Embedded newform 201.3.n.a.7.11

• Label: 201.3.n.a.7.11
• 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			7.11
Character	\(\chi\)	\(=\)	201.7
Dual form			201.3.n.a.115.11

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.65710 + 3.21433i) q^{2} +(0.487975 - 1.66189i) q^{3} +(-5.26571 + 7.39465i) q^{4} +(2.82046 + 2.44394i) q^{5} +(6.15049 - 1.18541i) q^{6} +(1.70167 + 0.0810603i) q^{7} +(-18.1765 - 2.61339i) q^{8} +(-2.52376 - 1.62192i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 220 q - 26 q^{4} + 33 q^{6} + 15 q^{7} - 33 q^{8} + 66 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.65710	+	3.21433i	0.828551	+	1.60717i	0.794122	+	0.607758i	\(0.207931\pi\)
							0.0344292	+	0.999407i	\(0.489039\pi\)
\(3\)	0.487975	−	1.66189i	0.162658	−	0.553964i
							\(5\)	2.82046	+	2.44394i	0.564092	+	0.488789i	0.889594	−	0.456753i	\(-0.150988\pi\)
−0.325501	+	0.945542i	\(0.605533\pi\)
\(7\)	1.70167	+	0.0810603i	0.243095	+	0.0115800i	0.168776	−	0.985654i	\(-0.446019\pi\)
							0.0743192	+	0.997235i	\(0.476322\pi\)
\(11\)	−3.78345	+	19.6304i	−0.343950	+	1.78458i	0.237903	+	0.971289i	\(0.423540\pi\)
							−0.581853	+	0.813294i	\(0.697672\pi\)
\(13\)	0.0520294	−	0.129963i	0.00400226	−	0.00999717i	−0.926353	−	0.376656i	\(-0.877074\pi\)
							0.930356	+	0.366658i	\(0.119498\pi\)
\(17\)	3.60645	+	5.06455i	0.212144	+	0.297915i	0.906913	−	0.421319i	\(-0.138433\pi\)
							−0.694769	+	0.719233i	\(0.744493\pi\)
\(19\)	−1.04033	−	21.8393i	−0.0547544	−	1.14944i	−0.844344	−	0.535801i	\(-0.820010\pi\)
							0.789590	−	0.613635i	\(-0.210293\pi\)
\(23\)	27.1749	−	25.9112i	1.18152	−	1.12657i	0.192113	−	0.981373i	\(-0.438466\pi\)
							0.989402	−	0.145200i	\(-0.0463826\pi\)
\(29\)	8.69528	+	15.0607i	0.299837	+	0.519334i	0.976099	−	0.217328i	\(-0.0697343\pi\)
							−0.676261	+	0.736662i	\(0.736401\pi\)
\(31\)	6.77117	+	16.9136i	0.218425	+	0.545599i	0.996595	−	0.0824548i	\(-0.0262760\pi\)
							−0.778170	+	0.628054i	\(0.783852\pi\)
\(37\)	12.6761	−	21.9557i	0.342599	−	0.593398i	−0.642316	−	0.766440i	\(-0.722026\pi\)
							0.984914	+	0.173042i	\(0.0553596\pi\)
\(41\)	−4.34420	−	45.4946i	−0.105956	−	1.10962i	−0.879506	−	0.475888i	\(-0.842127\pi\)
							0.773550	−	0.633735i	\(-0.218479\pi\)
\(43\)	32.5145	+	14.8489i	0.756152	+	0.345323i	0.755909	−	0.654677i	\(-0.227195\pi\)
							0.000243096		1.00000i	\(0.499923\pi\)
\(47\)	4.76266	+	19.6319i	0.101333	+	0.417701i	0.999771	−	0.0213997i	\(-0.00681227\pi\)
							−0.898438	+	0.439101i	\(0.855297\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	201.3.n.a.7.11	✓	220
67.48	odd	66	inner	201.3.n.a.115.11	yes	220

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
201.3.n.a.7.11	✓	220	1.1	even	1	trivial
201.3.n.a.115.11	yes	220	67.48	odd	66	inner