Embedded newform 201.3.n.a.7.4

• Label: 201.3.n.a.7.4
• Level: $201$
• Weight: $3$
• Character: 201.7
• Analytic conductor: $5.477$
• Analytic rank: $0$
• Dimension: $220$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 201 = 3 \cdot 67 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	201.n (of order \(66\), degree \(20\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.47685331364\)
Analytic rank:	\(0\)
Dimension:	\(220\)
Relative dimension:	\(11\) over \(\Q(\zeta_{66})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{66}]$

Embedding invariants

Embedding label			7.4
Character	\(\chi\)	\(=\)	201.7
Dual form			201.3.n.a.115.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.781729 - 1.51634i) q^{2} +(0.487975 - 1.66189i) q^{3} +(0.632031 - 0.887563i) q^{4} +(-5.35383 - 4.63912i) q^{5} +(-2.90146 + 0.559211i) q^{6} +(5.80206 + 0.276386i) q^{7} +(-8.59443 - 1.23569i) q^{8} +(-2.52376 - 1.62192i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 220 q - 26 q^{4} + 33 q^{6} + 15 q^{7} - 33 q^{8} + 66 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/201\mathbb{Z}\right)^\times\).

\(n\)	\(68\)	\(136\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{23}{66}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.781729	−	1.51634i	−0.390865	−	0.758172i	0.608509	−	0.793547i	\(-0.291768\pi\)
							−0.999374	+	0.0353749i	\(0.988737\pi\)
\(3\)	0.487975	−	1.66189i	0.162658	−	0.553964i
							\(5\)	−5.35383	−	4.63912i	−1.07077	−	0.927824i	−0.0731833	−	0.997319i	\(-0.523316\pi\)
−0.997582	+	0.0694949i	\(0.977861\pi\)
\(7\)	5.80206	+	0.276386i	0.828866	+	0.0394838i	0.457726	−	0.889093i	\(-0.348664\pi\)
							0.371140	+	0.928577i	\(0.378967\pi\)
\(11\)	1.82287	−	9.45796i	0.165716	−	0.859814i	−0.799958	−	0.600056i	\(-0.795145\pi\)
							0.965674	−	0.259758i	\(-0.0836429\pi\)
\(13\)	−4.89889	+	12.2368i	−0.376838	+	0.941296i	0.611804	+	0.791009i	\(0.290444\pi\)
							−0.988642	+	0.150287i	\(0.951980\pi\)
\(17\)	8.83258	+	12.4036i	0.519564	+	0.729625i	0.988099	−	0.153820i	\(-0.0491575\pi\)
							−0.468535	+	0.883445i	\(0.655218\pi\)
\(19\)	−0.946777	−	19.8753i	−0.0498304	−	1.04607i	−0.875898	−	0.482497i	\(-0.839730\pi\)
							0.826067	−	0.563571i	\(-0.190573\pi\)
\(23\)	−7.82168	+	7.45796i	−0.340073	+	0.324259i	−0.840800	−	0.541347i	\(-0.817915\pi\)
							0.500727	+	0.865606i	\(0.333066\pi\)
\(29\)	−12.7080	−	22.0108i	−0.438205	−	0.758994i	0.559346	−	0.828934i	\(-0.311052\pi\)
							−0.997551	+	0.0699404i	\(0.977719\pi\)
\(31\)	2.15515	+	5.38330i	0.0695210	+	0.173655i	0.959059	−	0.283206i	\(-0.0913980\pi\)
							−0.889538	+	0.456861i	\(0.848974\pi\)
\(37\)	9.79054	−	16.9577i	0.264609	−	0.458317i	−0.702852	−	0.711336i	\(-0.748090\pi\)
							0.967461	+	0.253020i	\(0.0814237\pi\)
\(41\)	5.62885	+	58.9480i	0.137289	+	1.43776i	0.757251	+	0.653124i	\(0.226542\pi\)
							−0.619962	+	0.784632i	\(0.712852\pi\)
\(43\)	−26.9546	−	12.3098i	−0.626852	−	0.286274i	0.0765503	−	0.997066i	\(-0.475609\pi\)
							−0.703402	+	0.710792i	\(0.748337\pi\)
\(47\)	−16.5589	−	68.2567i	−0.352317	−	1.45227i	−0.822027	−	0.569449i	\(-0.807157\pi\)
							0.469710	−	0.882821i	\(-0.344358\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	201.3.n.a.7.4	✓	220
67.48	odd	66	inner	201.3.n.a.115.4	yes	220

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
201.3.n.a.7.4	✓	220	1.1	even	1	trivial
201.3.n.a.115.4	yes	220	67.48	odd	66	inner