Base field 4.4.13525.1
Generator \(w\), with minimal polynomial \(x^{4} - x^{3} - 12x^{2} + 8x + 29\); narrow class number \(1\) and class number \(1\).
Form
| Weight: | $[2, 2, 2, 2]$ |
| Level: | $[25, 5, \frac{2}{5}w^{3} - \frac{14}{5}w - \frac{3}{5}]$ |
| Dimension: | $8$ |
| CM: | no |
| Base change: | yes |
| Newspace dimension: | $27$ |
Hecke eigenvalues ($q$-expansion)
The Hecke eigenvalue field is $\Q(e)$ where $e$ is a root of the defining polynomial:
| \(x^{8} - x^{7} - 38x^{6} + 24x^{5} + 367x^{4} - 275x^{3} - 738x^{2} + 796x - 200\) |
Show full eigenvalues Hide large eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| 5 | $[5, 5, -w + 2]$ | $\phantom{-}1$ |
| 5 | $[5, 5, -\frac{3}{5}w^{3} - w^{2} + \frac{26}{5}w + \frac{42}{5}]$ | $\phantom{-}1$ |
| 9 | $[9, 3, \frac{2}{5}w^{3} - \frac{19}{5}w - \frac{3}{5}]$ | $\phantom{-}e$ |
| 9 | $[9, 3, -\frac{1}{5}w^{3} + \frac{2}{5}w + \frac{4}{5}]$ | $\phantom{-}e$ |
| 11 | $[11, 11, -\frac{2}{5}w^{3} - w^{2} + \frac{14}{5}w + \frac{28}{5}]$ | $\phantom{-}\frac{113}{1440}e^{7} - \frac{61}{1440}e^{6} - \frac{121}{40}e^{5} + \frac{67}{120}e^{4} + \frac{42683}{1440}e^{3} - \frac{1443}{160}e^{2} - \frac{257}{4}e + \frac{2521}{72}$ |
| 11 | $[11, 11, \frac{1}{5}w^{3} - w^{2} - \frac{7}{5}w + \frac{36}{5}]$ | $\phantom{-}\frac{113}{1440}e^{7} - \frac{61}{1440}e^{6} - \frac{121}{40}e^{5} + \frac{67}{120}e^{4} + \frac{42683}{1440}e^{3} - \frac{1443}{160}e^{2} - \frac{257}{4}e + \frac{2521}{72}$ |
| 16 | $[16, 2, 2]$ | $-\frac{37}{480}e^{7} + \frac{17}{480}e^{6} + \frac{119}{40}e^{5} - \frac{11}{40}e^{4} - \frac{14023}{480}e^{3} + \frac{957}{160}e^{2} + \frac{1277}{20}e - \frac{653}{24}$ |
| 29 | $[29, 29, -w]$ | $-\frac{23}{720}e^{7} - \frac{1}{360}e^{6} + \frac{49}{40}e^{5} + \frac{67}{120}e^{4} - \frac{8447}{720}e^{3} - \frac{19}{5}e^{2} + \frac{517}{20}e - \frac{25}{9}$ |
| 29 | $[29, 29, \frac{1}{5}w^{3} - \frac{12}{5}w + \frac{1}{5}]$ | $-\frac{23}{720}e^{7} - \frac{1}{360}e^{6} + \frac{49}{40}e^{5} + \frac{67}{120}e^{4} - \frac{8447}{720}e^{3} - \frac{19}{5}e^{2} + \frac{517}{20}e - \frac{25}{9}$ |
| 41 | $[41, 41, -\frac{2}{5}w^{3} - w^{2} + \frac{14}{5}w + \frac{38}{5}]$ | $\phantom{-}\frac{1}{16}e^{7} - \frac{19}{8}e^{5} - \frac{7}{8}e^{4} + \frac{353}{16}e^{3} + \frac{39}{8}e^{2} - \frac{165}{4}e + \frac{21}{2}$ |
| 41 | $[41, 41, -w^{2} + 10]$ | $-\frac{1}{288}e^{7} + \frac{43}{1440}e^{6} + \frac{1}{10}e^{5} - \frac{59}{60}e^{4} - \frac{1211}{1440}e^{3} + \frac{1217}{160}e^{2} + \frac{11}{5}e - \frac{451}{72}$ |
