Embedded newform 77.4.f.b.36.3

• Label: 77.4.f.b.36.3
• 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			36.3
Character	\(\chi\)	\(=\)	77.36
Dual form			77.4.f.b.15.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.08555 + 1.51524i) q^{2} +(-1.27325 - 3.91867i) q^{3} +(-0.418581 + 1.28826i) q^{4} +(0.733295 + 0.532770i) q^{5} +(8.59314 + 6.24328i) q^{6} +(-2.16312 + 6.65740i) q^{7} +(-7.45191 - 22.9346i) q^{8} +(8.10867 - 5.89129i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q + 8 q^{2} - 18 q^{3} - 34 q^{4} - 24 q^{5} + 30 q^{6} + 70 q^{7} - 72 q^{8} - 136 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.08555	+	1.51524i	−0.737352	+	0.535717i	−0.891881	−	0.452271i	\(-0.850614\pi\)
							0.154529	+	0.987988i	\(0.450614\pi\)
\(3\)	−1.27325	−	3.91867i	−0.245038	−	0.754148i	−0.995630	−	0.0933829i	\(-0.970232\pi\)
							0.750593	−	0.660765i	\(-0.229768\pi\)
\(5\)	0.733295	+	0.532770i	0.0655879	+	0.0476524i	0.620096	−	0.784526i	\(-0.287094\pi\)
							−0.554508	+	0.832178i	\(0.687094\pi\)
\(7\)	−2.16312	+	6.65740i	−0.116797	+	0.359466i
							\(11\)	11.5028	−	34.6220i	0.315293	−	0.948994i
\(13\)	58.7010	−	42.6488i	1.25236	−	0.909896i								0.254007	−	0.967202i	\(-0.418251\pi\)
							0.998357	+	0.0573068i	\(0.0182513\pi\)
\(17\)	15.4736	+	11.2422i	0.220758	+	0.160390i	0.692668	−	0.721257i	\(-0.256435\pi\)
							−0.471910	+	0.881647i	\(0.656435\pi\)
\(19\)	−5.67667	−	17.4710i	−0.0685430	−	0.210954i	0.910918	−	0.412588i	\(-0.135375\pi\)
							−0.979461	+	0.201634i	\(0.935375\pi\)
\(23\)	113.500			1.02897			0.514484	−	0.857500i	\(-0.327983\pi\)
							0.514484	+	0.857500i	\(0.327983\pi\)
\(29\)	47.0769	−	144.888i	0.301447	−	0.927757i	−0.679533	−	0.733645i	\(-0.737817\pi\)
							0.980979	−	0.194112i	\(-0.0621826\pi\)
\(31\)	−216.164	+	157.052i	−1.25239	+	0.909915i	−0.998358	−	0.0572804i	\(-0.981757\pi\)
							−0.254033	+	0.967196i	\(0.581757\pi\)
\(37\)	−128.704	+	396.112i	−0.571862	+	1.76001i	0.0747630	+	0.997201i	\(0.476180\pi\)
							−0.646625	+	0.762808i	\(0.723820\pi\)
\(41\)	−88.6091	−	272.711i	−0.337522	−	1.03879i	−0.965466	−	0.260528i	\(-0.916103\pi\)
							0.627944	−	0.778259i	\(-0.283897\pi\)
\(43\)	164.598			0.583743			0.291871	−	0.956458i	\(-0.405722\pi\)
							0.291871	+	0.956458i	\(0.405722\pi\)
\(47\)	30.0699	+	92.5455i	0.0933222	+	0.287216i	0.986813	−	0.161866i	\(-0.0517512\pi\)
							−0.893491	+	0.449082i	\(0.851751\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	77.4.f.b.36.3	yes	40
11.2	odd	10		847.4.a.r.1.5		20
11.4	even	5	inner	77.4.f.b.15.3	✓	40
11.9	even	5		847.4.a.q.1.16		20

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
77.4.f.b.15.3	✓	40	11.4	even	5	inner
77.4.f.b.36.3	yes	40	1.1	even	1	trivial
847.4.a.q.1.16		20	11.9	even	5
847.4.a.r.1.5		20	11.2	odd	10