Embedded newform 201.3.o.b.17.11

• Label: 201.3.o.b.17.11
• 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			17.11
Character	\(\chi\)	\(=\)	201.17
Dual form			201.3.o.b.71.11

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.223275 - 2.33824i) q^{2} +(-1.10838 + 2.78774i) q^{3} +(-1.48980 + 0.287135i) q^{4} +(-3.05677 - 0.439497i) q^{5} +(6.76587 + 1.96923i) q^{6} +(2.18219 + 3.06445i) q^{7} +(-1.64299 - 5.59553i) q^{8} +(-6.54298 - 6.17976i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 840 q - 16 q^{3} - 126 q^{4} - 25 q^{6} - 34 q^{7} - 24 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.223275	−	2.33824i	−0.111637	−	1.16912i	−0.861582	−	0.507619i	\(-0.830526\pi\)
							0.749944	−	0.661501i	\(-0.230080\pi\)
\(3\)	−1.10838	+	2.78774i	−0.369461	+	0.929246i
							\(5\)	−3.05677	−	0.439497i	−0.611353	−	0.0878993i	−0.170318	−	0.985389i	\(-0.554480\pi\)
−0.441035	+	0.897490i	\(0.645389\pi\)
\(7\)	2.18219	+	3.06445i	0.311741	+	0.437779i	0.940486	−	0.339833i	\(-0.110370\pi\)
							−0.628745	+	0.777612i	\(0.716431\pi\)
\(11\)	9.13388	−	11.6147i	0.830353	−	1.05588i	−0.167197	−	0.985924i	\(-0.553472\pi\)
							0.997550	−	0.0699563i	\(-0.0222860\pi\)
\(13\)	−2.45368	−	10.1142i	−0.188745	−	0.778016i	−0.984929	−	0.172961i	\(-0.944666\pi\)
							0.796184	−	0.605055i	\(-0.206849\pi\)
\(17\)	3.59197	−	18.6369i	0.211292	−	1.09629i	−0.709045	−	0.705163i	\(-0.750874\pi\)
							0.920338	−	0.391125i	\(-0.127914\pi\)
\(19\)	3.61076	−	5.07061i	0.190040	−	0.266874i	−0.708552	−	0.705658i	\(-0.750651\pi\)
							0.898592	+	0.438784i	\(0.144591\pi\)
\(23\)	4.42782	−	8.58878i	0.192514	−	0.373425i	−0.772746	−	0.634715i	\(-0.781118\pi\)
							0.965260	+	0.261290i	\(0.0841478\pi\)
\(29\)	−13.1720	−	7.60484i	−0.454206	−	0.262236i	0.255399	−	0.966836i	\(-0.417793\pi\)
							−0.709605	+	0.704600i	\(0.751126\pi\)
\(31\)	0.0191240	−	0.0788301i	0.000616903	−	0.00254291i	−0.971502	−	0.237030i	\(-0.923826\pi\)
							0.972119	+	0.234487i	\(0.0753411\pi\)
\(37\)	29.6314	+	51.3231i	0.800848	+	1.38711i	0.919058	+	0.394121i	\(0.128951\pi\)
							−0.118210	+	0.992989i	\(0.537716\pi\)
\(41\)	−32.3615	+	11.2004i	−0.789305	+	0.273181i	−0.691833	−	0.722057i	\(-0.743197\pi\)
							−0.0974716	+	0.995238i	\(0.531075\pi\)
\(43\)	−4.31339	+	4.97792i	−0.100311	+	0.115765i	−0.803685	−	0.595054i	\(-0.797130\pi\)
							0.703374	+	0.710820i	\(0.251676\pi\)
\(47\)	22.1539	−	1.05532i	0.471361	−	0.0224537i	0.189442	−	0.981892i	\(-0.439332\pi\)
							0.281919	+	0.959438i	\(0.409029\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	201.3.o.b.17.11	✓	840
3.2	odd	2	inner	201.3.o.b.17.32	yes	840
67.4	even	33	inner	201.3.o.b.71.32	yes	840
201.71	odd	66	inner	201.3.o.b.71.11	yes	840

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
201.3.o.b.17.11	✓	840	1.1	even	1	trivial
201.3.o.b.17.32	yes	840	3.2	odd	2	inner
201.3.o.b.71.11	yes	840	201.71	odd	66	inner
201.3.o.b.71.32	yes	840	67.4	even	33	inner