Base field 4.4.16997.1
Generator \(w\), with minimal polynomial \(x^{4} - 6x^{2} - x + 5\); narrow class number \(1\) and class number \(1\).
Form
| Weight: | $[2, 2, 2, 2]$ |
| Level: | $[13, 13, -w^{2} + 3]$ |
| Dimension: | $13$ |
| CM: | no |
| Base change: | no |
| Newspace dimension: | $26$ |
Hecke eigenvalues ($q$-expansion)
The Hecke eigenvalue field is $\Q(e)$ where $e$ is a root of the defining polynomial:
| \(x^{13} - 40x^{11} - 7x^{10} + 584x^{9} + 274x^{8} - 3874x^{7} - 2989x^{6} + 11027x^{5} + 11035x^{4} - 9108x^{3} - 9394x^{2} + 691x + 133\) |
Show full eigenvalues Hide large eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| 5 | $[5, 5, w]$ | $\phantom{-}\frac{397241452766}{51119573161757}e^{12} - \frac{923496980306}{51119573161757}e^{11} - \frac{14276502552002}{51119573161757}e^{10} + \frac{31210651654142}{51119573161757}e^{9} + \frac{179010735972087}{51119573161757}e^{8} - \frac{333339644405821}{51119573161757}e^{7} - \frac{1017078029497001}{51119573161757}e^{6} + \frac{1426972292546288}{51119573161757}e^{5} + \frac{2548959204484532}{51119573161757}e^{4} - \frac{2374462773510714}{51119573161757}e^{3} - \frac{1952406615229471}{51119573161757}e^{2} + \frac{1338095940935911}{51119573161757}e + \frac{38203493163051}{51119573161757}$ |
| 5 | $[5, 5, -w^{2} + w + 2]$ | $\phantom{-}e$ |
| 7 | $[7, 7, -w^{2} + 2]$ | $\phantom{-}\frac{2376793337886}{51119573161757}e^{12} - \frac{4606801722967}{51119573161757}e^{11} - \frac{85381037146775}{51119573161757}e^{10} + \frac{152007629674266}{51119573161757}e^{9} + \frac{1073661707126261}{51119573161757}e^{8} - \frac{1522349873111369}{51119573161757}e^{7} - \frac{6125934869666048}{51119573161757}e^{6} + \frac{5671843343519696}{51119573161757}e^{5} + \frac{15355036486252268}{51119573161757}e^{4} - \frac{6658879280334025}{51119573161757}e^{3} - \frac{11307902281811172}{51119573161757}e^{2} + \frac{2027394774294945}{51119573161757}e + \frac{111528657278233}{51119573161757}$ |
| 13 | $[13, 13, -w^{2} + 3]$ | $-1$ |
| 13 | $[13, 13, w^{2} - w - 4]$ | $-\frac{3384084507805}{51119573161757}e^{12} + \frac{7209419227780}{51119573161757}e^{11} + \frac{123265088372270}{51119573161757}e^{10} - \frac{234256890312004}{51119573161757}e^{9} - \frac{1579406207904499}{51119573161757}e^{8} + \frac{2304002084009739}{51119573161757}e^{7} + \frac{9250153371621999}{51119573161757}e^{6} - \frac{8208090464015474}{51119573161757}e^{5} - \frac{23891124129002744}{51119573161757}e^{4} + \frac{8193861479222971}{51119573161757}e^{3} + \frac{18224935660646151}{51119573161757}e^{2} - \frac{1437018868051388}{51119573161757}e - \frac{574762180884137}{51119573161757}$ |
| 16 | $[16, 2, 2]$ | $\phantom{-}\frac{483575384427}{51119573161757}e^{12} - \frac{1535586042396}{51119573161757}e^{11} - \frac{15897280989795}{51119573161757}e^{10} + \frac{50214706438021}{51119573161757}e^{9} + \frac{171554339020006}{51119573161757}e^{8} - \frac{513352470808680}{51119573161757}e^{7} - \frac{806838145954883}{51119573161757}e^{6} + \frac{2093055440512763}{51119573161757}e^{5} + \frac{1638554289424318}{51119573161757}e^{4} - \frac{3301189783115799}{51119573161757}e^{3} - \frac{858230936787462}{51119573161757}e^{2} + \frac{1677219148397756}{51119573161757}e - \frac{163102923989126}{51119573161757}$ |
