Embedded newform 209.3.l.a.39.12

• Label: 209.3.l.a.39.12
• Level: $209$
• Weight: $3$
• Character: 209.39
• Analytic conductor: $5.695$
• Analytic rank: $0$
• Dimension: $144$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 209 = 11 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	209.l (of order \(10\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.69483752513\)
Analytic rank:	\(0\)
Dimension:	\(144\)
Relative dimension:	\(36\) over \(\Q(\zeta_{10})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{10}]$

Embedding invariants

Embedding label			39.12
Character	\(\chi\)	\(=\)	209.39
Dual form			209.3.l.a.134.12

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.70220 + 0.553079i) q^{2} +(1.61487 + 1.17327i) q^{3} +(-0.644475 + 0.468239i) q^{4} +(-0.908993 + 2.79759i) q^{5} +(-3.39775 - 1.10399i) q^{6} +(-6.55700 - 9.02494i) q^{7} +(5.04613 - 6.94540i) q^{8} +(-1.54991 - 4.77014i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 144 q + 2 q^{3} + 84 q^{4} + 4 q^{5} - 15 q^{7} - 40 q^{8} - 110 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(78\)	\(134\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{9}{10}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.70220	+	0.553079i	−0.851101	+	0.276539i	−0.701907	−	0.712269i	\(-0.747668\pi\)
							−0.149194	+	0.988808i	\(0.547668\pi\)
\(3\)	1.61487	+	1.17327i	0.538290	+	0.391091i	0.823450	−	0.567389i	\(-0.192047\pi\)
							−0.285159	+	0.958480i	\(0.592047\pi\)
\(5\)	−0.908993	+	2.79759i	−0.181799	+	0.559519i	−0.999879	−	0.0155860i	\(-0.995039\pi\)
							0.818080	+	0.575105i	\(0.195039\pi\)
\(7\)	−6.55700	−	9.02494i	−0.936715	−	1.28928i	−0.957182	−	0.289487i	\(-0.906515\pi\)
							0.0204672	−	0.999791i	\(-0.493485\pi\)
\(11\)	9.19185	−	6.04234i	0.835623	−	0.549303i
							\(13\)	0.181180	−	0.0588691i	0.0139370	−	0.00452839i	−0.302040	−	0.953295i	\(-0.597668\pi\)
0.315977	+	0.948767i	\(0.397668\pi\)
\(17\)	24.9392	+	8.10324i	1.46701	+	0.476661i	0.930204	−	0.367043i	\(-0.119630\pi\)
							0.536808	+	0.843704i	\(0.319630\pi\)
\(19\)	2.56210	−	3.52642i	0.134847	−	0.185601i
							\(23\)	−20.5791			−0.894745			−0.447373	−	0.894348i	\(-0.647640\pi\)
−0.447373	+	0.894348i	\(0.647640\pi\)
\(29\)	−28.6719	−	39.4635i	−0.988687	−	1.36081i	−0.932015	−	0.362419i	\(-0.881951\pi\)
							−0.0566719	−	0.998393i	\(-0.518049\pi\)
\(31\)	3.94325	+	12.1361i	0.127202	+	0.391486i	0.994296	−	0.106658i	\(-0.0340150\pi\)
							−0.867094	+	0.498144i	\(0.834015\pi\)
\(37\)	31.9775	−	23.2330i	0.864256	−	0.627919i	−0.0647836	−	0.997899i	\(-0.520636\pi\)
							0.929039	+	0.369981i	\(0.120636\pi\)
\(41\)	25.7969	−	35.5064i	0.629193	−	0.866010i	−0.368789	−	0.929513i	\(-0.620227\pi\)
							0.997982	+	0.0635034i	\(0.0202274\pi\)
\(43\)		−	54.9807i		−	1.27862i	−0.768949	−	0.639310i	\(-0.779220\pi\)
							0.768949	−	0.639310i	\(-0.220780\pi\)
\(47\)	−58.2862	−	42.3474i	−1.24013	−	0.901008i	−0.242524	−	0.970145i	\(-0.577975\pi\)
							−0.997607	+	0.0691373i	\(0.977975\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	209.3.l.a.39.12	✓	144
11.2	odd	10	inner	209.3.l.a.134.12	yes	144

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
209.3.l.a.39.12	✓	144	1.1	even	1	trivial
209.3.l.a.134.12	yes	144	11.2	odd	10	inner