Embedded newform 270.2.r.a.257.2

• Label: 270.2.r.a.257.2
• 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			257.2
Character	\(\chi\)	\(=\)	270.257
Dual form			270.2.r.a.83.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.819152 - 0.573576i) q^{2} +(-1.67974 - 0.422447i) q^{3} +(0.342020 + 0.939693i) q^{4} +(1.00216 - 1.99892i) q^{5} +(1.13366 + 1.30951i) q^{6} +(-0.0561590 - 0.120433i) q^{7} +(0.258819 - 0.965926i) q^{8} +(2.64308 + 1.41920i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 216 q + 12 q^{6}+O(q^{10}) \)

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
							\(3\)	−1.67974	−	0.422447i	−0.969800	−	0.243900i
\(5\)	1.00216	−	1.99892i	0.448178	−	0.893944i
							\(7\)	−0.0561590	−	0.120433i	−0.0212261	−	0.0455196i	0.895409	−	0.445245i	\(-0.146883\pi\)
−0.916635	+	0.399725i	\(0.869106\pi\)
\(11\)	−1.11291	−	1.32632i	−0.335556	−	0.399900i	0.571711	−	0.820455i	\(-0.306280\pi\)
							−0.907267	+	0.420555i	\(0.861835\pi\)
\(13\)	−1.22280	−	1.74633i	−0.339143	−	0.484346i	0.612964	−	0.790110i	\(-0.289977\pi\)
							−0.952107	+	0.305765i	\(0.901088\pi\)
\(17\)	−1.68152	−	6.27550i	−0.407827	−	1.52203i	−0.798781	−	0.601622i	\(-0.794521\pi\)
							0.390953	−	0.920411i	\(-0.372145\pi\)
\(19\)	−6.47014	+	3.73554i	−1.48435	+	0.856991i	−0.999842	−	0.0177955i	\(-0.994335\pi\)
							−0.484509	+	0.874786i	\(0.661002\pi\)
\(23\)	−1.60127	−	0.746684i	−0.333888	−	0.155694i	0.248443	−	0.968647i	\(-0.420081\pi\)
							−0.582330	+	0.812952i	\(0.697859\pi\)
\(29\)	−0.637912	−	3.61778i	−0.118457	−	0.671805i	−0.984980	−	0.172667i	\(-0.944761\pi\)
							0.866523	−	0.499137i	\(-0.166350\pi\)
\(31\)	6.03428	−	2.19630i	1.08379	−	0.394467i	0.262473	−	0.964939i	\(-0.415462\pi\)
							0.821317	+	0.570472i	\(0.193240\pi\)
\(37\)	−9.06304	+	2.42843i	−1.48995	+	0.399232i	−0.909722	−	0.415218i	\(-0.863705\pi\)
							−0.580233	+	0.814451i	\(0.697039\pi\)
\(41\)	5.12420	+	0.903535i	0.800266	+	0.141109i	0.558802	−	0.829301i	\(-0.311261\pi\)
							0.241464	+	0.970410i	\(0.422372\pi\)
\(43\)	−0.171770	+	1.96334i	−0.0261947	+	0.299407i	0.971785	+	0.235869i	\(0.0757938\pi\)
							−0.997979	+	0.0635374i	\(0.979762\pi\)
\(47\)	2.27592	−	1.06128i	0.331977	−	0.154803i	−0.249481	−	0.968380i	\(-0.580260\pi\)
							0.581458	+	0.813576i	\(0.302482\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	270.2.r.a.257.2	yes	216
3.2	odd	2		810.2.s.a.557.13		216
5.3	odd	4	inner	270.2.r.a.203.7	yes	216
15.8	even	4		810.2.s.a.233.11		216
27.2	odd	18	inner	270.2.r.a.137.7	yes	216
27.25	even	9		810.2.s.a.737.11		216
135.83	even	36	inner	270.2.r.a.83.2	✓	216
135.133	odd	36		810.2.s.a.413.13		216

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
270.2.r.a.83.2	✓	216	135.83	even	36	inner
270.2.r.a.137.7	yes	216	27.2	odd	18	inner
270.2.r.a.203.7	yes	216	5.3	odd	4	inner
270.2.r.a.257.2	yes	216	1.1	even	1	trivial
810.2.s.a.233.11		216	15.8	even	4
810.2.s.a.413.13		216	135.133	odd	36
810.2.s.a.557.13		216	3.2	odd	2
810.2.s.a.737.11		216	27.25	even	9