Embedded newform 270.3.q.a.13.2

• Label: 270.3.q.a.13.2
• Level: $270$
• Weight: $3$
• Character: 270.13
• Analytic conductor: $7.357$
• Analytic rank: $0$
• Dimension: $216$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 270 = 2 \cdot 3^{3} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	270.q (of order \(36\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.35696713773\)
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			13.2
Character	\(\chi\)	\(=\)	270.13
Dual form			270.3.q.a.187.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.15846 + 0.811160i) q^{2} +(-2.96929 + 0.428152i) q^{3} +(0.684040 + 1.87939i) q^{4} +(0.872329 - 4.92332i) q^{5} +(-3.78709 - 1.91257i) q^{6} +(-8.20650 + 3.82675i) q^{7} +(-0.732051 + 2.73205i) q^{8} +(8.63337 - 2.54261i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 216 q - 36 q^{6} + 18 q^{7} + 216 q^{8}+O(q^{10}) \)

Character values

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

\(n\)	\(191\)	\(217\)
\(\chi(n)\)	\(e\left(\frac{4}{9}\right)\)	\(e\left(\frac{3}{4}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.15846	+	0.811160i	0.579228	+	0.405580i
							\(3\)	−2.96929	+	0.428152i	−0.989763	+	0.142717i
\(5\)	0.872329	−	4.92332i	0.174466	−	0.984663i
							\(7\)	−8.20650	+	3.82675i	−1.17236	+	0.546679i	−0.908501	−	0.417883i	\(-0.862772\pi\)
−0.263856	+	0.964562i	\(0.584994\pi\)
\(11\)	13.1850	−	11.0635i	1.19864	−	1.00578i	0.198971	−	0.980005i	\(-0.436240\pi\)
							0.999668	−	0.0257717i	\(-0.00820430\pi\)
\(13\)	4.74491	−	3.32242i	0.364993	−	0.255571i	−0.376664	−	0.926350i	\(-0.622929\pi\)
							0.741657	+	0.670779i	\(0.234040\pi\)
\(17\)	−6.50023	−	24.2592i	−0.382366	−	1.42701i	−0.842277	−	0.539045i	\(-0.818785\pi\)
							0.459910	−	0.887965i	\(-0.347881\pi\)
\(19\)	21.4508	−	12.3846i	1.12899	−	0.651823i	0.185310	−	0.982680i	\(-0.440671\pi\)
							0.943681	+	0.330857i	\(0.107338\pi\)
\(23\)	6.62924	+	3.09126i	0.288228	+	0.134403i	0.561355	−	0.827575i	\(-0.310280\pi\)
							−0.273127	+	0.961978i	\(0.588058\pi\)
\(29\)	32.5696	−	5.74291i	1.12309	−	0.198031i	0.418894	−	0.908035i	\(-0.362418\pi\)
							0.704198	+	0.710004i	\(0.251307\pi\)
\(31\)	−23.9303	+	8.70991i	−0.771945	+	0.280965i	−0.697810	−	0.716283i	\(-0.745842\pi\)
							−0.0741352	+	0.997248i	\(0.523620\pi\)
\(37\)	−9.99401	−	37.2982i	−0.270108	−	1.00806i	−0.959049	−	0.283241i	\(-0.908591\pi\)
							0.688940	−	0.724818i	\(-0.258076\pi\)
\(41\)	3.58246	−	20.3171i	0.0873770	−	0.495539i	−0.909441	−	0.415832i	\(-0.863490\pi\)
							0.996818	−	0.0797072i	\(-0.0253985\pi\)
\(43\)	80.3250	+	7.02753i	1.86802	+	0.163431i	0.964322	−	0.264733i	\(-0.0852837\pi\)
							0.903701	+	0.428163i	\(0.140839\pi\)
\(47\)	−26.2047	+	12.2194i	−0.557546	+	0.259988i	−0.680902	−	0.732374i	\(-0.738412\pi\)
							0.123356	+	0.992362i	\(0.460634\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	270.3.q.a.13.2	✓	216
5.2	odd	4	inner	270.3.q.a.67.10	yes	216
27.25	even	9	inner	270.3.q.a.133.10	yes	216
135.52	odd	36	inner	270.3.q.a.187.2	yes	216

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
270.3.q.a.13.2	✓	216	1.1	even	1	trivial
270.3.q.a.67.10	yes	216	5.2	odd	4	inner
270.3.q.a.133.10	yes	216	27.25	even	9	inner
270.3.q.a.187.2	yes	216	135.52	odd	36	inner