Embedded newform 483.2.bf.a.388.14

• Label: 483.2.bf.a.388.14
• 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			388.14
Character	\(\chi\)	\(=\)	483.388
Dual form			483.2.bf.a.61.14

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.451440 + 0.355016i) q^{2} +(-0.814576 + 0.580057i) q^{3} +(-0.393756 + 1.62309i) q^{4} +(0.598753 + 0.115400i) q^{5} +(0.161803 - 0.551049i) q^{6} +(-1.36900 - 2.26403i) q^{7} +(-0.875621 - 1.91734i) q^{8} +(0.327068 - 0.945001i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 640 q - 4 q^{2} + 36 q^{4} + 24 q^{8} - 32 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.451440	+	0.355016i	−0.319216	+	0.251034i	−0.764890	−	0.644161i	\(-0.777207\pi\)
							0.445674	+	0.895196i	\(0.352964\pi\)
\(3\)	−0.814576	+	0.580057i	−0.470296	+	0.334896i
							\(5\)	0.598753	+	0.115400i	0.267770	+	0.0516085i	0.321369	−	0.946954i	\(-0.395857\pi\)
−0.0535986	+	0.998563i	\(0.517069\pi\)
\(7\)	−1.36900	−	2.26403i	−0.517434	−	0.855723i
							\(11\)	−1.70470	+	2.16770i	−0.513986	+	0.653586i	−0.971895	−	0.235413i	\(-0.924356\pi\)
0.457910	+	0.888999i	\(0.348598\pi\)
\(13\)	−2.52621	+	3.93087i	−0.700646	+	1.09023i	0.290426	+	0.956897i	\(0.406203\pi\)
							−0.991072	+	0.133329i	\(0.957433\pi\)
\(17\)	0.524184	−	0.499808i	0.127133	−	0.121221i	−0.623845	−	0.781548i	\(-0.714431\pi\)
							0.750979	+	0.660326i	\(0.229582\pi\)
\(19\)	−3.03709	−	2.89586i	−0.696756	−	0.664355i	0.256491	−	0.966547i	\(-0.417433\pi\)
							−0.953247	+	0.302191i	\(0.902282\pi\)
\(23\)	−2.08493	−	4.31892i	−0.434738	−	0.900557i
							\(29\)	3.39452	+	0.996720i	0.630346	+	0.185086i	0.581272	−	0.813709i	\(-0.302555\pi\)
0.0490732	+	0.998795i	\(0.484373\pi\)
\(31\)	0.714409	−	7.48163i	0.128312	−	1.34374i	−0.670388	−	0.742011i	\(-0.733872\pi\)
							0.798700	−	0.601730i	\(-0.205522\pi\)
\(37\)	−2.76488	−	0.956933i	−0.454543	−	0.157319i	0.0901969	−	0.995924i	\(-0.471250\pi\)
							−0.544740	+	0.838605i	\(0.683372\pi\)
\(41\)	2.08159	+	1.80371i	0.325089	+	0.281692i	0.802080	−	0.597217i	\(-0.203727\pi\)
							−0.476990	+	0.878909i	\(0.658272\pi\)
\(43\)	−4.55925	−	2.08214i	−0.695279	−	0.317523i	0.0362413	−	0.999343i	\(-0.488462\pi\)
							−0.731520	+	0.681820i	\(0.761189\pi\)
\(47\)	2.68378	−	1.54948i	0.391470	−	0.226015i	−0.291327	−	0.956624i	\(-0.594097\pi\)
							0.682797	+	0.730608i	\(0.260763\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	483.2.bf.a.388.14	yes	640
7.5	odd	6	inner	483.2.bf.a.250.19	yes	640
23.15	odd	22	inner	483.2.bf.a.199.19	yes	640
161.61	even	66	inner	483.2.bf.a.61.14	✓	640

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
483.2.bf.a.61.14	✓	640	161.61	even	66	inner
483.2.bf.a.199.19	yes	640	23.15	odd	22	inner
483.2.bf.a.250.19	yes	640	7.5	odd	6	inner
483.2.bf.a.388.14	yes	640	1.1	even	1	trivial