Embedded newform 483.2.y.a.16.4

• Label: 483.2.y.a.16.4
• 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			16.4
Character	\(\chi\)	\(=\)	483.16
Dual form			483.2.y.a.151.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.461078 - 1.33220i) q^{2} +(0.888835 + 0.458227i) q^{3} +(0.00994599 - 0.00782161i) q^{4} +(3.91624 + 0.373956i) q^{5} +(0.200626 - 1.39538i) q^{6} +(-1.00683 - 2.44669i) q^{7} +(-2.38689 - 1.53396i) q^{8} +(0.580057 + 0.814576i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 320 q + 2 q^{2} - 16 q^{3} + 18 q^{4} - 2 q^{5} - 18 q^{6} + 2 q^{7} - 12 q^{8} + 16 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.461078	−	1.33220i	−0.326032	−	0.942007i	−0.982273	−	0.187456i	\(-0.939976\pi\)
							0.656241	−	0.754551i	\(-0.272145\pi\)
\(3\)	0.888835	+	0.458227i	0.513169	+	0.264557i
							\(5\)	3.91624	+	0.373956i	1.75140	+	0.167238i	0.920903	−	0.389791i	\(-0.127453\pi\)
0.830493	+	0.557029i	\(0.188059\pi\)
\(7\)	−1.00683	−	2.44669i	−0.380545	−	0.924763i
							\(11\)	0.718534	−	2.07607i	0.216646	−	0.625958i	−0.783344	−	0.621588i	\(-0.786488\pi\)
0.999990	−	0.00436999i	\(-0.00139102\pi\)
\(13\)	−2.77780	−	0.815635i	−0.770422	−	0.226216i	−0.127181	−	0.991880i	\(-0.540593\pi\)
							−0.643242	+	0.765663i	\(0.722411\pi\)
\(17\)	−5.91972	+	2.36990i	−1.43574	+	0.574784i	−0.953537	−	0.301274i	\(-0.902588\pi\)
							−0.482204	+	0.876059i	\(0.660164\pi\)
\(19\)	5.31114	+	2.12626i	1.21846	+	0.487797i	0.889669	−	0.456606i	\(-0.150935\pi\)
							0.328789	+	0.944403i	\(0.393359\pi\)
\(23\)	3.80601	−	2.91792i	0.793609	−	0.608428i
							\(29\)	0.814101	−	5.66220i	0.151175	−	1.05144i	−0.763080	−	0.646304i	\(-0.776314\pi\)
0.914255	−	0.405140i	\(-0.132777\pi\)
\(31\)	−0.517648	+	10.8668i	−0.0929723	+	1.95173i	0.166644	+	0.986017i	\(0.446707\pi\)
							−0.259617	+	0.965712i	\(0.583596\pi\)
\(37\)	0.577496	+	0.810980i	0.0949398	+	0.133324i	0.859314	−	0.511449i	\(-0.170891\pi\)
							−0.764374	+	0.644773i	\(0.776952\pi\)
\(41\)	3.70836	+	8.12017i	0.579148	+	1.26816i	0.941782	+	0.336225i	\(0.109150\pi\)
							−0.362634	+	0.931932i	\(0.618122\pi\)
\(43\)	−2.57104	+	1.65231i	−0.392080	+	0.251975i	−0.721796	−	0.692105i	\(-0.756683\pi\)
							0.329716	+	0.944080i	\(0.393047\pi\)
\(47\)	−1.20253	+	2.08284i	−0.175407	+	0.303813i	−0.940302	−	0.340341i	\(-0.889457\pi\)
							0.764895	+	0.644155i	\(0.222791\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	483.2.y.a.16.4	✓	320
7.4	even	3	inner	483.2.y.a.361.13	yes	320
23.13	even	11	inner	483.2.y.a.289.13	yes	320
161.151	even	33	inner	483.2.y.a.151.4	yes	320

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
483.2.y.a.16.4	✓	320	1.1	even	1	trivial
483.2.y.a.151.4	yes	320	161.151	even	33	inner
483.2.y.a.289.13	yes	320	23.13	even	11	inner
483.2.y.a.361.13	yes	320	7.4	even	3	inner