Embedded newform 483.2.bf.a.388.15

• Label: 483.2.bf.a.388.15
• 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			388.15
Character	\(\chi\)	\(=\)	483.388
Dual form			483.2.bf.a.61.15

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.423819 + 0.333295i) q^{2} +(0.814576 - 0.580057i) q^{3} +(-0.402981 + 1.66111i) q^{4} +(2.64705 + 0.510177i) q^{5} +(-0.151903 + 0.517333i) q^{6} +(2.13968 + 1.55621i) q^{7} +(-0.830810 - 1.81922i) q^{8} +(0.327068 - 0.945001i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 640 q - 4 q^{2} + 36 q^{4} + 24 q^{8} - 32 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.423819	+	0.333295i	−0.299685	+	0.235675i	−0.756666	−	0.653801i	\(-0.773173\pi\)
							0.456981	+	0.889476i	\(0.348931\pi\)
\(3\)	0.814576	−	0.580057i	0.470296	−	0.334896i
							\(5\)	2.64705	+	0.510177i	1.18380	+	0.228158i	0.742902	−	0.669400i	\(-0.233449\pi\)
0.440895	+	0.897558i	\(0.354661\pi\)
\(7\)	2.13968	+	1.55621i	0.808721	+	0.588192i
							\(11\)	3.14186	−	3.99521i	0.947308	−	1.20460i	−0.0314346	−	0.999506i	\(-0.510008\pi\)
0.978742	−	0.205094i	\(-0.0657500\pi\)
\(13\)	−0.763946	+	1.18872i	−0.211881	+	0.329693i	−0.930885	−	0.365313i	\(-0.880962\pi\)
							0.719004	+	0.695006i	\(0.244598\pi\)
\(17\)	−2.02290	+	1.92883i	−0.490624	+	0.467809i	−0.894541	−	0.446987i	\(-0.852497\pi\)
							0.403916	+	0.914796i	\(0.367649\pi\)
\(19\)	−1.09217	−	1.04138i	−0.250560	−	0.238908i	0.554382	−	0.832263i	\(-0.312955\pi\)
							−0.804941	+	0.593354i	\(0.797803\pi\)
\(23\)	−0.824588	+	4.72441i	−0.171939	+	0.985108i
							\(29\)	2.58978	+	0.760429i	0.480911	+	0.141208i	0.513197	−	0.858271i	\(-0.328461\pi\)
−0.0322869	+	0.999479i	\(0.510279\pi\)
\(31\)	0.211355	−	2.21341i	0.0379604	−	0.397540i	−0.956445	−	0.291912i	\(-0.905709\pi\)
							0.994406	−	0.105628i	\(-0.0336853\pi\)
\(37\)	−4.81860	−	1.66773i	−0.792173	−	0.274174i	−0.0991344	−	0.995074i	\(-0.531607\pi\)
							−0.693039	+	0.720900i	\(0.743729\pi\)
\(41\)	2.53334	+	2.19515i	0.395641	+	0.342825i	0.829849	−	0.557988i	\(-0.188426\pi\)
							−0.434208	+	0.900813i	\(0.642972\pi\)
\(43\)	−0.960109	−	0.438467i	−0.146415	−	0.0668656i	0.340860	−	0.940114i	\(-0.389282\pi\)
							−0.487275	+	0.873249i	\(0.662009\pi\)
\(47\)	4.74062	−	2.73700i	0.691491	−	0.399232i	−0.112680	−	0.993631i	\(-0.535943\pi\)
							0.804170	+	0.594399i	\(0.202610\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	483.2.bf.a.388.15	yes	640
7.5	odd	6	inner	483.2.bf.a.250.18	yes	640
23.15	odd	22	inner	483.2.bf.a.199.18	yes	640
161.61	even	66	inner	483.2.bf.a.61.15	✓	640

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
483.2.bf.a.61.15	✓	640	161.61	even	66	inner
483.2.bf.a.199.18	yes	640	23.15	odd	22	inner
483.2.bf.a.250.18	yes	640	7.5	odd	6	inner
483.2.bf.a.388.15	yes	640	1.1	even	1	trivial