Embedded newform 483.2.y.b.4.1

• Label: 483.2.y.b.4.1
• Level: $483$
• Weight: $2$
• Character: 483.4
• 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			4.1
Character	\(\chi\)	\(=\)	483.4
Dual form			483.2.y.b.121.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.48158 + 2.08058i) q^{2} +(0.235759 - 0.971812i) q^{3} +(-1.47962 - 4.27509i) q^{4} +(0.193923 - 4.07094i) q^{5} +(1.67264 + 1.93033i) q^{6} +(-2.46944 - 0.949661i) q^{7} +(6.18541 + 1.81620i) q^{8} +(-0.888835 - 0.458227i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 320 q + 2 q^{2} + 16 q^{3} + 18 q^{4} - 2 q^{5} + 18 q^{6} + 2 q^{7} - 12 q^{8} + 16 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(323\)	\(346\)	\(442\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{2}{11}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.48158	+	2.08058i	−1.04763	+	1.47120i	−0.173917	+	0.984760i	\(0.555642\pi\)
							−0.873717	+	0.486435i	\(0.838297\pi\)
\(3\)	0.235759	−	0.971812i	0.136115	−	0.561076i
							\(5\)	0.193923	−	4.07094i	0.0867249	−	1.82058i	−0.367865	−	0.929879i	\(-0.619911\pi\)
0.454590	−	0.890701i	\(-0.349786\pi\)
\(7\)	−2.46944	−	0.949661i	−0.933361	−	0.358938i
							\(11\)	−0.448336	−	0.629599i	−0.135178	−	0.189831i	0.741438	−	0.671021i	\(-0.234144\pi\)
−0.876616	+	0.481190i	\(0.840205\pi\)
\(13\)	−0.607778	+	4.22719i	−0.168567	+	1.17241i	0.713281	+	0.700879i	\(0.247209\pi\)
							−0.881848	+	0.471534i	\(0.843701\pi\)
\(17\)	−4.50344	+	0.867966i	−1.09224	+	0.210513i	−0.703402	−	0.710792i	\(-0.748337\pi\)
							−0.388841	+	0.921305i	\(0.627125\pi\)
\(19\)	7.00991	+	1.35105i	1.60818	+	0.309952i	0.912713	−	0.408602i	\(-0.133984\pi\)
							0.695471	+	0.718554i	\(0.255196\pi\)
\(23\)	−4.57574	−	1.43618i	−0.954108	−	0.299464i
							\(29\)	−3.32466	−	3.83686i	−0.617373	−	0.712486i	0.357833	−	0.933786i	\(-0.383516\pi\)
−0.975206	+	0.221299i	\(0.928970\pi\)
\(31\)	−3.55639	+	3.39101i	−0.638747	+	0.609044i	−0.938627	−	0.344933i	\(-0.887902\pi\)
							0.299881	+	0.953977i	\(0.403053\pi\)
\(37\)	2.25055	+	1.16024i	0.369988	+	0.190742i	0.633186	−	0.774000i	\(-0.281747\pi\)
							−0.263198	+	0.964742i	\(0.584777\pi\)
\(41\)	4.44012	+	2.85349i	0.693430	+	0.445641i	0.839304	−	0.543663i	\(-0.182963\pi\)
							−0.145873	+	0.989303i	\(0.546599\pi\)
\(43\)	−2.69708	+	0.791933i	−0.411300	+	0.120769i	−0.480836	−	0.876811i	\(-0.659667\pi\)
							0.0695351	+	0.997580i	\(0.477848\pi\)
\(47\)	−5.72446	−	9.91505i	−0.834998	−	1.44626i	−0.894032	−	0.448003i	\(-0.852135\pi\)
							0.0590340	−	0.998256i	\(-0.481198\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	483.2.y.b.4.1	✓	320
7.2	even	3	inner	483.2.y.b.142.16	yes	320
23.6	even	11	inner	483.2.y.b.466.16	yes	320
161.121	even	33	inner	483.2.y.b.121.1	yes	320

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
483.2.y.b.4.1	✓	320	1.1	even	1	trivial
483.2.y.b.121.1	yes	320	161.121	even	33	inner
483.2.y.b.142.16	yes	320	7.2	even	3	inner
483.2.y.b.466.16	yes	320	23.6	even	11	inner