Embedded newform 201.4.j.b.5.12

• Label: 201.4.j.b.5.12
• 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			5.12
Character	\(\chi\)	\(=\)	201.5
Dual form			201.4.j.b.161.12

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.79597 + 3.22672i) q^{2} +(-3.97281 - 3.34915i) q^{3} +(-1.45576 - 10.1250i) q^{4} +(-12.6136 + 3.70369i) q^{5} +(21.9146 - 3.45504i) q^{6} +(19.5475 + 16.9380i) q^{7} +(8.00668 + 5.14558i) q^{8} +(4.56644 + 26.6110i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 640 q - 11 q^{3} - 290 q^{4} + 35 q^{6} - 22 q^{7} - 33 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.79597	+	3.22672i	−0.988524	+	1.14082i	0.00151108	+	0.999999i	\(0.499519\pi\)
							−0.990035	+	0.140819i	\(0.955026\pi\)
\(3\)	−3.97281	−	3.34915i	−0.764568	−	0.644544i
							\(5\)	−12.6136	+	3.70369i	−1.12820	+	0.331268i	−0.791998	−	0.610523i	\(-0.790959\pi\)
−0.336198	+	0.941791i	\(0.609141\pi\)
\(7\)	19.5475	+	16.9380i	1.05546	+	0.914565i	0.996491	−	0.0836948i	\(-0.0266721\pi\)
							0.0589726	+	0.998260i	\(0.481218\pi\)
\(11\)	48.1994	−	14.1526i	1.32115	−	0.387925i	0.456243	−	0.889855i	\(-0.349195\pi\)
							0.864908	+	0.501931i	\(0.167377\pi\)
\(13\)	36.1717	+	56.2843i	0.771710	+	1.20080i	0.975113	+	0.221710i	\(0.0711637\pi\)
							−0.203402	+	0.979095i	\(0.565200\pi\)
\(17\)	−5.06099	−	0.727660i	−0.0722041	−	0.0103814i	0.106118	−	0.994354i	\(-0.466158\pi\)
							−0.178322	+	0.983972i	\(0.557067\pi\)
\(19\)	−71.6043	−	82.6358i	−0.864587	−	0.997787i	−0.999976	−	0.00699416i	\(-0.997774\pi\)
							0.135388	−	0.990793i	\(-0.456772\pi\)
\(23\)	−186.379	+	85.1162i	−1.68968	+	0.771651i	−0.690864	+	0.722985i	\(0.742770\pi\)
							−0.998815	+	0.0486660i	\(0.984503\pi\)
\(29\)			180.275i			1.15435i	0.816621	+	0.577175i	\(0.195845\pi\)
							−0.816621	+	0.577175i	\(0.804155\pi\)
\(31\)	1.91893	−	2.98592i	0.0111178	−	0.0172996i	−0.835650	−	0.549262i	\(-0.814909\pi\)
							0.846768	+	0.531962i	\(0.178545\pi\)
\(37\)	144.763			0.643212			0.321606	−	0.946874i	\(-0.395777\pi\)
							0.321606	+	0.946874i	\(0.395777\pi\)
\(41\)	32.8034	−	228.153i	0.124952	−	0.869061i	−0.826866	−	0.562399i	\(-0.809879\pi\)
							0.951818	−	0.306663i	\(-0.0992123\pi\)
\(43\)	−420.839	−	60.5075i	−1.49250	−	0.214588i	−0.652752	−	0.757572i	\(-0.726386\pi\)
							−0.839743	+	0.542983i	\(0.817295\pi\)
\(47\)	295.899	−	135.133i	0.918326	−	0.419385i	0.100557	−	0.994931i	\(-0.467938\pi\)
							0.817769	+	0.575546i	\(0.195210\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	201.4.j.b.5.12	✓	640
3.2	odd	2	inner	201.4.j.b.5.53	yes	640
67.27	odd	22	inner	201.4.j.b.161.53	yes	640
201.161	even	22	inner	201.4.j.b.161.12	yes	640

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
201.4.j.b.5.12	✓	640	1.1	even	1	trivial
201.4.j.b.5.53	yes	640	3.2	odd	2	inner
201.4.j.b.161.12	yes	640	201.161	even	22	inner
201.4.j.b.161.53	yes	640	67.27	odd	22	inner