Embedded newform 77.4.f.b.36.1

• Label: 77.4.f.b.36.1
• 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			36.1
Character	\(\chi\)	\(=\)	77.36
Dual form			77.4.f.b.15.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-4.23611 + 3.07771i) q^{2} +(-2.01219 - 6.19288i) q^{3} +(6.00015 - 18.4666i) q^{4} +(-14.2597 - 10.3603i) q^{5} +(27.5837 + 20.0407i) q^{6} +(-2.16312 + 6.65740i) q^{7} +(18.4731 + 56.8544i) q^{8} +(-12.4593 + 9.05225i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q + 8 q^{2} - 18 q^{3} - 34 q^{4} - 24 q^{5} + 30 q^{6} + 70 q^{7} - 72 q^{8} - 136 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−4.23611	+	3.07771i	−1.49769	+	1.08814i	−0.526400	+	0.850237i	\(0.676458\pi\)
							−0.971290	+	0.237898i	\(0.923542\pi\)
\(3\)	−2.01219	−	6.19288i	−0.387246	−	1.19182i	−0.934838	−	0.355074i	\(-0.884456\pi\)
							0.547593	−	0.836745i	\(-0.315544\pi\)
\(5\)	−14.2597	−	10.3603i	−1.27543	−	0.926655i	−0.276026	−	0.961150i	\(-0.589018\pi\)
							−0.999405	+	0.0344955i	\(0.989018\pi\)
\(7\)	−2.16312	+	6.65740i	−0.116797	+	0.359466i
							\(11\)	−18.3432	+	31.5361i	−0.502788	+	0.864410i
\(13\)	27.0156	−	19.6280i	0.576368	−	0.418756i								−0.261045	−	0.965327i	\(-0.584067\pi\)
							0.837413	+	0.546571i	\(0.184067\pi\)
\(17\)	−68.3035	−	49.6254i	−0.974473	−	0.707996i	−0.0180063	−	0.999838i	\(-0.505732\pi\)
							−0.956467	+	0.291842i	\(0.905732\pi\)
\(19\)	32.6572	+	100.509i	0.394320	+	1.21359i	0.929490	+	0.368847i	\(0.120247\pi\)
							−0.535170	+	0.844744i	\(0.679753\pi\)
\(23\)	126.631			1.14802			0.574010	−	0.818848i	\(-0.305387\pi\)
							0.574010	+	0.818848i	\(0.305387\pi\)
\(29\)	−62.5299	+	192.447i	−0.400397	+	1.23229i	0.524282	+	0.851545i	\(0.324334\pi\)
							−0.924678	+	0.380749i	\(0.875666\pi\)
\(31\)	32.2140	−	23.4048i	0.186639	−	0.135601i	−0.490542	−	0.871417i	\(-0.663201\pi\)
							0.677181	+	0.735816i	\(0.263201\pi\)
\(37\)	2.49384	−	7.67524i	0.0110807	−	0.0341028i	−0.945363	−	0.326019i	\(-0.894293\pi\)
							0.956444	+	0.291916i	\(0.0942928\pi\)
\(41\)	80.0895	+	246.490i	0.305070	+	0.938910i	0.979651	+	0.200709i	\(0.0643245\pi\)
							−0.674581	+	0.738201i	\(0.735675\pi\)
\(43\)	−256.168			−0.908494			−0.454247	−	0.890876i	\(-0.650092\pi\)
							−0.454247	+	0.890876i	\(0.650092\pi\)
\(47\)	−48.6504	−	149.731i	−0.150987	−	0.464690i	0.846745	−	0.531999i	\(-0.178559\pi\)
							−0.997732	+	0.0673083i	\(0.978559\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	77.4.f.b.36.1	yes	40
11.2	odd	10		847.4.a.r.1.1		20
11.4	even	5	inner	77.4.f.b.15.1	✓	40
11.9	even	5		847.4.a.q.1.20		20

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
77.4.f.b.15.1	✓	40	11.4	even	5	inner
77.4.f.b.36.1	yes	40	1.1	even	1	trivial
847.4.a.q.1.20		20	11.9	even	5
847.4.a.r.1.1		20	11.2	odd	10