Embedded newform 201.2.m.b.16.3

• Label: 201.2.m.b.16.3
• 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			16.3
Character	\(\chi\)	\(=\)	201.16
Dual form			201.2.m.b.88.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.774865 - 0.149343i) q^{2} +(0.415415 + 0.909632i) q^{3} +(-1.27862 - 0.511883i) q^{4} +(1.13149 - 0.332236i) q^{5} +(-0.186043 - 0.766882i) q^{6} +(0.783813 + 2.26468i) q^{7} +(2.24202 + 1.44086i) q^{8} +(-0.654861 + 0.755750i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 120 q + 2 q^{2} - 12 q^{3} + 8 q^{4} + 2 q^{5} - 9 q^{6} + 5 q^{7} - 33 q^{8} - 12 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.774865	−	0.149343i	−0.547912	−	0.105601i	−0.0922198	−	0.995739i	\(-0.529396\pi\)
							−0.455693	+	0.890137i	\(0.650608\pi\)
\(3\)	0.415415	+	0.909632i	0.239840	+	0.525176i
							\(5\)	1.13149	−	0.332236i	0.506018	−	0.148580i	−0.0187521	−	0.999824i	\(-0.505969\pi\)
0.524770	+	0.851244i	\(0.324151\pi\)
\(7\)	0.783813	+	2.26468i	0.296254	+	0.855969i	0.990656	+	0.136382i	\(0.0435475\pi\)
							−0.694403	+	0.719587i	\(0.744331\pi\)
\(11\)	0.0664521	−	0.273919i	0.0200361	−	0.0825898i	−0.960916	−	0.276841i	\(-0.910713\pi\)
							0.980952	+	0.194251i	\(0.0622277\pi\)
\(13\)	3.86961	+	1.99492i	1.07324	+	0.553292i	0.901911	−	0.431923i	\(-0.142165\pi\)
							0.171326	+	0.985214i	\(0.445195\pi\)
\(17\)	3.40210	−	1.36200i	0.825131	−	0.330332i	0.0796192	−	0.996825i	\(-0.474630\pi\)
							0.745511	+	0.666493i	\(0.232205\pi\)
\(19\)	−2.42725	+	7.01307i	−0.556849	+	1.60891i	0.217437	+	0.976074i	\(0.430230\pi\)
							−0.774286	+	0.632835i	\(0.781891\pi\)
\(23\)	2.02497	−	2.84367i	0.422235	−	0.592946i	−0.547591	−	0.836746i	\(-0.684455\pi\)
							0.969825	+	0.243801i	\(0.0783942\pi\)
\(29\)	2.61360	−	4.52689i	0.485333	−	0.840622i	−0.514525	−	0.857476i	\(-0.672032\pi\)
							0.999858	+	0.0168535i	\(0.00536488\pi\)
\(31\)	−0.231120	+	0.119151i	−0.0415104	+	0.0214001i	−0.478857	−	0.877893i	\(-0.658949\pi\)
							0.437346	+	0.899293i	\(0.355918\pi\)
\(37\)	−1.55267	−	2.68930i	−0.255257	−	0.442118i	0.709708	−	0.704495i	\(-0.248827\pi\)
							−0.964965	+	0.262378i	\(0.915493\pi\)
\(41\)	−0.0276262	−	0.0217255i	−0.00431449	−	0.00339295i	0.615999	−	0.787747i	\(-0.288752\pi\)
							−0.620314	+	0.784354i	\(0.712995\pi\)
\(43\)	1.21733	−	8.46671i	0.185641	−	1.29116i	−0.657495	−	0.753459i	\(-0.728384\pi\)
							0.843135	−	0.537701i	\(-0.180707\pi\)
\(47\)	−13.1042	−	1.25130i	−1.91144	−	0.182520i	−0.929127	−	0.369761i	\(-0.879439\pi\)
							−0.982314	+	0.187241i	\(0.940046\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	201.2.m.b.16.3	✓	120
3.2	odd	2		603.2.z.d.217.4		120
67.21	even	33	inner	201.2.m.b.88.3	yes	120
201.155	odd	66		603.2.z.d.289.4		120

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
201.2.m.b.16.3	✓	120	1.1	even	1	trivial
201.2.m.b.88.3	yes	120	67.21	even	33	inner
603.2.z.d.217.4		120	3.2	odd	2
603.2.z.d.289.4		120	201.155	odd	66