Base field 4.4.19429.1
Generator \(w\), with minimal polynomial \(x^{4} - x^{3} - 7x^{2} - x + 5\); narrow class number \(1\) and class number \(1\).
Form
| Weight: | $[2, 2, 2, 2]$ |
| Level: | $[13, 13, -w^{3} + 2w^{2} + 5w - 2]$ |
| Dimension: | $15$ |
| CM: | no |
| Base change: | no |
| Newspace dimension: | $34$ |
Hecke eigenvalues ($q$-expansion)
The Hecke eigenvalue field is $\Q(e)$ where $e$ is a root of the defining polynomial:
| \(x^{15} + 9x^{14} + 8x^{13} - 153x^{12} - 470x^{11} + 479x^{10} + 3859x^{9} + 3270x^{8} - 9199x^{7} - 18908x^{6} - 3620x^{5} + 19857x^{4} + 20777x^{3} + 7302x^{2} + 279x - 222\) |
Show full eigenvalues Hide large eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| 3 | $[3, 3, -w^{3} + 2w^{2} + 5w - 3]$ | $\phantom{-}e$ |
| 5 | $[5, 5, w]$ | $\phantom{-}\frac{435505617}{811044511}e^{14} + \frac{6801625159}{1622089022}e^{13} - \frac{1076475709}{1622089022}e^{12} - \frac{65841405896}{811044511}e^{11} - \frac{126694412848}{811044511}e^{10} + \frac{356215280596}{811044511}e^{9} + \frac{2513217022417}{1622089022}e^{8} - \frac{96111283201}{1622089022}e^{7} - \frac{3928351421870}{811044511}e^{6} - \frac{3625942311123}{811044511}e^{5} + \frac{2644667661033}{811044511}e^{4} + \frac{5548303038369}{811044511}e^{3} + \frac{5125479741441}{1622089022}e^{2} + \frac{326836350671}{1622089022}e - \frac{83235055849}{811044511}$ |
| 7 | $[7, 7, -w^{3} + 2w^{2} + 4w - 3]$ | $\phantom{-}\frac{324267435}{1622089022}e^{14} + \frac{2813453893}{1622089022}e^{13} + \frac{1380276071}{1622089022}e^{12} - \frac{25928085432}{811044511}e^{11} - \frac{65984926531}{811044511}e^{10} + \frac{236588683777}{1622089022}e^{9} + \frac{1192487158943}{1622089022}e^{8} + \frac{402284594469}{1622089022}e^{7} - \frac{1743530735514}{811044511}e^{6} - \frac{2168273468360}{811044511}e^{5} + \frac{853895888244}{811044511}e^{4} + \frac{5845380525281}{1622089022}e^{3} + \frac{3229635389325}{1622089022}e^{2} + \frac{306826660933}{1622089022}e - \frac{56146365761}{811044511}$ |
| 13 | $[13, 13, -w^{2} + w + 4]$ | $-\frac{545407735}{1622089022}e^{14} - \frac{2152707235}{811044511}e^{13} + \frac{175047062}{811044511}e^{12} + \frac{41371657603}{811044511}e^{11} + \frac{82616767230}{811044511}e^{10} - \frac{437503915383}{1622089022}e^{9} - \frac{807100634565}{811044511}e^{8} - \frac{19819010328}{811044511}e^{7} + \frac{2486920757561}{811044511}e^{6} + \frac{2440421363437}{811044511}e^{5} - \frac{1557188558279}{811044511}e^{4} - \frac{7250590776411}{1622089022}e^{3} - \frac{1803844446666}{811044511}e^{2} - \frac{156332452968}{811044511}e + \frac{59356191193}{811044511}$ |
| 13 | $[13, 13, -w^{3} + 2w^{2} + 5w - 2]$ | $\phantom{-}1$ |
| 16 | $[16, 2, 2]$ | $-\frac{1636670875}{1622089022}e^{14} - \frac{12767001459}{1622089022}e^{13} + \frac{1042127553}{811044511}e^{12} + \frac{123492219819}{811044511}e^{11} + \frac{237055343238}{811044511}e^{10} - \frac{1333055531511}{1622089022}e^{9} - \frac{4697622534065}{1622089022}e^{8} + \frac{74695106633}{811044511}e^{7} + \frac{7311125932064}{811044511}e^{6} + \frac{6846932108035}{811044511}e^{5} - \frac{4788408162488}{811044511}e^{4} - \frac{20845622143349}{1622089022}e^{3} - \frac{9964294479695}{1622089022}e^{2} - \frac{365935293813}{811044511}e + \frac{162025579201}{811044511}$ |
