Embedded newform 201.3.n.a.7.6

• Label: 201.3.n.a.7.6
• 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			7.6
Character	\(\chi\)	\(=\)	201.7
Dual form			201.3.n.a.115.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.131330 - 0.254745i) q^{2} +(0.487975 - 1.66189i) q^{3} +(2.27258 - 3.19139i) q^{4} +(4.91546 + 4.25927i) q^{5} +(-0.487444 + 0.0939471i) q^{6} +(9.56392 + 0.455585i) q^{7} +(-2.24620 - 0.322955i) q^{8} +(-2.52376 - 1.62192i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 220 q - 26 q^{4} + 33 q^{6} + 15 q^{7} - 33 q^{8} + 66 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.131330	−	0.254745i	−0.0656651	−	0.127372i	0.853718	−	0.520735i	\(-0.174342\pi\)
							−0.919383	+	0.393363i	\(0.871312\pi\)
\(3\)	0.487975	−	1.66189i	0.162658	−	0.553964i
							\(5\)	4.91546	+	4.25927i	0.983092	+	0.851854i	0.988968	−	0.148129i	\(-0.0473250\pi\)
−0.00587607	+	0.999983i	\(0.501870\pi\)
\(7\)	9.56392	+	0.455585i	1.36627	+	0.0650836i	0.717851	−	0.696197i	\(-0.245126\pi\)
							0.648423	+	0.761280i	\(0.275429\pi\)
\(11\)	−2.94023	+	15.2553i	−0.267293	+	1.38685i	0.561584	+	0.827420i	\(0.310192\pi\)
							−0.828878	+	0.559430i	\(0.811020\pi\)
\(13\)	−1.26409	+	3.15756i	−0.0972380	+	0.242889i	−0.969009	−	0.247024i	\(-0.920547\pi\)
							0.871771	+	0.489913i	\(0.162972\pi\)
\(17\)	−3.03933	−	4.26814i	−0.178784	−	0.251067i	0.715453	−	0.698661i	\(-0.246220\pi\)
							−0.894237	+	0.447594i	\(0.852281\pi\)
\(19\)	0.0760234	+	1.59593i	0.00400123	+	0.0839962i	0.999985	−	0.00554172i	\(-0.00176399\pi\)
							−0.995983	+	0.0895380i	\(0.971461\pi\)
\(23\)	4.23450	−	4.03758i	0.184109	−	0.175547i	−0.592420	−	0.805629i	\(-0.701827\pi\)
							0.776528	+	0.630082i	\(0.216979\pi\)
\(29\)	−21.5196	−	37.2731i	−0.742056	−	1.28528i	−0.951558	−	0.307470i	\(-0.900518\pi\)
							0.209502	−	0.977808i	\(-0.432816\pi\)
\(31\)	−16.2266	−	40.5320i	−0.523438	−	1.30748i	−0.920876	−	0.389856i	\(-0.872525\pi\)
							0.397438	−	0.917629i	\(-0.369899\pi\)
\(37\)	−16.8581	+	29.1990i	−0.455623	+	0.789163i	−0.998724	−	0.0505049i	\(-0.983917\pi\)
							0.543100	+	0.839668i	\(0.317250\pi\)
\(41\)	4.39083	+	45.9829i	0.107093	+	1.12153i	0.876046	+	0.482228i	\(0.160172\pi\)
							−0.768953	+	0.639306i	\(0.779222\pi\)
\(43\)	−35.3405	−	16.1395i	−0.821873	−	0.375337i	−0.0403287	−	0.999186i	\(-0.512841\pi\)
							−0.781544	+	0.623850i	\(0.785568\pi\)
\(47\)	2.13482	+	8.79984i	0.0454217	+	0.187231i	0.990003	−	0.141047i	\(-0.0450469\pi\)
							−0.944581	+	0.328278i	\(0.893532\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	201.3.n.a.7.6	✓	220
67.48	odd	66	inner	201.3.n.a.115.6	yes	220

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
201.3.n.a.7.6	✓	220	1.1	even	1	trivial
201.3.n.a.115.6	yes	220	67.48	odd	66	inner