Embedded newform 483.2.m.a.137.2

• Label: 483.2.m.a.137.2
• 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			137.2
Character	\(\chi\)	\(=\)	483.137
Dual form			483.2.m.a.275.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.35341 + 1.35874i) q^{2} +(1.22585 + 1.22364i) q^{3} +(2.69235 - 4.66329i) q^{4} +(1.88300 + 3.26145i) q^{5} +(-4.54753 - 1.21410i) q^{6} +(0.393331 + 2.61635i) q^{7} +9.19786i q^{8} +(0.00542953 + 3.00000i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 120 q - 2 q^{3} + 52 q^{4} - 24 q^{6} + 2 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.35341	+	1.35874i	−1.66411	+	0.960774i	−0.693390	+	0.720563i	\(0.743884\pi\)
							−0.970721	+	0.240212i	\(0.922783\pi\)
\(3\)	1.22585	+	1.22364i	0.707746	+	0.706467i
							\(5\)	1.88300	+	3.26145i	0.842102	+	1.45856i	0.888115	+	0.459622i	\(0.152015\pi\)
−0.0460131	+	0.998941i	\(0.514652\pi\)
\(7\)	0.393331	+	2.61635i	0.148665	+	0.988888i
							\(11\)	0.717113	−	1.24208i	0.216218	−	0.374500i	−0.737431	−	0.675423i	\(-0.763961\pi\)
0.953649	+	0.300923i	\(0.0972946\pi\)
\(13\)	−0.667422			−0.185110			−0.0925548	−	0.995708i	\(-0.529503\pi\)
							−0.0925548	+	0.995708i	\(0.529503\pi\)
\(17\)	1.07276	−	1.85808i	0.260183	−	0.450651i	−0.706107	−	0.708105i	\(-0.749550\pi\)
							0.966290	+	0.257454i	\(0.0828837\pi\)
\(19\)	5.01385	−	2.89475i	1.15026	−	0.664100i	0.201306	−	0.979528i	\(-0.435482\pi\)
							0.948949	+	0.315428i	\(0.102148\pi\)
\(23\)	−0.365917	+	4.78185i	−0.0762991	+	0.997085i
							\(29\)		−	2.79298i		−	0.518643i	−0.965791	−	0.259321i	\(-0.916501\pi\)
0.965791	−	0.259321i	\(-0.0834989\pi\)
\(31\)	2.13907	−	3.70498i	0.384189	−	0.665435i	−0.607467	−	0.794345i	\(-0.707814\pi\)
							0.991656	+	0.128910i	\(0.0411478\pi\)
\(37\)	6.99265	−	4.03721i	1.14959	−	0.663713i	0.200799	−	0.979632i	\(-0.435646\pi\)
							0.948786	+	0.315919i	\(0.102313\pi\)
\(41\)		−	7.47298i		−	1.16708i	−0.812083	−	0.583542i	\(-0.801667\pi\)
							0.812083	−	0.583542i	\(-0.198333\pi\)
\(43\)		−	7.30328i		−	1.11374i	−0.830600	−	0.556870i	\(-0.812002\pi\)
							0.830600	−	0.556870i	\(-0.187998\pi\)
\(47\)	−3.96856	+	2.29125i	−0.578873	+	0.334213i	−0.760685	−	0.649121i	\(-0.775137\pi\)
							0.181812	+	0.983333i	\(0.441804\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	483.2.m.a.137.2	yes	120
3.2	odd	2	inner	483.2.m.a.137.59	yes	120
7.2	even	3	inner	483.2.m.a.275.60	yes	120
21.2	odd	6	inner	483.2.m.a.275.1	yes	120
23.22	odd	2	inner	483.2.m.a.137.1	✓	120
69.68	even	2	inner	483.2.m.a.137.60	yes	120
161.114	odd	6	inner	483.2.m.a.275.59	yes	120
483.275	even	6	inner	483.2.m.a.275.2	yes	120

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
483.2.m.a.137.1	✓	120	23.22	odd	2	inner
483.2.m.a.137.2	yes	120	1.1	even	1	trivial
483.2.m.a.137.59	yes	120	3.2	odd	2	inner
483.2.m.a.137.60	yes	120	69.68	even	2	inner
483.2.m.a.275.1	yes	120	21.2	odd	6	inner
483.2.m.a.275.2	yes	120	483.275	even	6	inner
483.2.m.a.275.59	yes	120	161.114	odd	6	inner
483.2.m.a.275.60	yes	120	7.2	even	3	inner