Embedded newform 201.3.n.a.115.11

• Label: 201.3.n.a.115.11
• Level: $201$
• Weight: $3$
• Character: 201.115
• Analytic conductor: $5.477$
• Analytic rank: $0$
• Dimension: $220$
• 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			115.11
Character	\(\chi\)	\(=\)	201.115
Dual form			201.3.n.a.7.11

$q$-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 \(\left(\mathbb{Z}/201\mathbb{Z}\right)^\times\).

\(n\)	\(68\)	\(136\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{43}{66}\right)\)

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

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
201.3.n.a.7.11	✓	220	67.7	odd	66	inner
201.3.n.a.115.11	yes	220	1.1	even	1	trivial