Embedded newform 201.4.m.b.4.3

• Label: 201.4.m.b.4.3
• Level: $201$
• Weight: $4$
• Character: 201.4
• Analytic conductor: $11.859$
• Analytic rank: $0$
• Dimension: $360$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 201 = 3 \cdot 67 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	201.m (of order \(33\), degree \(20\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(11.8593839112\)
Analytic rank:	\(0\)
Dimension:	\(360\)
Relative dimension:	\(18\) over \(\Q(\zeta_{33})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{33}]$

Embedding invariants

Embedding label			4.3
Character	\(\chi\)	\(=\)	201.4
Dual form			201.4.m.b.151.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-4.30544 - 0.411120i) q^{2} +(-2.52376 - 1.62192i) q^{3} +(10.5124 + 2.02609i) q^{4} +(2.02353 + 14.0740i) q^{5} +(10.1991 + 8.02066i) q^{6} +(12.0391 - 16.9066i) q^{7} +(-11.2287 - 3.29705i) q^{8} +(3.73874 + 8.18669i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 360 q - 2 q^{2} + 108 q^{3} + 90 q^{4} + 4 q^{5} - 27 q^{6} - 22 q^{7} + 183 q^{8} - 324 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−4.30544	−	0.411120i	−1.52220	−	0.145353i	−0.699833	−	0.714306i	\(-0.746742\pi\)
							−0.822370	+	0.568954i	\(0.807348\pi\)
\(3\)	−2.52376	−	1.62192i	−0.485698	−	0.312139i
							\(5\)	2.02353	+	14.0740i	0.180990	+	1.25882i	0.854427	+	0.519571i	\(0.173908\pi\)
−0.673437	+	0.739245i	\(0.735183\pi\)
\(7\)	12.0391	−	16.9066i	0.650051	−	0.912869i	−0.349661	−	0.936876i	\(-0.613703\pi\)
							0.999712	+	0.0240074i	\(0.00764252\pi\)
\(11\)	6.34772	−	4.99190i	0.173992	−	0.136829i	−0.527366	−	0.849638i	\(-0.676820\pi\)
							0.701357	+	0.712810i	\(0.252578\pi\)
\(13\)	−4.40305	+	18.1496i	−0.0939373	+	0.387215i	−0.999299	−	0.0374254i	\(-0.988084\pi\)
							0.905362	+	0.424640i	\(0.139599\pi\)
\(17\)	−84.7294	+	16.3303i	−1.20882	+	0.232980i	−0.753560	−	0.657379i	\(-0.771665\pi\)
							−0.455258	+	0.890360i	\(0.650453\pi\)
\(19\)	−20.2388	−	28.4214i	−0.244373	−	0.343174i	0.674112	−	0.738629i	\(-0.264526\pi\)
							−0.918485	+	0.395455i	\(0.870587\pi\)
\(23\)	140.373	−	72.3672i	1.27260	−	0.656069i	0.316353	−	0.948641i	\(-0.397541\pi\)
							0.956243	+	0.292572i	\(0.0945111\pi\)
\(29\)	139.181	+	241.069i	0.891218	+	1.54363i	0.838417	+	0.545029i	\(0.183481\pi\)
							0.0528005	+	0.998605i	\(0.483185\pi\)
\(31\)	−32.8875	−	135.564i	−0.190541	−	0.785420i	−0.984200	−	0.177058i	\(-0.943342\pi\)
							0.793660	−	0.608362i	\(-0.208173\pi\)
\(37\)	37.0329	−	64.1429i	0.164545	−	0.285001i	−0.771948	−	0.635685i	\(-0.780718\pi\)
							0.936494	+	0.350684i	\(0.114051\pi\)
\(41\)	−125.106	+	361.469i	−0.476542	+	1.37688i	0.409092	+	0.912493i	\(0.365846\pi\)
							−0.885634	+	0.464385i	\(0.846276\pi\)
\(43\)	208.039	+	240.090i	0.737806	+	0.851473i	0.993327	−	0.115330i	\(-0.0367927\pi\)
							−0.255522	+	0.966803i	\(0.582247\pi\)
\(47\)	−13.3051	+	279.309i	−0.0412926	+	0.866838i	0.879289	+	0.476288i	\(0.158018\pi\)
							−0.920582	+	0.390550i	\(0.872285\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	201.4.m.b.4.3	✓	360
67.17	even	33	inner	201.4.m.b.151.3	yes	360

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
201.4.m.b.4.3	✓	360	1.1	even	1	trivial
201.4.m.b.151.3	yes	360	67.17	even	33	inner