Base field 4.4.14272.1
Generator \(w\), with minimal polynomial \(x^{4} - 2x^{3} - 5x^{2} + 2x + 3\); narrow class number \(2\) and class number \(1\).
Form
| Weight: | $[2, 2, 2, 2]$ |
| Level: | $[13, 13, w^{3} - 2w^{2} - 4w + 2]$ |
| Dimension: | $9$ |
| CM: | no |
| Base change: | no |
| Newspace dimension: | $18$ |
Hecke eigenvalues ($q$-expansion)
The Hecke eigenvalue field is $\Q(e)$ where $e$ is a root of the defining polynomial:
| \(x^{9} - 3x^{8} - 13x^{7} + 36x^{6} + 47x^{5} - 110x^{4} - 44x^{3} + 100x^{2} + 10x - 25\) |
Show full eigenvalues Hide large eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| 3 | $[3, 3, w]$ | $\phantom{-}e$ |
| 4 | $[4, 2, -w^{3} + 3w^{2} + w - 2]$ | $-\frac{4}{35}e^{8} + \frac{16}{105}e^{7} + \frac{206}{105}e^{6} - \frac{26}{15}e^{5} - \frac{1124}{105}e^{4} + \frac{97}{21}e^{3} + \frac{671}{35}e^{2} - \frac{29}{7}e - \frac{169}{21}$ |
| 11 | $[11, 11, -w^{3} + 3w^{2} + 2w - 5]$ | $\phantom{-}\frac{34}{105}e^{8} - \frac{19}{35}e^{7} - \frac{179}{35}e^{6} + \frac{77}{15}e^{5} + \frac{871}{35}e^{4} - \frac{122}{21}e^{3} - \frac{1207}{35}e^{2} - \frac{58}{21}e + \frac{263}{21}$ |
| 13 | $[13, 13, w^{3} - 2w^{2} - 4w + 2]$ | $-1$ |
| 13 | $[13, 13, -w + 2]$ | $-\frac{4}{35}e^{8} + \frac{16}{105}e^{7} + \frac{206}{105}e^{6} - \frac{26}{15}e^{5} - \frac{1124}{105}e^{4} + \frac{97}{21}e^{3} + \frac{671}{35}e^{2} - \frac{22}{7}e - \frac{211}{21}$ |
| 17 | $[17, 17, w^{3} - 3w^{2} - 2w + 2]$ | $\phantom{-}\frac{46}{105}e^{8} - \frac{73}{105}e^{7} - \frac{743}{105}e^{6} + \frac{103}{15}e^{5} + \frac{3737}{105}e^{4} - \frac{73}{7}e^{3} - \frac{1843}{35}e^{2} + \frac{8}{21}e + \frac{137}{7}$ |
| 19 | $[19, 19, -w^{2} + w + 4]$ | $\phantom{-}\frac{16}{105}e^{8} - \frac{11}{35}e^{7} - \frac{76}{35}e^{6} + \frac{53}{15}e^{5} + \frac{309}{35}e^{4} - \frac{176}{21}e^{3} - \frac{323}{35}e^{2} + \frac{11}{21}e + \frac{41}{21}$ |
| 19 | $[19, 19, w^{3} - 2w^{2} - 3w + 2]$ | $\phantom{-}\frac{16}{105}e^{8} - \frac{11}{35}e^{7} - \frac{76}{35}e^{6} + \frac{53}{15}e^{5} + \frac{309}{35}e^{4} - \frac{197}{21}e^{3} - \frac{288}{35}e^{2} + \frac{116}{21}e + \frac{41}{21}$ |
| 23 | $[23, 23, w^{2} - 2w - 1]$ | $-\frac{88}{105}e^{8} + \frac{199}{105}e^{7} + \frac{1289}{105}e^{6} - \frac{319}{15}e^{5} - \frac{5711}{105}e^{4} + \frac{374}{7}e^{3} + \frac{2494}{35}e^{2} - \frac{638}{21}e - \frac{165}{7}$ |
| 27 | $[27, 3, w^{3} - 2w^{2} - 5w - 1]$ | $\phantom{-}\frac{53}{105}e^{8} - \frac{43}{35}e^{7} - \frac{243}{35}e^{6} + \frac{199}{15}e^{5} + \frac{982}{35}e^{4} - \frac{625}{21}e^{3} - \frac{1164}{35}e^{2} + \frac{253}{21}e + \frac{292}{21}$ |
