Embedded newform 201.2.m.b.16.1

• Label: 201.2.m.b.16.1
• 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			16.1
Character	\(\chi\)	\(=\)	201.16
Dual form			201.2.m.b.88.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.69052 - 0.518556i) q^{2} +(0.415415 + 0.909632i) q^{3} +(5.11329 + 2.04705i) q^{4} +(4.04867 - 1.18880i) q^{5} +(-0.645989 - 2.66280i) q^{6} +(0.854798 + 2.46978i) q^{7} +(-8.08577 - 5.19641i) q^{8} +(-0.654861 + 0.755750i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 120 q + 2 q^{2} - 12 q^{3} + 8 q^{4} + 2 q^{5} - 9 q^{6} + 5 q^{7} - 33 q^{8} - 12 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.69052	−	0.518556i	−1.90249	−	0.366675i	−0.904303	−	0.426892i	\(-0.859609\pi\)
							−0.998185	+	0.0602171i	\(0.980821\pi\)
\(3\)	0.415415	+	0.909632i	0.239840	+	0.525176i
							\(5\)	4.04867	−	1.18880i	1.81062	−	0.531645i	0.811974	−	0.583694i	\(-0.198393\pi\)
0.998645	+	0.0520488i	\(0.0165751\pi\)
\(7\)	0.854798	+	2.46978i	0.323083	+	0.933488i	0.983233	+	0.182354i	\(0.0583717\pi\)
							−0.660150	+	0.751134i	\(0.729507\pi\)
\(11\)	0.0680798	−	0.280629i	0.0205268	−	0.0846128i	−0.960627	−	0.277842i	\(-0.910381\pi\)
							0.981154	+	0.193229i	\(0.0618960\pi\)
\(13\)	−0.871878	−	0.449484i	−0.241815	−	0.124665i	0.333059	−	0.942906i	\(-0.391919\pi\)
							−0.574875	+	0.818241i	\(0.694949\pi\)
\(17\)	−1.17426	+	0.470104i	−0.284800	+	0.114017i	−0.509664	−	0.860373i	\(-0.670230\pi\)
							0.224864	+	0.974390i	\(0.427806\pi\)
\(19\)	1.09978	−	3.17761i	0.252307	−	0.728994i	−0.745723	−	0.666256i	\(-0.767896\pi\)
							0.998030	−	0.0627374i	\(-0.0199831\pi\)
\(23\)	−2.86745	+	4.02677i	−0.597904	+	0.839639i	−0.997011	−	0.0772656i	\(-0.975381\pi\)
							0.399106	+	0.916905i	\(0.369320\pi\)
\(29\)	−0.967171	+	1.67519i	−0.179599	+	0.311075i	−0.941743	−	0.336333i	\(-0.890813\pi\)
							0.762144	+	0.647407i	\(0.224147\pi\)
\(31\)	1.23947	−	0.638989i	0.222615	−	0.114766i	−0.343312	−	0.939221i	\(-0.611549\pi\)
							0.565927	+	0.824456i	\(0.308519\pi\)
\(37\)	−3.60938	−	6.25163i	−0.593379	−	1.02776i	−0.993774	−	0.111419i	\(-0.964460\pi\)
							0.400395	−	0.916343i	\(-0.368873\pi\)
\(41\)	1.99161	+	1.56622i	0.311038	+	0.244603i	0.761459	−	0.648213i	\(-0.224484\pi\)
							−0.450421	+	0.892816i	\(0.648726\pi\)
\(43\)	−1.57416	+	10.9485i	−0.240058	+	1.66964i	0.411783	+	0.911282i	\(0.364906\pi\)
							−0.651841	+	0.758356i	\(0.726003\pi\)
\(47\)	3.51512	+	0.335654i	0.512733	+	0.0489601i	0.348216	−	0.937414i	\(-0.386787\pi\)
							0.164517	+	0.986374i	\(0.447393\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	201.2.m.b.16.1	✓	120
3.2	odd	2		603.2.z.d.217.6		120
67.21	even	33	inner	201.2.m.b.88.1	yes	120
201.155	odd	66		603.2.z.d.289.6		120

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
201.2.m.b.16.1	✓	120	1.1	even	1	trivial
201.2.m.b.88.1	yes	120	67.21	even	33	inner
603.2.z.d.217.6		120	3.2	odd	2
603.2.z.d.289.6		120	201.155	odd	66