Embedded newform 270.2.r.a.173.11

• Label: 270.2.r.a.173.11
• Level: $270$
• Weight: $2$
• Character: 270.173
• 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			173.11
Character	\(\chi\)	\(=\)	270.173
Dual form			270.2.r.a.167.11

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.0871557 + 0.996195i) q^{2} +(-1.31970 + 1.12178i) q^{3} +(-0.984808 + 0.173648i) q^{4} +(2.16824 - 0.546565i) q^{5} +(-1.23253 - 1.21691i) q^{6} +(2.54517 + 3.63487i) q^{7} +(-0.258819 - 0.965926i) q^{8} +(0.483211 - 2.96083i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 216 q + 12 q^{6}+O(q^{10}) \)

Character values

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

\(n\)	\(191\)	\(217\)
\(\chi(n)\)	\(e\left(\frac{13}{18}\right)\)	\(e\left(\frac{3}{4}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.0871557	+	0.996195i	0.0616284	+	0.704416i
							\(3\)	−1.31970	+	1.12178i	−0.761929	+	0.647661i
\(5\)	2.16824	−	0.546565i	0.969667	−	0.244431i
							\(7\)	2.54517	+	3.63487i	0.961983	+	1.37385i	0.927244	+	0.374457i	\(0.122171\pi\)
0.0347383	+	0.999396i	\(0.488940\pi\)
\(11\)	1.06118	+	2.91557i	0.319959	+	0.879079i	0.990538	+	0.137241i	\(0.0438234\pi\)
							−0.670579	+	0.741838i	\(0.733954\pi\)
\(13\)	−5.06827	−	0.443416i	−1.40568	−	0.122981i	−0.641077	−	0.767477i	\(-0.721512\pi\)
							−0.764608	+	0.644495i	\(0.777067\pi\)
\(17\)	−0.778460	+	2.90525i	−0.188804	+	0.704627i	0.804980	+	0.593302i	\(0.202176\pi\)
							−0.993784	+	0.111325i	\(0.964491\pi\)
\(19\)	−4.83096	−	2.78915i	−1.10830	−	0.639876i	−0.169910	−	0.985460i	\(-0.554348\pi\)
							−0.938388	+	0.345584i	\(0.887681\pi\)
\(23\)	4.66812	+	3.26865i	0.973370	+	0.681561i	0.948051	−	0.318118i	\(-0.103051\pi\)
							0.0253193	+	0.999679i	\(0.491940\pi\)
\(29\)	0.945635	+	0.793482i	0.175600	+	0.147346i	0.726352	−	0.687323i	\(-0.241214\pi\)
							−0.550752	+	0.834669i	\(0.685659\pi\)
\(31\)	−0.863767	−	4.89867i	−0.155137	−	0.879827i	−0.958660	−	0.284553i	\(-0.908155\pi\)
							0.803523	−	0.595274i	\(-0.202956\pi\)
\(37\)	0.578116	+	0.154906i	0.0950417	+	0.0254663i	0.306026	−	0.952023i	\(-0.401000\pi\)
							−0.210985	+	0.977489i	\(0.567667\pi\)
\(41\)	0.230916	+	0.275195i	0.0360631	+	0.0429783i	0.783775	−	0.621045i	\(-0.213292\pi\)
							−0.747711	+	0.664024i	\(0.768847\pi\)
\(43\)	1.80897	+	0.843535i	0.275865	+	0.128638i	0.555628	−	0.831431i	\(-0.312478\pi\)
							−0.279763	+	0.960069i	\(0.590256\pi\)
\(47\)	6.60916	−	4.62779i	0.964046	−	0.675032i	0.0182633	−	0.999833i	\(-0.494186\pi\)
							0.945782	+	0.324801i	\(0.105297\pi\)

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
270.2.r.a.113.7	✓	216	27.5	odd	18	inner
270.2.r.a.167.11	yes	216	135.32	even	36	inner
270.2.r.a.173.11	yes	216	1.1	even	1	trivial
270.2.r.a.227.7	yes	216	5.2	odd	4	inner
810.2.s.a.197.15		216	15.2	even	4
810.2.s.a.287.2		216	135.22	odd	36
810.2.s.a.683.2		216	3.2	odd	2
810.2.s.a.773.15		216	27.22	even	9