Embedded newform 507.2.q.a.55.4

• Label: 507.2.q.a.55.4
• Level: $507$
• Weight: $2$
• Character: 507.55
• 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			55.4
Character	\(\chi\)	\(=\)	507.55
Dual form			507.2.q.a.295.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.18444 + 1.45068i) q^{2} +(0.996757 + 0.0804666i) q^{3} +(-0.301514 - 1.47691i) q^{4} +(0.266003 + 2.19074i) q^{5} +(-1.29733 + 1.35067i) q^{6} +(-0.690655 + 0.436744i) q^{7} +(-0.816908 - 0.428746i) q^{8} +(0.987050 + 0.160411i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 360 q + q^{2} - 15 q^{3} + 15 q^{4} + 2 q^{5} - q^{6} + 2 q^{7} + 6 q^{8} + 15 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(170\)	\(340\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{28}{39}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.18444	+	1.45068i	−0.837528	+	1.02579i	0.161713	+	0.986838i	\(0.448298\pi\)
							−0.999241	+	0.0389476i	\(0.987599\pi\)
\(3\)	0.996757	+	0.0804666i	0.575478	+	0.0464574i
							\(5\)	0.266003	+	2.19074i	0.118960	+	0.979727i	0.922906	+	0.385024i	\(0.125807\pi\)
−0.803946	+	0.594702i	\(0.797270\pi\)
\(7\)	−0.690655	+	0.436744i	−0.261043	+	0.165074i	−0.658576	−	0.752514i	\(-0.728841\pi\)
							0.397533	+	0.917588i	\(0.369867\pi\)
\(11\)	−4.18193	+	0.679630i	−1.26090	+	0.204916i	−0.753877	−	0.657016i	\(-0.771818\pi\)
							−0.507024	+	0.861932i	\(0.669254\pi\)
\(13\)	−2.97288	−	2.04009i	−0.824529	−	0.565820i
							\(17\)	−2.90665	+	1.83806i	−0.704967	+	0.445794i	−0.838252	−	0.545284i	\(-0.816422\pi\)
0.133284	+	0.991078i	\(0.457448\pi\)
\(19\)	−0.845977	+	1.46528i	−0.194080	+	0.336157i	−0.946599	−	0.322414i	\(-0.895506\pi\)
							0.752518	+	0.658571i	\(0.228839\pi\)
\(23\)	2.34548	+	4.06249i	0.489067	+	0.847088i	0.999921	−	0.0125789i	\(-0.00400410\pi\)
							−0.510854	+	0.859667i	\(0.670671\pi\)
\(29\)	−2.19327	+	2.68627i	−0.407281	+	0.498828i	−0.936993	−	0.349349i	\(-0.886403\pi\)
							0.529712	+	0.848178i	\(0.322300\pi\)
\(31\)	−6.36143	−	1.56795i	−1.14255	−	0.281612i	−0.377764	−	0.925902i	\(-0.623307\pi\)
							−0.764782	+	0.644290i	\(0.777153\pi\)
\(37\)	0.854022	+	2.94843i	0.140400	+	0.484718i	0.999639	−	0.0268562i	\(-0.00854961\pi\)
							−0.859239	+	0.511574i	\(0.829062\pi\)
\(41\)	1.63292	+	0.131823i	0.255019	+	0.0205873i	0.207315	−	0.978274i	\(-0.433527\pi\)
							0.0477041	+	0.998862i	\(0.484810\pi\)
\(43\)	1.06821	−	3.68789i	0.162901	−	0.562398i	−0.836984	−	0.547227i	\(-0.815683\pi\)
							0.999885	−	0.0151714i	\(-0.00482939\pi\)
\(47\)	1.18546	+	1.05023i	0.172917	+	0.153192i	0.745143	−	0.666905i	\(-0.232381\pi\)
							−0.572225	+	0.820096i	\(0.693920\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	507.2.q.a.55.4	✓	360
169.126	even	39	inner	507.2.q.a.295.4	yes	360

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
507.2.q.a.55.4	✓	360	1.1	even	1	trivial
507.2.q.a.295.4	yes	360	169.126	even	39	inner