Embedded newform 507.2.q.b.490.7

• Label: 507.2.q.b.490.7
• Level: $507$
• Weight: $2$
• Character: 507.490
• 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			490.7
Character	\(\chi\)	\(=\)	507.490
Dual form			507.2.q.b.328.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.0988984 - 0.484436i) q^{2} +(0.987050 + 0.160411i) q^{3} +(1.61506 - 0.688113i) q^{4} +(0.576785 - 0.142165i) q^{5} +(-0.0199086 - 0.494027i) q^{6} +(1.32650 - 2.79554i) q^{7} +(-1.05481 - 1.52815i) q^{8} +(0.948536 + 0.316668i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 384 q + q^{2} + 16 q^{3} + 17 q^{4} + 6 q^{5} + q^{6} - 3 q^{7} - 18 q^{8} + 16 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(170\)	\(340\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{17}{39}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.0988984	−	0.484436i	−0.0699317	−	0.342548i	0.929739	−	0.368220i	\(-0.120033\pi\)
							−0.999670	+	0.0256717i	\(0.991828\pi\)
\(3\)	0.987050	+	0.160411i	0.569874	+	0.0926135i
							\(5\)	0.576785	−	0.142165i	0.257946	−	0.0635780i	−0.108222	−	0.994127i	\(-0.534516\pi\)
0.366168	+	0.930549i	\(0.380670\pi\)
\(7\)	1.32650	−	2.79554i	0.501369	−	1.05661i	−0.482196	−	0.876063i	\(-0.660161\pi\)
							0.983566	−	0.180551i	\(-0.0577879\pi\)
\(11\)	−0.422764	+	0.141139i	−0.127468	+	0.0425551i	−0.379686	−	0.925115i	\(-0.623968\pi\)
							0.252218	+	0.967670i	\(0.418840\pi\)
\(13\)	−2.93382	−	2.09588i	−0.813695	−	0.581292i
							\(17\)	−1.94545	+	4.09995i	−0.471841	+	0.994385i	0.518415	+	0.855129i	\(0.326522\pi\)
−0.990256	+	0.139256i	\(0.955529\pi\)
\(19\)	0.0631291	+	0.109343i	0.0144828	+	0.0250850i	0.873176	−	0.487405i	\(-0.162056\pi\)
							−0.858693	+	0.512490i	\(0.828723\pi\)
\(23\)	0.186753	−	0.323466i	0.0389407	−	0.0674474i	−0.845898	−	0.533344i	\(-0.820935\pi\)
							0.884839	+	0.465897i	\(0.154268\pi\)
\(29\)	1.64744	+	8.06971i	0.305923	+	1.49851i	0.784316	+	0.620362i	\(0.213014\pi\)
							−0.478393	+	0.878146i	\(0.658781\pi\)
\(31\)	3.99727	+	2.09793i	0.717932	+	0.376800i	0.783787	−	0.621030i	\(-0.213285\pi\)
							−0.0658555	+	0.997829i	\(0.520978\pi\)
\(37\)	2.94855	−	1.86455i	0.484738	−	0.306530i	−0.269650	−	0.962958i	\(-0.586908\pi\)
							0.754388	+	0.656428i	\(0.227934\pi\)
\(41\)	6.77561	+	1.10114i	1.05817	+	0.171970i	0.664505	−	0.747284i	\(-0.268642\pi\)
							0.393667	+	0.919253i	\(0.371206\pi\)
\(43\)	0.513833	+	0.324928i	0.0783587	+	0.0495511i	0.573071	−	0.819506i	\(-0.305752\pi\)
							−0.494712	+	0.869057i	\(0.664726\pi\)
\(47\)	0.123780	+	1.01942i	0.0180552	+	0.148698i	0.998948	−	0.0458491i	\(-0.0145993\pi\)
							−0.980893	+	0.194547i	\(0.937676\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	507.2.q.b.490.7	yes	384
169.159	even	39	inner	507.2.q.b.328.7	✓	384

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
507.2.q.b.328.7	✓	384	169.159	even	39	inner
507.2.q.b.490.7	yes	384	1.1	even	1	trivial