Embedded newform 507.2.t.b.4.14

• Label: 507.2.t.b.4.14
• 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			4.14
Character	\(\chi\)	\(=\)	507.4
Dual form			507.2.t.b.127.14

$q$-expansion

\(f(q)\)	\(=\)	\(q+(2.27767 + 0.0917871i) q^{2} +(-0.278217 + 0.960518i) q^{3} +(3.18585 + 0.257189i) q^{4} +(-1.29798 - 0.492260i) q^{5} +(-0.721851 + 2.16221i) q^{6} +(1.28355 + 3.01261i) q^{7} +(2.70692 + 0.328679i) q^{8} +(-0.845190 - 0.534466i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 360 q - 15 q^{3} - 14 q^{4} + 3 q^{7} + 15 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	2.27767	+	0.0917871i	1.61056	+	0.0649032i	0.829116	−	0.559076i	\(-0.188844\pi\)
							0.781441	+	0.623979i	\(0.214485\pi\)
\(3\)	−0.278217	+	0.960518i	−0.160629	+	0.554555i
							\(5\)	−1.29798	−	0.492260i	−0.580476	−	0.220146i	0.0468504	−	0.998902i	\(-0.485082\pi\)
−0.627326	+	0.778756i	\(0.715851\pi\)
\(7\)	1.28355	+	3.01261i	0.485138	+	1.13866i	0.965221	+	0.261437i	\(0.0841963\pi\)
							−0.480083	+	0.877223i	\(0.659393\pi\)
\(11\)	1.23690	+	1.95600i	0.372938	+	0.589755i	0.978551	−	0.206005i	\(-0.0660462\pi\)
							−0.605613	+	0.795760i	\(0.707072\pi\)
\(13\)	2.73525	+	2.34913i	0.758621	+	0.651533i
							\(17\)	4.22227	−	1.79894i	1.02405	−	0.436307i	0.186377	−	0.982478i	\(-0.440326\pi\)
0.837673	+	0.546171i	\(0.183915\pi\)
\(19\)	−1.50270	+	0.867585i	−0.344743	+	0.199038i	−0.662368	−	0.749179i	\(-0.730448\pi\)
							0.317624	+	0.948217i	\(0.397115\pi\)
\(23\)	4.00318	−	6.93371i	0.834720	−	1.44578i	−0.0595375	−	0.998226i	\(-0.518963\pi\)
							0.894258	−	0.447552i	\(-0.147704\pi\)
\(29\)	0.0925207	−	2.29588i	0.0171807	−	0.426334i	−0.967595	−	0.252506i	\(-0.918745\pi\)
							0.984776	−	0.173828i	\(-0.0556137\pi\)
\(31\)	−2.88098	−	3.25196i	−0.517440	−	0.584069i	0.430471	−	0.902604i	\(-0.358347\pi\)
							−0.947911	+	0.318536i	\(0.896809\pi\)
\(37\)	−11.7498	+	2.39873i	−1.93165	+	0.394349i	−0.932171	+	0.362018i	\(0.882088\pi\)
							−0.999477	+	0.0323306i	\(0.989707\pi\)
\(41\)	−1.82716	−	0.529244i	−0.285355	−	0.0826540i	0.132454	−	0.991189i	\(-0.457714\pi\)
							−0.417808	+	0.908535i	\(0.637202\pi\)
\(43\)	0.284648	−	1.39430i	0.0434084	−	0.212628i	−0.952301	−	0.305159i	\(-0.901290\pi\)
							0.995710	+	0.0925306i	\(0.0294956\pi\)
\(47\)	8.22934	−	5.68030i	1.20037	−	0.828558i	0.211508	−	0.977376i	\(-0.432162\pi\)
							0.988864	+	0.148819i	\(0.0475471\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	507.2.t.b.4.14	✓	360
169.127	even	78	inner	507.2.t.b.127.14	yes	360

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
507.2.t.b.4.14	✓	360	1.1	even	1	trivial
507.2.t.b.127.14	yes	360	169.127	even	78	inner