Embedded newform 201.4.p.a.2.6

• Label: 201.4.p.a.2.6
• 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			2.6
Character	\(\chi\)	\(=\)	201.2
Dual form			201.4.p.a.101.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.239166 + 5.02071i) q^{2} +(2.74669 - 4.41086i) q^{3} +(-17.1866 - 1.64112i) q^{4} +(3.46556 - 3.99947i) q^{5} +(21.4887 + 14.8453i) q^{6} +(6.74721 + 13.0878i) q^{7} +(6.62737 - 46.0944i) q^{8} +(-11.9113 - 24.2306i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1320 q - 22 q^{3} + 214 q^{4} + q^{6} + 22 q^{7} + 48 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.239166	+	5.02071i	−0.0845579	+	1.77509i	0.415979	+	0.909374i	\(0.363439\pi\)
							−0.500537	+	0.865715i	\(0.666864\pi\)
\(3\)	2.74669	−	4.41086i	0.528601	−	0.848870i
							\(5\)	3.46556	−	3.99947i	0.309969	−	0.357724i	−0.579295	−	0.815118i	\(-0.696672\pi\)
0.889264	+	0.457395i	\(0.151217\pi\)
\(7\)	6.74721	+	13.0878i	0.364315	+	0.706673i	0.997744	−	0.0671276i	\(-0.0213835\pi\)
							−0.633429	+	0.773801i	\(0.718353\pi\)
\(11\)	−23.1836	−	66.9845i	−0.635465	−	1.83605i	−0.541903	−	0.840441i	\(-0.682296\pi\)
							−0.0935613	−	0.995614i	\(-0.529825\pi\)
\(13\)	−32.6315	−	41.4944i	−0.696182	−	0.885267i	0.301490	−	0.953469i	\(-0.402516\pi\)
							−0.997671	+	0.0682025i	\(0.978274\pi\)
\(17\)	−0.591539	−	6.19487i	−0.00843936	−	0.0883810i	0.990271	−	0.139153i	\(-0.0444379\pi\)
							−0.998710	+	0.0507718i	\(0.983832\pi\)
\(19\)	−73.9538	−	38.1258i	−0.892956	−	0.460351i	−0.0502572	−	0.998736i	\(-0.516004\pi\)
							−0.842699	+	0.538386i	\(0.819034\pi\)
\(23\)	165.723	−	40.2039i	1.50242	−	0.364482i	0.601640	−	0.798767i	\(-0.294514\pi\)
							0.900775	+	0.434285i	\(0.142999\pi\)
\(29\)	136.406	+	78.7538i	0.873444	+	0.504283i	0.868491	−	0.495704i	\(-0.165090\pi\)
							0.00495302	+	0.999988i	\(0.498423\pi\)
\(31\)	44.1007	−	56.0787i	0.255507	−	0.324904i	−0.641329	−	0.767266i	\(-0.721617\pi\)
							0.896837	+	0.442362i	\(0.145859\pi\)
\(37\)	−50.2890	−	87.1030i	−0.223445	−	0.387018i	0.732407	−	0.680867i	\(-0.238397\pi\)
							−0.955852	+	0.293850i	\(0.905064\pi\)
\(41\)	−61.1954	−	85.9370i	−0.233101	−	0.327344i	0.681413	−	0.731899i	\(-0.261366\pi\)
							−0.914514	+	0.404555i	\(0.867426\pi\)
\(43\)	−130.852	−	59.7579i	−0.464062	−	0.211930i	0.169642	−	0.985506i	\(-0.445739\pi\)
							−0.633704	+	0.773576i	\(0.718466\pi\)
\(47\)	265.369	+	278.311i	0.823577	+	0.863743i	0.992545	−	0.121879i	\(-0.0388920\pi\)
							−0.168968	+	0.985622i	\(0.554043\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	201.4.p.a.2.6	✓	1320
3.2	odd	2	inner	201.4.p.a.2.61	yes	1320
67.34	odd	66	inner	201.4.p.a.101.61	yes	1320
201.101	even	66	inner	201.4.p.a.101.6	yes	1320

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
201.4.p.a.2.6	✓	1320	1.1	even	1	trivial
201.4.p.a.2.61	yes	1320	3.2	odd	2	inner
201.4.p.a.101.6	yes	1320	201.101	even	66	inner
201.4.p.a.101.61	yes	1320	67.34	odd	66	inner