.357202\pi\)
							0.997190	+	0.0749165i	\(0.0238690\pi\)
\(37\)			3784.89i			0.0747219i	0.999302	+	0.0373610i	\(0.0118951\pi\)
							−0.999302	+	0.0373610i	\(0.988105\pi\)
\(41\)	−72790.0	+	86747.7i	−1.05614	+	1.25865i	−0.0912938	+	0.995824i	\(0.529100\pi\)
							−0.964842	+	0.262830i	\(0.915344\pi\)
\(43\)	−12238.4	+	4454.42i	−0.153929	+	0.0560255i	−0.417835	−	0.908523i	\(-0.637211\pi\)
							0.263907	+	0.964548i	\(0.414989\pi\)
\(47\)	−24532.5	−	139131.i	−0.236291	−	1.34008i	−0.839877	−	0.542777i	\(-0.817373\pi\)
							0.603585	−	0.797298i	\(-0.293738\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	19.7.f.a.2.5	✓	54
19.10	odd	18	inner	19.7.f.a.10.5	yes	54

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
19.7.f.a.2.5	✓	54	1.1	even	1	trivial
19.7.f.a.10.5	yes	54	19.10	odd	18	inner