Embedded newform 201.3.o.b.17.7

• Label: 201.3.o.b.17.7
• 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			17.7
Character	\(\chi\)	\(=\)	201.17
Dual form			201.3.o.b.71.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.287750 - 3.01346i) q^{2} +(2.06232 - 2.17873i) q^{3} +(-5.07042 + 0.977244i) q^{4} +(-5.16381 - 0.742444i) q^{5} +(-7.15894 - 5.58777i) q^{6} +(-0.342776 - 0.481361i) q^{7} +(0.992495 + 3.38013i) q^{8} +(-0.493707 - 8.98645i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 840 q - 16 q^{3} - 126 q^{4} - 25 q^{6} - 34 q^{7} - 24 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.287750	−	3.01346i	−0.143875	−	1.50673i	−0.722634	−	0.691231i	\(-0.757069\pi\)
							0.578759	−	0.815499i	\(-0.303537\pi\)
\(3\)	2.06232	−	2.17873i	0.687439	−	0.726243i
							\(5\)	−5.16381	−	0.742444i	−1.03276	−	0.148489i	−0.394967	−	0.918695i	\(-0.629244\pi\)
−0.637795	+	0.770207i	\(0.720153\pi\)
\(7\)	−0.342776	−	0.481361i	−0.0489680	−	0.0687659i	0.789366	−	0.613923i	\(-0.210409\pi\)
							−0.838334	+	0.545157i	\(0.816470\pi\)
\(11\)	−2.76141	+	3.51141i	−0.251037	+	0.319220i	−0.895196	−	0.445672i	\(-0.852965\pi\)
							0.644159	+	0.764891i	\(0.277207\pi\)
\(13\)	0.422770	+	1.74268i	0.0325208	+	0.134052i	0.985758	−	0.168171i	\(-0.0537862\pi\)
							−0.953237	+	0.302224i	\(0.902271\pi\)
\(17\)	−0.0442460	+	0.229570i	−0.00260270	+	0.0135041i	−0.983207	−	0.182495i	\(-0.941583\pi\)
							0.980604	+	0.195999i	\(0.0627949\pi\)
\(19\)	−1.63055	+	2.28978i	−0.0858182	+	0.120515i	−0.855253	−	0.518210i	\(-0.826598\pi\)
							0.769435	+	0.638725i	\(0.220538\pi\)
\(23\)	14.7173	−	28.5475i	0.639881	−	1.24120i	−0.316367	−	0.948637i	\(-0.602463\pi\)
							0.956248	−	0.292559i	\(-0.0945066\pi\)
\(29\)	17.0179	+	9.82531i	0.586825	+	0.338804i	0.763841	−	0.645404i	\(-0.223311\pi\)
							−0.177016	+	0.984208i	\(0.556644\pi\)
\(31\)	8.20433	−	33.8187i	0.264656	−	1.09093i	−0.671966	−	0.740582i	\(-0.734550\pi\)
							0.936621	−	0.350343i	\(-0.113935\pi\)
\(37\)	−14.6726	−	25.4138i	−0.396558	−	0.686858i	0.596741	−	0.802434i	\(-0.296462\pi\)
							−0.993299	+	0.115576i	\(0.963129\pi\)
\(41\)	33.1474	−	11.4724i	0.808474	−	0.279816i	0.108603	−	0.994085i	\(-0.465362\pi\)
							0.699871	+	0.714270i	\(0.253241\pi\)
\(43\)	−20.4312	+	23.5789i	−0.475145	+	0.548346i	−0.941835	−	0.336074i	\(-0.890901\pi\)
							0.466691	+	0.884421i	\(0.345446\pi\)
\(47\)	−6.51969	+	0.310571i	−0.138717	+	0.00660789i	−0.116825	−	0.993152i	\(-0.537272\pi\)
							−0.0218913	+	0.999760i	\(0.506969\pi\)

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
201.3.o.b.17.7	✓	840	1.1	even	1	trivial
201.3.o.b.17.36	yes	840	3.2	odd	2	inner
201.3.o.b.71.7	yes	840	201.71	odd	66	inner
201.3.o.b.71.36	yes	840	67.4	even	33	inner