Embedded newform 180.2.x.a.43.29

• Label: 180.2.x.a.43.29
• Level: $180$
• Weight: $2$
• Character: 180.43
• Analytic conductor: $1.437$
• Analytic rank: $0$
• Dimension: $128$
• CM: no
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 180 = 2^{2} \cdot 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	180.x (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.43730723638\)
Analytic rank:	\(0\)
Dimension:	\(128\)
Relative dimension:	\(32\) over \(\Q(\zeta_{12})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			43.29
Character	\(\chi\)	\(=\)	180.43
Dual form			180.2.x.a.67.29

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.33961 - 0.453270i) q^{2} +(-1.51730 + 0.835348i) q^{3} +(1.58909 - 1.21441i) q^{4} +(1.01645 - 1.99169i) q^{5} +(-1.65394 + 1.80678i) q^{6} +(-0.00373871 + 0.00100179i) q^{7} +(1.57830 - 2.34712i) q^{8} +(1.60439 - 2.53494i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 128 q - 2 q^{2} - 4 q^{5} - 8 q^{6} - 8 q^{8}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/180\mathbb{Z}\right)^\times\).

\(n\)	\(37\)	\(91\)	\(101\)
\(\chi(n)\)	\(e\left(\frac{3}{4}\right)\)	\(-1\)	\(e\left(\frac{2}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.33961	−	0.453270i	0.947245	−	0.320510i
							\(3\)	−1.51730	+	0.835348i	−0.876012	+	0.482288i
\(5\)	1.01645	−	1.99169i	0.454569	−	0.890711i
							\(7\)	−0.00373871	+	0.00100179i	−0.00141310	+	0.000378639i	−0.259526	−	0.965736i	\(-0.583566\pi\)
0.258112	+	0.966115i	\(0.416900\pi\)
\(11\)	3.58713	+	2.07103i	1.08156	+	0.624439i	0.931317	−	0.364210i	\(-0.118661\pi\)
							0.150243	+	0.988649i	\(0.451994\pi\)
\(13\)	−0.767553	+	2.86455i	−0.212881	+	0.794483i	0.774021	+	0.633160i	\(0.218243\pi\)
							−0.986902	+	0.161322i	\(0.948424\pi\)
\(17\)	−2.07784	+	2.07784i	−0.503949	+	0.503949i	−0.912663	−	0.408713i	\(-0.865978\pi\)
							0.408713	+	0.912663i	\(0.365978\pi\)
\(19\)	−7.31525			−1.67823			−0.839117	−	0.543951i	\(-0.816928\pi\)
							−0.839117	+	0.543951i	\(0.816928\pi\)
\(23\)	−3.73983	−	1.00208i	−0.779808	−	0.208949i	−0.153108	−	0.988209i	\(-0.548928\pi\)
							−0.626700	+	0.779260i	\(0.715595\pi\)
\(29\)	4.99228	+	2.88229i	0.927042	+	0.535228i	0.885875	−	0.463924i	\(-0.153559\pi\)
							0.0411674	+	0.999152i	\(0.486892\pi\)
\(31\)	1.77642	−	1.02562i	0.319054	−	0.184206i	−0.331917	−	0.943309i	\(-0.607695\pi\)
							0.650971	+	0.759103i	\(0.274362\pi\)
\(37\)	−6.09468	+	6.09468i	−1.00196	+	1.00196i	−0.00196150	+	0.999998i	\(0.500624\pi\)
							−0.999998	+	0.00196150i	\(0.999376\pi\)
\(41\)	1.82477	+	3.16060i	0.284981	+	0.493602i	0.972605	−	0.232465i	\(-0.0746792\pi\)
							−0.687623	+	0.726068i	\(0.741346\pi\)
\(43\)	−0.259538	−	0.968609i	−0.0395792	−	0.147711i	0.943309	−	0.331917i	\(-0.107695\pi\)
							−0.982888	+	0.184206i	\(0.941029\pi\)
\(47\)	4.55740	−	1.22115i	0.664765	−	0.178123i	0.0893696	−	0.995999i	\(-0.471515\pi\)
							0.575396	+	0.817875i	\(0.304848\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	180.2.x.a.43.29	yes	128
3.2	odd	2		540.2.y.a.343.4		128
4.3	odd	2	inner	180.2.x.a.43.2	yes	128
5.2	odd	4	inner	180.2.x.a.7.20	yes	128
5.3	odd	4		900.2.bf.e.7.13		128
5.4	even	2		900.2.bf.e.43.4		128
9.4	even	3	inner	180.2.x.a.103.14	yes	128
9.5	odd	6		540.2.y.a.523.19		128
12.11	even	2		540.2.y.a.343.31		128
15.2	even	4		540.2.y.a.127.13		128
20.3	even	4		900.2.bf.e.7.19		128
20.7	even	4	inner	180.2.x.a.7.14	✓	128
20.19	odd	2		900.2.bf.e.43.31		128
36.23	even	6		540.2.y.a.523.13		128
36.31	odd	6	inner	180.2.x.a.103.20	yes	128
45.4	even	6		900.2.bf.e.643.19		128
45.13	odd	12		900.2.bf.e.607.31		128
45.22	odd	12	inner	180.2.x.a.67.2	yes	128
45.32	even	12		540.2.y.a.307.31		128
60.47	odd	4		540.2.y.a.127.19		128
180.67	even	12	inner	180.2.x.a.67.29	yes	128
180.103	even	12		900.2.bf.e.607.4		128
180.139	odd	6		900.2.bf.e.643.13		128
180.167	odd	12		540.2.y.a.307.4		128

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
180.2.x.a.7.14	✓	128	20.7	even	4	inner
180.2.x.a.7.20	yes	128	5.2	odd	4	inner
180.2.x.a.43.2	yes	128	4.3	odd	2	inner
180.2.x.a.43.29	yes	128	1.1	even	1	trivial
180.2.x.a.67.2	yes	128	45.22	odd	12	inner
180.2.x.a.67.29	yes	128	180.67	even	12	inner
180.2.x.a.103.14	yes	128	9.4	even	3	inner
180.2.x.a.103.20	yes	128	36.31	odd	6	inner
540.2.y.a.127.13		128	15.2	even	4
540.2.y.a.127.19		128	60.47	odd	4
540.2.y.a.307.4		128	180.167	odd	12
540.2.y.a.307.31		128	45.32	even	12
540.2.y.a.343.4		128	3.2	odd	2
540.2.y.a.343.31		128	12.11	even	2
540.2.y.a.523.13		128	36.23	even	6
540.2.y.a.523.19		128	9.5	odd	6
900.2.bf.e.7.13		128	5.3	odd	4
900.2.bf.e.7.19		128	20.3	even	4
900.2.bf.e.43.4		128	5.4	even	2
900.2.bf.e.43.31		128	20.19	odd	2
900.2.bf.e.607.4		128	180.103	even	12
900.2.bf.e.607.31		128	45.13	odd	12
900.2.bf.e.643.13		128	180.139	odd	6
900.2.bf.e.643.19		128	45.4	even	6