Base field 4.4.15952.1
Generator \(w\), with minimal polynomial \(x^{4} - 6x^{2} - 2x + 1\); narrow class number \(2\) and class number \(1\).
Form
| Weight: | $[2, 2, 2, 2]$ |
| Level: | $[13, 13, -w^{3} + w^{2} + 4w + 1]$ |
| Dimension: | $14$ |
| 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:
| \(x^{14} + 3x^{13} - 16x^{12} - 52x^{11} + 88x^{10} + 332x^{9} - 189x^{8} - 990x^{7} + 65x^{6} + 1421x^{5} + 281x^{4} - 867x^{3} - 285x^{2} + 129x + 34\) |
Show full eigenvalues Hide large eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| 2 | $[2, 2, w + 1]$ | $\phantom{-}e$ |
| 3 | $[3, 3, -w^{3} + w^{2} + 5w - 2]$ | $\phantom{-}\frac{162}{10427}e^{13} + \frac{951}{10427}e^{12} + \frac{717}{10427}e^{11} - \frac{12738}{10427}e^{10} - \frac{37368}{10427}e^{9} + \frac{50794}{10427}e^{8} + \frac{245324}{10427}e^{7} - \frac{52093}{10427}e^{6} - \frac{595274}{10427}e^{5} - \frac{46094}{10427}e^{4} + \frac{542697}{10427}e^{3} + \frac{79735}{10427}e^{2} - \frac{127987}{10427}e - \frac{23042}{10427}$ |
| 11 | $[11, 11, -w^{3} + 5w + 1]$ | $-\frac{2463}{10427}e^{13} - \frac{4611}{10427}e^{12} + \frac{47606}{10427}e^{11} + \frac{83602}{10427}e^{10} - \frac{349444}{10427}e^{9} - \frac{561786}{10427}e^{8} + \frac{1228012}{10427}e^{7} + \frac{1752256}{10427}e^{6} - \frac{2103431}{10427}e^{5} - \frac{2559762}{10427}e^{4} + \frac{1532418}{10427}e^{3} + \frac{1493153}{10427}e^{2} - \frac{271792}{10427}e - \frac{162144}{10427}$ |
| 11 | $[11, 11, -w + 2]$ | $\phantom{-}\frac{129}{10427}e^{13} + \frac{178}{10427}e^{12} - \frac{3484}{10427}e^{11} - \frac{5509}{10427}e^{10} + \frac{32806}{10427}e^{9} + \frac{52805}{10427}e^{8} - \frac{133293}{10427}e^{7} - \frac{196921}{10427}e^{6} + \frac{227684}{10427}e^{5} + \frac{258341}{10427}e^{4} - \frac{117587}{10427}e^{3} - \frac{35950}{10427}e^{2} - \frac{15217}{10427}e - \frac{51174}{10427}$ |
| 13 | $[13, 13, -w^{3} + w^{2} + 4w + 1]$ | $\phantom{-}1$ |
| 17 | $[17, 17, -w^{3} + w^{2} + 6w - 1]$ | $-\frac{612}{10427}e^{13} - \frac{117}{10427}e^{12} + \frac{11194}{10427}e^{11} - \frac{538}{10427}e^{10} - \frac{77799}{10427}e^{9} + \frac{25920}{10427}e^{8} + \frac{263057}{10427}e^{7} - \frac{163515}{10427}e^{6} - \frac{439036}{10427}e^{5} + \frac{384990}{10427}e^{4} + \frac{278508}{10427}e^{3} - \frac{351039}{10427}e^{2} - \frac{5404}{10427}e + \frac{75462}{10427}$ |
| 17 | $[17, 17, -w^{2} - w + 3]$ | $\phantom{-}\frac{4884}{10427}e^{13} + \frac{10134}{10427}e^{12} - \frac{87288}{10427}e^{11} - \frac{173170}{10427}e^{10} + \frac{583452}{10427}e^{9} + \frac{1078736}{10427}e^{8} - \frac{1855388}{10427}e^{7} - \frac{3079333}{10427}e^{6} + \frac{2877855}{10427}e^{5} + \frac{4086071}{10427}e^{4} - \frac{1866245}{10427}e^{3} - \frac{2158143}{10427}e^{2} + \frac{253915}{10427}e + \frac{201276}{10427}$ |
