Embedded newform 180.2.x.a.7.7

• Label: 180.2.x.a.7.7
• Level: $180$
• Weight: $2$
• Character: 180.7
• 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			7.7
Character	\(\chi\)	\(=\)	180.7
Dual form			180.2.x.a.103.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.16528 - 0.801326i) q^{2} +(-1.70815 - 0.286731i) q^{3} +(0.715754 + 1.86754i) q^{4} +(1.09795 - 1.94795i) q^{5} +(1.76071 + 1.70291i) q^{6} +(-0.0500694 - 0.186862i) q^{7} +(0.662452 - 2.74976i) q^{8} +(2.83557 + 0.979559i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 128 q - 2 q^{2} - 4 q^{5} - 8 q^{6} - 8 q^{8}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.16528	−	0.801326i	−0.823977	−	0.566623i
							\(3\)	−1.70815	−	0.286731i	−0.986202	−	0.165544i
\(5\)	1.09795	−	1.94795i	0.491019	−	0.871149i
							\(7\)	−0.0500694	−	0.186862i	−0.0189245	−	0.0706271i	0.955818	−	0.293960i	\(-0.0949731\pi\)
−0.974742	+	0.223332i	\(0.928306\pi\)
\(11\)	−4.51273	−	2.60542i	−1.36064	−	0.785565i	−0.370929	−	0.928661i	\(-0.620961\pi\)
							−0.989709	+	0.143097i	\(0.954294\pi\)
\(13\)	−6.23458	−	1.67055i	−1.72916	−	0.463328i	−0.749172	−	0.662375i	\(-0.769549\pi\)
							−0.979990	+	0.199047i	\(0.936215\pi\)
\(17\)	−0.305124	−	0.305124i	−0.0740033	−	0.0740033i	0.669136	−	0.743140i	\(-0.266664\pi\)
							−0.743140	+	0.669136i	\(0.766664\pi\)
\(19\)	2.39244			0.548863			0.274431	−	0.961607i	\(-0.411510\pi\)
							0.274431	+	0.961607i	\(0.411510\pi\)
\(23\)	1.13783	−	4.24643i	0.237253	−	0.885441i	−0.739867	−	0.672753i	\(-0.765112\pi\)
							0.977120	−	0.212688i	\(-0.0682218\pi\)
\(29\)	−2.28164	−	1.31730i	−0.423689	−	0.244617i	0.272965	−	0.962024i	\(-0.411996\pi\)
							−0.696654	+	0.717407i	\(0.745329\pi\)
\(31\)	0.124207	−	0.0717112i	0.0223083	−	0.0128797i	−0.488804	−	0.872393i	\(-0.662567\pi\)
							0.511113	+	0.859514i	\(0.329233\pi\)
\(37\)	2.83375	+	2.83375i	0.465866	+	0.465866i	0.900572	−	0.434706i	\(-0.143148\pi\)
							−0.434706	+	0.900572i	\(0.643148\pi\)
\(41\)	−1.76356	−	3.05458i	−0.275422	−	0.477045i	0.694819	−	0.719184i	\(-0.255484\pi\)
							−0.970242	+	0.242139i	\(0.922151\pi\)
\(43\)	7.96962	−	2.13545i	1.21536	−	0.325654i	0.406495	−	0.913653i	\(-0.366751\pi\)
							0.808861	+	0.588000i	\(0.200084\pi\)
\(47\)	2.07353	+	7.73851i	0.302455	+	1.12878i	0.935114	+	0.354347i	\(0.115297\pi\)
							−0.632659	+	0.774431i	\(0.718036\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	180.2.x.a.7.7	✓	128
3.2	odd	2		540.2.y.a.127.26		128
4.3	odd	2	inner	180.2.x.a.7.13	yes	128
5.2	odd	4		900.2.bf.e.43.23		128
5.3	odd	4	inner	180.2.x.a.43.10	yes	128
5.4	even	2		900.2.bf.e.7.26		128
9.4	even	3	inner	180.2.x.a.67.28	yes	128
9.5	odd	6		540.2.y.a.307.5		128
12.11	even	2		540.2.y.a.127.20		128
15.8	even	4		540.2.y.a.343.23		128
20.3	even	4	inner	180.2.x.a.43.28	yes	128
20.7	even	4		900.2.bf.e.43.5		128
20.19	odd	2		900.2.bf.e.7.20		128
36.23	even	6		540.2.y.a.307.23		128
36.31	odd	6	inner	180.2.x.a.67.10	yes	128
45.4	even	6		900.2.bf.e.607.5		128
45.13	odd	12	inner	180.2.x.a.103.13	yes	128
45.22	odd	12		900.2.bf.e.643.20		128
45.23	even	12		540.2.y.a.523.20		128
60.23	odd	4		540.2.y.a.343.5		128
180.23	odd	12		540.2.y.a.523.26		128
180.67	even	12		900.2.bf.e.643.26		128
180.103	even	12	inner	180.2.x.a.103.7	yes	128
180.139	odd	6		900.2.bf.e.607.23		128

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
180.2.x.a.7.7	✓	128	1.1	even	1	trivial
180.2.x.a.7.13	yes	128	4.3	odd	2	inner
180.2.x.a.43.10	yes	128	5.3	odd	4	inner
180.2.x.a.43.28	yes	128	20.3	even	4	inner
180.2.x.a.67.10	yes	128	36.31	odd	6	inner
180.2.x.a.67.28	yes	128	9.4	even	3	inner
180.2.x.a.103.7	yes	128	180.103	even	12	inner
180.2.x.a.103.13	yes	128	45.13	odd	12	inner
540.2.y.a.127.20		128	12.11	even	2
540.2.y.a.127.26		128	3.2	odd	2
540.2.y.a.307.5		128	9.5	odd	6
540.2.y.a.307.23		128	36.23	even	6
540.2.y.a.343.5		128	60.23	odd	4
540.2.y.a.343.23		128	15.8	even	4
540.2.y.a.523.20		128	45.23	even	12
540.2.y.a.523.26		128	180.23	odd	12
900.2.bf.e.7.20		128	20.19	odd	2
900.2.bf.e.7.26		128	5.4	even	2
900.2.bf.e.43.5		128	20.7	even	4
900.2.bf.e.43.23		128	5.2	odd	4
900.2.bf.e.607.5		128	45.4	even	6
900.2.bf.e.607.23		128	180.139	odd	6
900.2.bf.e.643.20		128	45.22	odd	12
900.2.bf.e.643.26		128	180.67	even	12