Embedded newform 201.3.o.b.71.36

• Label: 201.3.o.b.71.36
• Level: $201$
• Weight: $3$
• Character: 201.71
• Analytic conductor: $5.477$
• Analytic rank: $0$
• Dimension: $840$
• 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			71.36
Character	\(\chi\)	\(=\)	201.71
Dual form			201.3.o.b.17.36

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.287750 - 3.01346i) q^{2} +(2.83856 + 0.970872i) q^{3} +(-5.07042 - 0.977244i) q^{4} +(5.16381 - 0.742444i) q^{5} +(3.74248 - 8.27451i) q^{6} +(-0.342776 + 0.481361i) q^{7} +(-0.992495 + 3.38013i) q^{8} +(7.11481 + 5.51175i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 840 q - 16 q^{3} - 126 q^{4} - 25 q^{6} - 34 q^{7} - 24 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.287750	−	3.01346i	0.143875	−	1.50673i	−0.578759	−	0.815499i	\(-0.696463\pi\)
							0.722634	−	0.691231i	\(-0.242931\pi\)
\(3\)	2.83856	+	0.970872i	0.946186	+	0.323624i
							\(5\)	5.16381	−	0.742444i	1.03276	−	0.148489i	0.394967	−	0.918695i	\(-0.370756\pi\)
0.637795	+	0.770207i	\(0.279847\pi\)
\(7\)	−0.342776	+	0.481361i	−0.0489680	+	0.0687659i	−0.838334	−	0.545157i	\(-0.816470\pi\)
							0.789366	+	0.613923i	\(0.210409\pi\)
\(11\)	2.76141	+	3.51141i	0.251037	+	0.319220i	0.895196	−	0.445672i	\(-0.147035\pi\)
							−0.644159	+	0.764891i	\(0.722793\pi\)
\(13\)	0.422770	−	1.74268i	0.0325208	−	0.134052i	−0.953237	−	0.302224i	\(-0.902271\pi\)
							0.985758	+	0.168171i	\(0.0537862\pi\)
\(17\)	0.0442460	+	0.229570i	0.00260270	+	0.0135041i	0.983207	−	0.182495i	\(-0.0584172\pi\)
							−0.980604	+	0.195999i	\(0.937205\pi\)
\(19\)	−1.63055	−	2.28978i	−0.0858182	−	0.120515i	0.769435	−	0.638725i	\(-0.220538\pi\)
							−0.855253	+	0.518210i	\(0.826598\pi\)
\(23\)	−14.7173	−	28.5475i	−0.639881	−	1.24120i	−0.956248	−	0.292559i	\(-0.905493\pi\)
							0.316367	−	0.948637i	\(-0.397537\pi\)
\(29\)	−17.0179	+	9.82531i	−0.586825	+	0.338804i	−0.763841	−	0.645404i	\(-0.776689\pi\)
							0.177016	+	0.984208i	\(0.443356\pi\)
\(31\)	8.20433	+	33.8187i	0.264656	+	1.09093i	0.936621	+	0.350343i	\(0.113935\pi\)
							−0.671966	+	0.740582i	\(0.734550\pi\)
\(37\)	−14.6726	+	25.4138i	−0.396558	+	0.686858i	−0.993299	−	0.115576i	\(-0.963129\pi\)
							0.596741	+	0.802434i	\(0.296462\pi\)
\(41\)	−33.1474	−	11.4724i	−0.808474	−	0.279816i	−0.108603	−	0.994085i	\(-0.534638\pi\)
							−0.699871	+	0.714270i	\(0.746759\pi\)
\(43\)	−20.4312	−	23.5789i	−0.475145	−	0.548346i	0.466691	−	0.884421i	\(-0.345446\pi\)
							−0.941835	+	0.336074i	\(0.890901\pi\)
\(47\)	6.51969	+	0.310571i	0.138717	+	0.00660789i	0.116825	−	0.993152i	\(-0.462728\pi\)
							0.0218913	+	0.999760i	\(0.493031\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	201.3.o.b.71.36	yes	840
3.2	odd	2	inner	201.3.o.b.71.7	yes	840
67.17	even	33	inner	201.3.o.b.17.7	✓	840
201.17	odd	66	inner	201.3.o.b.17.36	yes	840

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
201.3.o.b.17.7	✓	840	67.17	even	33	inner
201.3.o.b.17.36	yes	840	201.17	odd	66	inner
201.3.o.b.71.7	yes	840	3.2	odd	2	inner
201.3.o.b.71.36	yes	840	1.1	even	1	trivial