Embedded newform 201.2.m.b.16.2

• Label: 201.2.m.b.16.2
• Level: $201$
• Weight: $2$
• Character: 201.16
• 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			16.2
Character	\(\chi\)	\(=\)	201.16
Dual form			201.2.m.b.88.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.13211 - 0.410930i) q^{2} +(0.415415 + 0.909632i) q^{3} +(2.52028 + 1.00897i) q^{4} +(-2.10182 + 0.617149i) q^{5} +(-0.511914 - 2.11014i) q^{6} +(-1.23081 - 3.55620i) q^{7} +(-1.30558 - 0.839048i) q^{8} +(-0.654861 + 0.755750i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 120 q + 2 q^{2} - 12 q^{3} + 8 q^{4} + 2 q^{5} - 9 q^{6} + 5 q^{7} - 33 q^{8} - 12 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(68\)	\(136\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{2}{33}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.13211	−	0.410930i	−1.50763	−	0.290571i	−0.632536	−	0.774531i	\(-0.717986\pi\)
							−0.875091	+	0.483959i	\(0.839198\pi\)
\(3\)	0.415415	+	0.909632i	0.239840	+	0.525176i
							\(5\)	−2.10182	+	0.617149i	−0.939961	+	0.275997i	−0.715601	−	0.698509i	\(-0.753847\pi\)
−0.224360	+	0.974506i	\(0.572029\pi\)
\(7\)	−1.23081	−	3.55620i	−0.465203	−	1.34412i	−0.896954	−	0.442123i	\(-0.854225\pi\)
							0.431751	−	0.901993i	\(-0.357896\pi\)
\(11\)	0.914020	−	3.76764i	0.275587	−	1.13599i	−0.650750	−	0.759292i	\(-0.725545\pi\)
							0.926337	−	0.376695i	\(-0.122939\pi\)
\(13\)	5.86983	+	3.02611i	1.62800	+	0.839291i	0.998035	+	0.0626660i	\(0.0199603\pi\)
							0.629963	+	0.776625i	\(0.283070\pi\)
\(17\)	−0.568428	+	0.227564i	−0.137864	+	0.0551925i	−0.439569	−	0.898209i	\(-0.644869\pi\)
							0.301704	+	0.953401i	\(0.402444\pi\)
\(19\)	2.56956	−	7.42426i	0.589498	−	1.70324i	−0.114846	−	0.993383i	\(-0.536637\pi\)
							0.704344	−	0.709859i	\(-0.251241\pi\)
\(23\)	3.46292	−	4.86298i	0.722068	−	1.01400i	−0.276527	−	0.961006i	\(-0.589184\pi\)
							0.998595	−	0.0529961i	\(-0.0168771\pi\)
\(29\)	2.74279	−	4.75065i	0.509323	−	0.882173i	−0.490619	−	0.871374i	\(-0.663229\pi\)
							0.999942	−	0.0107985i	\(-0.00343734\pi\)
\(31\)	−2.97651	+	1.53450i	−0.534596	+	0.275604i	−0.704313	−	0.709889i	\(-0.748745\pi\)
							0.169717	+	0.985493i	\(0.445715\pi\)
\(37\)	−2.78019	−	4.81543i	−0.457060	−	0.791652i	0.541744	−	0.840544i	\(-0.317764\pi\)
							−0.998804	+	0.0488919i	\(0.984431\pi\)
\(41\)	−5.36499	−	4.21908i	−0.837871	−	0.658909i	0.104114	−	0.994565i	\(-0.466799\pi\)
							−0.941984	+	0.335656i	\(0.891042\pi\)
\(43\)	−1.30913	+	9.10517i	−0.199640	+	1.38853i	0.605691	+	0.795700i	\(0.292897\pi\)
							−0.805331	+	0.592825i	\(0.798012\pi\)
\(47\)	5.90962	+	0.564300i	0.862006	+	0.0823116i	0.516688	−	0.856174i	\(-0.327165\pi\)
							0.345319	+	0.938486i	\(0.387771\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	201.2.m.b.16.2	✓	120
3.2	odd	2		603.2.z.d.217.5		120
67.21	even	33	inner	201.2.m.b.88.2	yes	120
201.155	odd	66		603.2.z.d.289.5		120

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
201.2.m.b.16.2	✓	120	1.1	even	1	trivial
201.2.m.b.88.2	yes	120	67.21	even	33	inner
603.2.z.d.217.5		120	3.2	odd	2
603.2.z.d.289.5		120	201.155	odd	66