Embedded newform 77.4.f.b.36.8

• Label: 77.4.f.b.36.8
• Level: $77$
• Weight: $4$
• Character: 77.36
• Analytic conductor: $4.543$
• Analytic rank: $0$
• Dimension: $40$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 77 = 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	77.f (of order \(5\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.54314707044\)
Analytic rank:	\(0\)
Dimension:	\(40\)
Relative dimension:	\(10\) over \(\Q(\zeta_{5})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{5}]$

Embedding invariants

Embedding label			36.8
Character	\(\chi\)	\(=\)	77.36
Dual form			77.4.f.b.15.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(3.62073 - 2.63062i) q^{2} +(-0.489110 - 1.50533i) q^{3} +(3.71743 - 11.4411i) q^{4} +(-16.0185 - 11.6381i) q^{5} +(-5.73088 - 4.16373i) q^{6} +(-2.16312 + 6.65740i) q^{7} +(-5.57330 - 17.1529i) q^{8} +(19.8167 - 14.3977i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q + 8 q^{2} - 18 q^{3} - 34 q^{4} - 24 q^{5} + 30 q^{6} + 70 q^{7} - 72 q^{8} - 136 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/77\mathbb{Z}\right)^\times\).

\(n\)	\(45\)	\(57\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{4}{5}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	3.62073	−	2.63062i	1.28012	−	0.930064i	0.280566	−	0.959835i	\(-0.409478\pi\)
							0.999557	+	0.0297708i	\(0.00947776\pi\)
\(3\)	−0.489110	−	1.50533i	−0.0941293	−	0.289700i	0.892896	−	0.450262i	\(-0.148669\pi\)
							−0.987026	+	0.160562i	\(0.948669\pi\)
\(5\)	−16.0185	−	11.6381i	−1.43273	−	1.04094i	−0.989499	−	0.144537i	\(-0.953831\pi\)
							−0.443235	−	0.896405i	\(-0.646169\pi\)
\(7\)	−2.16312	+	6.65740i	−0.116797	+	0.359466i
							\(11\)	34.7313	+	11.1686i	0.951989	+	0.306132i
\(13\)	36.5036	−	26.5214i	0.778790	−	0.565824i								−0.125826	−	0.992052i	\(-0.540158\pi\)
							0.904616	+	0.426229i	\(0.140158\pi\)
\(17\)	61.3228	+	44.5536i	0.874880	+	0.635637i	0.931892	−	0.362736i	\(-0.118157\pi\)
							−0.0570120	+	0.998373i	\(0.518157\pi\)
\(19\)	−24.6641	−	75.9082i	−0.297807	−	0.916555i	−0.982264	−	0.187502i	\(-0.939961\pi\)
							0.684458	−	0.729053i	\(-0.260039\pi\)
\(23\)	−175.226			−1.58857			−0.794285	−	0.607545i	\(-0.792154\pi\)
							−0.794285	+	0.607545i	\(0.792154\pi\)
\(29\)	−34.8337	+	107.207i	−0.223050	+	0.686478i	0.775434	+	0.631429i	\(0.217531\pi\)
							−0.998484	+	0.0550484i	\(0.982469\pi\)
\(31\)	−96.2768	+	69.9492i	−0.557801	+	0.405266i	−0.830653	−	0.556790i	\(-0.812033\pi\)
							0.272853	+	0.962056i	\(0.412033\pi\)
\(37\)	78.8675	−	242.729i	0.350426	−	1.07850i	−0.608189	−	0.793792i	\(-0.708104\pi\)
							0.958615	−	0.284707i	\(-0.0918962\pi\)
\(41\)	55.8491	+	171.886i	0.212736	+	0.654734i	0.999307	+	0.0372335i	\(0.0118545\pi\)
							−0.786571	+	0.617500i	\(0.788145\pi\)
\(43\)	308.087			1.09262			0.546312	−	0.837582i	\(-0.316031\pi\)
							0.546312	+	0.837582i	\(0.316031\pi\)
\(47\)	155.666	+	479.091i	0.483112	+	1.48687i	0.834697	+	0.550709i	\(0.185643\pi\)
							−0.351585	+	0.936156i	\(0.614357\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	77.4.f.b.36.8	yes	40
11.2	odd	10		847.4.a.r.1.17		20
11.4	even	5	inner	77.4.f.b.15.8	✓	40
11.9	even	5		847.4.a.q.1.4		20

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
77.4.f.b.15.8	✓	40	11.4	even	5	inner
77.4.f.b.36.8	yes	40	1.1	even	1	trivial
847.4.a.q.1.4		20	11.9	even	5
847.4.a.r.1.17		20	11.2	odd	10