Embedded newform 270.2.r.a.257.6

• Label: 270.2.r.a.257.6
• 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			257.6
Character	\(\chi\)	\(=\)	270.257
Dual form			270.2.r.a.83.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.819152 - 0.573576i) q^{2} +(0.618740 - 1.61776i) q^{3} +(0.342020 + 0.939693i) q^{4} +(1.91317 - 1.15748i) q^{5} +(-1.43475 + 0.970300i) q^{6} +(-0.957323 - 2.05299i) q^{7} +(0.258819 - 0.965926i) q^{8} +(-2.23432 - 2.00195i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 216 q + 12 q^{6}+O(q^{10}) \)

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
							\(3\)	0.618740	−	1.61776i	0.357230	−	0.934017i
\(5\)	1.91317	−	1.15748i	0.855598	−	0.517641i
							\(7\)	−0.957323	−	2.05299i	−0.361834	−	0.775956i	−0.999982	−	0.00604179i	\(-0.998077\pi\)
0.638148	−	0.769914i	\(-0.279701\pi\)
\(11\)	3.39033	+	4.04043i	1.02222	+	1.21824i	0.975653	+	0.219321i	\(0.0703843\pi\)
							0.0465691	+	0.998915i	\(0.485171\pi\)
\(13\)	−2.24888	−	3.21174i	−0.623728	−	0.890775i	0.375624	−	0.926772i	\(-0.377428\pi\)
							−0.999352	+	0.0359967i	\(0.988539\pi\)
\(17\)	0.889884	+	3.32109i	0.215829	+	0.805483i	0.985873	+	0.167493i	\(0.0535673\pi\)
							−0.770045	+	0.637990i	\(0.779766\pi\)
\(19\)	3.09115	−	1.78467i	0.709157	−	0.409432i	−0.101592	−	0.994826i	\(-0.532393\pi\)
							0.810749	+	0.585394i	\(0.199060\pi\)
\(23\)	−4.68275	−	2.18360i	−0.976420	−	0.455312i	−0.132150	−	0.991230i	\(-0.542188\pi\)
							−0.844270	+	0.535918i	\(0.819966\pi\)
\(29\)	−0.136517	−	0.774225i	−0.0253505	−	0.143770i	0.969505	−	0.245070i	\(-0.0788109\pi\)
							−0.994856	+	0.101300i	\(0.967700\pi\)
\(31\)	−10.3110	+	3.75290i	−1.85191	+	0.674041i	−0.867690	+	0.497106i	\(0.834396\pi\)
							−0.984222	+	0.176935i	\(0.943382\pi\)
\(37\)	8.66041	−	2.32055i	1.42376	−	0.381496i	0.536946	−	0.843617i	\(-0.319578\pi\)
							0.886817	+	0.462121i	\(0.152911\pi\)
\(41\)	2.74726	+	0.484417i	0.429050	+	0.0756532i	0.384003	−	0.923332i	\(-0.374545\pi\)
							0.0450470	+	0.998985i	\(0.485656\pi\)
\(43\)	−0.583752	+	6.67232i	−0.0890213	+	1.01752i	0.811975	+	0.583692i	\(0.198392\pi\)
							−0.900996	+	0.433826i	\(0.857163\pi\)
\(47\)	5.28675	−	2.46525i	0.771152	−	0.359594i	0.00311126	−	0.999995i	\(-0.499010\pi\)
							0.768041	+	0.640401i	\(0.221232\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	270.2.r.a.257.6	yes	216
3.2	odd	2		810.2.s.a.557.11		216
5.3	odd	4	inner	270.2.r.a.203.9	yes	216
15.8	even	4		810.2.s.a.233.13		216
27.2	odd	18	inner	270.2.r.a.137.9	yes	216
27.25	even	9		810.2.s.a.737.13		216
135.83	even	36	inner	270.2.r.a.83.6	✓	216
135.133	odd	36		810.2.s.a.413.11		216

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
270.2.r.a.83.6	✓	216	135.83	even	36	inner
270.2.r.a.137.9	yes	216	27.2	odd	18	inner
270.2.r.a.203.9	yes	216	5.3	odd	4	inner
270.2.r.a.257.6	yes	216	1.1	even	1	trivial
810.2.s.a.233.13		216	15.8	even	4
810.2.s.a.413.11		216	135.133	odd	36
810.2.s.a.557.11		216	3.2	odd	2
810.2.s.a.737.13		216	27.25	even	9