Embedded newform 201.4.j.b.5.20

• Label: 201.4.j.b.5.20
• Level: $201$
• Weight: $4$
• Character: 201.5
• Analytic conductor: $11.859$
• Analytic rank: $0$
• Dimension: $640$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 201 = 3 \cdot 67 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	201.j (of order \(22\), degree \(10\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(11.8593839112\)
Analytic rank:	\(0\)
Dimension:	\(640\)
Relative dimension:	\(64\) over \(\Q(\zeta_{22})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{22}]$

Embedding invariants

Embedding label			5.20
Character	\(\chi\)	\(=\)	201.5
Dual form			201.4.j.b.161.20

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.71809 + 1.98278i) q^{2} +(3.65642 - 3.69196i) q^{3} +(0.158929 + 1.10537i) q^{4} +(-18.6302 + 5.47032i) q^{5} +(1.03829 + 13.5930i) q^{6} +(-6.55071 - 5.67622i) q^{7} +(-20.1216 - 12.9314i) q^{8} +(-0.261161 - 26.9987i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 640 q - 11 q^{3} - 290 q^{4} + 35 q^{6} - 22 q^{7} - 33 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.71809	+	1.98278i	−0.607437	+	0.701019i	−0.973271	−	0.229661i	\(-0.926238\pi\)
							0.365834	+	0.930680i	\(0.380784\pi\)
\(3\)	3.65642	−	3.69196i	0.703679	−	0.710518i
							\(5\)	−18.6302	+	5.47032i	−1.66634	+	0.489280i	−0.972898	−	0.231236i	\(-0.925723\pi\)
−0.693438	+	0.720517i	\(0.743905\pi\)
\(7\)	−6.55071	−	5.67622i	−0.353705	−	0.306487i	0.459826	−	0.888009i	\(-0.347912\pi\)
							−0.813531	+	0.581522i	\(0.802457\pi\)
\(11\)	50.9459	−	14.9591i	1.39643	−	0.410030i	0.504976	−	0.863133i	\(-0.331501\pi\)
							0.891458	+	0.453103i	\(0.149683\pi\)
\(13\)	19.9393	+	31.0262i	0.425398	+	0.661932i	0.986113	−	0.166077i	\(-0.0531100\pi\)
							−0.560715	+	0.828009i	\(0.689474\pi\)
\(17\)	83.3524	+	11.9843i	1.18917	+	0.170977i	0.708363	−	0.705848i	\(-0.249434\pi\)
							0.480809	+	0.876825i	\(0.340343\pi\)
\(19\)	−11.1081	−	12.8194i	−0.134125	−	0.154788i	0.684714	−	0.728812i	\(-0.259927\pi\)
							−0.818839	+	0.574024i	\(0.805382\pi\)
\(23\)	111.688	−	51.0062i	1.01255	−	0.462414i	0.161146	−	0.986931i	\(-0.448481\pi\)
							0.851401	+	0.524516i	\(0.175754\pi\)
\(29\)			46.9953i			0.300924i	0.988616	+	0.150462i	\(0.0480761\pi\)
							−0.988616	+	0.150462i	\(0.951924\pi\)
\(31\)	81.0809	−	126.164i	0.469760	−	0.730961i	−0.522835	−	0.852434i	\(-0.675126\pi\)
							0.992595	+	0.121473i	\(0.0387619\pi\)
\(37\)	−251.681			−1.11827			−0.559137	−	0.829075i	\(-0.688868\pi\)
							−0.559137	+	0.829075i	\(0.688868\pi\)
\(41\)	−46.8664	+	325.963i	−0.178519	+	1.24163i	0.681672	+	0.731658i	\(0.261253\pi\)
							−0.860191	+	0.509972i	\(0.829656\pi\)
\(43\)	45.4089	+	6.52881i	0.161042	+	0.0231543i	0.222364	−	0.974964i	\(-0.428623\pi\)
							−0.0613223	+	0.998118i	\(0.519532\pi\)
\(47\)	139.156	−	63.5505i	0.431872	−	0.197230i	−0.187604	−	0.982245i	\(-0.560072\pi\)
							0.619477	+	0.785015i	\(0.287345\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	201.4.j.b.5.20	✓	640
3.2	odd	2	inner	201.4.j.b.5.45	yes	640
67.27	odd	22	inner	201.4.j.b.161.45	yes	640
201.161	even	22	inner	201.4.j.b.161.20	yes	640

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
201.4.j.b.5.20	✓	640	1.1	even	1	trivial
201.4.j.b.5.45	yes	640	3.2	odd	2	inner
201.4.j.b.161.20	yes	640	201.161	even	22	inner
201.4.j.b.161.45	yes	640	67.27	odd	22	inner