Embedded newform 270.2.r.a.203.13

• Label: 270.2.r.a.203.13
• Level: $270$
• Weight: $2$
• Character: 270.203
• Analytic conductor: $2.156$
• Analytic rank: $0$
• Dimension: $216$
• CM: no
• 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			203.13
Character	\(\chi\)	\(=\)	270.203
Dual form			270.2.r.a.137.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.573576 - 0.819152i) q^{2} +(-0.308872 + 1.70429i) q^{3} +(-0.342020 - 0.939693i) q^{4} +(-2.07393 + 0.835951i) q^{5} +(1.21891 + 1.23055i) q^{6} +(-3.83305 + 1.78738i) q^{7} +(-0.965926 - 0.258819i) q^{8} +(-2.80920 - 1.05281i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 216 q + 12 q^{6}+O(q^{10}) \)

Character values

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

\(n\)	\(191\)	\(217\)
\(\chi(n)\)	\(e\left(\frac{17}{18}\right)\)	\(e\left(\frac{3}{4}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.573576	−	0.819152i	0.405580	−	0.579228i
							\(3\)	−0.308872	+	1.70429i	−0.178327	+	0.983971i
\(5\)	−2.07393	+	0.835951i	−0.927490	+	0.373849i
							\(7\)	−3.83305	+	1.78738i	−1.44876	+	0.675567i	−0.978364	−	0.206890i	\(-0.933666\pi\)
−0.470393	+	0.882457i	\(0.655888\pi\)
\(11\)	3.34414	+	3.98539i	1.00830	+	1.20164i	0.979374	+	0.202058i	\(0.0647629\pi\)
							0.0289216	+	0.999582i	\(0.490793\pi\)
\(13\)	0.307904	−	0.215597i	0.0853972	−	0.0597958i	−0.530098	−	0.847936i	\(-0.677845\pi\)
							0.615495	+	0.788141i	\(0.288956\pi\)
\(17\)	−5.90088	+	1.58114i	−1.43117	+	0.383482i	−0.889433	−	0.457065i	\(-0.848901\pi\)
							−0.541739	+	0.840546i	\(0.682234\pi\)
\(19\)	2.93211	−	1.69286i	0.672672	−	0.388368i	−0.124416	−	0.992230i	\(-0.539706\pi\)
							0.797088	+	0.603863i	\(0.206372\pi\)
\(23\)	0.472247	−	1.01274i	0.0984704	−	0.211170i	−0.850901	−	0.525326i	\(-0.823943\pi\)
							0.949372	+	0.314155i	\(0.101721\pi\)
\(29\)	1.75816	+	9.97103i	0.326483	+	1.85157i	0.499045	+	0.866576i	\(0.333684\pi\)
							−0.172563	+	0.984999i	\(0.555205\pi\)
\(31\)	−3.35678	+	1.22177i	−0.602896	+	0.219436i	−0.625392	−	0.780311i	\(-0.715061\pi\)
							0.0224959	+	0.999747i	\(0.492839\pi\)
\(37\)	−1.51118	−	5.63980i	−0.248436	−	0.927178i	−0.971625	−	0.236527i	\(-0.923991\pi\)
							0.723188	−	0.690651i	\(-0.242676\pi\)
\(41\)	4.77146	+	0.841337i	0.745177	+	0.131395i	0.533329	−	0.845908i	\(-0.320941\pi\)
							0.211848	+	0.977303i	\(0.432052\pi\)
\(43\)	6.52010	+	0.570435i	0.994306	+	0.0869905i	0.572680	−	0.819779i	\(-0.305904\pi\)
							0.421626	+	0.906770i	\(0.361459\pi\)
\(47\)	3.10175	+	6.65173i	0.452437	+	0.970254i	0.991755	+	0.128152i	\(0.0409044\pi\)
							−0.539318	+	0.842102i	\(0.681318\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	270.2.r.a.203.13	yes	216
3.2	odd	2		810.2.s.a.233.8		216
5.2	odd	4	inner	270.2.r.a.257.18	yes	216
15.2	even	4		810.2.s.a.557.6		216
27.2	odd	18	inner	270.2.r.a.83.18	✓	216
27.25	even	9		810.2.s.a.413.6		216
135.2	even	36	inner	270.2.r.a.137.13	yes	216
135.52	odd	36		810.2.s.a.737.8		216

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
270.2.r.a.83.18	✓	216	27.2	odd	18	inner
270.2.r.a.137.13	yes	216	135.2	even	36	inner
270.2.r.a.203.13	yes	216	1.1	even	1	trivial
270.2.r.a.257.18	yes	216	5.2	odd	4	inner
810.2.s.a.233.8		216	3.2	odd	2
810.2.s.a.413.6		216	27.25	even	9
810.2.s.a.557.6		216	15.2	even	4
810.2.s.a.737.8		216	135.52	odd	36