Embedded newform 180.2.x.a.43.15

• Label: 180.2.x.a.43.15
• 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			43.15
Character	\(\chi\)	\(=\)	180.43
Dual form			180.2.x.a.67.15

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.0967939 + 1.41090i) q^{2} +(-1.60183 - 0.658900i) q^{3} +(-1.98126 - 0.273133i) q^{4} +(1.07991 - 1.95801i) q^{5} +(1.08469 - 2.19624i) q^{6} +(2.10544 - 0.564152i) q^{7} +(0.577136 - 2.76892i) q^{8} +(2.13170 + 2.11089i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 128 q - 2 q^{2} - 4 q^{5} - 8 q^{6} - 8 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.0967939	+	1.41090i	−0.0684436	+	0.997655i
							\(3\)	−1.60183	−	0.658900i	−0.924815	−	0.380416i
\(5\)	1.07991	−	1.95801i	0.482948	−	0.875649i
							\(7\)	2.10544	−	0.564152i	0.795782	−	0.213229i	0.162051	−	0.986782i	\(-0.448189\pi\)
0.633731	+	0.773553i	\(0.281522\pi\)
\(11\)	0.321601	+	0.185677i	0.0969664	+	0.0559836i	0.547699	−	0.836675i	\(-0.315504\pi\)
							−0.450733	+	0.892659i	\(0.648837\pi\)
\(13\)	1.32948	−	4.96169i	0.368732	−	1.37613i	−0.493559	−	0.869712i	\(-0.664304\pi\)
							0.862291	−	0.506414i	\(-0.169029\pi\)
\(17\)	−1.43808	+	1.43808i	−0.348786	+	0.348786i	−0.859657	−	0.510871i	\(-0.829323\pi\)
							0.510871	+	0.859657i	\(0.329323\pi\)
\(19\)	7.37002			1.69080			0.845400	−	0.534134i	\(-0.179362\pi\)
							0.845400	+	0.534134i	\(0.179362\pi\)
\(23\)	−6.69069	−	1.79276i	−1.39510	−	0.373817i	−0.518520	−	0.855065i	\(-0.673517\pi\)
							−0.876584	+	0.481248i	\(0.840184\pi\)
\(29\)	−0.694157	−	0.400772i	−0.128902	−	0.0744215i	0.434163	−	0.900835i	\(-0.357044\pi\)
							−0.563064	+	0.826413i	\(0.690378\pi\)
\(31\)	−0.847809	+	0.489483i	−0.152271	+	0.0879137i	−0.574200	−	0.818715i	\(-0.694687\pi\)
							0.421929	+	0.906629i	\(0.361353\pi\)
\(37\)	2.56806	−	2.56806i	0.422186	−	0.422186i	−0.463770	−	0.885956i	\(-0.653504\pi\)
							0.885956	+	0.463770i	\(0.153504\pi\)
\(41\)	3.35859	+	5.81724i	0.524523	+	0.908500i	0.999592	+	0.0285521i	\(0.00908965\pi\)
							−0.475069	+	0.879948i	\(0.657577\pi\)
\(43\)	1.21206	+	4.52346i	0.184837	+	0.689822i	0.994665	+	0.103155i	\(0.0328937\pi\)
							−0.809828	+	0.586667i	\(0.800440\pi\)
\(47\)	1.24898	−	0.334664i	0.182183	−	0.0488157i	−0.166574	−	0.986029i	\(-0.553271\pi\)
							0.348757	+	0.937213i	\(0.386604\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	180.2.x.a.43.15	yes	128
3.2	odd	2		540.2.y.a.343.18		128
4.3	odd	2	inner	180.2.x.a.43.22	yes	128
5.2	odd	4	inner	180.2.x.a.7.1	✓	128
5.3	odd	4		900.2.bf.e.7.32		128
5.4	even	2		900.2.bf.e.43.18		128
9.4	even	3	inner	180.2.x.a.103.6	yes	128
9.5	odd	6		540.2.y.a.523.27		128
12.11	even	2		540.2.y.a.343.11		128
15.2	even	4		540.2.y.a.127.32		128
20.3	even	4		900.2.bf.e.7.27		128
20.7	even	4	inner	180.2.x.a.7.6	yes	128
20.19	odd	2		900.2.bf.e.43.11		128
36.23	even	6		540.2.y.a.523.32		128
36.31	odd	6	inner	180.2.x.a.103.1	yes	128
45.4	even	6		900.2.bf.e.643.27		128
45.13	odd	12		900.2.bf.e.607.11		128
45.22	odd	12	inner	180.2.x.a.67.22	yes	128
45.32	even	12		540.2.y.a.307.11		128
60.47	odd	4		540.2.y.a.127.27		128
180.67	even	12	inner	180.2.x.a.67.15	yes	128
180.103	even	12		900.2.bf.e.607.18		128
180.139	odd	6		900.2.bf.e.643.32		128
180.167	odd	12		540.2.y.a.307.18		128

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
180.2.x.a.7.1	✓	128	5.2	odd	4	inner
180.2.x.a.7.6	yes	128	20.7	even	4	inner
180.2.x.a.43.15	yes	128	1.1	even	1	trivial
180.2.x.a.43.22	yes	128	4.3	odd	2	inner
180.2.x.a.67.15	yes	128	180.67	even	12	inner
180.2.x.a.67.22	yes	128	45.22	odd	12	inner
180.2.x.a.103.1	yes	128	36.31	odd	6	inner
180.2.x.a.103.6	yes	128	9.4	even	3	inner
540.2.y.a.127.27		128	60.47	odd	4
540.2.y.a.127.32		128	15.2	even	4
540.2.y.a.307.11		128	45.32	even	12
540.2.y.a.307.18		128	180.167	odd	12
540.2.y.a.343.11		128	12.11	even	2
540.2.y.a.343.18		128	3.2	odd	2
540.2.y.a.523.27		128	9.5	odd	6
540.2.y.a.523.32		128	36.23	even	6
900.2.bf.e.7.27		128	20.3	even	4
900.2.bf.e.7.32		128	5.3	odd	4
900.2.bf.e.43.11		128	20.19	odd	2
900.2.bf.e.43.18		128	5.4	even	2
900.2.bf.e.607.11		128	45.13	odd	12
900.2.bf.e.607.18		128	180.103	even	12
900.2.bf.e.643.27		128	45.4	even	6
900.2.bf.e.643.32		128	180.139	odd	6