Base field 4.4.8725.1
Generator \(w\), with minimal polynomial \(x^{4} - x^{3} - 10x^{2} + 2x + 19\); narrow class number \(1\) and class number \(1\).
Form
| Weight: | $[2, 2, 2, 2]$ |
| Level: | $[31, 31, \frac{2}{3}w^{3} - \frac{7}{3}w^{2} - \frac{10}{3}w + \frac{23}{3}]$ |
| Dimension: | $10$ |
| CM: | no |
| Base change: | no |
| Newspace dimension: | $21$ |
Hecke eigenvalues ($q$-expansion)
The Hecke eigenvalue field is $\Q(e)$ where $e$ is a root of the defining polynomial:
| \(x^{10} - 7x^{9} - 11x^{8} + 175x^{7} - 255x^{6} - 798x^{5} + 2296x^{4} - 1152x^{3} - 960x^{2} + 544x + 32\) |
Show full eigenvalues Hide large eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| 9 | $[9, 3, \frac{1}{3}w^{3} - \frac{2}{3}w^{2} - \frac{8}{3}w + \frac{10}{3}]$ | $\phantom{-}e$ |
| 9 | $[9, 3, w + 1]$ | $\phantom{-}\frac{71381}{859192}e^{9} - \frac{392689}{859192}e^{8} - \frac{6445}{4072}e^{7} + \frac{10331035}{859192}e^{6} - \frac{2868399}{859192}e^{5} - \frac{29052369}{429596}e^{4} + \frac{36638661}{429596}e^{3} + \frac{2702721}{214798}e^{2} - \frac{2793500}{107399}e + \frac{375740}{107399}$ |
| 11 | $[11, 11, w^{2} - w - 6]$ | $\phantom{-}\frac{17443}{322197}e^{9} - \frac{222411}{859192}e^{8} - \frac{14953}{12216}e^{7} + \frac{18112777}{2577576}e^{6} + \frac{7502167}{2577576}e^{5} - \frac{112961255}{2577576}e^{4} + \frac{5695447}{214798}e^{3} + \frac{7998937}{214798}e^{2} - \frac{712110}{107399}e - \frac{1320266}{322197}$ |
| 11 | $[11, 11, \frac{2}{3}w^{3} - \frac{7}{3}w^{2} - \frac{7}{3}w + \frac{23}{3}]$ | $\phantom{-}\frac{61487}{859192}e^{9} - \frac{301845}{859192}e^{8} - \frac{6131}{4072}e^{7} + \frac{8061957}{859192}e^{6} + \frac{717395}{859192}e^{5} - \frac{23839607}{429596}e^{4} + \frac{5779095}{107399}e^{3} + \frac{3168247}{107399}e^{2} - \frac{2700562}{107399}e - \frac{122108}{107399}$ |
| 16 | $[16, 2, 2]$ | $-\frac{3463}{2577576}e^{9} - \frac{13767}{859192}e^{8} + \frac{1829}{12216}e^{7} + \frac{670525}{2577576}e^{6} - \frac{8369597}{2577576}e^{5} + \frac{574856}{322197}e^{4} + \frac{1913058}{107399}e^{3} - \frac{4813245}{214798}e^{2} - \frac{789392}{107399}e + \frac{1218505}{322197}$ |
| 19 | $[19, 19, w]$ | $-\frac{108757}{1288788}e^{9} + \frac{48070}{107399}e^{8} + \frac{2441}{1527}e^{7} - \frac{7551511}{644394}e^{6} + \frac{2377217}{644394}e^{5} + \frac{83380603}{1288788}e^{4} - \frac{19297669}{214798}e^{3} - \frac{569397}{107399}e^{2} + \frac{3646733}{107399}e - \frac{292498}{322197}$ |