| 19 | $[19, 19, -w^{2} + w + 1]$ | $\phantom{-}\frac{5174326477375}{51119573161757}e^{12} - \frac{9602574558824}{51119573161757}e^{11} - \frac{188328658379274}{51119573161757}e^{10} + \frac{313807243367501}{51119573161757}e^{9} + \frac{2412586387738278}{51119573161757}e^{8} - \frac{3073084294545438}{51119573161757}e^{7} - \frac{14051159995588673}{51119573161757}e^{6} + \frac{10748215210925168}{51119573161757}e^{5} + \frac{35855992372029950}{51119573161757}e^{4} - \frac{9911694198075750}{51119573161757}e^{3} - \frac{26779400825729887}{51119573161757}e^{2} + \frac{1153921485635057}{51119573161757}e + \frac{697345109288758}{51119573161757}$ |
| 23 | $[23, 23, -w^{3} + 4w + 2]$ | $\phantom{-}\frac{3138498228088}{51119573161757}e^{12} - \frac{5441249698532}{51119573161757}e^{11} - \frac{115339218244481}{51119573161757}e^{10} + \frac{178238641629368}{51119573161757}e^{9} + \frac{1497203464217785}{51119573161757}e^{8} - \frac{1742369004304203}{51119573161757}e^{7} - \frac{8821305722007927}{51119573161757}e^{6} + \frac{5999565254722979}{51119573161757}e^{5} + \frac{22648346313306563}{51119573161757}e^{4} - \frac{5056838046133790}{51119573161757}e^{3} - \frac{16801310364394646}{51119573161757}e^{2} + \frac{157485110717672}{51119573161757}e - \frac{33454640718494}{51119573161757}$ |
| 25 | $[25, 5, -w^{3} + w^{2} + 3w - 1]$ | $-\frac{2112048358245}{51119573161757}e^{12} + \frac{3804617446292}{51119573161757}e^{11} + \frac{77333781103667}{51119573161757}e^{10} - \frac{121685045845508}{51119573161757}e^{9} - \frac{1001507477584330}{51119573161757}e^{8} + \frac{1134825033487880}{51119573161757}e^{7} + \frac{5937998073626249}{51119573161757}e^{6} - \frac{3472118254286137}{51119573161757}e^{5} - \frac{15500494141019391}{51119573161757}e^{4} + \frac{1336212360165940}{51119573161757}e^{3} + \frac{11798673063161602}{51119573161757}e^{2} + \frac{1442077552165566}{51119573161757}e - \frac{191790768150743}{51119573161757}$ |
| 29 | $[29, 29, w^{3} - w^{2} - 4w + 1]$ | $\phantom{-}\frac{2880010179084}{51119573161757}e^{12} - \frac{5052567755622}{51119573161757}e^{11} - \frac{103586300897990}{51119573161757}e^{10} + \frac{165823675844409}{51119573161757}e^{9} + \frac{1305468244566671}{51119573161757}e^{8} - \frac{1622350977551514}{51119573161757}e^{7} - \frac{7440250163225747}{51119573161757}e^{6} + \frac{5682348242830867}{51119573161757}e^{5} + \frac{18478297615391747}{51119573161757}e^{4} - \frac{5353941745747463}{51119573161757}e^{3} - \frac{13190070615416447}{51119573161757}e^{2} + \frac{712436254970164}{51119573161757}e + \frac{185432669131708}{51119573161757}$ |
| 29 | $[29, 29, -w + 3]$ | $\phantom{-}\frac{8878920643045}{51119573161757}e^{12} - \frac{17592718175118}{51119573161757}e^{11} - \frac{323267169013367}{51119573161757}e^{10} + \frac{570543135632856}{51119573161757}e^{9} + \frac{4141280410762886}{51119573161757}e^{8} - \frac{5544891634462524}{51119573161757}e^{7} - \frac{24183160456851801}{51119573161757}e^{6} + \frac{19128610225225241}{51119573161757}e^{5} + \frac{61955191780354506}{51119573161757}e^{4} - \frac{16839040433726348}{51119573161757}e^{3} - \frac{46055062864377660}{51119573161757}e^{2} + \frac{1339309272619149}{51119573161757}e + \frac{752488073277586}{51119573161757}$ |
