Embedded newform 177.3.h.a.71.1

• Label: 177.3.h.a.71.1
• Level: $177$
• Weight: $3$
• Character: 177.71
• Analytic conductor: $4.823$
• Analytic rank: $0$
• Dimension: $1064$
• CM: no
• Inner twists: $4$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(4.82290067918\)
Analytic rank:	\(0\)
Dimension:	\(1064\)
Relative dimension:	\(38\) over \(\Q(\zeta_{58})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{58}]$

Embedding invariants

Embedding label			71.1
Character	\(\chi\)	\(=\)	177.71
Dual form			177.3.h.a.5.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.23800 - 3.67424i) q^{2} +(2.23781 + 1.99805i) q^{3} +(-8.78304 + 6.67669i) q^{4} +(2.27487 - 0.906393i) q^{5} +(4.57091 - 10.6958i) q^{6} +(0.939802 + 0.434799i) q^{7} +(22.5687 + 15.3019i) q^{8} +(1.01559 + 8.94251i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1064 q - 29 q^{3} + 18 q^{4} - 21 q^{6} - 46 q^{7} - 49 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(61\)	\(119\)
\(\chi(n)\)	\(e\left(\frac{26}{29}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.23800	−	3.67424i	−0.618998	−	1.83712i	−0.542510	−	0.840050i	\(-0.682526\pi\)
							−0.0764882	−	0.997070i	\(-0.524371\pi\)
\(3\)	2.23781	+	1.99805i	0.745937	+	0.666017i
							\(5\)	2.27487	−	0.906393i	0.454975	−	0.181279i	−0.131381	−	0.991332i	\(-0.541941\pi\)
0.586356	+	0.810053i	\(0.300562\pi\)
\(7\)	0.939802	+	0.434799i	0.134257	+	0.0621141i	0.485868	−	0.874032i	\(-0.338504\pi\)
							−0.351610	+	0.936146i	\(0.614366\pi\)
\(11\)	17.5370	−	4.86912i	1.59427	−	0.442647i	0.646845	−	0.762622i	\(-0.276088\pi\)
							0.947426	+	0.319975i	\(0.103674\pi\)
\(13\)	−4.59322	−	8.66373i	−0.353324	−	0.666441i	0.641603	−	0.767037i	\(-0.278270\pi\)
							−0.994927	+	0.100596i	\(0.967925\pi\)
\(17\)	−4.21342	−	9.10717i	−0.247849	−	0.535716i	0.743224	−	0.669043i	\(-0.233296\pi\)
							−0.991072	+	0.133327i	\(0.957434\pi\)
\(19\)	30.5596	+	6.72667i	1.60840	+	0.354035i	0.926171	−	0.377104i	\(-0.123080\pi\)
							0.682228	+	0.731139i	\(0.261011\pi\)
\(23\)	13.2646	−	2.17462i	0.576722	−	0.0945488i	0.133640	−	0.991030i	\(-0.457333\pi\)
							0.443082	+	0.896481i	\(0.353885\pi\)
\(29\)	6.34262	−	18.8242i	0.218711	−	0.649111i	−0.780987	−	0.624547i	\(-0.785284\pi\)
							0.999698	−	0.0245641i	\(-0.00781978\pi\)
\(31\)	−12.7737	+	2.81171i	−0.412056	+	0.0907004i	−0.416161	−	0.909291i	\(-0.636625\pi\)
							0.00410432	+	0.999992i	\(0.498694\pi\)
\(37\)	3.23099	+	4.76536i	0.0873241	+	0.128793i	0.868842	−	0.495089i	\(-0.164865\pi\)
							−0.781518	+	0.623883i	\(0.785554\pi\)
\(41\)	−13.4943	−	2.21228i	−0.329130	−	0.0539581i	−0.00504840	−	0.999987i	\(-0.501607\pi\)
							−0.324082	+	0.946029i	\(0.605055\pi\)
\(43\)	−17.8628	+	64.3362i	−0.415415	+	1.49619i	0.399567	+	0.916704i	\(0.369160\pi\)
							−0.814982	+	0.579486i	\(0.803253\pi\)
\(47\)	36.7159	+	14.6290i	0.781190	+	0.311255i	0.726426	−	0.687245i	\(-0.241180\pi\)
							0.0547643	+	0.998499i	\(0.482559\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	177.3.h.a.71.1	yes	1064
3.2	odd	2	inner	177.3.h.a.71.38	yes	1064
59.5	even	29	inner	177.3.h.a.5.38	yes	1064
177.5	odd	58	inner	177.3.h.a.5.1	✓	1064

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.3.h.a.5.1	✓	1064	177.5	odd	58	inner
177.3.h.a.5.38	yes	1064	59.5	even	29	inner
177.3.h.a.71.1	yes	1064	1.1	even	1	trivial
177.3.h.a.71.38	yes	1064	3.2	odd	2	inner