Base field 4.4.8725.1
Generator \(w\), with minimal polynomial \(x^{4} - x^{3} - 10x^{2} + 2x + 19\); narrow class number \(1\) and class number \(1\).
Form
| Weight: | $[2, 2, 2, 2]$ |
| Level: | $[19,19,\frac{2}{3}w^{3} - \frac{4}{3}w^{2} - \frac{13}{3}w + \frac{17}{3}]$ |
| Dimension: | $7$ |
| CM: | no |
| Base change: | no |
| Newspace dimension: | $10$ |
Hecke eigenvalues ($q$-expansion)
The Hecke eigenvalue field is $\Q(e)$ where $e$ is a root of the defining polynomial:
| \(x^{7} - 3x^{6} - 29x^{5} + 27x^{4} + 248x^{3} + 284x^{2} + 64x - 16\) |
Show full eigenvalues Hide large eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| 9 | $[9, 3, \frac{1}{3}w^{3} - \frac{2}{3}w^{2} - \frac{8}{3}w + \frac{10}{3}]$ | $-\frac{17}{64}e^{6} + \frac{35}{32}e^{5} + \frac{411}{64}e^{4} - \frac{223}{16}e^{3} - \frac{785}{16}e^{2} - \frac{109}{4}e + \frac{3}{4}$ |
| 9 | $[9, 3, w + 1]$ | $\phantom{-}e$ |
| 11 | $[11, 11, w^{2} - w - 6]$ | $-\frac{3}{16}e^{6} + \frac{3}{4}e^{5} + \frac{75}{16}e^{4} - \frac{39}{4}e^{3} - 37e^{2} - 15e + 7$ |
| 11 | $[11, 11, \frac{2}{3}w^{3} - \frac{7}{3}w^{2} - \frac{7}{3}w + \frac{23}{3}]$ | $-\frac{13}{64}e^{6} + \frac{27}{32}e^{5} + \frac{311}{64}e^{4} - \frac{175}{16}e^{3} - \frac{581}{16}e^{2} - \frac{71}{4}e + \frac{7}{4}$ |
| 16 | $[16, 2, 2]$ | $-\frac{5}{64}e^{6} + \frac{11}{32}e^{5} + \frac{119}{64}e^{4} - \frac{81}{16}e^{3} - \frac{221}{16}e^{2} + \frac{3}{4}e + \frac{15}{4}$ |
| 19 | $[19, 19, w]$ | $-\frac{7}{64}e^{6} + \frac{17}{32}e^{5} + \frac{149}{64}e^{4} - \frac{125}{16}e^{3} - \frac{251}{16}e^{2} + \frac{17}{4}e + \frac{13}{4}$ |
| 19 | $[19, 19, \frac{1}{3}w^{3} - \frac{2}{3}w^{2} - \frac{8}{3}w + \frac{7}{3}]$ | $\phantom{-}\frac{21}{64}e^{6} - \frac{47}{32}e^{5} - \frac{471}{64}e^{4} + \frac{315}{16}e^{3} + \frac{841}{16}e^{2} + \frac{69}{4}e - \frac{11}{4}$ |
| 19 | $[19, 19, \frac{1}{3}w^{3} - \frac{2}{3}w^{2} - \frac{2}{3}w + \frac{10}{3}]$ | $-\frac{1}{4}e^{6} + e^{5} + \frac{25}{4}e^{4} - 13e^{3} - \frac{97}{2}e^{2} - \frac{47}{2}e + 4$ |
| 19 | $[19, 19, -\frac{2}{3}w^{3} + \frac{4}{3}w^{2} + \frac{13}{3}w - \frac{17}{3}]$ | $-1$ |
| 25 | $[25, 5, \frac{2}{3}w^{3} - \frac{4}{3}w^{2} - \frac{10}{3}w + \frac{11}{3}]$ | $\phantom{-}\frac{15}{64}e^{6} - \frac{33}{32}e^{5} - \frac{349}{64}e^{4} + \frac{225}{16}e^{3} + \frac{655}{16}e^{2} + \frac{39}{4}e - \frac{29}{4}$ |
