Newform orbit 89.4.g.a

• Label: 89.4.g.a
• Level: $89$
• Weight: $4$
• Character orbit: 89.g
• Analytic conductor: $5.251$
• Analytic rank: $0$
• Dimension: $420$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 89 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	89.g (of order \(44\), degree \(20\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.25116999051\)
Analytic rank:	\(0\)
Dimension:	\(420\)
Relative dimension:	\(21\) over \(\Q(\zeta_{44})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{44}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 420 q - 14 q^{2} - 14 q^{3} - 170 q^{4} - 22 q^{5} - 32 q^{6} - 20 q^{7} + 62 q^{8} + 330 q^{9}+O(q^{10}) \)

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	−3.63633	+	4.19655i	0.714007	−	0.534499i	−3.24960	−	22.6015i	2.15350	−	3.35091i	−0.353314	+	4.93998i	0.248138	+	1.14067i	69.2941	+	44.5326i	−7.38266	+	25.1430i	6.23142	+	21.2223i
5.2	−2.81427	+	3.24784i	3.23400	−	2.42095i	−1.48983	−	10.3620i	−4.45421	+	6.93089i	−1.23851	+	17.3167i	−4.02249	−	18.4911i	8.92445	+	5.73540i	−3.00897	+	10.2476i	−9.97506	−	33.9719i
5.3	−2.74368	+	3.16638i	7.84288	−	5.87110i	−1.35964	−	9.45647i	0.300640	−	0.467806i	−2.92822	+	40.9419i	1.76982	+	8.13575i	5.47625	+	3.51937i	19.4341	−	66.1864i	0.656387	+	2.23545i
5.4	−2.68429	+	3.09784i	−3.75849	+	2.81357i	−1.25266	−	8.71243i	−11.7774	+	18.3260i	1.37290	−	19.1956i	2.07819	+	9.55329i	2.76561	+	1.77735i	−1.39673	+	4.75681i	−25.1570	−	85.6768i
5.5	−2.60204	+	3.00292i	−5.48322	+	4.10469i	−1.10837	−	7.70885i	4.56428	−	7.10216i	1.94154	−	27.1462i	−2.98676	−	13.7299i	−0.708225	−	0.455149i	5.61048	−	19.1076i	9.45074	+	32.1863i
5.6	−2.28717	+	2.63953i	−0.386777	+	0.289538i	−0.597472	−	4.15551i	6.20386	−	9.65339i	0.120380	−	1.68313i	7.78672	+	35.7950i	−11.1702	−	7.17863i	−7.54101	+	25.6823i	11.2912	+	38.4542i
5.7	−1.29080	+	1.48966i	2.95935	−	2.21534i	0.585593	+	4.07289i	9.17049	−	14.2696i	−0.519815	+	7.26796i	−5.54646	−	25.4967i	−20.0886	−	12.9102i	−3.75677	+	12.7944i	9.41953	+	32.0800i
5.8	−1.19665	+	1.38101i	3.75165	−	2.80845i	0.663308	+	4.61341i	−5.05753	+	7.86967i	−0.610920	+	8.54177i	0.977722	+	4.49452i	−19.4629	−	12.5080i	−1.41930	+	4.83371i	−4.81598	−	16.4017i
5.9	−0.586766	+	0.677164i	−7.69126	+	5.75761i	1.02426	+	7.12390i	−1.80029	+	2.80130i	0.614127	−	8.58661i	4.64468	+	21.3513i	−11.4553	−	7.36185i	18.3987	−	62.6602i	−0.840594	−	2.86280i
5.10	−0.514581	+	0.593858i	−3.43084	+	2.56829i	1.05064	+	7.30740i	−4.62619	+	7.19849i	0.240242	−	3.35903i	−4.77569	−	21.9535i	−10.1686	−	6.53494i	−2.43226	+	8.28353i	−1.89433	−	6.45151i
5.11	0.00569842	−	0.00657633i	−2.45836	+	1.84031i	1.13851	+	7.91850i	4.86749	−	7.57396i	−0.00190632	+	0.0266538i	0.853865	+	3.92515i	0.117125	+	0.0752718i	−4.94997	+	16.8580i	−0.0220718	−	0.0751698i
5.12	0.444562	−	0.513052i	6.93485	−	5.19136i	1.07293	+	7.46240i	6.74138	−	10.4898i	0.419532	−	5.86582i	3.36673	+	15.4766i	8.87437	+	5.70321i	13.5351	−	46.0963i	−2.38485	−	8.12204i
5.13	0.888831	−	1.02577i	2.72130	−	2.03714i	0.876344	+	6.09511i	−6.75457	+	10.5103i	0.329148	−	4.60209i	4.14185	+	19.0398i	16.1656	+	10.3890i	−4.35124	+	14.8190i	4.77745	+	16.2705i
5.14	1.28480	−	1.48274i	−2.79755	+	2.09422i	0.590717	+	4.10852i	7.73082	−	12.0294i	−0.489113	+	6.83869i	2.17497	+	9.99818i	20.0548	+	12.8884i	−4.16625	+	14.1889i	−7.90388	−	26.9182i
5.15	1.71794	−	1.98261i	5.40650	−	4.04726i	0.159103	+	1.10658i	−1.33480	+	2.07698i	1.26392	−	17.6719i	−6.58817	−	30.2853i	20.1226	+	12.9320i	5.24319	−	17.8567i	1.82474	+	6.21450i
5.16	1.75195	−	2.02186i	−1.99481	+	1.49329i	0.119935	+	0.834166i	−7.87308	+	12.2508i	−0.475574	+	6.64939i	−2.67216	−	12.2837i	19.9016	+	12.7900i	−5.85745	+	19.9487i	10.9761	+	37.3810i
5.17	2.24618	−	2.59222i	−7.51680	+	5.62701i	−0.535805	−	3.72661i	7.49185	−	11.6575i	−2.29759	+	32.1245i	−6.36703	−	29.2688i	12.2203	+	7.85352i	17.2323	−	58.6879i	−13.3910	−	45.6054i
5.18	2.62980	−	3.03495i	−5.20401	+	3.89567i	−1.15656	−	8.04405i	−5.07640	+	7.89904i	−1.86233	+	26.0387i	4.86511	+	22.3646i	−0.428268	−	0.275231i	4.29866	−	14.6399i	10.6233	+	36.1795i
5.19	2.89971	−	3.34644i	2.28774	−	1.71258i	−1.65185	−	11.4888i	5.25614	−	8.17872i	0.902727	−	12.6218i	1.13935	+	5.23751i	−13.4362	−	8.63494i	−5.30596	+	18.0704i	−12.1283	−	41.3053i
5.20	3.28222	−	3.78788i	7.69008	−	5.75672i	−2.43657	−	16.9467i	−10.1131	+	15.7363i	3.43474	−	48.0239i	4.54988	+	20.9155i	−38.4581	−	24.7155i	18.3907	−	62.6330i	26.4139	+	89.9574i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
89.g	even	44	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	89.4.g.a	✓	420
89.g	even	44	1	inner	89.4.g.a	✓	420

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
89.4.g.a	✓	420	1.a	even	1	1	trivial
89.4.g.a	✓	420	89.g	even	44	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(89, [\chi])\).