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: | no |
| Newspace dimension: | $76$ |
Hecke eigenvalues ($q$-expansion)
The Hecke eigenvalue field is $\Q(e)$ where $e$ is a root of the defining polynomial:
| \(x^{8} + 8x^{7} - 32x^{6} - 304x^{5} + 280x^{4} + 3296x^{3} - 224x^{2} - 7936x - 3872\) |
Show full eigenvalues Hide large eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| 2 | $[2, 2, -w - 11]$ | $-\frac{1}{1914}e^{7} - \frac{7}{1914}e^{6} + \frac{25}{957}e^{5} + \frac{160}{957}e^{4} - \frac{299}{638}e^{3} - \frac{4027}{1914}e^{2} + \frac{923}{319}e + \frac{382}{87}$ |
| 3 | $[3, 3, w]$ | $\phantom{-}1$ |
| 7 | $[7, 7, w + 2]$ | $-\frac{1}{1914}e^{7} - \frac{7}{1914}e^{6} + \frac{25}{957}e^{5} + \frac{160}{957}e^{4} - \frac{299}{638}e^{3} - \frac{4027}{1914}e^{2} + \frac{1242}{319}e + \frac{382}{87}$ |
| 7 | $[7, 7, w + 5]$ | $\phantom{-}\frac{1}{1914}e^{7} + \frac{7}{1914}e^{6} - \frac{25}{957}e^{5} - \frac{160}{957}e^{4} + \frac{299}{638}e^{3} + \frac{4027}{1914}e^{2} - \frac{1242}{319}e - \frac{556}{87}$ |
| 17 | $[17, 17, w + 2]$ | $\phantom{-}\frac{7}{5104}e^{7} + \frac{17}{1914}e^{6} - \frac{93}{1276}e^{5} - \frac{763}{1914}e^{4} + \frac{433}{319}e^{3} + \frac{3151}{638}e^{2} - \frac{7325}{957}e - \frac{1043}{87}$ |
| 17 | $[17, 17, w + 15]$ | $\phantom{-}\frac{7}{5104}e^{7} + \frac{79}{7656}e^{6} - \frac{41}{638}e^{5} - \frac{917}{1914}e^{4} + \frac{312}{319}e^{3} + \frac{3657}{638}e^{2} - \frac{5059}{957}e - \frac{745}{87}$ |
| 19 | $[19, 19, w + 3]$ | $\phantom{-}\frac{1}{1914}e^{7} - \frac{2}{957}e^{6} - \frac{2}{33}e^{5} - \frac{23}{1914}e^{4} + \frac{2525}{1914}e^{3} + \frac{1783}{1914}e^{2} - \frac{2562}{319}e - \frac{138}{29}$ |
| 19 | $[19, 19, w + 16]$ | $-\frac{1}{1914}e^{7} - \frac{3}{319}e^{6} - \frac{8}{957}e^{5} + \frac{617}{1914}e^{4} + \frac{731}{1914}e^{3} - \frac{6271}{1914}e^{2} - \frac{78}{319}e + \frac{524}{87}$ |
| 23 | $[23, 23, -w - 10]$ | $-\frac{1}{957}e^{7} - \frac{15}{2552}e^{6} + \frac{233}{3828}e^{5} + \frac{1291}{3828}e^{4} - \frac{941}{957}e^{3} - \frac{9407}{1914}e^{2} + \frac{4295}{957}e + \frac{1004}{87}$ |
| 23 | $[23, 23, w - 10]$ | $-\frac{1}{957}e^{7} - \frac{67}{7656}e^{6} + \frac{167}{3828}e^{5} + \frac{423}{1276}e^{4} - \frac{853}{957}e^{3} - \frac{6701}{1914}e^{2} + \frac{6781}{957}e + \frac{524}{87}$ |
| 25 | $[25, 5, -5]$ | $\phantom{-}\frac{1}{116}e^{6} + \frac{3}{58}e^{5} - \frac{27}{116}e^{4} - \frac{37}{29}e^{3} + \frac{51}{29}e^{2} + \frac{180}{29}e - \frac{100}{29}$ |
| 29 | $[29, 29, w + 6]$ | $\phantom{-}\frac{13}{15312}e^{7} + \frac{5}{957}e^{6} - \frac{179}{3828}e^{5} - \frac{1205}{3828}e^{4} + \frac{1063}{1914}e^{3} + \frac{8297}{1914}e^{2} - \frac{349}{319}e - \frac{201}{29}$ |
| 29 | $[29, 29, w + 23]$ | $\phantom{-}\frac{13}{15312}e^{7} + \frac{17}{2552}e^{6} - \frac{73}{1914}e^{5} - \frac{875}{3828}e^{4} + \frac{1613}{1914}e^{3} + \frac{4073}{1914}e^{2} - \frac{1933}{319}e - \frac{421}{87}$ |
| 37 | $[37, 37, 9w + 100]$ | $\phantom{-}\frac{1}{348}e^{6} + \frac{1}{58}e^{5} - \frac{9}{116}e^{4} - \frac{37}{87}e^{3} + \frac{17}{29}e^{2} + \frac{60}{29}e + \frac{103}{87}$ |
| 37 | $[37, 37, -2w - 23]$ | $\phantom{-}\frac{1}{348}e^{6} + \frac{1}{58}e^{5} - \frac{9}{116}e^{4} - \frac{37}{87}e^{3} + \frac{17}{29}e^{2} + \frac{60}{29}e + \frac{103}{87}$ |
| 41 | $[41, 41, w]$ | $\phantom{-}\frac{7}{7656}e^{7} + \frac{49}{7656}e^{6} - \frac{175}{3828}e^{5} - \frac{280}{957}e^{4} + \frac{142}{319}e^{3} + \frac{2447}{957}e^{2} + \frac{538}{319}e - \frac{16}{87}$ |
| 53 | $[53, 53, w + 21]$ | $\phantom{-}\frac{1}{638}e^{7} + \frac{53}{3828}e^{6} - \frac{39}{638}e^{5} - \frac{739}{1276}e^{4} + \frac{1877}{1914}e^{3} + \frac{2041}{319}e^{2} - \frac{2428}{319}e - \frac{608}{87}$ |
| 53 | $[53, 53, w + 32]$ | $\phantom{-}\frac{1}{638}e^{7} + \frac{31}{3828}e^{6} - \frac{61}{638}e^{5} - \frac{541}{1276}e^{4} + \frac{3505}{1914}e^{3} + \frac{1986}{319}e^{2} - \frac{3110}{319}e - \frac{1684}{87}$ |
| 59 | $[59, 59, -w - 8]$ | $-\frac{5}{7656}e^{7} - \frac{1}{319}e^{6} + \frac{79}{1914}e^{5} + \frac{41}{319}e^{4} - \frac{268}{319}e^{3} - \frac{1399}{957}e^{2} + \frac{3574}{957}e + \frac{186}{29}$ |
| 59 | $[59, 59, w - 8]$ | $-\frac{5}{7656}e^{7} - \frac{23}{3828}e^{6} + \frac{23}{957}e^{5} + \frac{277}{957}e^{4} - \frac{26}{319}e^{3} - \frac{2917}{957}e^{2} - \frac{958}{957}e - \frac{38}{87}$ |
Atkin-Lehner eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| $3$ | $[3, 3, w]$ | $-1$ |