Embedded newform 177.2.f.a.32.4

• Label: 177.2.f.a.32.4
• Level: $177$
• Weight: $2$
• Character: 177.32
• 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			32.4
Character	\(\chi\)	\(=\)	177.32
Dual form			177.2.f.a.83.4

$q$-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 \(\left(\mathbb{Z}/177\mathbb{Z}\right)^\times\).

\(n\)	\(61\)	\(119\)
\(\chi(n)\)	\(e\left(\frac{5}{58}\right)\)	\(-1\)

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

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.2.f.a.32.4	✓	504	1.1	even	1	trivial
177.2.f.a.32.15	yes	504	3.2	odd	2	inner
177.2.f.a.83.4	yes	504	177.83	even	58	inner
177.2.f.a.83.15	yes	504	59.24	odd	58	inner