Embedded newform 507.2.t.b.4.13

• Label: 507.2.t.b.4.13
• Level: $507$
• Weight: $2$
• Character: 507.4
• 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.t (of order \(78\), degree \(24\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.04841538248\)
Analytic rank:	\(0\)
Dimension:	\(360\)
Relative dimension:	\(15\) over \(\Q(\zeta_{78})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{78}]$

Embedding invariants

Embedding label			4.13
Character	\(\chi\)	\(=\)	507.4
Dual form			507.2.t.b.127.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.92708 + 0.0776588i) q^{2} +(-0.278217 + 0.960518i) q^{3} +(1.71410 + 0.138377i) q^{4} +(3.62781 + 1.37585i) q^{5} +(-0.610741 + 1.82939i) q^{6} +(1.17937 + 2.76808i) q^{7} +(-0.536697 - 0.0651668i) q^{8} +(-0.845190 - 0.534466i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 360 q - 15 q^{3} - 14 q^{4} + 3 q^{7} + 15 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(170\)	\(340\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{1}{78}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.92708	+	0.0776588i	1.36265	+	0.0549131i	0.710780	−	0.703415i	\(-0.248342\pi\)
							0.651874	+	0.758328i	\(0.273983\pi\)
\(3\)	−0.278217	+	0.960518i	−0.160629	+	0.554555i
							\(5\)	3.62781	+	1.37585i	1.62240	+	0.615297i	0.987546	−	0.157332i	\(-0.0502892\pi\)
0.634858	+	0.772629i	\(0.281058\pi\)
\(7\)	1.17937	+	2.76808i	0.445759	+	1.04623i	0.979656	+	0.200685i	\(0.0643168\pi\)
							−0.533897	+	0.845550i	\(0.679273\pi\)
\(11\)	−1.78622	−	2.82468i	−0.538566	−	0.851674i	0.460792	−	0.887508i	\(-0.347566\pi\)
							−0.999358	+	0.0358342i	\(0.988591\pi\)
\(13\)	−2.71174	−	2.37623i	−0.752101	−	0.659047i
							\(17\)	−2.15601	+	0.918591i	−0.522910	+	0.222791i	−0.637259	−	0.770650i	\(-0.719932\pi\)
0.114349	+	0.993441i	\(0.463522\pi\)
\(19\)	1.03386	−	0.596898i	0.237183	−	0.136938i	−0.376698	−	0.926336i	\(-0.622941\pi\)
							0.613881	+	0.789398i	\(0.289607\pi\)
\(23\)	4.67428	−	8.09610i	0.974655	−	1.68815i	0.293588	−	0.955932i	\(-0.405151\pi\)
							0.681067	−	0.732221i	\(-0.261516\pi\)
\(29\)	−0.242867	+	6.02667i	−0.0450992	+	1.11912i	0.805260	+	0.592922i	\(0.202026\pi\)
							−0.850359	+	0.526203i	\(0.823615\pi\)
\(31\)	0.596932	+	0.673797i	0.107212	+	0.121017i	0.799667	−	0.600444i	\(-0.205009\pi\)
							−0.692455	+	0.721461i	\(0.743471\pi\)
\(37\)	9.40469	−	1.91998i	1.54612	−	0.315643i	0.650168	−	0.759791i	\(-0.274699\pi\)
							0.895953	+	0.444148i	\(0.146494\pi\)
\(41\)	−3.14715	−	0.911583i	−0.491502	−	0.142365i	0.0232062	−	0.999731i	\(-0.492613\pi\)
							−0.514708	+	0.857365i	\(0.672100\pi\)
\(43\)	1.20677	−	5.91115i	0.184031	−	0.901442i	−0.777950	−	0.628326i	\(-0.783741\pi\)
							0.961981	−	0.273116i	\(-0.0880543\pi\)
\(47\)	0.455761	−	0.314589i	0.0664795	−	0.0458875i	−0.534361	−	0.845256i	\(-0.679448\pi\)
							0.600841	+	0.799369i	\(0.294832\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	507.2.t.b.4.13	✓	360
169.127	even	78	inner	507.2.t.b.127.13	yes	360

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
507.2.t.b.4.13	✓	360	1.1	even	1	trivial
507.2.t.b.127.13	yes	360	169.127	even	78	inner