Embedded newform 270.2.r.a.113.1

• Label: 270.2.r.a.113.1
• Level: $270$
• Weight: $2$
• Character: 270.113
• 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			113.1
Character	\(\chi\)	\(=\)	270.113
Dual form			270.2.r.a.227.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.996195 - 0.0871557i) q^{2} +(-1.67699 - 0.433251i) q^{3} +(0.984808 + 0.173648i) q^{4} +(2.20857 + 0.349587i) q^{5} +(1.63285 + 0.577761i) q^{6} +(-1.10212 - 0.771712i) q^{7} +(-0.965926 - 0.258819i) q^{8} +(2.62459 + 1.45311i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 216 q + 12 q^{6}+O(q^{10}) \)

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
							\(3\)	−1.67699	−	0.433251i	−0.968210	−	0.250137i
\(5\)	2.20857	+	0.349587i	0.987703	+	0.156340i
							\(7\)	−1.10212	−	0.771712i	−0.416562	−	0.291680i	0.346445	−	0.938070i	\(-0.387389\pi\)
−0.763007	+	0.646391i	\(0.776278\pi\)
\(11\)	−1.46618	+	4.02828i	−0.442068	+	1.21457i	0.496061	+	0.868288i	\(0.334779\pi\)
							−0.938129	+	0.346285i	\(0.887443\pi\)
\(13\)	−0.298865	−	3.41605i	−0.0828903	−	0.947441i	−0.917797	−	0.397050i	\(-0.870034\pi\)
							0.834907	−	0.550391i	\(-0.185521\pi\)
\(17\)	7.43061	−	1.99103i	1.80219	−	0.482895i	0.807871	−	0.589359i	\(-0.200620\pi\)
							0.994317	+	0.106464i	\(0.0339530\pi\)
\(19\)	6.12582	−	3.53674i	1.40536	−	0.811384i	0.410423	−	0.911895i	\(-0.365381\pi\)
							0.994936	+	0.100511i	\(0.0320477\pi\)
\(23\)	2.78440	+	3.97654i	0.580588	+	0.829166i	0.996786	−	0.0801079i	\(-0.0255265\pi\)
							−0.416198	+	0.909274i	\(0.636638\pi\)
\(29\)	0.00478023	−	0.00401109i	0.000887667	−	0.000744841i	−0.642344	−	0.766417i	\(-0.722038\pi\)
							0.643231	+	0.765672i	\(0.277593\pi\)
\(31\)	0.582373	−	3.30280i	0.104597	−	0.593200i	−0.886783	−	0.462186i	\(-0.847065\pi\)
							0.991380	−	0.131015i	\(-0.0418235\pi\)
\(37\)	−1.24815	−	4.65816i	−0.205195	−	0.765797i	−0.989390	−	0.145283i	\(-0.953591\pi\)
							0.784196	−	0.620514i	\(-0.213076\pi\)
\(41\)	0.705833	−	0.841179i	0.110233	−	0.131370i	−0.708107	−	0.706105i	\(-0.750451\pi\)
							0.818339	+	0.574735i	\(0.194895\pi\)
\(43\)	3.14516	+	6.74482i	0.479632	+	1.02857i	0.986075	+	0.166301i	\(0.0531824\pi\)
							−0.506443	+	0.862274i	\(0.669040\pi\)
\(47\)	−2.04873	+	2.92589i	−0.298838	+	0.426785i	−0.940288	−	0.340380i	\(-0.889444\pi\)
							0.641450	+	0.767165i	\(0.278333\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	270.2.r.a.113.1	✓	216
3.2	odd	2		810.2.s.a.773.11		216
5.2	odd	4	inner	270.2.r.a.167.14	yes	216
15.2	even	4		810.2.s.a.287.7		216
27.11	odd	18	inner	270.2.r.a.173.14	yes	216
27.16	even	9		810.2.s.a.683.7		216
135.92	even	36	inner	270.2.r.a.227.1	yes	216
135.97	odd	36		810.2.s.a.197.11		216

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
270.2.r.a.113.1	✓	216	1.1	even	1	trivial
270.2.r.a.167.14	yes	216	5.2	odd	4	inner
270.2.r.a.173.14	yes	216	27.11	odd	18	inner
270.2.r.a.227.1	yes	216	135.92	even	36	inner
810.2.s.a.197.11		216	135.97	odd	36
810.2.s.a.287.7		216	15.2	even	4
810.2.s.a.683.7		216	27.16	even	9
810.2.s.a.773.11		216	3.2	odd	2