Base field 4.4.5725.1
Generator \(w\), with minimal polynomial \(x^{4} - x^{3} - 8x^{2} + 6x + 11\); narrow class number \(1\) and class number \(1\).
Form
| Weight: | $[2, 2, 2, 2]$ |
| Level: | $[31,31,\frac{2}{3}w^{3} - \frac{7}{3}w + \frac{2}{3}]$ |
| Dimension: | $7$ |
| CM: | no |
| Base change: | no |
| Newspace dimension: | $11$ |
Hecke eigenvalues ($q$-expansion)
The Hecke eigenvalue field is $\Q(e)$ where $e$ is a root of the defining polynomial:
| \(x^{7} - 2x^{6} - 41x^{5} + 46x^{4} + 456x^{3} - 168x^{2} - 1208x - 496\) |
Show full eigenvalues Hide large eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| 9 | $[9, 3, \frac{1}{3}w^{3} - \frac{8}{3}w - \frac{2}{3}]$ | $-\frac{2733}{285556}e^{6} + \frac{3871}{142778}e^{5} + \frac{99441}{285556}e^{4} - \frac{82167}{142778}e^{3} - \frac{236521}{71389}e^{2} + \frac{86802}{71389}e + \frac{444954}{71389}$ |
| 9 | $[9, 3, -w + 1]$ | $\phantom{-}e$ |
| 11 | $[11, 11, w]$ | $\phantom{-}\frac{876}{71389}e^{6} - \frac{6243}{285556}e^{5} - \frac{71505}{142778}e^{4} + \frac{151843}{285556}e^{3} + \frac{726465}{142778}e^{2} - \frac{218804}{71389}e - \frac{566404}{71389}$ |
| 11 | $[11, 11, -\frac{1}{3}w^{3} + \frac{2}{3}w + \frac{5}{3}]$ | $-\frac{741}{285556}e^{6} + \frac{1363}{142778}e^{5} + \frac{15207}{285556}e^{4} - \frac{9846}{71389}e^{3} + \frac{8358}{71389}e^{2} + \frac{11310}{71389}e - \frac{88276}{71389}$ |
| 11 | $[11, 11, \frac{1}{3}w^{3} - \frac{8}{3}w + \frac{1}{3}]$ | $-\frac{741}{285556}e^{6} + \frac{1363}{142778}e^{5} + \frac{15207}{285556}e^{4} - \frac{9846}{71389}e^{3} + \frac{8358}{71389}e^{2} + \frac{11310}{71389}e - \frac{88276}{71389}$ |
| 11 | $[11, 11, -\frac{2}{3}w^{3} + \frac{13}{3}w + \frac{4}{3}]$ | $\phantom{-}\frac{305}{142778}e^{6} - \frac{9277}{285556}e^{5} - \frac{2791}{142778}e^{4} + \frac{277803}{285556}e^{3} - \frac{27923}{142778}e^{2} - \frac{407586}{71389}e + \frac{66504}{71389}$ |
| 16 | $[16, 2, 2]$ | $\phantom{-}\frac{397}{285556}e^{6} - \frac{895}{71389}e^{5} - \frac{4101}{285556}e^{4} + \frac{55349}{142778}e^{3} - \frac{82657}{142778}e^{2} - \frac{178896}{71389}e + \frac{99223}{71389}$ |
| 25 | $[25, 5, \frac{2}{3}w^{3} - \frac{10}{3}w - \frac{1}{3}]$ | $-\frac{378}{71389}e^{6} + \frac{1227}{285556}e^{5} + \frac{14694}{71389}e^{4} - \frac{26893}{285556}e^{3} - \frac{236707}{142778}e^{2} + \frac{534}{71389}e + \frac{33174}{71389}$ |
| 29 | $[29, 29, -w - 3]$ | $\phantom{-}\frac{5757}{285556}e^{6} - \frac{2549}{71389}e^{5} - \frac{216993}{285556}e^{4} + \frac{54530}{71389}e^{3} + \frac{473228}{71389}e^{2} - \frac{230648}{71389}e - \frac{511302}{71389}$ |