| 31 | $[31, 31, -w^{3} + w^{2} + 5w - 2]$ | $-\frac{6249623915653}{51119573161757}e^{12} + \frac{11882396412203}{51119573161757}e^{11} + \frac{226716406202432}{51119573161757}e^{10} - \frac{386679799192851}{51119573161757}e^{9} - \frac{2891645788726759}{51119573161757}e^{8} + \frac{3762089753285752}{51119573161757}e^{7} + \frac{16788517416213804}{51119573161757}e^{6} - \frac{12987801009766040}{51119573161757}e^{5} - \frac{42786657586258710}{51119573161757}e^{4} + \frac{11317914163861189}{51119573161757}e^{3} + \frac{31809771221053471}{51119573161757}e^{2} - \frac{224466885768200}{51119573161757}e - \frac{408021743177138}{51119573161757}$ |
| 37 | $[37, 37, w^{3} - 4w - 1]$ | $-\frac{2213527398186}{51119573161757}e^{12} + \frac{4194685395627}{51119573161757}e^{11} + \frac{82857655931752}{51119573161757}e^{10} - \frac{133890691592611}{51119573161757}e^{9} - \frac{1101920969923644}{51119573161757}e^{8} + \frac{1252972032274090}{51119573161757}e^{7} + \frac{6694105323898894}{51119573161757}e^{6} - \frac{3757155511748005}{51119573161757}e^{5} - \frac{17720395454699688}{51119573161757}e^{4} + \frac{699527443718932}{51119573161757}e^{3} + \frac{13472803772370198}{51119573161757}e^{2} + \frac{2919066060586690}{51119573161757}e - \frac{320862078992967}{51119573161757}$ |
| 37 | $[37, 37, w^{3} - 3w + 1]$ | $\phantom{-}\frac{6093721547398}{51119573161757}e^{12} - \frac{11340378822212}{51119573161757}e^{11} - \frac{220027875386638}{51119573161757}e^{10} + \frac{368294459734752}{51119573161757}e^{9} + \frac{2788864646021312}{51119573161757}e^{8} - \frac{3556512416450267}{51119573161757}e^{7} - \frac{16072384805882107}{51119573161757}e^{6} + \frac{12087364120780692}{51119573161757}e^{5} + \frac{40615068317497635}{51119573161757}e^{4} - \frac{9976924800941422}{51119573161757}e^{3} - \frac{29759761895118044}{51119573161757}e^{2} - \frac{163460561686138}{51119573161757}e + \frac{517535509164818}{51119573161757}$ |
| 53 | $[53, 53, -w^{3} + 2w^{2} + 4w - 6]$ | $-\frac{9407835642272}{51119573161757}e^{12} + \frac{17005002368083}{51119573161757}e^{11} + \frac{341998143195434}{51119573161757}e^{10} - \frac{551515078677909}{51119573161757}e^{9} - \frac{4377370948935894}{51119573161757}e^{8} + \frac{5296851899066031}{51119573161757}e^{7} + \frac{25494367831067012}{51119573161757}e^{6} - \frac{17626029483016532}{51119573161757}e^{5} - \frac{65033092650768849}{51119573161757}e^{4} + \frac{12810796405821946}{51119573161757}e^{3} + \frac{48094114332191455}{51119573161757}e^{2} + \frac{1898620965627047}{51119573161757}e - \frac{827785409685030}{51119573161757}$ |
