Embedded newform 177.2.f.a.83.1

• Label: 177.2.f.a.83.1
• Level: $177$
• Weight: $2$
• Character: 177.83
• Analytic conductor: $1.413$
• Analytic rank: $0$
• Dimension: $504$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 177 = 3 \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	177.f (of order \(58\), degree \(28\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.41335211578\)
Analytic rank:	\(0\)
Dimension:	\(504\)
Relative dimension:	\(18\) over \(\Q(\zeta_{58})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{58}]$

Embedding invariants

Embedding label			83.1
Character	\(\chi\)	\(=\)	177.83
Dual form			177.2.f.a.32.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.706265 - 2.54374i) q^{2} +(1.58615 + 0.695787i) q^{3} +(-4.25808 + 2.56200i) q^{4} +(3.25381 - 0.176417i) q^{5} +(0.649655 - 4.52617i) q^{6} +(-0.210117 - 1.28165i) q^{7} +(5.69118 + 5.39097i) q^{8} +(2.03176 + 2.20725i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 504 q - 27 q^{3} - 70 q^{4} - 29 q^{6} - 58 q^{7} - 19 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/177\mathbb{Z}\right)^\times\).

\(n\)	\(61\)	\(119\)
\(\chi(n)\)	\(e\left(\frac{53}{58}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.706265	−	2.54374i	−0.499405	−	1.79869i	−0.595540	−	0.803326i	\(-0.703062\pi\)
							0.0961350	−	0.995368i	\(-0.469352\pi\)
\(3\)	1.58615	+	0.695787i	0.915766	+	0.401713i
							\(5\)	3.25381	−	0.176417i	1.45515	−	0.0788960i	0.690496	−	0.723336i	\(-0.257392\pi\)
0.764654	+	0.644440i	\(0.222910\pi\)
\(7\)	−0.210117	−	1.28165i	−0.0794166	−	0.484420i	−0.996349	−	0.0853767i	\(-0.972791\pi\)
							0.916932	−	0.399043i	\(-0.130658\pi\)
\(11\)	−1.93353	−	3.64703i	−0.582981	−	1.09962i	−0.983085	−	0.183150i	\(-0.941371\pi\)
							0.400104	−	0.916470i	\(-0.368974\pi\)
\(13\)	−1.56476	−	0.623457i	−0.433986	−	0.172916i	0.142906	−	0.989736i	\(-0.454355\pi\)
							−0.576892	+	0.816821i	\(0.695735\pi\)
\(17\)	3.37883	+	0.553932i	0.819488	+	0.134348i	0.556921	−	0.830566i	\(-0.311983\pi\)
							0.262567	+	0.964914i	\(0.415431\pi\)
\(19\)	−5.39585	+	6.35248i	−1.23789	+	1.45736i	−0.395525	+	0.918455i	\(0.629437\pi\)
							−0.842367	+	0.538904i	\(0.818839\pi\)
\(23\)	−0.326929	−	0.248525i	−0.0681695	−	0.0518211i	0.570535	−	0.821274i	\(-0.306736\pi\)
							−0.638704	+	0.769453i	\(0.720529\pi\)
\(29\)	−6.81956	−	1.89344i	−1.26636	−	0.351603i	−0.431524	−	0.902101i	\(-0.642024\pi\)
							−0.834836	+	0.550498i	\(0.814438\pi\)
\(31\)	−0.0480452	+	0.0408100i	−0.00862918	+	0.00732969i	−0.651690	−	0.758485i	\(-0.725940\pi\)
							0.643061	+	0.765815i	\(0.277664\pi\)
\(37\)	3.21386	+	3.39283i	0.528356	+	0.557778i	0.934469	−	0.356044i	\(-0.115875\pi\)
							−0.406113	+	0.913823i	\(0.633116\pi\)
\(41\)	2.06688	+	2.71894i	0.322793	+	0.424626i	0.928656	−	0.370942i	\(-0.120965\pi\)
							−0.605863	+	0.795569i	\(0.707172\pi\)
\(43\)	−4.01672	−	2.12953i	−0.612544	−	0.324750i	0.133092	−	0.991104i	\(-0.457509\pi\)
							−0.745636	+	0.666353i	\(0.767854\pi\)
\(47\)	0.111152	−	2.05008i	0.0162132	−	0.299034i	−0.979169	−	0.203046i	\(-0.934916\pi\)
							0.995382	−	0.0959887i	\(-0.0306013\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	177.2.f.a.83.1	yes	504
3.2	odd	2	inner	177.2.f.a.83.18	yes	504
59.32	odd	58	inner	177.2.f.a.32.18	yes	504
177.32	even	58	inner	177.2.f.a.32.1	✓	504

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.2.f.a.32.1	✓	504	177.32	even	58	inner
177.2.f.a.32.18	yes	504	59.32	odd	58	inner
177.2.f.a.83.1	yes	504	1.1	even	1	trivial
177.2.f.a.83.18	yes	504	3.2	odd	2	inner