Embedded newform 483.2.bf.a.61.27

• Label: 483.2.bf.a.61.27
• Level: $483$
• Weight: $2$
• Character: 483.61
• 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			61.27
Character	\(\chi\)	\(=\)	483.61
Dual form			483.2.bf.a.388.27

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.54285 + 1.21331i) q^{2} +(-0.814576 - 0.580057i) q^{3} +(0.436742 + 1.80028i) q^{4} +(-4.04503 + 0.779615i) q^{5} +(-0.552978 - 1.88327i) q^{6} +(0.0404792 - 2.64544i) q^{7} +(0.120271 - 0.263357i) q^{8} +(0.327068 + 0.945001i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 640 q - 4 q^{2} + 36 q^{4} + 24 q^{8} - 32 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(323\)	\(346\)	\(442\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{5}{6}\right)\)	\(e\left(\frac{17}{22}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.54285	+	1.21331i	1.09096	+	0.857939i	0.990570	−	0.137010i	\(-0.0437492\pi\)
							0.100388	+	0.994948i	\(0.467992\pi\)
\(3\)	−0.814576	−	0.580057i	−0.470296	−	0.334896i
							\(5\)	−4.04503	+	0.779615i	−1.80899	+	0.348655i	−0.978157	−	0.207866i	\(-0.933348\pi\)
−0.830834	+	0.556520i	\(0.812136\pi\)
\(7\)	0.0404792	−	2.64544i	0.0152997	−	0.999883i
							\(11\)	−3.14530	−	3.99958i	−0.948344	−	1.20592i	−0.978474	−	0.206371i	\(-0.933834\pi\)
0.0301297	−	0.999546i	\(-0.490408\pi\)
\(13\)	0.309335	+	0.481334i	0.0857940	+	0.133498i	0.881498	−	0.472187i	\(-0.156535\pi\)
							−0.795704	+	0.605685i	\(0.792899\pi\)
\(17\)	−2.05258	−	1.95713i	−0.497824	−	0.474674i	0.399070	−	0.916920i	\(-0.369333\pi\)
							−0.896893	+	0.442247i	\(0.854182\pi\)
\(19\)	−3.00021	+	2.86069i	−0.688295	+	0.656288i	−0.951215	−	0.308530i	\(-0.900163\pi\)
							0.262920	+	0.964818i	\(0.415315\pi\)
\(23\)	−4.78074	−	0.380220i	−0.996852	−	0.0792814i
							\(29\)	−1.31746	+	0.386842i	−0.244647	+	0.0718348i	−0.401757	−	0.915746i	\(-0.631600\pi\)
0.157110	+	0.987581i	\(0.449782\pi\)
\(31\)	0.329319	+	3.44879i	0.0591475	+	0.619420i	0.975641	+	0.219372i	\(0.0704007\pi\)
							−0.916494	+	0.400049i	\(0.868993\pi\)
\(37\)	7.06231	−	2.44429i	1.16104	−	0.401839i	0.322440	−	0.946590i	\(-0.395497\pi\)
							0.838598	+	0.544751i	\(0.183376\pi\)
\(41\)	5.88229	−	5.09703i	0.918660	−	0.796023i	−0.0606985	−	0.998156i	\(-0.519333\pi\)
							0.979358	+	0.202133i	\(0.0647874\pi\)
\(43\)	−10.9164	+	4.98535i	−1.66473	+	0.760258i	−0.664821	+	0.747003i	\(0.731492\pi\)
							−0.999912	+	0.0132556i	\(0.995780\pi\)
\(47\)	−10.5973	−	6.11834i	−1.54577	−	0.892452i	−0.998457	−	0.0555257i	\(-0.982317\pi\)
							−0.547315	−	0.836927i	\(-0.684350\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	483.2.bf.a.61.27	✓	640
7.3	odd	6	inner	483.2.bf.a.199.6	yes	640
23.20	odd	22	inner	483.2.bf.a.250.6	yes	640
161.66	even	66	inner	483.2.bf.a.388.27	yes	640

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
483.2.bf.a.61.27	✓	640	1.1	even	1	trivial
483.2.bf.a.199.6	yes	640	7.3	odd	6	inner
483.2.bf.a.250.6	yes	640	23.20	odd	22	inner
483.2.bf.a.388.27	yes	640	161.66	even	66	inner