Embedded newform 201.2.m.a.4.1

• Label: 201.2.m.a.4.1
• Level: $201$
• Weight: $2$
• Character: 201.4
• Analytic conductor: $1.605$
• Analytic rank: $0$
• Dimension: $100$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 201 = 3 \cdot 67 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	201.m (of order \(33\), degree \(20\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.60499308063\)
Analytic rank:	\(0\)
Dimension:	\(100\)
Relative dimension:	\(5\) over \(\Q(\zeta_{33})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{33}]$

Embedding invariants

Embedding label			4.1
Character	\(\chi\)	\(=\)	201.4
Dual form			201.2.m.a.151.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.41489 - 0.230594i) q^{2} +(-0.841254 - 0.540641i) q^{3} +(3.81465 + 0.735214i) q^{4} +(-0.381162 - 2.65104i) q^{5} +(1.90686 + 1.49958i) q^{6} +(-0.198983 + 0.279433i) q^{7} +(-4.38721 - 1.28820i) q^{8} +(0.415415 + 0.909632i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 100 q + 2 q^{2} + 10 q^{3} + 6 q^{4} - 2 q^{5} + 9 q^{6} + q^{7} - 45 q^{8} - 10 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.41489	−	0.230594i	−1.70758	−	0.163054i	−0.804915	−	0.593390i	\(-0.797789\pi\)
							−0.902669	+	0.430336i	\(0.858395\pi\)
\(3\)	−0.841254	−	0.540641i	−0.485698	−	0.312139i
							\(5\)	−0.381162	−	2.65104i	−0.170461	−	1.18558i	−0.877913	−	0.478821i	\(-0.841064\pi\)
0.707452	−	0.706762i	\(-0.249845\pi\)
\(7\)	−0.198983	+	0.279433i	−0.0752085	+	0.105616i	−0.850467	−	0.526029i	\(-0.823680\pi\)
							0.775258	+	0.631644i	\(0.217620\pi\)
\(11\)	2.89092	−	2.27344i	0.871644	−	0.685468i	−0.0786232	−	0.996904i	\(-0.525052\pi\)
							0.950267	+	0.311436i	\(0.100810\pi\)
\(13\)	−0.0921200	+	0.379724i	−0.0255495	+	0.105316i	−0.983159	−	0.182754i	\(-0.941499\pi\)
							0.957609	+	0.288071i	\(0.0930138\pi\)
\(17\)	−5.84868	+	1.12724i	−1.41851	+	0.273396i	−0.840137	−	0.542374i	\(-0.817526\pi\)
							−0.578376	+	0.815770i	\(0.696313\pi\)
\(19\)	−4.83662	−	6.79209i	−1.10960	−	1.55821i	−0.792555	−	0.609800i	\(-0.791250\pi\)
							−0.317042	−	0.948411i	\(-0.602690\pi\)
\(23\)	−6.68359	+	3.44563i	−1.39362	+	0.718464i	−0.981284	−	0.192566i	\(-0.938319\pi\)
							−0.412341	+	0.911030i	\(0.635289\pi\)
\(29\)	−0.0825400	−	0.142963i	−0.0153273	−	0.0265477i	0.858260	−	0.513215i	\(-0.171546\pi\)
							−0.873587	+	0.486667i	\(0.838212\pi\)
\(31\)	−0.671114	−	2.76637i	−0.120536	−	0.496854i	−0.999790	−	0.0205016i	\(-0.993474\pi\)
							0.879254	−	0.476353i	\(-0.158041\pi\)
\(37\)	1.33771	−	2.31698i	0.219918	−	0.380910i	−0.734865	−	0.678214i	\(-0.762754\pi\)
							0.954783	+	0.297304i	\(0.0960876\pi\)
\(41\)	−1.69599	+	4.90023i	−0.264869	+	0.765288i	0.731632	+	0.681700i	\(0.238759\pi\)
							−0.996500	+	0.0835876i	\(0.973362\pi\)
\(43\)	−2.18240	−	2.51862i	−0.332812	−	0.384086i	0.564536	−	0.825408i	\(-0.309055\pi\)
							−0.897349	+	0.441322i	\(0.854510\pi\)
\(47\)	0.304597	−	6.39427i	0.0444300	−	0.932701i	−0.861104	−	0.508428i	\(-0.830227\pi\)
							0.905534	−	0.424273i	\(-0.139470\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	201.2.m.a.4.1	✓	100
3.2	odd	2		603.2.z.b.406.5		100
67.17	even	33	inner	201.2.m.a.151.1	yes	100
201.17	odd	66		603.2.z.b.352.5		100

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
201.2.m.a.4.1	✓	100	1.1	even	1	trivial
201.2.m.a.151.1	yes	100	67.17	even	33	inner
603.2.z.b.352.5		100	201.17	odd	66
603.2.z.b.406.5		100	3.2	odd	2