Embedded newform 483.2.bf.a.388.10

• Label: 483.2.bf.a.388.10
• 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			388.10
Character	\(\chi\)	\(=\)	483.388
Dual form			483.2.bf.a.61.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.23347 + 0.970013i) q^{2} +(0.814576 - 0.580057i) q^{3} +(0.109009 - 0.449342i) q^{4} +(-2.51427 - 0.484586i) q^{5} +(-0.442094 + 1.50563i) q^{6} +(1.06433 + 2.42223i) q^{7} +(-1.00233 - 2.19479i) q^{8} +(0.327068 - 0.945001i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 640 q - 4 q^{2} + 36 q^{4} + 24 q^{8} - 32 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.23347	+	0.970013i	−0.872196	+	0.685903i	−0.950398	−	0.311035i	\(-0.899324\pi\)
							0.0782025	+	0.996937i	\(0.475082\pi\)
\(3\)	0.814576	−	0.580057i	0.470296	−	0.334896i
							\(5\)	−2.51427	−	0.484586i	−1.12442	−	0.216713i	−0.407047	−	0.913407i	\(-0.633442\pi\)
−0.717369	+	0.696694i	\(0.754654\pi\)
\(7\)	1.06433	+	2.42223i	0.402279	+	0.915517i
							\(11\)	−0.810377	+	1.03048i	−0.244338	+	0.310701i	−0.892718	−	0.450616i	\(-0.851204\pi\)
0.648380	+	0.761317i	\(0.275447\pi\)
\(13\)	−1.00254	+	1.55998i	−0.278053	+	0.432659i	−0.951990	−	0.306130i	\(-0.900966\pi\)
							0.673936	+	0.738789i	\(0.264602\pi\)
\(17\)	−0.643367	+	0.613449i	−0.156039	+	0.148783i	−0.764077	−	0.645125i	\(-0.776805\pi\)
							0.608038	+	0.793908i	\(0.291957\pi\)
\(19\)	−6.28289	−	5.99072i	−1.44139	−	1.37437i	−0.783851	−	0.620949i	\(-0.786747\pi\)
							−0.657543	−	0.753417i	\(-0.728404\pi\)
\(23\)	−2.08926	−	4.31683i	−0.435641	−	0.900121i
							\(29\)	−4.05949	−	1.19197i	−0.753829	−	0.221344i	−0.117830	−	0.993034i	\(-0.537594\pi\)
−0.635999	+	0.771690i	\(0.719412\pi\)
\(31\)	−0.653507	+	6.84384i	−0.117373	+	1.22919i	0.724408	+	0.689371i	\(0.242113\pi\)
							−0.841782	+	0.539818i	\(0.818493\pi\)
\(37\)	−5.43957	−	1.88265i	−0.894260	−	0.309507i	−0.158990	−	0.987280i	\(-0.550824\pi\)
							−0.735271	+	0.677774i	\(0.762945\pi\)
\(41\)	−7.43089	−	6.43891i	−1.16051	−	1.00559i	−0.999827	−	0.0185824i	\(-0.994085\pi\)
							−0.160683	−	0.987006i	\(-0.551370\pi\)
\(43\)	1.54580	+	0.705944i	0.235733	+	0.107655i	0.529779	−	0.848136i	\(-0.322275\pi\)
							−0.294047	+	0.955791i	\(0.595002\pi\)
\(47\)	−6.26001	+	3.61422i	−0.913116	+	0.527188i	−0.881432	−	0.472310i	\(-0.843420\pi\)
							−0.0316835	+	0.999498i	\(0.510087\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	483.2.bf.a.388.10	yes	640
7.5	odd	6	inner	483.2.bf.a.250.23	yes	640
23.15	odd	22	inner	483.2.bf.a.199.23	yes	640
161.61	even	66	inner	483.2.bf.a.61.10	✓	640

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
483.2.bf.a.61.10	✓	640	161.61	even	66	inner
483.2.bf.a.199.23	yes	640	23.15	odd	22	inner
483.2.bf.a.250.23	yes	640	7.5	odd	6	inner
483.2.bf.a.388.10	yes	640	1.1	even	1	trivial