Base field 4.4.18097.1
Generator \(w\), with minimal polynomial \(x^{4} - x^{3} - 7x^{2} + 6x + 4\); narrow class number \(2\) and class number \(1\).
Form
| Weight: | $[2, 2, 2, 2]$ |
| Level: | $[16, 4, -w^{3} + 6w]$ |
| Dimension: | $16$ |
| CM: | no |
| Base change: | no |
| Newspace dimension: | $28$ |
Hecke eigenvalues ($q$-expansion)
The Hecke eigenvalue field is $\Q(e)$ where $e$ is a root of the defining polynomial:
| \(x^{16} - 32x^{14} + 396x^{12} - 2398x^{10} + 7332x^{8} - 10340x^{6} + 5125x^{4} - 616x^{2} + 16\) |
Show full eigenvalues Hide large eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| 3 | $[3, 3, -w + 1]$ | $\phantom{-}e$ |
| 4 | $[4, 2, w]$ | $\phantom{-}0$ |
| 4 | $[4, 2, -\frac{1}{2}w^{3} + \frac{1}{2}w^{2} + \frac{7}{2}w - 3]$ | $\phantom{-}\frac{195113}{3937168}e^{15} - \frac{388405}{246073}e^{13} + \frac{4774234}{246073}e^{11} - \frac{228883817}{1968584}e^{9} + \frac{171723517}{492146}e^{7} - \frac{116053553}{246073}e^{5} + \frac{791911257}{3937168}e^{3} - \frac{7347781}{984292}e$ |
| 7 | $[7, 7, -\frac{1}{2}w^{3} + \frac{1}{2}w^{2} + \frac{5}{2}w - 2]$ | $\phantom{-}\frac{703}{246073}e^{14} - \frac{91669}{984292}e^{12} + \frac{1134441}{984292}e^{10} - \frac{6561085}{984292}e^{8} + \frac{17045363}{984292}e^{6} - \frac{13306071}{984292}e^{4} - \frac{5656301}{984292}e^{2} + \frac{190073}{246073}$ |
| 13 | $[13, 13, \frac{1}{2}w^{3} + \frac{1}{2}w^{2} - \frac{7}{2}w - 1]$ | $-\frac{12739}{492146}e^{14} + \frac{821287}{984292}e^{12} - \frac{10258775}{984292}e^{10} + \frac{62798855}{984292}e^{8} - \frac{193835105}{984292}e^{6} + \frac{272117801}{984292}e^{4} - \frac{122433547}{984292}e^{2} + \frac{1486427}{246073}$ |
| 17 | $[17, 17, \frac{1}{2}w^{3} - \frac{1}{2}w^{2} - \frac{5}{2}w]$ | $\phantom{-}\frac{315}{246073}e^{15} - \frac{7556}{246073}e^{13} + \frac{47010}{246073}e^{11} + \frac{160324}{246073}e^{9} - \frac{2879465}{246073}e^{7} + \frac{11042464}{246073}e^{5} - \frac{14714822}{246073}e^{3} + \frac{4066790}{246073}e$ |
| 27 | $[27, 3, -w^{3} + 7w + 1]$ | $-\frac{37443}{984292}e^{15} + \frac{1209849}{984292}e^{13} - \frac{15184769}{984292}e^{11} + \frac{93955947}{984292}e^{9} - \frac{297845491}{984292}e^{7} + \frac{451089923}{984292}e^{5} - \frac{134118711}{492146}e^{3} + \frac{11986965}{246073}e$ |
| 31 | $[31, 31, w + 3]$ | $\phantom{-}\frac{32679}{984292}e^{14} - \frac{525523}{492146}e^{12} + \frac{6516257}{492146}e^{10} - \frac{19619565}{246073}e^{8} + \frac{117328439}{492146}e^{6} - \frac{155369747}{492146}e^{4} + \frac{126834933}{984292}e^{2} - \frac{1935761}{246073}$ |
| 31 | $[31, 31, -w^{2} + 5]$ | $\phantom{-}\frac{35891}{984292}e^{15} - \frac{287101}{246073}e^{13} + \frac{3559592}{246073}e^{11} - \frac{43376783}{492146}e^{9} + \frac{67320567}{246073}e^{7} - \frac{98403499}{246073}e^{5} + \frac{216018727}{984292}e^{3} - \frac{9586686}{246073}e$ |
| 37 | $[37, 37, w^{2} - 3]$ | $-\frac{172447}{984292}e^{15} + \frac{2742431}{492146}e^{13} - \frac{33631705}{492146}e^{11} + \frac{100382466}{246073}e^{9} - \frac{598499197}{492146}e^{7} + \frac{799022559}{492146}e^{5} - \frac{664805701}{984292}e^{3} + \frac{8378084}{246073}e$ |
