Embedded newform 540.2.y.a.127.17

• Label: 540.2.y.a.127.17
• 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			127.17
Character	\(\chi\)	\(=\)	540.127
Dual form			540.2.y.a.523.17

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.168283 - 1.40417i) q^{2} +(-1.94336 - 0.472596i) q^{4} +(-2.09202 - 0.789601i) q^{5} +(1.04292 + 3.89223i) q^{7} +(-0.990638 + 2.64927i) q^{8} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 128 q + 2 q^{2} + 4 q^{5} + 8 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.168283	−	1.40417i	0.118994	−	0.992895i
							\(3\)		0			0
\(5\)	−2.09202	−	0.789601i	−0.935578	−	0.353120i
							\(7\)	1.04292	+	3.89223i	0.394187	+	1.47113i	0.823160	+	0.567809i	\(0.192209\pi\)
−0.428973	+	0.903317i	\(0.641125\pi\)
\(11\)	3.25799	+	1.88100i	0.982320	+	0.567143i	0.902970	−	0.429704i	\(-0.141382\pi\)
							0.0793505	+	0.996847i	\(0.474715\pi\)
\(13\)	−1.05959	−	0.283916i	−0.293877	−	0.0787440i	0.108869	−	0.994056i	\(-0.465277\pi\)
							−0.402745	+	0.915312i	\(0.631944\pi\)
\(17\)	0.945812	+	0.945812i	0.229393	+	0.229393i	0.812439	−	0.583046i	\(-0.198139\pi\)
							−0.583046	+	0.812439i	\(0.698139\pi\)
\(19\)	4.90169			1.12453			0.562263	−	0.826959i	\(-0.309931\pi\)
							0.562263	+	0.826959i	\(0.309931\pi\)
\(23\)	−0.252236	+	0.941359i	−0.0525949	+	0.196287i	−0.987224	−	0.159336i	\(-0.949065\pi\)
							0.934629	+	0.355623i	\(0.115731\pi\)
\(29\)	−5.62912	−	3.24997i	−1.04530	−	0.603505i	−0.123971	−	0.992286i	\(-0.539563\pi\)
							−0.921330	+	0.388781i	\(0.872896\pi\)
\(31\)	−4.24124	+	2.44868i	−0.761749	+	0.439796i	−0.829924	−	0.557877i	\(-0.811616\pi\)
							0.0681740	+	0.997673i	\(0.478283\pi\)
\(37\)	7.13011	+	7.13011i	1.17218	+	1.17218i	0.981688	+	0.190496i	\(0.0610095\pi\)
							0.190496	+	0.981688i	\(0.438991\pi\)
\(41\)	1.20263	+	2.08301i	0.187819	+	0.325312i	0.944523	−	0.328446i	\(-0.106525\pi\)
							−0.756704	+	0.653758i	\(0.773192\pi\)
\(43\)	1.71286	−	0.458958i	0.261208	−	0.0699905i	−0.125838	−	0.992051i	\(-0.540162\pi\)
							0.387046	+	0.922060i	\(0.373495\pi\)
\(47\)	1.32619	+	4.94940i	0.193444	+	0.721945i	0.992664	+	0.120905i	\(0.0385798\pi\)
							−0.799220	+	0.601039i	\(0.794754\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	540.2.y.a.127.17		128
3.2	odd	2		180.2.x.a.7.16	yes	128
4.3	odd	2	inner	540.2.y.a.127.22		128
5.3	odd	4	inner	540.2.y.a.343.2		128
9.4	even	3	inner	540.2.y.a.307.25		128
9.5	odd	6		180.2.x.a.67.8	yes	128
12.11	even	2		180.2.x.a.7.11	✓	128
15.2	even	4		900.2.bf.e.43.2		128
15.8	even	4		180.2.x.a.43.31	yes	128
15.14	odd	2		900.2.bf.e.7.17		128
20.3	even	4	inner	540.2.y.a.343.25		128
36.23	even	6		180.2.x.a.67.31	yes	128
36.31	odd	6	inner	540.2.y.a.307.2		128
45.13	odd	12	inner	540.2.y.a.523.22		128
45.14	odd	6		900.2.bf.e.607.25		128
45.23	even	12		180.2.x.a.103.11	yes	128
45.32	even	12		900.2.bf.e.643.22		128
60.23	odd	4		180.2.x.a.43.8	yes	128
60.47	odd	4		900.2.bf.e.43.25		128
60.59	even	2		900.2.bf.e.7.22		128
180.23	odd	12		180.2.x.a.103.16	yes	128
180.59	even	6		900.2.bf.e.607.2		128
180.103	even	12	inner	540.2.y.a.523.17		128
180.167	odd	12		900.2.bf.e.643.17		128

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
180.2.x.a.7.11	✓	128	12.11	even	2
180.2.x.a.7.16	yes	128	3.2	odd	2
180.2.x.a.43.8	yes	128	60.23	odd	4
180.2.x.a.43.31	yes	128	15.8	even	4
180.2.x.a.67.8	yes	128	9.5	odd	6
180.2.x.a.67.31	yes	128	36.23	even	6
180.2.x.a.103.11	yes	128	45.23	even	12
180.2.x.a.103.16	yes	128	180.23	odd	12
540.2.y.a.127.17		128	1.1	even	1	trivial
540.2.y.a.127.22		128	4.3	odd	2	inner
540.2.y.a.307.2		128	36.31	odd	6	inner
540.2.y.a.307.25		128	9.4	even	3	inner
540.2.y.a.343.2		128	5.3	odd	4	inner
540.2.y.a.343.25		128	20.3	even	4	inner
540.2.y.a.523.17		128	180.103	even	12	inner
540.2.y.a.523.22		128	45.13	odd	12	inner
900.2.bf.e.7.17		128	15.14	odd	2
900.2.bf.e.7.22		128	60.59	even	2
900.2.bf.e.43.2		128	15.2	even	4
900.2.bf.e.43.25		128	60.47	odd	4
900.2.bf.e.607.2		128	180.59	even	6
900.2.bf.e.607.25		128	45.14	odd	6
900.2.bf.e.643.17		128	180.167	odd	12
900.2.bf.e.643.22		128	45.32	even	12