| 23 | $[23, 23, w^{3} - 6w]$ | $-\frac{2130}{10427}e^{13} - \frac{4394}{10427}e^{12} + \frac{36915}{10427}e^{11} + \frac{71321}{10427}e^{10} - \frac{238570}{10427}e^{9} - \frac{413351}{10427}e^{8} + \frac{739407}{10427}e^{7} + \frac{1078063}{10427}e^{6} - \frac{1142014}{10427}e^{5} - \frac{1303635}{10427}e^{4} + \frac{748510}{10427}e^{3} + \frac{625359}{10427}e^{2} - \frac{73192}{10427}e - \frac{29932}{10427}$ |
| 27 | $[27, 3, w^{3} + w^{2} - 5w - 4]$ | $-\frac{4497}{10427}e^{13} - \frac{9600}{10427}e^{12} + \frac{76836}{10427}e^{11} + \frac{156643}{10427}e^{10} - \frac{485034}{10427}e^{9} - \frac{909894}{10427}e^{8} + \frac{1455509}{10427}e^{7} + \frac{2342592}{10427}e^{6} - \frac{2194803}{10427}e^{5} - \frac{2664574}{10427}e^{4} + \frac{1523911}{10427}e^{3} + \frac{1059728}{10427}e^{2} - \frac{351701}{10427}e - \frac{21134}{10427}$ |
| 41 | $[41, 41, -w^{3} + w^{2} + 5w]$ | $\phantom{-}\frac{6312}{10427}e^{13} + \frac{10407}{10427}e^{12} - \frac{116883}{10427}e^{11} - \frac{178866}{10427}e^{10} + \frac{817118}{10427}e^{9} + \frac{1122526}{10427}e^{8} - \frac{2736814}{10427}e^{7} - \frac{3229575}{10427}e^{6} + \frac{4500087}{10427}e^{5} + \frac{4313877}{10427}e^{4} - \frac{3172998}{10427}e^{3} - \frac{2298336}{10427}e^{2} + \frac{561956}{10427}e + \frac{192030}{10427}$ |
| 41 | $[41, 41, 2w^{3} - 11w - 4]$ | $-\frac{1239}{10427}e^{13} - \frac{4377}{10427}e^{12} + \frac{14791}{10427}e^{11} + \frac{63824}{10427}e^{10} - \frac{37441}{10427}e^{9} - \frac{300816}{10427}e^{8} - \frac{90554}{10427}e^{7} + \frac{494382}{10427}e^{6} + \frac{422107}{10427}e^{5} - \frac{76518}{10427}e^{4} - \frac{390535}{10427}e^{3} - \frac{265541}{10427}e^{2} + \frac{10118}{10427}e - \frac{258}{10427}$ |
| 53 | $[53, 53, 2w^{3} - 2w^{2} - 11w + 6]$ | $-\frac{349}{10427}e^{13} + \frac{1620}{10427}e^{12} + \frac{10234}{10427}e^{11} - \frac{26238}{10427}e^{10} - \frac{100232}{10427}e^{9} + \frac{155239}{10427}e^{8} + \frac{420994}{10427}e^{7} - \frac{445362}{10427}e^{6} - \frac{802860}{10427}e^{5} + \frac{693769}{10427}e^{4} + \frac{717178}{10427}e^{3} - \frac{519792}{10427}e^{2} - \frac{306964}{10427}e + \frac{44362}{10427}$ |
| 59 | $[59, 59, 2w^{3} - 11w - 2]$ | $\phantom{-}\frac{10256}{10427}e^{13} + \frac{21588}{10427}e^{12} - \frac{176278}{10427}e^{11} - \frac{354975}{10427}e^{10} + \frac{1113425}{10427}e^{9} + \frac{2081602}{10427}e^{8} - \frac{3280466}{10427}e^{7} - \frac{5415133}{10427}e^{6} + \frac{4636947}{10427}e^{5} + \frac{6257354}{10427}e^{4} - \frac{2729498}{10427}e^{3} - \frac{2701921}{10427}e^{2} + \frac{392876}{10427}e + \frac{199264}{10427}$ |
| 67 | $[67, 67, w^{3} - 7w - 1]$ | $\phantom{-}\frac{5296}{10427}e^{13} + \frac{7146}{10427}e^{12} - \frac{98981}{10427}e^{11} - \frac{117129}{10427}e^{10} + \frac{700433}{10427}e^{9} + \frac{677941}{10427}e^{8} - \frac{2384884}{10427}e^{7} - \frac{1692049}{10427}e^{6} + \frac{4021546}{10427}e^{5} + \frac{1736179}{10427}e^{4} - \frac{2968451}{10427}e^{3} - \frac{518880}{10427}e^{2} + \frac{583516}{10427}e - \frac{33940}{10427}$ |
