Embedded newform 201.2.p.b.2.2

• Label: 201.2.p.b.2.2
• 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			2.2
Character	\(\chi\)	\(=\)	201.2
Dual form			201.2.p.b.101.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.111707 + 2.34502i) q^{2} +(1.26924 - 1.17858i) q^{3} +(-3.49572 - 0.333801i) q^{4} +(2.49839 - 2.88330i) q^{5} +(2.62201 + 3.10804i) q^{6} +(-0.0230737 - 0.0447567i) q^{7} +(0.505048 - 3.51269i) q^{8} +(0.221918 - 2.99178i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 400 q - 22 q^{3} - 20 q^{4} - 20 q^{6} - 38 q^{7} - 30 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.111707	+	2.34502i	−0.0789890	+	1.65818i	0.517590	+	0.855629i	\(0.326829\pi\)
							−0.596579	+	0.802554i	\(0.703474\pi\)
\(3\)	1.26924	−	1.17858i	0.732794	−	0.680451i
							\(5\)	2.49839	−	2.88330i	1.11732	−	1.28945i	0.164340	−	0.986404i	\(-0.447451\pi\)
0.952976	−	0.303047i	\(-0.0980039\pi\)
\(7\)	−0.0230737	−	0.0447567i	−0.00872103	−	0.0169164i	0.884435	−	0.466664i	\(-0.154544\pi\)
							−0.893156	+	0.449748i	\(0.851514\pi\)
\(11\)	1.32872	+	3.83910i	0.400626	+	1.15753i	0.947345	+	0.320215i	\(0.103755\pi\)
							−0.546719	+	0.837316i	\(0.684123\pi\)
\(13\)	−1.97340	−	2.50938i	−0.547323	−	0.695978i	0.431167	−	0.902272i	\(-0.358102\pi\)
							−0.978490	+	0.206294i	\(0.933860\pi\)
\(17\)	0.593528	+	6.21570i	0.143952	+	1.50753i	0.722209	+	0.691675i	\(0.243127\pi\)
							−0.578257	+	0.815854i	\(0.696267\pi\)
\(19\)	0.719813	+	0.371089i	0.165136	+	0.0851337i	0.538813	−	0.842425i	\(-0.318873\pi\)
							−0.373677	+	0.927559i	\(0.621903\pi\)
\(23\)	−5.80192	+	1.40753i	−1.20978	+	0.293490i	−0.789413	−	0.613862i	\(-0.789615\pi\)
							−0.420370	+	0.907353i	\(0.638100\pi\)
\(29\)	−2.67490	−	1.54435i	−0.496716	−	0.286779i	0.230640	−	0.973039i	\(-0.425918\pi\)
							−0.727356	+	0.686260i	\(0.759251\pi\)
\(31\)	−3.30270	+	4.19972i	−0.593182	+	0.754292i	−0.986258	−	0.165214i	\(-0.947169\pi\)
							0.393076	+	0.919506i	\(0.371411\pi\)
\(37\)	2.01217	+	3.48517i	0.330798	+	0.572959i	0.982669	−	0.185371i	\(-0.0593488\pi\)
							−0.651871	+	0.758330i	\(0.726015\pi\)
\(41\)	5.19569	+	7.29633i	0.811431	+	1.13949i	0.987885	+	0.155186i	\(0.0495977\pi\)
							−0.176455	+	0.984309i	\(0.556463\pi\)
\(43\)	4.57705	+	2.09027i	0.697993	+	0.318763i	0.732620	−	0.680638i	\(-0.238297\pi\)
							−0.0346274	+	0.999400i	\(0.511024\pi\)
\(47\)	2.97767	+	3.12289i	0.434337	+	0.455520i	0.904130	−	0.427257i	\(-0.140520\pi\)
							−0.469793	+	0.882777i	\(0.655671\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	201.2.p.b.2.2	✓	400
3.2	odd	2	inner	201.2.p.b.2.19	yes	400
67.34	odd	66	inner	201.2.p.b.101.19	yes	400
201.101	even	66	inner	201.2.p.b.101.2	yes	400

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
201.2.p.b.2.2	✓	400	1.1	even	1	trivial
201.2.p.b.2.19	yes	400	3.2	odd	2	inner
201.2.p.b.101.2	yes	400	201.101	even	66	inner
201.2.p.b.101.19	yes	400	67.34	odd	66	inner