Embedded newform 207.3.n.a.2.8

• Label: 207.3.n.a.2.8
• Level: $207$
• Weight: $3$
• Character: 207.2
• Analytic conductor: $5.640$
• Analytic rank: $0$
• Dimension: $920$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 207 = 3^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	207.n (of order \(66\), degree \(20\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.64034147226\)
Analytic rank:	\(0\)
Dimension:	\(920\)
Relative dimension:	\(46\) over \(\Q(\zeta_{66})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{66}]$

Embedding invariants

Embedding label			2.8
Character	\(\chi\)	\(=\)	207.2
Dual form			207.3.n.a.104.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.35044 - 2.61949i) q^{2} +(-0.490983 + 2.95955i) q^{3} +(-2.71780 + 3.81661i) q^{4} +(-0.602504 + 0.146166i) q^{5} +(8.41554 - 2.71057i) q^{6} +(-1.83383 - 5.29850i) q^{7} +(1.99936 + 0.287465i) q^{8} +(-8.51787 - 2.90618i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 920 q - 27 q^{2} - 20 q^{3} - 97 q^{4} - 45 q^{5} - 29 q^{6} - 9 q^{7} - 24 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/207\mathbb{Z}\right)^\times\).

\(n\)	\(28\)	\(47\)
\(\chi(n)\)	\(e\left(\frac{1}{11}\right)\)	\(e\left(\frac{1}{6}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.35044	−	2.61949i	−0.675220	−	1.30974i	−0.938585	−	0.345047i	\(-0.887863\pi\)
							0.263366	−	0.964696i	\(-0.415167\pi\)
\(3\)	−0.490983	+	2.95955i	−0.163661	+	0.986517i
							\(5\)	−0.602504	+	0.146166i	−0.120501	+	0.0292332i	−0.295556	−	0.955325i	\(-0.595505\pi\)
0.175055	+	0.984559i	\(0.443990\pi\)
\(7\)	−1.83383	−	5.29850i	−0.261976	−	0.756929i	−0.996893	−	0.0787720i	\(-0.974900\pi\)
							0.734917	−	0.678157i	\(-0.237221\pi\)
\(11\)	13.0225	−	0.620340i	1.18387	−	0.0563945i	0.553607	−	0.832778i	\(-0.313251\pi\)
							0.630260	+	0.776384i	\(0.282948\pi\)
\(13\)	−6.73753	+	19.4668i	−0.518272	+	1.49745i	0.317440	+	0.948278i	\(0.397177\pi\)
							−0.835711	+	0.549169i	\(0.814944\pi\)
\(17\)	20.3697	+	9.30255i	1.19822	+	0.547209i	0.911696	−	0.410865i	\(-0.134773\pi\)
							0.286523	+	0.958073i	\(0.407500\pi\)
\(19\)	11.4958	+	25.1724i	0.605045	+	1.32486i	0.925912	+	0.377739i	\(0.123298\pi\)
							−0.320868	+	0.947124i	\(0.603974\pi\)
\(23\)	22.9907	−	0.655720i	0.999594	−	0.0285096i
							\(29\)	−23.4554	+	16.7025i	−0.808808	+	0.575950i	−0.907928	−	0.419126i	\(-0.862337\pi\)
0.0991202	+	0.995075i	\(0.468397\pi\)
\(31\)	−54.6236	+	21.8680i	−1.76205	+	0.705420i	−0.763857	+	0.645386i	\(0.776697\pi\)
							−0.998196	+	0.0600336i	\(0.980879\pi\)
\(37\)	22.3335	−	6.55771i	0.603609	−	0.177236i	0.0343719	−	0.999409i	\(-0.489057\pi\)
							0.569237	+	0.822174i	\(0.307239\pi\)
\(41\)	17.9178	−	4.34682i	0.437020	−	0.106020i	−0.0112120	−	0.999937i	\(-0.503569\pi\)
							0.448232	+	0.893917i	\(0.352054\pi\)
\(43\)	59.5967	+	23.8589i	1.38597	+	0.554858i	0.940302	−	0.340340i	\(-0.110542\pi\)
							0.445667	+	0.895199i	\(0.352967\pi\)
\(47\)	−13.4068	−	7.74044i	−0.285252	−	0.164690i	0.350547	−	0.936545i	\(-0.385996\pi\)
							−0.635799	+	0.771855i	\(0.719329\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	207.3.n.a.2.8	✓	920
9.5	odd	6	inner	207.3.n.a.140.8	yes	920
23.12	even	11	inner	207.3.n.a.173.8	yes	920
207.104	odd	66	inner	207.3.n.a.104.8	yes	920

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
207.3.n.a.2.8	✓	920	1.1	even	1	trivial
207.3.n.a.104.8	yes	920	207.104	odd	66	inner
207.3.n.a.140.8	yes	920	9.5	odd	6	inner
207.3.n.a.173.8	yes	920	23.12	even	11	inner