Embedded newform 201.3.n.a.7.8

• Label: 201.3.n.a.7.8
• Level: $201$
• Weight: $3$
• Character: 201.7
• Analytic conductor: $5.477$
• Analytic rank: $0$
• Dimension: $220$
• 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:	\(220\)
Relative dimension:	\(11\) over \(\Q(\zeta_{66})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{66}]$

Embedding invariants

Embedding label			7.8
Character	\(\chi\)	\(=\)	201.7
Dual form			201.3.n.a.115.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.06534 + 2.06647i) q^{2} +(0.487975 - 1.66189i) q^{3} +(-0.815135 + 1.14470i) q^{4} +(6.29564 + 5.45520i) q^{5} +(3.95411 - 0.762093i) q^{6} +(-3.64687 - 0.173722i) q^{7} +(5.97115 + 0.858522i) q^{8} +(-2.52376 - 1.62192i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 220 q - 26 q^{4} + 33 q^{6} + 15 q^{7} - 33 q^{8} + 66 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.06534	+	2.06647i	0.532671	+	1.03324i	0.989615	+	0.143742i	\(0.0459135\pi\)
							−0.456944	+	0.889495i	\(0.651056\pi\)
\(3\)	0.487975	−	1.66189i	0.162658	−	0.553964i
							\(5\)	6.29564	+	5.45520i	1.25913	+	1.09104i	0.991849	+	0.127421i	\(0.0406699\pi\)
0.267279	+	0.963619i	\(0.413876\pi\)
\(7\)	−3.64687	−	0.173722i	−0.520982	−	0.0248174i	−0.214556	−	0.976712i	\(-0.568830\pi\)
							−0.306426	+	0.951894i	\(0.599133\pi\)
\(11\)	2.11655	−	10.9817i	0.192414	−	0.998337i	−0.749379	−	0.662142i	\(-0.769648\pi\)
							0.941792	−	0.336195i	\(-0.109140\pi\)
\(13\)	−0.449998	+	1.12404i	−0.0346152	+	0.0864647i	−0.944668	−	0.328027i	\(-0.893616\pi\)
							0.910053	+	0.414492i	\(0.136041\pi\)
\(17\)	1.87615	+	2.63468i	0.110362	+	0.154981i	0.866063	−	0.499936i	\(-0.166643\pi\)
							−0.755701	+	0.654917i	\(0.772704\pi\)
\(19\)	0.650943	+	13.6650i	0.0342602	+	0.719209i	0.949021	+	0.315213i	\(0.102076\pi\)
							−0.914761	+	0.403996i	\(0.867621\pi\)
\(23\)	−14.4076	+	13.7376i	−0.626418	+	0.597288i	−0.935319	−	0.353806i	\(-0.884887\pi\)
							0.308901	+	0.951094i	\(0.400039\pi\)
\(29\)	3.35135	+	5.80472i	0.115564	+	0.200163i	0.918005	−	0.396569i	\(-0.129799\pi\)
							−0.802441	+	0.596731i	\(0.796466\pi\)
\(31\)	−19.7993	−	49.4563i	−0.638688	−	1.59537i	−0.794285	−	0.607545i	\(-0.792154\pi\)
							0.155597	−	0.987821i	\(-0.450270\pi\)
\(37\)	23.0295	−	39.8883i	0.622419	−	1.07806i	−0.366615	−	0.930373i	\(-0.619483\pi\)
							0.989034	−	0.147688i	\(-0.0471833\pi\)
\(41\)	−2.39410	−	25.0722i	−0.0583927	−	0.611516i	−0.976549	−	0.215297i	\(-0.930928\pi\)
							0.918156	−	0.396219i	\(-0.129678\pi\)
\(43\)	−31.4735	−	14.3735i	−0.731942	−	0.334267i	0.0143394	−	0.999897i	\(-0.495435\pi\)
							−0.746282	+	0.665630i	\(0.768163\pi\)
\(47\)	−6.51545	−	26.8570i	−0.138627	−	0.571426i	−0.998176	−	0.0603735i	\(-0.980771\pi\)
							0.859549	−	0.511053i	\(-0.170744\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	201.3.n.a.7.8	✓	220
67.48	odd	66	inner	201.3.n.a.115.8	yes	220

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
201.3.n.a.7.8	✓	220	1.1	even	1	trivial
201.3.n.a.115.8	yes	220	67.48	odd	66	inner