Embedded newform 270.2.r.a.113.18

• Label: 270.2.r.a.113.18
• Level: $270$
• Weight: $2$
• Character: 270.113
• Analytic conductor: $2.156$
• Analytic rank: $0$
• Dimension: $216$
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 270 = 2 \cdot 3^{3} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	270.r (of order \(36\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.15596085457\)
Analytic rank:	\(0\)
Dimension:	\(216\)
Relative dimension:	\(18\) over \(\Q(\zeta_{36})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{36}]$

Embedding invariants

Embedding label			113.18
Character	\(\chi\)	\(=\)	270.113
Dual form			270.2.r.a.227.18

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.996195 + 0.0871557i) q^{2} +(1.50372 - 0.859551i) q^{3} +(0.984808 + 0.173648i) q^{4} +(-1.39703 + 1.74594i) q^{5} +(1.57291 - 0.725222i) q^{6} +(0.564372 + 0.395177i) q^{7} +(0.965926 + 0.258819i) q^{8} +(1.52234 - 2.58505i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 216 q + 12 q^{6}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/270\mathbb{Z}\right)^\times\).

\(n\)	\(191\)	\(217\)
\(\chi(n)\)	\(e\left(\frac{5}{18}\right)\)	\(e\left(\frac{3}{4}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.996195	+	0.0871557i	0.704416	+	0.0616284i
							\(3\)	1.50372	−	0.859551i	0.868173	−	0.496262i
\(5\)	−1.39703	+	1.74594i	−0.624772	+	0.780807i
							\(7\)	0.564372	+	0.395177i	0.213312	+	0.149363i	0.675350	−	0.737497i	\(-0.263993\pi\)
−0.462038	+	0.886860i	\(0.652881\pi\)
\(11\)	1.76146	−	4.83957i	0.531100	−	1.45919i	−0.326663	−	0.945141i	\(-0.605924\pi\)
							0.857763	−	0.514045i	\(-0.171853\pi\)
\(13\)	0.446561	+	5.10422i	0.123854	+	1.41565i	0.762915	+	0.646499i	\(0.223767\pi\)
							−0.639061	+	0.769156i	\(0.720677\pi\)
\(17\)	0.221302	−	0.0592977i	0.0536736	−	0.0143818i	−0.231882	−	0.972744i	\(-0.574488\pi\)
							0.285556	+	0.958362i	\(0.407822\pi\)
\(19\)	−6.21536	+	3.58844i	−1.42590	+	0.823245i	−0.996794	−	0.0800089i	\(-0.974505\pi\)
							−0.429107	+	0.903253i	\(0.641172\pi\)
\(23\)	−4.94460	−	7.06162i	−1.03102	−	1.47245i	−0.874653	−	0.484749i	\(-0.838911\pi\)
							−0.156366	−	0.987699i	\(-0.549978\pi\)
\(29\)	−3.94345	+	3.30895i	−0.732281	+	0.614457i	−0.930752	−	0.365650i	\(-0.880847\pi\)
							0.198471	+	0.980107i	\(0.436402\pi\)
\(31\)	0.210869	−	1.19590i	0.0378731	−	0.214789i	−0.959998	−	0.280007i	\(-0.909663\pi\)
							0.997871	+	0.0652179i	\(0.0207743\pi\)
\(37\)	2.11922	+	7.90903i	0.348397	+	1.30024i	0.888593	+	0.458697i	\(0.151684\pi\)
							−0.540196	+	0.841539i	\(0.681650\pi\)
\(41\)	−5.46868	+	6.51731i	−0.854064	+	1.01783i	0.145531	+	0.989354i	\(0.453511\pi\)
							−0.999594	+	0.0284797i	\(0.990933\pi\)
\(43\)	−0.185563	−	0.397941i	−0.0282981	−	0.0606854i	0.891651	−	0.452724i	\(-0.149548\pi\)
							−0.919949	+	0.392039i	\(0.871770\pi\)
\(47\)	−2.03954	+	2.91277i	−0.297498	+	0.424871i	−0.939876	−	0.341517i	\(-0.889059\pi\)
							0.642377	+	0.766388i	\(0.277948\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	270.2.r.a.113.18	✓	216
3.2	odd	2		810.2.s.a.773.7		216
5.2	odd	4	inner	270.2.r.a.167.5	yes	216
15.2	even	4		810.2.s.a.287.10		216
27.11	odd	18	inner	270.2.r.a.173.5	yes	216
27.16	even	9		810.2.s.a.683.10		216
135.92	even	36	inner	270.2.r.a.227.18	yes	216
135.97	odd	36		810.2.s.a.197.7		216

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
270.2.r.a.113.18	✓	216	1.1	even	1	trivial
270.2.r.a.167.5	yes	216	5.2	odd	4	inner
270.2.r.a.173.5	yes	216	27.11	odd	18	inner
270.2.r.a.227.18	yes	216	135.92	even	36	inner
810.2.s.a.197.7		216	135.97	odd	36
810.2.s.a.287.10		216	15.2	even	4
810.2.s.a.683.10		216	27.16	even	9
810.2.s.a.773.7		216	3.2	odd	2