Embedded newform 201.2.i.a.64.1

• Label: 201.2.i.a.64.1
• Level: $201$
• Weight: $2$
• Character: 201.64
• Analytic conductor: $1.605$
• Analytic rank: $0$
• Dimension: $50$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 201 = 3 \cdot 67 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	201.i (of order \(11\), degree \(10\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.60499308063\)
Analytic rank:	\(0\)
Dimension:	\(50\)
Relative dimension:	\(5\) over \(\Q(\zeta_{11})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{11}]$

Embedding invariants

Embedding label			64.1
Character	\(\chi\)	\(=\)	201.64
Dual form			201.2.i.a.22.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.87424 - 0.550325i) q^{2} +(0.142315 - 0.989821i) q^{3} +(1.52739 + 0.981596i) q^{4} +(-1.74459 - 3.82012i) q^{5} +(-0.811455 + 1.77684i) q^{6} +(-0.0581100 - 0.0170626i) q^{7} +(0.235861 + 0.272198i) q^{8} +(-0.959493 - 0.281733i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 50 q - 2 q^{2} + 5 q^{3} - 6 q^{4} + 2 q^{5} - 9 q^{6} + 2 q^{7} + 27 q^{8} - 5 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(68\)	\(136\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{1}{11}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.87424	−	0.550325i	−1.32528	−	0.389139i	−0.458887	−	0.888494i	\(-0.651752\pi\)
							−0.866397	+	0.499356i	\(0.833570\pi\)
\(3\)	0.142315	−	0.989821i	0.0821655	−	0.571474i
							\(5\)	−1.74459	−	3.82012i	−0.780204	−	1.70841i	−0.702778	−	0.711409i	\(-0.748057\pi\)
−0.0774257	−	0.996998i	\(-0.524670\pi\)
\(7\)	−0.0581100	−	0.0170626i	−0.0219635	−	0.00644907i	0.270732	−	0.962655i	\(-0.412734\pi\)
							−0.292696	+	0.956206i	\(0.594552\pi\)
\(11\)	−0.281079	−	0.615478i	−0.0847486	−	0.185573i	0.862513	−	0.506035i	\(-0.168889\pi\)
							−0.947262	+	0.320461i	\(0.896162\pi\)
\(13\)	−1.51554	+	1.74903i	−0.420335	+	0.485092i	−0.925939	−	0.377673i	\(-0.876724\pi\)
							0.505604	+	0.862766i	\(0.331270\pi\)
\(17\)	−1.61555	+	1.03825i	−0.391828	+	0.251813i	−0.721690	−	0.692217i	\(-0.756634\pi\)
							0.329862	+	0.944029i	\(0.392998\pi\)
\(19\)	0.944928	−	0.277456i	0.216781	−	0.0636528i	−0.171538	−	0.985177i	\(-0.554874\pi\)
							0.388320	+	0.921525i	\(0.373056\pi\)
\(23\)	−0.204387	+	1.42154i	−0.0426177	+	0.296412i	0.957354	+	0.288917i	\(0.0932951\pi\)
							−0.999972	+	0.00749567i	\(0.997614\pi\)
\(29\)	−8.15004			−1.51342			−0.756712	−	0.653749i	\(-0.773195\pi\)
							−0.756712	+	0.653749i	\(0.773195\pi\)
\(31\)	−5.96914	−	6.88875i	−1.07209	−	1.23726i	−0.970160	−	0.242465i	\(-0.922044\pi\)
							−0.101929	−	0.994792i	\(-0.532501\pi\)
\(37\)	11.8225			1.94361			0.971804	−	0.235791i	\(-0.0757680\pi\)
							0.971804	+	0.235791i	\(0.0757680\pi\)
\(41\)	6.75833	−	4.34332i	1.05547	−	0.678312i	0.106708	−	0.994290i	\(-0.465969\pi\)
							0.948766	+	0.315978i	\(0.102333\pi\)
\(43\)	6.95696	−	4.47097i	1.06093	−	0.681816i	0.110851	−	0.993837i	\(-0.464642\pi\)
							0.950075	+	0.312021i	\(0.101006\pi\)
\(47\)	0.442865	−	3.08019i	0.0645985	−	0.449292i	−0.931693	−	0.363247i	\(-0.881668\pi\)
							0.996291	−	0.0860450i	\(-0.0274229\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	201.2.i.a.64.1	yes	50
3.2	odd	2		603.2.u.d.64.5		50
67.22	even	11	inner	201.2.i.a.22.1	✓	50
201.89	odd	22		603.2.u.d.424.5		50

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
201.2.i.a.22.1	✓	50	67.22	even	11	inner
201.2.i.a.64.1	yes	50	1.1	even	1	trivial
603.2.u.d.64.5		50	3.2	odd	2
603.2.u.d.424.5		50	201.89	odd	22