Embedded newform 177.2.f.a.83.4

• Label: 177.2.f.a.83.4
• 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			83.4
Character	\(\chi\)	\(=\)	177.83
Dual form			177.2.f.a.32.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.487885 - 1.75720i) q^{2} +(1.13622 - 1.30729i) q^{3} +(-1.13602 + 0.683518i) q^{4} +(-0.529541 + 0.0287109i) q^{5} +(-2.85152 - 1.35876i) q^{6} +(0.0261422 + 0.159460i) q^{7} +(-0.892635 - 0.845549i) q^{8} +(-0.418004 - 2.97074i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 504 q - 27 q^{3} - 70 q^{4} - 29 q^{6} - 58 q^{7} - 19 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.487885	−	1.75720i	−0.344987	−	1.24253i	−0.909011	−	0.416772i	\(-0.863161\pi\)
							0.564024	−	0.825758i	\(-0.309252\pi\)
\(3\)	1.13622	−	1.30729i	0.655997	−	0.754763i
							\(5\)	−0.529541	+	0.0287109i	−0.236818	+	0.0128399i	−0.172166	−	0.985068i	\(-0.555077\pi\)
−0.0646520	+	0.997908i	\(0.520594\pi\)
\(7\)	0.0261422	+	0.159460i	0.00988082	+	0.0602703i	0.991307	−	0.131571i	\(-0.0420023\pi\)
							−0.981426	+	0.191842i	\(0.938554\pi\)
\(11\)	−0.214767	−	0.405094i	−0.0647547	−	0.122140i	0.849022	−	0.528358i	\(-0.177192\pi\)
							−0.913776	+	0.406218i	\(0.866847\pi\)
\(13\)	1.99544	+	0.795056i	0.553435	+	0.220509i	0.630053	−	0.776552i	\(-0.283033\pi\)
							−0.0766182	+	0.997061i	\(0.524412\pi\)
\(17\)	2.58024	+	0.423008i	0.625799	+	0.102595i	0.466334	−	0.884609i	\(-0.345575\pi\)
							0.159466	+	0.987203i	\(0.449023\pi\)
\(19\)	−0.426347	+	0.501935i	−0.0978107	+	0.115152i	−0.808890	−	0.587961i	\(-0.799931\pi\)
							0.711079	+	0.703112i	\(0.248207\pi\)
\(23\)	4.86969	+	3.70184i	1.01540	+	0.771887i	0.973837	−	0.227249i	\(-0.0729730\pi\)
							0.0415634	+	0.999136i	\(0.486766\pi\)
\(29\)	4.99162	+	1.38592i	0.926921	+	0.257358i	0.698009	−	0.716089i	\(-0.254069\pi\)
							0.228912	+	0.973447i	\(0.426483\pi\)
\(31\)	3.15514	−	2.68000i	0.566681	−	0.481343i	−0.317647	−	0.948209i	\(-0.602893\pi\)
							0.884328	+	0.466866i	\(0.154617\pi\)
\(37\)	−0.125378	−	0.132360i	−0.0206120	−	0.0217598i	0.715608	−	0.698502i	\(-0.246150\pi\)
							−0.736220	+	0.676742i	\(0.763391\pi\)
\(41\)	−4.26748	−	5.61378i	−0.666469	−	0.876725i	0.331297	−	0.943526i	\(-0.392514\pi\)
							−0.997766	+	0.0668015i	\(0.978721\pi\)
\(43\)	5.67103	+	3.00659i	0.864824	+	0.458500i	0.840886	−	0.541213i	\(-0.182035\pi\)
							0.0239380	+	0.999713i	\(0.492380\pi\)
\(47\)	−0.327397	+	6.03848i	−0.0477557	+	0.880802i	0.872289	+	0.488990i	\(0.162635\pi\)
							−0.920045	+	0.391812i	\(0.871848\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	177.2.f.a.83.4	yes	504
3.2	odd	2	inner	177.2.f.a.83.15	yes	504
59.32	odd	58	inner	177.2.f.a.32.15	yes	504
177.32	even	58	inner	177.2.f.a.32.4	✓	504

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.2.f.a.32.4	✓	504	177.32	even	58	inner
177.2.f.a.32.15	yes	504	59.32	odd	58	inner
177.2.f.a.83.4	yes	504	1.1	even	1	trivial
177.2.f.a.83.15	yes	504	3.2	odd	2	inner