Embedded newform 483.2.m.a.137.14

• Label: 483.2.m.a.137.14
• 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			137.14
Character	\(\chi\)	\(=\)	483.137
Dual form			483.2.m.a.275.14

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.44748 + 0.835704i) q^{2} +(1.59432 + 0.676852i) q^{3} +(0.396802 - 0.687281i) q^{4} +(1.25144 + 2.16756i) q^{5} +(-2.87340 + 0.352653i) q^{6} +(-2.40151 - 1.11029i) q^{7} -2.01638i q^{8} +(2.08374 + 2.15824i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 120 q - 2 q^{3} + 52 q^{4} - 24 q^{6} + 2 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.44748	+	0.835704i	−1.02352	+	0.590932i	−0.915123	−	0.403175i	\(-0.867906\pi\)
							−0.108401	+	0.994107i	\(0.534573\pi\)
\(3\)	1.59432	+	0.676852i	0.920484	+	0.390781i
							\(5\)	1.25144	+	2.16756i	0.559661	+	0.969362i	0.997524	+	0.0703202i	\(0.0224021\pi\)
−0.437863	+	0.899042i	\(0.644265\pi\)
\(7\)	−2.40151	−	1.11029i	−0.907685	−	0.419651i
							\(11\)	−1.80972	+	3.13452i	−0.545651	+	0.945095i	0.452915	+	0.891554i	\(0.350384\pi\)
−0.998566	+	0.0535409i	\(0.982949\pi\)
\(13\)	−3.77658			−1.04744			−0.523718	−	0.851892i	\(-0.675455\pi\)
							−0.523718	+	0.851892i	\(0.675455\pi\)
\(17\)	0.0271760	−	0.0470702i	0.00659115	−	0.0114162i	−0.862711	−	0.505697i	\(-0.831235\pi\)
							0.869302	+	0.494281i	\(0.164569\pi\)
\(19\)	−6.51140	+	3.75936i	−1.49382	+	0.862455i	−0.999975	−	0.00709620i	\(-0.997741\pi\)
							−0.493842	+	0.869552i	\(0.664408\pi\)
\(23\)	4.68107	−	1.04286i	0.976071	−	0.217452i
							\(29\)			8.09750i			1.50367i	0.659353	+	0.751834i	\(0.270830\pi\)
−0.659353	+	0.751834i	\(0.729170\pi\)
\(31\)	2.75458	−	4.77107i	0.494736	−	0.856909i	−0.505245	−	0.862976i	\(-0.668598\pi\)
							0.999982	+	0.00606723i	\(0.00193127\pi\)
\(37\)	0.764929	−	0.441632i	0.125754	−	0.0726039i	−0.435804	−	0.900042i	\(-0.643536\pi\)
							0.561557	+	0.827438i	\(0.310202\pi\)
\(41\)			2.57974i			0.402888i	0.979500	+	0.201444i	\(0.0645634\pi\)
							−0.979500	+	0.201444i	\(0.935437\pi\)
\(43\)		−	2.37340i		−	0.361940i	−0.983489	−	0.180970i	\(-0.942076\pi\)
							0.983489	−	0.180970i	\(-0.0579237\pi\)
\(47\)	−3.62708	+	2.09409i	−0.529064	+	0.305455i	−0.740635	−	0.671907i	\(-0.765475\pi\)
							0.211571	+	0.977363i	\(0.432142\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	483.2.m.a.137.14	yes	120
3.2	odd	2	inner	483.2.m.a.137.47	yes	120
7.2	even	3	inner	483.2.m.a.275.48	yes	120
21.2	odd	6	inner	483.2.m.a.275.13	yes	120
23.22	odd	2	inner	483.2.m.a.137.13	✓	120
69.68	even	2	inner	483.2.m.a.137.48	yes	120
161.114	odd	6	inner	483.2.m.a.275.47	yes	120
483.275	even	6	inner	483.2.m.a.275.14	yes	120

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
483.2.m.a.137.13	✓	120	23.22	odd	2	inner
483.2.m.a.137.14	yes	120	1.1	even	1	trivial
483.2.m.a.137.47	yes	120	3.2	odd	2	inner
483.2.m.a.137.48	yes	120	69.68	even	2	inner
483.2.m.a.275.13	yes	120	21.2	odd	6	inner
483.2.m.a.275.14	yes	120	483.275	even	6	inner
483.2.m.a.275.47	yes	120	161.114	odd	6	inner
483.2.m.a.275.48	yes	120	7.2	even	3	inner