| 71 | $[71, 71, 3w^{3} - 2w^{2} - 15w + 1]$ | $-\frac{4269}{10427}e^{13} - \frac{7103}{10427}e^{12} + \frac{75528}{10427}e^{11} + \frac{119020}{10427}e^{10} - \frac{495918}{10427}e^{9} - \frac{717144}{10427}e^{8} + \frac{1540491}{10427}e^{7} + \frac{1933681}{10427}e^{6} - \frac{2381874}{10427}e^{5} - \frac{2339400}{10427}e^{4} + \frac{1750137}{10427}e^{3} + \frac{1110158}{10427}e^{2} - \frac{496688}{10427}e - \frac{154744}{10427}$ |
| 79 | $[79, 79, -3w^{3} + w^{2} + 16w + 3]$ | $-\frac{72}{10427}e^{13} + \frac{3053}{10427}e^{12} + \frac{3157}{10427}e^{11} - \frac{42998}{10427}e^{10} - \frac{25100}{10427}e^{9} + \frac{177855}{10427}e^{8} + \frac{13774}{10427}e^{7} - \frac{121667}{10427}e^{6} + \frac{322494}{10427}e^{5} - \frac{547206}{10427}e^{4} - \frac{706938}{10427}e^{3} + \frac{731526}{10427}e^{2} + \frac{336095}{10427}e - \frac{112566}{10427}$ |
| 89 | $[89, 89, -4w^{3} + 2w^{2} + 22w - 3]$ | $\phantom{-}\frac{744}{10427}e^{13} + \frac{3209}{10427}e^{12} - \frac{4817}{10427}e^{11} - \frac{38805}{10427}e^{10} - \frac{36065}{10427}e^{9} + \frac{101587}{10427}e^{8} + \frac{323408}{10427}e^{7} + \frac{231904}{10427}e^{6} - \frac{673553}{10427}e^{5} - \frac{1143942}{10427}e^{4} + \frac{298082}{10427}e^{3} + \frac{1126589}{10427}e^{2} + \frac{127809}{10427}e - \frac{192328}{10427}$ |
| 101 | $[101, 101, -2w^{3} + 13w + 6]$ | $\phantom{-}\frac{14964}{10427}e^{13} + \frac{20648}{10427}e^{12} - \frac{279020}{10427}e^{11} - \frac{347088}{10427}e^{10} + \frac{1970344}{10427}e^{9} + \frac{2110985}{10427}e^{8} - \frac{6703308}{10427}e^{7} - \frac{5825972}{10427}e^{6} + \frac{11333902}{10427}e^{5} + \frac{7403528}{10427}e^{4} - \frac{8499581}{10427}e^{3} - \frac{3742693}{10427}e^{2} + \frac{1863424}{10427}e + \frac{340870}{10427}$ |
| 101 | $[101, 101, w^{3} + w^{2} - 6w - 3]$ | $-\frac{13034}{10427}e^{13} - \frac{29786}{10427}e^{12} + \frac{220752}{10427}e^{11} + \frac{487675}{10427}e^{10} - \frac{1369355}{10427}e^{9} - \frac{2844107}{10427}e^{8} + \frac{3966580}{10427}e^{7} + \frac{7358469}{10427}e^{6} - \frac{5577108}{10427}e^{5} - \frac{8492302}{10427}e^{4} + \frac{3371766}{10427}e^{3} + \frac{3759084}{10427}e^{2} - \frac{548459}{10427}e - \frac{387276}{10427}$ |
| 101 | $[101, 101, -3w^{3} + 2w^{2} + 17w - 3]$ | $\phantom{-}\frac{592}{10427}e^{13} - \frac{5407}{10427}e^{12} - \frac{24799}{10427}e^{11} + \frac{80120}{10427}e^{10} + \frac{294428}{10427}e^{9} - \frac{381431}{10427}e^{8} - \frac{1450226}{10427}e^{7} + \frac{591403}{10427}e^{6} + \frac{3215308}{10427}e^{5} + \frac{144239}{10427}e^{4} - \frac{2994738}{10427}e^{3} - \frac{710902}{10427}e^{2} + \frac{818806}{10427}e + \frac{98334}{10427}$ |
Atkin-Lehner eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| $13$ | $[13, 13, -w^{3} + w^{2} + 4w + 1]$ | $-1$ |