Base field 5.5.160801.1
Generator \(w\), with minimal polynomial \(x^{5} - x^{4} - 5x^{3} + 4x^{2} + 3x - 1\); narrow class number \(1\) and class number \(1\).
Form
| Weight: | $[2, 2, 2, 2, 2]$ |
| Level: | $[23, 23, -w^{2} + 3]$ |
| Dimension: | $14$ |
| CM: | no |
| Base change: | no |
| Newspace dimension: | $39$ |
Hecke eigenvalues ($q$-expansion)
The Hecke eigenvalue field is $\Q(e)$ where $e$ is a root of the defining polynomial:
| \(x^{14} + 3x^{13} - 23x^{12} - 66x^{11} + 203x^{10} + 531x^{9} - 906x^{8} - 1928x^{7} + 2296x^{6} + 3044x^{5} - 3349x^{4} - 1252x^{3} + 2214x^{2} - 748x + 76\) |
Show full eigenvalues Hide large eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| 3 | $[3, 3, w^{4} - w^{3} - 5w^{2} + 3w + 3]$ | $\phantom{-}e$ |
| 9 | $[9, 3, -w^{4} + 5w^{2} - 3]$ | $\phantom{-}\frac{87265}{2753488}e^{13} + \frac{104137}{1376744}e^{12} - \frac{2271245}{2753488}e^{11} - \frac{4898289}{2753488}e^{10} + \frac{5798719}{688372}e^{9} + \frac{43131035}{2753488}e^{8} - \frac{119118121}{2753488}e^{7} - \frac{176657031}{2753488}e^{6} + \frac{324775019}{2753488}e^{5} + \frac{331566585}{2753488}e^{4} - \frac{223777233}{1376744}e^{3} - \frac{106403649}{1376744}e^{2} + \frac{31023785}{344186}e - \frac{8902813}{688372}$ |
| 9 | $[9, 3, -w^{4} + w^{3} + 5w^{2} - 3w - 2]$ | $-\frac{76753}{2753488}e^{13} - \frac{27161}{1376744}e^{12} + \frac{2152181}{2753488}e^{11} + \frac{1076593}{2753488}e^{10} - \frac{5798209}{688372}e^{9} - \frac{7318763}{2753488}e^{8} + \frac{120488913}{2753488}e^{7} + \frac{19111383}{2753488}e^{6} - \frac{308639635}{2753488}e^{5} - \frac{10029761}{2753488}e^{4} + \frac{174555245}{1376744}e^{3} - \frac{9288727}{1376744}e^{2} - \frac{14473387}{344186}e + \frac{3818017}{688372}$ |
| 13 | $[13, 13, -w^{4} + w^{3} + 4w^{2} - 3w - 1]$ | $\phantom{-}\frac{281475}{2753488}e^{13} + \frac{109219}{344186}e^{12} - \frac{6126669}{2753488}e^{11} - \frac{18866695}{2753488}e^{10} + \frac{24395563}{1376744}e^{9} + \frac{147833247}{2753488}e^{8} - \frac{179727811}{2753488}e^{7} - \frac{518457295}{2753488}e^{6} + \frac{329158197}{2753488}e^{5} + \frac{795529109}{2753488}e^{4} - \frac{161735311}{1376744}e^{3} - \frac{197623161}{1376744}e^{2} + \frac{11857449}{172093}e - \frac{3266217}{688372}$ |
| 17 | $[17, 17, w^{4} - w^{3} - 5w^{2} + 3w + 1]$ | $\phantom{-}\frac{38443}{688372}e^{13} + \frac{345187}{1376744}e^{12} - \frac{1420917}{1376744}e^{11} - \frac{931768}{172093}e^{10} + \frac{8484411}{1376744}e^{9} + \frac{58460721}{1376744}e^{8} - \frac{8969817}{688372}e^{7} - \frac{206378121}{1376744}e^{6} + \frac{6938649}{688372}e^{5} + \frac{325105455}{1376744}e^{4} - \frac{5170413}{172093}e^{3} - \frac{86871895}{688372}e^{2} + \frac{19949243}{344186}e - \frac{2158867}{344186}$ |
| 19 | $[19, 19, -w^{3} + w^{2} + 4w - 2]$ | $-\frac{158225}{1376744}e^{13} - \frac{167459}{688372}e^{12} + \frac{3953511}{1376744}e^{11} + \frac{7368029}{1376744}e^{10} - \frac{4785067}{172093}e^{9} - \frac{59583401}{1376744}e^{8} + \frac{184512777}{1376744}e^{7} + \frac{219633899}{1376744}e^{6} - \frac{471084497}{1376744}e^{5} - \frac{359808495}{1376744}e^{4} + \frac{305606371}{688372}e^{3} + \frac{87210007}{688372}e^{2} - \frac{40271091}{172093}e + \frac{16438763}{344186}$ |
| 23 | $[23, 23, -w^{2} + 3]$ | $-1$ |
