Embedded newform 201.4.j.b.161.6

• Label: 201.4.j.b.161.6
• Level: $201$
• Weight: $4$
• Character: 201.161
• Analytic conductor: $11.859$
• Analytic rank: $0$
• Dimension: $640$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 201 = 3 \cdot 67 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	201.j (of order \(22\), degree \(10\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(11.8593839112\)
Analytic rank:	\(0\)
Dimension:	\(640\)
Relative dimension:	\(64\) over \(\Q(\zeta_{22})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{22}]$

Embedding invariants

Embedding label			161.6
Character	\(\chi\)	\(=\)	201.161
Dual form			201.4.j.b.5.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-3.20732 - 3.70144i) q^{2} +(5.16210 - 0.593913i) q^{3} +(-2.27527 + 15.8248i) q^{4} +(-8.51736 - 2.50092i) q^{5} +(-18.7548 - 17.2024i) q^{6} +(14.3505 - 12.4348i) q^{7} +(32.9106 - 21.1503i) q^{8} +(26.2945 - 6.13167i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 640 q - 11 q^{3} - 290 q^{4} + 35 q^{6} - 22 q^{7} - 33 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(68\)	\(136\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{17}{22}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−3.20732	−	3.70144i	−1.13396	−	1.30866i	−0.945147	−	0.326645i	\(-0.894082\pi\)
							−0.188811	−	0.982013i	\(-0.560463\pi\)
\(3\)	5.16210	−	0.593913i	0.993446	−	0.114299i
							\(5\)	−8.51736	−	2.50092i	−0.761815	−	0.223689i	−0.122327	−	0.992490i	\(-0.539036\pi\)
−0.639489	+	0.768801i	\(0.720854\pi\)
\(7\)	14.3505	−	12.4348i	0.774856	−	0.671417i	−0.174837	−	0.984597i	\(-0.555940\pi\)
							0.949694	+	0.313180i	\(0.101394\pi\)
\(11\)	34.2415	+	10.0542i	0.938563	+	0.275587i	0.715018	−	0.699106i	\(-0.246419\pi\)
							0.223546	+	0.974693i	\(0.428237\pi\)
\(13\)	−1.35275	+	2.10493i	−0.0288605	+	0.0449078i	−0.855384	−	0.517994i	\(-0.826679\pi\)
							0.826524	+	0.562902i	\(0.190315\pi\)
\(17\)	−9.48322	+	1.36348i	−0.135295	+	0.0194525i	−0.209630	−	0.977781i	\(-0.567226\pi\)
							0.0743345	+	0.997233i	\(0.476317\pi\)
\(19\)	79.8001	−	92.0943i	0.963548	−	1.11199i	−0.0301100	−	0.999547i	\(-0.509586\pi\)
							0.993658	−	0.112447i	\(-0.0358688\pi\)
\(23\)	−19.1730	−	8.75600i	−0.173819	−	0.0793805i	0.326605	−	0.945161i	\(-0.394095\pi\)
							−0.500424	+	0.865780i	\(0.666823\pi\)
\(29\)			82.2888i			0.526919i	0.964671	+	0.263459i	\(0.0848635\pi\)
							−0.964671	+	0.263459i	\(0.915137\pi\)
\(31\)	−159.090	−	247.548i	−0.921720	−	1.43423i	−0.900698	−	0.434446i	\(-0.856944\pi\)
							−0.0210226	−	0.999779i	\(-0.506692\pi\)
\(37\)	−96.1312			−0.427132			−0.213566	−	0.976929i	\(-0.568508\pi\)
							−0.213566	+	0.976929i	\(0.568508\pi\)
\(41\)	−37.3929	−	260.073i	−0.142434	−	0.990649i	−0.928188	−	0.372111i	\(-0.878634\pi\)
							0.785755	−	0.618538i	\(-0.212275\pi\)
\(43\)	216.127	−	31.0743i	0.766489	−	0.110204i	0.252030	−	0.967719i	\(-0.418902\pi\)
							0.514459	+	0.857515i	\(0.327993\pi\)
\(47\)	−194.524	−	88.8363i	−0.603709	−	0.275705i	0.0900211	−	0.995940i	\(-0.471307\pi\)
							−0.693730	+	0.720235i	\(0.744034\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	201.4.j.b.161.6	yes	640
3.2	odd	2	inner	201.4.j.b.161.59	yes	640
67.5	odd	22	inner	201.4.j.b.5.59	yes	640
201.5	even	22	inner	201.4.j.b.5.6	✓	640

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
201.4.j.b.5.6	✓	640	201.5	even	22	inner
201.4.j.b.5.59	yes	640	67.5	odd	22	inner
201.4.j.b.161.6	yes	640	1.1	even	1	trivial
201.4.j.b.161.59	yes	640	3.2	odd	2	inner