Embedded newform 270.2.r.a.113.17

• Label: 270.2.r.a.113.17
• 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			113.17
Character	\(\chi\)	\(=\)	270.113
Dual form			270.2.r.a.227.17

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.996195 + 0.0871557i) q^{2} +(1.39244 + 1.03011i) q^{3} +(0.984808 + 0.173648i) q^{4} +(0.209292 - 2.22625i) q^{5} +(1.29736 + 1.14755i) q^{6} +(-0.734969 - 0.514631i) q^{7} +(0.965926 + 0.258819i) q^{8} +(0.877763 + 2.86872i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 216 q + 12 q^{6}+O(q^{10}) \)

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
							\(3\)	1.39244	+	1.03011i	0.803924	+	0.594732i
\(5\)	0.209292	−	2.22625i	0.0935982	−	0.995610i
							\(7\)	−0.734969	−	0.514631i	−0.277792	−	0.194512i	0.426375	−	0.904547i	\(-0.359790\pi\)
−0.704167	+	0.710034i	\(0.748679\pi\)
\(11\)	0.457015	−	1.25564i	0.137795	−	0.378589i	−0.851532	−	0.524303i	\(-0.824326\pi\)
							0.989327	+	0.145714i	\(0.0465480\pi\)
\(13\)	0.0942983	+	1.07783i	0.0261537	+	0.298938i	0.997995	+	0.0632997i	\(0.0201624\pi\)
							−0.971841	+	0.235638i	\(0.924282\pi\)
\(17\)	−3.60283	+	0.965374i	−0.873814	+	0.234138i	−0.667736	−	0.744398i	\(-0.732737\pi\)
							−0.206077	+	0.978536i	\(0.566070\pi\)
\(19\)	−4.47083	+	2.58124i	−1.02568	+	0.592176i	−0.915744	−	0.401763i	\(-0.868398\pi\)
							−0.109935	+	0.993939i	\(0.535064\pi\)
\(23\)	−0.187700	−	0.268064i	−0.0391382	−	0.0558952i	0.799106	−	0.601190i	\(-0.205307\pi\)
							−0.838244	+	0.545295i	\(0.816418\pi\)
\(29\)	3.73628	−	3.13512i	0.693811	−	0.582176i	−0.226195	−	0.974082i	\(-0.572629\pi\)
							0.920005	+	0.391906i	\(0.128184\pi\)
\(31\)	−1.54601	+	8.76783i	−0.277671	+	1.57475i	0.452678	+	0.891674i	\(0.350469\pi\)
							−0.730348	+	0.683075i	\(0.760642\pi\)
\(37\)	−2.74278	−	10.2362i	−0.450910	−	1.68282i	−0.699841	−	0.714298i	\(-0.746746\pi\)
							0.248931	−	0.968521i	\(-0.419921\pi\)
\(41\)	0.945418	−	1.12671i	0.147649	−	0.175962i	−0.687151	−	0.726515i	\(-0.741139\pi\)
							0.834800	+	0.550553i	\(0.185583\pi\)
\(43\)	0.475774	+	1.02030i	0.0725548	+	0.155594i	0.939240	−	0.343261i	\(-0.111532\pi\)
							−0.866685	+	0.498855i	\(0.833754\pi\)
\(47\)	4.63025	−	6.61269i	0.675392	−	0.964560i	−0.324434	−	0.945908i	\(-0.605174\pi\)
							0.999826	−	0.0186513i	\(-0.00593725\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	270.2.r.a.113.17	✓	216
3.2	odd	2		810.2.s.a.773.4		216
5.2	odd	4	inner	270.2.r.a.167.7	yes	216
15.2	even	4		810.2.s.a.287.16		216
27.11	odd	18	inner	270.2.r.a.173.7	yes	216
27.16	even	9		810.2.s.a.683.16		216
135.92	even	36	inner	270.2.r.a.227.17	yes	216
135.97	odd	36		810.2.s.a.197.4		216

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
270.2.r.a.113.17	✓	216	1.1	even	1	trivial
270.2.r.a.167.7	yes	216	5.2	odd	4	inner
270.2.r.a.173.7	yes	216	27.11	odd	18	inner
270.2.r.a.227.17	yes	216	135.92	even	36	inner
810.2.s.a.197.4		216	135.97	odd	36
810.2.s.a.287.16		216	15.2	even	4
810.2.s.a.683.16		216	27.16	even	9
810.2.s.a.773.4		216	3.2	odd	2