Embedded newform 270.2.r.a.23.9

• Label: 270.2.r.a.23.9
• Level: $270$
• Weight: $2$
• Character: 270.23
• 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			23.9
Character	\(\chi\)	\(=\)	270.23
Dual form			270.2.r.a.47.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.422618 - 0.906308i) q^{2} +(1.57567 + 0.719210i) q^{3} +(-0.642788 + 0.766044i) q^{4} +(-2.23484 - 0.0739999i) q^{5} +(-0.0140815 - 1.73199i) q^{6} +(-0.290253 + 3.31761i) q^{7} +(0.965926 + 0.258819i) q^{8} +(1.96547 + 2.26648i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 216 q + 12 q^{6}+O(q^{10}) \)

Character values

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

\(n\)	\(191\)	\(217\)
\(\chi(n)\)	\(e\left(\frac{11}{18}\right)\)	\(e\left(\frac{3}{4}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.422618	−	0.906308i	−0.298836	−	0.640856i
							\(3\)	1.57567	+	0.719210i	0.909714	+	0.415236i
\(5\)	−2.23484	−	0.0739999i	−0.999452	−	0.0330938i
							\(7\)	−0.290253	+	3.31761i	−0.109705	+	1.25394i	0.719300	+	0.694699i	\(0.244463\pi\)
−0.829006	+	0.559240i	\(0.811093\pi\)
\(11\)	5.15842	−	0.909569i	1.55532	−	0.274245i	0.671120	−	0.741349i	\(-0.265814\pi\)
							0.884203	+	0.467104i	\(0.154703\pi\)
\(13\)	2.89399	+	1.34949i	0.802649	+	0.374281i	0.780250	−	0.625468i	\(-0.215092\pi\)
							0.0223988	+	0.999749i	\(0.492870\pi\)
\(17\)	4.30809	−	1.15435i	1.04487	−	0.279971i	0.304737	−	0.952437i	\(-0.401431\pi\)
							0.740128	+	0.672466i	\(0.234765\pi\)
\(19\)	−4.29957	+	2.48236i	−0.986389	+	0.569492i	−0.904193	−	0.427124i	\(-0.859527\pi\)
							−0.0821962	+	0.996616i	\(0.526193\pi\)
\(23\)	−7.23494	+	0.632975i	−1.50859	+	0.131984i	−0.811226	−	0.584733i	\(-0.801199\pi\)
							−0.697363	+	0.716718i	\(0.745644\pi\)
\(29\)	−3.90691	−	1.42200i	−0.725495	−	0.264059i	−0.0472384	−	0.998884i	\(-0.515042\pi\)
							−0.678257	+	0.734825i	\(0.737264\pi\)
\(31\)	−2.63399	−	2.21018i	−0.473079	−	0.396960i	0.374837	−	0.927091i	\(-0.377699\pi\)
							−0.847916	+	0.530130i	\(0.822143\pi\)
\(37\)	−0.209441	−	0.781646i	−0.0344319	−	0.128502i	0.946571	−	0.322497i	\(-0.104522\pi\)
							−0.981003	+	0.193995i	\(0.937855\pi\)
\(41\)	−3.96621	−	10.8971i	−0.619418	−	1.70184i	−0.708407	−	0.705804i	\(-0.750586\pi\)
							0.0889892	−	0.996033i	\(-0.471636\pi\)
\(43\)	1.49834	−	2.13984i	0.228494	−	0.326323i	−0.688491	−	0.725245i	\(-0.741727\pi\)
							0.916985	+	0.398922i	\(0.130615\pi\)
\(47\)	10.2729	+	0.898762i	1.49846	+	0.131098i	0.806669	−	0.591004i	\(-0.201268\pi\)
							0.691787	+	0.722102i	\(0.256824\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	270.2.r.a.23.9	✓	216
3.2	odd	2		810.2.s.a.503.18		216
5.2	odd	4	inner	270.2.r.a.77.14	yes	216
15.2	even	4		810.2.s.a.17.9		216
27.7	even	9		810.2.s.a.143.9		216
27.20	odd	18	inner	270.2.r.a.263.14	yes	216
135.7	odd	36		810.2.s.a.467.18		216
135.47	even	36	inner	270.2.r.a.47.9	yes	216

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
270.2.r.a.23.9	✓	216	1.1	even	1	trivial
270.2.r.a.47.9	yes	216	135.47	even	36	inner
270.2.r.a.77.14	yes	216	5.2	odd	4	inner
270.2.r.a.263.14	yes	216	27.20	odd	18	inner
810.2.s.a.17.9		216	15.2	even	4
810.2.s.a.143.9		216	27.7	even	9
810.2.s.a.467.18		216	135.7	odd	36
810.2.s.a.503.18		216	3.2	odd	2