Embedded newform 270.2.r.a.113.2

• Label: 270.2.r.a.113.2
• 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			113.2
Character	\(\chi\)	\(=\)	270.113
Dual form			270.2.r.a.227.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.996195 - 0.0871557i) q^{2} +(-1.57309 - 0.724826i) q^{3} +(0.984808 + 0.173648i) q^{4} +(-1.77982 - 1.35360i) q^{5} +(1.50394 + 0.859172i) q^{6} +(-1.16174 - 0.813458i) q^{7} +(-0.965926 - 0.258819i) q^{8} +(1.94925 + 2.28044i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 216 q + 12 q^{6}+O(q^{10}) \)

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
							\(3\)	−1.57309	−	0.724826i	−0.908227	−	0.418479i
\(5\)	−1.77982	−	1.35360i	−0.795962	−	0.605347i
							\(7\)	−1.16174	−	0.813458i	−0.439096	−	0.307458i	0.333044	−	0.942911i	\(-0.391924\pi\)
−0.772139	+	0.635453i	\(0.780813\pi\)
\(11\)	0.290109	−	0.797067i	0.0874711	−	0.240325i	−0.888243	−	0.459375i	\(-0.848074\pi\)
							0.975714	+	0.219050i	\(0.0702957\pi\)
\(13\)	0.499139	+	5.70518i	0.138436	+	1.58233i	0.674577	+	0.738204i	\(0.264326\pi\)
							−0.536141	+	0.844128i	\(0.680119\pi\)
\(17\)	−6.26651	+	1.67911i	−1.51985	+	0.407243i	−0.919693	−	0.392638i	\(-0.871563\pi\)
							−0.600159	+	0.799881i	\(0.704896\pi\)
\(19\)	0.472451	−	0.272770i	0.108388	−	0.0625777i	−0.444826	−	0.895617i	\(-0.646735\pi\)
							0.553214	+	0.833039i	\(0.313401\pi\)
\(23\)	2.49736	+	3.56660i	0.520736	+	0.743688i	0.990260	−	0.139232i	\(-0.0444633\pi\)
							−0.469524	+	0.882920i	\(0.655574\pi\)
\(29\)	−2.18127	+	1.83030i	−0.405051	+	0.339878i	−0.822442	−	0.568849i	\(-0.807389\pi\)
							0.417391	+	0.908727i	\(0.362945\pi\)
\(31\)	−1.43289	+	8.12633i	−0.257355	+	1.45953i	0.532600	+	0.846367i	\(0.321215\pi\)
							−0.789955	+	0.613165i	\(0.789896\pi\)
\(37\)	0.0155816	+	0.0581515i	0.00256161	+	0.00956005i	0.967195	−	0.254036i	\(-0.0817581\pi\)
							−0.964633	+	0.263596i	\(0.915091\pi\)
\(41\)	−6.57900	+	7.84055i	−1.02747	+	1.22449i	−0.0533198	+	0.998577i	\(0.516980\pi\)
							−0.974148	+	0.225911i	\(0.927464\pi\)
\(43\)	−3.71506	−	7.96698i	−0.566542	−	1.21495i	−0.955853	−	0.293846i	\(-0.905065\pi\)
							0.389311	−	0.921106i	\(-0.372713\pi\)
\(47\)	4.34754	−	6.20893i	0.634154	−	0.905666i	−0.365530	−	0.930800i	\(-0.619112\pi\)
							0.999684	+	0.0251337i	\(0.00800116\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	270.2.r.a.113.2	✓	216
3.2	odd	2		810.2.s.a.773.17		216
5.2	odd	4	inner	270.2.r.a.167.13	yes	216
15.2	even	4		810.2.s.a.287.5		216
27.11	odd	18	inner	270.2.r.a.173.13	yes	216
27.16	even	9		810.2.s.a.683.5		216
135.92	even	36	inner	270.2.r.a.227.2	yes	216
135.97	odd	36		810.2.s.a.197.17		216

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
270.2.r.a.113.2	✓	216	1.1	even	1	trivial
270.2.r.a.167.13	yes	216	5.2	odd	4	inner
270.2.r.a.173.13	yes	216	27.11	odd	18	inner
270.2.r.a.227.2	yes	216	135.92	even	36	inner
810.2.s.a.197.17		216	135.97	odd	36
810.2.s.a.287.5		216	15.2	even	4
810.2.s.a.683.5		216	27.16	even	9
810.2.s.a.773.17		216	3.2	odd	2