Embedded newform 201.3.n.b.13.1

• Label: 201.3.n.b.13.1
• Level: $201$
• Weight: $3$
• Character: 201.13
• Analytic conductor: $5.477$
• Analytic rank: $0$
• Dimension: $240$
• 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:	\(240\)
Relative dimension:	\(12\) over \(\Q(\zeta_{66})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{66}]$

Embedding invariants

Embedding label			13.1
Character	\(\chi\)	\(=\)	201.13
Dual form			201.3.n.b.31.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.23952 - 2.84778i) q^{2} +(-1.30900 - 1.13425i) q^{3} +(-2.15136 + 8.86804i) q^{4} +(-3.98657 - 6.20322i) q^{5} +(-0.298577 + 6.26791i) q^{6} +(-2.90854 + 7.26517i) q^{7} +(16.8902 - 7.71351i) q^{8} +(0.426945 + 2.96946i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 240 q - 34 q^{4} - 33 q^{6} - 21 q^{7} - 33 q^{8} + 72 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(68\)	\(136\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{19}{66}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.23952	−	2.84778i	−1.11976	−	1.42389i	−0.893068	−	0.449921i	\(-0.851452\pi\)
							−0.226690	−	0.973967i	\(-0.572790\pi\)
\(3\)	−1.30900	−	1.13425i	−0.436332	−	0.378084i
							\(5\)	−3.98657	−	6.20322i	−0.797314	−	1.24064i	−0.966907	−	0.255130i	\(-0.917882\pi\)
0.169593	−	0.985514i	\(-0.445755\pi\)
\(7\)	−2.90854	+	7.26517i	−0.415505	+	1.03788i	0.562031	+	0.827116i	\(0.310020\pi\)
							−0.977537	+	0.210766i	\(0.932404\pi\)
\(11\)	−5.74901	+	0.273859i	−0.522637	+	0.0248963i	−0.307243	−	0.951631i	\(-0.599407\pi\)
							−0.215394	+	0.976527i	\(0.569104\pi\)
\(13\)	−0.293647	−	3.07521i	−0.0225882	−	0.236554i	−0.999714	−	0.0238945i	\(-0.992393\pi\)
							0.977126	−	0.212660i	\(-0.0682127\pi\)
\(17\)	4.62169	+	19.0509i	0.271864	+	1.12064i	0.929947	+	0.367694i	\(0.119853\pi\)
							−0.658082	+	0.752946i	\(0.728632\pi\)
\(19\)	21.7696	−	8.71524i	1.14577	−	0.458697i	0.280425	−	0.959876i	\(-0.409524\pi\)
							0.865344	+	0.501179i	\(0.167100\pi\)
\(23\)	−22.4508	−	4.32704i	−0.976122	−	0.188132i	−0.323843	−	0.946111i	\(-0.604975\pi\)
							−0.652279	+	0.757979i	\(0.726187\pi\)
\(29\)	−1.25270	+	2.16974i	−0.0431966	+	0.0748186i	−0.886815	−	0.462124i	\(-0.847088\pi\)
							0.843619	+	0.536943i	\(0.180421\pi\)
\(31\)	4.87897	−	51.0949i	0.157386	−	1.64822i	−0.479702	−	0.877431i	\(-0.659255\pi\)
							0.637088	−	0.770791i	\(-0.280139\pi\)
\(37\)	18.3303	+	31.7490i	0.495413	+	0.858081i	0.999986	−	0.00528843i	\(-0.00168337\pi\)
							−0.504573	+	0.863369i	\(0.668350\pi\)
\(41\)	41.0298	+	43.0309i	1.00073	+	1.04953i	0.998772	+	0.0495349i	\(0.0157739\pi\)
							0.00195534	+	0.999998i	\(0.499378\pi\)
\(43\)	−19.7924	+	67.4066i	−0.460288	+	1.56760i	0.323289	+	0.946300i	\(0.395211\pi\)
							−0.783577	+	0.621295i	\(0.786607\pi\)
\(47\)	−20.7690	−	60.0081i	−0.441893	−	1.27677i	−0.917720	−	0.397228i	\(-0.869972\pi\)
							0.475826	−	0.879539i	\(-0.342149\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	201.3.n.b.13.1	✓	240
67.31	odd	66	inner	201.3.n.b.31.1	yes	240

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
201.3.n.b.13.1	✓	240	1.1	even	1	trivial
201.3.n.b.31.1	yes	240	67.31	odd	66	inner