Embedded newform 19.9.f.a.2.12

• Label: 19.9.f.a.2.12
• Level: $19$
• Weight: $9$
• Character: 19.2
• 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			2.12
Character	\(\chi\)	\(=\)	19.2
Dual form			19.9.f.a.10.12

$q$-expansion

\(f(q)\)	\(=\)	\(q+(28.1724 + 4.96755i) q^{2} +(-19.0758 + 22.7336i) q^{3} +(528.446 + 192.339i) q^{4} +(353.774 - 128.763i) q^{5} +(-650.341 + 545.701i) q^{6} +(44.1972 + 76.5518i) q^{7} +(7589.89 + 4382.03i) q^{8} +(986.373 + 5594.00i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 72 q - 6 q^{2} + 162 q^{3} - 720 q^{4} - 6 q^{5} - 720 q^{6} + 3282 q^{7} - 9 q^{8} + 8562 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	28.1724	+	4.96755i	1.76077	+	0.310472i	0.958204	−	0.286085i	\(-0.0923540\pi\)
							0.802571	+	0.596557i	\(0.203465\pi\)
\(3\)	−19.0758	+	22.7336i	−0.235503	+	0.280662i	−0.870833	−	0.491579i	\(-0.836420\pi\)
							0.635329	+	0.772241i	\(0.280864\pi\)
\(5\)	353.774	−	128.763i	0.566038	−	0.206021i	−0.0431200	−	0.999070i	\(-0.513730\pi\)
							0.609158	+	0.793049i	\(0.291508\pi\)
\(7\)	44.1972	+	76.5518i	0.0184078	+	0.0318833i	0.875083	−	0.483974i	\(-0.160807\pi\)
							−0.856675	+	0.515857i	\(0.827474\pi\)
\(11\)	−288.517	+	499.725i	−0.0197061	+	0.0341319i	−0.875710	−	0.482837i	\(-0.839606\pi\)
							0.856004	+	0.516969i	\(0.172940\pi\)
\(13\)	−30539.3	−	36395.3i	−1.06927	−	1.27430i	−0.959913	−	0.280299i	\(-0.909566\pi\)
							−0.109353	−	0.994003i	\(-0.534878\pi\)
\(17\)	6085.18	−	34510.7i	0.0728580	−	0.413198i	−0.926464	−	0.376383i	\(-0.877168\pi\)
							0.999322	−	0.0368152i	\(-0.0117213\pi\)
\(19\)	82212.1	−	101117.i	0.630843	−	0.775911i
							\(23\)	−291803.	−	106208.i	−1.04275	−	0.379529i	−0.236826	−	0.971552i	\(-0.576107\pi\)
−0.805921	+	0.592023i	\(0.798329\pi\)
\(29\)	628489.	−	110820.i	0.888599	−	0.156684i	0.289328	−	0.957230i	\(-0.406568\pi\)
							0.599271	+	0.800546i	\(0.295457\pi\)
\(31\)	−216033.	+	124727.i	−0.233923	+	0.135056i	−0.612381	−	0.790563i	\(-0.709788\pi\)
							0.378457	+	0.925619i	\(0.376455\pi\)
\(37\)			1.33035e6i			0.709840i	0.934897	+	0.354920i	\(0.115492\pi\)
							−0.934897	+	0.354920i	\(0.884508\pi\)
\(41\)	−2.18357e6	+	2.60228e6i	−0.772737	+	0.920912i	−0.998581	−	0.0532511i	\(-0.983042\pi\)
							0.225844	+	0.974164i	\(0.427486\pi\)
\(43\)	4.02216e6	−	1.46394e6i	1.17648	−	0.428204i	0.321523	−	0.946902i	\(-0.395805\pi\)
							0.854958	+	0.518697i	\(0.173583\pi\)
\(47\)	900407.	+	5.10646e6i	0.184522	+	1.04647i	0.926568	+	0.376127i	\(0.122744\pi\)
							−0.742047	+	0.670348i	\(0.766145\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	19.9.f.a.2.12	✓	72
19.10	odd	18	inner	19.9.f.a.10.12	yes	72

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
19.9.f.a.2.12	✓	72	1.1	even	1	trivial
19.9.f.a.10.12	yes	72	19.10	odd	18	inner