Embedded newform 77.4.f.b.36.6

• Label: 77.4.f.b.36.6
• 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			36.6
Character	\(\chi\)	\(=\)	77.36
Dual form			77.4.f.b.15.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.06715 - 0.775331i) q^{2} +(2.12611 + 6.54350i) q^{3} +(-1.93446 + 5.95366i) q^{4} +(-8.06046 - 5.85627i) q^{5} +(7.34226 + 5.33447i) q^{6} +(-2.16312 + 6.65740i) q^{7} +(5.81262 + 17.8894i) q^{8} +(-16.4536 + 11.9543i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q + 8 q^{2} - 18 q^{3} - 34 q^{4} - 24 q^{5} + 30 q^{6} + 70 q^{7} - 72 q^{8} - 136 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.06715	−	0.775331i	0.377295	−	0.274121i	−0.382935	−	0.923775i	\(-0.625087\pi\)
							0.760229	+	0.649655i	\(0.225087\pi\)
\(3\)	2.12611	+	6.54350i	0.409171	+	1.25930i	0.917362	+	0.398054i	\(0.130314\pi\)
							−0.508191	+	0.861244i	\(0.669686\pi\)
\(5\)	−8.06046	−	5.85627i	−0.720950	−	0.523801i	0.165738	−	0.986170i	\(-0.446999\pi\)
							−0.886687	+	0.462369i	\(0.846999\pi\)
\(7\)	−2.16312	+	6.65740i	−0.116797	+	0.359466i
							\(11\)	−29.4414	+	21.5454i	−0.806992	+	0.590562i
\(13\)	24.2371	−	17.6093i	0.517090	−	0.375688i								−0.298417	−	0.954436i	\(-0.596459\pi\)
							0.815506	+	0.578748i	\(0.196459\pi\)
\(17\)	73.5201	+	53.4154i	1.04890	+	0.762068i	0.972002	−	0.234972i	\(-0.0754997\pi\)
							0.0768939	+	0.997039i	\(0.475500\pi\)
\(19\)	21.0053	+	64.6475i	0.253628	+	0.780588i	0.994097	+	0.108497i	\(0.0346036\pi\)
							−0.740469	+	0.672091i	\(0.765396\pi\)
\(23\)	200.229			1.81524			0.907621	−	0.419790i	\(-0.137896\pi\)
							0.907621	+	0.419790i	\(0.137896\pi\)
\(29\)	73.6081	−	226.542i	0.471334	−	1.45062i	−0.379506	−	0.925189i	\(-0.623906\pi\)
							0.850839	−	0.525426i	\(-0.176094\pi\)
\(31\)	−134.529	+	97.7414i	−0.779426	+	0.566286i	−0.904807	−	0.425822i	\(-0.859985\pi\)
							0.125381	+	0.992109i	\(0.459985\pi\)
\(37\)	72.9850	−	224.625i	0.324288	−	0.998056i	−0.647473	−	0.762088i	\(-0.724174\pi\)
							0.971761	−	0.235967i	\(-0.0758259\pi\)
\(41\)	39.5817	+	121.820i	0.150771	+	0.464027i	0.997708	−	0.0676672i	\(-0.0215556\pi\)
							−0.846936	+	0.531694i	\(0.821556\pi\)
\(43\)	−360.013			−1.27678			−0.638389	−	0.769714i	\(-0.720399\pi\)
							−0.638389	+	0.769714i	\(0.720399\pi\)
\(47\)	148.434	+	456.832i	0.460666	+	1.41778i	0.864353	+	0.502886i	\(0.167728\pi\)
							−0.403687	+	0.914897i	\(0.632272\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	77.4.f.b.36.6	yes	40
11.2	odd	10		847.4.a.r.1.12		20
11.4	even	5	inner	77.4.f.b.15.6	✓	40
11.9	even	5		847.4.a.q.1.9		20

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
77.4.f.b.15.6	✓	40	11.4	even	5	inner
77.4.f.b.36.6	yes	40	1.1	even	1	trivial
847.4.a.q.1.9		20	11.9	even	5
847.4.a.r.1.12		20	11.2	odd	10