Embedded newform 270.2.r.a.257.8

• Label: 270.2.r.a.257.8
• 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			257.8
Character	\(\chi\)	\(=\)	270.257
Dual form			270.2.r.a.83.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.819152 - 0.573576i) q^{2} +(1.50129 + 0.863781i) q^{3} +(0.342020 + 0.939693i) q^{4} +(0.160152 - 2.23033i) q^{5} +(-0.734343 - 1.56867i) q^{6} +(1.92341 + 4.12477i) q^{7} +(0.258819 - 0.965926i) q^{8} +(1.50776 + 2.59358i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 216 q + 12 q^{6}+O(q^{10}) \)

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
							\(3\)	1.50129	+	0.863781i	0.866772	+	0.498704i
\(5\)	0.160152	−	2.23033i	0.0716221	−	0.997432i
							\(7\)	1.92341	+	4.12477i	0.726981	+	1.55902i	0.825834	+	0.563913i	\(0.190705\pi\)
−0.0988531	+	0.995102i	\(0.531517\pi\)
\(11\)	−0.0108394	−	0.0129179i	−0.00326820	−	0.00389489i	0.764408	−	0.644733i	\(-0.223031\pi\)
							−0.767676	+	0.640838i	\(0.778587\pi\)
\(13\)	−1.70362	−	2.43302i	−0.472499	−	0.674799i	0.510190	−	0.860062i	\(-0.329575\pi\)
							−0.982689	+	0.185263i	\(0.940686\pi\)
\(17\)	0.126008	+	0.470270i	0.0305615	+	0.114057i	0.979522	−	0.201339i	\(-0.0645293\pi\)
							−0.948960	+	0.315396i	\(0.897863\pi\)
\(19\)	2.44011	−	1.40880i	0.559799	−	0.323200i	−0.193266	−	0.981146i	\(-0.561908\pi\)
							0.753065	+	0.657946i	\(0.228575\pi\)
\(23\)	5.98777	+	2.79214i	1.24854	+	0.582202i	0.930645	−	0.365925i	\(-0.119247\pi\)
							0.317892	+	0.948127i	\(0.397025\pi\)
\(29\)	−1.01729	−	5.76933i	−0.188906	−	1.07134i	−0.920834	−	0.389956i	\(-0.872490\pi\)
							0.731928	−	0.681382i	\(-0.238621\pi\)
\(31\)	−4.13283	+	1.50423i	−0.742279	+	0.270167i	−0.685353	−	0.728211i	\(-0.740352\pi\)
							−0.0569257	+	0.998378i	\(0.518130\pi\)
\(37\)	9.53706	−	2.55545i	1.56788	−	0.420113i	0.632734	−	0.774369i	\(-0.281933\pi\)
							0.935148	+	0.354256i	\(0.115266\pi\)
\(41\)	−9.02018	−	1.59050i	−1.40871	−	0.248394i	−0.582995	−	0.812475i	\(-0.698119\pi\)
							−0.825719	+	0.564081i	\(0.809231\pi\)
\(43\)	0.278251	−	3.18042i	0.0424328	−	0.485010i	−0.945011	−	0.327039i	\(-0.893949\pi\)
							0.987444	−	0.157971i	\(-0.0504952\pi\)
\(47\)	−2.19479	+	1.02345i	−0.320143	+	0.149285i	−0.576046	−	0.817417i	\(-0.695405\pi\)
							0.255903	+	0.966702i	\(0.417627\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	270.2.r.a.257.8	yes	216
3.2	odd	2		810.2.s.a.557.14		216
5.3	odd	4	inner	270.2.r.a.203.4	yes	216
15.8	even	4		810.2.s.a.233.10		216
27.2	odd	18	inner	270.2.r.a.137.4	yes	216
27.25	even	9		810.2.s.a.737.10		216
135.83	even	36	inner	270.2.r.a.83.8	✓	216
135.133	odd	36		810.2.s.a.413.14		216

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
270.2.r.a.83.8	✓	216	135.83	even	36	inner
270.2.r.a.137.4	yes	216	27.2	odd	18	inner
270.2.r.a.203.4	yes	216	5.3	odd	4	inner
270.2.r.a.257.8	yes	216	1.1	even	1	trivial
810.2.s.a.233.10		216	15.8	even	4
810.2.s.a.413.14		216	135.133	odd	36
810.2.s.a.557.14		216	3.2	odd	2
810.2.s.a.737.10		216	27.25	even	9