| 17 | $[17, 17, -w + 2]$ | $\phantom{-}\frac{481022950}{811044511}e^{14} + \frac{7433785589}{1622089022}e^{13} - \frac{1643717011}{1622089022}e^{12} - \frac{72095323758}{811044511}e^{11} - \frac{134670710641}{811044511}e^{10} + \frac{392351563992}{811044511}e^{9} + \frac{2690012800907}{1622089022}e^{8} - \frac{152879254947}{1622089022}e^{7} - \frac{4190668905388}{811044511}e^{6} - \frac{3882070134299}{811044511}e^{5} + \frac{2723409770492}{811044511}e^{4} + \frac{5949606526747}{811044511}e^{3} + \frac{5787325518843}{1622089022}e^{2} + \frac{466317155371}{1622089022}e - \frac{94009140799}{811044511}$ |
| 19 | $[19, 19, -w^{3} + 2w^{2} + 3w - 2]$ | $\phantom{-}\frac{529461401}{1622089022}e^{14} + \frac{4114594035}{1622089022}e^{13} - \frac{312504129}{811044511}e^{12} - \frac{39558677466}{811044511}e^{11} - \frac{77015479446}{811044511}e^{10} + \frac{418932304595}{1622089022}e^{9} + \frac{1518038768285}{1622089022}e^{8} + \frac{16386244695}{811044511}e^{7} - \frac{2338525280468}{811044511}e^{6} - \frac{2326910987844}{811044511}e^{5} + \frac{1435311970424}{811044511}e^{4} + \frac{6925537935557}{1622089022}e^{3} + \frac{3499485515767}{1622089022}e^{2} + \frac{149830899646}{811044511}e - \frac{61495507967}{811044511}$ |
| 27 | $[27, 3, w^{3} - 3w^{2} - 4w + 7]$ | $\phantom{-}\frac{197654297}{1622089022}e^{14} + \frac{1297608839}{1622089022}e^{13} - \frac{1003196073}{811044511}e^{12} - \frac{14147044367}{811044511}e^{11} - \frac{10426726234}{811044511}e^{10} + \frac{206396013651}{1622089022}e^{9} + \frac{329417898885}{1622089022}e^{8} - \frac{277128062666}{811044511}e^{7} - \frac{664313927439}{811044511}e^{6} + \frac{117999027805}{811044511}e^{5} + \frac{887178599159}{811044511}e^{4} + \frac{599274286523}{1622089022}e^{3} - \frac{466415101207}{1622089022}e^{2} - \frac{74903893074}{811044511}e + \frac{12420255578}{811044511}$ |
| 31 | $[31, 31, w^{3} - 3w^{2} - 3w + 7]$ | $\phantom{-}\frac{263589937}{1622089022}e^{14} + \frac{933548402}{811044511}e^{13} - \frac{1534205031}{1622089022}e^{12} - \frac{18893379724}{811044511}e^{11} - \frac{25375127556}{811044511}e^{10} + \frac{232004108195}{1622089022}e^{9} + \frac{288731540126}{811044511}e^{8} - \frac{303856532097}{1622089022}e^{7} - \frac{964107611279}{811044511}e^{6} - \frac{573772650511}{811044511}e^{5} + \frac{782489053623}{811044511}e^{4} + \frac{2230486757195}{1622089022}e^{3} + \frac{441008520577}{811044511}e^{2} + \frac{30974104397}{1622089022}e - \frac{14852852405}{811044511}$ |
