Embedded newform 201.3.n.a.13.5

• Label: 201.3.n.a.13.5
• 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			13.5
Character	\(\chi\)	\(=\)	201.13
Dual form			201.3.n.a.31.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.377872 - 0.480504i) q^{2} +(1.30900 + 1.13425i) q^{3} +(0.854939 - 3.52411i) q^{4} +(-1.13061 - 1.75926i) q^{5} +(0.0503788 - 1.05758i) q^{6} +(3.34306 - 8.35055i) q^{7} +(-4.24059 + 1.93661i) q^{8} +(0.426945 + 2.96946i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 220 q - 26 q^{4} + 33 q^{6} + 15 q^{7} - 33 q^{8} + 66 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.377872	−	0.480504i	−0.188936	−	0.240252i	0.682352	−	0.731024i	\(-0.260957\pi\)
							−0.871288	+	0.490772i	\(0.836715\pi\)
\(3\)	1.30900	+	1.13425i	0.436332	+	0.378084i
							\(5\)	−1.13061	−	1.75926i	−0.226121	−	0.351852i	0.709592	−	0.704613i	\(-0.248879\pi\)
−0.935714	+	0.352761i	\(0.885243\pi\)
\(7\)	3.34306	−	8.35055i	0.477580	−	1.19294i	−0.472612	−	0.881271i	\(-0.656689\pi\)
							0.950192	−	0.311666i	\(-0.100887\pi\)
\(11\)	−11.6625	+	0.555553i	−1.06023	+	0.0505049i	−0.570442	−	0.821338i	\(-0.693228\pi\)
							−0.489786	+	0.871843i	\(0.662925\pi\)
\(13\)	0.00965924	+	0.101156i	0.000743019	+	0.00778124i	0.995836	−	0.0911647i	\(-0.0290590\pi\)
							−0.995093	+	0.0989459i	\(0.968453\pi\)
\(17\)	−3.65928	−	15.0838i	−0.215252	−	0.887280i	−0.972265	−	0.233880i	\(-0.924858\pi\)
							0.757014	−	0.653399i	\(-0.226658\pi\)
\(19\)	−0.247502	+	0.0990850i	−0.0130264	+	0.00521500i	−0.378167	−	0.925738i	\(-0.623445\pi\)
							0.365140	+	0.930953i	\(0.381021\pi\)
\(23\)	36.5234	+	7.03930i	1.58797	+	0.306056i	0.905334	−	0.424700i	\(-0.139621\pi\)
							0.682638	+	0.730757i	\(0.260833\pi\)
\(29\)	−3.12808	+	5.41800i	−0.107865	+	0.186828i	−0.914905	−	0.403669i	\(-0.867735\pi\)
							0.807040	+	0.590497i	\(0.201068\pi\)
\(31\)	4.29971	−	45.0286i	0.138700	−	1.45254i	−0.611458	−	0.791277i	\(-0.709417\pi\)
							0.750158	−	0.661259i	\(-0.229977\pi\)
\(37\)	26.4247	+	45.7689i	0.714181	+	1.23700i	0.963275	+	0.268518i	\(0.0865338\pi\)
							−0.249094	+	0.968479i	\(0.580133\pi\)
\(41\)	−20.9466	−	21.9682i	−0.510892	−	0.535809i	0.416877	−	0.908963i	\(-0.363125\pi\)
							−0.927769	+	0.373154i	\(0.878276\pi\)
\(43\)	4.59768	−	15.6582i	0.106923	−	0.364145i	−0.888598	−	0.458686i	\(-0.848320\pi\)
							0.995521	+	0.0945410i	\(0.0301384\pi\)
\(47\)	11.4203	+	32.9968i	0.242985	+	0.702059i	0.998878	+	0.0473638i	\(0.0150820\pi\)
							−0.755893	+	0.654696i	\(0.772797\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	201.3.n.a.13.5	✓	220
67.31	odd	66	inner	201.3.n.a.31.5	yes	220

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
201.3.n.a.13.5	✓	220	1.1	even	1	trivial
201.3.n.a.31.5	yes	220	67.31	odd	66	inner