Base field \(\Q(\sqrt{157}) \)
Generator \(w\), with minimal polynomial \(x^{2} - x - 39\); narrow class number \(1\) and class number \(1\).
Form
| Weight: | $[2, 2]$ |
| Level: | $[16, 4, 4]$ |
| Dimension: | $8$ |
| CM: | no |
| Base change: | no |
| Newspace dimension: | $36$ |
Hecke eigenvalues ($q$-expansion)
The Hecke eigenvalue field is $\Q(e)$ where $e$ is a root of the defining polynomial:
| \(x^{8} - 3x^{7} - 12x^{6} + 43x^{5} + 20x^{4} - 156x^{3} + 103x^{2} + 28x - 23\) |
Show full eigenvalues Hide large eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| 3 | $[3, 3, w + 6]$ | $\phantom{-}e$ |
| 3 | $[3, 3, -w + 7]$ | $\phantom{-}\frac{15}{37}e^{7} - \frac{18}{37}e^{6} - \frac{205}{37}e^{5} + \frac{276}{37}e^{4} + \frac{708}{37}e^{3} - 29e^{2} - \frac{120}{37}e + \frac{315}{37}$ |
| 4 | $[4, 2, 2]$ | $\phantom{-}0$ |
| 11 | $[11, 11, -3w - 17]$ | $\phantom{-}\frac{17}{37}e^{7} - \frac{13}{37}e^{6} - \frac{220}{37}e^{5} + \frac{224}{37}e^{4} + \frac{684}{37}e^{3} - 25e^{2} + \frac{160}{37}e + \frac{172}{37}$ |
| 11 | $[11, 11, 3w - 20]$ | $\phantom{-}\frac{10}{37}e^{7} - \frac{12}{37}e^{6} - \frac{149}{37}e^{5} + \frac{184}{37}e^{4} + \frac{583}{37}e^{3} - 20e^{2} - \frac{265}{37}e + \frac{284}{37}$ |
| 13 | $[13, 13, 2w - 13]$ | $-\frac{10}{37}e^{7} + \frac{12}{37}e^{6} + \frac{149}{37}e^{5} - \frac{184}{37}e^{4} - \frac{583}{37}e^{3} + 19e^{2} + \frac{228}{37}e - \frac{62}{37}$ |
| 13 | $[13, 13, 2w + 11]$ | $-\frac{5}{37}e^{7} + \frac{6}{37}e^{6} + \frac{56}{37}e^{5} - \frac{92}{37}e^{4} - \frac{125}{37}e^{3} + 10e^{2} - \frac{145}{37}e - \frac{142}{37}$ |
| 17 | $[17, 17, w + 7]$ | $\phantom{-}\frac{39}{37}e^{7} - \frac{32}{37}e^{6} - \frac{533}{37}e^{5} + \frac{540}{37}e^{4} + \frac{1863}{37}e^{3} - 61e^{2} - \frac{386}{37}e + \frac{671}{37}$ |
| 17 | $[17, 17, -w + 8]$ | $\phantom{-}\frac{24}{37}e^{7} - \frac{14}{37}e^{6} - \frac{328}{37}e^{5} + \frac{264}{37}e^{4} + \frac{1155}{37}e^{3} - 32e^{2} - \frac{229}{37}e + \frac{356}{37}$ |
| 19 | $[19, 19, -w - 4]$ | $\phantom{-}\frac{6}{37}e^{7} + \frac{15}{37}e^{6} - \frac{82}{37}e^{5} - \frac{193}{37}e^{4} + \frac{372}{37}e^{3} + 18e^{2} - \frac{640}{37}e - \frac{170}{37}$ |
| 19 | $[19, 19, -w + 5]$ | $-\frac{39}{37}e^{7} + \frac{32}{37}e^{6} + \frac{533}{37}e^{5} - \frac{503}{37}e^{4} - \frac{1826}{37}e^{3} + 53e^{2} + \frac{201}{37}e - \frac{264}{37}$ |
| 25 | $[25, 5, 5]$ | $-\frac{33}{37}e^{7} + \frac{47}{37}e^{6} + \frac{451}{37}e^{5} - \frac{696}{37}e^{4} - \frac{1528}{37}e^{3} + 71e^{2} + \frac{5}{37}e - \frac{730}{37}$ |
| 31 | $[31, 31, -6w - 35]$ | $-e^{6} + 13e^{4} - 3e^{3} - 43e^{2} + 19e + 11$ |
| 31 | $[31, 31, -6w + 41]$ | $\phantom{-}\frac{3}{37}e^{7} - \frac{11}{37}e^{6} - \frac{41}{37}e^{5} + \frac{107}{37}e^{4} + \frac{112}{37}e^{3} - 6e^{2} + \frac{124}{37}e + \frac{63}{37}$ |
| 37 | $[37, 37, -w - 1]$ | $\phantom{-}\frac{21}{37}e^{7} - \frac{40}{37}e^{6} - \frac{287}{37}e^{5} + \frac{601}{37}e^{4} + \frac{895}{37}e^{3} - 61e^{2} + \frac{535}{37}e + \frac{404}{37}$ |
| 37 | $[37, 37, w - 2]$ | $\phantom{-}\frac{12}{37}e^{7} - \frac{7}{37}e^{6} - \frac{164}{37}e^{5} + \frac{95}{37}e^{4} + \frac{559}{37}e^{3} - 10e^{2} - \frac{96}{37}e + \frac{252}{37}$ |
| 47 | $[47, 47, 3w + 16]$ | $-\frac{5}{37}e^{7} + \frac{6}{37}e^{6} + \frac{93}{37}e^{5} - \frac{55}{37}e^{4} - \frac{495}{37}e^{3} + 4e^{2} + \frac{632}{37}e - \frac{68}{37}$ |
| 47 | $[47, 47, -3w + 19]$ | $-\frac{46}{37}e^{7} + \frac{70}{37}e^{6} + \frac{604}{37}e^{5} - \frac{1061}{37}e^{4} - \frac{1890}{37}e^{3} + 109e^{2} - \frac{483}{37}e - \frac{818}{37}$ |
| 49 | $[49, 7, -7]$ | $-\frac{39}{37}e^{7} + \frac{32}{37}e^{6} + \frac{533}{37}e^{5} - \frac{540}{37}e^{4} - \frac{1826}{37}e^{3} + 61e^{2} + \frac{127}{37}e - \frac{634}{37}$ |
| 67 | $[67, 67, 3w - 22]$ | $\phantom{-}\frac{26}{37}e^{7} - \frac{9}{37}e^{6} - \frac{343}{37}e^{5} + \frac{175}{37}e^{4} + \frac{1131}{37}e^{3} - 19e^{2} - \frac{23}{37}e - \frac{120}{37}$ |
Atkin-Lehner eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| $4$ | $[4, 2, 2]$ | $1$ |