Embedded newform 209.2.j.b.111.2

• Label: 209.2.j.b.111.2
• Level: $209$
• Weight: $2$
• Character: 209.111
• Analytic conductor: $1.669$
• Analytic rank: $0$
• Dimension: $42$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 209 = 11 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	209.j (of order \(9\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.66887340224\)
Analytic rank:	\(0\)
Dimension:	\(42\)
Relative dimension:	\(7\) over \(\Q(\zeta_{9})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{9}]$

Embedding invariants

Embedding label			111.2
Character	\(\chi\)	\(=\)	209.111
Dual form			209.2.j.b.177.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.28771 - 1.08051i) q^{2} +(-1.02863 + 0.374390i) q^{3} +(0.143380 + 0.813151i) q^{4} +(-0.0882338 + 0.500399i) q^{5} +(1.72910 + 0.629342i) q^{6} +(0.0395283 + 0.0684651i) q^{7} +(-0.986992 + 1.70952i) q^{8} +(-1.38023 + 1.15815i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 42 q - 3 q^{2} + 3 q^{4} - 6 q^{5} + 12 q^{6} - 6 q^{7} - 15 q^{8}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/209\mathbb{Z}\right)^\times\).

\(n\)	\(78\)	\(134\)
\(\chi(n)\)	\(e\left(\frac{2}{9}\right)\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.28771	−	1.08051i	−0.910545	−	0.764038i	0.0616773	−	0.998096i	\(-0.480355\pi\)
							−0.972223	+	0.234058i	\(0.924799\pi\)
\(3\)	−1.02863	+	0.374390i	−0.593879	+	0.216154i	−0.621434	−	0.783466i	\(-0.713450\pi\)
							0.0275559	+	0.999620i	\(0.491228\pi\)
\(5\)	−0.0882338	+	0.500399i	−0.0394593	+	0.223785i	−0.998160	−	0.0606312i	\(-0.980689\pi\)
							0.958701	+	0.284416i	\(0.0917998\pi\)
\(7\)	0.0395283	+	0.0684651i	0.0149403	+	0.0258774i	0.873399	−	0.487006i	\(-0.161911\pi\)
							−0.858459	+	0.512883i	\(0.828578\pi\)
\(11\)	−0.500000	+	0.866025i	−0.150756	+	0.261116i
							\(13\)	3.43796	+	1.25131i	0.953518	+	0.347052i	0.771490	−	0.636241i	\(-0.219512\pi\)
0.182028	+	0.983293i	\(0.441734\pi\)
\(17\)	2.75007	+	2.30758i	0.666991	+	0.559672i	0.912173	−	0.409805i	\(-0.134403\pi\)
							−0.245182	+	0.969477i	\(0.578848\pi\)
\(19\)	−1.99861	+	3.87370i	−0.458513	+	0.888688i
							\(23\)	0.963469	+	5.46411i	0.200897	+	1.13934i	0.903766	+	0.428026i	\(0.140791\pi\)
−0.702869	+	0.711319i	\(0.748098\pi\)
\(29\)	−3.09472	+	2.59678i	−0.574675	+	0.482210i	−0.883194	−	0.469009i	\(-0.844611\pi\)
							0.308518	+	0.951218i	\(0.400167\pi\)
\(31\)	−3.24816	−	5.62598i	−0.583387	−	1.01046i	−0.995074	−	0.0991313i	\(-0.968394\pi\)
							0.411687	−	0.911325i	\(-0.364940\pi\)
\(37\)	−1.71356			−0.281707			−0.140854	−	0.990030i	\(-0.544985\pi\)
							−0.140854	+	0.990030i	\(0.544985\pi\)
\(41\)	−10.3928	+	3.78266i	−1.62308	+	0.590753i	−0.983965	−	0.178360i	\(-0.942921\pi\)
							−0.639114	+	0.769112i	\(0.720699\pi\)
\(43\)	−0.804230	+	4.56102i	−0.122644	+	0.695549i	0.860035	+	0.510234i	\(0.170441\pi\)
							−0.982679	+	0.185314i	\(0.940670\pi\)
\(47\)	−5.30916	+	4.45491i	−0.774420	+	0.649816i	−0.941837	−	0.336070i	\(-0.890902\pi\)
							0.167416	+	0.985886i	\(0.446458\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	209.2.j.b.111.2	✓	42
19.5	even	9		3971.2.a.t.1.5		21
19.6	even	9	inner	209.2.j.b.177.2	yes	42
19.14	odd	18		3971.2.a.s.1.17		21

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
209.2.j.b.111.2	✓	42	1.1	even	1	trivial
209.2.j.b.177.2	yes	42	19.6	even	9	inner
3971.2.a.s.1.17		21	19.14	odd	18
3971.2.a.t.1.5		21	19.5	even	9