Embedded newform 540.2.y.a.127.18

• Label: 540.2.y.a.127.18
• Level: $540$
• Weight: $2$
• Character: 540.127
• Analytic conductor: $4.312$
• Analytic rank: $0$
• Dimension: $128$
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 540 = 2^{2} \cdot 3^{3} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	540.y (of order \(12\), degree \(4\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.31192170915\)
Analytic rank:	\(0\)
Dimension:	\(128\)
Relative dimension:	\(32\) over \(\Q(\zeta_{12})\)
Twist minimal:	no (minimal twist has level 180)
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			127.18
Character	\(\chi\)	\(=\)	540.127
Dual form			540.2.y.a.523.18

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.241682 - 1.39341i) q^{2} +(-1.88318 - 0.673524i) q^{4} +(1.26760 - 1.84206i) q^{5} +(-1.09131 - 4.07282i) q^{7} +(-1.39362 + 2.46126i) q^{8} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 128 q + 2 q^{2} + 4 q^{5} + 8 q^{8}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/540\mathbb{Z}\right)^\times\).

\(n\)	\(217\)	\(271\)	\(461\)
\(\chi(n)\)	\(e\left(\frac{1}{4}\right)\)	\(-1\)	\(e\left(\frac{2}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.241682	−	1.39341i	0.170895	−	0.985289i
							\(3\)		0			0
\(5\)	1.26760	−	1.84206i	0.566889	−	0.823794i
							\(7\)	−1.09131	−	4.07282i	−0.412476	−	1.53938i	−0.789837	−	0.613317i	\(-0.789835\pi\)
0.377361	−	0.926066i	\(-0.376832\pi\)
\(11\)	0.447973	+	0.258637i	0.135069	+	0.0779821i	0.566012	−	0.824397i	\(-0.308486\pi\)
							−0.430943	+	0.902379i	\(0.641819\pi\)
\(13\)	−1.88061	−	0.503908i	−0.521588	−	0.139759i	−0.0115869	−	0.999933i	\(-0.503688\pi\)
							−0.510001	+	0.860174i	\(0.670355\pi\)
\(17\)	4.37652	+	4.37652i	1.06146	+	1.06146i	0.997983	+	0.0634784i	\(0.0202194\pi\)
							0.0634784	+	0.997983i	\(0.479781\pi\)
\(19\)	−5.33020			−1.22283			−0.611416	−	0.791309i	\(-0.709400\pi\)
							−0.611416	+	0.791309i	\(0.709400\pi\)
\(23\)	0.996062	−	3.71735i	0.207693	−	0.775122i	−0.780918	−	0.624633i	\(-0.785249\pi\)
							0.988612	−	0.150489i	\(-0.0480847\pi\)
\(29\)	−3.49078	−	2.01540i	−0.648222	−	0.374251i	0.139553	−	0.990215i	\(-0.455434\pi\)
							−0.787775	+	0.615964i	\(0.788767\pi\)
\(31\)	−1.01483	+	0.585914i	−0.182269	+	0.105233i	−0.588358	−	0.808600i	\(-0.700226\pi\)
							0.406089	+	0.913833i	\(0.366892\pi\)
\(37\)	−1.89681	−	1.89681i	−0.311833	−	0.311833i	0.533786	−	0.845619i	\(-0.320769\pi\)
							−0.845619	+	0.533786i	\(0.820769\pi\)
\(41\)	0.924526	+	1.60133i	0.144387	+	0.250085i	0.929144	−	0.369718i	\(-0.120546\pi\)
							−0.784757	+	0.619803i	\(0.787212\pi\)
\(43\)	−1.13787	+	0.304891i	−0.173524	+	0.0464955i	−0.344535	−	0.938774i	\(-0.611963\pi\)
							0.171011	+	0.985269i	\(0.445297\pi\)
\(47\)	−0.627826	−	2.34308i	−0.0915779	−	0.341773i	0.904901	−	0.425623i	\(-0.139945\pi\)
							−0.996478	+	0.0838499i	\(0.973278\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	540.2.y.a.127.18		128
3.2	odd	2		180.2.x.a.7.15	yes	128
4.3	odd	2	inner	540.2.y.a.127.23		128
5.3	odd	4	inner	540.2.y.a.343.3		128
9.4	even	3	inner	540.2.y.a.307.24		128
9.5	odd	6		180.2.x.a.67.9	yes	128
12.11	even	2		180.2.x.a.7.10	✓	128
15.2	even	4		900.2.bf.e.43.3		128
15.8	even	4		180.2.x.a.43.30	yes	128
15.14	odd	2		900.2.bf.e.7.18		128
20.3	even	4	inner	540.2.y.a.343.24		128
36.23	even	6		180.2.x.a.67.30	yes	128
36.31	odd	6	inner	540.2.y.a.307.3		128
45.13	odd	12	inner	540.2.y.a.523.23		128
45.14	odd	6		900.2.bf.e.607.24		128
45.23	even	12		180.2.x.a.103.10	yes	128
45.32	even	12		900.2.bf.e.643.23		128
60.23	odd	4		180.2.x.a.43.9	yes	128
60.47	odd	4		900.2.bf.e.43.24		128
60.59	even	2		900.2.bf.e.7.23		128
180.23	odd	12		180.2.x.a.103.15	yes	128
180.59	even	6		900.2.bf.e.607.3		128
180.103	even	12	inner	540.2.y.a.523.18		128
180.167	odd	12		900.2.bf.e.643.18		128

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
180.2.x.a.7.10	✓	128	12.11	even	2
180.2.x.a.7.15	yes	128	3.2	odd	2
180.2.x.a.43.9	yes	128	60.23	odd	4
180.2.x.a.43.30	yes	128	15.8	even	4
180.2.x.a.67.9	yes	128	9.5	odd	6
180.2.x.a.67.30	yes	128	36.23	even	6
180.2.x.a.103.10	yes	128	45.23	even	12
180.2.x.a.103.15	yes	128	180.23	odd	12
540.2.y.a.127.18		128	1.1	even	1	trivial
540.2.y.a.127.23		128	4.3	odd	2	inner
540.2.y.a.307.3		128	36.31	odd	6	inner
540.2.y.a.307.24		128	9.4	even	3	inner
540.2.y.a.343.3		128	5.3	odd	4	inner
540.2.y.a.343.24		128	20.3	even	4	inner
540.2.y.a.523.18		128	180.103	even	12	inner
540.2.y.a.523.23		128	45.13	odd	12	inner
900.2.bf.e.7.18		128	15.14	odd	2
900.2.bf.e.7.23		128	60.59	even	2
900.2.bf.e.43.3		128	15.2	even	4
900.2.bf.e.43.24		128	60.47	odd	4
900.2.bf.e.607.3		128	180.59	even	6
900.2.bf.e.607.24		128	45.14	odd	6
900.2.bf.e.643.18		128	180.167	odd	12
900.2.bf.e.643.23		128	45.32	even	12