Base field 4.4.10273.1
Generator \(w\), with minimal polynomial \(x^{4} - 2x^{3} - 5x^{2} + x + 2\); narrow class number \(1\) and class number \(1\).
Form
| Weight: | $[2, 2, 2, 2]$ |
| Level: | $[13, 13, -w^{2} + 2w + 3]$ |
| Dimension: | $11$ |
| CM: | no |
| Base change: | no |
| Newspace dimension: | $17$ |
Hecke eigenvalues ($q$-expansion)
The Hecke eigenvalue field is $\Q(e)$ where $e$ is a root of the defining polynomial:
| \(x^{11} - 3x^{10} - 17x^{9} + 51x^{8} + 100x^{7} - 293x^{6} - 254x^{5} + 667x^{4} + 296x^{3} - 504x^{2} - 192x + 48\) |
Show full eigenvalues Hide large eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| 2 | $[2, 2, -w]$ | $\phantom{-}e$ |
| 3 | $[3, 3, w - 1]$ | $-\frac{3}{32}e^{10} + \frac{5}{32}e^{9} + \frac{51}{32}e^{8} - \frac{81}{32}e^{7} - \frac{71}{8}e^{6} + \frac{427}{32}e^{5} + \frac{279}{16}e^{4} - \frac{813}{32}e^{3} - \frac{47}{8}e^{2} + \frac{85}{8}e - 1$ |
| 8 | $[8, 2, w^{3} - 2w^{2} - 5w + 1]$ | $-\frac{1}{16}e^{10} + \frac{1}{16}e^{9} + \frac{19}{16}e^{8} - \frac{13}{16}e^{7} - \frac{63}{8}e^{6} + \frac{41}{16}e^{5} + 21e^{4} + \frac{21}{16}e^{3} - \frac{135}{8}e^{2} - \frac{41}{4}e - \frac{3}{2}$ |
| 13 | $[13, 13, -w^{2} + 2w + 3]$ | $-1$ |
| 17 | $[17, 17, -w^{2} + 3w + 3]$ | $-\frac{1}{16}e^{10} + \frac{7}{16}e^{9} + \frac{9}{16}e^{8} - \frac{115}{16}e^{7} + \frac{5}{4}e^{6} + \frac{625}{16}e^{5} - \frac{155}{8}e^{4} - \frac{1287}{16}e^{3} + \frac{155}{4}e^{2} + \frac{189}{4}e - 6$ |
| 17 | $[17, 17, -w^{2} + 2w + 1]$ | $-\frac{1}{8}e^{10} + \frac{1}{2}e^{9} + \frac{7}{4}e^{8} - 8e^{7} - \frac{53}{8}e^{6} + \frac{333}{8}e^{5} + \frac{13}{8}e^{4} - \frac{641}{8}e^{3} + \frac{175}{8}e^{2} + 43e - \frac{15}{2}$ |
| 27 | $[27, 3, w^{3} - w^{2} - 6w - 5]$ | $\phantom{-}\frac{3}{16}e^{10} - \frac{3}{8}e^{9} - \frac{13}{4}e^{8} + \frac{45}{8}e^{7} + \frac{313}{16}e^{6} - \frac{407}{16}e^{5} - \frac{789}{16}e^{4} + \frac{527}{16}e^{3} + \frac{717}{16}e^{2} + 5e - \frac{25}{4}$ |
| 29 | $[29, 29, w^{3} - 2w^{2} - 2w + 1]$ | $\phantom{-}\frac{1}{32}e^{10} - \frac{1}{32}e^{9} - \frac{27}{32}e^{8} + \frac{21}{32}e^{7} + \frac{123}{16}e^{6} - \frac{145}{32}e^{5} - 28e^{4} + \frac{355}{32}e^{3} + \frac{575}{16}e^{2} - \frac{61}{8}e - \frac{33}{4}$ |
| 47 | $[47, 47, w^{3} - 2w^{2} - 4w - 3]$ | $\phantom{-}\frac{1}{8}e^{10} - \frac{9}{4}e^{8} - \frac{1}{2}e^{7} + \frac{113}{8}e^{6} + \frac{51}{8}e^{5} - \frac{305}{8}e^{4} - \frac{183}{8}e^{3} + \frac{357}{8}e^{2} + 25e - \frac{21}{2}$ |
| 49 | $[49, 7, w^{3} - 3w^{2} - 3w + 1]$ | $\phantom{-}\frac{1}{8}e^{10} - \frac{9}{4}e^{8} - \frac{1}{2}e^{7} + \frac{113}{8}e^{6} + \frac{43}{8}e^{5} - \frac{305}{8}e^{4} - \frac{95}{8}e^{3} + \frac{357}{8}e^{2} - e - \frac{25}{2}$ |