| 19 | $[19, 19, \frac{1}{3}w^{3} - \frac{2}{3}w^{2} - \frac{8}{3}w + \frac{7}{3}]$ | $\phantom{-}\frac{30607}{322197}e^{9} - \frac{446033}{859192}e^{8} - \frac{22345}{12216}e^{7} + \frac{35408041}{2577576}e^{6} - \frac{8864621}{2577576}e^{5} - \frac{203240123}{2577576}e^{4} + \frac{41977831}{429596}e^{3} + \frac{2709192}{107399}e^{2} - \frac{4539758}{107399}e + \frac{662824}{322197}$ |
| 19 | $[19, 19, \frac{1}{3}w^{3} - \frac{2}{3}w^{2} - \frac{2}{3}w + \frac{10}{3}]$ | $-\frac{8870}{322197}e^{9} + \frac{71733}{429596}e^{8} + \frac{1499}{3054}e^{7} - \frac{2845181}{644394}e^{6} + \frac{649061}{322197}e^{5} + \frac{8172071}{322197}e^{4} - \frac{14402377}{429596}e^{3} - \frac{1813481}{214798}e^{2} + \frac{1278300}{107399}e + \frac{768976}{322197}$ |
| 19 | $[19, 19, -\frac{2}{3}w^{3} + \frac{4}{3}w^{2} + \frac{13}{3}w - \frac{17}{3}]$ | $\phantom{-}\frac{143183}{2577576}e^{9} - \frac{249223}{859192}e^{8} - \frac{13339}{12216}e^{7} + \frac{19612381}{2577576}e^{6} - \frac{3412709}{2577576}e^{5} - \frac{54468823}{1288788}e^{4} + \frac{5559071}{107399}e^{3} + \frac{444620}{107399}e^{2} - \frac{1622599}{107399}e + \frac{1896196}{322197}$ |
| 25 | $[25, 5, \frac{2}{3}w^{3} - \frac{4}{3}w^{2} - \frac{10}{3}w + \frac{11}{3}]$ | $\phantom{-}\frac{3595}{1288788}e^{9} + \frac{20365}{859192}e^{8} - \frac{2311}{12216}e^{7} - \frac{1468925}{2577576}e^{6} + \frac{9238585}{2577576}e^{5} + \frac{5508073}{2577576}e^{4} - \frac{8608235}{429596}e^{3} + \frac{865092}{107399}e^{2} + \frac{1875036}{107399}e - \frac{1100810}{322197}$ |
| 31 | $[31, 31, w + 3]$ | $\phantom{-}\frac{126025}{1288788}e^{9} - \frac{56680}{107399}e^{8} - \frac{6049}{3054}e^{7} + \frac{4544207}{322197}e^{6} - \frac{267079}{322197}e^{5} - \frac{107983609}{1288788}e^{4} + \frac{8890520}{107399}e^{3} + \frac{4856951}{107399}e^{2} - \frac{3018230}{107399}e - \frac{300578}{322197}$ |
| 31 | $[31, 31, -\frac{1}{3}w^{3} + \frac{2}{3}w^{2} + \frac{8}{3}w - \frac{16}{3}]$ | $\phantom{-}\frac{49495}{644394}e^{9} - \frac{37192}{107399}e^{8} - \frac{2656}{1527}e^{7} + \frac{6030455}{644394}e^{6} + \frac{2680667}{644394}e^{5} - \frac{18501911}{322197}e^{4} + \frac{8363117}{214798}e^{3} + \frac{9371081}{214798}e^{2} - \frac{2010956}{107399}e - \frac{569710}{322197}$ |
| 31 | $[31, 31, \frac{2}{3}w^{3} - \frac{7}{3}w^{2} - \frac{10}{3}w + \frac{23}{3}]$ | $-1$ |
| 31 | $[31, 31, -\frac{1}{3}w^{3} + \frac{5}{3}w^{2} + \frac{5}{3}w - \frac{25}{3}]$ | $-\frac{73331}{644394}e^{9} + \frac{239145}{429596}e^{8} + \frac{14855}{6108}e^{7} - \frac{19256303}{1288788}e^{6} - \frac{2969429}{1288788}e^{5} + \frac{115916023}{1288788}e^{4} - \frac{8562423}{107399}e^{3} - \frac{6297911}{107399}e^{2} + \frac{3527887}{107399}e + \frac{3989294}{322197}$ |
