Embedded newform 483.2.bf.a.145.4

• Label: 483.2.bf.a.145.4
• 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			145.4
Character	\(\chi\)	\(=\)	483.145
Dual form			483.2.bf.a.10.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.797371 - 2.30385i) q^{2} +(0.458227 - 0.888835i) q^{3} +(-3.09983 + 2.43773i) q^{4} +(1.48893 + 0.142176i) q^{5} +(-2.41312 - 0.346955i) q^{6} +(2.64327 - 0.114618i) q^{7} +(3.98605 + 2.56168i) q^{8} +(-0.580057 - 0.814576i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 640 q - 4 q^{2} + 36 q^{4} + 24 q^{8} - 32 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.797371	−	2.30385i	−0.563827	−	1.62907i	−0.761037	−	0.648709i	\(-0.775309\pi\)
							0.197210	−	0.980361i	\(-0.436812\pi\)
\(3\)	0.458227	−	0.888835i	0.264557	−	0.513169i
							\(5\)	1.48893	+	0.142176i	0.665871	+	0.0635830i	0.422518	−	0.906355i	\(-0.361146\pi\)
0.243354	+	0.969938i	\(0.421752\pi\)
\(7\)	2.64327	−	0.114618i	0.999061	−	0.0433217i
							\(11\)	5.25822	+	1.81989i	1.58541	+	0.548717i	0.970754	−	0.240076i	\(-0.0771723\pi\)
0.614659	+	0.788793i	\(0.289293\pi\)
\(13\)	1.14332	−	3.89380i	0.317101	−	1.07995i	−0.634580	−	0.772857i	\(-0.718827\pi\)
							0.951681	−	0.307088i	\(-0.0993547\pi\)
\(17\)	2.36184	−	0.945539i	0.572831	−	0.229327i	−0.0671173	−	0.997745i	\(-0.521380\pi\)
							0.639948	+	0.768418i	\(0.278956\pi\)
\(19\)	−4.40203	−	1.76231i	−1.00989	−	0.404301i	−0.193069	−	0.981185i	\(-0.561844\pi\)
							−0.816826	+	0.576885i	\(0.804268\pi\)
\(23\)	0.501961	+	4.76949i	0.104666	+	0.994507i
							\(29\)	−0.100916	+	0.701885i	−0.0187396	+	0.130337i	−0.997044	−	0.0768377i	\(-0.975518\pi\)
0.978304	+	0.207174i	\(0.0664268\pi\)
\(31\)	1.67480	+	0.0797805i	0.300803	+	0.0143290i	0.197441	−	0.980315i	\(-0.436737\pi\)
							0.103362	+	0.994644i	\(0.467040\pi\)
\(37\)	−0.543946	+	0.387342i	−0.0894242	+	0.0636787i	−0.623895	−	0.781509i	\(-0.714451\pi\)
							0.534470	+	0.845187i	\(0.320511\pi\)
\(41\)	−5.32488	+	2.43179i	−0.831607	+	0.379782i	−0.785277	−	0.619144i	\(-0.787480\pi\)
							−0.0463296	+	0.998926i	\(0.514752\pi\)
\(43\)	−4.30916	−	6.70519i	−0.657141	−	1.02253i	−0.996639	−	0.0819244i	\(-0.973893\pi\)
							0.339498	−	0.940607i	\(-0.389743\pi\)
\(47\)	1.90751	+	1.10130i	0.278239	+	0.160641i	0.632626	−	0.774458i	\(-0.281977\pi\)
							−0.354387	+	0.935099i	\(0.615310\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	483.2.bf.a.145.4	yes	640
7.3	odd	6	inner	483.2.bf.a.283.29	yes	640
23.10	odd	22	inner	483.2.bf.a.355.29	yes	640
161.10	even	66	inner	483.2.bf.a.10.4	✓	640

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
483.2.bf.a.10.4	✓	640	161.10	even	66	inner
483.2.bf.a.145.4	yes	640	1.1	even	1	trivial
483.2.bf.a.283.29	yes	640	7.3	odd	6	inner
483.2.bf.a.355.29	yes	640	23.10	odd	22	inner