Embedded newform 270.2.r.a.257.4

• Label: 270.2.r.a.257.4
• 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			257.4
Character	\(\chi\)	\(=\)	270.257
Dual form			270.2.r.a.83.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.819152 - 0.573576i) q^{2} +(0.0563218 + 1.73113i) q^{3} +(0.342020 + 0.939693i) q^{4} +(-1.67202 - 1.48471i) q^{5} +(0.946802 - 1.45037i) q^{6} +(-1.44286 - 3.09422i) q^{7} +(0.258819 - 0.965926i) q^{8} +(-2.99366 + 0.195001i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 216 q + 12 q^{6}+O(q^{10}) \)

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
							\(3\)	0.0563218	+	1.73113i	0.0325174	+	0.999471i
\(5\)	−1.67202	−	1.48471i	−0.747748	−	0.663983i
							\(7\)	−1.44286	−	3.09422i	−0.545350	−	1.16951i	−0.965103	−	0.261869i	\(-0.915661\pi\)
0.419753	−	0.907638i	\(-0.362117\pi\)
\(11\)	−0.893065	−	1.06431i	−0.269269	−	0.320903i	0.614418	−	0.788981i	\(-0.289391\pi\)
							−0.883687	+	0.468078i	\(0.844947\pi\)
\(13\)	0.995054	+	1.42108i	0.275978	+	0.394138i	0.933083	−	0.359662i	\(-0.117108\pi\)
							−0.657105	+	0.753799i	\(0.728219\pi\)
\(17\)	−1.93558	−	7.22369i	−0.469448	−	1.75200i	−0.641707	−	0.766950i	\(-0.721773\pi\)
							0.172259	−	0.985052i	\(-0.444893\pi\)
\(19\)	5.97284	−	3.44842i	1.37026	−	0.791122i	0.379303	−	0.925273i	\(-0.376164\pi\)
							0.990961	+	0.134150i	\(0.0428306\pi\)
\(23\)	−6.18670	−	2.88491i	−1.29002	−	0.601544i	−0.348120	−	0.937450i	\(-0.613180\pi\)
							−0.941896	+	0.335905i	\(0.890958\pi\)
\(29\)	0.522950	+	2.96580i	0.0971093	+	0.550734i	0.994081	+	0.108646i	\(0.0346513\pi\)
							−0.896971	+	0.442089i	\(0.854238\pi\)
\(31\)	−6.89042	+	2.50791i	−1.23756	+	0.450433i	−0.876179	−	0.481987i	\(-0.839915\pi\)
							−0.361377	+	0.932420i	\(0.617693\pi\)
\(37\)	−2.52024	+	0.675297i	−0.414325	+	0.111018i	−0.459959	−	0.887940i	\(-0.652136\pi\)
							0.0456338	+	0.998958i	\(0.485469\pi\)
\(41\)	7.98281	+	1.40758i	1.24671	+	0.219828i	0.757787	−	0.652502i	\(-0.226281\pi\)
							0.488918	+	0.872330i	\(0.337392\pi\)
\(43\)	0.492741	−	5.63206i	0.0751423	−	0.858881i	−0.861309	−	0.508082i	\(-0.830355\pi\)
							0.936451	−	0.350799i	\(-0.114090\pi\)
\(47\)	−6.42491	+	2.99598i	−0.937169	+	0.437009i	−0.830333	−	0.557268i	\(-0.811850\pi\)
							−0.106836	+	0.994277i	\(0.534072\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	270.2.r.a.257.4	yes	216
3.2	odd	2		810.2.s.a.557.17		216
5.3	odd	4	inner	270.2.r.a.203.1	yes	216
15.8	even	4		810.2.s.a.233.12		216
27.2	odd	18	inner	270.2.r.a.137.1	yes	216
27.25	even	9		810.2.s.a.737.12		216
135.83	even	36	inner	270.2.r.a.83.4	✓	216
135.133	odd	36		810.2.s.a.413.17		216

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
270.2.r.a.83.4	✓	216	135.83	even	36	inner
270.2.r.a.137.1	yes	216	27.2	odd	18	inner
270.2.r.a.203.1	yes	216	5.3	odd	4	inner
270.2.r.a.257.4	yes	216	1.1	even	1	trivial
810.2.s.a.233.12		216	15.8	even	4
810.2.s.a.413.17		216	135.133	odd	36
810.2.s.a.557.17		216	3.2	odd	2
810.2.s.a.737.12		216	27.25	even	9