Embedded newform 177.4.f.a.89.3

• Label: 177.4.f.a.89.3
• Level: $177$
• Weight: $4$
• Character: 177.89
• Analytic conductor: $10.443$
• Analytic rank: $0$
• Dimension: $1624$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 177 = 3 \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	177.f (of order \(58\), degree \(28\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(10.4433380710\)
Analytic rank:	\(0\)
Dimension:	\(1624\)
Relative dimension:	\(58\) over \(\Q(\zeta_{58})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{58}]$

Embedding invariants

Embedding label			89.3
Character	\(\chi\)	\(=\)	177.89
Dual form			177.4.f.a.2.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.282358 - 5.20779i) q^{2} +(-0.412985 - 5.17971i) q^{3} +(-19.0882 + 2.07597i) q^{4} +(-2.66012 - 7.89495i) q^{5} +(-26.8583 + 3.61327i) q^{6} +(2.27648 - 3.35756i) q^{7} +(9.45083 + 57.6475i) q^{8} +(-26.6589 + 4.27829i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1624 q - 21 q^{3} - 278 q^{4} - 29 q^{6} - 42 q^{7} - 25 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(61\)	\(119\)
\(\chi(n)\)	\(e\left(\frac{57}{58}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.282358	−	5.20779i	−0.0998287	−	1.84123i	−0.436613	−	0.899650i	\(-0.643822\pi\)
							0.336784	−	0.941582i	\(-0.390661\pi\)
\(3\)	−0.412985	−	5.17971i	−0.0794790	−	0.996837i
							\(5\)	−2.66012	−	7.89495i	−0.237928	−	0.706146i	−0.998388	−	0.0567605i	\(-0.981923\pi\)
0.760460	−	0.649385i	\(-0.224974\pi\)
\(7\)	2.27648	−	3.35756i	0.122918	−	0.181291i	−0.761251	−	0.648457i	\(-0.775415\pi\)
							0.884170	+	0.467166i	\(0.154725\pi\)
\(11\)	19.5208	+	4.29684i	0.535066	+	0.117777i	0.474280	−	0.880374i	\(-0.342708\pi\)
							0.0607858	+	0.998151i	\(0.480639\pi\)
\(13\)	−36.1212	−	30.6817i	−0.770633	−	0.654582i	0.173244	−	0.984879i	\(-0.444575\pi\)
							−0.943877	+	0.330297i	\(0.892851\pi\)
\(17\)	48.1112	−	32.6202i	0.686393	−	0.465386i	−0.167474	−	0.985876i	\(-0.553561\pi\)
							0.853868	+	0.520490i	\(0.174251\pi\)
\(19\)	39.6830	+	74.8501i	0.479153	+	0.903779i	0.998780	+	0.0493853i	\(0.0157262\pi\)
							−0.519627	+	0.854393i	\(0.673929\pi\)
\(23\)	−107.044	−	101.397i	−0.970440	−	0.919250i	0.0263220	−	0.999654i	\(-0.491620\pi\)
							−0.996762	+	0.0804036i	\(0.974379\pi\)
\(29\)	71.3607	+	3.86907i	0.456943	+	0.0247747i	0.281174	−	0.959657i	\(-0.409276\pi\)
							0.175769	+	0.984431i	\(0.443759\pi\)
\(31\)	−242.317	−	128.468i	−1.40392	−	0.744310i	−0.417518	−	0.908669i	\(-0.637100\pi\)
							−0.986401	+	0.164358i	\(0.947445\pi\)
\(37\)	−40.4009	−	6.62339i	−0.179510	−	0.0294292i	0.0713565	−	0.997451i	\(-0.477267\pi\)
							−0.250866	+	0.968022i	\(0.580715\pi\)
\(41\)	224.811	+	237.330i	0.856332	+	0.904019i	0.996283	−	0.0861399i	\(-0.0274532\pi\)
							−0.139951	+	0.990158i	\(0.544695\pi\)
\(43\)	−76.7406	−	348.636i	−0.272159	−	1.23643i	−0.892120	−	0.451798i	\(-0.850783\pi\)
							0.619961	−	0.784632i	\(-0.287148\pi\)
\(47\)	136.613	+	46.0303i	0.423980	+	0.142856i	0.523200	−	0.852210i	\(-0.324738\pi\)
							−0.0992197	+	0.995066i	\(0.531635\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	177.4.f.a.89.3	yes	1624
3.2	odd	2	inner	177.4.f.a.89.56	yes	1624
59.2	odd	58	inner	177.4.f.a.2.56	yes	1624
177.2	even	58	inner	177.4.f.a.2.3	✓	1624

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.4.f.a.2.3	✓	1624	177.2	even	58	inner
177.4.f.a.2.56	yes	1624	59.2	odd	58	inner
177.4.f.a.89.3	yes	1624	1.1	even	1	trivial
177.4.f.a.89.56	yes	1624	3.2	odd	2	inner