Embedded newform 201.3.n.b.13.6

• Label: 201.3.n.b.13.6
• Level: $201$
• Weight: $3$
• Character: 201.13
• Analytic conductor: $5.477$
• Analytic rank: $0$
• Dimension: $240$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 201 = 3 \cdot 67 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	201.n (of order \(66\), degree \(20\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.47685331364\)
Analytic rank:	\(0\)
Dimension:	\(240\)
Relative dimension:	\(12\) over \(\Q(\zeta_{66})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{66}]$

Embedding invariants

Embedding label			13.6
Character	\(\chi\)	\(=\)	201.13
Dual form			201.3.n.b.31.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.333348 - 0.423887i) q^{2} +(-1.30900 - 1.13425i) q^{3} +(0.874477 - 3.60464i) q^{4} +(-3.59519 - 5.59422i) q^{5} +(-0.0444427 + 0.932967i) q^{6} +(1.22122 - 3.05045i) q^{7} +(-3.78158 + 1.72699i) q^{8} +(0.426945 + 2.96946i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 240 q - 34 q^{4} - 33 q^{6} - 21 q^{7} - 33 q^{8} + 72 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/201\mathbb{Z}\right)^\times\).

\(n\)	\(68\)	\(136\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{19}{66}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.333348	−	0.423887i	−0.166674	−	0.211943i	0.695540	−	0.718487i	\(-0.255165\pi\)
							−0.862214	+	0.506544i	\(0.830923\pi\)
\(3\)	−1.30900	−	1.13425i	−0.436332	−	0.378084i
							\(5\)	−3.59519	−	5.59422i	−0.719037	−	1.11884i	−0.987814	−	0.155641i	\(-0.950256\pi\)
0.268777	−	0.963203i	\(-0.413381\pi\)
\(7\)	1.22122	−	3.05045i	0.174460	−	0.435779i	−0.815206	−	0.579171i	\(-0.803377\pi\)
							0.989666	+	0.143391i	\(0.0458008\pi\)
\(11\)	−2.10995	+	0.100509i	−0.191813	+	0.00913719i	−0.143269	−	0.989684i	\(-0.545761\pi\)
							−0.0485445	+	0.998821i	\(0.515458\pi\)
\(13\)	2.02773	+	21.2354i	0.155979	+	1.63349i	0.647104	+	0.762401i	\(0.275980\pi\)
							−0.491125	+	0.871089i	\(0.663414\pi\)
\(17\)	−1.98960	−	8.20126i	−0.117036	−	0.482427i	−0.999918	−	0.0128323i	\(-0.995915\pi\)
							0.882882	−	0.469595i	\(-0.155600\pi\)
\(19\)	13.0411	−	5.22087i	0.686374	−	0.274783i	−0.00214862	−	0.999998i	\(-0.500684\pi\)
							0.688522	+	0.725215i	\(0.258260\pi\)
\(23\)	−25.6338	−	4.94050i	−1.11451	−	0.214804i	−0.401432	−	0.915889i	\(-0.631488\pi\)
							−0.713078	+	0.701084i	\(0.752700\pi\)
\(29\)	1.76523	−	3.05747i	0.0608700	−	0.105430i	−0.833985	−	0.551788i	\(-0.813946\pi\)
							0.894855	+	0.446358i	\(0.147279\pi\)
\(31\)	1.14005	−	11.9392i	0.0367759	−	0.385134i	−0.958281	−	0.285828i	\(-0.907732\pi\)
							0.995057	−	0.0993065i	\(-0.0316624\pi\)
\(37\)	−21.9855	−	38.0800i	−0.594202	−	1.02919i	−0.993659	−	0.112437i	\(-0.964134\pi\)
							0.399456	−	0.916752i	\(-0.369199\pi\)
\(41\)	−26.3006	−	27.5833i	−0.641478	−	0.672762i	0.320072	−	0.947393i	\(-0.396293\pi\)
							−0.961550	+	0.274631i	\(0.911444\pi\)
\(43\)	−3.91423	+	13.3306i	−0.0910286	+	0.310015i	−0.992404	−	0.123024i	\(-0.960741\pi\)
							0.901375	+	0.433039i	\(0.142559\pi\)
\(47\)	9.59036	+	27.7095i	0.204050	+	0.589564i	0.999873	−	0.0159659i	\(-0.00508233\pi\)
							−0.795822	+	0.605530i	\(0.792961\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	201.3.n.b.13.6	✓	240
67.31	odd	66	inner	201.3.n.b.31.6	yes	240

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
201.3.n.b.13.6	✓	240	1.1	even	1	trivial
201.3.n.b.31.6	yes	240	67.31	odd	66	inner