Embedded newform 483.2.bf.a.178.4

• Label: 483.2.bf.a.178.4
• 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			178.4
Character	\(\chi\)	\(=\)	483.178
Dual form			483.2.bf.a.19.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.39951 + 0.229125i) q^{2} +(0.690079 + 0.723734i) q^{3} +(3.74129 - 0.721074i) q^{4} +(1.30345 + 0.671976i) q^{5} +(-1.82168 - 1.57849i) q^{6} +(-2.17813 - 1.50191i) q^{7} +(-4.18647 + 1.22926i) q^{8} +(-0.0475819 + 0.998867i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 640 q - 4 q^{2} + 36 q^{4} + 24 q^{8} - 32 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.39951	+	0.229125i	−1.69671	+	0.162016i	−0.898087	−	0.439817i	\(-0.855043\pi\)
							−0.798622	+	0.601833i	\(0.794437\pi\)
\(3\)	0.690079	+	0.723734i	0.398417	+	0.417848i
							\(5\)	1.30345	+	0.671976i	0.582922	+	0.300517i	0.724351	−	0.689431i	\(-0.242140\pi\)
−0.141429	+	0.989948i	\(0.545170\pi\)
\(7\)	−2.17813	−	1.50191i	−0.823256	−	0.567670i
							\(11\)	−0.0780550	+	0.817429i	−0.0235345	+	0.246464i	0.976048	+	0.217557i	\(0.0698088\pi\)
−0.999582	+	0.0289071i	\(0.990797\pi\)
\(13\)	5.79223	−	0.832796i	1.60647	−	0.230976i	0.720075	−	0.693897i	\(-0.244108\pi\)
							0.886400	+	0.462920i	\(0.153198\pi\)
\(17\)	2.49870	+	7.21951i	0.606023	+	1.75099i	0.658731	+	0.752379i	\(0.271094\pi\)
							−0.0527074	+	0.998610i	\(0.516785\pi\)
\(19\)	1.23391	−	3.56516i	0.283079	−	0.817903i	−0.710391	−	0.703808i	\(-0.751482\pi\)
							0.993470	−	0.114096i	\(-0.0363971\pi\)
\(23\)	−4.76857	−	0.510610i	−0.994316	−	0.106469i
							\(29\)	3.34156	−	3.85636i	0.620512	−	0.716109i	−0.355292	−	0.934755i	\(-0.615619\pi\)
0.975804	+	0.218646i	\(0.0701641\pi\)
\(31\)	8.23822	+	1.99857i	1.47963	+	0.358954i	0.892687	−	0.450677i	\(-0.148817\pi\)
							0.586940	+	0.809630i	\(0.300332\pi\)
\(37\)	2.58444	+	0.123112i	0.424879	+	0.0202395i	0.258933	−	0.965895i	\(-0.416629\pi\)
							0.165946	+	0.986135i	\(0.446932\pi\)
\(41\)	6.49868	+	10.1121i	1.01492	+	1.57925i	0.797632	+	0.603144i	\(0.206086\pi\)
							0.217291	+	0.976107i	\(0.430278\pi\)
\(43\)	−2.74150	+	9.33669i	−0.418075	+	1.42383i	0.434237	+	0.900799i	\(0.357018\pi\)
							−0.852312	+	0.523034i	\(0.824800\pi\)
\(47\)	−7.78999	+	4.49755i	−1.13629	+	0.656036i	−0.945509	−	0.325597i	\(-0.894435\pi\)
							−0.190779	+	0.981633i	\(0.561101\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	483.2.bf.a.178.4	yes	640
7.5	odd	6	inner	483.2.bf.a.40.29	yes	640
23.19	odd	22	inner	483.2.bf.a.157.29	yes	640
161.19	even	66	inner	483.2.bf.a.19.4	✓	640

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
483.2.bf.a.19.4	✓	640	161.19	even	66	inner
483.2.bf.a.40.29	yes	640	7.5	odd	6	inner
483.2.bf.a.157.29	yes	640	23.19	odd	22	inner
483.2.bf.a.178.4	yes	640	1.1	even	1	trivial