Embedded newform 483.2.bf.a.388.11

• Label: 483.2.bf.a.388.11
• 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			388.11
Character	\(\chi\)	\(=\)	483.388
Dual form			483.2.bf.a.61.11

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.03420 + 0.813301i) q^{2} +(0.814576 - 0.580057i) q^{3} +(-0.0634142 + 0.261397i) q^{4} +(-0.255961 - 0.0493325i) q^{5} +(-0.370671 + 1.26239i) q^{6} +(-2.61931 - 0.373143i) q^{7} +(-1.24012 - 2.71548i) q^{8} +(0.327068 - 0.945001i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 640 q - 4 q^{2} + 36 q^{4} + 24 q^{8} - 32 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.03420	+	0.813301i	−0.731287	+	0.575091i	−0.912853	−	0.408288i	\(-0.866126\pi\)
							0.181566	+	0.983379i	\(0.441883\pi\)
\(3\)	0.814576	−	0.580057i	0.470296	−	0.334896i
							\(5\)	−0.255961	−	0.0493325i	−0.114469	−	0.0220621i	0.131695	−	0.991290i	\(-0.457958\pi\)
−0.246164	+	0.969228i	\(0.579170\pi\)
\(7\)	−2.61931	−	0.373143i	−0.990005	−	0.141035i
							\(11\)	1.25293	−	1.59323i	0.377772	−	0.480376i	−0.559572	−	0.828782i	\(-0.689034\pi\)
0.937344	+	0.348406i	\(0.113277\pi\)
\(13\)	1.10216	−	1.71500i	0.305686	−	0.475656i	−0.654093	−	0.756414i	\(-0.726950\pi\)
							0.959779	+	0.280758i	\(0.0905859\pi\)
\(17\)	5.65102	−	5.38824i	1.37057	−	1.30684i	0.463361	−	0.886170i	\(-0.346643\pi\)
							0.907214	−	0.420670i	\(-0.138205\pi\)
\(19\)	0.566098	+	0.539774i	0.129872	+	0.123833i	0.752232	−	0.658899i	\(-0.228977\pi\)
							−0.622360	+	0.782731i	\(0.713826\pi\)
\(23\)	4.54762	+	1.52286i	0.948245	+	0.317539i
							\(29\)	2.59510	+	0.761990i	0.481898	+	0.141498i	0.513654	−	0.857998i	\(-0.328292\pi\)
−0.0317556	+	0.999496i	\(0.510110\pi\)
\(31\)	0.0699355	−	0.732398i	0.0125608	−	0.131543i	−0.987016	−	0.160620i	\(-0.948651\pi\)
							0.999577	+	0.0290771i	\(0.00925682\pi\)
\(37\)	−9.20070	−	3.18439i	−1.51259	−	0.523511i	−0.560187	−	0.828366i	\(-0.689271\pi\)
							−0.952399	+	0.304855i	\(0.901392\pi\)
\(41\)	−5.57319	−	4.82920i	−0.870387	−	0.754194i	0.100195	−	0.994968i	\(-0.468053\pi\)
							−0.970581	+	0.240773i	\(0.922599\pi\)
\(43\)	10.1601	+	4.63996i	1.54940	+	0.707588i	0.992415	−	0.122934i	\(-0.0392304\pi\)
							0.556986	+	0.830522i	\(0.311958\pi\)
\(47\)	8.06516	−	4.65642i	1.17642	−	0.679209i	0.221240	−	0.975219i	\(-0.428990\pi\)
							0.955185	+	0.296010i	\(0.0956562\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	483.2.bf.a.388.11	yes	640
7.5	odd	6	inner	483.2.bf.a.250.22	yes	640
23.15	odd	22	inner	483.2.bf.a.199.22	yes	640
161.61	even	66	inner	483.2.bf.a.61.11	✓	640

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
483.2.bf.a.61.11	✓	640	161.61	even	66	inner
483.2.bf.a.199.22	yes	640	23.15	odd	22	inner
483.2.bf.a.250.22	yes	640	7.5	odd	6	inner
483.2.bf.a.388.11	yes	640	1.1	even	1	trivial