Embedded newform 201.3.o.b.17.10

• Label: 201.3.o.b.17.10
• 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			17.10
Character	\(\chi\)	\(=\)	201.17
Dual form			201.3.o.b.71.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.238781 - 2.50063i) q^{2} +(2.20075 + 2.03880i) q^{3} +(-2.26841 + 0.437200i) q^{4} +(4.46322 + 0.641713i) q^{5} +(4.57279 - 5.99007i) q^{6} +(5.28893 + 7.42726i) q^{7} +(-1.19593 - 4.07295i) q^{8} +(0.686561 + 8.97377i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 840 q - 16 q^{3} - 126 q^{4} - 25 q^{6} - 34 q^{7} - 24 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.238781	−	2.50063i	−0.119390	−	1.25031i	−0.834384	−	0.551183i	\(-0.814177\pi\)
							0.714994	−	0.699131i	\(-0.246429\pi\)
\(3\)	2.20075	+	2.03880i	0.733582	+	0.679601i
							\(5\)	4.46322	+	0.641713i	0.892643	+	0.128343i	0.573347	−	0.819312i	\(-0.305645\pi\)
0.319296	+	0.947655i	\(0.396554\pi\)
\(7\)	5.28893	+	7.42726i	0.755561	+	1.06104i	0.995910	+	0.0903505i	\(0.0287987\pi\)
							−0.240349	+	0.970687i	\(0.577262\pi\)
\(11\)	−8.86246	+	11.2695i	−0.805678	+	1.02450i	0.193381	+	0.981124i	\(0.438055\pi\)
							−0.999059	+	0.0433792i	\(0.986188\pi\)
\(13\)	3.68079	+	15.1724i	0.283138	+	1.16711i	0.918667	+	0.395032i	\(0.129267\pi\)
							−0.635530	+	0.772077i	\(0.719218\pi\)
\(17\)	5.77683	−	29.9731i	0.339814	−	1.76312i	−0.262569	−	0.964913i	\(-0.584570\pi\)
							0.602382	−	0.798208i	\(-0.294218\pi\)
\(19\)	1.39061	−	1.95285i	0.0731903	−	0.102781i	−0.776358	−	0.630292i	\(-0.782935\pi\)
							0.849548	+	0.527511i	\(0.176875\pi\)
\(23\)	13.9719	−	27.1018i	0.607476	−	1.17834i	−0.361775	−	0.932265i	\(-0.617829\pi\)
							0.969251	−	0.246074i	\(-0.0791405\pi\)
\(29\)	−28.8791	−	16.6734i	−0.995831	−	0.574943i	−0.0888191	−	0.996048i	\(-0.528309\pi\)
							−0.907012	+	0.421104i	\(0.861643\pi\)
\(31\)	10.4738	−	43.1736i	0.337865	−	1.39270i	−0.508943	−	0.860800i	\(-0.669964\pi\)
							0.846808	−	0.531898i	\(-0.178521\pi\)
\(37\)	−6.38046	−	11.0513i	−0.172445	−	0.298683i	0.766829	−	0.641851i	\(-0.221833\pi\)
							−0.939274	+	0.343168i	\(0.888500\pi\)
\(41\)	−15.7196	+	5.44061i	−0.383405	+	0.132698i	−0.511970	−	0.859003i	\(-0.671084\pi\)
							0.128565	+	0.991701i	\(0.458963\pi\)
\(43\)	36.3226	−	41.9185i	0.844712	−	0.974849i	−0.155203	−	0.987883i	\(-0.549603\pi\)
							0.999915	+	0.0130334i	\(0.00414877\pi\)
\(47\)	−39.1277	+	1.86388i	−0.832504	+	0.0396570i	−0.459505	−	0.888175i	\(-0.651973\pi\)
							−0.372998	+	0.927832i	\(0.621670\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	201.3.o.b.17.10	✓	840
3.2	odd	2	inner	201.3.o.b.17.33	yes	840
67.4	even	33	inner	201.3.o.b.71.33	yes	840
201.71	odd	66	inner	201.3.o.b.71.10	yes	840

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
201.3.o.b.17.10	✓	840	1.1	even	1	trivial
201.3.o.b.17.33	yes	840	3.2	odd	2	inner
201.3.o.b.71.10	yes	840	201.71	odd	66	inner
201.3.o.b.71.33	yes	840	67.4	even	33	inner