Embedded newform 270.2.r.a.257.16

• Label: 270.2.r.a.257.16
• 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			257.16
Character	\(\chi\)	\(=\)	270.257
Dual form			270.2.r.a.83.16

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.819152 + 0.573576i) q^{2} +(1.17634 + 1.27131i) q^{3} +(0.342020 + 0.939693i) q^{4} +(2.18527 - 0.473911i) q^{5} +(0.234408 + 1.71612i) q^{6} +(-0.171895 - 0.368630i) q^{7} +(-0.258819 + 0.965926i) q^{8} +(-0.232451 + 2.99098i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 216 q + 12 q^{6}+O(q^{10}) \)

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
							\(3\)	1.17634	+	1.27131i	0.679160	+	0.733990i
\(5\)	2.18527	−	0.473911i	0.977283	−	0.211940i
							\(7\)	−0.171895	−	0.368630i	−0.0649702	−	0.139329i	0.871141	−	0.491032i	\(-0.163380\pi\)
−0.936111	+	0.351703i	\(0.885603\pi\)
\(11\)	−3.16016	−	3.76614i	−0.952825	−	1.13553i	−0.990675	−	0.136249i	\(-0.956495\pi\)
							0.0378496	−	0.999283i	\(-0.487949\pi\)
\(13\)	−2.75679	−	3.93710i	−0.764595	−	1.09196i	−0.992968	−	0.118382i	\(-0.962229\pi\)
							0.228373	−	0.973574i	\(-0.426659\pi\)
\(17\)	0.352969	+	1.31730i	0.0856075	+	0.319491i	0.995429	−	0.0955093i	\(-0.0304480\pi\)
							−0.909821	+	0.415001i	\(0.863781\pi\)
\(19\)	−0.763415	+	0.440758i	−0.175139	+	0.101117i	−0.585007	−	0.811028i	\(-0.698908\pi\)
							0.409868	+	0.912145i	\(0.365575\pi\)
\(23\)	0.893489	+	0.416641i	0.186305	+	0.0868756i	0.513533	−	0.858070i	\(-0.328337\pi\)
							−0.327227	+	0.944946i	\(0.606114\pi\)
\(29\)	1.09136	+	6.18939i	0.202660	+	1.14934i	0.901080	+	0.433653i	\(0.142776\pi\)
							−0.698420	+	0.715688i	\(0.746113\pi\)
\(31\)	−7.20395	+	2.62202i	−1.29387	+	0.470929i	−0.894995	−	0.446077i	\(-0.852821\pi\)
							−0.398873	+	0.917006i	\(0.630599\pi\)
\(37\)	−8.35530	+	2.23880i	−1.37360	+	0.368056i	−0.868793	−	0.495175i	\(-0.835104\pi\)
							−0.504810	+	0.863231i	\(0.668437\pi\)
\(41\)	10.9197	+	1.92544i	1.70537	+	0.300703i	0.939566	−	0.342369i	\(-0.111229\pi\)
							0.765806	+	0.643072i	\(0.222340\pi\)
\(43\)	1.07657	−	12.3052i	0.164175	−	1.87653i	−0.252666	−	0.967553i	\(-0.581307\pi\)
							0.416842	−	0.908979i	\(-0.363137\pi\)
\(47\)	3.46784	−	1.61708i	0.505836	−	0.235875i	−0.152911	−	0.988240i	\(-0.548865\pi\)
							0.658747	+	0.752365i	\(0.271087\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	270.2.r.a.257.16	yes	216
3.2	odd	2		810.2.s.a.557.1		216
5.3	odd	4	inner	270.2.r.a.203.12	yes	216
15.8	even	4		810.2.s.a.233.5		216
27.2	odd	18	inner	270.2.r.a.137.12	yes	216
27.25	even	9		810.2.s.a.737.5		216
135.83	even	36	inner	270.2.r.a.83.16	✓	216
135.133	odd	36		810.2.s.a.413.1		216

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
270.2.r.a.83.16	✓	216	135.83	even	36	inner
270.2.r.a.137.12	yes	216	27.2	odd	18	inner
270.2.r.a.203.12	yes	216	5.3	odd	4	inner
270.2.r.a.257.16	yes	216	1.1	even	1	trivial
810.2.s.a.233.5		216	15.8	even	4
810.2.s.a.413.1		216	135.133	odd	36
810.2.s.a.557.1		216	3.2	odd	2
810.2.s.a.737.5		216	27.25	even	9