Embedded newform 483.2.bf.a.178.20

• Label: 483.2.bf.a.178.20
• 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			178.20
Character	\(\chi\)	\(=\)	483.178
Dual form			483.2.bf.a.19.20

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.739640 - 0.0706270i) q^{2} +(0.690079 + 0.723734i) q^{3} +(-1.42178 + 0.274025i) q^{4} +(-3.42445 - 1.76543i) q^{5} +(0.561525 + 0.486564i) q^{6} +(2.62322 - 0.344549i) q^{7} +(-2.45806 + 0.721753i) q^{8} +(-0.0475819 + 0.998867i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 640 q - 4 q^{2} + 36 q^{4} + 24 q^{8} - 32 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.739640	−	0.0706270i	0.523004	−	0.0499408i	0.169784	−	0.985481i	\(-0.445693\pi\)
							0.353220	+	0.935540i	\(0.385087\pi\)
\(3\)	0.690079	+	0.723734i	0.398417	+	0.417848i
							\(5\)	−3.42445	−	1.76543i	−1.53146	−	0.789523i	−0.533061	−	0.846077i	\(-0.678958\pi\)
−0.998400	+	0.0565538i	\(0.981989\pi\)
\(7\)	2.62322	−	0.344549i	0.991484	−	0.130227i
							\(11\)	0.416487	−	4.36165i	0.125576	−	1.31509i	−0.684609	−	0.728911i	\(-0.740027\pi\)
0.810184	−	0.586176i	\(-0.199367\pi\)
\(13\)	−5.31123	+	0.763640i	−1.47307	+	0.211796i	−0.831610	−	0.555360i	\(-0.812581\pi\)
							−0.641461	+	0.767155i	\(0.721671\pi\)
\(17\)	−1.85101	−	5.34814i	−0.448936	−	1.29712i	−0.911781	−	0.410676i	\(-0.865293\pi\)
							0.462845	−	0.886439i	\(-0.346828\pi\)
\(19\)	0.381157	−	1.10128i	0.0874434	−	0.252651i	−0.892798	−	0.450458i	\(-0.851261\pi\)
							0.980241	+	0.197807i	\(0.0633819\pi\)
\(23\)	−3.27799	−	3.50069i	−0.683507	−	0.729944i
							\(29\)	−1.18291	+	1.36515i	−0.219660	+	0.253501i	−0.854875	−	0.518834i	\(-0.826366\pi\)
0.635215	+	0.772336i	\(0.280912\pi\)
\(31\)	−7.88041	−	1.91177i	−1.41536	−	0.343363i	−0.546103	−	0.837718i	\(-0.683889\pi\)
							−0.869260	+	0.494354i	\(0.835404\pi\)
\(37\)	−0.105084	−	0.00500575i	−0.0172757	−	0.000822941i	0.0389423	−	0.999241i	\(-0.487601\pi\)
							−0.0562180	+	0.998419i	\(0.517904\pi\)
\(41\)	5.99008	+	9.32075i	0.935494	+	1.45566i	0.889616	+	0.456709i	\(0.150972\pi\)
							0.0458775	+	0.998947i	\(0.485392\pi\)
\(43\)	−0.630678	+	2.14789i	−0.0961775	+	0.327551i	−0.993501	−	0.113823i	\(-0.963690\pi\)
							0.897324	+	0.441373i	\(0.145509\pi\)
\(47\)	4.42578	−	2.55522i	0.645566	−	0.372718i	−0.141190	−	0.989983i	\(-0.545093\pi\)
							0.786755	+	0.617265i	\(0.211759\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	483.2.bf.a.178.20	yes	640
7.5	odd	6	inner	483.2.bf.a.40.13	yes	640
23.19	odd	22	inner	483.2.bf.a.157.13	yes	640
161.19	even	66	inner	483.2.bf.a.19.20	✓	640

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
483.2.bf.a.19.20	✓	640	161.19	even	66	inner
483.2.bf.a.40.13	yes	640	7.5	odd	6	inner
483.2.bf.a.157.13	yes	640	23.19	odd	22	inner
483.2.bf.a.178.20	yes	640	1.1	even	1	trivial