Embedded newform 201.4.p.a.2.19

• Label: 201.4.p.a.2.19
• 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			2.19
Character	\(\chi\)	\(=\)	201.2
Dual form			201.4.p.a.101.19

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.134842 + 2.83067i) q^{2} +(-5.15314 + 0.667213i) q^{3} +(-0.0307503 - 0.00293630i) q^{4} +(-3.07901 + 3.55336i) q^{5} +(-1.19380 - 14.6768i) q^{6} +(11.0994 + 21.5299i) q^{7} +(-3.21397 + 22.3537i) q^{8} +(26.1097 - 6.87648i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1320 q - 22 q^{3} + 214 q^{4} + q^{6} + 22 q^{7} + 48 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.134842	+	2.83067i	−0.0476737	+	1.00079i	0.840651	+	0.541578i	\(0.182173\pi\)
							−0.888324	+	0.459216i	\(0.848130\pi\)
\(3\)	−5.15314	+	0.667213i	−0.991722	+	0.128405i
							\(5\)	−3.07901	+	3.55336i	−0.275395	+	0.317822i	−0.876551	−	0.481309i	\(-0.840161\pi\)
0.601156	+	0.799131i	\(0.294707\pi\)
\(7\)	11.0994	+	21.5299i	0.599313	+	1.16250i	0.972095	+	0.234586i	\(0.0753734\pi\)
							−0.372783	+	0.927919i	\(0.621596\pi\)
\(11\)	0.108066	+	0.312237i	0.00296211	+	0.00855845i	0.946472	−	0.322785i	\(-0.104619\pi\)
							−0.943510	+	0.331344i	\(0.892498\pi\)
\(13\)	30.2277	+	38.4376i	0.644896	+	0.820052i	0.993088	−	0.117371i	\(-0.0374466\pi\)
							−0.348192	+	0.937423i	\(0.613204\pi\)
\(17\)	0.172096	+	1.80227i	0.00245526	+	0.0257126i	0.996617	−	0.0821918i	\(-0.0261920\pi\)
							−0.994161	+	0.107904i	\(0.965586\pi\)
\(19\)	−78.0480	−	40.2366i	−0.942392	−	0.485837i	−0.0826576	−	0.996578i	\(-0.526341\pi\)
							−0.859735	+	0.510741i	\(0.829371\pi\)
\(23\)	−71.9736	+	17.4606i	−0.652502	+	0.158295i	−0.548327	−	0.836264i	\(-0.684735\pi\)
							−0.104174	+	0.994559i	\(0.533220\pi\)
\(29\)	−164.573	−	95.0165i	−1.05381	−	0.608418i	−0.130097	−	0.991501i	\(-0.541529\pi\)
							−0.923714	+	0.383083i	\(0.874862\pi\)
\(31\)	180.295	−	229.263i	1.04458	−	1.32829i	0.101820	−	0.994803i	\(-0.467533\pi\)
							0.942756	−	0.333484i	\(-0.108224\pi\)
\(37\)	−118.768	−	205.713i	−0.527714	−	0.914027i	−0.999478	−	0.0323023i	\(-0.989716\pi\)
							0.471764	−	0.881725i	\(-0.343617\pi\)
\(41\)	150.415	+	211.228i	0.572947	+	0.804591i	0.994776	−	0.102084i	\(-0.0325512\pi\)
							−0.421829	+	0.906675i	\(0.638612\pi\)
\(43\)	−218.756	−	99.9024i	−0.775813	−	0.354302i	−0.0121575	−	0.999926i	\(-0.503870\pi\)
							−0.763655	+	0.645624i	\(0.776597\pi\)
\(47\)	196.335	+	205.910i	0.609328	+	0.639045i	0.954046	−	0.299661i	\(-0.0968737\pi\)
							−0.344717	+	0.938707i	\(0.612025\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	201.4.p.a.2.19	✓	1320
3.2	odd	2	inner	201.4.p.a.2.48	yes	1320
67.34	odd	66	inner	201.4.p.a.101.48	yes	1320
201.101	even	66	inner	201.4.p.a.101.19	yes	1320

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
201.4.p.a.2.19	✓	1320	1.1	even	1	trivial
201.4.p.a.2.48	yes	1320	3.2	odd	2	inner
201.4.p.a.101.19	yes	1320	201.101	even	66	inner
201.4.p.a.101.48	yes	1320	67.34	odd	66	inner