Embedded newform 201.4.m.b.4.8

• Label: 201.4.m.b.4.8
• Level: $201$
• Weight: $4$
• Character: 201.4
• Analytic conductor: $11.859$
• Analytic rank: $0$
• Dimension: $360$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 201 = 3 \cdot 67 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	201.m (of order \(33\), degree \(20\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(11.8593839112\)
Analytic rank:	\(0\)
Dimension:	\(360\)
Relative dimension:	\(18\) over \(\Q(\zeta_{33})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{33}]$

Embedding invariants

Embedding label			4.8
Character	\(\chi\)	\(=\)	201.4
Dual form			201.4.m.b.151.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.65101 - 0.157652i) q^{2} +(-2.52376 - 1.62192i) q^{3} +(-5.15445 - 0.993439i) q^{4} +(0.854289 + 5.94171i) q^{5} +(3.91105 + 3.07569i) q^{6} +(2.57806 - 3.62038i) q^{7} +(21.0841 + 6.19086i) q^{8} +(3.73874 + 8.18669i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 360 q - 2 q^{2} + 108 q^{3} + 90 q^{4} + 4 q^{5} - 27 q^{6} - 22 q^{7} + 183 q^{8} - 324 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.65101	−	0.157652i	−0.583720	−	0.0557385i	−0.200981	−	0.979595i	\(-0.564413\pi\)
							−0.382739	+	0.923857i	\(0.625019\pi\)
\(3\)	−2.52376	−	1.62192i	−0.485698	−	0.312139i
							\(5\)	0.854289	+	5.94171i	0.0764099	+	0.531443i	0.991693	+	0.128630i	\(0.0410581\pi\)
−0.915283	+	0.402812i	\(0.868033\pi\)
\(7\)	2.57806	−	3.62038i	0.139202	−	0.195482i	−0.739090	−	0.673607i	\(-0.764744\pi\)
							0.878292	+	0.478125i	\(0.158683\pi\)
\(11\)	−27.0955	+	21.3081i	−0.742691	+	0.584059i	−0.916175	−	0.400778i	\(-0.868740\pi\)
							0.173484	+	0.984837i	\(0.444497\pi\)
\(13\)	−8.58744	+	35.3979i	−0.183210	+	0.755201i	0.803848	+	0.594835i	\(0.202783\pi\)
							−0.987058	+	0.160366i	\(0.948733\pi\)
\(17\)	50.3473	−	9.70364i	0.718295	−	0.138440i	0.183009	−	0.983111i	\(-0.441416\pi\)
							0.535285	+	0.844671i	\(0.320204\pi\)
\(19\)	−62.6379	−	87.9626i	−0.756322	−	1.06211i	−0.995832	−	0.0912115i	\(-0.970926\pi\)
							0.239510	−	0.970894i	\(-0.423013\pi\)
\(23\)	77.9119	−	40.1664i	0.706337	−	0.364142i	−0.0673331	−	0.997731i	\(-0.521449\pi\)
							0.773670	+	0.633588i	\(0.218419\pi\)
\(29\)	−98.2432	−	170.162i	−0.629079	−	1.08960i	−0.987737	−	0.156128i	\(-0.950099\pi\)
							0.358658	−	0.933469i	\(-0.383235\pi\)
\(31\)	−62.0090	−	255.605i	−0.359263	−	1.48090i	−0.809036	−	0.587759i	\(-0.800010\pi\)
							0.449773	−	0.893143i	\(-0.351505\pi\)
\(37\)	−145.300	+	251.666i	−0.645597	+	1.11821i	0.338566	+	0.940943i	\(0.390058\pi\)
							−0.984163	+	0.177264i	\(0.943275\pi\)
\(41\)	112.058	−	323.770i	0.426842	−	1.23328i	−0.502668	−	0.864480i	\(-0.667648\pi\)
							0.929509	−	0.368798i	\(-0.120231\pi\)
\(43\)	44.2513	+	51.0687i	0.156936	+	0.181114i	0.828773	−	0.559586i	\(-0.189040\pi\)
							−0.671836	+	0.740700i	\(0.734494\pi\)
\(47\)	20.0607	−	421.127i	0.0622587	−	1.30697i	−0.724313	−	0.689471i	\(-0.757843\pi\)
							0.786572	−	0.617499i	\(-0.211854\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	201.4.m.b.4.8	✓	360
67.17	even	33	inner	201.4.m.b.151.8	yes	360

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
201.4.m.b.4.8	✓	360	1.1	even	1	trivial
201.4.m.b.151.8	yes	360	67.17	even	33	inner