Embedded newform 786.2.o.a.185.12

• Label: 786.2.o.a.185.12
• Level: $786$
• Weight: $2$
• Character: 786.185
• 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			185.12
Character	\(\chi\)	\(=\)	786.185
Dual form			786.2.o.a.17.12

$q$-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 \(\left(\mathbb{Z}/786\mathbb{Z}\right)^\times\).

\(n\)	\(133\)	\(263\)
\(\chi(n)\)	\(e\left(\frac{87}{130}\right)\)	\(-1\)

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

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
786.2.o.a.17.12	✓	1056	393.17	even	130	inner
786.2.o.a.185.12	yes	1056	1.1	even	1	trivial
786.2.o.b.17.3	yes	1056	131.17	odd	130
786.2.o.b.185.3	yes	1056	3.2	odd	2