Embedded newform 483.2.y.b.340.10

• Label: 483.2.y.b.340.10
• Level: $483$
• Weight: $2$
• Character: 483.340
• 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			340.10
Character	\(\chi\)	\(=\)	483.340
Dual form			483.2.y.b.331.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.212023 + 0.873970i) q^{2} +(-0.327068 - 0.945001i) q^{3} +(1.05880 - 0.545850i) q^{4} +(-3.10343 + 1.24243i) q^{5} +(0.756557 - 0.486209i) q^{6} +(-2.28739 - 1.32961i) q^{7} +(1.87941 + 2.16895i) q^{8} +(-0.786053 + 0.618159i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 320 q + 2 q^{2} + 16 q^{3} + 18 q^{4} - 2 q^{5} + 18 q^{6} + 2 q^{7} - 12 q^{8} + 16 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(323\)	\(346\)	\(442\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{6}{11}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.212023	+	0.873970i	0.149923	+	0.617990i	0.996341	+	0.0854709i	\(0.0272395\pi\)
							−0.846418	+	0.532519i	\(0.821245\pi\)
\(3\)	−0.327068	−	0.945001i	−0.188833	−	0.545596i
							\(5\)	−3.10343	+	1.24243i	−1.38790	+	0.555629i	−0.940832	−	0.338873i	\(-0.889954\pi\)
−0.447063	+	0.894502i	\(0.647530\pi\)
\(7\)	−2.28739	−	1.32961i	−0.864551	−	0.502546i
							\(11\)	−0.745498	+	3.07298i	−0.224776	+	0.926539i	0.741895	+	0.670516i	\(0.233927\pi\)
−0.966671	+	0.256023i	\(0.917588\pi\)
\(13\)	−1.43075	+	3.13290i	−0.396818	+	0.868911i	0.600765	+	0.799426i	\(0.294863\pi\)
							−0.997583	+	0.0694849i	\(0.977864\pi\)
\(17\)	0.290648	+	6.10144i	0.0704924	+	1.47982i	0.706269	+	0.707944i	\(0.250377\pi\)
							−0.635776	+	0.771873i	\(0.719320\pi\)
\(19\)	−0.0594907	+	1.24886i	−0.0136481	+	0.286509i	0.981717	+	0.190345i	\(0.0609605\pi\)
							−0.995365	+	0.0961645i	\(0.969343\pi\)
\(23\)	−4.76475	−	0.545102i	−0.993520	−	0.113662i
							\(29\)	5.94407	−	3.82002i	1.10379	−	0.709361i	0.143856	−	0.989599i	\(-0.454050\pi\)
0.959931	+	0.280238i	\(0.0904134\pi\)
\(31\)	−9.45636	+	1.82256i	−1.69841	+	0.327342i	−0.944221	−	0.329313i	\(-0.893183\pi\)
							−0.754191	+	0.656655i	\(0.771971\pi\)
\(37\)	5.18345	−	4.07631i	0.852153	−	0.670141i	−0.0933928	−	0.995629i	\(-0.529771\pi\)
							0.945546	+	0.325489i	\(0.105529\pi\)
\(41\)	−0.438978	+	3.05316i	−0.0685568	+	0.476823i	0.926402	+	0.376537i	\(0.122885\pi\)
							−0.994959	+	0.100287i	\(0.968024\pi\)
\(43\)	−3.34758	+	3.86332i	−0.510502	+	0.589150i	−0.951227	−	0.308492i	\(-0.900176\pi\)
							0.440725	+	0.897642i	\(0.354721\pi\)
\(47\)	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.340.10	yes	320
7.2	even	3	inner	483.2.y.b.478.10	yes	320
23.9	even	11	inner	483.2.y.b.193.10	✓	320
161.9	even	33	inner	483.2.y.b.331.10	yes	320

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
483.2.y.b.193.10	✓	320	23.9	even	11	inner
483.2.y.b.331.10	yes	320	161.9	even	33	inner
483.2.y.b.340.10	yes	320	1.1	even	1	trivial
483.2.y.b.478.10	yes	320	7.2	even	3	inner