| 31 | $[31, 31, w^{3} - 4w + 2]$ | $\phantom{-}\frac{96751}{2753488}e^{13} + \frac{95051}{688372}e^{12} - \frac{2131157}{2753488}e^{11} - \frac{8691163}{2753488}e^{10} + \frac{9081793}{1376744}e^{9} + \frac{74038183}{2753488}e^{8} - \frac{82795367}{2753488}e^{7} - \frac{293360743}{2753488}e^{6} + \frac{242956393}{2753488}e^{5} + \frac{534662693}{2753488}e^{4} - \frac{227346507}{1376744}e^{3} - \frac{160003693}{1376744}e^{2} + \frac{23389309}{172093}e - \frac{22066501}{688372}$ |
| 32 | $[32, 2, 2]$ | $\phantom{-}\frac{891529}{2753488}e^{13} + \frac{704929}{688372}e^{12} - \frac{19730775}{2753488}e^{11} - \frac{61640253}{2753488}e^{10} + \frac{81538163}{1376744}e^{9} + \frac{491954421}{2753488}e^{8} - \frac{654146225}{2753488}e^{7} - \frac{1774198225}{2753488}e^{6} + \frac{1439901715}{2753488}e^{5} + \frac{2838835127}{2753488}e^{4} - \frac{945735705}{1376744}e^{3} - \frac{729805447}{1376744}e^{2} + \frac{80819728}{172093}e - \frac{55145747}{688372}$ |
| 37 | $[37, 37, w^{3} - 3w - 1]$ | $-\frac{1748999}{2753488}e^{13} - \frac{383663}{172093}e^{12} + \frac{37183937}{2753488}e^{11} + \frac{134245451}{2753488}e^{10} - \frac{144527643}{1376744}e^{9} - \frac{1072243003}{2753488}e^{8} + \frac{1062828471}{2753488}e^{7} + \frac{3876690315}{2753488}e^{6} - \frac{2149333985}{2753488}e^{5} - \frac{6258353753}{2753488}e^{4} + \frac{1433408091}{1376744}e^{3} + \frac{1674075965}{1376744}e^{2} - \frac{140880980}{172093}e + \frac{84630349}{688372}$ |
| 53 | $[53, 53, -2w^{4} + w^{3} + 9w^{2} - 3w - 2]$ | $-\frac{578021}{1376744}e^{13} - \frac{1973201}{1376744}e^{12} + \frac{1519657}{172093}e^{11} + \frac{42619771}{1376744}e^{10} - \frac{92198823}{1376744}e^{9} - \frac{167173495}{688372}e^{8} + \frac{319775109}{1376744}e^{7} + \frac{588651949}{688372}e^{6} - \frac{571975543}{1376744}e^{5} - \frac{913115517}{688372}e^{4} + \frac{332556433}{688372}e^{3} + \frac{228873969}{344186}e^{2} - \frac{134212665}{344186}e + \frac{9610605}{172093}$ |
| 59 | $[59, 59, -w^{4} + 5w^{2} + w - 4]$ | $\phantom{-}\frac{1309775}{2753488}e^{13} + \frac{1189305}{688372}e^{12} - \frac{27022709}{2753488}e^{11} - \frac{103518147}{2753488}e^{10} + \frac{99099621}{1376744}e^{9} + \frac{821386343}{2753488}e^{8} - \frac{648292991}{2753488}e^{7} - \frac{2946920831}{2753488}e^{6} + \frac{1074530753}{2753488}e^{5} + \frac{4733482605}{2753488}e^{4} - \frac{647940079}{1376744}e^{3} - \frac{1299359941}{1376744}e^{2} + \frac{76993625}{172093}e - \frac{26126653}{688372}$ |
| 61 | $[61, 61, -w^{4} + w^{3} + 5w^{2} - 4w]$ | $-\frac{413633}{1376744}e^{13} - \frac{202769}{172093}e^{12} + \frac{8208985}{1376744}e^{11} + \frac{35286601}{1376744}e^{10} - \frac{27970083}{688372}e^{9} - \frac{279813251}{1376744}e^{8} + \frac{156915057}{1376744}e^{7} + \frac{1003617033}{1376744}e^{6} - \frac{194264057}{1376744}e^{5} - \frac{1617304373}{1376744}e^{4} + \frac{118687297}{688372}e^{3} + \frac{453570527}{688372}e^{2} - \frac{43076594}{172093}e + \frac{3954145}{344186}$ |
| 67 | $[67, 67, -w^{4} + 6w^{2} + 2w - 4]$ | $-\frac{145505}{2753488}e^{13} - \frac{260721}{688372}e^{12} + \frac{1404851}{2753488}e^{11} + \frac{22476933}{2753488}e^{10} + \frac{5823881}{1376744}e^{9} - \frac{175630609}{2753488}e^{8} - \frac{181180263}{2753488}e^{7} + \frac{616371097}{2753488}e^{6} + \frac{706858545}{2753488}e^{5} - \frac{969800915}{2753488}e^{4} - \frac{483809539}{1376744}e^{3} + \frac{284428075}{1376744}e^{2} + \frac{17938019}{172093}e - \frac{22501325}{688372}$ |
