Embedded newform 483.2.bf.a.10.7

• Label: 483.2.bf.a.10.7
• Level: $483$
• Weight: $2$
• Character: 483.10
• 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			10.7
Character	\(\chi\)	\(=\)	483.10
Dual form			483.2.bf.a.145.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.632527 + 1.82757i) q^{2} +(0.458227 + 0.888835i) q^{3} +(-1.36781 - 1.07566i) q^{4} +(0.843558 - 0.0805501i) q^{5} +(-1.91425 + 0.275227i) q^{6} +(-2.38264 + 1.15022i) q^{7} +(-0.422845 + 0.271746i) q^{8} +(-0.580057 + 0.814576i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 640 q - 4 q^{2} + 36 q^{4} + 24 q^{8} - 32 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(323\)	\(346\)	\(442\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{3}{22}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.632527	+	1.82757i	−0.447264	+	1.29229i	0.465952	+	0.884810i	\(0.345712\pi\)
							−0.913216	+	0.407476i	\(0.866409\pi\)
\(3\)	0.458227	+	0.888835i	0.264557	+	0.513169i
							\(5\)	0.843558	−	0.0805501i	0.377251	−	0.0360231i	0.0952914	−	0.995449i	\(-0.469622\pi\)
0.281959	+	0.959426i	\(0.409016\pi\)
\(7\)	−2.38264	+	1.15022i	−0.900554	+	0.434743i
							\(11\)	−1.15551	+	0.399927i	−0.348401	+	0.120583i	−0.495663	−	0.868515i	\(-0.665075\pi\)
0.147263	+	0.989097i	\(0.452954\pi\)
\(13\)	0.0600004	+	0.204343i	0.0166411	+	0.0566744i	0.967395	−	0.253272i	\(-0.0815069\pi\)
							−0.950754	+	0.309947i	\(0.899689\pi\)
\(17\)	1.79588	+	0.718962i	0.435565	+	0.174374i	0.579074	−	0.815275i	\(-0.303414\pi\)
							−0.143509	+	0.989649i	\(0.545839\pi\)
\(19\)	−5.96701	+	2.38883i	−1.36893	+	0.548035i	−0.935555	−	0.353180i	\(-0.885100\pi\)
							−0.433371	+	0.901216i	\(0.642676\pi\)
\(23\)	3.32915	+	3.45207i	0.694176	+	0.719805i
							\(29\)	−0.0808512	−	0.562332i	−0.0150137	−	0.104422i	0.980937	−	0.194328i	\(-0.0622526\pi\)
−0.995950	+	0.0899055i	\(0.971344\pi\)
\(31\)	−8.09949	+	0.385826i	−1.45471	+	0.0692965i	−0.759968	−	0.649961i	\(-0.774785\pi\)
							−0.694744	+	0.719257i	\(0.744482\pi\)
\(37\)	3.80059	+	2.70639i	0.624813	+	0.444927i	0.848108	−	0.529824i	\(-0.177742\pi\)
							−0.223295	+	0.974751i	\(0.571681\pi\)
\(41\)	7.47575	+	3.41406i	1.16752	+	0.533186i	0.902345	−	0.431015i	\(-0.141844\pi\)
							0.265171	+	0.964202i	\(0.414572\pi\)
\(43\)	2.30466	−	3.58612i	0.351457	−	0.546877i	−0.619846	−	0.784723i	\(-0.712805\pi\)
							0.971303	+	0.237846i	\(0.0764413\pi\)
\(47\)	4.76860	−	2.75315i	0.695572	−	0.401589i	−0.110124	−	0.993918i	\(-0.535125\pi\)
							0.805696	+	0.592329i	\(0.201791\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	483.2.bf.a.10.7	✓	640
7.5	odd	6	inner	483.2.bf.a.355.26	yes	640
23.7	odd	22	inner	483.2.bf.a.283.26	yes	640
161.145	even	66	inner	483.2.bf.a.145.7	yes	640

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
483.2.bf.a.10.7	✓	640	1.1	even	1	trivial
483.2.bf.a.145.7	yes	640	161.145	even	66	inner
483.2.bf.a.283.26	yes	640	23.7	odd	22	inner
483.2.bf.a.355.26	yes	640	7.5	odd	6	inner