Embedded newform 360.2.bd.b.59.50

• Label: 360.2.bd.b.59.50
• Level: $360$
• Weight: $2$
• Character: 360.59
• Analytic conductor: $2.875$
• Analytic rank: $0$
• Dimension: $128$
• CM: no
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 360 = 2^{3} \cdot 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	360.bd (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.87461447277\)
Analytic rank:	\(0\)
Dimension:	\(128\)
Relative dimension:	\(64\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			59.50
Character	\(\chi\)	\(=\)	360.59
Dual form			360.2.bd.b.299.50

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.958742 - 1.03962i) q^{2} +(-0.0381767 + 1.73163i) q^{3} +(-0.161627 - 1.99346i) q^{4} +(-0.289425 - 2.21726i) q^{5} +(1.76364 + 1.69988i) q^{6} +(-2.06690 - 3.57998i) q^{7} +(-2.22740 - 1.74318i) q^{8} +(-2.99709 - 0.132216i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 128 q - 10 q^{4} + 4 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/360\mathbb{Z}\right)^\times\).

\(n\)	\(181\)	\(217\)	\(271\)	\(281\)
\(\chi(n)\)	\(-1\)	\(-1\)	\(-1\)	\(e\left(\frac{5}{6}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.958742	−	1.03962i	0.677933	−	0.735124i
							\(3\)	−0.0381767	+	1.73163i	−0.0220413	+	0.999757i
\(5\)	−0.289425	−	2.21726i	−0.129435	−	0.991588i
							\(7\)	−2.06690	−	3.57998i	−0.781216	−	1.35310i	−0.931234	−	0.364422i	\(-0.881267\pi\)
0.150018	−	0.988683i	\(-0.452067\pi\)
\(11\)	1.04488	−	0.603262i	0.315043	−	0.181890i	−0.334138	−	0.942524i	\(-0.608445\pi\)
							0.649181	+	0.760634i	\(0.275112\pi\)
\(13\)	−0.444938	+	0.770655i	−0.123404	+	0.213741i	−0.921108	−	0.389308i	\(-0.872714\pi\)
							0.797704	+	0.603049i	\(0.206048\pi\)
\(17\)	−0.456672			−0.110759			−0.0553796	−	0.998465i	\(-0.517637\pi\)
							−0.0553796	+	0.998465i	\(0.517637\pi\)
\(19\)	7.68779			1.76370			0.881850	−	0.471531i	\(-0.156298\pi\)
							0.881850	+	0.471531i	\(0.156298\pi\)
\(23\)	4.65266	+	2.68621i	0.970146	+	0.560114i	0.899281	−	0.437372i	\(-0.144091\pi\)
							0.0708653	+	0.997486i	\(0.477424\pi\)
\(29\)	−3.54452	−	6.13928i	−0.658200	−	1.14004i	−0.981081	−	0.193597i	\(-0.937985\pi\)
							0.322881	−	0.946440i	\(-0.395349\pi\)
\(31\)	7.17070	+	4.14000i	1.28790	+	0.743567i	0.978279	−	0.207295i	\(-0.0664659\pi\)
							0.309617	+	0.950861i	\(0.399799\pi\)
\(37\)	4.24133			0.697270			0.348635	−	0.937259i	\(-0.386645\pi\)
							0.348635	+	0.937259i	\(0.386645\pi\)
\(41\)	−5.17921	−	2.99022i	−0.808856	−	0.466994i	0.0377021	−	0.999289i	\(-0.487996\pi\)
							−0.846559	+	0.532296i	\(0.821330\pi\)
\(43\)	3.70454	−	2.13881i	0.564936	−	0.326166i	−0.190188	−	0.981748i	\(-0.560910\pi\)
							0.755124	+	0.655581i	\(0.227576\pi\)
\(47\)	−3.27309	+	1.88972i	−0.477429	+	0.275644i	−0.719344	−	0.694654i	\(-0.755558\pi\)
							0.241916	+	0.970297i	\(0.422224\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	360.2.bd.b.59.50	yes	128
5.4	even	2	inner	360.2.bd.b.59.15	yes	128
8.3	odd	2	inner	360.2.bd.b.59.6	✓	128
9.2	odd	6	inner	360.2.bd.b.299.59	yes	128
40.19	odd	2	inner	360.2.bd.b.59.59	yes	128
45.29	odd	6	inner	360.2.bd.b.299.6	yes	128
72.11	even	6	inner	360.2.bd.b.299.15	yes	128
360.299	even	6	inner	360.2.bd.b.299.50	yes	128

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
360.2.bd.b.59.6	✓	128	8.3	odd	2	inner
360.2.bd.b.59.15	yes	128	5.4	even	2	inner
360.2.bd.b.59.50	yes	128	1.1	even	1	trivial
360.2.bd.b.59.59	yes	128	40.19	odd	2	inner
360.2.bd.b.299.6	yes	128	45.29	odd	6	inner
360.2.bd.b.299.15	yes	128	72.11	even	6	inner
360.2.bd.b.299.50	yes	128	360.299	even	6	inner
360.2.bd.b.299.59	yes	128	9.2	odd	6	inner