Newform orbit 592.2.bl.a

• Label: 592.2.bl.a
• Level: $592$
• Weight: $2$
• Character orbit: 592.bl
• Analytic conductor: $4.727$
• Analytic rank: $0$
• Dimension: $296$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 592 = 2^{4} \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	592.bl (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.72714379966\)
Analytic rank:	\(0\)
Dimension:	\(296\)
Relative dimension:	\(74\) over \(\Q(\zeta_{12})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 296 q - 6 q^{3} - 12 q^{4} - 6 q^{5} - 4 q^{7}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
51.1	−1.41384	−	0.0324319i	0.129674	−	0.483951i	1.99790	+	0.0917070i	0.327392	−	0.189020i	−0.199034	+	0.680024i	−0.635574	−	1.10085i	−2.82173	−	0.194455i	2.38068	+	1.37449i	−0.469011	+	0.256627i
51.2	−1.41352	+	0.0443189i	0.0273586	−	0.102104i	1.99607	−	0.125291i	3.43045	−	1.98057i	−0.0341468	+	0.145538i	−1.61675	−	2.80030i	−2.81593	+	0.265565i	2.58840	+	1.49441i	−4.76123	+	2.95161i
51.3	−1.40679	−	0.144693i	−0.552822	+	2.06316i	1.95813	+	0.407105i	2.54336	−	1.46841i	1.07623	−	2.82245i	2.37518	+	4.11393i	−2.69577	−	0.856039i	−1.35294	−	0.781122i	−3.79044	+	1.69774i
51.4	−1.40654	+	0.147133i	0.286836	−	1.07049i	1.95670	−	0.413898i	−3.56489	+	2.05819i	−0.245942	+	1.54788i	−0.00191601	−	0.00331863i	−2.69128	+	0.870060i	1.53441	+	0.885891i	4.71132	−	3.41944i
51.5	−1.40488	−	0.162222i	0.663867	−	2.47758i	1.94737	+	0.455803i	−1.07939	+	0.623185i	−1.33457	+	3.37301i	2.26443	+	3.92211i	−2.66188	−	0.956254i	−3.09963	−	1.78957i	1.61750	−	0.700399i
51.6	−1.36124	−	0.383429i	−0.655391	+	2.44595i	1.70596	+	1.04388i	1.22682	−	0.708307i	1.82999	−	3.07824i	−0.990259	−	1.71518i	−1.92198	−	2.07509i	−2.95506	−	1.70611i	−1.94159	+	0.493779i
51.7	−1.34319	+	0.442540i	0.751509	−	2.80467i	1.60832	−	1.18883i	1.61400	−	0.931844i	0.231760	+	4.09978i	0.0928006	+	0.160735i	−1.63417	+	2.30857i	−4.70333	−	2.71547i	−1.75553	+	1.96590i
51.8	−1.33188	+	0.475498i	−0.430251	+	1.60572i	1.54780	−	1.26661i	−0.303976	+	0.175501i	−0.190474	−	2.34320i	−0.209310	−	0.362536i	−1.45922	+	2.42295i	0.204864	+	0.118278i	0.321409	−	0.378286i
51.9	−1.28384	+	0.593089i	−0.109465	+	0.408530i	1.29649	−	1.52286i	0.0413160	−	0.0238538i	−0.101758	−	0.589409i	2.08451	+	3.61047i	−0.761296	+	2.72405i	2.44316	+	1.41056i	−0.0388957	+	0.0551285i
51.10	−1.28137	−	0.598400i	0.767320	−	2.86368i	1.28384	+	1.53355i	−1.94098	+	1.12063i	−2.69685	+	3.21028i	−2.06845	−	3.58266i	−0.727398	−	2.73329i	−5.01379	−	2.89471i	3.15771	−	0.274458i
51.11	−1.27069	−	0.620773i	0.166554	−	0.621587i	1.22928	+	1.57761i	1.08801	−	0.628166i	−0.597501	+	0.686449i	0.442359	+	0.766189i	−0.582689	−	2.76776i	2.23945	+	1.29295i	−1.77247	+	0.122790i
51.12	−1.26991	−	0.622361i	−0.455855	+	1.70127i	1.22533	+	1.58068i	−2.61123	+	1.50759i	1.63770	−	1.87675i	−1.95644	−	3.38866i	−0.572305	−	2.76992i	−0.0884496	−	0.0510664i	4.25429	−	0.289378i
51.13	−1.25418	+	0.653476i	−0.811153	+	3.02726i	1.14594	−	1.63915i	−0.241412	+	0.139379i	−0.960912	−	4.32681i	−1.65877	−	2.87307i	−0.366065	+	2.80464i	−5.90829	−	3.41115i	0.211693	−	0.332563i
51.14	−1.22537	−	0.706025i	−0.0976689	+	0.364505i	1.00306	+	1.73028i	−1.29645	+	0.748507i	0.377030	−	0.377697i	1.29798	+	2.24817i	−0.00749447	−	2.82842i	2.47475	+	1.42880i	2.11710	−	0.00186989i
51.15	−1.16062	−	0.808054i	0.679548	−	2.53611i	0.694098	+	1.87569i	2.93382	−	1.69384i	−2.83801	+	2.39436i	0.314679	+	0.545040i	0.710074	−	2.73784i	−3.37199	−	1.94682i	−4.77378	−	0.404771i
51.16	−1.11179	+	0.874033i	0.260643	−	0.972734i	0.472134	−	1.94347i	2.77715	−	1.60339i	0.560422	+	1.30928i	0.627049	+	1.08608i	1.17375	+	2.57339i	1.71980	+	0.992927i	−1.68618	+	4.20995i
51.17	−1.04406	+	0.953911i	−0.521820	+	1.94746i	0.180107	−	1.99187i	−3.57739	+	2.06541i	−1.31289	−	2.53103i	1.63731	+	2.83590i	1.71203	+	2.25143i	−0.922225	−	0.532447i	1.76478	−	5.56891i
51.18	−1.03842	+	0.960039i	−0.0380764	+	0.142103i	0.156652	−	1.99386i	−1.99008	+	1.14897i	−0.0968850	−	0.184118i	−2.39045	−	4.14038i	1.75151	+	2.22086i	2.57933	+	1.48918i	0.963487	−	3.10367i
51.19	−0.936454	+	1.05974i	0.510515	−	1.90527i	−0.246107	−	1.98480i	−1.97858	+	1.14233i	1.54102	+	2.32521i	0.307813	+	0.533148i	2.33384	+	1.59787i	−0.771351	−	0.445339i	0.642271	−	3.16653i
51.20	−0.927755	−	1.06737i	−0.469723	+	1.75303i	−0.278541	+	1.98051i	−2.20934	+	1.27556i	2.30691	−	1.12502i	1.57589	+	2.72952i	2.37235	−	1.54012i	−0.254397	−	0.146876i	3.41122	+	1.17477i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
592.bl	even	12	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	592.2.bl.a	yes	296
16.f	odd	4	1		592.2.bf.a	✓	296
37.g	odd	12	1		592.2.bf.a	✓	296
592.bl	even	12	1	inner	592.2.bl.a	yes	296

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
592.2.bf.a	✓	296	16.f	odd	4	1
592.2.bf.a	✓	296	37.g	odd	12	1
592.2.bl.a	yes	296	1.a	even	1	1	trivial
592.2.bl.a	yes	296	592.bl	even	12	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(592, [\chi])\).