| 31 | $[31, 31, -w^{3} + w^{2} + 6w + 1]$ | $-\frac{1935771071}{1622089022}e^{14} - \frac{7539448178}{811044511}e^{13} + \frac{2731887721}{1622089022}e^{12} + \frac{146325140696}{811044511}e^{11} + \frac{277990989783}{811044511}e^{10} - \frac{1595464911519}{1622089022}e^{9} - \frac{2768525909066}{811044511}e^{8} + \frac{337064027573}{1622089022}e^{7} + \frac{8680234484234}{811044511}e^{6} + \frac{7844338675880}{811044511}e^{5} - \frac{5918567469509}{811044511}e^{4} - \frac{24189791689443}{1622089022}e^{3} - \frac{5542351539600}{811044511}e^{2} - \frac{743641813743}{1622089022}e + \frac{170500011088}{811044511}$ |
| 41 | $[41, 41, w^{2} - w - 1]$ | $\phantom{-}\frac{72942842}{811044511}e^{14} + \frac{788038073}{811044511}e^{13} + \frac{1165067605}{811044511}e^{12} - \frac{13932952602}{811044511}e^{11} - \frac{48544142799}{811044511}e^{10} + \frac{53083916665}{811044511}e^{9} + \frac{409275666260}{811044511}e^{8} + \frac{213686002661}{811044511}e^{7} - \frac{1190225608684}{811044511}e^{6} - \frac{1527798606778}{811044511}e^{5} + \frac{679535389962}{811044511}e^{4} + \frac{1967722400082}{811044511}e^{3} + \frac{922459967312}{811044511}e^{2} + \frac{61063528019}{811044511}e - \frac{28988421090}{811044511}$ |
| 43 | $[43, 43, 2w^{3} - 5w^{2} - 6w + 4]$ | $-\frac{287577811}{1622089022}e^{14} - \frac{1330742707}{811044511}e^{13} - \frac{1149162488}{811044511}e^{12} + \frac{23832375545}{811044511}e^{11} + \frac{70001374084}{811044511}e^{10} - \frac{192721968089}{1622089022}e^{9} - \frac{609140443303}{811044511}e^{8} - \frac{309801899375}{811044511}e^{7} + \frac{1737611147804}{811044511}e^{6} + \frac{2421232132712}{811044511}e^{5} - \frac{744716757509}{811044511}e^{4} - \frac{6283083692943}{1622089022}e^{3} - \frac{1765012232701}{811044511}e^{2} - \frac{150644774807}{811044511}e + \frac{66110662381}{811044511}$ |
| 47 | $[47, 47, -w^{3} + 2w^{2} + 5w - 1]$ | $\phantom{-}\frac{532768587}{1622089022}e^{14} + \frac{1976761417}{811044511}e^{13} - \frac{2051331333}{1622089022}e^{12} - \frac{39433612012}{811044511}e^{11} - \frac{62914671055}{811044511}e^{10} + \frac{465920098935}{1622089022}e^{9} + \frac{671193035512}{811044511}e^{8} - \frac{451548236147}{1622089022}e^{7} - \frac{2205568241498}{811044511}e^{6} - \frac{1505354564419}{811044511}e^{5} + \frac{1787843428974}{811044511}e^{4} + \frac{5300995996067}{1622089022}e^{3} + \frac{981672443824}{811044511}e^{2} + \frac{52753684257}{1622089022}e - \frac{28044214058}{811044511}$ |
| 53 | $[53, 53, -w - 3]$ | $\phantom{-}\frac{558291333}{811044511}e^{14} + \frac{8539143211}{1622089022}e^{13} - \frac{2458222605}{1622089022}e^{12} - \frac{83210114702}{811044511}e^{11} - \frac{150496882328}{811044511}e^{10} + \frac{459399462151}{811044511}e^{9} + \frac{3043179886551}{1622089022}e^{8} - \frac{306158148899}{1622089022}e^{7} - \frac{4778786442869}{811044511}e^{6} - \frac{4268709668929}{811044511}e^{5} + \frac{3207051236573}{811044511}e^{4} + \frac{6679183259438}{811044511}e^{3} + \frac{6303138688987}{1622089022}e^{2} + \frac{414101300255}{1622089022}e - \frac{108428071503}{811044511}$ |
