Embedded newform 201.2.p.b.2.17

• Label: 201.2.p.b.2.17
• Level: $201$
• Weight: $2$
• Character: 201.2
• Analytic conductor: $1.605$
• Analytic rank: $0$
• Dimension: $400$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 201 = 3 \cdot 67 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	201.p (of order \(66\), degree \(20\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.60499308063\)
Analytic rank:	\(0\)
Dimension:	\(400\)
Relative dimension:	\(20\) over \(\Q(\zeta_{66})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{66}]$

Embedding invariants

Embedding label			2.17
Character	\(\chi\)	\(=\)	201.2
Dual form			201.2.p.b.101.17

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.0942093 - 1.97770i) q^{2} +(-0.549085 - 1.64271i) q^{3} +(-1.91146 - 0.182522i) q^{4} +(2.22606 - 2.56901i) q^{5} +(-3.30051 + 0.931165i) q^{6} +(0.731283 + 1.41849i) q^{7} +(0.0224982 - 0.156478i) q^{8} +(-2.39701 + 1.80398i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 400 q - 22 q^{3} - 20 q^{4} - 20 q^{6} - 38 q^{7} - 30 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.0942093	−	1.97770i	0.0666160	−	1.39844i	−0.680012	−	0.733201i	\(-0.738026\pi\)
							0.746628	−	0.665241i	\(-0.231671\pi\)
\(3\)	−0.549085	−	1.64271i	−0.317015	−	0.948421i
							\(5\)	2.22606	−	2.56901i	0.995523	−	1.14889i	0.00667279	−	0.999978i	\(-0.497876\pi\)
0.988850	−	0.148917i	\(-0.0475786\pi\)
\(7\)	0.731283	+	1.41849i	0.276399	+	0.536139i	0.985647	−	0.168820i	\(-0.0539956\pi\)
							−0.709248	+	0.704959i	\(0.750965\pi\)
\(11\)	0.360437	+	1.04141i	0.108676	+	0.313998i	0.986206	−	0.165524i	\(-0.0529314\pi\)
							−0.877530	+	0.479522i	\(0.840810\pi\)
\(13\)	3.82524	+	4.86419i	1.06093	+	1.34908i	0.934041	+	0.357166i	\(0.116257\pi\)
							0.126890	+	0.991917i	\(0.459501\pi\)
\(17\)	0.266225	+	2.78804i	0.0645691	+	0.676198i	0.968582	+	0.248693i	\(0.0800012\pi\)
							−0.904013	+	0.427505i	\(0.859393\pi\)
\(19\)	−0.525233	−	0.270776i	−0.120497	−	0.0621204i	0.396924	−	0.917851i	\(-0.370078\pi\)
							−0.517421	+	0.855731i	\(0.673108\pi\)
\(23\)	−7.91517	+	1.92020i	−1.65043	+	0.400389i	−0.949730	−	0.313071i	\(-0.898642\pi\)
							−0.700696	+	0.713460i	\(0.747127\pi\)
\(29\)	5.75948	+	3.32523i	1.06951	+	0.617481i	0.928047	−	0.372463i	\(-0.121487\pi\)
							0.141461	+	0.989944i	\(0.454820\pi\)
\(31\)	3.07053	−	3.90450i	0.551483	−	0.701268i	−0.427775	−	0.903885i	\(-0.640702\pi\)
							0.979258	+	0.202617i	\(0.0649447\pi\)
\(37\)	−3.41718	−	5.91874i	−0.561782	−	0.973034i	−0.997341	−	0.0728743i	\(-0.976783\pi\)
							0.435560	−	0.900160i	\(-0.356551\pi\)
\(41\)	−2.59932	−	3.65024i	−0.405946	−	0.570072i	0.560105	−	0.828422i	\(-0.310761\pi\)
							−0.966051	+	0.258350i	\(0.916821\pi\)
\(43\)	3.68352	+	1.68221i	0.561732	+	0.256534i	0.675983	−	0.736918i	\(-0.263719\pi\)
							−0.114251	+	0.993452i	\(0.536447\pi\)
\(47\)	−3.82758	−	4.01426i	−0.558311	−	0.585539i	0.382718	−	0.923865i	\(-0.374988\pi\)
							−0.941029	+	0.338326i	\(0.890139\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	201.2.p.b.2.17	yes	400
3.2	odd	2	inner	201.2.p.b.2.4	✓	400
67.34	odd	66	inner	201.2.p.b.101.4	yes	400
201.101	even	66	inner	201.2.p.b.101.17	yes	400

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
201.2.p.b.2.4	✓	400	3.2	odd	2	inner
201.2.p.b.2.17	yes	400	1.1	even	1	trivial
201.2.p.b.101.4	yes	400	67.34	odd	66	inner
201.2.p.b.101.17	yes	400	201.101	even	66	inner