Embedded newform 201.4.j.b.5.9

• Label: 201.4.j.b.5.9
• Level: $201$
• Weight: $4$
• Character: 201.5
• Analytic conductor: $11.859$
• Analytic rank: $0$
• Dimension: $640$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 201 = 3 \cdot 67 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	201.j (of order \(22\), degree \(10\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(11.8593839112\)
Analytic rank:	\(0\)
Dimension:	\(640\)
Relative dimension:	\(64\) over \(\Q(\zeta_{22})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{22}]$

Embedding invariants

Embedding label			5.9
Character	\(\chi\)	\(=\)	201.5
Dual form			201.4.j.b.161.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.89391 + 3.33975i) q^{2} +(-2.77933 + 4.39037i) q^{3} +(-1.64070 - 11.4113i) q^{4} +(10.1030 - 2.96651i) q^{5} +(-6.61962 - 21.9876i) q^{6} +(-7.38800 - 6.40174i) q^{7} +(13.1183 + 8.43060i) q^{8} +(-11.5507 - 24.4045i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 640 q - 11 q^{3} - 290 q^{4} + 35 q^{6} - 22 q^{7} - 33 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(68\)	\(136\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{5}{22}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.89391	+	3.33975i	−1.02315	+	1.18078i	−0.0397752	+	0.999209i	\(0.512664\pi\)
							−0.983378	+	0.181573i	\(0.941881\pi\)
\(3\)	−2.77933	+	4.39037i	−0.534882	+	0.844927i
							\(5\)	10.1030	−	2.96651i	0.903639	−	0.265332i	0.203278	−	0.979121i	\(-0.434841\pi\)
0.700361	+	0.713789i	\(0.253022\pi\)
\(7\)	−7.38800	−	6.40174i	−0.398914	−	0.345661i	0.432185	−	0.901785i	\(-0.357743\pi\)
							−0.831100	+	0.556124i	\(0.812288\pi\)
\(11\)	37.1804	−	10.9171i	1.01912	−	0.299240i	0.270840	−	0.962624i	\(-0.412699\pi\)
							0.748279	+	0.663384i	\(0.230880\pi\)
\(13\)	34.9242	+	54.3431i	0.745095	+	1.15939i	0.982188	+	0.187903i	\(0.0601692\pi\)
							−0.237092	+	0.971487i	\(0.576194\pi\)
\(17\)	4.03126	+	0.579608i	0.0575132	+	0.00826915i	0.171011	−	0.985269i	\(-0.445297\pi\)
							−0.113498	+	0.993538i	\(0.536206\pi\)
\(19\)	15.6788	+	18.0943i	0.189314	+	0.218480i	0.842470	−	0.538743i	\(-0.181101\pi\)
							−0.653156	+	0.757224i	\(0.726555\pi\)
\(23\)	134.128	−	61.2542i	1.21598	−	0.555321i	0.299003	−	0.954252i	\(-0.403346\pi\)
							0.916981	+	0.398931i	\(0.130619\pi\)
\(29\)		−	36.3600i		−	0.232823i	−0.993201	−	0.116412i	\(-0.962861\pi\)
							0.993201	−	0.116412i	\(-0.0371392\pi\)
\(31\)	−87.5809	+	136.278i	−0.507419	+	0.789559i	−0.996580	−	0.0826342i	\(-0.973667\pi\)
							0.489161	+	0.872194i	\(0.337303\pi\)
\(37\)	0.638551			0.00283722			0.00141861	−	0.999999i	\(-0.499548\pi\)
							0.00141861	+	0.999999i	\(0.499548\pi\)
\(41\)	6.05161	−	42.0899i	0.0230513	−	0.160325i	−0.975044	−	0.222010i	\(-0.928738\pi\)
							0.998096	+	0.0616848i	\(0.0196473\pi\)
\(43\)	−163.912	−	23.5669i	−0.581309	−	0.0835797i	−0.154614	−	0.987975i	\(-0.549413\pi\)
							−0.426696	+	0.904395i	\(0.640322\pi\)
\(47\)	467.586	−	213.539i	1.45116	−	0.662721i	0.475039	−	0.879965i	\(-0.342434\pi\)
							0.976117	+	0.217243i	\(0.0697065\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	201.4.j.b.5.9	✓	640
3.2	odd	2	inner	201.4.j.b.5.56	yes	640
67.27	odd	22	inner	201.4.j.b.161.56	yes	640
201.161	even	22	inner	201.4.j.b.161.9	yes	640

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
201.4.j.b.5.9	✓	640	1.1	even	1	trivial
201.4.j.b.5.56	yes	640	3.2	odd	2	inner
201.4.j.b.161.9	yes	640	201.161	even	22	inner
201.4.j.b.161.56	yes	640	67.27	odd	22	inner