Embedded newform 201.3.n.b.13.7

• Label: 201.3.n.b.13.7
• 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			13.7
Character	\(\chi\)	\(=\)	201.13
Dual form			201.3.n.b.31.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.127437 + 0.162049i) q^{2} +(-1.30900 - 1.13425i) q^{3} +(0.933016 - 3.84594i) q^{4} +(2.68870 + 4.18370i) q^{5} +(0.0169901 - 0.356667i) q^{6} +(1.06344 - 2.65636i) q^{7} +(1.49223 - 0.681481i) q^{8} +(0.426945 + 2.96946i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 240 q - 34 q^{4} - 33 q^{6} - 21 q^{7} - 33 q^{8} + 72 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.127437	+	0.162049i	0.0637184	+	0.0810245i	0.816868	−	0.576825i	\(-0.195709\pi\)
							−0.753149	+	0.657850i	\(0.771466\pi\)
\(3\)	−1.30900	−	1.13425i	−0.436332	−	0.378084i
							\(5\)	2.68870	+	4.18370i	0.537740	+	0.836740i	0.998715	−	0.0506729i	\(-0.0161366\pi\)
−0.460975	+	0.887413i	\(0.652500\pi\)
\(7\)	1.06344	−	2.65636i	0.151921	−	0.379479i	−0.832811	−	0.553557i	\(-0.813270\pi\)
							0.984732	+	0.174078i	\(0.0556943\pi\)
\(11\)	9.85173	−	0.469296i	0.895612	−	0.0426632i	0.405274	−	0.914195i	\(-0.367176\pi\)
							0.490338	+	0.871532i	\(0.336873\pi\)
\(13\)	−1.32871	−	13.9149i	−0.102209	−	1.07038i	−0.890471	−	0.455039i	\(-0.849625\pi\)
							0.788263	−	0.615339i	\(-0.210981\pi\)
\(17\)	−3.57852	−	14.7509i	−0.210501	−	0.867697i	−0.974844	−	0.222887i	\(-0.928452\pi\)
							0.764343	−	0.644810i	\(-0.223063\pi\)
\(19\)	7.28438	−	2.91623i	0.383389	−	0.153486i	−0.171959	−	0.985104i	\(-0.555010\pi\)
							0.555347	+	0.831619i	\(0.312585\pi\)
\(23\)	3.22175	+	0.620941i	0.140076	+	0.0269974i	0.258807	−	0.965929i	\(-0.416671\pi\)
							−0.118731	+	0.992926i	\(0.537883\pi\)
\(29\)	10.0626	−	17.4289i	0.346986	−	0.600997i	−0.638727	−	0.769434i	\(-0.720539\pi\)
							0.985712	+	0.168437i	\(0.0538719\pi\)
\(31\)	−0.0383598	+	0.401722i	−0.00123741	+	0.0129588i	−0.996070	−	0.0885746i	\(-0.971769\pi\)
							0.994832	+	0.101533i	\(0.0323749\pi\)
\(37\)	7.57029	+	13.1121i	0.204603	+	0.354382i	0.950006	−	0.312232i	\(-0.101076\pi\)
							−0.745403	+	0.666614i	\(0.767743\pi\)
\(41\)	17.4663	+	18.3181i	0.426007	+	0.446783i	0.901398	−	0.432991i	\(-0.142542\pi\)
							−0.475391	+	0.879774i	\(0.657693\pi\)
\(43\)	−20.8826	+	71.1197i	−0.485643	+	1.65395i	0.243739	+	0.969841i	\(0.421626\pi\)
							−0.729382	+	0.684106i	\(0.760192\pi\)
\(47\)	10.3664	+	29.9517i	0.220562	+	0.637271i	0.999943	+	0.0107205i	\(0.00341250\pi\)
							−0.779381	+	0.626550i	\(0.784466\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	201.3.n.b.13.7	✓	240
67.31	odd	66	inner	201.3.n.b.31.7	yes	240

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
201.3.n.b.13.7	✓	240	1.1	even	1	trivial
201.3.n.b.31.7	yes	240	67.31	odd	66	inner