| 41 | $[41, 41, \frac{1}{2}w^{3} + \frac{1}{2}w^{2} - \frac{3}{2}w - 2]$ | $-\frac{273103}{1968584}e^{15} + \frac{2189267}{492146}e^{13} - \frac{27162521}{492146}e^{11} + \frac{330042145}{984292}e^{9} - \frac{506651391}{492146}e^{7} + \frac{718497137}{492146}e^{5} - \frac{1436760659}{1968584}e^{3} + \frac{43012313}{492146}e$ |
| 47 | $[47, 47, w^{3} - 5w - 3]$ | $\phantom{-}\frac{39963}{984292}e^{15} - \frac{1270297}{984292}e^{13} + \frac{15560849}{984292}e^{11} - \frac{92673355}{984292}e^{9} + \frac{274809771}{984292}e^{7} - \frac{362750211}{984292}e^{5} + \frac{75751569}{492146}e^{3} - \frac{5329823}{246073}e$ |
| 53 | $[53, 53, \frac{1}{2}w^{3} - \frac{1}{2}w^{2} - \frac{7}{2}w]$ | $\phantom{-}\frac{329441}{984292}e^{15} - \frac{10521711}{984292}e^{13} + \frac{129795067}{984292}e^{11} - \frac{781670201}{984292}e^{9} + \frac{2364621065}{984292}e^{7} - \frac{3251273465}{984292}e^{5} + \frac{735731687}{492146}e^{3} - \frac{23938777}{246073}e$ |
| 61 | $[61, 61, w^{3} - w^{2} - 5w + 3]$ | $\phantom{-}\frac{56897}{984292}e^{14} - \frac{1842109}{984292}e^{12} + \frac{23103249}{984292}e^{10} - \frac{141918411}{984292}e^{8} + \frac{440009843}{984292}e^{6} - \frac{627027151}{984292}e^{4} + \frac{77031942}{246073}e^{2} - \frac{6012540}{246073}$ |
| 83 | $[83, 83, w^{3} + w^{2} - 6w - 7]$ | $-\frac{187829}{984292}e^{15} + \frac{5995229}{984292}e^{13} - \frac{73859409}{984292}e^{11} + \frac{443572539}{984292}e^{9} - \frac{1333635243}{984292}e^{7} + \frac{1804846131}{984292}e^{5} - \frac{192056836}{246073}e^{3} + \frac{4938011}{246073}e$ |
| 83 | $[83, 83, -w^{3} + 5w + 1]$ | $\phantom{-}\frac{4969}{984292}e^{14} - \frac{91625}{492146}e^{12} + \frac{1308203}{492146}e^{10} - \frac{4521991}{246073}e^{8} + \frac{30594843}{492146}e^{6} - \frac{43144555}{492146}e^{4} + \frac{23616551}{984292}e^{2} + \frac{1943485}{246073}$ |
| 83 | $[83, 83, 2w - 1]$ | $\phantom{-}\frac{45788}{246073}e^{15} - \frac{1460800}{246073}e^{13} + \frac{18000341}{246073}e^{11} - \frac{108332063}{246073}e^{9} + \frac{328131660}{246073}e^{7} - \frac{455296774}{246073}e^{5} + \frac{217367216}{246073}e^{3} - \frac{22699339}{246073}e$ |
| 83 | $[83, 83, -\frac{3}{2}w^{3} - \frac{1}{2}w^{2} + \frac{19}{2}w + 2]$ | $-\frac{57685}{984292}e^{14} + \frac{1801641}{984292}e^{12} - \frac{21627233}{984292}e^{10} + \frac{126178015}{984292}e^{8} - \frac{368577651}{984292}e^{6} + \frac{489806455}{984292}e^{4} - \frac{54621859}{246073}e^{2} + \frac{4180060}{246073}$ |
| 89 | $[89, 89, \frac{1}{2}w^{3} - \frac{1}{2}w^{2} - \frac{7}{2}w - 2]$ | $\phantom{-}\frac{14145}{246073}e^{14} - \frac{456478}{246073}e^{12} + \frac{5696608}{246073}e^{10} - \frac{34679970}{246073}e^{8} + \frac{105440234}{246073}e^{6} - \frac{142711936}{246073}e^{4} + \frac{58880769}{246073}e^{2} - \frac{4561292}{246073}$ |
| 89 | $[89, 89, w^{3} - 5w + 5]$ | $\phantom{-}\frac{12594}{246073}e^{14} - \frac{798707}{492146}e^{12} + \frac{9770221}{492146}e^{10} - \frac{58314033}{492146}e^{8} + \frac{175128467}{492146}e^{6} - \frac{241631057}{492146}e^{4} + \frac{115161969}{492146}e^{2} - \frac{2602870}{246073}$ |
Atkin-Lehner eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| $4$ | $[4, 2, w]$ | $1$ |