Embedded newform 180.2.x.a.43.4

• Label: 180.2.x.a.43.4
• 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			43.4
Character	\(\chi\)	\(=\)	180.43
Dual form			180.2.x.a.67.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.27862 - 0.604268i) q^{2} +(-1.19621 - 1.25263i) q^{3} +(1.26972 + 1.54525i) q^{4} +(-0.946126 - 2.02604i) q^{5} +(0.772562 + 2.32447i) q^{6} +(-2.24052 + 0.600346i) q^{7} +(-0.689735 - 2.74304i) q^{8} +(-0.138181 + 2.99682i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 128 q - 2 q^{2} - 4 q^{5} - 8 q^{6} - 8 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.27862	−	0.604268i	−0.904118	−	0.427282i
							\(3\)	−1.19621	−	1.25263i	−0.690630	−	0.723208i
\(5\)	−0.946126	−	2.02604i	−0.423120	−	0.906073i
							\(7\)	−2.24052	+	0.600346i	−0.846838	+	0.226910i	−0.656046	−	0.754721i	\(-0.727772\pi\)
−0.190792	+	0.981630i	\(0.561106\pi\)
\(11\)	−0.696845	−	0.402323i	−0.210107	−	0.121305i	0.391254	−	0.920283i	\(-0.372041\pi\)
							−0.601361	+	0.798978i	\(0.705375\pi\)
\(13\)	−0.205830	+	0.768168i	−0.0570870	+	0.213051i	−0.988577	−	0.150715i	\(-0.951843\pi\)
							0.931490	+	0.363766i	\(0.118509\pi\)
\(17\)	−3.72182	+	3.72182i	−0.902673	+	0.902673i	−0.995667	−	0.0929939i	\(-0.970356\pi\)
							0.0929939	+	0.995667i	\(0.470356\pi\)
\(19\)	−5.35708			−1.22900			−0.614499	−	0.788918i	\(-0.710642\pi\)
							−0.614499	+	0.788918i	\(0.710642\pi\)
\(23\)	0.127597	+	0.0341896i	0.0266059	+	0.00712903i	0.272097	−	0.962270i	\(-0.412283\pi\)
							−0.245492	+	0.969399i	\(0.578949\pi\)
\(29\)	−4.50748	−	2.60239i	−0.837017	−	0.483252i	0.0192321	−	0.999815i	\(-0.493878\pi\)
							−0.856249	+	0.516563i	\(0.827211\pi\)
\(31\)	6.83666	−	3.94715i	1.22790	−	0.708929i	0.261310	−	0.965255i	\(-0.415845\pi\)
							0.966590	+	0.256326i	\(0.0825121\pi\)
\(37\)	4.43358	−	4.43358i	0.728876	−	0.728876i	−0.241520	−	0.970396i	\(-0.577646\pi\)
							0.970396	+	0.241520i	\(0.0776458\pi\)
\(41\)	−5.49152	−	9.51160i	−0.857632	−	1.48546i	−0.874182	−	0.485599i	\(-0.838602\pi\)
							0.0165497	−	0.999863i	\(-0.494732\pi\)
\(43\)	−2.77403	−	10.3528i	−0.423036	−	1.57879i	−0.768175	−	0.640240i	\(-0.778835\pi\)
							0.345138	−	0.938552i	\(-0.387832\pi\)
\(47\)	−7.44461	+	1.99478i	−1.08591	+	0.290968i	−0.757014	−	0.653398i	\(-0.773343\pi\)
							−0.328894	+	0.944367i	\(0.606676\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	180.2.x.a.43.4	yes	128
3.2	odd	2		540.2.y.a.343.29		128
4.3	odd	2	inner	180.2.x.a.43.23	yes	128
5.2	odd	4	inner	180.2.x.a.7.21	✓	128
5.3	odd	4		900.2.bf.e.7.12		128
5.4	even	2		900.2.bf.e.43.29		128
9.4	even	3	inner	180.2.x.a.103.26	yes	128
9.5	odd	6		540.2.y.a.523.7		128
12.11	even	2		540.2.y.a.343.10		128
15.2	even	4		540.2.y.a.127.12		128
20.3	even	4		900.2.bf.e.7.7		128
20.7	even	4	inner	180.2.x.a.7.26	yes	128
20.19	odd	2		900.2.bf.e.43.10		128
36.23	even	6		540.2.y.a.523.12		128
36.31	odd	6	inner	180.2.x.a.103.21	yes	128
45.4	even	6		900.2.bf.e.643.7		128
45.13	odd	12		900.2.bf.e.607.10		128
45.22	odd	12	inner	180.2.x.a.67.23	yes	128
45.32	even	12		540.2.y.a.307.10		128
60.47	odd	4		540.2.y.a.127.7		128
180.67	even	12	inner	180.2.x.a.67.4	yes	128
180.103	even	12		900.2.bf.e.607.29		128
180.139	odd	6		900.2.bf.e.643.12		128
180.167	odd	12		540.2.y.a.307.29		128

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
180.2.x.a.7.21	✓	128	5.2	odd	4	inner
180.2.x.a.7.26	yes	128	20.7	even	4	inner
180.2.x.a.43.4	yes	128	1.1	even	1	trivial
180.2.x.a.43.23	yes	128	4.3	odd	2	inner
180.2.x.a.67.4	yes	128	180.67	even	12	inner
180.2.x.a.67.23	yes	128	45.22	odd	12	inner
180.2.x.a.103.21	yes	128	36.31	odd	6	inner
180.2.x.a.103.26	yes	128	9.4	even	3	inner
540.2.y.a.127.7		128	60.47	odd	4
540.2.y.a.127.12		128	15.2	even	4
540.2.y.a.307.10		128	45.32	even	12
540.2.y.a.307.29		128	180.167	odd	12
540.2.y.a.343.10		128	12.11	even	2
540.2.y.a.343.29		128	3.2	odd	2
540.2.y.a.523.7		128	9.5	odd	6
540.2.y.a.523.12		128	36.23	even	6
900.2.bf.e.7.7		128	20.3	even	4
900.2.bf.e.7.12		128	5.3	odd	4
900.2.bf.e.43.10		128	20.19	odd	2
900.2.bf.e.43.29		128	5.4	even	2
900.2.bf.e.607.10		128	45.13	odd	12
900.2.bf.e.607.29		128	180.103	even	12
900.2.bf.e.643.7		128	45.4	even	6
900.2.bf.e.643.12		128	180.139	odd	6