Embedded newform 177.2.e.b.64.4

• Label: 177.2.e.b.64.4
• Level: $177$
• Weight: $2$
• Character: 177.64
• Analytic conductor: $1.413$
• Analytic rank: $0$
• Dimension: $140$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 177 = 3 \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	177.e (of order \(29\), degree \(28\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.41335211578\)
Analytic rank:	\(0\)
Dimension:	\(140\)
Relative dimension:	\(5\) over \(\Q(\zeta_{29})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{29}]$

Embedding invariants

Embedding label			64.4
Character	\(\chi\)	\(=\)	177.64
Dual form			177.2.e.b.130.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.32319 + 0.445835i) q^{2} +(0.856857 + 0.515554i) q^{3} +(-0.0401232 - 0.0305009i) q^{4} +(-1.32107 + 3.31564i) q^{5} +(0.903933 + 1.06419i) q^{6} +(3.77207 - 1.74515i) q^{7} +(-1.60664 - 2.36962i) q^{8} +(0.468408 + 0.883512i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 140 q + q^{2} + 5 q^{3} - q^{4} + 2 q^{5} - q^{6} - 2 q^{7} - 3 q^{8} - 5 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(61\)	\(119\)
\(\chi(n)\)	\(e\left(\frac{3}{29}\right)\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.32319	+	0.445835i	0.935637	+	0.315253i	0.745500	−	0.666505i	\(-0.232211\pi\)
							0.190136	+	0.981758i	\(0.439107\pi\)
\(3\)	0.856857	+	0.515554i	0.494707	+	0.297655i
							\(5\)	−1.32107	+	3.31564i	−0.590801	+	1.48280i	0.264354	+	0.964426i	\(0.414841\pi\)
−0.855155	+	0.518373i	\(0.826538\pi\)
\(7\)	3.77207	−	1.74515i	1.42571	−	0.659603i	0.452974	−	0.891524i	\(-0.350363\pi\)
							0.972735	+	0.231920i	\(0.0745008\pi\)
\(11\)	−0.526267	+	1.89544i	−0.158676	+	0.571498i	0.840867	+	0.541242i	\(0.182046\pi\)
							−0.999542	+	0.0302553i	\(0.990368\pi\)
\(13\)	0.345743	−	0.652140i	0.0958918	−	0.180871i	−0.830925	−	0.556384i	\(-0.812188\pi\)
							0.926817	+	0.375513i	\(0.122533\pi\)
\(17\)	−6.27360	−	2.90248i	−1.52157	−	0.703954i	−0.531834	−	0.846849i	\(-0.678497\pi\)
							−0.989738	+	0.142894i	\(0.954359\pi\)
\(19\)	2.03210	−	0.447298i	0.466195	−	0.102617i	0.0243397	−	0.999704i	\(-0.492252\pi\)
							0.441855	+	0.897087i	\(0.354321\pi\)
\(23\)	1.23610	−	7.53989i	0.257745	−	1.57217i	−0.467695	−	0.883890i	\(-0.654916\pi\)
							0.725440	−	0.688285i	\(-0.241636\pi\)
\(29\)	0.855008	−	0.288086i	0.158771	−	0.0534962i	−0.238794	−	0.971070i	\(-0.576752\pi\)
							0.397565	+	0.917574i	\(0.369855\pi\)
\(31\)	−0.00573740	−	0.00126290i	−0.00103047	−	0.000226823i	0.214455	−	0.976734i	\(-0.431202\pi\)
							−0.215486	+	0.976507i	\(0.569133\pi\)
\(37\)	−4.72610	+	6.97048i	−0.776967	+	1.14594i	0.208822	+	0.977954i	\(0.433037\pi\)
							−0.985789	+	0.167987i	\(0.946273\pi\)
\(41\)	0.866161	+	5.28335i	0.135272	+	0.825121i	0.964085	+	0.265595i	\(0.0855684\pi\)
							−0.828813	+	0.559526i	\(0.810983\pi\)
\(43\)	−1.53390	−	5.52460i	−0.233917	−	0.842494i	−0.983549	−	0.180639i	\(-0.942183\pi\)
							0.749632	−	0.661855i	\(-0.230230\pi\)
\(47\)	3.13424	+	7.86636i	0.457176	+	1.14743i	0.959239	+	0.282595i	\(0.0911953\pi\)
							−0.502063	+	0.864831i	\(0.667425\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	177.2.e.b.64.4	✓	140
3.2	odd	2		531.2.i.b.64.2		140
59.12	even	29	inner	177.2.e.b.130.4	yes	140
177.71	odd	58		531.2.i.b.307.2		140

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.2.e.b.64.4	✓	140	1.1	even	1	trivial
177.2.e.b.130.4	yes	140	59.12	even	29	inner
531.2.i.b.64.2		140	3.2	odd	2
531.2.i.b.307.2		140	177.71	odd	58