Embedded newform 201.2.j.a.5.3

• Label: 201.2.j.a.5.3
• Level: $201$
• Weight: $2$
• Character: 201.5
• Analytic conductor: $1.605$
• Analytic rank: $0$
• Dimension: $200$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 201 = 3 \cdot 67 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	201.j (of order \(22\), degree \(10\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.60499308063\)
Analytic rank:	\(0\)
Dimension:	\(200\)
Relative dimension:	\(20\) over \(\Q(\zeta_{22})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{22}]$

Embedding invariants

Embedding label			5.3
Character	\(\chi\)	\(=\)	201.5
Dual form			201.2.j.a.161.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.45043 + 1.67388i) q^{2} +(1.46447 + 0.924840i) q^{3} +(-0.413514 - 2.87605i) q^{4} +(1.41694 - 0.416050i) q^{5} +(-3.67218 + 1.10994i) q^{6} +(1.28570 + 1.11407i) q^{7} +(1.68743 + 1.08444i) q^{8} +(1.28934 + 2.70880i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 200 q - 11 q^{3} - 34 q^{4} - 7 q^{6} - 22 q^{7} + 3 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(68\)	\(136\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{5}{22}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.45043	+	1.67388i	−1.02561	+	1.18361i	−0.0427810	+	0.999084i	\(0.513622\pi\)
							−0.982827	+	0.184530i	\(0.940924\pi\)
\(3\)	1.46447	+	0.924840i	0.845512	+	0.533956i
							\(5\)	1.41694	−	0.416050i	0.633673	−	0.186063i	0.0509075	−	0.998703i	\(-0.483789\pi\)
0.582766	+	0.812640i	\(0.301970\pi\)
\(7\)	1.28570	+	1.11407i	0.485950	+	0.421078i	0.863068	−	0.505087i	\(-0.168540\pi\)
							−0.377119	+	0.926165i	\(0.623085\pi\)
\(11\)	−0.102054	+	0.0299658i	−0.0307705	+	0.00903502i	−0.297082	−	0.954852i	\(-0.596013\pi\)
							0.266311	+	0.963887i	\(0.414195\pi\)
\(13\)	2.08751	+	3.24823i	0.578971	+	0.900896i	0.999981	−	0.00615955i	\(-0.00196066\pi\)
							−0.421010	+	0.907056i	\(0.638324\pi\)
\(17\)	−2.54527	−	0.365955i	−0.617319	−	0.0887570i	−0.173441	−	0.984844i	\(-0.555489\pi\)
							−0.443878	+	0.896087i	\(0.646398\pi\)
\(19\)	−3.83423	−	4.42493i	−0.879632	−	1.01515i	−0.999749	−	0.0223894i	\(-0.992873\pi\)
							0.120118	−	0.992760i	\(-0.461673\pi\)
\(23\)	−0.771996	+	0.352559i	−0.160972	+	0.0735136i	−0.494272	−	0.869307i	\(-0.664565\pi\)
							0.333299	+	0.942821i	\(0.391838\pi\)
\(29\)		−	7.47177i		−	1.38747i	−0.720229	−	0.693736i	\(-0.755963\pi\)
							0.720229	−	0.693736i	\(-0.244037\pi\)
\(31\)	1.82266	−	2.83611i	0.327359	−	0.509380i	−0.638094	−	0.769959i	\(-0.720277\pi\)
							0.965453	+	0.260578i	\(0.0839132\pi\)
\(37\)	−2.92864			−0.481466			−0.240733	−	0.970591i	\(-0.577388\pi\)
							−0.240733	+	0.970591i	\(0.577388\pi\)
\(41\)	−0.663277	+	4.61320i	−0.103587	+	0.720460i	0.870151	+	0.492786i	\(0.164021\pi\)
							−0.973737	+	0.227675i	\(0.926888\pi\)
\(43\)	6.63560	+	0.954055i	1.01192	+	0.145492i	0.628277	−	0.777990i	\(-0.283760\pi\)
							0.383642	+	0.923482i	\(0.374669\pi\)
\(47\)	11.7876	−	5.38322i	1.71940	−	0.785223i	0.723959	−	0.689843i	\(-0.242321\pi\)
							0.995441	−	0.0953801i	\(-0.0304067\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	201.2.j.a.5.3	✓	200
3.2	odd	2	inner	201.2.j.a.5.18	yes	200
67.27	odd	22	inner	201.2.j.a.161.18	yes	200
201.161	even	22	inner	201.2.j.a.161.3	yes	200

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
201.2.j.a.5.3	✓	200	1.1	even	1	trivial
201.2.j.a.5.18	yes	200	3.2	odd	2	inner
201.2.j.a.161.3	yes	200	201.161	even	22	inner
201.2.j.a.161.18	yes	200	67.27	odd	22	inner