Embedded newform 180.2.x.a.103.31

• Label: 180.2.x.a.103.31
• Level: $180$
• Weight: $2$
• Character: 180.103
• 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			103.31
Character	\(\chi\)	\(=\)	180.103
Dual form			180.2.x.a.7.31

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.41143 + 0.0886778i) q^{2} +(0.119021 + 1.72796i) q^{3} +(1.98427 + 0.250325i) q^{4} +(0.395606 - 2.20079i) q^{5} +(0.0147586 + 2.44945i) q^{6} +(0.331032 - 1.23543i) q^{7} +(2.77846 + 0.529277i) q^{8} +(-2.97167 + 0.411327i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 128 q - 2 q^{2} - 4 q^{5} - 8 q^{6} - 8 q^{8}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.41143	+	0.0886778i	0.998032	+	0.0627047i
							\(3\)	0.119021	+	1.72796i	0.0687169	+	0.997636i
\(5\)	0.395606	−	2.20079i	0.176920	−	0.984225i
							\(7\)	0.331032	−	1.23543i	0.125118	−	0.466948i	−0.874726	−	0.484619i	\(-0.838958\pi\)
0.999844	+	0.0176708i	\(0.00562508\pi\)
\(11\)	−4.09029	+	2.36153i	−1.23327	+	0.712028i	−0.967710	−	0.252067i	\(-0.918890\pi\)
							−0.265558	+	0.964095i	\(0.585556\pi\)
\(13\)	−1.67443	+	0.448661i	−0.464402	+	0.124436i	−0.483430	−	0.875383i	\(-0.660609\pi\)
							0.0190283	+	0.999819i	\(0.493943\pi\)
\(17\)	−2.37189	+	2.37189i	−0.575267	+	0.575267i	−0.933595	−	0.358329i	\(-0.883347\pi\)
							0.358329	+	0.933595i	\(0.383347\pi\)
\(19\)	2.26465			0.519546			0.259773	−	0.965670i	\(-0.416352\pi\)
							0.259773	+	0.965670i	\(0.416352\pi\)
\(23\)	−2.26005	−	8.43461i	−0.471252	−	1.75874i	−0.635280	−	0.772282i	\(-0.719115\pi\)
							0.164028	−	0.986456i	\(-0.447551\pi\)
\(29\)	0.111045	−	0.0641116i	0.0206205	−	0.0119052i	−0.489654	−	0.871917i	\(-0.662877\pi\)
							0.510275	+	0.860011i	\(0.329544\pi\)
\(31\)	5.75638	+	3.32345i	1.03388	+	0.596909i	0.918093	−	0.396365i	\(-0.129729\pi\)
							0.115784	+	0.993274i	\(0.463062\pi\)
\(37\)	−0.433255	+	0.433255i	−0.0712267	+	0.0712267i	−0.741823	−	0.670596i	\(-0.766038\pi\)
							0.670596	+	0.741823i	\(0.266038\pi\)
\(41\)	3.06037	−	5.30071i	0.477949	−	0.827833i	−0.521731	−	0.853110i	\(-0.674714\pi\)
							0.999680	+	0.0252774i	\(0.00804691\pi\)
\(43\)	4.10111	+	1.09889i	0.625414	+	0.167579i	0.557588	−	0.830118i	\(-0.311727\pi\)
							0.0678260	+	0.997697i	\(0.478394\pi\)
\(47\)	−1.24315	+	4.63950i	−0.181332	+	0.676740i	0.814054	+	0.580789i	\(0.197256\pi\)
							−0.995386	+	0.0959510i	\(0.969411\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	180.2.x.a.103.31	yes	128
3.2	odd	2		540.2.y.a.523.2		128
4.3	odd	2	inner	180.2.x.a.103.29	yes	128
5.2	odd	4	inner	180.2.x.a.67.16	yes	128
5.3	odd	4		900.2.bf.e.607.17		128
5.4	even	2		900.2.bf.e.643.2		128
9.2	odd	6		540.2.y.a.343.21		128
9.7	even	3	inner	180.2.x.a.43.12	yes	128
12.11	even	2		540.2.y.a.523.4		128
15.2	even	4		540.2.y.a.307.17		128
20.3	even	4		900.2.bf.e.607.21		128
20.7	even	4	inner	180.2.x.a.67.12	yes	128
20.19	odd	2		900.2.bf.e.643.4		128
36.7	odd	6	inner	180.2.x.a.43.16	yes	128
36.11	even	6		540.2.y.a.343.17		128
45.2	even	12		540.2.y.a.127.4		128
45.7	odd	12	inner	180.2.x.a.7.29	✓	128
45.34	even	6		900.2.bf.e.43.21		128
45.43	odd	12		900.2.bf.e.7.4		128
60.47	odd	4		540.2.y.a.307.21		128
180.7	even	12	inner	180.2.x.a.7.31	yes	128
180.43	even	12		900.2.bf.e.7.2		128
180.47	odd	12		540.2.y.a.127.2		128
180.79	odd	6		900.2.bf.e.43.17		128

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
180.2.x.a.7.29	✓	128	45.7	odd	12	inner
180.2.x.a.7.31	yes	128	180.7	even	12	inner
180.2.x.a.43.12	yes	128	9.7	even	3	inner
180.2.x.a.43.16	yes	128	36.7	odd	6	inner
180.2.x.a.67.12	yes	128	20.7	even	4	inner
180.2.x.a.67.16	yes	128	5.2	odd	4	inner
180.2.x.a.103.29	yes	128	4.3	odd	2	inner
180.2.x.a.103.31	yes	128	1.1	even	1	trivial
540.2.y.a.127.2		128	180.47	odd	12
540.2.y.a.127.4		128	45.2	even	12
540.2.y.a.307.17		128	15.2	even	4
540.2.y.a.307.21		128	60.47	odd	4
540.2.y.a.343.17		128	36.11	even	6
540.2.y.a.343.21		128	9.2	odd	6
540.2.y.a.523.2		128	3.2	odd	2
540.2.y.a.523.4		128	12.11	even	2
900.2.bf.e.7.2		128	180.43	even	12
900.2.bf.e.7.4		128	45.43	odd	12
900.2.bf.e.43.17		128	180.79	odd	6
900.2.bf.e.43.21		128	45.34	even	6
900.2.bf.e.607.17		128	5.3	odd	4
900.2.bf.e.607.21		128	20.3	even	4
900.2.bf.e.643.2		128	5.4	even	2
900.2.bf.e.643.4		128	20.19	odd	2