Embedded newform 19.7.f.a.3.5

• Label: 19.7.f.a.3.5
• 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			3.5
Character	\(\chi\)	\(=\)	19.3
Dual form			19.7.f.a.13.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.521164 + 0.621099i) q^{2} +(-6.41153 - 17.6155i) q^{3} +(10.9993 + 62.3803i) q^{4} +(35.4222 - 200.889i) q^{5} +(14.2825 + 5.19839i) q^{6} +(-216.833 - 375.565i) q^{7} +(-89.4152 - 51.6239i) q^{8} +(289.247 - 242.707i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 54 q - 6 q^{2} - 36 q^{3} + 192 q^{4} - 6 q^{5} + 720 q^{6} - 219 q^{7} - 9 q^{8} - 2076 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.521164	+	0.621099i	−0.0651455	+	0.0776374i	−0.797633	−	0.603143i	\(-0.793915\pi\)
							0.732488	+	0.680780i	\(0.238359\pi\)
\(3\)	−6.41153	−	17.6155i	−0.237464	−	0.652427i	−0.999985	−	0.00546372i	\(-0.998261\pi\)
							0.762521	−	0.646963i	\(-0.223961\pi\)
\(5\)	35.4222	−	200.889i	0.283378	−	1.60711i	−0.427646	−	0.903946i	\(-0.640657\pi\)
							0.711024	−	0.703168i	\(-0.248232\pi\)
\(7\)	−216.833	−	375.565i	−0.632166	−	1.09494i	−0.987108	−	0.160055i	\(-0.948833\pi\)
							0.354942	−	0.934888i	\(-0.384500\pi\)
\(11\)	429.346	−	743.648i	0.322574	−	0.558714i	−0.658445	−	0.752629i	\(-0.728785\pi\)
							0.981018	+	0.193915i	\(0.0621187\pi\)
\(13\)	−678.193	+	1863.32i	−0.308691	+	0.848120i	0.684222	+	0.729274i	\(0.260142\pi\)
							−0.992913	+	0.118847i	\(0.962080\pi\)
\(17\)	1925.46	+	1615.65i	0.391910	+	0.328852i	0.817357	−	0.576132i	\(-0.195438\pi\)
							−0.425446	+	0.904984i	\(0.639883\pi\)
\(19\)	4672.08	+	5021.71i	0.681160	+	0.732134i
							\(23\)	−1600.39	−	9076.26i	−0.131535	−	0.745974i	−0.977210	−	0.212275i	\(-0.931913\pi\)
0.845675	−	0.533699i	\(-0.179198\pi\)
\(29\)	17608.4	+	20984.8i	0.721980	+	0.860422i	0.994822	−	0.101637i	\(-0.0324080\pi\)
							−0.272842	+	0.962059i	\(0.587964\pi\)
\(31\)	44557.3	−	25725.2i	1.49566	−	0.863522i	0.495676	−	0.868508i	\(-0.334921\pi\)
							0.999988	+	0.00498626i	\(0.00158718\pi\)
\(37\)		−	39236.8i		−	0.774620i	−0.921950	−	0.387310i	\(-0.873404\pi\)
							0.921950	−	0.387310i	\(-0.126596\pi\)
\(41\)	26614.1	+	73121.8i	0.386154	+	1.06095i	0.968718	+	0.248165i	\(0.0798276\pi\)
							−0.582563	+	0.812785i	\(0.697950\pi\)
\(43\)	−8386.92	+	47564.6i	−0.105487	+	0.598244i	0.885538	+	0.464566i	\(0.153790\pi\)
							−0.991025	+	0.133678i	\(0.957321\pi\)
\(47\)	−74919.9	+	62865.2i	−0.721612	+	0.605504i	−0.927831	−	0.373002i	\(-0.878328\pi\)
							0.206219	+	0.978506i	\(0.433884\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	19.7.f.a.3.5	✓	54
19.13	odd	18	inner	19.7.f.a.13.5	yes	54

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
19.7.f.a.3.5	✓	54	1.1	even	1	trivial
19.7.f.a.13.5	yes	54	19.13	odd	18	inner