Base field 4.4.14725.1
Generator \(w\), with minimal polynomial \(x^{4} - x^{3} - 13x^{2} + 11x + 31\); narrow class number \(2\) and class number \(1\).
Form
| Weight: | $[2, 2, 2, 2]$ |
| Level: | $[25, 5, -\frac{2}{3}w^{3} - \frac{2}{3}w^{2} + \frac{16}{3}w + \frac{13}{3}]$ |
| Dimension: | $16$ |
| CM: | no |
| Base change: | yes |
| Newspace dimension: | $38$ |
Hecke eigenvalues ($q$-expansion)
The Hecke eigenvalue field is $\Q(e)$ where $e$ is a root of the defining polynomial:
| \(x^{16} - 72x^{14} + 2106x^{12} - 32624x^{10} + 292869x^{8} - 1561008x^{6} + 4827200x^{4} - 7925760x^{2} + 5308416\) |
Show full eigenvalues Hide large eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| 9 | $[9, 3, -\frac{1}{3}w^{3} - \frac{1}{3}w^{2} + \frac{11}{3}w + \frac{11}{3}]$ | $\phantom{-}e$ |
| 9 | $[9, 3, w - 2]$ | $\phantom{-}e$ |
| 11 | $[11, 11, -\frac{1}{3}w^{3} - \frac{1}{3}w^{2} + \frac{5}{3}w + \frac{8}{3}]$ | $\phantom{-}\frac{119}{1437696}e^{15} - \frac{461}{89856}e^{13} + \frac{9635}{79872}e^{11} - \frac{839}{624}e^{9} + \frac{262133}{36864}e^{7} - \frac{2385713}{179712}e^{5} - \frac{83773}{5616}e^{3} + \frac{8519}{156}e$ |
| 11 | $[11, 11, \frac{2}{3}w^{3} + \frac{2}{3}w^{2} - \frac{19}{3}w - \frac{13}{3}]$ | $\phantom{-}\frac{119}{1437696}e^{15} - \frac{461}{89856}e^{13} + \frac{9635}{79872}e^{11} - \frac{839}{624}e^{9} + \frac{262133}{36864}e^{7} - \frac{2385713}{179712}e^{5} - \frac{83773}{5616}e^{3} + \frac{8519}{156}e$ |
| 16 | $[16, 2, 2]$ | $-\frac{547}{718848}e^{14} + \frac{4699}{89856}e^{12} - \frac{171005}{119808}e^{10} + \frac{895333}{44928}e^{8} - \frac{2819641}{18432}e^{6} + \frac{28886401}{44928}e^{4} - \frac{5126545}{3744}e^{2} + \frac{14902}{13}$ |
| 19 | $[19, 19, w + 2]$ | $\phantom{-}\frac{1045}{1916928}e^{15} - \frac{2987}{79872}e^{13} + \frac{2933923}{2875392}e^{11} - \frac{5134565}{359424}e^{9} + \frac{48910951}{442368}e^{7} - \frac{169669795}{359424}e^{5} + \frac{92510375}{89856}e^{3} - \frac{276937}{312}e$ |
| 25 | $[25, 5, -\frac{2}{3}w^{3} - \frac{2}{3}w^{2} + \frac{16}{3}w + \frac{13}{3}]$ | $-1$ |
| 29 | $[29, 29, -\frac{2}{3}w^{3} - \frac{2}{3}w^{2} + \frac{19}{3}w + \frac{19}{3}]$ | $-\frac{349}{718848}e^{14} + \frac{3017}{89856}e^{12} - \frac{332041}{359424}e^{10} + \frac{585071}{44928}e^{8} - \frac{5581525}{55296}e^{6} + \frac{2130257}{4992}e^{4} - \frac{375639}{416}e^{2} + \frac{9547}{13}$ |
| 29 | $[29, 29, -\frac{1}{3}w^{3} - \frac{1}{3}w^{2} + \frac{5}{3}w + \frac{14}{3}]$ | $-\frac{349}{718848}e^{14} + \frac{3017}{89856}e^{12} - \frac{332041}{359424}e^{10} + \frac{585071}{44928}e^{8} - \frac{5581525}{55296}e^{6} + \frac{2130257}{4992}e^{4} - \frac{375639}{416}e^{2} + \frac{9547}{13}$ |
| 29 | $[29, 29, \frac{1}{3}w^{3} + \frac{1}{3}w^{2} - \frac{11}{3}w - \frac{8}{3}]$ | $-\frac{47}{179712}e^{15} + \frac{1583}{89856}e^{13} - \frac{14129}{29952}e^{11} + \frac{292235}{44928}e^{9} - \frac{25653}{512}e^{7} + \frac{19459975}{89856}e^{5} - \frac{1826405}{3744}e^{3} + \frac{17219}{39}e$ |
