Embedded newform 270.2.r.a.257.14

• Label: 270.2.r.a.257.14
• Level: $270$
• Weight: $2$
• Character: 270.257
• 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			257.14
Character	\(\chi\)	\(=\)	270.257
Dual form			270.2.r.a.83.14

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.819152 + 0.573576i) q^{2} +(0.264030 + 1.71181i) q^{3} +(0.342020 + 0.939693i) q^{4} +(-2.23178 - 0.138341i) q^{5} +(-0.765573 + 1.55367i) q^{6} +(0.660295 + 1.41601i) q^{7} +(-0.258819 + 0.965926i) q^{8} +(-2.86058 + 0.903936i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 216 q + 12 q^{6}+O(q^{10}) \)

Character values

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

\(n\)	\(191\)	\(217\)
\(\chi(n)\)	\(e\left(\frac{17}{18}\right)\)	\(e\left(\frac{1}{4}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.819152	+	0.573576i	0.579228	+	0.405580i
							\(3\)	0.264030	+	1.71181i	0.152438	+	0.988313i
\(5\)	−2.23178	−	0.138341i	−0.998084	−	0.0618682i
							\(7\)	0.660295	+	1.41601i	0.249568	+	0.535200i	0.990685	−	0.136176i	\(-0.0434812\pi\)
−0.741117	+	0.671376i	\(0.765703\pi\)
\(11\)	2.49909	+	2.97830i	0.753504	+	0.897991i	0.997419	−	0.0718058i	\(-0.0228762\pi\)
							−0.243915	+	0.969797i	\(0.578432\pi\)
\(13\)	−2.53803	−	3.62468i	−0.703922	−	1.00530i	−0.998761	−	0.0497575i	\(-0.984155\pi\)
							0.294840	−	0.955547i	\(-0.404734\pi\)
\(17\)	1.52587	+	5.69463i	0.370078	+	1.38115i	0.860405	+	0.509612i	\(0.170211\pi\)
							−0.490326	+	0.871539i	\(0.663122\pi\)
\(19\)	5.43314	−	3.13682i	1.24645	−	0.719636i	0.276048	−	0.961144i	\(-0.410975\pi\)
							0.970399	+	0.241507i	\(0.0776418\pi\)
\(23\)	−1.84860	−	0.862017i	−0.385460	−	0.179743i	0.220222	−	0.975450i	\(-0.429322\pi\)
							−0.605682	+	0.795707i	\(0.707100\pi\)
\(29\)	−0.651227	−	3.69329i	−0.120930	−	0.685827i	−0.983642	−	0.180132i	\(-0.942347\pi\)
							0.862713	−	0.505694i	\(-0.168764\pi\)
\(31\)	4.48984	−	1.63417i	0.806400	−	0.293505i	0.0942638	−	0.995547i	\(-0.469950\pi\)
							0.712136	+	0.702042i	\(0.247728\pi\)
\(37\)	5.27901	−	1.41451i	0.867864	−	0.232544i	0.202700	−	0.979241i	\(-0.435028\pi\)
							0.665164	+	0.746697i	\(0.268362\pi\)
\(41\)	1.36266	+	0.240274i	0.212812	+	0.0375245i	0.279037	−	0.960280i	\(-0.409985\pi\)
							−0.0662257	+	0.997805i	\(0.521096\pi\)
\(43\)	−0.741688	+	8.47753i	−0.113106	+	1.29281i	0.701446	+	0.712722i	\(0.252538\pi\)
							−0.814553	+	0.580089i	\(0.803018\pi\)
\(47\)	−2.58064	+	1.20337i	−0.376425	+	0.175530i	−0.601617	−	0.798785i	\(-0.705477\pi\)
							0.225192	+	0.974314i	\(0.427699\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	270.2.r.a.257.14	yes	216
3.2	odd	2		810.2.s.a.557.9		216
5.3	odd	4	inner	270.2.r.a.203.10	yes	216
15.8	even	4		810.2.s.a.233.3		216
27.2	odd	18	inner	270.2.r.a.137.10	yes	216
27.25	even	9		810.2.s.a.737.3		216
135.83	even	36	inner	270.2.r.a.83.14	✓	216
135.133	odd	36		810.2.s.a.413.9		216

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
270.2.r.a.83.14	✓	216	135.83	even	36	inner
270.2.r.a.137.10	yes	216	27.2	odd	18	inner
270.2.r.a.203.10	yes	216	5.3	odd	4	inner
270.2.r.a.257.14	yes	216	1.1	even	1	trivial
810.2.s.a.233.3		216	15.8	even	4
810.2.s.a.413.9		216	135.133	odd	36
810.2.s.a.557.9		216	3.2	odd	2
810.2.s.a.737.3		216	27.25	even	9