Embedded newform 201.2.f.b.38.6

• Label: 201.2.f.b.38.6
• Level: $201$
• Weight: $2$
• Character: 201.38
• Analytic conductor: $1.605$
• Analytic rank: $0$
• Dimension: $40$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 201 = 3 \cdot 67 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	201.f (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.60499308063\)
Analytic rank:	\(0\)
Dimension:	\(40\)
Relative dimension:	\(20\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			38.6
Character	\(\chi\)	\(=\)	201.38
Dual form			201.2.f.b.164.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.725905 + 1.25730i) q^{2} +(1.60534 + 0.650289i) q^{3} +(-0.0538753 - 0.0933147i) q^{4} +0.670871 q^{5} +(-1.98294 + 1.54635i) q^{6} +(4.01999 - 2.32094i) q^{7} -2.74719 q^{8} +(2.15425 + 2.08787i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q - 24 q^{4} - 2 q^{6} - 6 q^{7} + 8 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.725905	+	1.25730i	−0.513292	+	0.889048i	0.486589	+	0.873631i	\(0.338241\pi\)
							−0.999881	+	0.0154171i	\(0.995092\pi\)
\(3\)	1.60534	+	0.650289i	0.926845	+	0.375445i
							\(5\)	0.670871			0.300023			0.150011	−	0.988684i	\(-0.452069\pi\)
0.150011	+	0.988684i	\(0.452069\pi\)
\(7\)	4.01999	−	2.32094i	1.51941	−	0.877234i	0.519676	−	0.854364i	\(-0.326053\pi\)
							0.999738	−	0.0228704i	\(-0.00728050\pi\)
\(11\)	0.0803974	+	0.139252i	0.0242407	+	0.0419862i	0.877891	−	0.478860i	\(-0.158950\pi\)
							−0.853651	+	0.520846i	\(0.825617\pi\)
\(13\)	−3.16018	−	1.82453i	−0.876477	−	0.506034i	−0.00698204	−	0.999976i	\(-0.502222\pi\)
							−0.869495	+	0.493941i	\(0.835556\pi\)
\(17\)	−5.25073	−	3.03151i	−1.27349	−	0.735249i	−0.297846	−	0.954614i	\(-0.596268\pi\)
							−0.975643	+	0.219365i	\(0.929601\pi\)
\(19\)	−1.97432	+	3.41963i	−0.452941	+	0.784517i	−0.998567	−	0.0535120i	\(-0.982958\pi\)
							0.545626	+	0.838029i	\(0.316292\pi\)
\(23\)	0.184328	+	0.106422i	0.0384351	+	0.0221905i	0.519094	−	0.854717i	\(-0.326269\pi\)
							−0.480659	+	0.876907i	\(0.659603\pi\)
\(29\)	−0.260404	+	0.150344i	−0.0483558	+	0.0279182i	−0.523983	−	0.851729i	\(-0.675554\pi\)
							0.475627	+	0.879647i	\(0.342221\pi\)
\(31\)	−4.10749	+	2.37146i	−0.737727	+	0.425927i	−0.821242	−	0.570580i	\(-0.806719\pi\)
							0.0835155	+	0.996506i	\(0.473385\pi\)
\(37\)	2.00444	−	3.47179i	0.329528	−	0.570759i	−0.652890	−	0.757452i	\(-0.726444\pi\)
							0.982418	+	0.186694i	\(0.0597771\pi\)
\(41\)	1.85818	+	3.21846i	0.290199	+	0.502639i	0.973857	−	0.227163i	\(-0.0729452\pi\)
							−0.683658	+	0.729803i	\(0.739612\pi\)
\(43\)			0.0314408i			0.00479468i	0.999997	+	0.00239734i	\(0.000763097\pi\)
							−0.999997	+	0.00239734i	\(0.999237\pi\)
\(47\)	2.71954	−	1.57013i	0.396685	−	0.229026i	−0.288367	−	0.957520i	\(-0.593112\pi\)
							0.685053	+	0.728493i	\(0.259779\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	201.2.f.b.38.6	✓	40
3.2	odd	2	inner	201.2.f.b.38.15	yes	40
67.30	odd	6	inner	201.2.f.b.164.15	yes	40
201.164	even	6	inner	201.2.f.b.164.6	yes	40

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
201.2.f.b.38.6	✓	40	1.1	even	1	trivial
201.2.f.b.38.15	yes	40	3.2	odd	2	inner
201.2.f.b.164.6	yes	40	201.164	even	6	inner
201.2.f.b.164.15	yes	40	67.30	odd	6	inner