Embedded newform 483.2.m.a.137.4

• Label: 483.2.m.a.137.4
• 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			137.4
Character	\(\chi\)	\(=\)	483.137
Dual form			483.2.m.a.275.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.20025 + 1.27031i) q^{2} +(-1.66604 - 0.473602i) q^{3} +(2.22739 - 3.85796i) q^{4} +(0.823724 + 1.42673i) q^{5} +(4.26733 - 1.07436i) q^{6} +(2.24449 - 1.40082i) q^{7} +6.23669i q^{8} +(2.55140 + 1.57808i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 120 q - 2 q^{3} + 52 q^{4} - 24 q^{6} + 2 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.20025	+	1.27031i	−1.55581	+	0.898247i	−0.558159	+	0.829734i	\(0.688492\pi\)
							−0.997650	+	0.0685134i	\(0.978174\pi\)
\(3\)	−1.66604	−	0.473602i	−0.961891	−	0.273434i
							\(5\)	0.823724	+	1.42673i	0.368380	+	0.638054i	0.989313	−	0.145811i	\(-0.0465791\pi\)
−0.620932	+	0.783864i	\(0.713246\pi\)
\(7\)	2.24449	−	1.40082i	0.848336	−	0.529459i
							\(11\)	3.00223	−	5.20002i	0.905207	−	1.56786i	0.0845678	−	0.996418i	\(-0.473049\pi\)
0.820639	−	0.571447i	\(-0.193618\pi\)
\(13\)	−5.26057			−1.45902			−0.729510	−	0.683971i	\(-0.760252\pi\)
							−0.729510	+	0.683971i	\(0.760252\pi\)
\(17\)	−1.95649	+	3.38875i	−0.474520	+	0.821892i	−0.999574	−	0.0291763i	\(-0.990712\pi\)
							0.525055	+	0.851069i	\(0.324045\pi\)
\(19\)	−2.21354	+	1.27799i	−0.507822	+	0.293191i	−0.731938	−	0.681372i	\(-0.761384\pi\)
							0.224116	+	0.974562i	\(0.428050\pi\)
\(23\)	−1.24591	−	4.63117i	−0.259790	−	0.965665i
							\(29\)		−	0.910062i		−	0.168994i	−0.996424	−	0.0844972i	\(-0.973072\pi\)
0.996424	−	0.0844972i	\(-0.0269284\pi\)
\(31\)	0.961637	−	1.66560i	0.172715	−	0.299151i	−0.766653	−	0.642062i	\(-0.778079\pi\)
							0.939368	+	0.342910i	\(0.111413\pi\)
\(37\)	6.44580	−	3.72149i	1.05968	−	0.611809i	0.134339	−	0.990935i	\(-0.457109\pi\)
							0.925345	+	0.379127i	\(0.123776\pi\)
\(41\)		−	2.55938i		−	0.399708i	−0.979826	−	0.199854i	\(-0.935953\pi\)
							0.979826	−	0.199854i	\(-0.0640468\pi\)
\(43\)			2.27409i			0.346795i	0.984852	+	0.173398i	\(0.0554746\pi\)
							−0.984852	+	0.173398i	\(0.944525\pi\)
\(47\)	2.96598	−	1.71241i	0.432633	−	0.249781i	−0.267835	−	0.963465i	\(-0.586308\pi\)
							0.700468	+	0.713684i	\(0.252975\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	483.2.m.a.137.4	yes	120
3.2	odd	2	inner	483.2.m.a.137.57	yes	120
7.2	even	3	inner	483.2.m.a.275.58	yes	120
21.2	odd	6	inner	483.2.m.a.275.3	yes	120
23.22	odd	2	inner	483.2.m.a.137.3	✓	120
69.68	even	2	inner	483.2.m.a.137.58	yes	120
161.114	odd	6	inner	483.2.m.a.275.57	yes	120
483.275	even	6	inner	483.2.m.a.275.4	yes	120

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
483.2.m.a.137.3	✓	120	23.22	odd	2	inner
483.2.m.a.137.4	yes	120	1.1	even	1	trivial
483.2.m.a.137.57	yes	120	3.2	odd	2	inner
483.2.m.a.137.58	yes	120	69.68	even	2	inner
483.2.m.a.275.3	yes	120	21.2	odd	6	inner
483.2.m.a.275.4	yes	120	483.275	even	6	inner
483.2.m.a.275.57	yes	120	161.114	odd	6	inner
483.2.m.a.275.58	yes	120	7.2	even	3	inner