Embedded newform 201.3.n.b.13.3

• Label: 201.3.n.b.13.3
• Level: $201$
• Weight: $3$
• Character: 201.13
• Analytic conductor: $5.477$
• Analytic rank: $0$
• Dimension: $240$
• CM: no
• Inner twists: $2$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(5.47685331364\)
Analytic rank:	\(0\)
Dimension:	\(240\)
Relative dimension:	\(12\) over \(\Q(\zeta_{66})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{66}]$

Embedding invariants

Embedding label			13.3
Character	\(\chi\)	\(=\)	201.13
Dual form			201.3.n.b.31.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.77293 - 2.25446i) q^{2} +(-1.30900 - 1.13425i) q^{3} +(-0.996290 + 4.10676i) q^{4} +(0.859686 + 1.33770i) q^{5} +(-0.236371 + 4.96203i) q^{6} +(-1.21984 + 3.04701i) q^{7} +(0.589306 - 0.269127i) q^{8} +(0.426945 + 2.96946i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 240 q - 34 q^{4} - 33 q^{6} - 21 q^{7} - 33 q^{8} + 72 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(68\)	\(136\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{19}{66}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.77293	−	2.25446i	−0.886465	−	1.12723i	−0.991153	−	0.132726i	\(-0.957627\pi\)
							0.104688	−	0.994505i	\(-0.466616\pi\)
\(3\)	−1.30900	−	1.13425i	−0.436332	−	0.378084i
							\(5\)	0.859686	+	1.33770i	0.171937	+	0.267540i	0.916519	−	0.399992i	\(-0.130987\pi\)
−0.744581	+	0.667532i	\(0.767351\pi\)
\(7\)	−1.21984	+	3.04701i	−0.174263	+	0.435288i	−0.989627	−	0.143661i	\(-0.954113\pi\)
							0.815364	+	0.578949i	\(0.196537\pi\)
\(11\)	16.1651	−	0.770039i	1.46955	−	0.0700035i	0.702545	−	0.711639i	\(-0.252047\pi\)
							0.767010	+	0.641636i	\(0.221744\pi\)
\(13\)	1.82895	+	19.1536i	0.140688	+	1.47336i	0.739874	+	0.672746i	\(0.234885\pi\)
							−0.599186	+	0.800610i	\(0.704509\pi\)
\(17\)	−2.10366	−	8.67141i	−0.123745	−	0.510083i	−0.999621	−	0.0275461i	\(-0.991231\pi\)
							0.875876	−	0.482537i	\(-0.160284\pi\)
\(19\)	−16.4422	+	6.58248i	−0.865381	+	0.346446i	−0.761517	−	0.648145i	\(-0.775545\pi\)
							−0.103864	+	0.994591i	\(0.533121\pi\)
\(23\)	19.5577	+	3.76944i	0.850336	+	0.163889i	0.595758	−	0.803164i	\(-0.296852\pi\)
							0.254577	+	0.967052i	\(0.418064\pi\)
\(29\)	18.4988	−	32.0409i	0.637891	−	1.10486i	−0.348004	−	0.937493i	\(-0.613141\pi\)
							0.985895	−	0.167366i	\(-0.0535261\pi\)
\(31\)	−3.99936	+	41.8832i	−0.129012	+	1.35107i	0.666669	+	0.745354i	\(0.267719\pi\)
							−0.795681	+	0.605716i	\(0.792887\pi\)
\(37\)	1.19379	+	2.06770i	0.0322646	+	0.0558839i	0.881707	−	0.471798i	\(-0.156395\pi\)
							−0.849442	+	0.527682i	\(0.823061\pi\)
\(41\)	−0.200111	−	0.209870i	−0.00488076	−	0.00511879i	0.721289	−	0.692634i	\(-0.243550\pi\)
							−0.726170	+	0.687515i	\(0.758701\pi\)
\(43\)	2.59025	−	8.82158i	0.0602384	−	0.205153i	−0.923875	−	0.382694i	\(-0.874996\pi\)
							0.984113	+	0.177541i	\(0.0568144\pi\)
\(47\)	20.0447	+	57.9154i	0.426484	+	1.23224i	0.929775	+	0.368127i	\(0.120001\pi\)
							−0.503292	+	0.864117i	\(0.667878\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	201.3.n.b.13.3	✓	240
67.31	odd	66	inner	201.3.n.b.31.3	yes	240

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
201.3.n.b.13.3	✓	240	1.1	even	1	trivial
201.3.n.b.31.3	yes	240	67.31	odd	66	inner