Newform orbit 89.4.f.a

• Label: 89.4.f.a
• Level: $89$
• Weight: $4$
• Character orbit: 89.f
• Analytic conductor: $5.251$
• Analytic rank: $0$
• Dimension: $220$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 89 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	89.f (of order \(22\), degree \(10\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.25116999051\)
Analytic rank:	\(0\)
Dimension:	\(220\)
Relative dimension:	\(22\) over \(\Q(\zeta_{22})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{22}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 220 q - 13 q^{2} - 33 q^{3} - 97 q^{4} + q^{5} - 11 q^{6} - 11 q^{7} - 41 q^{8} + 43 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
11.1	−3.60628	−	4.16187i	−5.42504	+	0.780003i	−3.17738	+	22.0992i	−12.5657	+	8.07549i	22.8105	+	19.7654i	−11.0371	−	17.1741i	66.3704	−	42.6537i	2.91638	−	0.856325i	78.9246	+	23.1744i
11.2	−3.51383	−	4.05518i	7.58268	−	1.09023i	−2.95894	+	20.5799i	3.79569	−	2.43934i	−31.0653	−	26.9183i	15.2878	+	23.7883i	57.7405	−	37.1076i	30.4022	−	8.92689i	−23.2294	−	6.82076i
11.3	−3.04684	−	3.51624i	−7.51401	+	1.08035i	−1.94219	+	13.5082i	15.3687	−	9.87684i	26.6927	+	23.1294i	15.2405	+	23.7147i	22.1032	−	14.2049i	29.3869	−	8.62877i	−81.5552	−	23.9468i
11.4	−2.91716	−	3.36658i	2.22175	−	0.319439i	−1.68553	+	11.7231i	3.43882	−	2.21000i	−7.55661	−	6.54784i	−8.52288	−	13.2619i	14.4041	−	9.25698i	−21.0722	+	6.18735i	−17.4717	−	5.13016i
11.5	−2.38570	−	2.75325i	−3.02346	+	0.434708i	−0.750279	+	5.21830i	−2.62884	+	1.68946i	8.40994	+	7.28725i	2.86440	+	4.45709i	−8.36073	+	5.37312i	−16.9540	+	4.97814i	10.9231	+	3.20732i
11.6	−2.07824	−	2.39841i	3.42363	−	0.492244i	−0.294800	+	2.05038i	−13.2425	+	8.51046i	−8.29572	−	7.18828i	11.1198	+	17.3027i	−15.8278	+	10.1719i	−14.4274	+	4.23626i	47.9327	+	14.0743i
11.7	−1.97475	−	2.27898i	10.1597	−	1.46075i	−0.155605	+	1.08226i	−4.45231	+	2.86133i	−23.3919	−	20.2692i	−18.2213	−	28.3529i	−17.5208	+	11.2600i	75.1796	−	22.0747i	15.3131	+	4.49634i
11.8	−1.59210	−	1.83738i	−5.97435	+	0.858982i	0.297335	−	2.06801i	9.61284	−	6.17780i	11.0900	+	9.60956i	−13.1893	−	20.5229i	−20.6351	+	13.2614i	9.04875	−	2.65695i	−26.6555	−	7.82676i
11.9	−1.31873	−	1.52190i	−9.97474	+	1.43415i	0.561399	−	3.90462i	−12.6439	+	8.12574i	15.3366	+	13.2893i	−2.58457	−	4.02167i	−20.2354	+	13.0045i	71.5323	−	21.0038i	29.0405	+	8.52705i
11.10	−1.01563	−	1.17210i	5.06569	−	0.728336i	0.796205	−	5.53773i	13.0474	−	8.38509i	−5.99855	−	5.19778i	4.55056	+	7.08081i	−17.7371	+	11.3989i	−0.775583	+	0.227732i	−23.0796	−	6.77677i
11.11	−0.394916	−	0.455758i	−3.67321	+	0.528128i	1.08676	−	7.55860i	2.45033	−	1.57473i	1.69131	+	1.46553i	3.96494	+	6.16956i	−7.93264	+	5.09800i	−12.6928	+	3.72693i	−1.68537	−	0.494869i
11.12	−0.0488520	−	0.0563782i	1.24844	−	0.179499i	1.13773	−	7.91306i	−13.9594	+	8.97118i	−0.0711088	−	0.0616162i	−11.9245	−	18.5549i	−1.00376	+	0.645077i	−24.3799	+	7.15859i	1.18773	+	0.348748i
11.13	0.617707	+	0.712872i	−4.88904	+	0.702939i	1.01189	−	7.03788i	−3.43980	+	2.21063i	−3.52110	−	3.05105i	19.4470	+	30.2601i	11.9904	−	7.70573i	−2.49767	+	0.733383i	−3.70068	−	1.08662i
11.14	0.841476	+	0.971115i	8.22301	−	1.18229i	0.903536	−	6.28423i	−5.49250	+	3.52982i	8.06760	+	6.99062i	9.61917	+	14.9677i	15.5109	−	9.96825i	40.3137	−	11.8372i	−8.04967	−	2.36360i
11.15	1.15146	+	1.32886i	−0.588539	+	0.0846191i	0.698521	−	4.85832i	8.18563	−	5.26058i	−0.790126	−	0.684648i	−13.8113	−	21.4907i	19.0939	−	12.2709i	−25.5671	+	7.50718i	16.4160	+	4.82017i
11.16	1.58379	+	1.82779i	5.82002	−	0.836793i	0.306091	−	2.12891i	4.29060	−	2.75740i	10.7472	+	9.31247i	−5.76809	−	8.97532i	20.6526	−	13.2726i	7.26611	−	2.13352i	11.8353	+	3.47517i
11.17	1.75788	+	2.02871i	−9.32455	+	1.34067i	0.113026	−	0.786112i	7.25873	−	4.66490i	−19.1113	−	16.5600i	−4.86592	−	7.57152i	19.8593	−	12.7628i	59.2436	−	17.3955i	22.2237	+	6.52547i
11.18	2.34196	+	2.70277i	−5.03668	+	0.724165i	−0.681650	+	4.74098i	−11.5856	+	7.44558i	−13.7530	−	11.9170i	−4.25047	−	6.61387i	9.65825	−	6.20698i	−1.06257	+	0.311998i	−47.2566	−	13.8758i
11.19	2.56495	+	2.96010i	−0.191455	+	0.0275271i	−1.04476	+	7.26644i	15.8553	−	10.1896i	−0.572556	−	0.496122i	12.7962	+	19.9113i	2.17087	−	1.39513i	−25.8704	+	7.59624i	70.8303	+	20.7977i
11.20	2.78858	+	3.21819i	2.75310	−	0.395835i	−1.44206	+	10.0298i	−12.5212	+	8.04690i	8.95110	+	7.75617i	7.53958	+	11.7318i	−7.64071	+	4.91038i	−18.4835	+	5.42723i	−60.8128	−	17.8563i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
89.f	even	22	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	89.4.f.a	✓	220
89.f	even	22	1	inner	89.4.f.a	✓	220

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
89.4.f.a	✓	220	1.a	even	1	1	trivial
89.4.f.a	✓	220	89.f	even	22	1	inner

Hecke kernels

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