Embedded newform 483.2.bf.a.178.30

• Label: 483.2.bf.a.178.30
• Level: $483$
• Weight: $2$
• Character: 483.178
• 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			178.30
Character	\(\chi\)	\(=\)	483.178
Dual form			483.2.bf.a.19.30

$q$-expansion

\(f(q)\)	\(=\)	\(q+(2.45475 - 0.234400i) q^{2} +(-0.690079 - 0.723734i) q^{3} +(4.00700 - 0.772287i) q^{4} +(2.61077 + 1.34595i) q^{5} +(-1.86362 - 1.61483i) q^{6} +(-0.495214 - 2.59899i) q^{7} +(4.92311 - 1.44556i) q^{8} +(-0.0475819 + 0.998867i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 640 q - 4 q^{2} + 36 q^{4} + 24 q^{8} - 32 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(323\)	\(346\)	\(442\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{7}{22}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	2.45475	−	0.234400i	1.73577	−	0.165746i	0.821327	−	0.570458i	\(-0.193234\pi\)
							0.914444	+	0.404712i	\(0.132628\pi\)
\(3\)	−0.690079	−	0.723734i	−0.398417	−	0.417848i
							\(5\)	2.61077	+	1.34595i	1.16757	+	0.601925i	0.929319	−	0.369278i	\(-0.120395\pi\)
0.238253	+	0.971203i	\(0.423425\pi\)
\(7\)	−0.495214	−	2.59899i	−0.187173	−	0.982327i
							\(11\)	−0.297978	+	3.12057i	−0.0898439	+	0.940888i	0.832373	+	0.554215i	\(0.186982\pi\)
−0.922217	+	0.386672i	\(0.873624\pi\)
\(13\)	−4.04049	+	0.580935i	−1.12063	+	0.161122i	−0.677633	−	0.735400i	\(-0.736994\pi\)
							−0.442998	+	0.896523i	\(0.646085\pi\)
\(17\)	−2.36060	−	6.82051i	−0.572530	−	1.65422i	−0.743341	−	0.668913i	\(-0.766760\pi\)
							0.170811	−	0.985304i	\(-0.445361\pi\)
\(19\)	−2.13615	+	6.17200i	−0.490066	+	1.41595i	0.380903	+	0.924615i	\(0.375613\pi\)
							−0.870969	+	0.491338i	\(0.836508\pi\)
\(23\)	3.52825	+	3.24830i	0.735690	+	0.677318i
							\(29\)	5.51281	−	6.36212i	1.02370	−	1.18142i	0.0404482	−	0.999182i	\(-0.487121\pi\)
0.983255	−	0.182235i	\(-0.0583331\pi\)
\(31\)	−0.108785	−	0.0263909i	−0.0195383	−	0.00473995i	0.225978	−	0.974132i	\(-0.427442\pi\)
							−0.245516	+	0.969392i	\(0.578957\pi\)
\(37\)	2.39453	+	0.114065i	0.393658	+	0.0187522i	0.243478	−	0.969906i	\(-0.421712\pi\)
							0.150180	+	0.988659i	\(0.452015\pi\)
\(41\)	−3.67872	−	5.72421i	−0.574520	−	0.893971i	0.425419	−	0.904996i	\(-0.360127\pi\)
							−0.999939	+	0.0110258i	\(0.996490\pi\)
\(43\)	0.578938	−	1.97168i	0.0882872	−	0.300678i	−0.903498	−	0.428592i	\(-0.859010\pi\)
							0.991785	+	0.127914i	\(0.0408280\pi\)
\(47\)	−7.51213	+	4.33713i	−1.09576	+	0.632636i	−0.935103	−	0.354375i	\(-0.884694\pi\)
							−0.160654	+	0.987011i	\(0.551360\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	483.2.bf.a.178.30	yes	640
7.5	odd	6	inner	483.2.bf.a.40.3	yes	640
23.19	odd	22	inner	483.2.bf.a.157.3	yes	640
161.19	even	66	inner	483.2.bf.a.19.30	✓	640

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
483.2.bf.a.19.30	✓	640	161.19	even	66	inner
483.2.bf.a.40.3	yes	640	7.5	odd	6	inner
483.2.bf.a.157.3	yes	640	23.19	odd	22	inner
483.2.bf.a.178.30	yes	640	1.1	even	1	trivial