Embedded newform 201.4.f.a.38.4

• Label: 201.4.f.a.38.4
• 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			38.4
Character	\(\chi\)	\(=\)	201.38
Dual form			201.4.f.a.164.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.57563 + 4.46113i) q^{2} +(-4.36337 + 2.82152i) q^{3} +(-9.26778 - 16.0523i) q^{4} -10.1201 q^{5} +(-1.34870 - 26.7328i) q^{6} +(9.96980 - 5.75607i) q^{7} +54.2715 q^{8} +(11.0781 - 24.6227i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 132 q - 258 q^{4} - 23 q^{6} - 66 q^{7} - 70 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.57563	+	4.46113i	−0.910624	+	1.57725i	−0.0974398	+	0.995241i	\(0.531065\pi\)
							−0.813184	+	0.582006i	\(0.802268\pi\)
\(3\)	−4.36337	+	2.82152i	−0.839732	+	0.543001i
							\(5\)	−10.1201			−0.905169			−0.452585	−	0.891721i	\(-0.649498\pi\)
−0.452585	+	0.891721i	\(0.649498\pi\)
\(7\)	9.96980	−	5.75607i	0.538319	−	0.310798i	−0.206079	−	0.978535i	\(-0.566070\pi\)
							0.744397	+	0.667737i	\(0.232737\pi\)
\(11\)	15.3253	+	26.5442i	0.420069	+	0.727581i	0.995946	−	0.0899556i	\(-0.0286725\pi\)
							−0.575877	+	0.817537i	\(0.695339\pi\)
\(13\)	−7.48665	−	4.32242i	−0.159725	−	0.0922172i	0.418007	−	0.908444i	\(-0.362729\pi\)
							−0.577732	+	0.816227i	\(0.696062\pi\)
\(17\)	−64.4812	−	37.2282i	−0.919940	−	0.531128i	−0.0363241	−	0.999340i	\(-0.511565\pi\)
							−0.883616	+	0.468212i	\(0.844898\pi\)
\(19\)	24.1142	−	41.7670i	0.291167	−	0.504316i	−0.682919	−	0.730494i	\(-0.739290\pi\)
							0.974086	+	0.226178i	\(0.0726232\pi\)
\(23\)	36.3703	+	20.9984i	0.329728	+	0.190368i	0.655720	−	0.755004i	\(-0.272365\pi\)
							−0.325993	+	0.945372i	\(0.605698\pi\)
\(29\)	218.692	−	126.262i	1.40035	−	0.808490i	0.405918	−	0.913910i	\(-0.366952\pi\)
							0.994428	+	0.105420i	\(0.0336186\pi\)
\(31\)	70.1310	−	40.4901i	0.406319	−	0.234588i	−0.282888	−	0.959153i	\(-0.591292\pi\)
							0.689207	+	0.724565i	\(0.257959\pi\)
\(37\)	−48.2772	+	83.6186i	−0.214506	+	0.371536i	−0.953120	−	0.302594i	\(-0.902148\pi\)
							0.738614	+	0.674129i	\(0.235481\pi\)
\(41\)	−167.246	−	289.679i	−0.637061	−	1.10342i	−0.986075	−	0.166304i	\(-0.946817\pi\)
							0.349014	−	0.937118i	\(-0.386517\pi\)
\(43\)			304.034i			1.07825i	0.842226	+	0.539125i	\(0.181245\pi\)
							−0.842226	+	0.539125i	\(0.818755\pi\)
\(47\)	−265.594	+	153.341i	−0.824273	+	0.475894i	−0.851888	−	0.523724i	\(-0.824542\pi\)
							0.0276146	+	0.999619i	\(0.491209\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	201.4.f.a.38.4	✓	132
3.2	odd	2	inner	201.4.f.a.38.63	yes	132
67.30	odd	6	inner	201.4.f.a.164.63	yes	132
201.164	even	6	inner	201.4.f.a.164.4	yes	132

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
201.4.f.a.38.4	✓	132	1.1	even	1	trivial
201.4.f.a.38.63	yes	132	3.2	odd	2	inner
201.4.f.a.164.4	yes	132	201.164	even	6	inner
201.4.f.a.164.63	yes	132	67.30	odd	6	inner