Embedded newform 507.2.q.b.16.2

• Label: 507.2.q.b.16.2
• 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			16.2
Character	\(\chi\)	\(=\)	507.16
Dual form			507.2.q.b.412.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.48128 - 0.200310i) q^{2} +(-0.845190 - 0.534466i) q^{3} +(4.14255 + 0.673230i) q^{4} +(1.36223 + 1.20683i) q^{5} +(1.99010 + 1.49546i) q^{6} +(-0.288513 + 0.300373i) q^{7} +(-5.30995 - 1.30878i) q^{8} +(0.428693 + 0.903450i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 384 q + q^{2} + 16 q^{3} + 17 q^{4} + 6 q^{5} + q^{6} - 3 q^{7} - 18 q^{8} + 16 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.48128	−	0.200310i	−1.75453	−	0.141641i	−0.839065	−	0.544031i	\(-0.816897\pi\)
							−0.915468	+	0.402391i	\(0.868179\pi\)
\(3\)	−0.845190	−	0.534466i	−0.487971	−	0.308574i
							\(5\)	1.36223	+	1.20683i	0.609208	+	0.539711i	0.910335	−	0.413873i	\(-0.135824\pi\)
−0.301127	+	0.953584i	\(0.597363\pi\)
\(7\)	−0.288513	+	0.300373i	−0.109048	+	0.113530i	−0.773489	−	0.633810i	\(-0.781490\pi\)
							0.664441	+	0.747341i	\(0.268670\pi\)
\(11\)	−1.77558	+	3.74196i	−0.535359	+	1.12824i	0.438004	+	0.898973i	\(0.355686\pi\)
							−0.973363	+	0.229271i	\(0.926366\pi\)
\(13\)	0.792477	+	3.51738i	0.219794	+	0.975546i
							\(17\)	−3.70811	+	3.86055i	−0.899349	+	0.936321i	−0.998271	−	0.0587741i	\(-0.981281\pi\)
0.0989228	+	0.995095i	\(0.468460\pi\)
\(19\)	3.81106	−	6.60095i	0.874317	−	1.51436i	0.0168294	−	0.999858i	\(-0.494643\pi\)
							0.857488	−	0.514504i	\(-0.172024\pi\)
\(23\)	−3.75359	−	6.50141i	−0.782678	−	1.35564i	−0.930376	−	0.366606i	\(-0.880520\pi\)
							0.147698	−	0.989032i	\(-0.452814\pi\)
\(29\)	−6.01096	−	0.485255i	−1.11621	−	0.0901095i	−0.491433	−	0.870915i	\(-0.663527\pi\)
							−0.624774	+	0.780806i	\(0.714809\pi\)
\(31\)	−0.756841	+	6.23315i	−0.135933	+	1.11951i	0.751879	+	0.659301i	\(0.229148\pi\)
							−0.887812	+	0.460206i	\(0.847776\pi\)
\(37\)	−8.29979	+	3.53621i	−1.36448	+	0.581349i	−0.945195	−	0.326507i	\(-0.894128\pi\)
							−0.419282	+	0.907856i	\(0.637718\pi\)
\(41\)	4.53244	+	2.86614i	0.707847	+	0.447616i	0.839273	−	0.543710i	\(-0.182981\pi\)
							−0.131425	+	0.991326i	\(0.541955\pi\)
\(43\)	0.270753	+	0.115357i	0.0412895	+	0.0175918i	0.412513	−	0.910952i	\(-0.364651\pi\)
							−0.371223	+	0.928544i	\(0.621062\pi\)
\(47\)	−1.26330	+	3.33106i	−0.184272	+	0.485884i	−0.995039	−	0.0994838i	\(-0.968281\pi\)
							0.810768	+	0.585368i	\(0.199050\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	507.2.q.b.16.2	✓	384
169.74	even	39	inner	507.2.q.b.412.2	yes	384

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
507.2.q.b.16.2	✓	384	1.1	even	1	trivial
507.2.q.b.412.2	yes	384	169.74	even	39	inner