Embedded newform 201.4.p.a.2.16

• Label: 201.4.p.a.2.16
• 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			2.16
Character	\(\chi\)	\(=\)	201.2
Dual form			201.4.p.a.101.16

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.160415 + 3.36752i) q^{2} +(1.06419 - 5.08601i) q^{3} +(-3.35071 - 0.319954i) q^{4} +(-6.11753 + 7.06001i) q^{5} +(16.9565 + 4.39956i) q^{6} +(-0.0381180 - 0.0739387i) q^{7} +(-2.22338 + 15.4640i) q^{8} +(-24.7350 - 10.8250i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1320 q - 22 q^{3} + 214 q^{4} + q^{6} + 22 q^{7} + 48 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.160415	+	3.36752i	−0.0567152	+	1.19060i	0.773736	+	0.633508i	\(0.218386\pi\)
							−0.830451	+	0.557091i	\(0.811917\pi\)
\(3\)	1.06419	−	5.08601i	0.204803	−	0.978803i
							\(5\)	−6.11753	+	7.06001i	−0.547169	+	0.631466i	−0.960221	−	0.279241i	\(-0.909917\pi\)
0.413052	+	0.910707i	\(0.364463\pi\)
\(7\)	−0.0381180	−	0.0739387i	−0.00205818	−	0.00399231i	0.887804	−	0.460222i	\(-0.152230\pi\)
							−0.889862	+	0.456229i	\(0.849200\pi\)
\(11\)	−2.81811	−	8.14239i	−0.0772447	−	0.223184i	0.899767	−	0.436371i	\(-0.143737\pi\)
							−0.977011	+	0.213188i	\(0.931615\pi\)
\(13\)	26.0518	+	33.1276i	0.555805	+	0.706764i	0.980043	−	0.198787i	\(-0.0637002\pi\)
							−0.424237	+	0.905551i	\(0.639458\pi\)
\(17\)	5.90147	+	61.8030i	0.0841951	+	0.881731i	0.934623	+	0.355640i	\(0.115737\pi\)
							−0.850428	+	0.526091i	\(0.823657\pi\)
\(19\)	−67.0025	−	34.5422i	−0.809023	−	0.417080i	0.00351120	−	0.999994i	\(-0.498882\pi\)
							−0.812534	+	0.582913i	\(0.801913\pi\)
\(23\)	−194.477	+	47.1796i	−1.76310	+	0.427723i	−0.980773	−	0.195150i	\(-0.937481\pi\)
							−0.782323	+	0.622873i	\(0.785965\pi\)
\(29\)	29.5367	+	17.0530i	0.189132	+	0.109195i	0.591576	−	0.806249i	\(-0.298506\pi\)
							−0.402444	+	0.915444i	\(0.631839\pi\)
\(31\)	−167.694	+	213.240i	−0.971571	+	1.23545i	0.000314323		1.00000i	\(0.499900\pi\)
							−0.971886	+	0.235453i	\(0.924342\pi\)
\(37\)	4.91882	+	8.51965i	0.0218554	+	0.0378546i	0.876746	−	0.480953i	\(-0.159709\pi\)
							−0.854891	+	0.518808i	\(0.826376\pi\)
\(41\)	−258.637	−	363.205i	−0.985178	−	1.38349i	−0.922733	−	0.385441i	\(-0.874049\pi\)
							−0.0624454	−	0.998048i	\(-0.519890\pi\)
\(43\)	359.757	+	164.295i	1.27587	+	0.582670i	0.934066	−	0.357101i	\(-0.116235\pi\)
							0.341804	+	0.939771i	\(0.388962\pi\)
\(47\)	96.9138	+	101.640i	0.300773	+	0.315441i	0.856726	−	0.515772i	\(-0.172495\pi\)
							−0.555953	+	0.831214i	\(0.687646\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	201.4.p.a.2.16	✓	1320
3.2	odd	2	inner	201.4.p.a.2.51	yes	1320
67.34	odd	66	inner	201.4.p.a.101.51	yes	1320
201.101	even	66	inner	201.4.p.a.101.16	yes	1320

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
201.4.p.a.2.16	✓	1320	1.1	even	1	trivial
201.4.p.a.2.51	yes	1320	3.2	odd	2	inner
201.4.p.a.101.16	yes	1320	201.101	even	66	inner
201.4.p.a.101.51	yes	1320	67.34	odd	66	inner