Embedded newform 201.4.j.b.5.13

• Label: 201.4.j.b.5.13
• 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			5.13
Character	\(\chi\)	\(=\)	201.5
Dual form			201.4.j.b.161.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.41929 + 2.79201i) q^{2} +(-2.24694 + 4.68522i) q^{3} +(-0.803837 - 5.59081i) q^{4} +(-11.7964 + 3.46373i) q^{5} +(-7.64518 - 17.6084i) q^{6} +(12.9935 + 11.2589i) q^{7} +(-7.30882 - 4.69710i) q^{8} +(-16.9025 - 21.0548i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 640 q - 11 q^{3} - 290 q^{4} + 35 q^{6} - 22 q^{7} - 33 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.41929	+	2.79201i	−0.855349	+	0.987125i	−0.999997	−	0.00235790i	\(-0.999249\pi\)
							0.144648	+	0.989483i	\(0.453795\pi\)
\(3\)	−2.24694	+	4.68522i	−0.432424	+	0.901671i
							\(5\)	−11.7964	+	3.46373i	−1.05510	+	0.309806i	−0.762877	−	0.646544i	\(-0.776214\pi\)
−0.292225	+	0.956350i	\(0.594396\pi\)
\(7\)	12.9935	+	11.2589i	0.701581	+	0.607924i	0.930834	−	0.365443i	\(-0.119083\pi\)
							−0.229252	+	0.973367i	\(0.573628\pi\)
\(11\)	−23.3159	+	6.84616i	−0.639091	+	0.187654i	−0.585195	−	0.810893i	\(-0.698982\pi\)
							−0.0538958	+	0.998547i	\(0.517164\pi\)
\(13\)	−13.3546	−	20.7801i	−0.284915	−	0.443336i	0.669067	−	0.743203i	\(-0.266694\pi\)
							−0.953981	+	0.299867i	\(0.903058\pi\)
\(17\)	−7.27025	−	1.04530i	−0.103723	−	0.0149131i	0.0902577	−	0.995918i	\(-0.471231\pi\)
							−0.193981	+	0.981005i	\(0.562140\pi\)
\(19\)	−21.3594	−	24.6501i	−0.257905	−	0.297638i	0.612000	−	0.790858i	\(-0.290365\pi\)
							−0.869905	+	0.493220i	\(0.835820\pi\)
\(23\)	13.4233	−	6.13020i	0.121693	−	0.0555754i	−0.353638	−	0.935382i	\(-0.615056\pi\)
							0.475331	+	0.879807i	\(0.342328\pi\)
\(29\)			97.3470i			0.623341i	0.950190	+	0.311670i	\(0.100888\pi\)
							−0.950190	+	0.311670i	\(0.899112\pi\)
\(31\)	−47.4784	+	73.8778i	−0.275076	+	0.428027i	−0.951114	−	0.308840i	\(-0.900059\pi\)
							0.676038	+	0.736867i	\(0.263696\pi\)
\(37\)	24.8791			0.110543			0.0552715	−	0.998471i	\(-0.482398\pi\)
							0.0552715	+	0.998471i	\(0.482398\pi\)
\(41\)	−50.5589	+	351.645i	−0.192585	+	1.33946i	0.632549	+	0.774520i	\(0.282009\pi\)
							−0.825134	+	0.564937i	\(0.808901\pi\)
\(43\)	436.338	+	62.7360i	1.54746	+	0.222492i	0.862489	−	0.506075i	\(-0.168904\pi\)
							0.684975	+	0.728567i	\(0.259813\pi\)
\(47\)	13.2037	−	6.02994i	0.0409779	−	0.0187140i	−0.394821	−	0.918758i	\(-0.629193\pi\)
							0.435799	+	0.900044i	\(0.356466\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	201.4.j.b.5.13	✓	640
3.2	odd	2	inner	201.4.j.b.5.52	yes	640
67.27	odd	22	inner	201.4.j.b.161.52	yes	640
201.161	even	22	inner	201.4.j.b.161.13	yes	640

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
201.4.j.b.5.13	✓	640	1.1	even	1	trivial
201.4.j.b.5.52	yes	640	3.2	odd	2	inner
201.4.j.b.161.13	yes	640	201.161	even	22	inner
201.4.j.b.161.52	yes	640	67.27	odd	22	inner