Embedded newform 201.2.m.b.4.6

• Label: 201.2.m.b.4.6
• Level: $201$
• Weight: $2$
• Character: 201.4
• Analytic conductor: $1.605$
• Analytic rank: $0$
• Dimension: $120$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 201 = 3 \cdot 67 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	201.m (of order \(33\), degree \(20\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.60499308063\)
Analytic rank:	\(0\)
Dimension:	\(120\)
Relative dimension:	\(6\) over \(\Q(\zeta_{33})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{33}]$

Embedding invariants

Embedding label			4.6
Character	\(\chi\)	\(=\)	201.4
Dual form			201.2.m.b.151.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(2.11489 + 0.201947i) q^{2} +(0.841254 + 0.540641i) q^{3} +(2.46811 + 0.475689i) q^{4} +(-0.349341 - 2.42972i) q^{5} +(1.66998 + 1.31328i) q^{6} +(-0.760089 + 1.06740i) q^{7} +(1.04680 + 0.307370i) q^{8} +(0.415415 + 0.909632i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 120 q + 2 q^{2} - 12 q^{3} + 8 q^{4} + 2 q^{5} - 9 q^{6} + 5 q^{7} - 33 q^{8} - 12 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	2.11489	+	0.201947i	1.49545	+	0.142798i	0.810478	−	0.585770i	\(-0.199208\pi\)
							0.684973	+	0.728568i	\(0.259814\pi\)
\(3\)	0.841254	+	0.540641i	0.485698	+	0.312139i
							\(5\)	−0.349341	−	2.42972i	−0.156230	−	1.08660i	−0.905502	−	0.424342i	\(-0.860505\pi\)
0.749272	−	0.662263i	\(-0.230404\pi\)
\(7\)	−0.760089	+	1.06740i	−0.287287	+	0.403438i	−0.932848	−	0.360271i	\(-0.882684\pi\)
							0.645561	+	0.763709i	\(0.276624\pi\)
\(11\)	−4.41890	+	3.47506i	−1.33235	+	1.04777i	−0.338052	+	0.941128i	\(0.609768\pi\)
							−0.994297	+	0.106643i	\(0.965990\pi\)
\(13\)	0.652926	−	2.69140i	0.181089	−	0.746459i	−0.806739	−	0.590908i	\(-0.798770\pi\)
							0.987828	−	0.155551i	\(-0.0497153\pi\)
\(17\)	−0.243866	+	0.0470014i	−0.0591463	+	0.0113995i	−0.218739	−	0.975784i	\(-0.570194\pi\)
							0.159592	+	0.987183i	\(0.448982\pi\)
\(19\)	2.96359	+	4.16178i	0.679894	+	0.954778i	0.999973	+	0.00738662i	\(0.00235126\pi\)
							−0.320079	+	0.947391i	\(0.603709\pi\)
\(23\)	1.55044	−	0.799310i	0.323290	−	0.166668i	−0.288940	−	0.957347i	\(-0.593303\pi\)
							0.612230	+	0.790680i	\(0.290273\pi\)
\(29\)	−0.276924	−	0.479646i	−0.0514235	−	0.0890680i	0.839168	−	0.543872i	\(-0.183042\pi\)
							−0.890591	+	0.454804i	\(0.849709\pi\)
\(31\)	0.996776	+	4.10877i	0.179026	+	0.737956i	0.988553	+	0.150873i	\(0.0482086\pi\)
							−0.809527	+	0.587083i	\(0.800276\pi\)
\(37\)	4.31421	−	7.47243i	0.709252	−	1.22846i	−0.255883	−	0.966708i	\(-0.582366\pi\)
							0.965135	−	0.261752i	\(-0.0843004\pi\)
\(41\)	2.70027	−	7.80191i	0.421711	−	1.21845i	−0.511550	−	0.859254i	\(-0.670928\pi\)
							0.933261	−	0.359200i	\(-0.116950\pi\)
\(43\)	−5.43173	−	6.26855i	−0.828331	−	0.955945i	0.171240	−	0.985229i	\(-0.445223\pi\)
							−0.999571	+	0.0292840i	\(0.990677\pi\)
\(47\)	0.204689	−	4.29696i	0.0298570	−	0.626775i	−0.933504	−	0.358567i	\(-0.883265\pi\)
							0.963361	−	0.268208i	\(-0.0864316\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	201.2.m.b.4.6	✓	120
3.2	odd	2		603.2.z.d.406.1		120
67.17	even	33	inner	201.2.m.b.151.6	yes	120
201.17	odd	66		603.2.z.d.352.1		120

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
201.2.m.b.4.6	✓	120	1.1	even	1	trivial
201.2.m.b.151.6	yes	120	67.17	even	33	inner
603.2.z.d.352.1		120	201.17	odd	66
603.2.z.d.406.1		120	3.2	odd	2