Embedded newform 507.2.q.b.16.9

• Label: 507.2.q.b.16.9
• 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			16.9
Character	\(\chi\)	\(=\)	507.16
Dual form			507.2.q.b.412.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.0277086 + 0.00223687i) q^{2} +(-0.845190 - 0.534466i) q^{3} +(-1.97334 - 0.320699i) q^{4} +(0.676752 + 0.599550i) q^{5} +(-0.0222235 - 0.0166999i) q^{6} +(-0.402242 + 0.418779i) q^{7} +(-0.107943 - 0.0266056i) q^{8} +(0.428693 + 0.903450i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 384 q + q^{2} + 16 q^{3} + 17 q^{4} + 6 q^{5} + q^{6} - 3 q^{7} - 18 q^{8} + 16 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.0277086	+	0.00223687i	0.0195930	+	0.00158171i	0.0902592	−	0.995918i	\(-0.471230\pi\)
							−0.0706662	+	0.997500i	\(0.522513\pi\)
\(3\)	−0.845190	−	0.534466i	−0.487971	−	0.308574i
							\(5\)	0.676752	+	0.599550i	0.302653	+	0.268127i	0.800756	−	0.598991i	\(-0.204431\pi\)
−0.498103	+	0.867118i	\(0.665970\pi\)
\(7\)	−0.402242	+	0.418779i	−0.152033	+	0.158284i	−0.792864	−	0.609399i	\(-0.791411\pi\)
							0.640830	+	0.767683i	\(0.278590\pi\)
\(11\)	−0.912090	+	1.92219i	−0.275005	+	0.579562i	−0.993219	−	0.116261i	\(-0.962909\pi\)
							0.718213	+	0.695823i	\(0.244960\pi\)
\(13\)	3.31516	−	1.41764i	0.919461	−	0.393182i
							\(17\)	4.44533	−	4.62808i	1.07815	−	1.12247i	0.0861445	−	0.996283i	\(-0.472545\pi\)
0.992006	−	0.126191i	\(-0.0402752\pi\)
\(19\)	1.16027	−	2.00966i	0.266185	−	0.461046i	−0.701688	−	0.712484i	\(-0.747570\pi\)
							0.967873	+	0.251438i	\(0.0809034\pi\)
\(23\)	0.130762	+	0.226487i	0.0272658	+	0.0472257i	0.879336	−	0.476201i	\(-0.157987\pi\)
							−0.852071	+	0.523427i	\(0.824653\pi\)
\(29\)	3.01090	+	0.243065i	0.559110	+	0.0451360i	0.356792	−	0.934184i	\(-0.383870\pi\)
							0.202318	+	0.979320i	\(0.435153\pi\)
\(31\)	0.367054	−	3.02296i	0.0659248	−	0.542940i	−0.921907	−	0.387412i	\(-0.873369\pi\)
							0.987832	−	0.155528i	\(-0.0497078\pi\)
\(37\)	−3.92427	+	1.67198i	−0.645146	+	0.274871i	−0.689655	−	0.724138i	\(-0.742238\pi\)
							0.0445089	+	0.999009i	\(0.485828\pi\)
\(41\)	6.99227	+	4.42165i	1.09201	+	0.690545i	0.953976	−	0.299882i	\(-0.0969474\pi\)
							0.138034	+	0.990428i	\(0.455922\pi\)
\(43\)	6.36721	+	2.71282i	0.970991	+	0.413701i	0.818438	−	0.574595i	\(-0.194840\pi\)
							0.152553	+	0.988295i	\(0.451251\pi\)
\(47\)	4.16030	−	10.9698i	0.606843	−	1.60011i	−0.180485	−	0.983578i	\(-0.557767\pi\)
							0.787328	−	0.616535i	\(-0.211464\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	507.2.q.b.16.9	✓	384
169.74	even	39	inner	507.2.q.b.412.9	yes	384

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
507.2.q.b.16.9	✓	384	1.1	even	1	trivial
507.2.q.b.412.9	yes	384	169.74	even	39	inner