Embedded newform 77.4.f.a.36.8

• Label: 77.4.f.a.36.8
• 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			36.8
Character	\(\chi\)	\(=\)	77.36
Dual form			77.4.f.a.15.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(3.88476 - 2.82244i) q^{2} +(-0.0845386 - 0.260183i) q^{3} +(4.65302 - 14.3205i) q^{4} +(1.75855 + 1.27766i) q^{5} +(-1.06276 - 0.772142i) q^{6} +(2.16312 - 6.65740i) q^{7} +(-10.4722 - 32.2302i) q^{8} +(21.7829 - 15.8262i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q - 2 q^{2} + 18 q^{3} - 48 q^{4} + 16 q^{5} + 30 q^{6} - 56 q^{7} + 62 q^{8} - 46 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	3.88476	−	2.82244i	1.37347	−	0.997883i	0.376012	−	0.926615i	\(-0.377295\pi\)
							0.997457	−	0.0712685i	\(-0.0227047\pi\)
\(3\)	−0.0845386	−	0.260183i	−0.0162695	−	0.0500722i	0.942592	−	0.333946i	\(-0.108380\pi\)
							−0.958862	+	0.283874i	\(0.908380\pi\)
\(5\)	1.75855	+	1.27766i	0.157289	+	0.114277i	0.663646	−	0.748047i	\(-0.269008\pi\)
							−0.506357	+	0.862324i	\(0.669008\pi\)
\(7\)	2.16312	−	6.65740i	0.116797	−	0.359466i
							\(11\)	−34.0645	+	13.0619i	−0.933711	+	0.358028i
\(13\)	−32.1323	+	23.3455i	−0.685530	+	0.498067i								−0.875188	−	0.483783i	\(-0.839262\pi\)
							0.189658	+	0.981850i	\(0.439262\pi\)
\(17\)	55.6270	+	40.4153i	0.793619	+	0.576598i	0.909035	−	0.416719i	\(-0.136820\pi\)
							−0.115416	+	0.993317i	\(0.536820\pi\)
\(19\)	43.8569	+	134.978i	0.529551	+	1.62979i	0.755136	+	0.655568i	\(0.227571\pi\)
							−0.225585	+	0.974223i	\(0.572429\pi\)
\(23\)	−11.3654			−0.103037			−0.0515186	−	0.998672i	\(-0.516406\pi\)
							−0.0515186	+	0.998672i	\(0.516406\pi\)
\(29\)	7.26467	−	22.3584i	0.0465178	−	0.143167i	−0.925100	−	0.379724i	\(-0.876019\pi\)
							0.971618	+	0.236557i	\(0.0760190\pi\)
\(31\)	−38.2849	+	27.8156i	−0.221812	+	0.161156i	−0.693142	−	0.720801i	\(-0.743774\pi\)
							0.471330	+	0.881957i	\(0.343774\pi\)
\(37\)	−23.4233	+	72.0895i	−0.104075	+	0.320309i	−0.989512	−	0.144449i	\(-0.953859\pi\)
							0.885437	+	0.464759i	\(0.153859\pi\)
\(41\)	−34.8016	−	107.108i	−0.132563	−	0.407988i	0.862640	−	0.505819i	\(-0.168810\pi\)
							−0.995203	+	0.0978307i	\(0.968810\pi\)
\(43\)	−150.839			−0.534949			−0.267474	−	0.963565i	\(-0.586189\pi\)
							−0.267474	+	0.963565i	\(0.586189\pi\)
\(47\)	−100.661	−	309.801i	−0.312401	−	0.961472i	−0.976811	−	0.214103i	\(-0.931317\pi\)
							0.664410	−	0.747368i	\(-0.268683\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	77.4.f.a.36.8	yes	32
11.2	odd	10		847.4.a.p.1.15		16
11.4	even	5	inner	77.4.f.a.15.8	✓	32
11.9	even	5		847.4.a.o.1.2		16

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
77.4.f.a.15.8	✓	32	11.4	even	5	inner
77.4.f.a.36.8	yes	32	1.1	even	1	trivial
847.4.a.o.1.2		16	11.9	even	5
847.4.a.p.1.15		16	11.2	odd	10