| 49 | $[49, 7, -2w^{3} + 5w^{2} + 7w - 3]$ | $-\frac{1}{32}e^{10} - \frac{7}{32}e^{9} + \frac{19}{32}e^{8} + \frac{115}{32}e^{7} - \frac{55}{16}e^{6} - \frac{591}{32}e^{5} + \frac{21}{4}e^{4} + \frac{973}{32}e^{3} + \frac{37}{16}e^{2} - \frac{83}{8}e - \frac{11}{4}$ |
| 59 | $[59, 59, -2w^{3} + 6w^{2} + 5w - 7]$ | $\phantom{-}\frac{1}{16}e^{10} + \frac{1}{8}e^{9} - \frac{3}{2}e^{8} - \frac{19}{8}e^{7} + \frac{207}{16}e^{6} + \frac{243}{16}e^{5} - \frac{763}{16}e^{4} - \frac{563}{16}e^{3} + \frac{1067}{16}e^{2} + 17e - \frac{63}{4}$ |
| 61 | $[61, 61, 2w^{3} - 4w^{2} - 9w - 3]$ | $\phantom{-}\frac{1}{8}e^{10} + \frac{1}{4}e^{9} - 3e^{8} - \frac{19}{4}e^{7} + \frac{207}{8}e^{6} + \frac{251}{8}e^{5} - \frac{755}{8}e^{4} - \frac{667}{8}e^{3} + \frac{1011}{8}e^{2} + 70e - \frac{47}{2}$ |
| 71 | $[71, 71, w^{3} - 3w^{2} - 3w + 7]$ | $\phantom{-}\frac{1}{8}e^{10} - \frac{9}{4}e^{8} - \frac{1}{2}e^{7} + \frac{113}{8}e^{6} + \frac{51}{8}e^{5} - \frac{297}{8}e^{4} - \frac{175}{8}e^{3} + \frac{285}{8}e^{2} + 20e + \frac{3}{2}$ |
| 73 | $[73, 73, 2w^{3} - 6w^{2} - 3w + 5]$ | $-\frac{7}{16}e^{10} + \frac{15}{16}e^{9} + \frac{117}{16}e^{8} - \frac{235}{16}e^{7} - \frac{329}{8}e^{6} + \frac{1175}{16}e^{5} + \frac{183}{2}e^{4} - \frac{2069}{16}e^{3} - \frac{533}{8}e^{2} + \frac{195}{4}e + \frac{17}{2}$ |
| 73 | $[73, 73, w^{3} - 3w^{2} - 4w + 5]$ | $\phantom{-}\frac{1}{4}e^{10} - \frac{1}{8}e^{9} - \frac{37}{8}e^{8} + \frac{9}{8}e^{7} + \frac{239}{8}e^{6} + \frac{5}{4}e^{5} - \frac{641}{8}e^{4} - \frac{47}{2}e^{3} + \frac{627}{8}e^{2} + \frac{67}{2}e - \frac{19}{2}$ |
| 83 | $[83, 83, -2w^{3} + 5w^{2} + 8w - 3]$ | $\phantom{-}\frac{3}{16}e^{10} - \frac{5}{8}e^{9} - \frac{5}{2}e^{8} + \frac{79}{8}e^{7} + \frac{125}{16}e^{6} - \frac{807}{16}e^{5} + \frac{143}{16}e^{4} + \frac{1511}{16}e^{3} - \frac{847}{16}e^{2} - 45e + \frac{75}{4}$ |
| 89 | $[89, 89, -2w^{3} + 6w^{2} + 4w - 7]$ | $\phantom{-}\frac{7}{8}e^{9} - \frac{9}{8}e^{8} - \frac{119}{8}e^{7} + \frac{133}{8}e^{6} + \frac{169}{2}e^{5} - \frac{607}{8}e^{4} - \frac{735}{4}e^{3} + \frac{905}{8}e^{2} + \frac{235}{2}e - \frac{21}{2}$ |
| 89 | $[89, 89, w^{3} - 2w^{2} - 6w + 3]$ | $\phantom{-}\frac{1}{8}e^{9} + \frac{1}{8}e^{8} - \frac{17}{8}e^{7} - \frac{13}{8}e^{6} + \frac{23}{2}e^{5} + \frac{47}{8}e^{4} - \frac{81}{4}e^{3} - \frac{57}{8}e^{2} + \frac{9}{2}e + \frac{21}{2}$ |
| 97 | $[97, 97, -3w - 1]$ | $-\frac{3}{8}e^{10} + \frac{17}{16}e^{9} + \frac{93}{16}e^{8} - \frac{273}{16}e^{7} - \frac{451}{16}e^{6} + \frac{709}{8}e^{5} + \frac{701}{16}e^{4} - \frac{331}{2}e^{3} + \frac{17}{16}e^{2} + \frac{283}{4}e - \frac{49}{4}$ |
Atkin-Lehner eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| $13$ | $[13, 13, -w^{2} + 2w + 3]$ | $1$ |