Embedded newform 201.3.n.b.31.8

• Label: 201.3.n.b.31.8
• Level: $201$
• Weight: $3$
• Character: 201.31
• 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			31.8
Character	\(\chi\)	\(=\)	201.31
Dual form			201.3.n.b.13.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.774122 - 0.984377i) q^{2} +(-1.30900 + 1.13425i) q^{3} +(0.573304 + 2.36319i) q^{4} +(-3.17987 + 4.94798i) q^{5} +(0.103208 + 2.16660i) q^{6} +(-3.85408 - 9.62704i) q^{7} +(7.32662 + 3.34595i) q^{8} +(0.426945 - 2.96946i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 240 q - 34 q^{4} - 33 q^{6} - 21 q^{7} - 33 q^{8} + 72 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(68\)	\(136\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{47}{66}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.774122	−	0.984377i	0.387061	−	0.492188i	−0.553006	−	0.833177i	\(-0.686519\pi\)
							0.940067	+	0.340989i	\(0.110762\pi\)
\(3\)	−1.30900	+	1.13425i	−0.436332	+	0.378084i
							\(5\)	−3.17987	+	4.94798i	−0.635975	+	0.989596i	0.362367	+	0.932035i	\(0.381969\pi\)
−0.998342	+	0.0575605i	\(0.981668\pi\)
\(7\)	−3.85408	−	9.62704i	−0.550583	−	1.37529i	−0.898999	−	0.437951i	\(-0.855704\pi\)
							0.348416	−	0.937340i	\(-0.386720\pi\)
\(11\)	−18.9185	−	0.901198i	−1.71986	−	0.0819271i	−0.835723	−	0.549151i	\(-0.814951\pi\)
							−0.884139	+	0.467224i	\(0.845254\pi\)
\(13\)	−1.50548	+	15.7661i	−0.115806	+	1.21278i	0.731564	+	0.681773i	\(0.238791\pi\)
							−0.847370	+	0.531003i	\(0.821815\pi\)
\(17\)	−2.92808	+	12.0697i	−0.172240	+	0.709982i	0.818534	+	0.574458i	\(0.194787\pi\)
							−0.990774	+	0.135524i	\(0.956728\pi\)
\(19\)	−32.6822	−	13.0840i	−1.72012	−	0.688630i	−0.720129	−	0.693840i	\(-0.755917\pi\)
							−0.999986	+	0.00520956i	\(0.998342\pi\)
\(23\)	23.9924	−	4.62415i	1.04315	−	0.201050i	0.361227	−	0.932478i	\(-0.382358\pi\)
							0.681919	+	0.731428i	\(0.261146\pi\)
\(29\)	26.6148	+	46.0982i	0.917752	+	1.58959i	0.802821	+	0.596220i	\(0.203331\pi\)
							0.114931	+	0.993373i	\(0.463335\pi\)
\(31\)	−2.04871	−	21.4550i	−0.0660874	−	0.692098i	−0.966432	−	0.256921i	\(-0.917292\pi\)
							0.900345	−	0.435177i	\(-0.143314\pi\)
\(37\)	6.85823	−	11.8788i	0.185357	−	0.321049i	−0.758339	−	0.651860i	\(-0.773989\pi\)
							0.943697	+	0.330811i	\(0.107322\pi\)
\(41\)	−6.66911	+	6.99436i	−0.162661	+	0.170594i	−0.800021	−	0.599972i	\(-0.795178\pi\)
							0.637359	+	0.770567i	\(0.280027\pi\)
\(43\)	−0.121864	−	0.415032i	−0.00283405	−	0.00965190i	0.958064	−	0.286555i	\(-0.0925101\pi\)
							−0.960898	+	0.276903i	\(0.910692\pi\)
\(47\)	−7.71877	+	22.3019i	−0.164229	+	0.474509i	−0.996848	−	0.0793377i	\(-0.974719\pi\)
							0.832619	+	0.553847i	\(0.186841\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	201.3.n.b.31.8	yes	240
67.13	odd	66	inner	201.3.n.b.13.8	✓	240

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
201.3.n.b.13.8	✓	240	67.13	odd	66	inner
201.3.n.b.31.8	yes	240	1.1	even	1	trivial