Embedded newform 201.4.f.a.38.10

• Label: 201.4.f.a.38.10
• 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			38.10
Character	\(\chi\)	\(=\)	201.38
Dual form			201.4.f.a.164.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.13016 + 3.68954i) q^{2} +(1.62492 + 4.93555i) q^{3} +(-5.07514 - 8.79039i) q^{4} -3.57613 q^{5} +(-21.6712 - 4.51829i) q^{6} +(10.1310 - 5.84914i) q^{7} +9.16083 q^{8} +(-21.7193 + 16.0397i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 132 q - 258 q^{4} - 23 q^{6} - 66 q^{7} - 70 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.13016	+	3.68954i	−0.753124	+	1.30445i	0.193177	+	0.981164i	\(0.438121\pi\)
							−0.946301	+	0.323285i	\(0.895213\pi\)
\(3\)	1.62492	+	4.93555i	0.312716	+	0.949847i
							\(5\)	−3.57613			−0.319859			−0.159930	−	0.987128i	\(-0.551127\pi\)
−0.159930	+	0.987128i	\(0.551127\pi\)
\(7\)	10.1310	−	5.84914i	0.547023	−	0.315824i	−0.200897	−	0.979612i	\(-0.564386\pi\)
							0.747920	+	0.663788i	\(0.231052\pi\)
\(11\)	−24.1758	−	41.8737i	−0.662662	−	1.14776i	−0.979913	−	0.199423i	\(-0.936093\pi\)
							0.317251	−	0.948341i	\(-0.397240\pi\)
\(13\)	9.95654	+	5.74841i	0.212419	+	0.122640i	0.602435	−	0.798168i	\(-0.294197\pi\)
							−0.390016	+	0.920808i	\(0.627531\pi\)
\(17\)	−95.3103	−	55.0274i	−1.35977	−	0.785065i	−0.370180	−	0.928960i	\(-0.620704\pi\)
							−0.989593	+	0.143894i	\(0.954037\pi\)
\(19\)	70.0294	−	121.295i	0.845571	−	1.46457i	−0.0395526	−	0.999217i	\(-0.512593\pi\)
							0.885124	−	0.465355i	\(-0.154073\pi\)
\(23\)	76.3532	+	44.0825i	0.692206	+	0.399645i	0.804438	−	0.594037i	\(-0.202467\pi\)
							−0.112232	+	0.993682i	\(0.535800\pi\)
\(29\)	−226.195	+	130.594i	−1.44839	+	0.836229i	−0.998386	−	0.0567972i	\(-0.981911\pi\)
							−0.450005	+	0.893026i	\(0.648578\pi\)
\(31\)	61.9458	−	35.7644i	0.358897	−	0.207209i	−0.309700	−	0.950834i	\(-0.600229\pi\)
							0.668597	+	0.743625i	\(0.266895\pi\)
\(37\)	−41.4405	+	71.7770i	−0.184129	+	0.318921i	−0.943283	−	0.331991i	\(-0.892280\pi\)
							0.759154	+	0.650911i	\(0.225613\pi\)
\(41\)	−84.9751	−	147.181i	−0.323680	−	0.560631i	0.657564	−	0.753399i	\(-0.271587\pi\)
							−0.981244	+	0.192768i	\(0.938254\pi\)
\(43\)			437.043i			1.54996i	0.631983	+	0.774982i	\(0.282241\pi\)
							−0.631983	+	0.774982i	\(0.717759\pi\)
\(47\)	548.387	−	316.611i	1.70192	−	0.982607i	0.758112	−	0.652124i	\(-0.226122\pi\)
							0.943812	−	0.330482i	\(-0.107211\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	201.4.f.a.38.10	✓	132
3.2	odd	2	inner	201.4.f.a.38.57	yes	132
67.30	odd	6	inner	201.4.f.a.164.57	yes	132
201.164	even	6	inner	201.4.f.a.164.10	yes	132

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
201.4.f.a.38.10	✓	132	1.1	even	1	trivial
201.4.f.a.38.57	yes	132	3.2	odd	2	inner
201.4.f.a.164.10	yes	132	201.164	even	6	inner
201.4.f.a.164.57	yes	132	67.30	odd	6	inner