Embedded newform 180.2.x.a.67.16

• Label: 180.2.x.a.67.16
• Level: $180$
• Weight: $2$
• Character: 180.67
• Analytic conductor: $1.437$
• Analytic rank: $0$
• Dimension: $128$
• 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			67.16
Character	\(\chi\)	\(=\)	180.67
Dual form			180.2.x.a.43.16

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.0886778 + 1.41143i) q^{2} +(1.72796 - 0.119021i) q^{3} +(-1.98427 - 0.250325i) q^{4} +(1.70814 - 1.44300i) q^{5} +(0.0147586 + 2.44945i) q^{6} +(1.23543 + 0.331032i) q^{7} +(0.529277 - 2.77846i) q^{8} +(2.97167 - 0.411327i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 128 q - 2 q^{2} - 4 q^{5} - 8 q^{6} - 8 q^{8}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.0886778	+	1.41143i	−0.0627047	+	0.998032i
							\(3\)	1.72796	−	0.119021i	0.997636	−	0.0687169i
\(5\)	1.70814	−	1.44300i	0.763904	−	0.645330i
							\(7\)	1.23543	+	0.331032i	0.466948	+	0.125118i	0.484619	−	0.874726i	\(-0.338958\pi\)
−0.0176708	+	0.999844i	\(0.505625\pi\)
\(11\)	−4.09029	+	2.36153i	−1.23327	+	0.712028i	−0.967710	−	0.252067i	\(-0.918890\pi\)
							−0.265558	+	0.964095i	\(0.585556\pi\)
\(13\)	0.448661	+	1.67443i	0.124436	+	0.464402i	0.999819	−	0.0190283i	\(-0.00605726\pi\)
							−0.875383	+	0.483430i	\(0.839391\pi\)
\(17\)	−2.37189	−	2.37189i	−0.575267	−	0.575267i	0.358329	−	0.933595i	\(-0.383347\pi\)
							−0.933595	+	0.358329i	\(0.883347\pi\)
\(19\)	−2.26465			−0.519546			−0.259773	−	0.965670i	\(-0.583648\pi\)
							−0.259773	+	0.965670i	\(0.583648\pi\)
\(23\)	−8.43461	+	2.26005i	−1.75874	+	0.471252i	−0.986456	−	0.164028i	\(-0.947551\pi\)
							−0.772282	+	0.635280i	\(0.780885\pi\)
\(29\)	−0.111045	+	0.0641116i	−0.0206205	+	0.0119052i	−0.510275	−	0.860011i	\(-0.670456\pi\)
							0.489654	+	0.871917i	\(0.337123\pi\)
\(31\)	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	0.670596	−	0.741823i	\(-0.266038\pi\)
							−0.741823	+	0.670596i	\(0.766038\pi\)
\(41\)	3.06037	−	5.30071i	0.477949	−	0.827833i	−0.521731	−	0.853110i	\(-0.674714\pi\)
							0.999680	+	0.0252774i	\(0.00804691\pi\)
\(43\)	1.09889	−	4.10111i	0.167579	−	0.625414i	−0.830118	−	0.557588i	\(-0.811727\pi\)
							0.997697	−	0.0678260i	\(-0.0216063\pi\)
\(47\)	−4.63950	−	1.24315i	−0.676740	−	0.181332i	−0.0959510	−	0.995386i	\(-0.530589\pi\)
							−0.580789	+	0.814054i	\(0.697256\pi\)

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
180.2.x.a.7.29	✓	128	9.7	even	3	inner
180.2.x.a.7.31	yes	128	36.7	odd	6	inner
180.2.x.a.43.12	yes	128	45.43	odd	12	inner
180.2.x.a.43.16	yes	128	180.43	even	12	inner
180.2.x.a.67.12	yes	128	4.3	odd	2	inner
180.2.x.a.67.16	yes	128	1.1	even	1	trivial
180.2.x.a.103.29	yes	128	20.3	even	4	inner
180.2.x.a.103.31	yes	128	5.3	odd	4	inner
540.2.y.a.127.2		128	36.11	even	6
540.2.y.a.127.4		128	9.2	odd	6
540.2.y.a.307.17		128	3.2	odd	2
540.2.y.a.307.21		128	12.11	even	2
540.2.y.a.343.17		128	180.83	odd	12
540.2.y.a.343.21		128	45.38	even	12
540.2.y.a.523.2		128	15.8	even	4
540.2.y.a.523.4		128	60.23	odd	4
900.2.bf.e.7.2		128	180.79	odd	6
900.2.bf.e.7.4		128	45.34	even	6
900.2.bf.e.43.17		128	180.7	even	12
900.2.bf.e.43.21		128	45.7	odd	12
900.2.bf.e.607.17		128	5.4	even	2
900.2.bf.e.607.21		128	20.19	odd	2
900.2.bf.e.643.2		128	5.2	odd	4
900.2.bf.e.643.4		128	20.7	even	4