Embedded newform 201.2.i.a.64.4

• Label: 201.2.i.a.64.4
• 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			64.4
Character	\(\chi\)	\(=\)	201.64
Dual form			201.2.i.a.22.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.907634 + 0.266505i) q^{2} +(0.142315 - 0.989821i) q^{3} +(-0.929732 - 0.597503i) q^{4} +(1.48348 + 3.24837i) q^{5} +(0.392963 - 0.860468i) q^{6} +(4.49742 + 1.32056i) q^{7} +(-1.92355 - 2.21990i) q^{8} +(-0.959493 - 0.281733i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 50 q - 2 q^{2} + 5 q^{3} - 6 q^{4} + 2 q^{5} - 9 q^{6} + 2 q^{7} + 27 q^{8} - 5 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.907634	+	0.266505i	0.641794	+	0.188448i	0.586406	−	0.810017i	\(-0.300542\pi\)
							0.0553880	+	0.998465i	\(0.482360\pi\)
\(3\)	0.142315	−	0.989821i	0.0821655	−	0.571474i
							\(5\)	1.48348	+	3.24837i	0.663433	+	1.45272i	0.879288	+	0.476291i	\(0.158019\pi\)
−0.215855	+	0.976425i	\(0.569254\pi\)
\(7\)	4.49742	+	1.32056i	1.69987	+	0.499126i	0.980668	−	0.195679i	\(-0.0626912\pi\)
							0.719198	+	0.694805i	\(0.244509\pi\)
\(11\)	−1.52114	−	3.33083i	−0.458641	−	1.00428i	−0.987795	−	0.155758i	\(-0.950218\pi\)
							0.529154	−	0.848526i	\(-0.322509\pi\)
\(13\)	1.07662	−	1.24249i	0.298601	−	0.344604i	−0.586546	−	0.809916i	\(-0.699513\pi\)
							0.885147	+	0.465312i	\(0.154058\pi\)
\(17\)	−2.35497	+	1.51345i	−0.571164	+	0.367065i	−0.794117	−	0.607764i	\(-0.792066\pi\)
							0.222954	+	0.974829i	\(0.428430\pi\)
\(19\)	1.54634	−	0.454045i	0.354754	−	0.104165i	−0.0994997	−	0.995038i	\(-0.531724\pi\)
							0.454253	+	0.890872i	\(0.349906\pi\)
\(23\)	−0.635483	+	4.41988i	−0.132507	+	0.921610i	0.809763	+	0.586757i	\(0.199596\pi\)
							−0.942271	+	0.334853i	\(0.891313\pi\)
\(29\)	−8.45931			−1.57085			−0.785427	−	0.618954i	\(-0.787557\pi\)
							−0.785427	+	0.618954i	\(0.787557\pi\)
\(31\)	−5.21068	−	6.01345i	−0.935867	−	1.08005i	−0.996641	−	0.0818961i	\(-0.973902\pi\)
							0.0607743	−	0.998152i	\(-0.480643\pi\)
\(37\)	−9.10163			−1.49630			−0.748149	−	0.663530i	\(-0.769057\pi\)
							−0.748149	+	0.663530i	\(0.769057\pi\)
\(41\)	5.87487	−	3.77555i	0.917500	−	0.589642i	0.00556882	−	0.999984i	\(-0.498227\pi\)
							0.911931	+	0.410343i	\(0.134591\pi\)
\(43\)	3.40181	−	2.18621i	0.518771	−	0.333394i	−0.254915	−	0.966964i	\(-0.582047\pi\)
							0.773686	+	0.633570i	\(0.218411\pi\)
\(47\)	−0.523158	+	3.63865i	−0.0763105	+	0.530751i	0.915429	+	0.402480i	\(0.131852\pi\)
							−0.991739	+	0.128271i	\(0.959057\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	201.2.i.a.64.4	yes	50
3.2	odd	2		603.2.u.d.64.2		50
67.22	even	11	inner	201.2.i.a.22.4	✓	50
201.89	odd	22		603.2.u.d.424.2		50

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
201.2.i.a.22.4	✓	50	67.22	even	11	inner
201.2.i.a.64.4	yes	50	1.1	even	1	trivial
603.2.u.d.64.2		50	3.2	odd	2
603.2.u.d.424.2		50	201.89	odd	22