Newform orbit 177.3.h.a

• Label: 177.3.h.a
• Level: $177$
• Weight: $3$
• Character orbit: 177.h
• 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:	$\mathrm{SU}(2)[C_{58}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 1064q - 29q^{3} + 18q^{4} - 21q^{6} - 46q^{7} - 49q^{9} + O(q^{10}) \)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
5.1	−1.23800	+	3.67424i	2.23781	−	1.99805i	−8.78304	−	6.67669i	2.27487	+	0.906393i	4.57091	+	10.6958i	0.939802	−	0.434799i	22.5687	−	15.3019i	1.01559	−	8.94251i	−6.14659	+	7.23632i
5.2	−1.21095	+	3.59398i	−2.84830	+	0.941900i	−8.26594	−	6.28360i	1.37514	+	0.547907i	0.0639885	−	11.3773i	8.82700	−	4.08381i	20.0368	−	13.5853i	7.22565	−	5.36563i	−3.63441	+	4.27875i
5.3	−1.15362	+	3.42383i	0.846084	+	2.87822i	−7.20737	−	5.47890i	−6.99239	−	2.78602i	−10.8306	−	0.423531i	0.665705	−	0.307988i	15.1117	−	10.2460i	−7.56829	+	4.87043i	17.6054	−	20.7267i
5.4	−1.00783	+	2.99114i	−2.99774	−	0.116447i	−4.74683	−	3.60844i	−2.80678	−	1.11832i	3.36953	−	8.84930i	−12.2974	+	5.68939i	5.12741	−	3.47647i	8.97288	+	0.698155i	6.17382	−	7.26839i
5.5	−0.999960	+	2.96778i	−1.56678	−	2.55836i	−4.62340	−	3.51462i	4.69371	+	1.87015i	9.15935	−	2.09161i	−2.98209	+	1.37966i	4.68548	−	3.17683i	−4.09038	+	8.01678i	−10.2437	+	12.0598i
5.6	−0.964344	+	2.86207i	−0.263179	+	2.98843i	−4.07712	−	3.09934i	5.65262	+	2.25221i	−8.29931	−	3.63511i	−4.32408	+	2.00053i	2.80324	−	1.90065i	−8.86147	−	1.57299i	−11.8970	+	14.0063i
5.7	−0.912087	+	2.70698i	2.96240	−	0.473468i	−3.31146	−	2.51731i	−6.32194	−	2.51889i	−1.42030	+	8.45101i	−6.08665	+	2.81599i	0.377434	−	0.255907i	8.55166	−	2.80521i	12.5848	−	14.8159i
5.8	−0.892372	+	2.64847i	0.713097	−	2.91402i	−3.03367	−	2.30614i	−2.87136	−	1.14405i	7.08133	+	4.48900i	2.33765	−	1.08151i	−0.437890	+	0.296897i	−7.98299	−	4.15595i	5.59230	−	6.58377i
5.9	−0.869361	+	2.58017i	2.78047	+	1.12649i	−2.71713	−	2.06551i	3.05628	+	1.21773i	−5.32377	+	6.19477i	9.38908	−	4.34385i	−1.32265	+	0.896782i	6.46204	+	6.26435i	−5.79898	+	6.82709i
5.10	−0.792761	+	2.35283i	−2.47356	−	1.69749i	−1.72297	−	1.30976i	−7.18639	−	2.86332i	5.95485	−	4.47417i	10.9013	−	5.04348i	−3.77238	+	2.55774i	3.23704	+	8.39771i	12.4340	−	14.6384i
5.11	−0.561287	+	1.66584i	2.68858	−	1.33099i	0.724387	+	0.550665i	7.94437	+	3.16532i	0.708151	+	5.22582i	−11.0072	+	5.09247i	−7.14376	+	4.84359i	5.45694	−	7.15694i	−9.73200	+	11.4574i
5.12	−0.551931	+	1.63807i	−1.84525	+	2.36538i	0.805720	+	0.612492i	−3.83005	−	1.52603i	−2.85622	−	4.32819i	−0.768529	+	0.355559i	−7.17084	+	4.86195i	−2.19007	−	8.72947i	4.61367	−	5.43163i
5.13	−0.528130	+	1.56743i	−2.80203	+	1.07173i	1.00644	+	0.765078i	6.70916	+	2.67318i	−0.200026	−	4.95802i	6.64296	−	3.07336i	−7.20679	+	4.88632i	6.70279	−	6.00604i	−7.73334	+	9.10439i
5.14	−0.386111	+	1.14594i	1.94700	+	2.28237i	2.02029	+	1.53578i	−1.21804	−	0.485310i	−3.36720	+	1.34988i	−3.89579	+	1.80238i	−6.54344	+	4.43656i	−1.41841	+	8.88753i	1.02643	−	1.20841i
5.15	−0.322077	+	0.955892i	−0.589478	−	2.94152i	2.37438	+	1.80495i	5.46847	+	2.17884i	3.00163	+	0.383919i	3.90866	−	1.80834i	−5.82961	+	3.95258i	−8.30503	+	3.46792i	−3.84400	+	4.52551i
5.16	−0.225119	+	0.668129i	0.0215190	−	2.99992i	2.78865	+	2.11988i	−5.48264	−	2.18448i	1.99949	+	0.689717i	−11.1057	+	5.13804i	−4.37833	+	2.96858i	−8.99907	−	0.129111i	2.69377	−	3.17135i
5.17	−0.221328	+	0.656877i	−2.72451	−	1.25582i	2.80187	+	2.12993i	−2.12339	−	0.846038i	1.42792	−	1.51172i	−1.48778	+	0.688319i	−4.31412	+	2.92505i	5.84586	+	6.84295i	1.02571	−	1.20756i
5.18	−0.192085	+	0.570088i	2.47232	−	1.69930i	2.89627	+	2.20169i	1.33026	+	0.530023i	0.493857	+	1.73585i	7.94838	−	3.67731i	−3.80316	+	2.57861i	3.22474	−	8.40244i	−0.557682	+	0.656554i
5.19	−0.0718854	+	0.213348i	2.81983	−	1.02398i	3.14402	+	2.39002i	−7.60675	−	3.03081i	0.0157589	+	0.675216i	4.35627	−	2.01543i	−1.48128	+	1.00433i	6.90294	−	5.77490i	1.19343	−	1.40502i
5.20	0.0718854	−	0.213348i	0.416138	+	2.97100i	3.14402	+	2.39002i	7.60675	+	3.03081i	0.663771	+	0.124789i	4.35627	−	2.01543i	1.48128	−	1.00433i	−8.65366	+	2.47269i	1.19343	−	1.40502i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
59.c	even	29	1	inner
177.h	odd	58	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	177.3.h.a	✓	1064
3.b	odd	2	1	inner	177.3.h.a	✓	1064
59.c	even	29	1	inner	177.3.h.a	✓	1064
177.h	odd	58	1	inner	177.3.h.a	✓	1064

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
177.3.h.a	✓	1064	1.a	even	1	1	trivial
177.3.h.a	✓	1064	3.b	odd	2	1	inner
177.3.h.a	✓	1064	59.c	even	29	1	inner
177.3.h.a	✓	1064	177.h	odd	58	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(177, [\chi])\).