Embedded newform 483.2.m.a.137.18

• Label: 483.2.m.a.137.18
• Level: $483$
• Weight: $2$
• Character: 483.137
• Analytic conductor: $3.857$
• Analytic rank: $0$
• Dimension: $120$
• CM: no
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 483 = 3 \cdot 7 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	483.m (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.85677441763\)
Analytic rank:	\(0\)
Dimension:	\(120\)
Relative dimension:	\(60\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			137.18
Character	\(\chi\)	\(=\)	483.137
Dual form			483.2.m.a.275.18

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.19221 + 0.688323i) q^{2} +(1.08950 - 1.34647i) q^{3} +(-0.0524225 + 0.0907985i) q^{4} +(2.03563 + 3.52582i) q^{5} +(-0.372109 + 2.35521i) q^{6} +(1.00571 - 2.44715i) q^{7} -2.89763i q^{8} +(-0.625968 - 2.93397i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 120 q - 2 q^{3} + 52 q^{4} - 24 q^{6} + 2 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/483\mathbb{Z}\right)^\times\).

\(n\)	\(323\)	\(346\)	\(442\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{2}{3}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.19221	+	0.688323i	−0.843020	+	0.486718i	−0.858290	−	0.513165i	\(-0.828473\pi\)
							0.0152695	+	0.999883i	\(0.495139\pi\)
\(3\)	1.08950	−	1.34647i	0.629025	−	0.777385i
							\(5\)	2.03563	+	3.52582i	0.910363	+	1.57680i	0.813552	+	0.581492i	\(0.197531\pi\)
0.0968111	+	0.995303i	\(0.469136\pi\)
\(7\)	1.00571	−	2.44715i	0.380124	−	0.924935i
							\(11\)	1.91796	−	3.32201i	0.578287	−	1.00162i	−0.417389	−	0.908728i	\(-0.637055\pi\)
0.995676	−	0.0928949i	\(-0.0296121\pi\)
\(13\)	1.10613			0.306785			0.153392	−	0.988165i	\(-0.450980\pi\)
							0.153392	+	0.988165i	\(0.450980\pi\)
\(17\)	0.724889	−	1.25554i	0.175811	−	0.304514i	−0.764630	−	0.644469i	\(-0.777078\pi\)
							0.940442	+	0.339955i	\(0.110412\pi\)
\(19\)	2.47623	−	1.42965i	0.568086	−	0.327985i	−0.188298	−	0.982112i	\(-0.560297\pi\)
							0.756385	+	0.654127i	\(0.226964\pi\)
\(23\)	−3.35321	+	3.42870i	−0.699193	+	0.714933i
							\(29\)			6.79586i			1.26196i	0.775800	+	0.630980i	\(0.217347\pi\)
−0.775800	+	0.630980i	\(0.782653\pi\)
\(31\)	4.00533	−	6.93744i	0.719379	−	1.24600i	−0.241867	−	0.970310i	\(-0.577760\pi\)
							0.961246	−	0.275692i	\(-0.0889070\pi\)
\(37\)	7.48837	−	4.32342i	1.23108	−	0.710765i	0.263826	−	0.964570i	\(-0.415016\pi\)
							0.967255	+	0.253805i	\(0.0816822\pi\)
\(41\)			5.73676i			0.895932i	0.894051	+	0.447966i	\(0.147851\pi\)
							−0.894051	+	0.447966i	\(0.852149\pi\)
\(43\)			5.85587i			0.893011i	0.894781	+	0.446506i	\(0.147332\pi\)
							−0.894781	+	0.446506i	\(0.852668\pi\)
\(47\)	−1.55372	+	0.897041i	−0.226634	+	0.130847i	−0.609018	−	0.793156i	\(-0.708436\pi\)
							0.382385	+	0.924003i	\(0.375103\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	483.2.m.a.137.18	yes	120
3.2	odd	2	inner	483.2.m.a.137.43	yes	120
7.2	even	3	inner	483.2.m.a.275.44	yes	120
21.2	odd	6	inner	483.2.m.a.275.17	yes	120
23.22	odd	2	inner	483.2.m.a.137.17	✓	120
69.68	even	2	inner	483.2.m.a.137.44	yes	120
161.114	odd	6	inner	483.2.m.a.275.43	yes	120
483.275	even	6	inner	483.2.m.a.275.18	yes	120

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
483.2.m.a.137.17	✓	120	23.22	odd	2	inner
483.2.m.a.137.18	yes	120	1.1	even	1	trivial
483.2.m.a.137.43	yes	120	3.2	odd	2	inner
483.2.m.a.137.44	yes	120	69.68	even	2	inner
483.2.m.a.275.17	yes	120	21.2	odd	6	inner
483.2.m.a.275.18	yes	120	483.275	even	6	inner
483.2.m.a.275.43	yes	120	161.114	odd	6	inner
483.2.m.a.275.44	yes	120	7.2	even	3	inner