Embedded newform 77.4.f.b.15.7

• Label: 77.4.f.b.15.7
• 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			15.7
Character	\(\chi\)	\(=\)	77.15
Dual form			77.4.f.b.36.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(2.07504 + 1.50761i) q^{2} +(0.186723 - 0.574674i) q^{3} +(-0.439213 - 1.35176i) q^{4} +(11.4299 - 8.30432i) q^{5} +(1.25384 - 0.910969i) q^{6} +(-2.16312 - 6.65740i) q^{7} +(7.46730 - 22.9820i) q^{8} +(21.5481 + 15.6556i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q + 8 q^{2} - 18 q^{3} - 34 q^{4} - 24 q^{5} + 30 q^{6} + 70 q^{7} - 72 q^{8} - 136 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	2.07504	+	1.50761i	0.733638	+	0.533019i	0.890712	−	0.454567i	\(-0.150206\pi\)
							−0.157074	+	0.987587i	\(0.550206\pi\)
\(3\)	0.186723	−	0.574674i	0.0359349	−	0.110596i	−0.931480	−	0.363792i	\(-0.881482\pi\)
							0.967415	+	0.253196i	\(0.0814818\pi\)
\(5\)	11.4299	−	8.30432i	1.02232	−	0.742761i	0.0555654	−	0.998455i	\(-0.482304\pi\)
							0.966758	+	0.255694i	\(0.0823039\pi\)
\(7\)	−2.16312	−	6.65740i	−0.116797	−	0.359466i
							\(11\)	−17.4346	+	32.0474i	−0.477883	+	0.878423i
\(13\)	−0.0838156	−	0.0608956i	−0.00178818	−	0.00129919i								0.586891	−	0.809666i	\(-0.300352\pi\)
							−0.588679	+	0.808367i	\(0.700352\pi\)
\(17\)	23.8259	−	17.3105i	0.339919	−	0.246966i	−0.404709	−	0.914446i	\(-0.632627\pi\)
							0.744628	+	0.667480i	\(0.232627\pi\)
\(19\)	−24.1919	+	74.4552i	−0.292106	+	0.899010i	0.692072	+	0.721828i	\(0.256698\pi\)
							−0.984178	+	0.177181i	\(0.943302\pi\)
\(23\)	−127.073			−1.15202			−0.576012	−	0.817441i	\(-0.695392\pi\)
							−0.576012	+	0.817441i	\(0.695392\pi\)
\(29\)	−45.8420	−	141.087i	−0.293540	−	0.903422i	−0.983708	−	0.179773i	\(-0.942464\pi\)
							0.690168	−	0.723649i	\(-0.257536\pi\)
\(31\)	177.076	+	128.653i	1.02593	+	0.745379i	0.967489	−	0.252912i	\(-0.0813883\pi\)
							0.0584370	+	0.998291i	\(0.481388\pi\)
\(37\)	5.49329	+	16.9066i	0.0244079	+	0.0751198i	0.962518	−	0.271216i	\(-0.0874258\pi\)
							−0.938111	+	0.346336i	\(0.887426\pi\)
\(41\)	−123.213	+	379.212i	−0.469334	+	1.44446i	0.384115	+	0.923285i	\(0.374507\pi\)
							−0.853449	+	0.521176i	\(0.825493\pi\)
\(43\)	−434.860			−1.54222			−0.771111	−	0.636700i	\(-0.780299\pi\)
							−0.771111	+	0.636700i	\(0.780299\pi\)
\(47\)	−4.05659	+	12.4849i	−0.0125897	+	0.0387471i	−0.957154	−	0.289579i	\(-0.906485\pi\)
							0.944564	+	0.328326i	\(0.106485\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	77.4.f.b.15.7	✓	40
11.3	even	5	inner	77.4.f.b.36.7	yes	40
11.5	even	5		847.4.a.q.1.7		20
11.6	odd	10		847.4.a.r.1.14		20

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
77.4.f.b.15.7	✓	40	1.1	even	1	trivial
77.4.f.b.36.7	yes	40	11.3	even	5	inner
847.4.a.q.1.7		20	11.5	even	5
847.4.a.r.1.14		20	11.6	odd	10