Embedded newform 77.4.f.b.15.2

• Label: 77.4.f.b.15.2
• 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			15.2
Character	\(\chi\)	\(=\)	77.15
Dual form			77.4.f.b.36.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-3.13241 - 2.27583i) q^{2} +(2.57135 - 7.91381i) q^{3} +(2.16046 + 6.64921i) q^{4} +(-14.0082 + 10.1776i) q^{5} +(-26.0650 + 18.9373i) q^{6} +(-2.16312 - 6.65740i) q^{7} +(-1.20677 + 3.71406i) q^{8} +(-34.1731 - 24.8282i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q + 8 q^{2} - 18 q^{3} - 34 q^{4} - 24 q^{5} + 30 q^{6} + 70 q^{7} - 72 q^{8} - 136 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−3.13241	−	2.27583i	−1.10747	−	0.804627i	−0.125210	−	0.992130i	\(-0.539961\pi\)
							−0.982264	+	0.187503i	\(0.939961\pi\)
\(3\)	2.57135	−	7.91381i	0.494857	−	1.52301i	−0.322322	−	0.946630i	\(-0.604463\pi\)
							0.817179	−	0.576384i	\(-0.195537\pi\)
\(5\)	−14.0082	+	10.1776i	−1.25293	+	0.910308i	−0.998388	−	0.0567529i	\(-0.981925\pi\)
							−0.254544	+	0.967061i	\(0.581925\pi\)
\(7\)	−2.16312	−	6.65740i	−0.116797	−	0.359466i
							\(11\)	35.0140	+	10.2478i	0.959739	+	0.280893i
\(13\)	−45.3386	−	32.9404i	−0.967282	−	0.702772i								−0.0124518	−	0.999922i	\(-0.503964\pi\)
							−0.954831	+	0.297151i	\(0.903964\pi\)
\(17\)	−35.4809	+	25.7784i	−0.506198	+	0.367775i	−0.811380	−	0.584520i	\(-0.801283\pi\)
							0.305181	+	0.952294i	\(0.401283\pi\)
\(19\)	−25.9213	+	79.7774i	−0.312987	+	0.963274i	0.663589	+	0.748098i	\(0.269033\pi\)
							−0.976575	+	0.215176i	\(0.930967\pi\)
\(23\)	−57.9547			−0.525409			−0.262704	−	0.964876i	\(-0.584614\pi\)
							−0.262704	+	0.964876i	\(0.584614\pi\)
\(29\)	8.59234	+	26.4445i	0.0550192	+	0.169332i	0.974790	−	0.223124i	\(-0.0716255\pi\)
							−0.919771	+	0.392456i	\(0.871626\pi\)
\(31\)	−191.187	−	138.906i	−1.10768	−	0.804780i	−0.125387	−	0.992108i	\(-0.540017\pi\)
							−0.982297	+	0.187328i	\(0.940017\pi\)
\(37\)	−50.0500	−	154.038i	−0.222383	−	0.684424i	−0.998547	−	0.0538927i	\(-0.982837\pi\)
							0.776164	−	0.630531i	\(-0.217163\pi\)
\(41\)	−128.061	+	394.132i	−0.487801	+	1.50130i	0.340082	+	0.940396i	\(0.389545\pi\)
							−0.827883	+	0.560901i	\(0.810455\pi\)
\(43\)	−370.184			−1.31285			−0.656425	−	0.754392i	\(-0.727932\pi\)
							−0.656425	+	0.754392i	\(0.727932\pi\)
\(47\)	36.8279	−	113.345i	0.114296	−	0.351766i	−0.877504	−	0.479570i	\(-0.840793\pi\)
							0.991799	+	0.127804i	\(0.0407928\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	77.4.f.b.15.2	✓	40
11.3	even	5	inner	77.4.f.b.36.2	yes	40
11.5	even	5		847.4.a.q.1.17		20
11.6	odd	10		847.4.a.r.1.4		20

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
77.4.f.b.15.2	✓	40	1.1	even	1	trivial
77.4.f.b.36.2	yes	40	11.3	even	5	inner
847.4.a.q.1.17		20	11.5	even	5
847.4.a.r.1.4		20	11.6	odd	10