Embedded newform 201.3.o.b.17.5

• Label: 201.3.o.b.17.5
• Level: $201$
• Weight: $3$
• Character: 201.17
• Analytic conductor: $5.477$
• Analytic rank: $0$
• Dimension: $840$
• CM: no
• Inner twists: $4$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(5.47685331364\)
Analytic rank:	\(0\)
Dimension:	\(840\)
Relative dimension:	\(42\) over \(\Q(\zeta_{66})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{66}]$

Embedding invariants

Embedding label			17.5
Character	\(\chi\)	\(=\)	201.17
Dual form			201.3.o.b.71.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.315296 - 3.30193i) q^{2} +(-0.380346 + 2.97579i) q^{3} +(-6.87561 + 1.32517i) q^{4} +(1.00410 + 0.144367i) q^{5} +(9.94578 + 0.317620i) q^{6} +(-4.51680 - 6.34296i) q^{7} +(2.80549 + 9.55461i) q^{8} +(-8.71067 - 2.26366i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 840 q - 16 q^{3} - 126 q^{4} - 25 q^{6} - 34 q^{7} - 24 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(68\)	\(136\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{32}{33}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.315296	−	3.30193i	−0.157648	−	1.65096i	−0.635190	−	0.772356i	\(-0.719078\pi\)
							0.477542	−	0.878609i	\(-0.341528\pi\)
\(3\)	−0.380346	+	2.97579i	−0.126782	+	0.991931i
							\(5\)	1.00410	+	0.144367i	0.200819	+	0.0288734i	0.241990	−	0.970279i	\(-0.422200\pi\)
−0.0411711	+	0.999152i	\(0.513109\pi\)
\(7\)	−4.51680	−	6.34296i	−0.645257	−	0.906136i	0.354323	−	0.935123i	\(-0.384711\pi\)
							−0.999580	+	0.0289866i	\(0.990772\pi\)
\(11\)	−8.10788	+	10.3100i	−0.737080	+	0.937274i	−0.999618	−	0.0276480i	\(-0.991198\pi\)
							0.262537	+	0.964922i	\(0.415441\pi\)
\(13\)	0.984731	+	4.05912i	0.0757485	+	0.312240i	0.997064	−	0.0765770i	\(-0.0243991\pi\)
							−0.921315	+	0.388817i	\(0.872884\pi\)
\(17\)	−3.01761	+	15.6568i	−0.177506	+	0.920991i	0.778500	+	0.627645i	\(0.215981\pi\)
							−0.956006	+	0.293346i	\(0.905231\pi\)
\(19\)	−3.53270	+	4.96098i	−0.185931	+	0.261104i	−0.897012	−	0.442007i	\(-0.854267\pi\)
							0.711080	+	0.703111i	\(0.248206\pi\)
\(23\)	−14.6337	+	28.3854i	−0.636247	+	1.23415i	0.321602	+	0.946875i	\(0.395779\pi\)
							−0.957849	+	0.287272i	\(0.907252\pi\)
\(29\)	−26.4141	−	15.2502i	−0.910831	−	0.525869i	−0.0301326	−	0.999546i	\(-0.509593\pi\)
							−0.880698	+	0.473677i	\(0.842926\pi\)
\(31\)	5.99019	−	24.6919i	0.193232	−	0.796513i	−0.789843	−	0.613309i	\(-0.789838\pi\)
							0.983075	−	0.183204i	\(-0.0586468\pi\)
\(37\)	−20.7557	−	35.9499i	−0.560964	−	0.971619i	−0.997413	−	0.0718893i	\(-0.977097\pi\)
							0.436448	−	0.899729i	\(-0.356236\pi\)
\(41\)	67.8289	−	23.4758i	1.65436	−	0.572580i	0.669032	−	0.743234i	\(-0.266709\pi\)
							0.985331	+	0.170653i	\(0.0545879\pi\)
\(43\)	2.02993	−	2.34266i	0.0472076	−	0.0544805i	−0.731655	−	0.681675i	\(-0.761252\pi\)
							0.778862	+	0.627195i	\(0.215797\pi\)
\(47\)	−13.7454	+	0.654775i	−0.292456	+	0.0139314i	−0.193297	−	0.981140i	\(-0.561918\pi\)
							−0.0991586	+	0.995072i	\(0.531615\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	201.3.o.b.17.5	✓	840
3.2	odd	2	inner	201.3.o.b.17.38	yes	840
67.4	even	33	inner	201.3.o.b.71.38	yes	840
201.71	odd	66	inner	201.3.o.b.71.5	yes	840

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
201.3.o.b.17.5	✓	840	1.1	even	1	trivial
201.3.o.b.17.38	yes	840	3.2	odd	2	inner
201.3.o.b.71.5	yes	840	201.71	odd	66	inner
201.3.o.b.71.38	yes	840	67.4	even	33	inner