| 29 | $[29, 29, -w^{3} + 2w^{2} + 5w - 1]$ | $\phantom{-}\frac{94}{105}e^{8} - \frac{242}{105}e^{7} - \frac{1252}{105}e^{6} + \frac{129}{5}e^{5} + \frac{4663}{105}e^{4} - \frac{1363}{21}e^{3} - \frac{1447}{35}e^{2} + \frac{734}{21}e + \frac{268}{21}$ |
| 37 | $[37, 37, -w^{3} + 2w^{2} + 5w - 4]$ | $\phantom{-}\frac{6}{35}e^{8} - \frac{8}{35}e^{7} - \frac{103}{35}e^{6} + \frac{13}{5}e^{5} + \frac{562}{35}e^{4} - \frac{45}{7}e^{3} - \frac{1024}{35}e^{2} + \frac{12}{7}e + \frac{116}{7}$ |
| 41 | $[41, 41, 2w^{3} - 5w^{2} - 6w + 4]$ | $-\frac{8}{15}e^{8} + \frac{14}{15}e^{7} + \frac{124}{15}e^{6} - \frac{143}{15}e^{5} - \frac{601}{15}e^{4} + 18e^{3} + \frac{309}{5}e^{2} - \frac{22}{3}e - 26$ |
| 53 | $[53, 53, w^{3} - 2w^{2} - 3w - 2]$ | $\phantom{-}\frac{22}{105}e^{8} - \frac{37}{35}e^{7} - \frac{52}{35}e^{6} + \frac{191}{15}e^{5} - \frac{157}{35}e^{4} - \frac{809}{21}e^{3} + \frac{1074}{35}e^{2} + \frac{506}{21}e - \frac{361}{21}$ |
| 59 | $[59, 59, w - 4]$ | $-\frac{62}{105}e^{8} + \frac{47}{35}e^{7} + \frac{312}{35}e^{6} - \frac{226}{15}e^{5} - \frac{1473}{35}e^{4} + \frac{766}{21}e^{3} + \frac{2271}{35}e^{2} - \frac{292}{21}e - \frac{613}{21}$ |
| 67 | $[67, 67, 2w^{2} - 3w - 8]$ | $-\frac{118}{105}e^{8} + \frac{169}{105}e^{7} + \frac{1979}{105}e^{6} - \frac{244}{15}e^{5} - \frac{10376}{105}e^{4} + \frac{197}{7}e^{3} + \frac{5414}{35}e^{2} - \frac{278}{21}e - \frac{405}{7}$ |
| 73 | $[73, 73, 2w^{3} - 5w^{2} - 6w + 5]$ | $\phantom{-}\frac{13}{21}e^{8} - \frac{12}{7}e^{7} - \frac{60}{7}e^{6} + \frac{59}{3}e^{5} + \frac{241}{7}e^{4} - \frac{1093}{21}e^{3} - \frac{241}{7}e^{2} + \frac{739}{21}e + \frac{181}{21}$ |
| 89 | $[89, 89, -2w^{3} + 6w^{2} + 5w - 7]$ | $-\frac{89}{105}e^{8} + \frac{212}{105}e^{7} + \frac{1207}{105}e^{6} - \frac{317}{15}e^{5} - \frac{4603}{105}e^{4} + \frac{296}{7}e^{3} + \frac{1357}{35}e^{2} - \frac{283}{21}e - \frac{75}{7}$ |
| 97 | $[97, 97, -2w^{3} + 6w^{2} + 3w - 7]$ | $\phantom{-}\frac{11}{105}e^{8} - \frac{73}{105}e^{7} - \frac{8}{105}e^{6} + \frac{41}{5}e^{5} - \frac{988}{105}e^{4} - \frac{485}{21}e^{3} + \frac{992}{35}e^{2} + \frac{169}{21}e - \frac{205}{21}$ |
| 97 | $[97, 97, -w^{3} + 3w^{2} + 3w - 1]$ | $\phantom{-}\frac{1}{15}e^{8} - \frac{1}{5}e^{7} - \frac{1}{5}e^{6} + \frac{26}{15}e^{5} - \frac{31}{5}e^{4} - \frac{2}{3}e^{3} + \frac{147}{5}e^{2} - \frac{13}{3}e - \frac{46}{3}$ |
Atkin-Lehner eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| $13$ | $[13, 13, w^{3} - 2w^{2} - 4w + 2]$ | $1$ |