Embedded newform 507.2.q.a.16.1

• Label: 507.2.q.a.16.1
• Level: $507$
• Weight: $2$
• Character: 507.16
• Analytic conductor: $4.048$
• Analytic rank: $0$
• Dimension: $360$
• 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:	\(360\)
Relative dimension:	\(15\) over \(\Q(\zeta_{39})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{39}]$

Embedding invariants

Embedding label			16.1
Character	\(\chi\)	\(=\)	507.16
Dual form			507.2.q.a.412.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.74572 - 0.221657i) q^{2} +(0.845190 + 0.534466i) q^{3} +(5.51573 + 0.896393i) q^{4} +(0.154238 + 0.136643i) q^{5} +(-2.20218 - 1.65483i) q^{6} +(-1.20996 + 1.25970i) q^{7} +(-9.59673 - 2.36538i) q^{8} +(0.428693 + 0.903450i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 360 q + q^{2} - 15 q^{3} + 15 q^{4} + 2 q^{5} - q^{6} + 2 q^{7} + 6 q^{8} + 15 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.74572	−	0.221657i	−1.94152	−	0.156735i	−0.952499	−	0.304542i	\(-0.901497\pi\)
							−0.989016	+	0.147807i	\(0.952779\pi\)
\(3\)	0.845190	+	0.534466i	0.487971	+	0.308574i
							\(5\)	0.154238	+	0.136643i	0.0689775	+	0.0611088i	0.696907	−	0.717161i	\(-0.254559\pi\)
−0.627930	+	0.778270i	\(0.716097\pi\)
\(7\)	−1.20996	+	1.25970i	−0.457322	+	0.476123i	−0.909440	−	0.415835i	\(-0.863489\pi\)
							0.452118	+	0.891958i	\(0.350669\pi\)
\(11\)	2.35566	−	4.96445i	0.710258	−	1.49684i	−0.150927	−	0.988545i	\(-0.548226\pi\)
							0.861185	−	0.508292i	\(-0.169723\pi\)
\(13\)	2.58424	+	2.51430i	0.716740	+	0.697340i
							\(17\)	0.723060	−	0.752785i	0.175368	−	0.182577i	−0.627718	−	0.778441i	\(-0.716011\pi\)
0.803085	+	0.595864i	\(0.203190\pi\)
\(19\)	1.88880	−	3.27149i	0.433320	−	0.750531i	−0.563837	−	0.825886i	\(-0.690675\pi\)
							0.997157	+	0.0753544i	\(0.0240088\pi\)
\(23\)	3.39740	+	5.88447i	0.708406	+	1.22700i	0.965448	+	0.260596i	\(0.0839189\pi\)
							−0.257042	+	0.966400i	\(0.582748\pi\)
\(29\)	−3.05082	−	0.246288i	−0.566524	−	0.0457345i	−0.206113	−	0.978528i	\(-0.566081\pi\)
							−0.360411	+	0.932794i	\(0.617364\pi\)
\(31\)	−0.301889	+	2.48628i	−0.0542209	+	0.446549i	0.940164	+	0.340723i	\(0.110672\pi\)
							−0.994385	+	0.105826i	\(0.966251\pi\)
\(37\)	2.90009	−	1.23561i	0.476772	−	0.203133i	−0.140193	−	0.990124i	\(-0.544772\pi\)
							0.616965	+	0.786991i	\(0.288362\pi\)
\(41\)	4.67507	+	2.95633i	0.730123	+	0.461702i	0.847100	−	0.531433i	\(-0.178346\pi\)
							−0.116978	+	0.993135i	\(0.537321\pi\)
\(43\)	2.70543	+	1.15268i	0.412574	+	0.175781i	0.588273	−	0.808663i	\(-0.299808\pi\)
							−0.175699	+	0.984444i	\(0.556218\pi\)
\(47\)	−3.31564	+	8.74264i	−0.483636	+	1.27524i	0.442063	+	0.896984i	\(0.354247\pi\)
							−0.925699	+	0.378260i	\(0.876522\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	507.2.q.a.16.1	✓	360
169.74	even	39	inner	507.2.q.a.412.1	yes	360

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
507.2.q.a.16.1	✓	360	1.1	even	1	trivial
507.2.q.a.412.1	yes	360	169.74	even	39	inner