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: | $[9, 3, 3]$ |
| Dimension: | $7$ |
| CM: | no |
| Base change: | yes |
| Newspace dimension: | $19$ |
Hecke eigenvalues ($q$-expansion)
The Hecke eigenvalue field is $\Q(e)$ where $e$ is a root of the defining polynomial:
| \(x^{7} - 4x^{6} - 16x^{5} + 67x^{4} + 47x^{3} - 248x^{2} + 54x + 123\) |
Show full eigenvalues Hide large eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| 3 | $[3, 3, w + 6]$ | $\phantom{-}1$ |
| 3 | $[3, 3, -w + 7]$ | $\phantom{-}1$ |
| 4 | $[4, 2, 2]$ | $\phantom{-}e$ |
| 11 | $[11, 11, -3w - 17]$ | $\phantom{-}\frac{2}{19}e^{6} + \frac{4}{19}e^{5} - \frac{27}{19}e^{4} - \frac{66}{19}e^{3} + \frac{2}{19}e^{2} + \frac{124}{19}e + \frac{111}{19}$ |
| 11 | $[11, 11, 3w - 20]$ | $\phantom{-}\frac{2}{19}e^{6} + \frac{4}{19}e^{5} - \frac{27}{19}e^{4} - \frac{66}{19}e^{3} + \frac{2}{19}e^{2} + \frac{124}{19}e + \frac{111}{19}$ |
| 13 | $[13, 13, 2w - 13]$ | $-\frac{21}{38}e^{6} + \frac{15}{38}e^{5} + \frac{369}{38}e^{4} - \frac{100}{19}e^{3} - \frac{1427}{38}e^{2} + \frac{693}{38}e + \frac{953}{38}$ |
| 13 | $[13, 13, 2w + 11]$ | $-\frac{21}{38}e^{6} + \frac{15}{38}e^{5} + \frac{369}{38}e^{4} - \frac{100}{19}e^{3} - \frac{1427}{38}e^{2} + \frac{693}{38}e + \frac{953}{38}$ |
| 17 | $[17, 17, w + 7]$ | $\phantom{-}\frac{1}{38}e^{6} - \frac{17}{38}e^{5} - \frac{23}{38}e^{4} + \frac{126}{19}e^{3} + \frac{191}{38}e^{2} - \frac{527}{38}e - \frac{87}{38}$ |
| 17 | $[17, 17, -w + 8]$ | $\phantom{-}\frac{1}{38}e^{6} - \frac{17}{38}e^{5} - \frac{23}{38}e^{4} + \frac{126}{19}e^{3} + \frac{191}{38}e^{2} - \frac{527}{38}e - \frac{87}{38}$ |
| 19 | $[19, 19, -w - 4]$ | $\phantom{-}\frac{5}{19}e^{6} - \frac{9}{19}e^{5} - \frac{96}{19}e^{4} + \frac{139}{19}e^{3} + \frac{461}{19}e^{2} - \frac{412}{19}e - \frac{378}{19}$ |
| 19 | $[19, 19, -w + 5]$ | $\phantom{-}\frac{5}{19}e^{6} - \frac{9}{19}e^{5} - \frac{96}{19}e^{4} + \frac{139}{19}e^{3} + \frac{461}{19}e^{2} - \frac{412}{19}e - \frac{378}{19}$ |
| 25 | $[25, 5, 5]$ | $\phantom{-}\frac{3}{19}e^{6} + \frac{6}{19}e^{5} - \frac{50}{19}e^{4} - \frac{99}{19}e^{3} + \frac{155}{19}e^{2} + \frac{205}{19}e - \frac{90}{19}$ |
| 31 | $[31, 31, -6w - 35]$ | $\phantom{-}\frac{7}{19}e^{6} - \frac{5}{19}e^{5} - \frac{123}{19}e^{4} + \frac{73}{19}e^{3} + \frac{463}{19}e^{2} - \frac{288}{19}e - \frac{267}{19}$ |
| 31 | $[31, 31, -6w + 41]$ | $\phantom{-}\frac{7}{19}e^{6} - \frac{5}{19}e^{5} - \frac{123}{19}e^{4} + \frac{73}{19}e^{3} + \frac{463}{19}e^{2} - \frac{288}{19}e - \frac{267}{19}$ |
| 37 | $[37, 37, -w - 1]$ | $\phantom{-}\frac{7}{38}e^{6} - \frac{5}{38}e^{5} - \frac{123}{38}e^{4} + \frac{27}{19}e^{3} + \frac{501}{38}e^{2} - \frac{117}{38}e - \frac{495}{38}$ |
| 37 | $[37, 37, w - 2]$ | $\phantom{-}\frac{7}{38}e^{6} - \frac{5}{38}e^{5} - \frac{123}{38}e^{4} + \frac{27}{19}e^{3} + \frac{501}{38}e^{2} - \frac{117}{38}e - \frac{495}{38}$ |
| 47 | $[47, 47, 3w + 16]$ | $\phantom{-}\frac{10}{19}e^{6} + \frac{1}{19}e^{5} - \frac{173}{19}e^{4} - \frac{26}{19}e^{3} + \frac{618}{19}e^{2} - \frac{121}{19}e - \frac{357}{19}$ |
| 47 | $[47, 47, -3w + 19]$ | $\phantom{-}\frac{10}{19}e^{6} + \frac{1}{19}e^{5} - \frac{173}{19}e^{4} - \frac{26}{19}e^{3} + \frac{618}{19}e^{2} - \frac{121}{19}e - \frac{357}{19}$ |
| 49 | $[49, 7, -7]$ | $-\frac{37}{38}e^{6} + \frac{21}{38}e^{5} + \frac{623}{38}e^{4} - \frac{140}{19}e^{3} - \frac{2127}{38}e^{2} + \frac{1335}{38}e + \frac{1243}{38}$ |
| 67 | $[67, 67, 3w - 22]$ | $-\frac{2}{19}e^{6} - \frac{4}{19}e^{5} + \frac{27}{19}e^{4} + \frac{66}{19}e^{3} - \frac{2}{19}e^{2} - \frac{162}{19}e - \frac{35}{19}$ |
Atkin-Lehner eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| $3$ | $[3, 3, w + 6]$ | $-1$ |
| $3$ | $[3, 3, -w + 7]$ | $-1$ |