Embedded newform 201.3.n.a.13.8

• Label: 201.3.n.a.13.8
• Level: $201$
• Weight: $3$
• Character: 201.13
• Analytic conductor: $5.477$
• Analytic rank: $0$
• Dimension: $220$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 201 = 3 \cdot 67 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	201.n (of order \(66\), degree \(20\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.47685331364\)
Analytic rank:	\(0\)
Dimension:	\(220\)
Relative dimension:	\(11\) over \(\Q(\zeta_{66})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{66}]$

Embedding invariants

Embedding label			13.8
Character	\(\chi\)	\(=\)	201.13
Dual form			201.3.n.a.31.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.953733 + 1.21277i) q^{2} +(1.30900 + 1.13425i) q^{3} +(0.381831 - 1.57393i) q^{4} +(5.07833 + 7.90204i) q^{5} +(-0.127154 + 2.66929i) q^{6} +(0.437890 - 1.09380i) q^{7} +(7.88672 - 3.60174i) q^{8} +(0.426945 + 2.96946i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 220 q - 26 q^{4} + 33 q^{6} + 15 q^{7} - 33 q^{8} + 66 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(68\)	\(136\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{19}{66}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.953733	+	1.21277i	0.476867	+	0.606385i	0.963660	−	0.267132i	\(-0.0860760\pi\)
							−0.486793	+	0.873517i	\(0.661834\pi\)
\(3\)	1.30900	+	1.13425i	0.436332	+	0.378084i
							\(5\)	5.07833	+	7.90204i	1.01567	+	1.58041i	0.796425	+	0.604738i	\(0.206722\pi\)
0.219242	+	0.975671i	\(0.429642\pi\)
\(7\)	0.437890	−	1.09380i	0.0625558	−	0.156257i	−0.893797	−	0.448472i	\(-0.851968\pi\)
							0.956352	+	0.292216i	\(0.0943925\pi\)
\(11\)	−11.2137	+	0.534174i	−1.01943	+	0.0485613i	−0.550634	−	0.834747i	\(-0.685614\pi\)
							−0.468793	+	0.883308i	\(0.655311\pi\)
\(13\)	−2.32996	−	24.4005i	−0.179228	−	1.87696i	−0.421606	−	0.906779i	\(-0.638533\pi\)
							0.242378	−	0.970182i	\(-0.422073\pi\)
\(17\)	6.60260	+	27.2163i	0.388388	+	1.60096i	0.745240	+	0.666796i	\(0.232335\pi\)
							−0.356852	+	0.934161i	\(0.616150\pi\)
\(19\)	−1.71611	+	0.687025i	−0.0903214	+	0.0361592i	−0.416383	−	0.909189i	\(-0.636702\pi\)
							0.326062	+	0.945348i	\(0.394278\pi\)
\(23\)	2.77099	+	0.534065i	0.120478	+	0.0232202i	0.249134	−	0.968469i	\(-0.419854\pi\)
							−0.128656	+	0.991689i	\(0.541066\pi\)
\(29\)	−4.33177	+	7.50284i	−0.149371	+	0.258719i	−0.930995	−	0.365031i	\(-0.881058\pi\)
							0.781624	+	0.623750i	\(0.214392\pi\)
\(31\)	4.59014	−	48.0701i	0.148069	−	1.55065i	−0.550335	−	0.834944i	\(-0.685500\pi\)
							0.698404	−	0.715704i	\(-0.253894\pi\)
\(37\)	2.10441	+	3.64494i	0.0568758	+	0.0985118i	0.893061	−	0.449935i	\(-0.148553\pi\)
							−0.836186	+	0.548447i	\(0.815219\pi\)
\(41\)	−36.3601	−	38.1333i	−0.886831	−	0.930081i	0.111137	−	0.993805i	\(-0.464551\pi\)
							−0.997968	+	0.0637239i	\(0.979702\pi\)
\(43\)	−2.06027	+	7.01665i	−0.0479134	+	0.163178i	−0.979974	−	0.199123i	\(-0.936191\pi\)
							0.932061	+	0.362301i	\(0.118009\pi\)
\(47\)	−23.7117	−	68.5106i	−0.504505	−	1.45767i	−0.853771	−	0.520648i	\(-0.825690\pi\)
							0.349266	−	0.937024i	\(-0.386431\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	201.3.n.a.13.8	✓	220
67.31	odd	66	inner	201.3.n.a.31.8	yes	220

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
201.3.n.a.13.8	✓	220	1.1	even	1	trivial
201.3.n.a.31.8	yes	220	67.31	odd	66	inner