Embedded newform 201.3.o.b.17.20

• Label: 201.3.o.b.17.20
• Level: $201$
• Weight: $3$
• Character: 201.17
• Analytic conductor: $5.477$
• Analytic rank: $0$
• Dimension: $840$
• CM: no
• Inner twists: $4$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(5.47685331364\)
Analytic rank:	\(0\)
Dimension:	\(840\)
Relative dimension:	\(42\) over \(\Q(\zeta_{66})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{66}]$

Embedding invariants

Embedding label			17.20
Character	\(\chi\)	\(=\)	201.17
Dual form			201.3.o.b.71.20

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.0661166 - 0.692405i) q^{2} +(-0.991594 + 2.83139i) q^{3} +(3.45266 - 0.665446i) q^{4} +(8.44666 + 1.21445i) q^{5} +(2.02602 + 0.499383i) q^{6} +(3.65289 + 5.12977i) q^{7} +(-1.47288 - 5.01616i) q^{8} +(-7.03348 - 5.61517i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 840 q - 16 q^{3} - 126 q^{4} - 25 q^{6} - 34 q^{7} - 24 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(68\)	\(136\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{32}{33}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.0661166	−	0.692405i	−0.0330583	−	0.346202i	−0.996835	−	0.0794932i	\(-0.974670\pi\)
							0.963777	−	0.266709i	\(-0.0859363\pi\)
\(3\)	−0.991594	+	2.83139i	−0.330531	+	0.943795i
							\(5\)	8.44666	+	1.21445i	1.68933	+	0.242889i	0.918860	−	0.394584i	\(-0.129111\pi\)
0.770473	+	0.637473i	\(0.220020\pi\)
\(7\)	3.65289	+	5.12977i	0.521842	+	0.732824i	0.988435	−	0.151648i	\(-0.0484580\pi\)
							−0.466593	+	0.884472i	\(0.654519\pi\)
\(11\)	−7.24503	+	9.21281i	−0.658640	+	0.837528i	−0.994539	−	0.104370i	\(-0.966717\pi\)
							0.335899	+	0.941898i	\(0.390960\pi\)
\(13\)	−4.81901	−	19.8642i	−0.370693	−	1.52802i	−0.785937	−	0.618307i	\(-0.787819\pi\)
							0.415244	−	0.909710i	\(-0.363696\pi\)
\(17\)	−4.18740	+	21.7263i	−0.246318	+	1.27802i	0.622396	+	0.782703i	\(0.286160\pi\)
							−0.868713	+	0.495315i	\(0.835053\pi\)
\(19\)	10.2651	−	14.4153i	0.540270	−	0.758702i	−0.450717	−	0.892667i	\(-0.648831\pi\)
							0.990986	+	0.133965i	\(0.0427709\pi\)
\(23\)	−0.274574	+	0.532600i	−0.0119380	+	0.0231565i	−0.894729	−	0.446609i	\(-0.852631\pi\)
							0.882791	+	0.469766i	\(0.155662\pi\)
\(29\)	−5.43332	−	3.13693i	−0.187356	−	0.108170i	0.403388	−	0.915029i	\(-0.367832\pi\)
							−0.590744	+	0.806859i	\(0.701166\pi\)
\(31\)	−12.3536	+	50.9222i	−0.398503	+	1.64265i	0.320211	+	0.947346i	\(0.396246\pi\)
							−0.718714	+	0.695306i	\(0.755269\pi\)
\(37\)	−24.7379	−	42.8473i	−0.668593	−	1.15804i	−0.978298	−	0.207204i	\(-0.933564\pi\)
							0.309705	−	0.950833i	\(-0.399770\pi\)
\(41\)	−45.3021	+	15.6792i	−1.10493	+	0.382420i	−0.817818	−	0.575477i	\(-0.804816\pi\)
							−0.287112	+	0.957897i	\(0.592695\pi\)
\(43\)	15.8533	−	18.2957i	0.368681	−	0.425481i	−0.540848	−	0.841120i	\(-0.681897\pi\)
							0.909530	+	0.415639i	\(0.136442\pi\)
\(47\)	−17.5089	+	0.834051i	−0.372530	+	0.0177458i	−0.233012	−	0.972474i	\(-0.574858\pi\)
							−0.139517	+	0.990220i	\(0.544555\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	201.3.o.b.17.20	✓	840
3.2	odd	2	inner	201.3.o.b.17.23	yes	840
67.4	even	33	inner	201.3.o.b.71.23	yes	840
201.71	odd	66	inner	201.3.o.b.71.20	yes	840

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
201.3.o.b.17.20	✓	840	1.1	even	1	trivial
201.3.o.b.17.23	yes	840	3.2	odd	2	inner
201.3.o.b.71.20	yes	840	201.71	odd	66	inner
201.3.o.b.71.23	yes	840	67.4	even	33	inner