Embedded newform 507.2.q.b.16.11

• Label: 507.2.q.b.16.11
• 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			16.11
Character	\(\chi\)	\(=\)	507.16
Dual form			507.2.q.b.412.11

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.910598 + 0.0735111i) q^{2} +(-0.845190 - 0.534466i) q^{3} +(-1.15032 - 0.186944i) q^{4} +(3.23408 + 2.86515i) q^{5} +(-0.730339 - 0.548814i) q^{6} +(-1.27940 + 1.33200i) q^{7} +(-2.80776 - 0.692051i) q^{8} +(0.428693 + 0.903450i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 384 q + q^{2} + 16 q^{3} + 17 q^{4} + 6 q^{5} + q^{6} - 3 q^{7} - 18 q^{8} + 16 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.910598	+	0.0735111i	0.643890	+	0.0519802i	0.398099	−	0.917343i	\(-0.369670\pi\)
							0.245791	+	0.969323i	\(0.420952\pi\)
\(3\)	−0.845190	−	0.534466i	−0.487971	−	0.308574i
							\(5\)	3.23408	+	2.86515i	1.44632	+	1.28133i	0.894273	+	0.447523i	\(0.147694\pi\)
0.552052	+	0.833810i	\(0.313845\pi\)
\(7\)	−1.27940	+	1.33200i	−0.483568	+	0.503448i	−0.917630	−	0.397435i	\(-0.869900\pi\)
							0.434062	+	0.900883i	\(0.357080\pi\)
\(11\)	0.424127	−	0.893828i	0.127879	−	0.269499i	−0.829405	−	0.558647i	\(-0.811321\pi\)
							0.957284	+	0.289148i	\(0.0933719\pi\)
\(13\)	−0.603766	+	3.55464i	−0.167455	+	0.985880i
							\(17\)	−2.25266	+	2.34527i	−0.546350	+	0.568811i	−0.935932	−	0.352181i	\(-0.885440\pi\)
0.389582	+	0.920992i	\(0.372620\pi\)
\(19\)	−1.52309	+	2.63806i	−0.349420	+	0.605213i	−0.986146	−	0.165877i	\(-0.946955\pi\)
							0.636727	+	0.771090i	\(0.280288\pi\)
\(23\)	−0.353527	−	0.612328i	−0.0737156	−	0.127679i	0.826811	−	0.562479i	\(-0.190152\pi\)
							−0.900527	+	0.434800i	\(0.856819\pi\)
\(29\)	7.23593	+	0.584145i	1.34368	+	0.108473i	0.731280	−	0.682077i	\(-0.238923\pi\)
							0.612398	+	0.790550i	\(0.290205\pi\)
\(31\)	0.0198237	−	0.163263i	0.00356044	−	0.0293228i	−0.990820	−	0.135185i	\(-0.956837\pi\)
							0.994381	+	0.105862i	\(0.0337602\pi\)
\(37\)	6.37129	−	2.71455i	1.04743	−	0.446270i	0.201461	−	0.979497i	\(-0.435431\pi\)
							0.845972	+	0.533227i	\(0.179021\pi\)
\(41\)	−1.39615	−	0.882871i	−0.218042	−	0.137881i	0.420980	−	0.907070i	\(-0.361686\pi\)
							−0.639022	+	0.769189i	\(0.720661\pi\)
\(43\)	−0.668250	−	0.284715i	−0.101907	−	0.0434186i	0.340411	−	0.940277i	\(-0.389434\pi\)
							−0.442319	+	0.896858i	\(0.645844\pi\)
\(47\)	1.35726	−	3.57879i	0.197976	−	0.522020i	−0.798859	−	0.601518i	\(-0.794563\pi\)
							0.996835	+	0.0794985i	\(0.0253319\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	507.2.q.b.16.11	✓	384
169.74	even	39	inner	507.2.q.b.412.11	yes	384

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
507.2.q.b.16.11	✓	384	1.1	even	1	trivial
507.2.q.b.412.11	yes	384	169.74	even	39	inner