Embedded newform 483.2.y.b.193.10

• Label: 483.2.y.b.193.10
• Level: $483$
• Weight: $2$
• Character: 483.193
• 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			193.10
Character	\(\chi\)	\(=\)	483.193
Dual form			483.2.y.b.478.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.650869 + 0.620602i) q^{2} +(0.981929 - 0.189251i) q^{3} +(-0.0566806 - 1.18987i) q^{4} +(2.62769 - 2.06643i) q^{5} +(0.756557 + 0.486209i) q^{6} +(0.493066 + 2.59940i) q^{7} +(1.87941 - 2.16895i) q^{8} +(0.928368 - 0.371662i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 320 q + 2 q^{2} + 16 q^{3} + 18 q^{4} - 2 q^{5} + 18 q^{6} + 2 q^{7} - 12 q^{8} + 16 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(323\)	\(346\)	\(442\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{5}{11}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.650869	+	0.620602i	0.460234	+	0.438832i	0.884384	−	0.466760i	\(-0.154579\pi\)
							−0.424150	+	0.905592i	\(0.639427\pi\)
\(3\)	0.981929	−	0.189251i	0.566917	−	0.109264i
							\(5\)	2.62769	−	2.06643i	1.17514	−	0.924138i	0.176943	−	0.984221i	\(-0.443379\pi\)
0.998193	+	0.0600833i	\(0.0191366\pi\)
\(7\)	0.493066	+	2.59940i	0.186361	+	0.982481i
							\(11\)	−2.28853	+	2.18211i	−0.690019	+	0.657932i	−0.951632	−	0.307241i	\(-0.900594\pi\)
0.261613	+	0.965173i	\(0.415746\pi\)
\(13\)	−1.43075	−	3.13290i	−0.396818	−	0.868911i	−0.997583	−	0.0694849i	\(-0.977864\pi\)
							0.600765	−	0.799426i	\(-0.294863\pi\)
\(17\)	−5.42933	+	2.79901i	−1.31681	+	0.678860i	−0.966232	−	0.257675i	\(-0.917044\pi\)
							−0.350574	+	0.936535i	\(0.614013\pi\)
\(19\)	1.11129	+	0.572911i	0.254948	+	0.131435i	0.580964	−	0.813930i	\(-0.302676\pi\)
							−0.326015	+	0.945364i	\(0.605706\pi\)
\(23\)	1.91030	−	4.39895i	0.398326	−	0.917244i
							\(29\)	5.94407	+	3.82002i	1.10379	+	0.709361i	0.959931	−	0.280238i	\(-0.0904134\pi\)
0.143856	+	0.989599i	\(0.454050\pi\)
\(31\)	3.14979	+	9.10073i	0.565720	+	1.63454i	0.757304	+	0.653063i	\(0.226516\pi\)
							−0.191584	+	0.981476i	\(0.561362\pi\)
\(37\)	−6.12191	+	2.45084i	−1.00644	+	0.402916i	−0.815544	−	0.578695i	\(-0.803562\pi\)
							−0.190891	+	0.981611i	\(0.561138\pi\)
\(41\)	−0.438978	−	3.05316i	−0.0685568	−	0.476823i	−0.994959	−	0.100287i	\(-0.968024\pi\)
							0.926402	−	0.376537i	\(-0.122885\pi\)
\(43\)	−3.34758	−	3.86332i	−0.510502	−	0.589150i	0.440725	−	0.897642i	\(-0.354721\pi\)
							−0.951227	+	0.308492i	\(0.900176\pi\)
\(47\)	0.523563	+	0.906838i	0.0763695	+	0.132276i	0.901681	−	0.432402i	\(-0.142334\pi\)
							−0.825312	+	0.564678i	\(0.809000\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	483.2.y.b.193.10	✓	320
7.2	even	3	inner	483.2.y.b.331.10	yes	320
23.18	even	11	inner	483.2.y.b.340.10	yes	320
161.156	even	33	inner	483.2.y.b.478.10	yes	320

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
483.2.y.b.193.10	✓	320	1.1	even	1	trivial
483.2.y.b.331.10	yes	320	7.2	even	3	inner
483.2.y.b.340.10	yes	320	23.18	even	11	inner
483.2.y.b.478.10	yes	320	161.156	even	33	inner