Embedded newform 77.4.f.b.15.4

• Label: 77.4.f.b.15.4
• Level: $77$
• Weight: $4$
• Character: 77.15
• 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			15.4
Character	\(\chi\)	\(=\)	77.15
Dual form			77.4.f.b.36.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.91048 - 1.38804i) q^{2} +(1.19654 - 3.68258i) q^{3} +(-0.748877 - 2.30480i) q^{4} +(12.0553 - 8.75871i) q^{5} +(-7.39754 + 5.37463i) q^{6} +(-2.16312 - 6.65740i) q^{7} +(-7.60635 + 23.4100i) q^{8} +(9.71380 + 7.05749i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q + 8 q^{2} - 18 q^{3} - 34 q^{4} - 24 q^{5} + 30 q^{6} + 70 q^{7} - 72 q^{8} - 136 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.91048	−	1.38804i	−0.675456	−	0.490747i	0.196391	−	0.980526i	\(-0.437078\pi\)
							−0.871847	+	0.489778i	\(0.837078\pi\)
\(3\)	1.19654	−	3.68258i	0.230275	−	0.708712i	−0.767439	−	0.641122i	\(-0.778469\pi\)
							0.997713	−	0.0675899i	\(-0.0215309\pi\)
\(5\)	12.0553	−	8.75871i	1.07826	−	0.783403i	0.100883	−	0.994898i	\(-0.467833\pi\)
							0.977379	+	0.211495i	\(0.0678332\pi\)
\(7\)	−2.16312	−	6.65740i	−0.116797	−	0.359466i
							\(11\)	−8.51858	−	35.4744i	−0.233495	−	0.972358i
\(13\)	−24.9529	−	18.1293i	−0.532360	−	0.386782i								0.288880	−	0.957365i	\(-0.406717\pi\)
							−0.821240	+	0.570583i	\(0.806717\pi\)
\(17\)	−79.6738	+	57.8864i	−1.13669	+	0.825853i	−0.986655	−	0.162827i	\(-0.947939\pi\)
							−0.150035	+	0.988681i	\(0.547939\pi\)
\(19\)	0.0483790	−	0.148895i	0.000584153	−	0.00179784i	−0.950764	−	0.309916i	\(-0.899699\pi\)
							0.951348	+	0.308118i	\(0.0996991\pi\)
\(23\)	89.2319			0.808962			0.404481	−	0.914546i	\(-0.367452\pi\)
							0.404481	+	0.914546i	\(0.367452\pi\)
\(29\)	−23.0118	−	70.8230i	−0.147351	−	0.453500i	0.849955	−	0.526856i	\(-0.176629\pi\)
							−0.997306	+	0.0733555i	\(0.976629\pi\)
\(31\)	131.588	+	95.6043i	0.762384	+	0.553905i	0.899641	−	0.436631i	\(-0.143828\pi\)
							−0.137256	+	0.990536i	\(0.543828\pi\)
\(37\)	59.3009	+	182.510i	0.263487	+	0.810929i	0.992038	+	0.125938i	\(0.0401940\pi\)
							−0.728551	+	0.684991i	\(0.759806\pi\)
\(41\)	84.9163	−	261.345i	0.323456	−	0.995495i	−0.648677	−	0.761064i	\(-0.724677\pi\)
							0.972133	−	0.234431i	\(-0.0753227\pi\)
\(43\)	396.899			1.40759			0.703797	−	0.710401i	\(-0.251487\pi\)
							0.703797	+	0.710401i	\(0.251487\pi\)
\(47\)	−2.04949	+	6.30768i	−0.00636062	+	0.0195760i	−0.954186	−	0.299213i	\(-0.903276\pi\)
							0.947826	+	0.318789i	\(0.103276\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	77.4.f.b.15.4	✓	40
11.3	even	5	inner	77.4.f.b.36.4	yes	40
11.5	even	5		847.4.a.q.1.14		20
11.6	odd	10		847.4.a.r.1.7		20

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
77.4.f.b.15.4	✓	40	1.1	even	1	trivial
77.4.f.b.36.4	yes	40	11.3	even	5	inner
847.4.a.q.1.14		20	11.5	even	5
847.4.a.r.1.7		20	11.6	odd	10