Embedded newform 483.2.y.b.163.1

• Label: 483.2.y.b.163.1
• Level: $483$
• Weight: $2$
• Character: 483.163
• 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			163.1
Character	\(\chi\)	\(=\)	483.163
Dual form			483.2.y.b.403.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.35566 + 1.21442i) q^{2} +(-0.786053 + 0.618159i) q^{3} +(2.91417 - 4.09238i) q^{4} +(2.56846 - 2.44902i) q^{5} +(1.10096 - 2.41077i) q^{6} +(0.193345 + 2.63868i) q^{7} +(-1.14055 + 7.93269i) q^{8} +(0.235759 - 0.971812i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 320 q + 2 q^{2} + 16 q^{3} + 18 q^{4} - 2 q^{5} + 18 q^{6} + 2 q^{7} - 12 q^{8} + 16 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.35566	+	1.21442i	−1.66570	+	0.858728i	−0.672793	+	0.739830i	\(0.734906\pi\)
							−0.992907	+	0.118897i	\(0.962064\pi\)
\(3\)	−0.786053	+	0.618159i	−0.453828	+	0.356894i
							\(5\)	2.56846	−	2.44902i	1.14865	−	1.09524i	0.154417	−	0.988006i	\(-0.450650\pi\)
0.994234	−	0.107231i	\(-0.0341984\pi\)
\(7\)	0.193345	+	2.63868i	0.0730776	+	0.997326i
							\(11\)	−4.35028	−	2.24272i	−1.31166	−	0.676207i	−0.346543	−	0.938034i	\(-0.612645\pi\)
−0.965115	+	0.261827i	\(0.915675\pi\)
\(13\)	2.03489	+	2.34839i	0.564378	+	0.651327i	0.964172	−	0.265278i	\(-0.0854639\pi\)
							−0.399794	+	0.916605i	\(0.630918\pi\)
\(17\)	5.00914	−	0.478315i	1.21490	−	0.116008i	0.532140	−	0.846656i	\(-0.321388\pi\)
							0.682755	+	0.730648i	\(0.260782\pi\)
\(19\)	3.28744	+	0.313913i	0.754191	+	0.0720165i	0.465067	−	0.885275i	\(-0.346030\pi\)
							0.289123	+	0.957292i	\(0.406636\pi\)
\(23\)	−3.34321	−	3.43845i	−0.697108	−	0.716966i
							\(29\)	−0.0709396	+	0.155336i	−0.0131732	+	0.0288452i	−0.916104	−	0.400940i	\(-0.868684\pi\)
0.902931	+	0.429785i	\(0.141411\pi\)
\(31\)	5.38015	−	2.15389i	0.966304	−	0.386850i	0.165788	−	0.986161i	\(-0.446983\pi\)
							0.800516	+	0.599312i	\(0.204559\pi\)
\(37\)	1.62049	−	6.67976i	0.266407	−	1.09815i	−0.668629	−	0.743596i	\(-0.733119\pi\)
							0.935037	−	0.354550i	\(-0.115366\pi\)
\(41\)	5.88412	+	1.72773i	0.918945	+	0.269826i	0.706801	−	0.707412i	\(-0.250137\pi\)
							0.212143	+	0.977239i	\(0.431956\pi\)
\(43\)	0.536783	+	3.73341i	0.0818587	+	0.569340i	0.988932	+	0.148368i	\(0.0474019\pi\)
							−0.907074	+	0.420972i	\(0.861689\pi\)
\(47\)	−1.46024	+	2.52922i	−0.212999	+	0.368924i	−0.952652	−	0.304064i	\(-0.901656\pi\)
							0.739653	+	0.672988i	\(0.234990\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	483.2.y.b.163.1	yes	320
7.4	even	3	inner	483.2.y.b.25.16	✓	320
23.12	even	11	inner	483.2.y.b.58.16	yes	320
161.81	even	33	inner	483.2.y.b.403.1	yes	320

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
483.2.y.b.25.16	✓	320	7.4	even	3	inner
483.2.y.b.58.16	yes	320	23.12	even	11	inner
483.2.y.b.163.1	yes	320	1.1	even	1	trivial
483.2.y.b.403.1	yes	320	161.81	even	33	inner