Embedded newform 201.4.f.a.38.6

• Label: 201.4.f.a.38.6
• Level: $201$
• Weight: $4$
• Character: 201.38
• Analytic conductor: $11.859$
• Analytic rank: $0$
• Dimension: $132$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 201 = 3 \cdot 67 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	201.f (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(11.8593839112\)
Analytic rank:	\(0\)
Dimension:	\(132\)
Relative dimension:	\(66\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			38.6
Character	\(\chi\)	\(=\)	201.38
Dual form			201.4.f.a.164.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.47054 + 4.27911i) q^{2} +(3.74376 - 3.60336i) q^{3} +(-8.20717 - 14.2152i) q^{4} +12.2051 q^{5} +(6.17004 + 24.9222i) q^{6} +(13.3204 - 7.69055i) q^{7} +41.5760 q^{8} +(1.03155 - 26.9803i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 132 q - 258 q^{4} - 23 q^{6} - 66 q^{7} - 70 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(68\)	\(136\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{1}{6}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.47054	+	4.27911i	−0.873469	+	1.51289i	−0.0150843	+	0.999886i	\(0.504802\pi\)
							−0.858385	+	0.513007i	\(0.828532\pi\)
\(3\)	3.74376	−	3.60336i	0.720488	−	0.693468i
							\(5\)	12.2051			1.09166			0.545830	−	0.837896i	\(-0.316215\pi\)
0.545830	+	0.837896i	\(0.316215\pi\)
\(7\)	13.3204	−	7.69055i	0.719235	−	0.415250i	−0.0952361	−	0.995455i	\(-0.530361\pi\)
							0.814471	+	0.580204i	\(0.197027\pi\)
\(11\)	−9.84403	−	17.0504i	−0.269826	−	0.467352i	0.698991	−	0.715131i	\(-0.253633\pi\)
							−0.968817	+	0.247778i	\(0.920300\pi\)
\(13\)	−71.9842	−	41.5601i	−1.53576	−	0.886669i	−0.999080	−	0.0428795i	\(-0.986347\pi\)
							−0.536675	−	0.843789i	\(-0.680320\pi\)
\(17\)	−25.0903	−	14.4859i	−0.357959	−	0.206668i	0.310226	−	0.950663i	\(-0.399595\pi\)
							−0.668185	+	0.743995i	\(0.732929\pi\)
\(19\)	21.9994	−	38.1040i	0.265632	−	0.460087i	−0.702097	−	0.712081i	\(-0.747753\pi\)
							0.967729	+	0.251994i	\(0.0810862\pi\)
\(23\)	−92.8253	−	53.5927i	−0.841540	−	0.485863i	0.0162476	−	0.999868i	\(-0.494828\pi\)
							−0.857787	+	0.514005i	\(0.828161\pi\)
\(29\)	95.7761	−	55.2963i	0.613282	−	0.354078i	−0.160967	−	0.986960i	\(-0.551461\pi\)
							0.774249	+	0.632881i	\(0.218128\pi\)
\(31\)	43.9138	−	25.3537i	0.254424	−	0.146892i	−0.367364	−	0.930077i	\(-0.619740\pi\)
							0.621788	+	0.783185i	\(0.286406\pi\)
\(37\)	−187.303	+	324.418i	−0.832228	+	1.44146i	0.0640397	+	0.997947i	\(0.479602\pi\)
							−0.896268	+	0.443514i	\(0.853732\pi\)
\(41\)	179.023	+	310.077i	0.681920	+	1.18112i	0.974394	+	0.224846i	\(0.0721879\pi\)
							−0.292475	+	0.956273i	\(0.594479\pi\)
\(43\)		−	277.557i		−	0.984349i	−0.870497	−	0.492174i	\(-0.836202\pi\)
							0.870497	−	0.492174i	\(-0.163798\pi\)
\(47\)	376.889	−	217.597i	1.16968	−	0.675315i	0.216075	−	0.976377i	\(-0.430675\pi\)
							0.953605	+	0.301062i	\(0.0973412\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	201.4.f.a.38.6	✓	132
3.2	odd	2	inner	201.4.f.a.38.61	yes	132
67.30	odd	6	inner	201.4.f.a.164.61	yes	132
201.164	even	6	inner	201.4.f.a.164.6	yes	132

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
201.4.f.a.38.6	✓	132	1.1	even	1	trivial
201.4.f.a.38.61	yes	132	3.2	odd	2	inner
201.4.f.a.164.6	yes	132	201.164	even	6	inner
201.4.f.a.164.61	yes	132	67.30	odd	6	inner