Embedded newform 201.2.p.b.2.3

• Label: 201.2.p.b.2.3
• Level: $201$
• Weight: $2$
• Character: 201.2
• Analytic conductor: $1.605$
• Analytic rank: $0$
• Dimension: $400$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 201 = 3 \cdot 67 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	201.p (of order \(66\), degree \(20\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.60499308063\)
Analytic rank:	\(0\)
Dimension:	\(400\)
Relative dimension:	\(20\) over \(\Q(\zeta_{66})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{66}]$

Embedding invariants

Embedding label			2.3
Character	\(\chi\)	\(=\)	201.2
Dual form			201.2.p.b.101.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.106680 + 2.23948i) q^{2} +(-0.226858 + 1.71713i) q^{3} +(-3.01294 - 0.287701i) q^{4} +(-0.703208 + 0.811545i) q^{5} +(-3.82127 - 0.691227i) q^{6} +(-1.71157 - 3.31998i) q^{7} +(0.327574 - 2.27833i) q^{8} +(-2.89707 - 0.779090i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 400 q - 22 q^{3} - 20 q^{4} - 20 q^{6} - 38 q^{7} - 30 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.106680	+	2.23948i	−0.0754338	+	1.58355i	0.571239	+	0.820783i	\(0.306463\pi\)
							−0.646673	+	0.762767i	\(0.723840\pi\)
\(3\)	−0.226858	+	1.71713i	−0.130977	+	0.991385i
							\(5\)	−0.703208	+	0.811545i	−0.314484	+	0.362934i	−0.890882	−	0.454235i	\(-0.849913\pi\)
0.576398	+	0.817169i	\(0.304458\pi\)
\(7\)	−1.71157	−	3.31998i	−0.646912	−	1.25483i	−0.953040	−	0.302844i	\(-0.902064\pi\)
							0.306128	−	0.951990i	\(-0.400966\pi\)
\(11\)	1.42265	+	4.11049i	0.428947	+	1.23936i	0.927932	+	0.372749i	\(0.121585\pi\)
							−0.498986	+	0.866610i	\(0.666294\pi\)
\(13\)	2.05155	+	2.60875i	0.568997	+	0.723538i	0.982354	−	0.187031i	\(-0.0598863\pi\)
							−0.413357	+	0.910569i	\(0.635644\pi\)
\(17\)	0.112288	+	1.17593i	0.0272337	+	0.285205i	0.998822	+	0.0485266i	\(0.0154526\pi\)
							−0.971588	+	0.236678i	\(0.923941\pi\)
\(19\)	2.56631	+	1.32302i	0.588751	+	0.303522i	0.726741	−	0.686911i	\(-0.241034\pi\)
							−0.137990	+	0.990434i	\(0.544064\pi\)
\(23\)	−3.30391	+	0.801520i	−0.688913	+	0.167128i	−0.564909	−	0.825153i	\(-0.691089\pi\)
							−0.124004	+	0.992282i	\(0.539574\pi\)
\(29\)	0.603941	+	0.348685i	0.112149	+	0.0647493i	0.555025	−	0.831833i	\(-0.312709\pi\)
							−0.442876	+	0.896583i	\(0.646042\pi\)
\(31\)	2.16526	−	2.75336i	0.388893	−	0.494517i	−0.551706	−	0.834039i	\(-0.686023\pi\)
							0.940598	+	0.339522i	\(0.110265\pi\)
\(37\)	1.75938	+	3.04734i	0.289241	+	0.500980i	0.973629	−	0.228138i	\(-0.0732639\pi\)
							−0.684388	+	0.729118i	\(0.739931\pi\)
\(41\)	−5.23146	−	7.34656i	−0.817017	−	1.14734i	−0.986814	−	0.161861i	\(-0.948250\pi\)
							0.169796	−	0.985479i	\(-0.445689\pi\)
\(43\)	−0.907557	−	0.414468i	−0.138401	−	0.0632057i	0.345011	−	0.938599i	\(-0.387875\pi\)
							−0.483412	+	0.875393i	\(0.660603\pi\)
\(47\)	8.68344	+	9.10693i	1.26661	+	1.32838i	0.920779	+	0.390086i	\(0.127555\pi\)
							0.345831	+	0.938297i	\(0.387597\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	201.2.p.b.2.3	✓	400
3.2	odd	2	inner	201.2.p.b.2.18	yes	400
67.34	odd	66	inner	201.2.p.b.101.18	yes	400
201.101	even	66	inner	201.2.p.b.101.3	yes	400

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
201.2.p.b.2.3	✓	400	1.1	even	1	trivial
201.2.p.b.2.18	yes	400	3.2	odd	2	inner
201.2.p.b.101.3	yes	400	201.101	even	66	inner
201.2.p.b.101.18	yes	400	67.34	odd	66	inner