Embedded newform 201.2.m.a.16.2

• Label: 201.2.m.a.16.2
• Level: $201$
• Weight: $2$
• Character: 201.16
• Analytic conductor: $1.605$
• Analytic rank: $0$
• Dimension: $100$
• 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:	\(100\)
Relative dimension:	\(5\) 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.a.88.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.41154 - 0.272052i) q^{2} +(-0.415415 - 0.909632i) q^{3} +(0.0616901 + 0.0246970i) q^{4} +(2.51664 - 0.738953i) q^{5} +(0.338907 + 1.39699i) q^{6} +(0.418058 + 1.20790i) q^{7} +(2.33827 + 1.50272i) q^{8} +(-0.654861 + 0.755750i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 100 q + 2 q^{2} + 10 q^{3} + 6 q^{4} - 2 q^{5} + 9 q^{6} + q^{7} - 45 q^{8} - 10 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.41154	−	0.272052i	−0.998108	−	0.192369i	−0.336068	−	0.941838i	\(-0.609097\pi\)
							−0.662040	+	0.749468i	\(0.730309\pi\)
\(3\)	−0.415415	−	0.909632i	−0.239840	−	0.525176i
							\(5\)	2.51664	−	0.738953i	1.12548	−	0.330470i	0.334548	−	0.942379i	\(-0.391416\pi\)
0.790928	+	0.611909i	\(0.209598\pi\)
\(7\)	0.418058	+	1.20790i	0.158011	+	0.456543i	0.996022	−	0.0891110i	\(-0.0284026\pi\)
							−0.838011	+	0.545654i	\(0.816281\pi\)
\(11\)	1.29598	−	5.34211i	0.390753	−	1.61071i	−0.348520	−	0.937301i	\(-0.613316\pi\)
							0.739273	−	0.673405i	\(-0.235169\pi\)
\(13\)	−3.50759	−	1.80829i	−0.972832	−	0.501530i	−0.102894	−	0.994692i	\(-0.532810\pi\)
							−0.869937	+	0.493163i	\(0.835841\pi\)
\(17\)	1.07338	−	0.429718i	0.260334	−	0.104222i	−0.237824	−	0.971308i	\(-0.576434\pi\)
							0.498158	+	0.867086i	\(0.334010\pi\)
\(19\)	0.613574	−	1.77281i	0.140764	−	0.406710i	−0.852480	−	0.522760i	\(-0.824902\pi\)
							0.993243	+	0.116051i	\(0.0370235\pi\)
\(23\)	4.63285	−	6.50592i	0.966015	−	1.35658i	0.0319447	−	0.999490i	\(-0.489830\pi\)
							0.934070	−	0.357089i	\(-0.116231\pi\)
\(29\)	−2.52190	+	4.36805i	−0.468305	+	0.811127i	−0.999344	−	0.0362200i	\(-0.988468\pi\)
							0.531039	+	0.847347i	\(0.321802\pi\)
\(31\)	−0.754806	+	0.389129i	−0.135567	+	0.0698897i	−0.524676	−	0.851302i	\(-0.675813\pi\)
							0.389109	+	0.921192i	\(0.372783\pi\)
\(37\)	1.87871	+	3.25403i	0.308858	+	0.534958i	0.978113	−	0.208075i	\(-0.0667197\pi\)
							−0.669254	+	0.743033i	\(0.733386\pi\)
\(41\)	−6.63629	−	5.21883i	−1.03641	−	0.815045i	−0.0533996	−	0.998573i	\(-0.517006\pi\)
							−0.983014	+	0.183528i	\(0.941248\pi\)
\(43\)	−0.560253	+	3.89665i	−0.0854378	+	0.594233i	0.901457	+	0.432869i	\(0.142499\pi\)
							−0.986895	+	0.161365i	\(0.948411\pi\)
\(47\)	10.4472	+	0.997583i	1.52387	+	0.145512i	0.823111	−	0.567881i	\(-0.192237\pi\)
							0.700764	+	0.713393i	\(0.252843\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	201.2.m.a.16.2	✓	100
3.2	odd	2		603.2.z.b.217.4		100
67.21	even	33	inner	201.2.m.a.88.2	yes	100
201.155	odd	66		603.2.z.b.289.4		100

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
201.2.m.a.16.2	✓	100	1.1	even	1	trivial
201.2.m.a.88.2	yes	100	67.21	even	33	inner
603.2.z.b.217.4		100	3.2	odd	2
603.2.z.b.289.4		100	201.155	odd	66