Embedded newform 77.4.f.a.15.2

• Label: 77.4.f.a.15.2
• Level: $77$
• Weight: $4$
• Character: 77.15
• 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			15.2
Character	\(\chi\)	\(=\)	77.15
Dual form			77.4.f.a.36.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-3.08826 - 2.24375i) q^{2} +(-2.65557 + 8.17302i) q^{3} +(2.03079 + 6.25011i) q^{4} +(9.24278 - 6.71527i) q^{5} +(26.5393 - 19.2819i) q^{6} +(2.16312 + 6.65740i) q^{7} +(-1.68477 + 5.18518i) q^{8} +(-37.9027 - 27.5379i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q - 2 q^{2} + 18 q^{3} - 48 q^{4} + 16 q^{5} + 30 q^{6} - 56 q^{7} + 62 q^{8} - 46 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−3.08826	−	2.24375i	−1.09186	−	0.793286i	−0.112151	−	0.993691i	\(-0.535774\pi\)
							−0.979713	+	0.200405i	\(0.935774\pi\)
\(3\)	−2.65557	+	8.17302i	−0.511066	+	1.57290i	0.279263	+	0.960215i	\(0.409910\pi\)
							−0.790329	+	0.612683i	\(0.790090\pi\)
\(5\)	9.24278	−	6.71527i	0.826700	−	0.600632i	−0.0919241	−	0.995766i	\(-0.529302\pi\)
							0.918624	+	0.395134i	\(0.129302\pi\)
\(7\)	2.16312	+	6.65740i	0.116797	+	0.359466i
							\(11\)	−24.0069	+	27.4713i	−0.658031	+	0.752991i
\(13\)	−59.7045	−	43.3779i	−1.27377	−	0.925450i								−0.274427	−	0.961608i	\(-0.588488\pi\)
							−0.999346	+	0.0361577i	\(0.988488\pi\)
\(17\)	−88.5916	+	64.3656i	−1.26392	+	0.918291i	−0.998943	−	0.0459645i	\(-0.985364\pi\)
							−0.264976	+	0.964255i	\(0.585364\pi\)
\(19\)	12.8765	−	39.6298i	0.155478	−	0.478511i	−0.842731	−	0.538334i	\(-0.819054\pi\)
							0.998209	+	0.0598235i	\(0.0190538\pi\)
\(23\)	−74.8635			−0.678701			−0.339350	−	0.940660i	\(-0.610207\pi\)
							−0.339350	+	0.940660i	\(0.610207\pi\)
\(29\)	−74.7424	−	230.033i	−0.478597	−	1.47297i	−0.841045	−	0.540966i	\(-0.818059\pi\)
							0.362448	−	0.932004i	\(-0.381941\pi\)
\(31\)	2.19930	+	1.59789i	0.0127421	+	0.00925771i	0.594138	−	0.804363i	\(-0.297493\pi\)
							−0.581396	+	0.813621i	\(0.697493\pi\)
\(37\)	63.1413	+	194.329i	0.280550	+	0.863445i	0.987697	+	0.156379i	\(0.0499820\pi\)
							−0.707147	+	0.707067i	\(0.750018\pi\)
\(41\)	−106.211	+	326.884i	−0.404570	+	1.24514i	0.516684	+	0.856176i	\(0.327166\pi\)
							−0.921254	+	0.388962i	\(0.872834\pi\)
\(43\)	6.07503			0.0215450			0.0107725	−	0.999942i	\(-0.496571\pi\)
							0.0107725	+	0.999942i	\(0.496571\pi\)
\(47\)	−29.2509	+	90.0251i	−0.0907806	+	0.279394i	−0.986131	−	0.165968i	\(-0.946925\pi\)
							0.895351	+	0.445362i	\(0.146925\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	77.4.f.a.15.2	✓	32
11.3	even	5	inner	77.4.f.a.36.2	yes	32
11.5	even	5		847.4.a.o.1.13		16
11.6	odd	10		847.4.a.p.1.4		16

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
77.4.f.a.15.2	✓	32	1.1	even	1	trivial
77.4.f.a.36.2	yes	32	11.3	even	5	inner
847.4.a.o.1.13		16	11.5	even	5
847.4.a.p.1.4		16	11.6	odd	10