| 29 | $[29, 29, w - 1]$ | $-\frac{47}{179712}e^{15} + \frac{1583}{89856}e^{13} - \frac{14129}{29952}e^{11} + \frac{292235}{44928}e^{9} - \frac{25653}{512}e^{7} + \frac{19459975}{89856}e^{5} - \frac{1826405}{3744}e^{3} + \frac{17219}{39}e$ |
| 31 | $[31, 31, w]$ | $-\frac{137}{1437696}e^{15} + \frac{277}{44928}e^{13} - \frac{111941}{718848}e^{11} + \frac{88643}{44928}e^{9} - \frac{1489697}{110592}e^{7} + \frac{2944807}{59904}e^{5} - \frac{110141}{1248}e^{3} + \frac{9905}{156}e$ |
| 31 | $[31, 31, \frac{1}{3}w^{3} - \frac{2}{3}w^{2} - \frac{8}{3}w + \frac{13}{3}]$ | $\phantom{-}\frac{7}{79872}e^{14} - \frac{599}{89856}e^{12} + \frac{74147}{359424}e^{10} - \frac{150377}{44928}e^{8} + \frac{1683047}{55296}e^{6} - \frac{6850549}{44928}e^{4} + \frac{1430465}{3744}e^{2} - \frac{4763}{13}$ |
| 31 | $[31, 31, -\frac{1}{3}w^{3} - \frac{1}{3}w^{2} + \frac{11}{3}w + \frac{5}{3}]$ | $-\frac{137}{1437696}e^{15} + \frac{277}{44928}e^{13} - \frac{111941}{718848}e^{11} + \frac{88643}{44928}e^{9} - \frac{1489697}{110592}e^{7} + \frac{2944807}{59904}e^{5} - \frac{110141}{1248}e^{3} + \frac{9905}{156}e$ |
| 41 | $[41, 41, \frac{1}{3}w^{3} + \frac{4}{3}w^{2} - \frac{8}{3}w - \frac{20}{3}]$ | $-\frac{149}{239616}e^{14} + \frac{1271}{29952}e^{12} - \frac{15385}{13312}e^{10} + \frac{243797}{14976}e^{8} - \frac{263205}{2048}e^{6} + \frac{1066369}{1872}e^{4} - \frac{543975}{416}e^{2} + \frac{15399}{13}$ |
| 41 | $[41, 41, \frac{2}{3}w^{3} + \frac{5}{3}w^{2} - \frac{16}{3}w - \frac{40}{3}]$ | $-\frac{149}{239616}e^{14} + \frac{1271}{29952}e^{12} - \frac{15385}{13312}e^{10} + \frac{243797}{14976}e^{8} - \frac{263205}{2048}e^{6} + \frac{1066369}{1872}e^{4} - \frac{543975}{416}e^{2} + \frac{15399}{13}$ |
| 49 | $[49, 7, -\frac{1}{3}w^{3} - \frac{4}{3}w^{2} + \frac{8}{3}w + \frac{38}{3}]$ | $-\frac{1279}{1437696}e^{14} + \frac{10651}{179712}e^{12} - \frac{372257}{239616}e^{10} + \frac{1849885}{89856}e^{8} - \frac{5447437}{36864}e^{6} + \frac{51097855}{89856}e^{4} - \frac{8073781}{7488}e^{2} + \frac{20339}{26}$ |
| 49 | $[49, 7, \frac{2}{3}w^{3} + \frac{5}{3}w^{2} - \frac{16}{3}w - \frac{43}{3}]$ | $-\frac{1279}{1437696}e^{14} + \frac{10651}{179712}e^{12} - \frac{372257}{239616}e^{10} + \frac{1849885}{89856}e^{8} - \frac{5447437}{36864}e^{6} + \frac{51097855}{89856}e^{4} - \frac{8073781}{7488}e^{2} + \frac{20339}{26}$ |
| 59 | $[59, 59, \frac{5}{3}w^{3} + \frac{8}{3}w^{2} - \frac{40}{3}w - \frac{49}{3}]$ | $\phantom{-}\frac{1117}{479232}e^{14} - \frac{28387}{179712}e^{12} + \frac{3056923}{718848}e^{10} - \frac{5281909}{89856}e^{8} + \frac{49854847}{110592}e^{6} - \frac{172397855}{89856}e^{4} + \frac{10469711}{2496}e^{2} - \frac{94199}{26}$ |
| 59 | $[59, 59, \frac{2}{3}w^{3} - \frac{1}{3}w^{2} - \frac{16}{3}w + \frac{11}{3}]$ | $\phantom{-}\frac{1117}{479232}e^{14} - \frac{28387}{179712}e^{12} + \frac{3056923}{718848}e^{10} - \frac{5281909}{89856}e^{8} + \frac{49854847}{110592}e^{6} - \frac{172397855}{89856}e^{4} + \frac{10469711}{2496}e^{2} - \frac{94199}{26}$ |
Atkin-Lehner eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| $25$ | $[25, 5, -\frac{2}{3}w^{3} - \frac{2}{3}w^{2} + \frac{16}{3}w + \frac{13}{3}]$ | $1$ |