Embedded newform 201.2.i.a.25.1

• Label: 201.2.i.a.25.1
• Level: $201$
• Weight: $2$
• Character: 201.25
• 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			25.1
Character	\(\chi\)	\(=\)	201.25
Dual form			201.2.i.a.193.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.381059 - 2.65032i) q^{2} +(0.654861 - 0.755750i) q^{3} +(-4.96003 + 1.45640i) q^{4} +(-2.85080 + 1.83210i) q^{5} +(-2.25252 - 1.44761i) q^{6} +(-0.558441 - 3.88404i) q^{7} +(3.52537 + 7.71949i) q^{8} +(-0.142315 - 0.989821i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 50 q - 2 q^{2} + 5 q^{3} - 6 q^{4} + 2 q^{5} - 9 q^{6} + 2 q^{7} + 27 q^{8} - 5 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(68\)	\(136\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{5}{11}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.381059	−	2.65032i	−0.269449	−	1.87406i	−0.453659	−	0.891175i	\(-0.649882\pi\)
							0.184210	−	0.982887i	\(-0.441027\pi\)
\(3\)	0.654861	−	0.755750i	0.378084	−	0.436332i
							\(5\)	−2.85080	+	1.83210i	−1.27492	+	0.819340i	−0.990252	−	0.139286i	\(-0.955519\pi\)
−0.284666	+	0.958627i	\(0.591883\pi\)
\(7\)	−0.558441	−	3.88404i	−0.211071	−	1.46803i	−0.769590	−	0.638538i	\(-0.779539\pi\)
							0.558519	−	0.829491i	\(-0.311370\pi\)
\(11\)	−2.19677	+	1.41178i	−0.662351	+	0.425667i	−0.828161	−	0.560491i	\(-0.810613\pi\)
							0.165810	+	0.986158i	\(0.446976\pi\)
\(13\)	0.236465	−	0.517787i	0.0655837	−	0.143608i	−0.874002	−	0.485923i	\(-0.838484\pi\)
							0.939585	+	0.342315i	\(0.111211\pi\)
\(17\)	−2.66471	−	0.782428i	−0.646286	−	0.189767i	−0.0578690	−	0.998324i	\(-0.518431\pi\)
							−0.588417	+	0.808557i	\(0.700249\pi\)
\(19\)	0.290390	−	2.01971i	0.0666201	−	0.463353i	−0.929017	−	0.370038i	\(-0.879345\pi\)
							0.995637	−	0.0933149i	\(-0.0297463\pi\)
\(23\)	5.15958	−	5.95447i	1.07585	−	1.24159i	0.106914	−	0.994268i	\(-0.465903\pi\)
							0.968933	−	0.247325i	\(-0.0795515\pi\)
\(29\)	−9.79012			−1.81798			−0.908990	−	0.416818i	\(-0.863145\pi\)
							−0.908990	+	0.416818i	\(0.863145\pi\)
\(31\)	−2.94246	−	6.44308i	−0.528481	−	1.15721i	−0.966128	−	0.258065i	\(-0.916915\pi\)
							0.437647	−	0.899147i	\(-0.355812\pi\)
\(37\)	0.00582462			0.000957561			0.000478781	−	1.00000i	\(-0.499848\pi\)
							0.000478781		1.00000i	\(0.499848\pi\)
\(41\)	5.68310	+	1.66871i	0.887551	+	0.260608i	0.693563	−	0.720396i	\(-0.256040\pi\)
							0.193987	+	0.981004i	\(0.437858\pi\)
\(43\)	4.23673	+	1.24402i	0.646095	+	0.189711i	0.588332	−	0.808620i	\(-0.299785\pi\)
							0.0577633	+	0.998330i	\(0.481603\pi\)
\(47\)	0.540991	−	0.624336i	0.0789116	−	0.0910688i	−0.714921	−	0.699205i	\(-0.753537\pi\)
							0.793832	+	0.608137i	\(0.208083\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	201.2.i.a.25.1	✓	50
3.2	odd	2		603.2.u.d.226.5		50
67.59	even	11	inner	201.2.i.a.193.1	yes	50
201.59	odd	22		603.2.u.d.595.5		50

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
201.2.i.a.25.1	✓	50	1.1	even	1	trivial
201.2.i.a.193.1	yes	50	67.59	even	11	inner
603.2.u.d.226.5		50	3.2	odd	2
603.2.u.d.595.5		50	201.59	odd	22