Embedded newform 201.4.p.a.2.11

• Label: 201.4.p.a.2.11
• 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			2.11
Character	\(\chi\)	\(=\)	201.2
Dual form			201.4.p.a.101.11

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.189440 + 3.97683i) q^{2} +(-0.298544 + 5.18757i) q^{3} +(-7.81552 - 0.746291i) q^{4} +(7.60791 - 8.78000i) q^{5} +(-20.5735 - 2.16999i) q^{6} +(14.2616 + 27.6636i) q^{7} +(-0.0843859 + 0.586917i) q^{8} +(-26.8217 - 3.09743i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1320 q - 22 q^{3} + 214 q^{4} + q^{6} + 22 q^{7} + 48 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.189440	+	3.97683i	−0.0669771	+	1.40602i	0.676097	+	0.736813i	\(0.263670\pi\)
							−0.743074	+	0.669209i	\(0.766633\pi\)
\(3\)	−0.298544	+	5.18757i	−0.0574548	+	0.998348i
							\(5\)	7.60791	−	8.78000i	0.680473	−	0.785307i	−0.305504	−	0.952191i	\(-0.598825\pi\)
0.985977	+	0.166884i	\(0.0533704\pi\)
\(7\)	14.2616	+	27.6636i	0.770054	+	1.49370i	0.866947	+	0.498401i	\(0.166079\pi\)
							−0.0968930	+	0.995295i	\(0.530890\pi\)
\(11\)	14.4813	+	41.8410i	0.396935	+	1.14687i	0.949600	+	0.313464i	\(0.101490\pi\)
							−0.552665	+	0.833403i	\(0.686389\pi\)
\(13\)	−49.9161	−	63.4735i	−1.06494	−	1.35418i	−0.931753	−	0.363093i	\(-0.881721\pi\)
							−0.133189	−	0.991091i	\(-0.542522\pi\)
\(17\)	3.16039	+	33.0971i	0.0450887	+	0.472190i	0.989614	+	0.143751i	\(0.0459165\pi\)
							−0.944525	+	0.328439i	\(0.893477\pi\)
\(19\)	37.8428	+	19.5093i	0.456933	+	0.235565i	0.671297	−	0.741189i	\(-0.265738\pi\)
							−0.214364	+	0.976754i	\(0.568768\pi\)
\(23\)	−103.503	+	25.1097i	−0.938346	+	0.227640i	−0.675637	−	0.737235i	\(-0.736131\pi\)
							−0.262709	+	0.964875i	\(0.584616\pi\)
\(29\)	190.723	+	110.114i	1.22125	+	0.705092i	0.965185	−	0.261566i	\(-0.0842391\pi\)
							0.256070	+	0.966658i	\(0.417572\pi\)
\(31\)	139.313	−	177.151i	0.807142	−	1.02636i	−0.191848	−	0.981425i	\(-0.561448\pi\)
							0.998990	−	0.0449397i	\(-0.0143096\pi\)
\(37\)	−97.5147	−	168.900i	−0.433279	−	0.750461i	0.563875	−	0.825860i	\(-0.309310\pi\)
							−0.997153	+	0.0753995i	\(0.975977\pi\)
\(41\)	−72.9099	−	102.388i	−0.277722	−	0.390007i	0.652029	−	0.758194i	\(-0.273918\pi\)
							−0.929752	+	0.368187i	\(0.879979\pi\)
\(43\)	−33.1263	−	15.1283i	−0.117482	−	0.0536520i	0.355807	−	0.934559i	\(-0.384206\pi\)
							−0.473289	+	0.880907i	\(0.656933\pi\)
\(47\)	148.175	+	155.401i	0.459863	+	0.482290i	0.912310	−	0.409501i	\(-0.134297\pi\)
							−0.452447	+	0.891791i	\(0.649449\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	201.4.p.a.2.11	✓	1320
3.2	odd	2	inner	201.4.p.a.2.56	yes	1320
67.34	odd	66	inner	201.4.p.a.101.56	yes	1320
201.101	even	66	inner	201.4.p.a.101.11	yes	1320

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
201.4.p.a.2.11	✓	1320	1.1	even	1	trivial
201.4.p.a.2.56	yes	1320	3.2	odd	2	inner
201.4.p.a.101.11	yes	1320	201.101	even	66	inner
201.4.p.a.101.56	yes	1320	67.34	odd	66	inner