Embedded newform 483.2.bf.a.388.29

• Label: 483.2.bf.a.388.29
• Level: $483$
• Weight: $2$
• Character: 483.388
• 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			388.29
Character	\(\chi\)	\(=\)	483.388
Dual form			483.2.bf.a.61.29

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.74563 - 1.37278i) q^{2} +(0.814576 - 0.580057i) q^{3} +(0.691183 - 2.84910i) q^{4} +(0.900327 + 0.173524i) q^{5} +(0.625658 - 2.13080i) q^{6} +(-2.36104 - 1.19395i) q^{7} +(-0.859559 - 1.88217i) q^{8} +(0.327068 - 0.945001i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 640 q - 4 q^{2} + 36 q^{4} + 24 q^{8} - 32 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.74563	−	1.37278i	1.23435	−	0.970700i	0.234347	−	0.972153i	\(-0.424705\pi\)
							0.999999	+	0.00145276i	\(0.000462427\pi\)
\(3\)	0.814576	−	0.580057i	0.470296	−	0.334896i
							\(5\)	0.900327	+	0.173524i	0.402639	+	0.0776022i	0.386550	−	0.922268i	\(-0.373667\pi\)
0.0160883	+	0.999871i	\(0.494879\pi\)
\(7\)	−2.36104	−	1.19395i	−0.892388	−	0.451269i
							\(11\)	1.96363	−	2.49696i	0.592056	−	0.752861i	−0.394030	−	0.919098i	\(-0.628919\pi\)
0.986086	+	0.166237i	\(0.0531617\pi\)
\(13\)	−1.26333	+	1.96579i	−0.350386	+	0.545211i	−0.971053	−	0.238863i	\(-0.923225\pi\)
							0.620668	+	0.784074i	\(0.286862\pi\)
\(17\)	0.328116	−	0.312858i	0.0795798	−	0.0758792i	−0.649245	−	0.760579i	\(-0.724915\pi\)
							0.728825	+	0.684700i	\(0.240067\pi\)
\(19\)	−0.682757	−	0.651007i	−0.156635	−	0.149351i	0.607709	−	0.794160i	\(-0.292089\pi\)
							−0.764344	+	0.644808i	\(0.776937\pi\)
\(23\)	1.31637	+	4.61163i	0.274483	+	0.961592i
							\(29\)	1.08409	+	0.318318i	0.201311	+	0.0591101i	0.380833	−	0.924644i	\(-0.375637\pi\)
−0.179523	+	0.983754i	\(0.557455\pi\)
\(31\)	−0.611021	+	6.39890i	−0.109743	+	1.14928i	0.757997	+	0.652258i	\(0.226178\pi\)
							−0.867740	+	0.497018i	\(0.834428\pi\)
\(37\)	8.69557	+	3.00957i	1.42954	+	0.494770i	0.928721	−	0.370780i	\(-0.120910\pi\)
							0.500823	+	0.865550i	\(0.333031\pi\)
\(41\)	4.93869	+	4.27940i	0.771294	+	0.668330i	0.948831	−	0.315784i	\(-0.102267\pi\)
							−0.177537	+	0.984114i	\(0.556813\pi\)
\(43\)	−5.82035	−	2.65806i	−0.887594	−	0.405351i	−0.0811785	−	0.996700i	\(-0.525868\pi\)
							−0.806416	+	0.591349i	\(0.798596\pi\)
\(47\)	−10.0744	+	5.81648i	−1.46951	+	0.848420i	−0.999415	−	0.0341974i	\(-0.989113\pi\)
							−0.470092	+	0.882618i	\(0.655779\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	483.2.bf.a.388.29	yes	640
7.5	odd	6	inner	483.2.bf.a.250.4	yes	640
23.15	odd	22	inner	483.2.bf.a.199.4	yes	640
161.61	even	66	inner	483.2.bf.a.61.29	✓	640

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
483.2.bf.a.61.29	✓	640	161.61	even	66	inner
483.2.bf.a.199.4	yes	640	23.15	odd	22	inner
483.2.bf.a.250.4	yes	640	7.5	odd	6	inner
483.2.bf.a.388.29	yes	640	1.1	even	1	trivial