Base field \(\Q(\sqrt{123}) \)
Generator \(w\), with minimal polynomial \(x^{2} - 123\); narrow class number \(4\) and class number \(2\).
Form
| Weight: | $[2, 2]$ |
| Level: | $[3, 3, w]$ |
| Dimension: | $8$ |
| CM: | no |
| Base change: | yes |
| Newspace dimension: | $76$ |
Hecke eigenvalues ($q$-expansion)
The Hecke eigenvalue field is $\Q(e)$ where $e$ is a root of the defining polynomial:
| \(x^{8} + 32x^{6} + 268x^{4} + 260x^{2} + 64\) |
Show full eigenvalues Hide large eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| 2 | $[2, 2, -w - 11]$ | $-\frac{11}{164}e^{6} - \frac{177}{82}e^{4} - \frac{727}{41}e^{2} - \frac{385}{41}$ |
| 3 | $[3, 3, w]$ | $\phantom{-}\frac{3}{656}e^{7} + \frac{13}{82}e^{5} + \frac{243}{164}e^{3} + \frac{351}{164}e$ |
| 7 | $[7, 7, w + 2]$ | $\phantom{-}e$ |
| 7 | $[7, 7, w + 5]$ | $\phantom{-}e$ |
| 17 | $[17, 17, w + 2]$ | $\phantom{-}\frac{1}{2}e^{3} + 8e$ |
| 17 | $[17, 17, w + 15]$ | $\phantom{-}\frac{1}{2}e^{3} + 8e$ |
| 19 | $[19, 19, w + 3]$ | $\phantom{-}\frac{4}{41}e^{7} + \frac{125}{41}e^{5} + \frac{968}{41}e^{3} + \frac{191}{41}e$ |
| 19 | $[19, 19, w + 16]$ | $\phantom{-}\frac{4}{41}e^{7} + \frac{125}{41}e^{5} + \frac{968}{41}e^{3} + \frac{191}{41}e$ |
| 23 | $[23, 23, -w - 10]$ | $\phantom{-}\frac{17}{82}e^{6} + \frac{521}{82}e^{4} + \frac{2016}{41}e^{2} + \frac{944}{41}$ |
| 23 | $[23, 23, w - 10]$ | $\phantom{-}\frac{17}{82}e^{6} + \frac{521}{82}e^{4} + \frac{2016}{41}e^{2} + \frac{944}{41}$ |
| 25 | $[25, 5, -5]$ | $-\frac{4}{41}e^{6} - \frac{125}{41}e^{4} - \frac{1009}{41}e^{2} - \frac{806}{41}$ |
| 29 | $[29, 29, w + 6]$ | $-\frac{10}{41}e^{7} - \frac{625}{82}e^{5} - \frac{4963}{82}e^{3} - \frac{1318}{41}e$ |
| 29 | $[29, 29, w + 23]$ | $-\frac{10}{41}e^{7} - \frac{625}{82}e^{5} - \frac{4963}{82}e^{3} - \frac{1318}{41}e$ |
| 37 | $[37, 37, 9w + 100]$ | $-\frac{31}{164}e^{6} - \frac{255}{41}e^{4} - \frac{2142}{41}e^{2} - \frac{1126}{41}$ |
| 37 | $[37, 37, -2w - 23]$ | $-\frac{31}{164}e^{6} - \frac{255}{41}e^{4} - \frac{2142}{41}e^{2} - \frac{1126}{41}$ |
| 41 | $[41, 41, w]$ | $-\frac{12}{41}e^{7} - \frac{375}{41}e^{5} - \frac{2945}{41}e^{3} - \frac{1188}{41}e$ |
| 53 | $[53, 53, w + 21]$ | $\phantom{-}\frac{23}{82}e^{7} + \frac{729}{82}e^{5} + \frac{2947}{41}e^{3} + \frac{1774}{41}e$ |
| 53 | $[53, 53, w + 32]$ | $\phantom{-}\frac{23}{82}e^{7} + \frac{729}{82}e^{5} + \frac{2947}{41}e^{3} + \frac{1774}{41}e$ |
| 59 | $[59, 59, -w - 8]$ | $-\frac{1}{41}e^{6} - \frac{21}{41}e^{4} - \frac{78}{41}e^{2} - \frac{140}{41}$ |
| 59 | $[59, 59, w - 8]$ | $-\frac{1}{41}e^{6} - \frac{21}{41}e^{4} - \frac{78}{41}e^{2} - \frac{140}{41}$ |
Atkin-Lehner eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| $3$ | $[3, 3, w]$ | $-\frac{3}{656}e^{7} - \frac{13}{82}e^{5} - \frac{243}{164}e^{3} - \frac{351}{164}e$ |