Embedded newform 507.2.q.b.16.10

• Label: 507.2.q.b.16.10
• Level: $507$
• Weight: $2$
• Character: 507.16
• Analytic conductor: $4.048$
• Analytic rank: $0$
• Dimension: $384$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 507 = 3 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	507.q (of order \(39\), degree \(24\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.04841538248\)
Analytic rank:	\(0\)
Dimension:	\(384\)
Relative dimension:	\(16\) over \(\Q(\zeta_{39})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{39}]$

Embedding invariants

Embedding label			16.10
Character	\(\chi\)	\(=\)	507.16
Dual form			507.2.q.b.412.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.655764 + 0.0529387i) q^{2} +(-0.845190 - 0.534466i) q^{3} +(-1.54688 - 0.251392i) q^{4} +(-1.61643 - 1.43203i) q^{5} +(-0.525951 - 0.395227i) q^{6} +(1.45558 - 1.51542i) q^{7} +(-2.27864 - 0.561634i) q^{8} +(0.428693 + 0.903450i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 384 q + q^{2} + 16 q^{3} + 17 q^{4} + 6 q^{5} + q^{6} - 3 q^{7} - 18 q^{8} + 16 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/507\mathbb{Z}\right)^\times\).

\(n\)	\(170\)	\(340\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{1}{39}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.655764	+	0.0529387i	0.463695	+	0.0374333i	0.310107	−	0.950702i	\(-0.399635\pi\)
							0.153588	+	0.988135i	\(0.450917\pi\)
\(3\)	−0.845190	−	0.534466i	−0.487971	−	0.308574i
							\(5\)	−1.61643	−	1.43203i	−0.722890	−	0.640424i	0.219241	−	0.975671i	\(-0.429642\pi\)
−0.942131	+	0.335246i	\(0.891180\pi\)
\(7\)	1.45558	−	1.51542i	0.550157	−	0.572774i	−0.386824	−	0.922153i	\(-0.626428\pi\)
							0.936981	+	0.349379i	\(0.113608\pi\)
\(11\)	−2.14574	+	4.52204i	−0.646964	+	1.36345i	0.269372	+	0.963036i	\(0.413184\pi\)
							−0.916336	+	0.400411i	\(0.868867\pi\)
\(13\)	0.547978	+	3.56367i	0.151982	+	0.988383i
							\(17\)	−1.62333	+	1.69006i	−0.393714	+	0.409900i	−0.888325	−	0.459214i	\(-0.848131\pi\)
0.494611	+	0.869114i	\(0.335310\pi\)
\(19\)	−1.40954	+	2.44140i	−0.323371	+	0.560096i	−0.981181	−	0.193088i	\(-0.938150\pi\)
							0.657810	+	0.753184i	\(0.271483\pi\)
\(23\)	0.954346	+	1.65298i	0.198995	+	0.344669i	0.948203	−	0.317666i	\(-0.102899\pi\)
							−0.749208	+	0.662335i	\(0.769566\pi\)
\(29\)	−5.71698	−	0.461522i	−1.06162	−	0.0857025i	−0.462700	−	0.886515i	\(-0.653119\pi\)
							−0.598916	+	0.800812i	\(0.704402\pi\)
\(31\)	0.304449	−	2.50736i	0.0546806	−	0.450335i	−0.939495	−	0.342563i	\(-0.888705\pi\)
							0.994176	−	0.107772i	\(-0.0343718\pi\)
\(37\)	−4.18034	+	1.78108i	−0.687244	+	0.292807i	−0.707220	−	0.706993i	\(-0.750051\pi\)
							0.0199763	+	0.999800i	\(0.493641\pi\)
\(41\)	−2.30738	−	1.45910i	−0.360352	−	0.227873i	0.342004	−	0.939698i	\(-0.388894\pi\)
							−0.702356	+	0.711825i	\(0.747869\pi\)
\(43\)	−9.34268	−	3.98054i	−1.42474	−	0.607027i	−0.464345	−	0.885654i	\(-0.653710\pi\)
							−0.960399	+	0.278627i	\(0.910121\pi\)
\(47\)	−3.00003	+	7.91042i	−0.437599	+	1.15385i	0.517007	+	0.855981i	\(0.327046\pi\)
							−0.954606	+	0.297872i	\(0.903723\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	507.2.q.b.16.10	✓	384
169.74	even	39	inner	507.2.q.b.412.10	yes	384

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
507.2.q.b.16.10	✓	384	1.1	even	1	trivial
507.2.q.b.412.10	yes	384	169.74	even	39	inner