Base field 6.6.820125.1
Generator \(w\), with minimal polynomial \(x^{6} - 9x^{4} - 4x^{3} + 9x^{2} + 3x - 1\); narrow class number \(1\) and class number \(1\).
Form
| Weight: | $[2, 2, 2, 2, 2, 2]$ |
| Level: | $[19,19,-\frac{2}{19}w^{5} - \frac{5}{19}w^{4} + \frac{15}{19}w^{3} + \frac{55}{19}w^{2} + \frac{34}{19}w - \frac{35}{19}]$ |
| Dimension: | $5$ |
| CM: | no |
| Base change: | no |
| Newspace dimension: | $6$ |
Hecke eigenvalues ($q$-expansion)
The Hecke eigenvalue field is $\Q(e)$ where $e$ is a root of the defining polynomial:
| \(x^{5} - 2x^{4} - 13x^{3} + 28x^{2} + 15x - 40\) |
Show full eigenvalues Hide large eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| 9 | $[9, 3, -\frac{6}{19}w^{5} + \frac{4}{19}w^{4} + \frac{45}{19}w^{3} - \frac{6}{19}w^{2} - \frac{12}{19}w + \frac{28}{19}]$ | $\phantom{-}e$ |
| 19 | $[19, 19, -\frac{20}{19}w^{5} + \frac{7}{19}w^{4} + \frac{169}{19}w^{3} + \frac{18}{19}w^{2} - \frac{116}{19}w + \frac{11}{19}]$ | $-e^{2} + 7$ |
| 19 | $[19, 19, -\frac{10}{19}w^{5} - \frac{6}{19}w^{4} + \frac{94}{19}w^{3} + \frac{85}{19}w^{2} - \frac{77}{19}w - \frac{42}{19}]$ | $\phantom{-}e^{4} - 13e^{2} + 2e + 25$ |
| 19 | $[19, 19, -\frac{16}{19}w^{5} - \frac{2}{19}w^{4} + \frac{139}{19}w^{3} + \frac{79}{19}w^{2} - \frac{108}{19}w - \frac{33}{19}]$ | $-6e^{4} + 2e^{3} + 80e^{2} - 32e - 134$ |
| 19 | $[19, 19, -\frac{4}{19}w^{5} + \frac{9}{19}w^{4} + \frac{30}{19}w^{3} - \frac{61}{19}w^{2} - \frac{8}{19}w + \frac{44}{19}]$ | $-4e^{4} + e^{3} + 54e^{2} - 19e - 92$ |
| 19 | $[19, 19, -\frac{8}{19}w^{5} - \frac{1}{19}w^{4} + \frac{79}{19}w^{3} + \frac{30}{19}w^{2} - \frac{111}{19}w - \frac{7}{19}]$ | $-e^{4} + 14e^{2} - 27$ |
| 19 | $[19, 19, \frac{2}{19}w^{5} + \frac{5}{19}w^{4} - \frac{15}{19}w^{3} - \frac{55}{19}w^{2} - \frac{34}{19}w + \frac{35}{19}]$ | $-1$ |
| 64 | $[64, 2, 2]$ | $\phantom{-}5e^{4} - e^{3} - 67e^{2} + 20e + 119$ |
| 71 | $[71, 71, \frac{16}{19}w^{5} - \frac{17}{19}w^{4} - \frac{139}{19}w^{3} + \frac{73}{19}w^{2} + \frac{165}{19}w - \frac{5}{19}]$ | $\phantom{-}e^{2} + 2e - 10$ |
| 71 | $[71, 71, -\frac{11}{19}w^{5} + \frac{20}{19}w^{4} + \frac{92}{19}w^{3} - \frac{125}{19}w^{2} - \frac{117}{19}w + \frac{64}{19}]$ | $-5e^{4} + 2e^{3} + 67e^{2} - 27e - 109$ |
| 71 | $[71, 71, \frac{12}{19}w^{5} - \frac{8}{19}w^{4} - \frac{109}{19}w^{3} + \frac{12}{19}w^{2} + \frac{138}{19}w + \frac{1}{19}]$ | $\phantom{-}4e^{4} - 2e^{3} - 54e^{2} + 26e + 90$ |
| 71 | $[71, 71, \frac{22}{19}w^{5} - \frac{21}{19}w^{4} - \frac{184}{19}w^{3} + \frac{79}{19}w^{2} + \frac{177}{19}w - \frac{52}{19}]$ | $\phantom{-}7e^{4} - 2e^{3} - 95e^{2} + 34e + 165$ |
| 71 | $[71, 71, w^{5} - w^{4} - 8w^{3} + 4w^{2} + 6w - 3]$ | $-3e^{4} + e^{3} + 41e^{2} - 17e - 75$ |
| 71 | $[71, 71, -\frac{24}{19}w^{5} + \frac{16}{19}w^{4} + \frac{199}{19}w^{3} - \frac{24}{19}w^{2} - \frac{162}{19}w - \frac{21}{19}]$ | $\phantom{-}4e^{4} - 2e^{3} - 54e^{2} + 26e + 90$ |
| 89 | $[89, 89, \frac{13}{19}w^{5} + \frac{4}{19}w^{4} - \frac{126}{19}w^{3} - \frac{82}{19}w^{2} + \frac{159}{19}w + \frac{66}{19}]$ | $\phantom{-}3e^{4} - 39e^{2} + 6e + 67$ |
| 89 | $[89, 89, -\frac{17}{19}w^{5} + \frac{5}{19}w^{4} + \frac{137}{19}w^{3} + \frac{40}{19}w^{2} - \frac{53}{19}w - \frac{41}{19}]$ | $-5e^{4} + 2e^{3} + 66e^{2} - 30e - 110$ |
| 89 | $[89, 89, -\frac{29}{19}w^{5} + \frac{13}{19}w^{4} + \frac{246}{19}w^{3} + \frac{9}{19}w^{2} - \frac{210}{19}w + \frac{15}{19}]$ | $\phantom{-}10e^{4} - 4e^{3} - 136e^{2} + 58e + 238$ |
| 89 | $[89, 89, \frac{3}{19}w^{5} + \frac{17}{19}w^{4} - \frac{32}{19}w^{3} - \frac{149}{19}w^{2} - \frac{13}{19}w + \frac{62}{19}]$ | $-e^{4} + 14e^{2} - 4e - 26$ |
| 89 | $[89, 89, \frac{7}{19}w^{5} - \frac{11}{19}w^{4} - \frac{62}{19}w^{3} + \frac{64}{19}w^{2} + \frac{71}{19}w - \frac{39}{19}]$ | $-12e^{4} + 3e^{3} + 162e^{2} - 57e - 284$ |
| 89 | $[89, 89, -\frac{9}{19}w^{5} + \frac{6}{19}w^{4} + \frac{77}{19}w^{3} - \frac{28}{19}w^{2} - \frac{56}{19}w + \frac{42}{19}]$ | $\phantom{-}3e^{4} - 39e^{2} + 6e + 67$ |
Atkin-Lehner eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| $19$ | $[19,19,-\frac{2}{19}w^{5} - \frac{5}{19}w^{4} + \frac{15}{19}w^{3} + \frac{55}{19}w^{2} + \frac{34}{19}w - \frac{35}{19}]$ | $1$ |