| 71 | $[71, 71, 2w^{4} - w^{3} - 9w^{2} + 4w + 5]$ | $\phantom{-}\frac{212915}{688372}e^{13} + \frac{1827761}{1376744}e^{12} - \frac{7743237}{1376744}e^{11} - \frac{9810713}{344186}e^{10} + \frac{42095333}{1376744}e^{9} + \frac{304779829}{1376744}e^{8} - \frac{18914839}{688372}e^{7} - \frac{1057339323}{1376744}e^{6} - \frac{61532141}{344186}e^{5} + \frac{1618016707}{1376744}e^{4} + \frac{118268751}{344186}e^{3} - \frac{428604137}{688372}e^{2} - \frac{9713265}{344186}e + \frac{14051409}{344186}$ |
| 79 | $[79, 79, 2w^{4} - w^{3} - 10w^{2} + 2w + 7]$ | $\phantom{-}\frac{1659071}{1376744}e^{13} + \frac{5960223}{1376744}e^{12} - \frac{4345926}{172093}e^{11} - \frac{130119065}{1376744}e^{10} + \frac{263480845}{1376744}e^{9} + \frac{518494429}{688372}e^{8} - \frac{926857087}{1376744}e^{7} - \frac{1870480619}{688372}e^{6} + \frac{1770038849}{1376744}e^{5} + \frac{3019644991}{688372}e^{4} - \frac{1176109485}{688372}e^{3} - \frac{409410149}{172093}e^{2} + \frac{500795115}{344186}e - \frac{34534483}{172093}$ |
| 83 | $[83, 83, -w^{4} + 2w^{3} + 5w^{2} - 7w - 2]$ | $-\frac{856655}{1376744}e^{13} - \frac{358821}{172093}e^{12} + \frac{18278857}{1376744}e^{11} + \frac{62231899}{1376744}e^{10} - \frac{71171873}{688372}e^{9} - \frac{491137071}{1376744}e^{8} + \frac{519574899}{1376744}e^{7} + \frac{1747482395}{1376744}e^{6} - \frac{1015642457}{1376744}e^{5} - \frac{2765376209}{1376744}e^{4} + \frac{635822881}{688372}e^{3} + \frac{725200561}{688372}e^{2} - \frac{121630977}{172093}e + \frac{33203909}{344186}$ |
| 83 | $[83, 83, -w^{4} + w^{3} + 4w^{2} - 3w + 3]$ | $\phantom{-}\frac{3864673}{2753488}e^{13} + \frac{1636395}{344186}e^{12} - \frac{82317667}{2753488}e^{11} - \frac{284372845}{2753488}e^{10} + \frac{319314993}{1376744}e^{9} + \frac{2249566793}{2753488}e^{8} - \frac{2310675937}{2753488}e^{7} - \frac{8022015341}{2753488}e^{6} + \frac{4433808539}{2753488}e^{5} + \frac{12702794859}{2753488}e^{4} - \frac{2721833805}{1376744}e^{3} - \frac{3313621803}{1376744}e^{2} + \frac{262730664}{172093}e - \frac{152002707}{688372}$ |
| 83 | $[83, 83, w^{4} - w^{3} - 5w^{2} + 4w - 1]$ | $-\frac{141245}{172093}e^{13} - \frac{2083963}{688372}e^{12} + \frac{11553383}{688372}e^{11} + \frac{22639321}{344186}e^{10} - \frac{83265137}{688372}e^{9} - \frac{358110185}{688372}e^{8} + \frac{130982847}{344186}e^{7} + \frac{1275931863}{688372}e^{6} - \frac{99839438}{172093}e^{5} - \frac{2019028093}{688372}e^{4} + \frac{226638121}{344186}e^{3} + \frac{535317929}{344186}e^{2} - \frac{118544501}{172093}e + \frac{11469761}{172093}$ |
| 83 | $[83, 83, -w^{4} + w^{3} + 4w^{2} - 4w - 2]$ | $-\frac{37575}{2753488}e^{13} - \frac{185567}{1376744}e^{12} + \frac{478635}{2753488}e^{11} + \frac{8669903}{2753488}e^{10} - \frac{271455}{688372}e^{9} - \frac{76561741}{2753488}e^{8} + \frac{3036439}{2753488}e^{7} + \frac{323959065}{2753488}e^{6} - \frac{79582205}{2753488}e^{5} - \frac{670930471}{2753488}e^{4} + \frac{168876311}{1376744}e^{3} + \frac{270968835}{1376744}e^{2} - \frac{45468053}{344186}e + \frac{3959591}{688372}$ |
Atkin-Lehner eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| $23$ | $[23, 23, -w^{2} + 3]$ | $1$ |