Embedded newform 201.4.p.a.2.1

• Label: 201.4.p.a.2.1
• Level: $201$
• Weight: $4$
• Character: 201.2
• Analytic conductor: $11.859$
• Analytic rank: $0$
• Dimension: $1320$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 201 = 3 \cdot 67 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	201.p (of order \(66\), degree \(20\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(11.8593839112\)
Analytic rank:	\(0\)
Dimension:	\(1320\)
Relative dimension:	\(66\) over \(\Q(\zeta_{66})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{66}]$

Embedding invariants

Embedding label			2.1
Character	\(\chi\)	\(=\)	201.2
Dual form			201.4.p.a.101.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.263188 + 5.52500i) q^{2} +(2.55934 + 4.52214i) q^{3} +(-22.4926 - 2.14778i) q^{4} +(12.6391 - 14.5863i) q^{5} +(-25.6584 + 12.9502i) q^{6} +(-13.6688 - 26.5138i) q^{7} +(11.4888 - 79.9067i) q^{8} +(-13.8995 + 23.1474i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1320 q - 22 q^{3} + 214 q^{4} + q^{6} + 22 q^{7} + 48 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.263188	+	5.52500i	−0.0930511	+	1.95338i	0.162782	+	0.986662i	\(0.447953\pi\)
							−0.255833	+	0.966721i	\(0.582350\pi\)
\(3\)	2.55934	+	4.52214i	0.492545	+	0.870287i
							\(5\)	12.6391	−	14.5863i	1.13047	−	1.30463i	0.183609	−	0.982999i	\(-0.441222\pi\)
0.946864	−	0.321636i	\(-0.104233\pi\)
\(7\)	−13.6688	−	26.5138i	−0.738047	−	1.43161i	−0.895950	−	0.444156i	\(-0.853504\pi\)
							0.157903	−	0.987455i	\(-0.449527\pi\)
\(11\)	−2.88005	−	8.32134i	−0.0789424	−	0.228089i	0.898623	−	0.438722i	\(-0.144569\pi\)
							−0.977565	+	0.210633i	\(0.932448\pi\)
\(13\)	−22.2209	−	28.2561i	−0.474074	−	0.602834i	0.488929	−	0.872323i	\(-0.337388\pi\)
							−0.963003	+	0.269489i	\(0.913145\pi\)
\(17\)	−0.832277	−	8.71599i	−0.0118739	−	0.124349i	0.987591	−	0.157045i	\(-0.0501968\pi\)
							−0.999465	+	0.0326958i	\(0.989591\pi\)
\(19\)	−28.3673	−	14.6244i	−0.342522	−	0.176582i	0.278379	−	0.960471i	\(-0.410203\pi\)
							−0.620901	+	0.783889i	\(0.713233\pi\)
\(23\)	13.6002	−	3.29937i	0.123297	−	0.0299116i	−0.173635	−	0.984810i	\(-0.555551\pi\)
							0.296933	+	0.954898i	\(0.404036\pi\)
\(29\)	−90.5247	−	52.2645i	−0.579656	−	0.334665i	0.181341	−	0.983420i	\(-0.441956\pi\)
							−0.760997	+	0.648756i	\(0.775290\pi\)
\(31\)	25.2866	−	32.1545i	0.146503	−	0.186294i	−0.707263	−	0.706951i	\(-0.750070\pi\)
							0.853766	+	0.520656i	\(0.174313\pi\)
\(37\)	100.731	+	174.472i	0.447571	+	0.775216i	0.998227	−	0.0595161i	\(-0.0189558\pi\)
							−0.550656	+	0.834732i	\(0.685622\pi\)
\(41\)	−238.726	−	335.244i	−0.909335	−	1.27698i	−0.960496	−	0.278292i	\(-0.910232\pi\)
							0.0511613	−	0.998690i	\(-0.483708\pi\)
\(43\)	−244.013	−	111.437i	−0.865388	−	0.395210i	−0.0672877	−	0.997734i	\(-0.521435\pi\)
							−0.798101	+	0.602524i	\(0.794162\pi\)
\(47\)	30.7221	+	32.2204i	0.0953463	+	0.0999963i	0.769678	−	0.638432i	\(-0.220417\pi\)
							−0.674332	+	0.738428i	\(0.735568\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	201.4.p.a.2.1	✓	1320
3.2	odd	2	inner	201.4.p.a.2.66	yes	1320
67.34	odd	66	inner	201.4.p.a.101.66	yes	1320
201.101	even	66	inner	201.4.p.a.101.1	yes	1320

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
201.4.p.a.2.1	✓	1320	1.1	even	1	trivial
201.4.p.a.2.66	yes	1320	3.2	odd	2	inner
201.4.p.a.101.1	yes	1320	201.101	even	66	inner
201.4.p.a.101.66	yes	1320	67.34	odd	66	inner