Embedded newform 201.3.n.a.7.2

• Label: 201.3.n.a.7.2
• Level: $201$
• Weight: $3$
• Character: 201.7
• Analytic conductor: $5.477$
• Analytic rank: $0$
• Dimension: $220$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 201 = 3 \cdot 67 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	201.n (of order \(66\), degree \(20\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.47685331364\)
Analytic rank:	\(0\)
Dimension:	\(220\)
Relative dimension:	\(11\) over \(\Q(\zeta_{66})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{66}]$

Embedding invariants

Embedding label			7.2
Character	\(\chi\)	\(=\)	201.7
Dual form			201.3.n.a.115.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.30082 - 2.52324i) q^{2} +(0.487975 - 1.66189i) q^{3} +(-2.35436 + 3.30624i) q^{4} +(4.87030 + 4.22014i) q^{5} +(-4.82811 + 0.930542i) q^{6} +(4.14875 + 0.197629i) q^{7} +(0.165362 + 0.0237755i) q^{8} +(-2.52376 - 1.62192i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 220 q - 26 q^{4} + 33 q^{6} + 15 q^{7} - 33 q^{8} + 66 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(68\)	\(136\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{23}{66}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.30082	−	2.52324i	−0.650409	−	1.26162i	−0.951390	−	0.307988i	\(-0.900344\pi\)
							0.300981	−	0.953630i	\(-0.402686\pi\)
\(3\)	0.487975	−	1.66189i	0.162658	−	0.553964i
							\(5\)	4.87030	+	4.22014i	0.974059	+	0.844027i	0.987776	−	0.155879i	\(-0.0498210\pi\)
−0.0137171	+	0.999906i	\(0.504366\pi\)
\(7\)	4.14875	+	0.197629i	0.592678	+	0.0282327i	0.341779	−	0.939780i	\(-0.388971\pi\)
							0.250899	+	0.968013i	\(0.419274\pi\)
\(11\)	3.62816	−	18.8247i	0.329833	−	1.71133i	−0.316766	−	0.948504i	\(-0.602597\pi\)
							0.646599	−	0.762830i	\(-0.276191\pi\)
\(13\)	3.78075	−	9.44385i	0.290827	−	0.726450i	−0.708949	−	0.705260i	\(-0.750830\pi\)
							0.999775	−	0.0211900i	\(-0.00674550\pi\)
\(17\)	−7.24590	−	10.1754i	−0.426230	−	0.598556i	0.544491	−	0.838767i	\(-0.316723\pi\)
							−0.970721	+	0.240211i	\(0.922783\pi\)
\(19\)	0.263334	+	5.52806i	0.0138597	+	0.290951i	0.995147	+	0.0984008i	\(0.0313727\pi\)
							−0.981287	+	0.192550i	\(0.938324\pi\)
\(23\)	9.38488	−	8.94846i	0.408038	−	0.389064i	−0.458073	−	0.888914i	\(-0.651460\pi\)
							0.866112	+	0.499851i	\(0.166612\pi\)
\(29\)	−7.59942	−	13.1626i	−0.262049	−	0.453882i	0.704737	−	0.709468i	\(-0.251065\pi\)
							−0.966786	+	0.255586i	\(0.917731\pi\)
\(31\)	12.8950	+	32.2101i	0.415967	+	1.03903i	0.977378	+	0.211498i	\(0.0678341\pi\)
							−0.561412	+	0.827537i	\(0.689742\pi\)
\(37\)	−13.0227	+	22.5559i	−0.351964	+	0.609619i	−0.986593	−	0.163197i	\(-0.947819\pi\)
							0.634630	+	0.772816i	\(0.281153\pi\)
\(41\)	−4.39045	−	45.9788i	−0.107084	−	1.12143i	−0.876075	−	0.482175i	\(-0.839847\pi\)
							0.768991	−	0.639260i	\(-0.220759\pi\)
\(43\)	8.54371	+	3.90178i	0.198691	+	0.0907391i	0.512276	−	0.858821i	\(-0.328803\pi\)
							−0.313585	+	0.949560i	\(0.601530\pi\)
\(47\)	12.0394	+	49.6269i	0.256157	+	1.05589i	0.943977	+	0.330011i	\(0.107052\pi\)
							−0.687820	+	0.725881i	\(0.741432\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	201.3.n.a.7.2	✓	220
67.48	odd	66	inner	201.3.n.a.115.2	yes	220

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
201.3.n.a.7.2	✓	220	1.1	even	1	trivial
201.3.n.a.115.2	yes	220	67.48	odd	66	inner