Embedded newform 201.2.p.b.2.14

• Label: 201.2.p.b.2.14
• 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			2.14
Character	\(\chi\)	\(=\)	201.2
Dual form			201.2.p.b.101.14

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.0765690 - 1.60738i) q^{2} +(1.71460 + 0.245273i) q^{3} +(-0.586870 - 0.0560393i) q^{4} +(-0.869862 + 1.00387i) q^{5} +(0.525532 - 2.73723i) q^{6} +(0.664215 + 1.28840i) q^{7} +(0.323015 - 2.24662i) q^{8} +(2.87968 + 0.841088i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 400 q - 22 q^{3} - 20 q^{4} - 20 q^{6} - 38 q^{7} - 30 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.0765690	−	1.60738i	0.0541425	−	1.13659i	−0.794377	−	0.607426i	\(-0.792202\pi\)
							0.848519	−	0.529165i	\(-0.177495\pi\)
\(3\)	1.71460	+	0.245273i	0.989923	+	0.141608i
							\(5\)	−0.869862	+	1.00387i	−0.389014	+	0.448946i	−0.916150	−	0.400835i	\(-0.868720\pi\)
0.527137	+	0.849781i	\(0.323266\pi\)
\(7\)	0.664215	+	1.28840i	0.251050	+	0.486968i	0.980360	−	0.197214i	\(-0.0631895\pi\)
							−0.729311	+	0.684183i	\(0.760159\pi\)
\(11\)	−1.76421	−	5.09735i	−0.531929	−	1.53691i	−0.815969	−	0.578096i	\(-0.803796\pi\)
							0.284039	−	0.958813i	\(-0.408325\pi\)
\(13\)	0.680610	+	0.865466i	0.188767	+	0.240037i	0.871220	−	0.490892i	\(-0.163329\pi\)
							−0.682453	+	0.730930i	\(0.739087\pi\)
\(17\)	0.527029	+	5.51930i	0.127823	+	1.33863i	0.800786	+	0.598950i	\(0.204415\pi\)
							−0.672963	+	0.739676i	\(0.734979\pi\)
\(19\)	−0.966076	−	0.498047i	−0.221633	−	0.114260i	0.343835	−	0.939030i	\(-0.388274\pi\)
							−0.565468	+	0.824770i	\(0.691304\pi\)
\(23\)	−4.33464	+	1.05157i	−0.903834	+	0.219268i	−0.660634	−	0.750708i	\(-0.729712\pi\)
							−0.243200	+	0.969976i	\(0.578197\pi\)
\(29\)	−6.33730	−	3.65884i	−1.17681	−	0.679430i	−0.221533	−	0.975153i	\(-0.571106\pi\)
							−0.955274	+	0.295723i	\(0.904440\pi\)
\(31\)	−2.78603	+	3.54273i	−0.500386	+	0.636292i	−0.968985	−	0.247118i	\(-0.920516\pi\)
							0.468599	+	0.883411i	\(0.344759\pi\)
\(37\)	−4.00886	−	6.94355i	−0.659052	−	1.14151i	−0.980861	−	0.194708i	\(-0.937624\pi\)
							0.321809	−	0.946805i	\(-0.395709\pi\)
\(41\)	3.90734	+	5.48710i	0.610225	+	0.856941i	0.997895	−	0.0648577i	\(-0.0206593\pi\)
							−0.387670	+	0.921798i	\(0.626720\pi\)
\(43\)	−0.399530	−	0.182459i	−0.0609277	−	0.0278248i	0.384718	−	0.923034i	\(-0.374299\pi\)
							−0.445646	+	0.895209i	\(0.647026\pi\)
\(47\)	3.60464	+	3.78044i	0.525791	+	0.551434i	0.932053	−	0.362323i	\(-0.118016\pi\)
							−0.406261	+	0.913757i	\(0.633168\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	201.2.p.b.2.14	yes	400
3.2	odd	2	inner	201.2.p.b.2.7	✓	400
67.34	odd	66	inner	201.2.p.b.101.7	yes	400
201.101	even	66	inner	201.2.p.b.101.14	yes	400

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
201.2.p.b.2.7	✓	400	3.2	odd	2	inner
201.2.p.b.2.14	yes	400	1.1	even	1	trivial
201.2.p.b.101.7	yes	400	67.34	odd	66	inner
201.2.p.b.101.14	yes	400	201.101	even	66	inner