| 41 | $[41, 41, \frac{11}{5}w^{3} + 3w^{2} - \frac{102}{5}w - \frac{149}{5}]$ | $-\frac{1}{288}e^{7} + \frac{43}{1440}e^{6} + \frac{1}{10}e^{5} - \frac{59}{60}e^{4} - \frac{1211}{1440}e^{3} + \frac{1217}{160}e^{2} + \frac{11}{5}e - \frac{451}{72}$ |
| 41 | $[41, 41, -\frac{1}{5}w^{3} + w^{2} + \frac{7}{5}w - \frac{26}{5}]$ | $\phantom{-}\frac{1}{16}e^{7} - \frac{19}{8}e^{5} - \frac{7}{8}e^{4} + \frac{353}{16}e^{3} + \frac{39}{8}e^{2} - \frac{165}{4}e + \frac{21}{2}$ |
| 49 | $[49, 7, \frac{1}{5}w^{3} + w^{2} - \frac{12}{5}w - \frac{19}{5}]$ | $-\frac{59}{720}e^{7} + \frac{43}{720}e^{6} + \frac{63}{20}e^{5} - \frac{61}{60}e^{4} - \frac{22649}{720}e^{3} + \frac{949}{80}e^{2} + \frac{149}{2}e - \frac{1495}{36}$ |
| 49 | $[49, 7, \frac{1}{5}w^{3} + w^{2} - \frac{12}{5}w - \frac{44}{5}]$ | $-\frac{59}{720}e^{7} + \frac{43}{720}e^{6} + \frac{63}{20}e^{5} - \frac{61}{60}e^{4} - \frac{22649}{720}e^{3} + \frac{949}{80}e^{2} + \frac{149}{2}e - \frac{1495}{36}$ |
| 59 | $[59, 59, -\frac{4}{5}w^{3} - w^{2} + \frac{33}{5}w + \frac{36}{5}]$ | $\phantom{-}\frac{83}{720}e^{7} - \frac{11}{144}e^{6} - \frac{22}{5}e^{5} + \frac{19}{15}e^{4} + \frac{30701}{720}e^{3} - \frac{1349}{80}e^{2} - \frac{917}{10}e + \frac{1975}{36}$ |
| 59 | $[59, 59, -2w^{3} - 3w^{2} + 17w + 26]$ | $\phantom{-}\frac{83}{720}e^{7} - \frac{11}{144}e^{6} - \frac{22}{5}e^{5} + \frac{19}{15}e^{4} + \frac{30701}{720}e^{3} - \frac{1349}{80}e^{2} - \frac{917}{10}e + \frac{1975}{36}$ |
| 61 | $[61, 61, -\frac{1}{5}w^{3} + w^{2} + \frac{12}{5}w - \frac{51}{5}]$ | $\phantom{-}\frac{31}{480}e^{7} - \frac{11}{480}e^{6} - \frac{97}{40}e^{5} - \frac{7}{40}e^{4} + \frac{10789}{480}e^{3} - \frac{111}{160}e^{2} - \frac{861}{20}e + \frac{527}{24}$ |
| 61 | $[61, 61, \frac{4}{5}w^{3} + w^{2} - \frac{28}{5}w - \frac{41}{5}]$ | $\phantom{-}\frac{31}{480}e^{7} - \frac{11}{480}e^{6} - \frac{97}{40}e^{5} - \frac{7}{40}e^{4} + \frac{10789}{480}e^{3} - \frac{111}{160}e^{2} - \frac{861}{20}e + \frac{527}{24}$ |
| 71 | $[71, 71, \frac{1}{5}w^{3} + w^{2} - \frac{7}{5}w - \frac{49}{5}]$ | $-\frac{13}{45}e^{7} + \frac{13}{90}e^{6} + 11e^{5} - \frac{4}{3}e^{4} - \frac{4729}{45}e^{3} + \frac{49}{2}e^{2} + \frac{1077}{5}e - \frac{968}{9}$ |
Atkin-Lehner eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| $5$ | $[5, 5, -w + 2]$ | $-1$ |
| $5$ | $[5, 5, -\frac{3}{5}w^{3} - w^{2} + \frac{26}{5}w + \frac{42}{5}]$ | $-1$ |