Embedded newform 201.4.p.a.2.7

• Label: 201.4.p.a.2.7
• 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			2.7
Character	\(\chi\)	\(=\)	201.2
Dual form			201.4.p.a.101.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.235330 + 4.94018i) q^{2} +(5.19611 - 0.0202274i) q^{3} +(-16.3863 - 1.56470i) q^{4} +(2.61014 - 3.01227i) q^{5} +(-1.12287 + 25.6745i) q^{6} +(8.63758 + 16.7546i) q^{7} +(5.95520 - 41.4193i) q^{8} +(26.9992 - 0.210208i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1320 q - 22 q^{3} + 214 q^{4} + q^{6} + 22 q^{7} + 48 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.235330	+	4.94018i	−0.0832017	+	1.74662i	0.443236	+	0.896405i	\(0.353830\pi\)
							−0.526438	+	0.850214i	\(0.676473\pi\)
\(3\)	5.19611	−	0.0202274i	0.999992	−	0.00389276i
							\(5\)	2.61014	−	3.01227i	0.233458	−	0.269425i	−0.626917	−	0.779086i	\(-0.715684\pi\)
0.860376	+	0.509661i	\(0.170229\pi\)
\(7\)	8.63758	+	16.7546i	0.466385	+	0.904661i	0.998324	+	0.0578781i	\(0.0184335\pi\)
							−0.531938	+	0.846783i	\(0.678536\pi\)
\(11\)	16.9609	+	49.0053i	0.464900	+	1.34324i	0.897245	+	0.441533i	\(0.145565\pi\)
							−0.432345	+	0.901708i	\(0.642314\pi\)
\(13\)	33.1317	+	42.1304i	0.706852	+	0.898836i	0.998332	−	0.0577415i	\(-0.0183899\pi\)
							−0.291479	+	0.956577i	\(0.594147\pi\)
\(17\)	−9.94869	−	104.187i	−0.141936	−	1.48642i	−0.733239	−	0.679972i	\(-0.761992\pi\)
							0.591302	−	0.806450i	\(-0.298614\pi\)
\(19\)	−107.706	−	55.5265i	−1.30050	−	0.670455i	−0.337850	−	0.941200i	\(-0.609700\pi\)
							−0.962651	+	0.270745i	\(0.912730\pi\)
\(23\)	−90.9315	+	22.0597i	−0.824370	+	0.199990i	−0.625688	−	0.780074i	\(-0.715182\pi\)
							−0.198683	+	0.980064i	\(0.563666\pi\)
\(29\)	35.2761	+	20.3666i	0.225883	+	0.130413i	0.608671	−	0.793423i	\(-0.291703\pi\)
							−0.382788	+	0.923836i	\(0.625036\pi\)
\(31\)	−52.0282	+	66.1592i	−0.301437	+	0.383308i	−0.913049	−	0.407850i	\(-0.866279\pi\)
							0.611612	+	0.791158i	\(0.290521\pi\)
\(37\)	60.0183	+	103.955i	0.266674	+	0.461893i	0.968001	−	0.250947i	\(-0.0807419\pi\)
							−0.701327	+	0.712840i	\(0.747409\pi\)
\(41\)	−24.3657	−	34.2169i	−0.0928120	−	0.130336i	0.765559	−	0.643366i	\(-0.222463\pi\)
							−0.858371	+	0.513030i	\(0.828523\pi\)
\(43\)	108.439	+	49.5225i	0.384577	+	0.175630i	0.598316	−	0.801260i	\(-0.295837\pi\)
							−0.213739	+	0.976891i	\(0.568564\pi\)
\(47\)	−362.906	−	380.605i	−1.12628	−	1.18121i	−0.981442	−	0.191758i	\(-0.938581\pi\)
							−0.144842	−	0.989455i	\(-0.546267\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	201.4.p.a.2.7	✓	1320
3.2	odd	2	inner	201.4.p.a.2.60	yes	1320
67.34	odd	66	inner	201.4.p.a.101.60	yes	1320
201.101	even	66	inner	201.4.p.a.101.7	yes	1320

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
201.4.p.a.2.7	✓	1320	1.1	even	1	trivial
201.4.p.a.2.60	yes	1320	3.2	odd	2	inner
201.4.p.a.101.7	yes	1320	201.101	even	66	inner
201.4.p.a.101.60	yes	1320	67.34	odd	66	inner