Embedded newform 201.4.p.a.2.8

• Label: 201.4.p.a.2.8
• 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			2.8
Character	\(\chi\)	\(=\)	201.2
Dual form			201.4.p.a.101.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.234299 + 4.91853i) q^{2} +(-4.94170 - 1.60611i) q^{3} +(-16.1733 - 1.54436i) q^{4} +(9.16856 - 10.5811i) q^{5} +(9.05754 - 23.9296i) q^{6} +(6.12309 + 11.8771i) q^{7} +(5.77918 - 40.1951i) q^{8} +(21.8408 + 15.8738i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1320 q - 22 q^{3} + 214 q^{4} + q^{6} + 22 q^{7} + 48 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.234299	+	4.91853i	−0.0828371	+	1.73896i	0.450249	+	0.892903i	\(0.351335\pi\)
							−0.533086	+	0.846061i	\(0.678968\pi\)
\(3\)	−4.94170	−	1.60611i	−0.951031	−	0.309096i
							\(5\)	9.16856	−	10.5811i	0.820061	−	0.946400i	−0.179240	−	0.983805i	\(-0.557364\pi\)
0.999300	+	0.0374051i	\(0.0119092\pi\)
\(7\)	6.12309	+	11.8771i	0.330616	+	0.641305i	0.994289	−	0.106717i	\(-0.0340340\pi\)
							−0.663674	+	0.748022i	\(0.731004\pi\)
\(11\)	−5.78233	−	16.7069i	−0.158494	−	0.457940i	0.837595	−	0.546292i	\(-0.183961\pi\)
							−0.996089	+	0.0883524i	\(0.971840\pi\)
\(13\)	19.4606	+	24.7462i	0.415186	+	0.527951i	0.947996	−	0.318283i	\(-0.103106\pi\)
							−0.532810	+	0.846235i	\(0.678864\pi\)
\(17\)	2.26892	+	23.7612i	0.0323703	+	0.338997i	0.997121	+	0.0758304i	\(0.0241608\pi\)
							−0.964750	+	0.263166i	\(0.915233\pi\)
\(19\)	71.6661	+	36.9465i	0.865333	+	0.446110i	0.832939	−	0.553365i	\(-0.186656\pi\)
							0.0323944	+	0.999475i	\(0.489687\pi\)
\(23\)	−76.5101	+	18.5611i	−0.693628	+	0.168272i	−0.567049	−	0.823684i	\(-0.691915\pi\)
							−0.126579	+	0.991956i	\(0.540400\pi\)
\(29\)	119.538	+	69.0151i	0.765434	+	0.441923i	0.831243	−	0.555909i	\(-0.187630\pi\)
							−0.0658097	+	0.997832i	\(0.520963\pi\)
\(31\)	−145.889	+	185.513i	−0.845239	+	1.07481i	0.151031	+	0.988529i	\(0.451741\pi\)
							−0.996271	+	0.0862805i	\(0.972502\pi\)
\(37\)	182.942	+	316.865i	0.812850	+	1.40790i	0.910861	+	0.412712i	\(0.135419\pi\)
							−0.0980113	+	0.995185i	\(0.531248\pi\)
\(41\)	−106.310	−	149.292i	−0.404948	−	0.568671i	0.560865	−	0.827907i	\(-0.310469\pi\)
							−0.965814	+	0.259236i	\(0.916529\pi\)
\(43\)	−11.1770	−	5.10435i	−0.0396389	−	0.0181025i	0.395497	−	0.918467i	\(-0.370572\pi\)
							−0.435136	+	0.900365i	\(0.643300\pi\)
\(47\)	190.676	+	199.975i	0.591765	+	0.620625i	0.949715	−	0.313115i	\(-0.101373\pi\)
							−0.357950	+	0.933741i	\(0.616524\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	201.4.p.a.2.8	✓	1320
3.2	odd	2	inner	201.4.p.a.2.59	yes	1320
67.34	odd	66	inner	201.4.p.a.101.59	yes	1320
201.101	even	66	inner	201.4.p.a.101.8	yes	1320

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
201.4.p.a.2.8	✓	1320	1.1	even	1	trivial
201.4.p.a.2.59	yes	1320	3.2	odd	2	inner
201.4.p.a.101.8	yes	1320	201.101	even	66	inner
201.4.p.a.101.59	yes	1320	67.34	odd	66	inner