Embedded newform 270.2.r.a.113.7

• Label: 270.2.r.a.113.7
• Level: $270$
• Weight: $2$
• Character: 270.113
• 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			113.7
Character	\(\chi\)	\(=\)	270.113
Dual form			270.2.r.a.227.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.996195 - 0.0871557i) q^{2} +(1.12178 - 1.31970i) q^{3} +(0.984808 + 0.173648i) q^{4} +(-1.30964 + 1.81241i) q^{5} +(-1.23253 + 1.21691i) q^{6} +(-3.63487 - 2.54517i) q^{7} +(-0.965926 - 0.258819i) q^{8} +(-0.483211 - 2.96083i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 216 q + 12 q^{6}+O(q^{10}) \)

Character values

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

\(n\)	\(191\)	\(217\)
\(\chi(n)\)	\(e\left(\frac{5}{18}\right)\)	\(e\left(\frac{3}{4}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.996195	−	0.0871557i	−0.704416	−	0.0616284i
							\(3\)	1.12178	−	1.31970i	0.647661	−	0.761929i
\(5\)	−1.30964	+	1.81241i	−0.585690	+	0.810535i
							\(7\)	−3.63487	−	2.54517i	−1.37385	−	0.961983i	−0.999396	−	0.0347383i	\(-0.988940\pi\)
−0.374457	−	0.927244i	\(-0.622171\pi\)
\(11\)	1.06118	−	2.91557i	0.319959	−	0.879079i	−0.670579	−	0.741838i	\(-0.733954\pi\)
							0.990538	−	0.137241i	\(-0.0438234\pi\)
\(13\)	−0.443416	−	5.06827i	−0.122981	−	1.40568i	−0.767477	−	0.641077i	\(-0.778488\pi\)
							0.644495	−	0.764608i	\(-0.277067\pi\)
\(17\)	−2.90525	+	0.778460i	−0.704627	+	0.188804i	−0.593302	−	0.804980i	\(-0.702176\pi\)
							−0.111325	+	0.993784i	\(0.535509\pi\)
\(19\)	4.83096	−	2.78915i	1.10830	−	0.639876i	0.169910	−	0.985460i	\(-0.445652\pi\)
							0.938388	+	0.345584i	\(0.112319\pi\)
\(23\)	3.26865	+	4.66812i	0.681561	+	0.973370i	0.999679	+	0.0253193i	\(0.00806026\pi\)
							−0.318118	+	0.948051i	\(0.603051\pi\)
\(29\)	−0.945635	+	0.793482i	−0.175600	+	0.147346i	−0.726352	−	0.687323i	\(-0.758786\pi\)
							0.550752	+	0.834669i	\(0.314341\pi\)
\(31\)	−0.863767	+	4.89867i	−0.155137	+	0.879827i	0.803523	+	0.595274i	\(0.202956\pi\)
							−0.958660	+	0.284553i	\(0.908155\pi\)
\(37\)	−0.154906	−	0.578116i	−0.0254663	−	0.0950417i	0.952023	−	0.306026i	\(-0.0989996\pi\)
							−0.977489	+	0.210985i	\(0.932333\pi\)
\(41\)	0.230916	−	0.275195i	0.0360631	−	0.0429783i	−0.747711	−	0.664024i	\(-0.768847\pi\)
							0.783775	+	0.621045i	\(0.213292\pi\)
\(43\)	0.843535	+	1.80897i	0.128638	+	0.275865i	0.960069	−	0.279763i	\(-0.0902558\pi\)
							−0.831431	+	0.555628i	\(0.812478\pi\)
\(47\)	4.62779	−	6.60916i	0.675032	−	0.964046i	−0.324801	−	0.945782i	\(-0.605297\pi\)
							0.999833	−	0.0182633i	\(-0.00581370\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	270.2.r.a.113.7	✓	216
3.2	odd	2		810.2.s.a.773.15		216
5.2	odd	4	inner	270.2.r.a.167.11	yes	216
15.2	even	4		810.2.s.a.287.2		216
27.11	odd	18	inner	270.2.r.a.173.11	yes	216
27.16	even	9		810.2.s.a.683.2		216
135.92	even	36	inner	270.2.r.a.227.7	yes	216
135.97	odd	36		810.2.s.a.197.15		216

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
270.2.r.a.113.7	✓	216	1.1	even	1	trivial
270.2.r.a.167.11	yes	216	5.2	odd	4	inner
270.2.r.a.173.11	yes	216	27.11	odd	18	inner
270.2.r.a.227.7	yes	216	135.92	even	36	inner
810.2.s.a.197.15		216	135.97	odd	36
810.2.s.a.287.2		216	15.2	even	4
810.2.s.a.683.2		216	27.16	even	9
810.2.s.a.773.15		216	3.2	odd	2