Embedded newform 201.3.n.b.7.6

• Label: 201.3.n.b.7.6
• Level: $201$
• Weight: $3$
• Character: 201.7
• 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			7.6
Character	\(\chi\)	\(=\)	201.7
Dual form			201.3.n.b.115.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.110765 - 0.214854i) q^{2} +(-0.487975 + 1.66189i) q^{3} +(2.28633 - 3.21071i) q^{4} +(-3.52548 - 3.05485i) q^{5} +(0.411115 - 0.0792359i) q^{6} +(-9.28062 - 0.442091i) q^{7} +(-1.90014 - 0.273199i) q^{8} +(-2.52376 - 1.62192i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 240 q - 34 q^{4} - 33 q^{6} - 21 q^{7} - 33 q^{8} + 72 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(68\)	\(136\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{23}{66}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.110765	−	0.214854i	−0.0553825	−	0.107427i	0.859520	−	0.511103i	\(-0.170763\pi\)
							−0.914902	+	0.403675i	\(0.867732\pi\)
\(3\)	−0.487975	+	1.66189i	−0.162658	+	0.553964i
							\(5\)	−3.52548	−	3.05485i	−0.705097	−	0.610970i	0.226694	−	0.973966i	\(-0.427208\pi\)
−0.931791	+	0.362996i	\(0.881754\pi\)
\(7\)	−9.28062	−	0.442091i	−1.32580	−	0.0631558i	−0.627308	−	0.778771i	\(-0.715843\pi\)
							−0.698495	+	0.715615i	\(0.746146\pi\)
\(11\)	−3.87723	+	20.1170i	−0.352475	+	1.82881i	0.181594	+	0.983374i	\(0.441874\pi\)
							−0.534069	+	0.845441i	\(0.679338\pi\)
\(13\)	−7.60150	+	18.9876i	−0.584731	+	1.46059i	0.280818	+	0.959761i	\(0.409394\pi\)
							−0.865548	+	0.500825i	\(0.833030\pi\)
\(17\)	0.931343	+	1.30789i	0.0547849	+	0.0769346i	0.841060	−	0.540942i	\(-0.181932\pi\)
							−0.786275	+	0.617877i	\(0.787993\pi\)
\(19\)	−1.58011	−	33.1706i	−0.0831636	−	1.74582i	−0.527138	−	0.849780i	\(-0.676735\pi\)
							0.443974	−	0.896039i	\(-0.353568\pi\)
\(23\)	−19.9959	+	19.0661i	−0.869388	+	0.828960i	−0.986433	−	0.164167i	\(-0.947507\pi\)
							0.117044	+	0.993127i	\(0.462658\pi\)
\(29\)	0.104380	+	0.180791i	0.00359931	+	0.00623418i	0.867819	−	0.496880i	\(-0.165521\pi\)
							−0.864220	+	0.503114i	\(0.832188\pi\)
\(31\)	3.34491	+	8.35518i	0.107900	+	0.269522i	0.972485	−	0.232965i	\(-0.0748427\pi\)
							−0.864585	+	0.502487i	\(0.832418\pi\)
\(37\)	18.9944	−	32.8992i	0.513361	−	0.889167i	−0.486519	−	0.873670i	\(-0.661734\pi\)
							0.999880	−	0.0154974i	\(-0.00493316\pi\)
\(41\)	−5.33130	−	55.8319i	−0.130032	−	1.36175i	−0.791219	−	0.611533i	\(-0.790553\pi\)
							0.661187	−	0.750221i	\(-0.270053\pi\)
\(43\)	−42.3945	−	19.3609i	−0.985919	−	0.450254i	−0.143836	−	0.989602i	\(-0.545944\pi\)
							−0.842083	+	0.539347i	\(0.818671\pi\)
\(47\)	−1.24729	−	5.14140i	−0.0265381	−	0.109391i	0.957002	−	0.290081i	\(-0.0936822\pi\)
							−0.983540	+	0.180690i	\(0.942167\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	201.3.n.b.7.6	✓	240
67.48	odd	66	inner	201.3.n.b.115.6	yes	240

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
201.3.n.b.7.6	✓	240	1.1	even	1	trivial
201.3.n.b.115.6	yes	240	67.48	odd	66	inner