| 59 | $[59, 59, -\frac{2}{3}w^{3} + \frac{1}{3}w^{2} + \frac{10}{3}w + \frac{1}{3}]$ | $\phantom{-}\frac{170525}{2577576}e^{9} - \frac{38837}{107399}e^{8} - \frac{4039}{3054}e^{7} + \frac{12407189}{1288788}e^{6} - \frac{1384231}{1288788}e^{5} - \frac{145416563}{2577576}e^{4} + \frac{25618777}{429596}e^{3} + \frac{2584415}{107399}e^{2} - \frac{2515624}{107399}e + \frac{2266522}{322197}$ |
| 59 | $[59, 59, \frac{5}{3}w^{3} - \frac{13}{3}w^{2} - \frac{25}{3}w + \frac{47}{3}]$ | $\phantom{-}\frac{6975}{859192}e^{9} - \frac{78827}{859192}e^{8} - \frac{59}{4072}e^{7} + \frac{2031709}{859192}e^{6} - \frac{3490505}{859192}e^{5} - \frac{5346085}{429596}e^{4} + \frac{12965797}{429596}e^{3} - \frac{836651}{214798}e^{2} - \frac{1703728}{107399}e - \frac{109368}{107399}$ |
| 61 | $[61, 61, \frac{1}{3}w^{3} + \frac{1}{3}w^{2} - \frac{11}{3}w - \frac{8}{3}]$ | $-\frac{78677}{859192}e^{9} + \frac{48936}{107399}e^{8} + \frac{1941}{1018}e^{7} - \frac{5170601}{429596}e^{6} - \frac{313613}{429596}e^{5} + \frac{58762259}{859192}e^{4} - \frac{28636091}{429596}e^{3} - \frac{1741633}{107399}e^{2} + \frac{355710}{107399}e - \frac{553464}{107399}$ |
| 61 | $[61, 61, -\frac{2}{3}w^{3} + \frac{7}{3}w^{2} + \frac{4}{3}w - \frac{26}{3}]$ | $\phantom{-}\frac{40960}{322197}e^{9} - \frac{293109}{429596}e^{8} - \frac{15323}{6108}e^{7} + \frac{23420843}{1288788}e^{6} - \frac{3873019}{1288788}e^{5} - \frac{137484757}{1288788}e^{4} + \frac{13156684}{107399}e^{3} + \frac{5383561}{107399}e^{2} - \frac{6538142}{107399}e + \frac{188968}{322197}$ |
| 71 | $[71, 71, \frac{2}{3}w^{3} - \frac{4}{3}w^{2} - \frac{13}{3}w + \frac{5}{3}]$ | $\phantom{-}\frac{26395}{644394}e^{9} - \frac{37791}{214798}e^{8} - \frac{3305}{3054}e^{7} + \frac{1601974}{322197}e^{6} + \frac{2059276}{322197}e^{5} - \frac{22566025}{644394}e^{4} - \frac{1097581}{214798}e^{3} + \frac{11849891}{214798}e^{2} + \frac{625854}{107399}e - \frac{2987242}{322197}$ |
| 71 | $[71, 71, -w^{3} + 3w^{2} + 5w - 10]$ | $-\frac{99125}{644394}e^{9} + \frac{346363}{429596}e^{8} + \frac{19211}{6108}e^{7} - \frac{27615755}{1288788}e^{6} + \frac{543835}{1288788}e^{5} + \frac{160662271}{1288788}e^{4} - \frac{26496391}{214798}e^{3} - \frac{5215377}{107399}e^{2} + \frac{2986346}{107399}e - \frac{1300810}{322197}$ |
Atkin-Lehner eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| $31$ | $[31, 31, \frac{2}{3}w^{3} - \frac{7}{3}w^{2} - \frac{10}{3}w + \frac{23}{3}]$ | $1$ |