| 29 | $[29, 29, -\frac{1}{3}w^{3} + \frac{8}{3}w - \frac{10}{3}]$ | $-\frac{107}{285556}e^{6} - \frac{3171}{285556}e^{5} + \frac{15491}{285556}e^{4} + \frac{54141}{285556}e^{3} - \frac{137763}{142778}e^{2} + \frac{106838}{71389}e + \frac{467226}{71389}$ |
| 31 | $[31, 31, w^{3} - 6w + 1]$ | $\phantom{-}\frac{771}{285556}e^{6} + \frac{1499}{285556}e^{5} - \frac{43569}{285556}e^{4} - \frac{12491}{285556}e^{3} + \frac{253423}{142778}e^{2} - \frac{60613}{71389}e - \frac{121450}{71389}$ |
| 31 | $[31, 31, w^{3} - 6w - 2]$ | $-1$ |
| 41 | $[41, 41, \frac{2}{3}w^{3} + w^{2} - \frac{13}{3}w - \frac{10}{3}]$ | $-\frac{3581}{285556}e^{6} + \frac{7297}{285556}e^{5} + \frac{130139}{285556}e^{4} - \frac{149577}{285556}e^{3} - \frac{594089}{142778}e^{2} + \frac{133561}{71389}e + \frac{823904}{71389}$ |
| 41 | $[41, 41, \frac{2}{3}w^{3} - \frac{13}{3}w + \frac{5}{3}]$ | $\phantom{-}\frac{2283}{285556}e^{6} + \frac{68}{71389}e^{5} - \frac{102345}{285556}e^{4} + \frac{7201}{142778}e^{3} + \frac{245065}{71389}e^{2} - \frac{203925}{71389}e - \frac{154624}{71389}$ |
| 59 | $[59, 59, \frac{2}{3}w^{3} - \frac{13}{3}w + \frac{8}{3}]$ | $-\frac{3053}{142778}e^{6} + \frac{5134}{71389}e^{5} + \frac{116413}{142778}e^{4} - \frac{138649}{71389}e^{3} - \frac{594150}{71389}e^{2} + \frac{773271}{71389}e + \frac{1318686}{71389}$ |
| 59 | $[59, 59, w^{3} + w^{2} - 6w - 4]$ | $-\frac{4831}{285556}e^{6} + \frac{10813}{142778}e^{5} + \frac{169665}{285556}e^{4} - \frac{163327}{71389}e^{3} - \frac{404191}{71389}e^{2} + \frac{1069425}{71389}e + \frac{1186178}{71389}$ |
| 79 | $[79, 79, \frac{2}{3}w^{3} + w^{2} - \frac{10}{3}w - \frac{19}{3}]$ | $-\frac{349}{285556}e^{6} + \frac{2585}{71389}e^{5} + \frac{15833}{285556}e^{4} - \frac{84119}{71389}e^{3} - \frac{120922}{71389}e^{2} + \frac{471234}{71389}e + \frac{807384}{71389}$ |
| 79 | $[79, 79, \frac{1}{3}w^{3} + w^{2} - \frac{2}{3}w - \frac{17}{3}]$ | $\phantom{-}\frac{5437}{285556}e^{6} - \frac{3835}{142778}e^{5} - \frac{200021}{285556}e^{4} + \frac{26289}{71389}e^{3} + \frac{896737}{142778}e^{2} - \frac{37898}{71389}e - \frac{653858}{71389}$ |
| 89 | $[89, 89, \frac{4}{3}w^{3} - \frac{23}{3}w - \frac{5}{3}]$ | $-\frac{6267}{285556}e^{6} + \frac{2440}{71389}e^{5} + \frac{270813}{285556}e^{4} - \frac{132151}{142778}e^{3} - \frac{734823}{71389}e^{2} + \frac{426298}{71389}e + \frac{1078306}{71389}$ |
| 89 | $[89, 89, \frac{1}{3}w^{3} + 2w^{2} - \frac{5}{3}w - \frac{32}{3}]$ | $\phantom{-}\frac{378}{71389}e^{6} - \frac{1227}{285556}e^{5} - \frac{14694}{71389}e^{4} + \frac{26893}{285556}e^{3} + \frac{379485}{142778}e^{2} - \frac{214701}{71389}e - \frac{747064}{71389}$ |
Atkin-Lehner eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| $31$ | $[31,31,\frac{2}{3}w^{3} - \frac{7}{3}w + \frac{2}{3}]$ | $1$ |