Embedded newform 786.2.o.a.17.12

• Label: 786.2.o.a.17.12
• Level: $786$
• Weight: $2$
• Character: 786.17
• Analytic conductor: $6.276$
• Analytic rank: $0$
• Dimension: $1056$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 786 = 2 \cdot 3 \cdot 131 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	786.o (of order \(130\), degree \(48\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(6.27624159887\)
Analytic rank:	\(0\)
Dimension:	\(1056\)
Relative dimension:	\(22\) over \(\Q(\zeta_{130})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{130}]$

Embedding invariants

Embedding label			17.12
Character	\(\chi\)	\(=\)	786.17
Dual form			786.2.o.a.185.12

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.861970 - 0.506960i) q^{2} +(0.115640 + 1.72819i) q^{3} +(0.485983 - 0.873968i) q^{4} +(-1.12802 + 1.40452i) q^{5} +(0.975799 + 1.43102i) q^{6} +(-1.22155 + 1.19238i) q^{7} +(-0.0241637 - 0.999708i) q^{8} +(-2.97325 + 0.399696i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1056 q - 22 q^{2} + q^{3} + 22 q^{4} - q^{6} - 2 q^{7} - 22 q^{8} + q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/786\mathbb{Z}\right)^\times\).

\(n\)	\(133\)	\(263\)
\(\chi(n)\)	\(e\left(\frac{43}{130}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.861970	−	0.506960i	0.609505	−	0.358475i
							\(3\)	0.115640	+	1.72819i	0.0667650	+	0.997769i
\(5\)	−1.12802	+	1.40452i	−0.504467	+	0.628120i								−0.965857	−	0.259077i	\(-0.916582\pi\)
							0.461390	+	0.887197i	\(0.347351\pi\)
\(7\)	−1.22155	+	1.19238i	−0.461701	+	0.450676i	−0.892017	−	0.452002i	\(-0.850710\pi\)
							0.430316	+	0.902679i	\(0.358402\pi\)
\(11\)	−3.68060	−	0.267305i	−1.10974	−	0.0805956i	−0.494680	−	0.869075i	\(-0.664715\pi\)
							−0.615061	+	0.788479i	\(0.710869\pi\)
\(13\)	−3.83443	+	0.559909i	−1.06348	+	0.155291i	−0.653594	−	0.756846i	\(-0.726739\pi\)
							−0.409887	+	0.912136i	\(0.634432\pi\)
\(17\)	2.51364	−	2.97936i	0.609647	−	0.722600i	−0.368705	−	0.929546i	\(-0.620199\pi\)
							0.978352	+	0.206946i	\(0.0663526\pi\)
\(19\)	−0.265851	+	1.07860i	−0.0609903	+	0.247448i	−0.993522	−	0.113641i	\(-0.963749\pi\)
							0.932532	+	0.361089i	\(0.117595\pi\)
\(23\)	−4.18772	−	1.47346i	−0.873200	−	0.307239i	−0.142382	−	0.989812i	\(-0.545476\pi\)
							−0.730817	+	0.682573i	\(0.760861\pi\)
\(29\)	1.65892	+	3.80827i	0.308054	+	0.707178i	0.999889	−	0.0148777i	\(-0.00473589\pi\)
							−0.691835	+	0.722056i	\(0.743197\pi\)
\(31\)	1.12262	−	3.76206i	0.201628	−	0.675685i	−0.795901	−	0.605427i	\(-0.793002\pi\)
							0.997529	−	0.0702581i	\(-0.0223823\pi\)
\(37\)	−1.98229	+	10.1254i	−0.325887	+	1.66461i	0.357016	+	0.934098i	\(0.383794\pi\)
							−0.682903	+	0.730509i	\(0.739283\pi\)
\(41\)	−4.03694	−	2.50751i	−0.630464	−	0.391608i	0.174732	−	0.984616i	\(-0.444094\pi\)
							−0.805196	+	0.593008i	\(0.797940\pi\)
\(43\)	12.4454	+	0.601982i	1.89791	+	0.0918014i	0.964029	−	0.265796i	\(-0.0856347\pi\)
							0.933876	+	0.357597i	\(0.116404\pi\)
\(47\)	2.35498	+	2.08633i	0.343509	+	0.304322i	0.817183	−	0.576378i	\(-0.195534\pi\)
							−0.473675	+	0.880700i	\(0.657073\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	786.2.o.a.17.12	✓	1056
3.2	odd	2		786.2.o.b.17.3	yes	1056
131.54	odd	130		786.2.o.b.185.3	yes	1056
393.185	even	130	inner	786.2.o.a.185.12	yes	1056

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
786.2.o.a.17.12	✓	1056	1.1	even	1	trivial
786.2.o.a.185.12	yes	1056	393.185	even	130	inner
786.2.o.b.17.3	yes	1056	3.2	odd	2
786.2.o.b.185.3	yes	1056	131.54	odd	130