Embedded newform 483.2.bf.a.10.26

• Label: 483.2.bf.a.10.26
• Level: $483$
• Weight: $2$
• Character: 483.10
• 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			10.26
Character	\(\chi\)	\(=\)	483.10
Dual form			483.2.bf.a.145.26

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.571977 - 1.65262i) q^{2} +(-0.458227 - 0.888835i) q^{3} +(-0.831885 - 0.654202i) q^{4} +(-0.0705413 + 0.00673588i) q^{5} +(-1.73100 + 0.248880i) q^{6} +(0.803186 - 2.52089i) q^{7} +(1.38541 - 0.890346i) q^{8} +(-0.580057 + 0.814576i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 640 q - 4 q^{2} + 36 q^{4} + 24 q^{8} - 32 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(323\)	\(346\)	\(442\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{3}{22}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.571977	−	1.65262i	0.404449	−	1.16858i	−0.540496	−	0.841347i	\(-0.681763\pi\)
							0.944944	−	0.327231i	\(-0.106116\pi\)
\(3\)	−0.458227	−	0.888835i	−0.264557	−	0.513169i
							\(5\)	−0.0705413	+	0.00673588i	−0.0315470	+	0.00301238i	−0.110818	−	0.993841i	\(-0.535347\pi\)
0.0792706	+	0.996853i	\(0.474741\pi\)
\(7\)	0.803186	−	2.52089i	0.303576	−	0.952807i
							\(11\)	4.41243	−	1.52716i	1.33040	−	0.460455i	0.432886	−	0.901449i	\(-0.357495\pi\)
0.897510	+	0.440994i	\(0.145374\pi\)
\(13\)	0.351518	+	1.19716i	0.0974937	+	0.332033i	0.993768	−	0.111467i	\(-0.0355549\pi\)
							−0.896274	+	0.443500i	\(0.853737\pi\)
\(17\)	−3.02630	−	1.21155i	−0.733985	−	0.293843i	−0.0256028	−	0.999672i	\(-0.508151\pi\)
							−0.708382	+	0.705829i	\(0.750575\pi\)
\(19\)	−3.72189	+	1.49002i	−0.853861	+	0.341834i	−0.756961	−	0.653460i	\(-0.773317\pi\)
							−0.0968996	+	0.995294i	\(0.530893\pi\)
\(23\)	−1.12811	+	4.66126i	−0.235227	+	0.971940i
							\(29\)	−0.615151	−	4.27847i	−0.114231	−	0.794491i	−0.963726	−	0.266895i	\(-0.914002\pi\)
0.849495	−	0.527597i	\(-0.176907\pi\)
\(31\)	−0.962215	+	0.0458359i	−0.172819	+	0.00823238i	−0.133813	−	0.991007i	\(-0.542722\pi\)
							−0.0390059	+	0.999239i	\(0.512419\pi\)
\(37\)	2.76191	+	1.96675i	0.454056	+	0.323332i	0.784101	−	0.620634i	\(-0.213125\pi\)
							−0.330045	+	0.943965i	\(0.607064\pi\)
\(41\)	9.24386	+	4.22153i	1.44365	+	0.659292i	0.974615	−	0.223889i	\(-0.0718752\pi\)
							0.469033	+	0.883180i	\(0.344602\pi\)
\(43\)	5.80251	−	9.02888i	0.884874	−	1.37689i	−0.0410401	−	0.999157i	\(-0.513067\pi\)
							0.925914	−	0.377734i	\(-0.123296\pi\)
\(47\)	1.47775	−	0.853178i	0.215552	−	0.124449i	−0.388337	−	0.921517i	\(-0.626950\pi\)
							0.603889	+	0.797069i	\(0.293617\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	483.2.bf.a.10.26	✓	640
7.5	odd	6	inner	483.2.bf.a.355.7	yes	640
23.7	odd	22	inner	483.2.bf.a.283.7	yes	640
161.145	even	66	inner	483.2.bf.a.145.26	yes	640

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
483.2.bf.a.10.26	✓	640	1.1	even	1	trivial
483.2.bf.a.145.26	yes	640	161.145	even	66	inner
483.2.bf.a.283.7	yes	640	23.7	odd	22	inner
483.2.bf.a.355.7	yes	640	7.5	odd	6	inner