Embedded newform 507.2.t.b.127.3

• Label: 507.2.t.b.127.3
• Level: $507$
• Weight: $2$
• Character: 507.127
• 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			127.3
Character	\(\chi\)	\(=\)	507.127
Dual form			507.2.t.b.4.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.85726 + 0.0748449i) q^{2} +(-0.278217 - 0.960518i) q^{3} +(1.45029 - 0.117079i) q^{4} +(-1.06783 + 0.404975i) q^{5} +(0.588611 + 1.76311i) q^{6} +(0.332955 - 0.781475i) q^{7} +(1.00563 - 0.122106i) q^{8} +(-0.845190 + 0.534466i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 360 q - 15 q^{3} - 14 q^{4} + 3 q^{7} + 15 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(170\)	\(340\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{77}{78}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.85726	+	0.0748449i	−1.31328	+	0.0529233i	−0.686989	−	0.726667i	\(-0.741068\pi\)
							−0.626289	+	0.779591i	\(0.715427\pi\)
\(3\)	−0.278217	−	0.960518i	−0.160629	−	0.554555i
							\(5\)	−1.06783	+	0.404975i	−0.477548	+	0.181110i	−0.581622	−	0.813459i	\(-0.697582\pi\)
0.104073	+	0.994570i	\(0.466812\pi\)
\(7\)	0.332955	−	0.781475i	0.125845	−	0.295370i	−0.845125	−	0.534568i	\(-0.820474\pi\)
							0.970971	+	0.239198i	\(0.0768846\pi\)
\(11\)	−2.33405	+	3.69100i	−0.703742	+	1.11288i	0.284306	+	0.958734i	\(0.408237\pi\)
							−0.988048	+	0.154146i	\(0.950737\pi\)
\(13\)	−0.346984	−	3.58882i	−0.0962361	−	0.995359i
							\(17\)	−2.27717	−	0.970211i	−0.552295	−	0.235311i	0.0977444	−	0.995212i	\(-0.468837\pi\)
−0.650039	+	0.759901i	\(0.725247\pi\)
\(19\)	4.47551	+	2.58394i	1.02675	+	0.592796i	0.916053	−	0.401058i	\(-0.131357\pi\)
							0.110700	+	0.993854i	\(0.464691\pi\)
\(23\)	−0.0828629	−	0.143523i	−0.0172781	−	0.0299266i	0.857257	−	0.514889i	\(-0.172167\pi\)
							−0.874535	+	0.484962i	\(0.838833\pi\)
\(29\)	0.381267	+	9.46104i	0.0707995	+	1.75687i	0.511560	+	0.859247i	\(0.329068\pi\)
							−0.440761	+	0.897625i	\(0.645291\pi\)
\(31\)	4.15421	−	4.68913i	0.746117	−	0.842192i	−0.245768	−	0.969329i	\(-0.579040\pi\)
							0.991885	+	0.127136i	\(0.0405786\pi\)
\(37\)	5.07334	+	1.03573i	0.834052	+	0.170273i	0.598029	−	0.801475i	\(-0.295951\pi\)
							0.236023	+	0.971747i	\(0.424156\pi\)
\(41\)	−0.363440	+	0.105272i	−0.0567599	+	0.0164407i	−0.306476	−	0.951878i	\(-0.599150\pi\)
							0.249716	+	0.968319i	\(0.419663\pi\)
\(43\)	0.404977	+	1.98371i	0.0617584	+	0.302513i	0.998924	−	0.0463715i	\(-0.0147658\pi\)
							−0.937166	+	0.348884i	\(0.886561\pi\)
\(47\)	10.1830	+	7.02879i	1.48534	+	1.02525i	0.987436	+	0.158020i	\(0.0505111\pi\)
							0.497901	+	0.867234i	\(0.334104\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	507.2.t.b.127.3	yes	360
169.4	even	78	inner	507.2.t.b.4.3	✓	360

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
507.2.t.b.4.3	✓	360	169.4	even	78	inner
507.2.t.b.127.3	yes	360	1.1	even	1	trivial