Embedded newform 77.4.f.a.36.7

• Label: 77.4.f.a.36.7
• Level: $77$
• Weight: $4$
• Character: 77.36
• Analytic conductor: $4.543$
• Analytic rank: $0$
• Dimension: $32$
• 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:	\(32\)
Relative dimension:	\(8\) over \(\Q(\zeta_{5})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{5}]$

Embedding invariants

Embedding label			36.7
Character	\(\chi\)	\(=\)	77.36
Dual form			77.4.f.a.15.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(2.46158 - 1.78845i) q^{2} +(1.79340 + 5.51951i) q^{3} +(0.388722 - 1.19636i) q^{4} +(2.07002 + 1.50396i) q^{5} +(14.2859 + 10.3793i) q^{6} +(2.16312 - 6.65740i) q^{7} +(6.33917 + 19.5100i) q^{8} +(-5.40527 + 3.92716i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q - 2 q^{2} + 18 q^{3} - 48 q^{4} + 16 q^{5} + 30 q^{6} - 56 q^{7} + 62 q^{8} - 46 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	2.46158	−	1.78845i	0.870301	−	0.632311i	−0.0603665	−	0.998176i	\(-0.519227\pi\)
							0.930668	+	0.365865i	\(0.119227\pi\)
\(3\)	1.79340	+	5.51951i	0.345140	+	1.06223i	0.961509	+	0.274773i	\(0.0886028\pi\)
							−0.616369	+	0.787457i	\(0.711397\pi\)
\(5\)	2.07002	+	1.50396i	0.185149	+	0.134518i	0.676499	−	0.736444i	\(-0.263496\pi\)
							−0.491350	+	0.870962i	\(0.663496\pi\)
\(7\)	2.16312	−	6.65740i	0.116797	−	0.359466i
							\(11\)	25.8547	+	25.7398i	0.708679	+	0.705531i
\(13\)	−0.799209	+	0.580660i	−0.0170508	+	0.0123882i								−0.596278	−	0.802778i	\(-0.703354\pi\)
							0.579227	+	0.815166i	\(0.303354\pi\)
\(17\)	−67.4626	−	49.0144i	−0.962475	−	0.699279i	−0.00875075	−	0.999962i	\(-0.502785\pi\)
							−0.953724	+	0.300683i	\(0.902785\pi\)
\(19\)	−49.8957	−	153.563i	−0.602466	−	1.85420i	−0.513352	−	0.858178i	\(-0.671597\pi\)
							−0.0891134	−	0.996021i	\(-0.528403\pi\)
\(23\)	−16.5642			−0.150169			−0.0750844	−	0.997177i	\(-0.523923\pi\)
							−0.0750844	+	0.997177i	\(0.523923\pi\)
\(29\)	51.6713	−	159.028i	0.330866	−	1.01830i	−0.637856	−	0.770156i	\(-0.720179\pi\)
							0.968723	−	0.248146i	\(-0.0798214\pi\)
\(31\)	−80.0390	+	58.1517i	−0.463723	+	0.336915i	−0.794990	−	0.606622i	\(-0.792524\pi\)
							0.331267	+	0.943537i	\(0.392524\pi\)
\(37\)	−88.8253	+	273.376i	−0.394670	+	1.21467i	0.534548	+	0.845138i	\(0.320482\pi\)
							−0.929218	+	0.369531i	\(0.879518\pi\)
\(41\)	50.0585	+	154.064i	0.190679	+	0.586848i	1.00000		0.000517828i	\(-0.000164830\pi\)
							−0.809321	+	0.587366i	\(0.800165\pi\)
\(43\)	333.459			1.18261			0.591303	−	0.806450i	\(-0.298614\pi\)
							0.591303	+	0.806450i	\(0.298614\pi\)
\(47\)	−112.721	−	346.921i	−0.349832	−	1.07667i	−0.958946	−	0.283590i	\(-0.908475\pi\)
							0.609113	−	0.793083i	\(-0.291525\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	77.4.f.a.36.7	yes	32
11.2	odd	10		847.4.a.p.1.12		16
11.4	even	5	inner	77.4.f.a.15.7	✓	32
11.9	even	5		847.4.a.o.1.5		16

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
77.4.f.a.15.7	✓	32	11.4	even	5	inner
77.4.f.a.36.7	yes	32	1.1	even	1	trivial
847.4.a.o.1.5		16	11.9	even	5
847.4.a.p.1.12		16	11.2	odd	10