Embedded newform 270.2.r.a.257.3

• Label: 270.2.r.a.257.3
• Level: $270$
• Weight: $2$
• Character: 270.257
• 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			257.3
Character	\(\chi\)	\(=\)	270.257
Dual form			270.2.r.a.83.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.819152 - 0.573576i) q^{2} +(-1.20282 + 1.24629i) q^{3} +(0.342020 + 0.939693i) q^{4} +(-1.67820 + 1.47771i) q^{5} +(1.70013 - 0.330990i) q^{6} +(1.22442 + 2.62578i) q^{7} +(0.258819 - 0.965926i) q^{8} +(-0.106460 - 2.99811i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 216 q + 12 q^{6}+O(q^{10}) \)

Character values

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

\(n\)	\(191\)	\(217\)
\(\chi(n)\)	\(e\left(\frac{17}{18}\right)\)	\(e\left(\frac{1}{4}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.819152	−	0.573576i	−0.579228	−	0.405580i
							\(3\)	−1.20282	+	1.24629i	−0.694447	+	0.719544i
\(5\)	−1.67820	+	1.47771i	−0.750516	+	0.660852i
							\(7\)	1.22442	+	2.62578i	0.462788	+	0.992453i	0.989783	+	0.142584i	\(0.0455412\pi\)
−0.526994	+	0.849869i	\(0.676681\pi\)
\(11\)	−4.04443	−	4.81996i	−1.21944	−	1.45327i	−0.852256	−	0.523124i	\(-0.824766\pi\)
							−0.367184	−	0.930148i	\(-0.619678\pi\)
\(13\)	−2.05773	−	2.93874i	−0.570712	−	0.815061i	0.425228	−	0.905086i	\(-0.360194\pi\)
							−0.995939	+	0.0900255i	\(0.971305\pi\)
\(17\)	1.42358	+	5.31289i	0.345270	+	1.28857i	0.892296	+	0.451450i	\(0.149093\pi\)
							−0.547026	+	0.837115i	\(0.684240\pi\)
\(19\)	−3.76766	+	2.17526i	−0.864360	+	0.499038i	−0.865470	−	0.500961i	\(-0.832980\pi\)
							0.00111017	+	0.999999i	\(0.499647\pi\)
\(23\)	−4.79087	−	2.23402i	−0.998965	−	0.465825i	−0.146838	−	0.989161i	\(-0.546910\pi\)
							−0.852127	+	0.523335i	\(0.824687\pi\)
\(29\)	−0.590150	−	3.34691i	−0.109588	−	0.621505i	−0.989288	−	0.145976i	\(-0.953368\pi\)
							0.879700	−	0.475529i	\(-0.157743\pi\)
\(31\)	−2.02399	+	0.736673i	−0.363520	+	0.132310i	−0.517321	−	0.855791i	\(-0.673071\pi\)
							0.153801	+	0.988102i	\(0.450848\pi\)
\(37\)	1.56666	−	0.419786i	0.257558	−	0.0690124i	−0.127730	−	0.991809i	\(-0.540769\pi\)
							0.385287	+	0.922797i	\(0.374102\pi\)
\(41\)	−2.83787	−	0.500392i	−0.443200	−	0.0781482i	−0.0524053	−	0.998626i	\(-0.516689\pi\)
							−0.390795	+	0.920478i	\(0.627800\pi\)
\(43\)	−0.447144	+	5.11088i	−0.0681889	+	0.779402i	0.882714	+	0.469912i	\(0.155714\pi\)
							−0.950902	+	0.309491i	\(0.899841\pi\)
\(47\)	−8.80682	+	4.10669i	−1.28461	+	0.599022i	−0.940463	−	0.339896i	\(-0.889608\pi\)
							−0.344143	+	0.938917i	\(0.611830\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	270.2.r.a.257.3	yes	216
3.2	odd	2		810.2.s.a.557.18		216
5.3	odd	4	inner	270.2.r.a.203.3	yes	216
15.8	even	4		810.2.s.a.233.15		216
27.2	odd	18	inner	270.2.r.a.137.3	yes	216
27.25	even	9		810.2.s.a.737.15		216
135.83	even	36	inner	270.2.r.a.83.3	✓	216
135.133	odd	36		810.2.s.a.413.18		216

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
270.2.r.a.83.3	✓	216	135.83	even	36	inner
270.2.r.a.137.3	yes	216	27.2	odd	18	inner
270.2.r.a.203.3	yes	216	5.3	odd	4	inner
270.2.r.a.257.3	yes	216	1.1	even	1	trivial
810.2.s.a.233.15		216	15.8	even	4
810.2.s.a.413.18		216	135.133	odd	36
810.2.s.a.557.18		216	3.2	odd	2
810.2.s.a.737.15		216	27.25	even	9