Embedded newform 201.2.i.b.40.2

• Label: 201.2.i.b.40.2
• Level: $201$
• Weight: $2$
• Character: 201.40
• Analytic conductor: $1.605$
• Analytic rank: $0$
• Dimension: $70$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 201 = 3 \cdot 67 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	201.i (of order \(11\), degree \(10\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.60499308063\)
Analytic rank:	\(0\)
Dimension:	\(70\)
Relative dimension:	\(7\) over \(\Q(\zeta_{11})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{11}]$

Embedding invariants

Embedding label			40.2
Character	\(\chi\)	\(=\)	201.40
Dual form			201.2.i.b.196.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.24364 - 1.43523i) q^{2} +(0.415415 - 0.909632i) q^{3} +(-0.228633 + 1.59018i) q^{4} +(-2.11340 - 0.620551i) q^{5} +(-1.82216 + 0.535034i) q^{6} +(-0.191393 - 0.220879i) q^{7} +(-0.628614 + 0.403985i) q^{8} +(-0.654861 - 0.755750i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 70 q - 2 q^{2} - 7 q^{3} - 12 q^{4} - 2 q^{5} + 9 q^{6} - 10 q^{7} + 15 q^{8} - 7 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(68\)	\(136\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{3}{11}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.24364	−	1.43523i	−0.879384	−	1.01486i	−0.999755	−	0.0221346i	\(-0.992954\pi\)
							0.120371	−	0.992729i	\(-0.461592\pi\)
\(3\)	0.415415	−	0.909632i	0.239840	−	0.525176i
							\(5\)	−2.11340	−	0.620551i	−0.945143	−	0.277519i	−0.227380	−	0.973806i	\(-0.573016\pi\)
−0.717763	+	0.696287i	\(0.754834\pi\)
\(7\)	−0.191393	−	0.220879i	−0.0723397	−	0.0834844i	0.718426	−	0.695603i	\(-0.244863\pi\)
							−0.790766	+	0.612119i	\(0.790317\pi\)
\(11\)	−2.71523	−	0.797265i	−0.818674	−	0.240384i	−0.154529	−	0.987988i	\(-0.549386\pi\)
							−0.664145	+	0.747604i	\(0.731204\pi\)
\(13\)	−1.84738	−	1.18724i	−0.512370	−	0.329280i	0.258777	−	0.965937i	\(-0.416680\pi\)
							−0.771147	+	0.636657i	\(0.780317\pi\)
\(17\)	0.0212108	+	0.147525i	0.00514439	+	0.0357800i	0.992232	−	0.124402i	\(-0.0397011\pi\)
							−0.987088	+	0.160182i	\(0.948792\pi\)
\(19\)	−0.530717	+	0.612480i	−0.121755	+	0.140513i	−0.813354	−	0.581769i	\(-0.802361\pi\)
							0.691599	+	0.722281i	\(0.256906\pi\)
\(23\)	1.10875	−	2.42782i	0.231190	−	0.506236i	−0.758111	−	0.652126i	\(-0.773877\pi\)
							0.989301	+	0.145890i	\(0.0466045\pi\)
\(29\)	1.28347			0.238334			0.119167	−	0.992874i	\(-0.461978\pi\)
							0.119167	+	0.992874i	\(0.461978\pi\)
\(31\)	−3.52653	+	2.26636i	−0.633383	+	0.407051i	−0.817561	−	0.575842i	\(-0.804674\pi\)
							0.184177	+	0.982893i	\(0.441038\pi\)
\(37\)	0.898599			0.147729			0.0738644	−	0.997268i	\(-0.476467\pi\)
							0.0738644	+	0.997268i	\(0.476467\pi\)
\(41\)	−0.327266	−	2.27618i	−0.0511103	−	0.355480i	−0.999288	−	0.0377385i	\(-0.987985\pi\)
							0.948177	−	0.317742i	\(-0.102924\pi\)
\(43\)	−1.70078	−	11.8292i	−0.259367	−	1.80393i	−0.537361	−	0.843352i	\(-0.680579\pi\)
							0.277995	−	0.960583i	\(-0.410330\pi\)
\(47\)	4.70248	−	10.2970i	0.685927	−	1.50197i	−0.170312	−	0.985390i	\(-0.554478\pi\)
							0.856239	−	0.516580i	\(-0.172795\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	201.2.i.b.40.2	✓	70
3.2	odd	2		603.2.u.e.442.6		70
67.62	even	11	inner	201.2.i.b.196.2	yes	70
201.62	odd	22		603.2.u.e.397.6		70

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
201.2.i.b.40.2	✓	70	1.1	even	1	trivial
201.2.i.b.196.2	yes	70	67.62	even	11	inner
603.2.u.e.397.6		70	201.62	odd	22
603.2.u.e.442.6		70	3.2	odd	2