Embedded newform 201.3.o.b.17.2

• Label: 201.3.o.b.17.2
• 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			17.2
Character	\(\chi\)	\(=\)	201.17
Dual form			201.3.o.b.71.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.365428 - 3.82693i) q^{2} +(2.09890 + 2.14350i) q^{3} +(-10.5841 + 2.03993i) q^{4} +(-7.82250 - 1.12471i) q^{5} +(7.43604 - 8.81565i) q^{6} +(5.16070 + 7.24719i) q^{7} +(7.34209 + 25.0049i) q^{8} +(-0.189206 + 8.99801i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 840 q - 16 q^{3} - 126 q^{4} - 25 q^{6} - 34 q^{7} - 24 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.365428	−	3.82693i	−0.182714	−	1.91347i	−0.366361	−	0.930473i	\(-0.619396\pi\)
							0.183647	−	0.982992i	\(-0.441210\pi\)
\(3\)	2.09890	+	2.14350i	0.699635	+	0.714501i
							\(5\)	−7.82250	−	1.12471i	−1.56450	−	0.224941i	−0.695050	−	0.718961i	\(-0.744618\pi\)
−0.869451	+	0.494020i	\(0.835527\pi\)
\(7\)	5.16070	+	7.24719i	0.737243	+	1.03531i	0.997562	+	0.0697903i	\(0.0222330\pi\)
							−0.260319	+	0.965523i	\(0.583828\pi\)
\(11\)	7.39673	−	9.40571i	0.672430	−	0.855065i	−0.323402	−	0.946261i	\(-0.604827\pi\)
							0.995833	+	0.0911967i	\(0.0290692\pi\)
\(13\)	1.41970	+	5.85208i	0.109208	+	0.450160i	0.999991	−	0.00426822i	\(-0.00135862\pi\)
							−0.890783	+	0.454429i	\(0.849843\pi\)
\(17\)	−1.71138	+	8.87950i	−0.100670	+	0.522324i	0.896227	+	0.443597i	\(0.146298\pi\)
							−0.996896	+	0.0787271i	\(0.974914\pi\)
\(19\)	−13.6227	+	19.1304i	−0.716985	+	1.00686i	0.281890	+	0.959447i	\(0.409039\pi\)
							−0.998875	+	0.0474180i	\(0.984901\pi\)
\(23\)	−7.75260	+	15.0379i	−0.337069	+	0.653824i	−0.995068	−	0.0991957i	\(-0.968373\pi\)
							0.657998	+	0.753019i	\(0.271403\pi\)
\(29\)	3.71806	+	2.14662i	0.128209	+	0.0740215i	0.562733	−	0.826639i	\(-0.309750\pi\)
							−0.434524	+	0.900660i	\(0.643083\pi\)
\(31\)	−11.6926	+	48.1975i	−0.377180	+	1.55476i	0.394591	+	0.918857i	\(0.370886\pi\)
							−0.771771	+	0.635901i	\(0.780629\pi\)
\(37\)	−15.7797	−	27.3313i	−0.426479	−	0.738684i	0.570078	−	0.821591i	\(-0.306913\pi\)
							−0.996557	+	0.0829068i	\(0.973580\pi\)
\(41\)	−3.19175	+	1.10468i	−0.0778476	+	0.0269433i	−0.365714	−	0.930727i	\(-0.619175\pi\)
							0.287867	+	0.957671i	\(0.407054\pi\)
\(43\)	9.97153	−	11.5078i	0.231896	−	0.267622i	−0.627861	−	0.778325i	\(-0.716069\pi\)
							0.859757	+	0.510703i	\(0.170615\pi\)
\(47\)	−68.8526	+	3.27986i	−1.46495	+	0.0697842i	−0.764826	−	0.644237i	\(-0.777175\pi\)
							−0.700124	+	0.714021i	\(0.746872\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	201.3.o.b.17.2	✓	840
3.2	odd	2	inner	201.3.o.b.17.41	yes	840
67.4	even	33	inner	201.3.o.b.71.41	yes	840
201.71	odd	66	inner	201.3.o.b.71.2	yes	840

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
201.3.o.b.17.2	✓	840	1.1	even	1	trivial
201.3.o.b.17.41	yes	840	3.2	odd	2	inner
201.3.o.b.71.2	yes	840	201.71	odd	66	inner
201.3.o.b.71.41	yes	840	67.4	even	33	inner