Embedded newform 270.2.r.a.257.7

• Label: 270.2.r.a.257.7
• 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			257.7
Character	\(\chi\)	\(=\)	270.257
Dual form			270.2.r.a.83.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.819152 - 0.573576i) q^{2} +(1.32654 - 1.11368i) q^{3} +(0.342020 + 0.939693i) q^{4} +(-1.31697 + 1.80709i) q^{5} +(-1.72542 + 0.151404i) q^{6} +(1.73056 + 3.71119i) q^{7} +(0.258819 - 0.965926i) q^{8} +(0.519419 - 2.95469i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 216 q + 12 q^{6}+O(q^{10}) \)

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
							\(3\)	1.32654	−	1.11368i	0.765878	−	0.642985i
\(5\)	−1.31697	+	1.80709i	−0.588968	+	0.808157i
							\(7\)	1.73056	+	3.71119i	0.654089	+	1.40270i	0.901068	+	0.433678i	\(0.142785\pi\)
−0.246978	+	0.969021i	\(0.579438\pi\)
\(11\)	2.47949	+	2.95495i	0.747596	+	0.890950i	0.996996	−	0.0774507i	\(-0.0246780\pi\)
							−0.249401	+	0.968400i	\(0.580234\pi\)
\(13\)	1.86602	+	2.66495i	0.517540	+	0.739124i	0.989820	−	0.142325i	\(-0.0454577\pi\)
							−0.472280	+	0.881449i	\(0.656569\pi\)
\(17\)	−1.32801	−	4.95618i	−0.322089	−	1.20205i	−0.917206	−	0.398413i	\(-0.869561\pi\)
							0.595118	−	0.803639i	\(-0.297105\pi\)
\(19\)	1.11122	−	0.641565i	0.254932	−	0.147185i	−0.367088	−	0.930186i	\(-0.619645\pi\)
							0.622021	+	0.783001i	\(0.286312\pi\)
\(23\)	−4.78933	−	2.23330i	−0.998645	−	0.465676i	−0.146629	−	0.989192i	\(-0.546842\pi\)
							−0.852016	+	0.523516i	\(0.824620\pi\)
\(29\)	0.692024	+	3.92466i	0.128506	+	0.728791i	0.979164	+	0.203072i	\(0.0650926\pi\)
							−0.850658	+	0.525719i	\(0.823796\pi\)
\(31\)	5.74539	−	2.09115i	1.03190	−	0.375582i	0.230097	−	0.973168i	\(-0.426096\pi\)
							0.801805	+	0.597586i	\(0.203873\pi\)
\(37\)	−3.87291	+	1.03774i	−0.636702	+	0.170604i	−0.562709	−	0.826655i	\(-0.690241\pi\)
							−0.0739930	+	0.997259i	\(0.523574\pi\)
\(41\)	8.38817	+	1.47906i	1.31001	+	0.230990i	0.784677	−	0.619905i	\(-0.212829\pi\)
							0.525335	+	0.850895i	\(0.323940\pi\)
\(43\)	0.0646079	−	0.738472i	0.00985261	−	0.112616i	−0.989676	−	0.143324i	\(-0.954221\pi\)
							0.999528	+	0.0307085i	\(0.00977635\pi\)
\(47\)	−1.53675	+	0.716598i	−0.224158	+	0.104527i	−0.531453	−	0.847088i	\(-0.678354\pi\)
							0.307295	+	0.951614i	\(0.400576\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	270.2.r.a.257.7	yes	216
3.2	odd	2		810.2.s.a.557.15		216
5.3	odd	4	inner	270.2.r.a.203.8	yes	216
15.8	even	4		810.2.s.a.233.17		216
27.2	odd	18	inner	270.2.r.a.137.8	yes	216
27.25	even	9		810.2.s.a.737.17		216
135.83	even	36	inner	270.2.r.a.83.7	✓	216
135.133	odd	36		810.2.s.a.413.15		216

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
270.2.r.a.83.7	✓	216	135.83	even	36	inner
270.2.r.a.137.8	yes	216	27.2	odd	18	inner
270.2.r.a.203.8	yes	216	5.3	odd	4	inner
270.2.r.a.257.7	yes	216	1.1	even	1	trivial
810.2.s.a.233.17		216	15.8	even	4
810.2.s.a.413.15		216	135.133	odd	36
810.2.s.a.557.15		216	3.2	odd	2
810.2.s.a.737.17		216	27.25	even	9