Embedded newform 201.2.f.b.38.12

• Label: 201.2.f.b.38.12
• 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			38.12
Character	\(\chi\)	\(=\)	201.38
Dual form			201.2.f.b.164.12

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.314267 - 0.544326i) q^{2} +(-1.35150 - 1.08326i) q^{3} +(0.802472 + 1.38992i) q^{4} -3.91059 q^{5} +(-1.01438 + 0.395223i) q^{6} +(-3.42502 + 1.97744i) q^{7} +2.26583 q^{8} +(0.653093 + 2.92805i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q - 24 q^{4} - 2 q^{6} - 6 q^{7} + 8 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.314267	−	0.544326i	0.222220	−	0.384897i	−0.733262	−	0.679947i	\(-0.762003\pi\)
							0.955482	+	0.295050i	\(0.0953362\pi\)
\(3\)	−1.35150	−	1.08326i	−0.780288	−	0.625421i
							\(5\)	−3.91059			−1.74887			−0.874434	−	0.485144i	\(-0.838767\pi\)
−0.874434	+	0.485144i	\(0.838767\pi\)
\(7\)	−3.42502	+	1.97744i	−1.29454	+	0.747402i	−0.979455	−	0.201663i	\(-0.935366\pi\)
							−0.315083	+	0.949064i	\(0.602032\pi\)
\(11\)	−0.0182624	−	0.0316314i	−0.00550632	−	0.00953722i	0.863259	−	0.504761i	\(-0.168419\pi\)
							−0.868765	+	0.495224i	\(0.835086\pi\)
\(13\)	−4.23271	−	2.44376i	−1.17394	−	0.677776i	−0.219337	−	0.975649i	\(-0.570389\pi\)
							−0.954606	+	0.297873i	\(0.903723\pi\)
\(17\)	2.19921	+	1.26971i	0.533387	+	0.307951i	0.742394	−	0.669963i	\(-0.233690\pi\)
							−0.209008	+	0.977914i	\(0.567023\pi\)
\(19\)	1.09219	−	1.89173i	0.250565	−	0.433992i	−0.713116	−	0.701046i	\(-0.752717\pi\)
							0.963682	+	0.267054i	\(0.0860502\pi\)
\(23\)	−3.21846	−	1.85818i	−0.671096	−	0.387458i	0.125396	−	0.992107i	\(-0.459980\pi\)
							−0.796492	+	0.604649i	\(0.793313\pi\)
\(29\)	−5.92804	+	3.42255i	−1.10081	+	0.635552i	−0.936433	−	0.350845i	\(-0.885894\pi\)
							−0.164376	+	0.986398i	\(0.552561\pi\)
\(31\)	−6.21555	+	3.58855i	−1.11635	+	0.644522i	−0.940465	−	0.339889i	\(-0.889610\pi\)
							−0.175880	+	0.984412i	\(0.556277\pi\)
\(37\)	−0.187039	+	0.323962i	−0.0307491	+	0.0532590i	−0.880990	−	0.473134i	\(-0.843123\pi\)
							0.850241	+	0.526393i	\(0.176456\pi\)
\(41\)	0.288218	+	0.499209i	0.0450122	+	0.0779634i	0.887654	−	0.460512i	\(-0.152334\pi\)
							−0.842642	+	0.538475i	\(0.819001\pi\)
\(43\)		−	4.64737i		−	0.708718i	−0.935109	−	0.354359i	\(-0.884699\pi\)
							0.935109	−	0.354359i	\(-0.115301\pi\)
\(47\)	−0.127317	+	0.0735065i	−0.0185711	+	0.0107220i	−0.509257	−	0.860615i	\(-0.670080\pi\)
							0.490686	+	0.871337i	\(0.336746\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	201.2.f.b.38.12	yes	40
3.2	odd	2	inner	201.2.f.b.38.9	✓	40
67.30	odd	6	inner	201.2.f.b.164.9	yes	40
201.164	even	6	inner	201.2.f.b.164.12	yes	40

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
201.2.f.b.38.9	✓	40	3.2	odd	2	inner
201.2.f.b.38.12	yes	40	1.1	even	1	trivial
201.2.f.b.164.9	yes	40	67.30	odd	6	inner
201.2.f.b.164.12	yes	40	201.164	even	6	inner