Embedded newform 209.3.o.a.34.4

• Label: 209.3.o.a.34.4
• Level: $209$
• Weight: $3$
• Character: 209.34
• Analytic conductor: $5.695$
• Analytic rank: $0$
• Dimension: $204$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 209 = 11 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	209.o (of order \(18\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.69483752513\)
Analytic rank:	\(0\)
Dimension:	\(204\)
Relative dimension:	\(34\) over \(\Q(\zeta_{18})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{18}]$

Embedding invariants

Embedding label			34.4
Character	\(\chi\)	\(=\)	209.34
Dual form			209.3.o.a.166.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.13663 + 3.12286i) q^{2} +(1.39965 - 0.246795i) q^{3} +(-5.39616 - 4.52792i) q^{4} +(-2.60598 + 2.18667i) q^{5} +(-0.820170 + 4.65142i) q^{6} +(-1.83554 + 3.17924i) q^{7} +(8.76130 - 5.05834i) q^{8} +(-6.55913 + 2.38733i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 204 q - 12 q^{4} + 36 q^{6}+O(q^{10}) \)

Character values

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

\(n\)	\(78\)	\(134\)
\(\chi(n)\)	\(e\left(\frac{11}{18}\right)\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.13663	+	3.12286i	−0.568314	+	1.56143i	0.238821	+	0.971064i	\(0.423239\pi\)
							−0.807135	+	0.590367i	\(0.798983\pi\)
\(3\)	1.39965	−	0.246795i	0.466549	−	0.0822652i	0.0645682	−	0.997913i	\(-0.479433\pi\)
							0.401981	+	0.915648i	\(0.368322\pi\)
\(5\)	−2.60598	+	2.18667i	−0.521195	+	0.437335i	−0.865048	−	0.501689i	\(-0.832712\pi\)
							0.343853	+	0.939023i	\(0.388268\pi\)
\(7\)	−1.83554	+	3.17924i	−0.262220	+	0.454178i	−0.966831	−	0.255415i	\(-0.917788\pi\)
							0.704612	+	0.709593i	\(0.251121\pi\)
\(11\)	−1.65831	−	2.87228i	−0.150756	−	0.261116i
							\(13\)	12.8377	+	2.26363i	0.987515	+	0.174126i	0.644003	−	0.765023i	\(-0.277272\pi\)
0.343512	+	0.939148i	\(0.388383\pi\)
\(17\)	−19.9725	−	7.26940i	−1.17485	−	0.427612i	−0.320472	−	0.947258i	\(-0.603842\pi\)
							−0.854382	+	0.519646i	\(0.826064\pi\)
\(19\)	−7.15663	−	17.6006i	−0.376665	−	0.926350i
							\(23\)	−30.7280	−	25.7839i	−1.33600	−	1.12104i	−0.982634	−	0.185552i	\(-0.940593\pi\)
−0.353366	−	0.935485i	\(-0.614963\pi\)
\(29\)	−3.27742	−	9.00464i	−0.113014	−	0.310505i	0.870272	−	0.492572i	\(-0.163943\pi\)
							−0.983286	+	0.182068i	\(0.941721\pi\)
\(31\)	50.6575	+	29.2471i	1.63411	+	0.943456i	0.982806	+	0.184643i	\(0.0591127\pi\)
							0.651308	+	0.758813i	\(0.274221\pi\)
\(37\)			16.9847i			0.459045i	0.973303	+	0.229522i	\(0.0737164\pi\)
							−0.973303	+	0.229522i	\(0.926284\pi\)
\(41\)	−7.84423	+	1.38315i	−0.191323	+	0.0337354i	−0.268488	−	0.963283i	\(-0.586524\pi\)
							0.0771656	+	0.997018i	\(0.475413\pi\)
\(43\)	−39.3055	+	32.9813i	−0.914083	+	0.767006i	−0.972891	−	0.231263i	\(-0.925714\pi\)
							0.0588087	+	0.998269i	\(0.481270\pi\)
\(47\)	−53.0693	+	19.3156i	−1.12913	+	0.410971i	−0.837978	−	0.545704i	\(-0.816262\pi\)
							−0.291156	+	0.956675i	\(0.594040\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	209.3.o.a.34.4	✓	204
19.14	odd	18	inner	209.3.o.a.166.4	yes	204

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
209.3.o.a.34.4	✓	204	1.1	even	1	trivial
209.3.o.a.166.4	yes	204	19.14	odd	18	inner