Embedded newform 483.2.y.a.16.1

• Label: 483.2.y.a.16.1
• Level: $483$
• Weight: $2$
• Character: 483.16
• Analytic conductor: $3.857$
• Analytic rank: $0$
• Dimension: $320$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 483 = 3 \cdot 7 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	483.y (of order \(33\), degree \(20\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.85677441763\)
Analytic rank:	\(0\)
Dimension:	\(320\)
Relative dimension:	\(16\) over \(\Q(\zeta_{33})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{33}]$

Embedding invariants

Embedding label			16.1
Character	\(\chi\)	\(=\)	483.16
Dual form			483.2.y.a.151.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.843018 - 2.43574i) q^{2} +(0.888835 + 0.458227i) q^{3} +(-3.65004 + 2.87043i) q^{4} +(-0.918397 - 0.0876963i) q^{5} +(0.366817 - 2.55126i) q^{6} +(-0.232163 - 2.63555i) q^{7} +(5.73200 + 3.68373i) q^{8} +(0.580057 + 0.814576i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 320 q + 2 q^{2} - 16 q^{3} + 18 q^{4} - 2 q^{5} - 18 q^{6} + 2 q^{7} - 12 q^{8} + 16 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(323\)	\(346\)	\(442\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{4}{11}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.843018	−	2.43574i	−0.596104	−	1.72233i	−0.687183	−	0.726484i	\(-0.741153\pi\)
							0.0910794	−	0.995844i	\(-0.470968\pi\)
\(3\)	0.888835	+	0.458227i	0.513169	+	0.264557i
							\(5\)	−0.918397	−	0.0876963i	−0.410720	−	0.0392190i	−0.112348	−	0.993669i	\(-0.535837\pi\)
−0.298371	+	0.954450i	\(0.596443\pi\)
\(7\)	−0.232163	−	2.63555i	−0.0877496	−	0.996143i
							\(11\)	0.363092	−	1.04908i	0.109476	−	0.316311i	−0.876934	−	0.480610i	\(-0.840415\pi\)
0.986411	+	0.164299i	\(0.0525363\pi\)
\(13\)	−6.64560	−	1.95132i	−1.84316	−	0.541200i	−0.999994	−	0.00354496i	\(-0.998872\pi\)
							−0.843165	−	0.537655i	\(-0.819310\pi\)
\(17\)	0.740192	−	0.296328i	0.179523	−	0.0718701i	−0.280160	−	0.959953i	\(-0.590387\pi\)
							0.459683	+	0.888083i	\(0.347963\pi\)
\(19\)	−3.45864	−	1.38463i	−0.793465	−	0.317656i	−0.0607168	−	0.998155i	\(-0.519339\pi\)
							−0.732749	+	0.680499i	\(0.761763\pi\)
\(23\)	−0.794994	+	4.72948i	−0.165768	+	0.986165i
							\(29\)	−0.378493	+	2.63248i	−0.0702844	+	0.488839i	0.924027	+	0.382327i	\(0.124877\pi\)
−0.994311	+	0.106512i	\(0.966032\pi\)
\(31\)	−0.114849	+	2.41098i	−0.0206275	+	0.433025i	0.964806	+	0.262963i	\(0.0846997\pi\)
							−0.985433	+	0.170062i	\(0.945603\pi\)
\(37\)	−3.26105	−	4.57950i	−0.536113	−	0.752866i	0.454323	−	0.890837i	\(-0.349881\pi\)
							−0.990436	+	0.137971i	\(0.955942\pi\)
\(41\)	−2.72824	−	5.97402i	−0.426080	−	0.932985i	−0.993947	−	0.109864i	\(-0.964958\pi\)
							0.567867	−	0.823120i	\(-0.307769\pi\)
\(43\)	−2.79788	+	1.79809i	−0.426672	+	0.274205i	−0.736304	−	0.676651i	\(-0.763431\pi\)
							0.309632	+	0.950857i	\(0.399794\pi\)
\(47\)	1.23563	−	2.14017i	0.180235	−	0.312176i	−0.761725	−	0.647900i	\(-0.775648\pi\)
							0.941961	+	0.335724i	\(0.108981\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	483.2.y.a.16.1	✓	320
7.4	even	3	inner	483.2.y.a.361.16	yes	320
23.13	even	11	inner	483.2.y.a.289.16	yes	320
161.151	even	33	inner	483.2.y.a.151.1	yes	320

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
483.2.y.a.16.1	✓	320	1.1	even	1	trivial
483.2.y.a.151.1	yes	320	161.151	even	33	inner
483.2.y.a.289.16	yes	320	23.13	even	11	inner
483.2.y.a.361.16	yes	320	7.4	even	3	inner