| 53 | $[53, 53, -w^{3} + 3w^{2} + 3w - 6]$ | $-\frac{648838029}{811044511}e^{14} - \frac{5294141833}{811044511}e^{13} - \frac{700736099}{811044511}e^{12} + \frac{100099653333}{811044511}e^{11} + \frac{220283449082}{811044511}e^{10} - \frac{501202146537}{811044511}e^{9} - \frac{2083665660690}{811044511}e^{8} - \frac{331191330598}{811044511}e^{7} + \frac{6285433983280}{811044511}e^{6} + \frac{6873136102742}{811044511}e^{5} - \frac{3584984940645}{811044511}e^{4} - \frac{9788841438246}{811044511}e^{3} - \frac{5081404591737}{811044511}e^{2} - \frac{418037539968}{811044511}e + \frac{178999903975}{811044511}$ |
| 59 | $[59, 59, 2w^{2} - 3w - 6]$ | $\phantom{-}\frac{925656515}{1622089022}e^{14} + \frac{7611078337}{1622089022}e^{13} + \frac{760929771}{811044511}e^{12} - \frac{71360481365}{811044511}e^{11} - \frac{162420492806}{811044511}e^{10} + \frac{694215776229}{1622089022}e^{9} + \frac{3037776647191}{1622089022}e^{8} + \frac{337880166657}{811044511}e^{7} - \frac{4527440295823}{811044511}e^{6} - \frac{5239443608814}{811044511}e^{5} + \frac{2403107359348}{811044511}e^{4} + \frac{14611501467809}{1622089022}e^{3} + \frac{7858883547803}{1622089022}e^{2} + \frac{356202148764}{811044511}e - \frac{130947804460}{811044511}$ |
| 59 | $[59, 59, w^{3} - w^{2} - 7w - 3]$ | $-\frac{328965366}{811044511}e^{14} - \frac{4962376159}{1622089022}e^{13} + \frac{1929007143}{1622089022}e^{12} + \frac{48845801420}{811044511}e^{11} + \frac{83844968653}{811044511}e^{10} - \frac{277928821722}{811044511}e^{9} - \frac{1734458387017}{1622089022}e^{8} + \frac{346392598331}{1622089022}e^{7} + \frac{2779006912057}{811044511}e^{6} + \frac{2221268770256}{811044511}e^{5} - \frac{2038213641410}{811044511}e^{4} - \frac{3627271638627}{811044511}e^{3} - \frac{3138525851463}{1622089022}e^{2} - \frac{195695579163}{1622089022}e + \frac{46247670337}{811044511}$ |
| 79 | $[79, 79, 2w^{3} - 4w^{2} - 8w + 3]$ | $-\frac{350205481}{811044511}e^{14} - \frac{5861915827}{1622089022}e^{13} - \frac{602606446}{811044511}e^{12} + \frac{55999668385}{811044511}e^{11} + \frac{124957286758}{811044511}e^{10} - \frac{290244631575}{811044511}e^{9} - \frac{2373225298203}{1622089022}e^{8} - \frac{89108198542}{811044511}e^{7} + \frac{3692207030608}{811044511}e^{6} + \frac{3543513791850}{811044511}e^{5} - \frac{2616970693792}{811044511}e^{4} - \frac{5266750609311}{811044511}e^{3} - \frac{4321196264713}{1622089022}e^{2} - \frac{45133123743}{811044511}e + \frac{69771700108}{811044511}$ |
| 79 | $[79, 79, w^{2} - 2w - 1]$ | $-\frac{962376445}{1622089022}e^{14} - \frac{7202613793}{1622089022}e^{13} + \frac{1609731754}{811044511}e^{12} + \frac{71228866708}{811044511}e^{11} + \frac{118396128789}{811044511}e^{10} - \frac{821627740089}{1622089022}e^{9} - \frac{2478036408977}{1622089022}e^{8} + \frac{308229423335}{811044511}e^{7} + \frac{3997414558384}{811044511}e^{6} + \frac{3049875858398}{811044511}e^{5} - \frac{3001464079925}{811044511}e^{4} - \frac{10189198569567}{1622089022}e^{3} - \frac{4311557121035}{1622089022}e^{2} - \frac{129404994862}{811044511}e + \frac{68710705113}{811044511}$ |
Atkin-Lehner eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| $13$ | $[13, 13, -w^{3} + 2w^{2} + 5w - 2]$ | $-1$ |