Embedded newform 483.2.bf.a.388.17

• Label: 483.2.bf.a.388.17
• 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			388.17
Character	\(\chi\)	\(=\)	483.388
Dual form			483.2.bf.a.61.17

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.0474955 - 0.0373509i) q^{2} +(0.814576 - 0.580057i) q^{3} +(-0.470657 + 1.94008i) q^{4} +(-2.52294 - 0.486256i) q^{5} +(0.0170231 - 0.0579752i) q^{6} +(-2.47853 - 0.925688i) q^{7} +(0.100311 + 0.219649i) q^{8} +(0.327068 - 0.945001i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 640 q - 4 q^{2} + 36 q^{4} + 24 q^{8} - 32 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.0474955	−	0.0373509i	0.0335844	−	0.0264111i	−0.601226	−	0.799079i	\(-0.705321\pi\)
							0.634810	+	0.772668i	\(0.281078\pi\)
\(3\)	0.814576	−	0.580057i	0.470296	−	0.334896i
							\(5\)	−2.52294	−	0.486256i	−1.12829	−	0.217460i	−0.409247	−	0.912424i	\(-0.634209\pi\)
−0.719045	+	0.694963i	\(0.755421\pi\)
\(7\)	−2.47853	−	0.925688i	−0.936796	−	0.349877i
							\(11\)	−0.645830	+	0.821240i	−0.194725	+	0.247613i	−0.873604	−	0.486637i	\(-0.838224\pi\)
0.678879	+	0.734250i	\(0.262466\pi\)
\(13\)	2.60918	−	4.05997i	0.723657	−	1.12603i	−0.263248	−	0.964728i	\(-0.584794\pi\)
							0.986905	−	0.161304i	\(-0.0515699\pi\)
\(17\)	−4.48017	+	4.27183i	−1.08660	+	1.03607i	−0.0873970	+	0.996174i	\(0.527855\pi\)
							−0.999204	+	0.0398982i	\(0.987297\pi\)
\(19\)	−4.83551	−	4.61065i	−1.10934	−	1.05775i	−0.997951	−	0.0639779i	\(-0.979621\pi\)
							−0.111390	−	0.993777i	\(-0.535530\pi\)
\(23\)	−3.72820	+	3.01670i	−0.777384	+	0.629026i
							\(29\)	−8.05452	−	2.36502i	−1.49569	−	0.439173i	−0.571336	−	0.820716i	\(-0.693575\pi\)
−0.924351	+	0.381543i	\(0.875393\pi\)
\(31\)	0.114168	−	1.19562i	0.0205052	−	0.214740i	−0.979412	−	0.201872i	\(-0.935298\pi\)
							0.999917	−	0.0128687i	\(-0.00409634\pi\)
\(37\)	8.56184	+	2.96328i	1.40756	+	0.487160i	0.922049	−	0.387072i	\(-0.126514\pi\)
							0.485508	+	0.874232i	\(0.338635\pi\)
\(41\)	3.91456	+	3.39198i	0.611351	+	0.529739i	0.904580	−	0.426304i	\(-0.140184\pi\)
							−0.293229	+	0.956042i	\(0.594730\pi\)
\(43\)	−1.99388	−	0.910575i	−0.304064	−	0.138861i	0.257540	−	0.966268i	\(-0.417088\pi\)
							−0.561604	+	0.827406i	\(0.689815\pi\)
\(47\)	8.34895	−	4.82027i	1.21782	−	0.703108i	0.253368	−	0.967370i	\(-0.418462\pi\)
							0.964451	+	0.264262i	\(0.0851283\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	483.2.bf.a.388.17	yes	640
7.5	odd	6	inner	483.2.bf.a.250.16	yes	640
23.15	odd	22	inner	483.2.bf.a.199.16	yes	640
161.61	even	66	inner	483.2.bf.a.61.17	✓	640

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
483.2.bf.a.61.17	✓	640	161.61	even	66	inner
483.2.bf.a.199.16	yes	640	23.15	odd	22	inner
483.2.bf.a.250.16	yes	640	7.5	odd	6	inner
483.2.bf.a.388.17	yes	640	1.1	even	1	trivial