| 31 | $[31, 31, w + 3]$ | $\phantom{-}\frac{11}{64}e^{6} - \frac{25}{32}e^{5} - \frac{241}{64}e^{4} + \frac{167}{16}e^{3} + \frac{419}{16}e^{2} + \frac{29}{4}e + \frac{15}{4}$ |
| 31 | $[31, 31, -\frac{1}{3}w^{3} + \frac{2}{3}w^{2} + \frac{8}{3}w - \frac{16}{3}]$ | $-\frac{9}{32}e^{6} + \frac{21}{16}e^{5} + \frac{191}{32}e^{4} - \frac{143}{8}e^{3} - \frac{321}{8}e^{2} - \frac{11}{2}e + \frac{3}{2}$ |
| 31 | $[31, 31, \frac{2}{3}w^{3} - \frac{7}{3}w^{2} - \frac{10}{3}w + \frac{23}{3}]$ | $-\frac{1}{8}e^{6} + \frac{1}{2}e^{5} + \frac{25}{8}e^{4} - \frac{13}{2}e^{3} - 25e^{2} - 11e + 10$ |
| 31 | $[31, 31, -\frac{1}{3}w^{3} + \frac{5}{3}w^{2} + \frac{5}{3}w - \frac{25}{3}]$ | $-\frac{13}{32}e^{6} + \frac{27}{16}e^{5} + \frac{311}{32}e^{4} - \frac{173}{8}e^{3} - \frac{583}{8}e^{2} - \frac{79}{2}e - \frac{7}{2}$ |
| 59 | $[59, 59, -\frac{2}{3}w^{3} + \frac{1}{3}w^{2} + \frac{10}{3}w + \frac{1}{3}]$ | $-\frac{3}{16}e^{6} + e^{5} + \frac{55}{16}e^{4} - \frac{29}{2}e^{3} - \frac{81}{4}e^{2} + 14e + 9$ |
| 59 | $[59, 59, \frac{5}{3}w^{3} - \frac{13}{3}w^{2} - \frac{25}{3}w + \frac{47}{3}]$ | $\phantom{-}\frac{7}{64}e^{6} - \frac{17}{32}e^{5} - \frac{149}{64}e^{4} + \frac{129}{16}e^{3} + \frac{239}{16}e^{2} - \frac{35}{4}e - \frac{13}{4}$ |
| 61 | $[61, 61, \frac{1}{3}w^{3} + \frac{1}{3}w^{2} - \frac{11}{3}w - \frac{8}{3}]$ | $\phantom{-}\frac{13}{64}e^{6} - \frac{31}{32}e^{5} - \frac{271}{64}e^{4} + \frac{207}{16}e^{3} + \frac{461}{16}e^{2} + \frac{33}{4}e + \frac{25}{4}$ |
| 61 | $[61, 61, -\frac{2}{3}w^{3} + \frac{7}{3}w^{2} + \frac{4}{3}w - \frac{26}{3}]$ | $\phantom{-}\frac{11}{32}e^{6} - \frac{19}{16}e^{5} - \frac{301}{32}e^{4} + \frac{109}{8}e^{3} + \frac{629}{8}e^{2} + \frac{119}{2}e - \frac{5}{2}$ |
| 71 | $[71, 71, \frac{2}{3}w^{3} - \frac{4}{3}w^{2} - \frac{13}{3}w + \frac{5}{3}]$ | $-\frac{41}{64}e^{6} + \frac{83}{32}e^{5} + \frac{1003}{64}e^{4} - \frac{525}{16}e^{3} - \frac{1929}{16}e^{2} - \frac{267}{4}e + \frac{3}{4}$ |
| 71 | $[71, 71, -w^{3} + 3w^{2} + 5w - 10]$ | $\phantom{-}\frac{13}{64}e^{6} - \frac{31}{32}e^{5} - \frac{271}{64}e^{4} + \frac{215}{16}e^{3} + \frac{453}{16}e^{2} - \frac{3}{4}e - \frac{15}{4}$ |
Atkin-Lehner eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| $19$ | $[19,19,\frac{2}{3}w^{3} - \frac{4}{3}w^{2} - \frac{13}{3}w + \frac{17}{3}]$ | $1$ |