Embedded newform 483.2.bf.a.61.11

• Label: 483.2.bf.a.61.11
• Level: $483$
• Weight: $2$
• Character: 483.61
• 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			61.11
Character	\(\chi\)	\(=\)	483.61
Dual form			483.2.bf.a.388.11

$q$-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 \(\left(\mathbb{Z}/483\mathbb{Z}\right)^\times\).

\(n\)	\(323\)	\(346\)	\(442\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{5}{6}\right)\)	\(e\left(\frac{17}{22}\right)\)

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

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
483.2.bf.a.61.11	✓	640	1.1	even	1	trivial
483.2.bf.a.199.22	yes	640	7.3	odd	6	inner
483.2.bf.a.250.22	yes	640	23.20	odd	22	inner
483.2.bf.a.388.11	yes	640	161.66	even	66	inner