| 59 | $[59, 59, w^{2} + w - 4]$ | $-\frac{1917128959126}{51119573161757}e^{12} + \frac{3324355526859}{51119573161757}e^{11} + \frac{66913307645627}{51119573161757}e^{10} - \frac{111232238747114}{51119573161757}e^{9} - \frac{807374633581663}{51119573161757}e^{8} + \frac{1126096710012490}{51119573161757}e^{7} + \frac{4371144835858026}{51119573161757}e^{6} - \frac{4366992824132272}{51119573161757}e^{5} - \frac{10481309484360591}{51119573161757}e^{4} + \frac{5955529824705162}{51119573161757}e^{3} + \frac{8017086729789433}{51119573161757}e^{2} - \frac{2372263325062739}{51119573161757}e - \frac{431078304020883}{51119573161757}$ |
| 61 | $[61, 61, -2w^{2} + w + 8]$ | $\phantom{-}\frac{623376677226}{51119573161757}e^{12} - \frac{1226087259787}{51119573161757}e^{11} - \frac{21371257437371}{51119573161757}e^{10} + \frac{43083537359615}{51119573161757}e^{9} + \frac{248260363640881}{51119573161757}e^{8} - \frac{481502975391695}{51119573161757}e^{7} - \frac{1241140834624288}{51119573161757}e^{6} + \frac{2229765079262624}{51119573161757}e^{5} + \frac{2512203812007837}{51119573161757}e^{4} - \frac{4097122907353163}{51119573161757}e^{3} - \frac{1025158293197221}{51119573161757}e^{2} + \frac{1895432509554405}{51119573161757}e - \frac{501351596137206}{51119573161757}$ |
| 73 | $[73, 73, -w^{3} + 5w - 1]$ | $-\frac{5541827185663}{51119573161757}e^{12} + \frac{8848458854734}{51119573161757}e^{11} + \frac{202718112170386}{51119573161757}e^{10} - \frac{286620555753484}{51119573161757}e^{9} - \frac{2619435625537526}{51119573161757}e^{8} + \frac{2700716055162293}{51119573161757}e^{7} + \frac{15373474860605578}{51119573161757}e^{6} - \frac{8387613169518246}{51119573161757}e^{5} - \frac{39378317301604353}{51119573161757}e^{4} + \frac{3398558971359292}{51119573161757}e^{3} + \frac{29300044334241153}{51119573161757}e^{2} + \frac{4082862979651687}{51119573161757}e - \frac{619780374511653}{51119573161757}$ |
| 79 | $[79, 79, 2w^{2} + w - 6]$ | $\phantom{-}\frac{2865946305674}{51119573161757}e^{12} - \frac{6697437555774}{51119573161757}e^{11} - \frac{103812420956919}{51119573161757}e^{10} + \frac{222504387355181}{51119573161757}e^{9} + \frac{1316439985856412}{51119573161757}e^{8} - \frac{2305701659724549}{51119573161757}e^{7} - \frac{7593814088582855}{51119573161757}e^{6} + \frac{9268786236552640}{51119573161757}e^{5} + \frac{19229722295991972}{51119573161757}e^{4} - \frac{13314640389384276}{51119573161757}e^{3} - \frac{14129341299944311}{51119573161757}e^{2} + \frac{6378556709885381}{51119573161757}e + \frac{84906666279636}{51119573161757}$ |
| 79 | $[79, 79, 2w^{3} - w^{2} - 9w + 3]$ | $\phantom{-}\frac{1973035521367}{51119573161757}e^{12} - \frac{4405384985703}{51119573161757}e^{11} - \frac{71106300911906}{51119573161757}e^{10} + \frac{146612663131481}{51119573161757}e^{9} + \frac{895733460379519}{51119573161757}e^{8} - \frac{1517149407929289}{51119573161757}e^{7} - \frac{5120163352165937}{51119573161757}e^{6} + \frac{6091795023260899}{51119573161757}e^{5} + \frac{12824986056326664}{51119573161757}e^{4} - \frac{8829385689342666}{51119573161757}e^{3} - \frac{9246567481697891}{51119573161757}e^{2} + \frac{4634525122315108}{51119573161757}e + \frac{18058374682953}{51119573161757}$ |
Atkin-Lehner eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| $13$ | $[13, 13, -w^{2} + 3]$ | $1$ |