Embedded newform 483.2.bf.a.145.1

• Label: 483.2.bf.a.145.1
• Level: $483$
• Weight: $2$
• Character: 483.145
• Analytic conductor: $3.857$
• Analytic rank: $0$
• Dimension: $640$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 483 = 3 \cdot 7 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	483.bf (of order \(66\), degree \(20\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.85677441763\)
Analytic rank:	\(0\)
Dimension:	\(640\)
Relative dimension:	\(32\) over \(\Q(\zeta_{66})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{66}]$

Embedding invariants

Embedding label			145.1
Character	\(\chi\)	\(=\)	483.145
Dual form			483.2.bf.a.10.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.853118 - 2.46492i) q^{2} +(0.458227 - 0.888835i) q^{3} +(-3.77593 + 2.96942i) q^{4} +(-2.73649 - 0.261303i) q^{5} +(-2.58183 - 0.371212i) q^{6} +(0.617353 - 2.57272i) q^{7} +(6.15210 + 3.95371i) q^{8} +(-0.580057 - 0.814576i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 640 q - 4 q^{2} + 36 q^{4} + 24 q^{8} - 32 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(323\)	\(346\)	\(442\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{5}{6}\right)\)	\(e\left(\frac{19}{22}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.853118	−	2.46492i	−0.603246	−	1.74296i	−0.667062	−	0.745002i	\(-0.732448\pi\)
							0.0638161	−	0.997962i	\(-0.479673\pi\)
\(3\)	0.458227	−	0.888835i	0.264557	−	0.513169i
							\(5\)	−2.73649	−	0.261303i	−1.22379	−	0.116858i	−0.536919	−	0.843634i	\(-0.680412\pi\)
−0.686875	+	0.726776i	\(0.741018\pi\)
\(7\)	0.617353	−	2.57272i	0.233338	−	0.972396i
							\(11\)	−2.34836	−	0.812774i	−0.708056	−	0.245061i	−0.0507792	−	0.998710i	\(-0.516170\pi\)
−0.657277	+	0.753649i	\(0.728292\pi\)
\(13\)	−1.00320	+	3.41660i	−0.278239	+	0.947595i	0.695232	+	0.718785i	\(0.255301\pi\)
							−0.973471	+	0.228810i	\(0.926517\pi\)
\(17\)	−7.11575	+	2.84872i	−1.72582	+	0.690916i	−0.725829	+	0.687876i	\(0.758543\pi\)
							−0.999995	+	0.00304018i	\(0.999032\pi\)
\(19\)	5.43666	+	2.17651i	1.24726	+	0.499326i	0.898948	−	0.438054i	\(-0.144332\pi\)
							0.348307	+	0.937380i	\(0.386757\pi\)
\(23\)	4.52020	−	1.60244i	0.942526	−	0.334132i
							\(29\)	0.235196	−	1.63583i	0.0436749	−	0.303766i	−0.956262	−	0.292512i	\(-0.905509\pi\)
0.999937	−	0.0112533i	\(-0.00358212\pi\)
\(31\)	2.55030	+	0.121486i	0.458047	+	0.0218194i	0.275338	−	0.961348i	\(-0.411210\pi\)
							0.182709	+	0.983167i	\(0.441513\pi\)
\(37\)	−4.95631	+	3.52937i	−0.814812	+	0.580225i	−0.909701	−	0.415263i	\(-0.863690\pi\)
							0.0948897	+	0.995488i	\(0.469750\pi\)
\(41\)	3.66051	−	1.67170i	0.571676	−	0.261076i	−0.108534	−	0.994093i	\(-0.534616\pi\)
							0.680210	+	0.733017i	\(0.261888\pi\)
\(43\)	−2.78260	−	4.32981i	−0.424343	−	0.660290i	0.561593	−	0.827413i	\(-0.310189\pi\)
							−0.985936	+	0.167123i	\(0.946552\pi\)
\(47\)	−4.77598	−	2.75742i	−0.696649	−	0.402210i	0.109449	−	0.993992i	\(-0.465091\pi\)
							−0.806098	+	0.591782i	\(0.798425\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	483.2.bf.a.145.1	yes	640
7.3	odd	6	inner	483.2.bf.a.283.32	yes	640
23.10	odd	22	inner	483.2.bf.a.355.32	yes	640
161.10	even	66	inner	483.2.bf.a.10.1	✓	640

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
483.2.bf.a.10.1	✓	640	161.10	even	66	inner
483.2.bf.a.145.1	yes	640	1.1	even	1	trivial
483.2.bf.a.283.32	yes	640	7.3	odd	6	inner
483.2.bf.a.355.32	yes	640	23.10	odd	22	inner