Embedded newform 201.3.o.b.17.3

• Label: 201.3.o.b.17.3
• Level: $201$
• Weight: $3$
• Character: 201.17
• Analytic conductor: $5.477$
• Analytic rank: $0$
• Dimension: $840$
• CM: no
• Inner twists: $4$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(5.47685331364\)
Analytic rank:	\(0\)
Dimension:	\(840\)
Relative dimension:	\(42\) over \(\Q(\zeta_{66})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{66}]$

Embedding invariants

Embedding label			17.3
Character	\(\chi\)	\(=\)	201.17
Dual form			201.3.o.b.71.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.339674 - 3.55723i) q^{2} +(-2.79427 + 1.09181i) q^{3} +(-8.61080 + 1.65960i) q^{4} +(4.72450 + 0.679280i) q^{5} +(4.83296 + 9.56901i) q^{6} +(4.33674 + 6.09010i) q^{7} +(4.80145 + 16.3522i) q^{8} +(6.61591 - 6.10162i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 840 q - 16 q^{3} - 126 q^{4} - 25 q^{6} - 34 q^{7} - 24 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(68\)	\(136\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{32}{33}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.339674	−	3.55723i	−0.169837	−	1.77862i	−0.532240	−	0.846594i	\(-0.678649\pi\)
							0.362402	−	0.932022i	\(-0.381957\pi\)
\(3\)	−2.79427	+	1.09181i	−0.931424	+	0.363936i
							\(5\)	4.72450	+	0.679280i	0.944899	+	0.135856i	0.597507	−	0.801864i	\(-0.296158\pi\)
0.347392	+	0.937720i	\(0.387067\pi\)
\(7\)	4.33674	+	6.09010i	0.619534	+	0.870014i	0.998464	−	0.0554116i	\(-0.0176471\pi\)
							−0.378929	+	0.925426i	\(0.623708\pi\)
\(11\)	3.26324	−	4.14954i	0.296658	−	0.377231i	−0.614759	−	0.788715i	\(-0.710747\pi\)
							0.911417	+	0.411484i	\(0.134989\pi\)
\(13\)	2.86860	+	11.8246i	0.220662	+	0.909581i	0.969158	+	0.246441i	\(0.0792613\pi\)
							−0.748496	+	0.663140i	\(0.769224\pi\)
\(17\)	−2.98418	+	15.4834i	−0.175540	+	0.910787i	0.782173	+	0.623061i	\(0.214111\pi\)
							−0.957713	+	0.287726i	\(0.907101\pi\)
\(19\)	14.2912	−	20.0692i	0.752170	−	1.05628i	−0.244080	−	0.969755i	\(-0.578486\pi\)
							0.996250	−	0.0865203i	\(-0.0275748\pi\)
\(23\)	16.1128	−	31.2544i	0.700556	−	1.35889i	−0.222770	−	0.974871i	\(-0.571510\pi\)
							0.923326	−	0.384017i	\(-0.125460\pi\)
\(29\)	36.1787	+	20.8878i	1.24754	+	0.720268i	0.970618	−	0.240624i	\(-0.0773520\pi\)
							0.276923	+	0.960892i	\(0.410685\pi\)
\(31\)	−4.70398	+	19.3901i	−0.151741	+	0.625486i	0.844243	+	0.535960i	\(0.180050\pi\)
							−0.995984	+	0.0895260i	\(0.971465\pi\)
\(37\)	5.13189	+	8.88870i	0.138700	+	0.240235i	0.927005	−	0.375050i	\(-0.122374\pi\)
							−0.788305	+	0.615285i	\(0.789041\pi\)
\(41\)	27.1140	−	9.38424i	0.661317	−	0.228884i	0.0242656	−	0.999706i	\(-0.492275\pi\)
							0.637051	+	0.770822i	\(0.280154\pi\)
\(43\)	31.2598	−	36.0758i	0.726973	−	0.838971i	−0.265155	−	0.964206i	\(-0.585423\pi\)
							0.992127	+	0.125235i	\(0.0399684\pi\)
\(47\)	69.6142	−	3.31613i	1.48115	−	0.0705561i	0.708647	−	0.705563i	\(-0.249306\pi\)
							0.772507	+	0.635007i	\(0.219003\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	201.3.o.b.17.3	✓	840
3.2	odd	2	inner	201.3.o.b.17.40	yes	840
67.4	even	33	inner	201.3.o.b.71.40	yes	840
201.71	odd	66	inner	201.3.o.b.71.3	yes	840

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
201.3.o.b.17.3	✓	840	1.1	even	1	trivial
201.3.o.b.17.40	yes	840	3.2	odd	2	inner
201.3.o.b.71.3	yes	840	201.71	odd	66	inner
201.3.o.b.71.40	yes	840	67.4	even	33	inner