Base field 4.4.17725.1
Generator \(w\), with minimal polynomial \(x^{4} - 2x^{3} - 12x^{2} + 13x + 41\); narrow class number \(1\) and class number \(1\).
Form
| Weight: | $[2, 2, 2, 2]$ |
| Level: | $[19, 19, w + 1]$ |
| Dimension: | $7$ |
| CM: | no |
| Base change: | no |
| Newspace dimension: | $32$ |
Hecke eigenvalues ($q$-expansion)
The Hecke eigenvalue field is $\Q(e)$ where $e$ is a root of the defining polynomial:
| \(x^{7} + 7x^{6} - 4x^{5} - 97x^{4} - 72x^{3} + 332x^{2} + 336x - 64\) |
Show full eigenvalues Hide large eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| 9 | $[9, 3, -w^{3} + 3w^{2} + 5w - 15]$ | $-\frac{37}{1616}e^{6} - \frac{265}{1616}e^{5} - \frac{13}{808}e^{4} + \frac{3017}{1616}e^{3} + \frac{1795}{808}e^{2} - \frac{1157}{202}e - \frac{374}{101}$ |
| 9 | $[9, 3, -w^{3} + 8w + 8]$ | $\phantom{-}e$ |
| 16 | $[16, 2, 2]$ | $\phantom{-}\frac{29}{808}e^{6} + \frac{235}{808}e^{5} + \frac{1}{202}e^{4} - \frac{2725}{808}e^{3} - \frac{205}{101}e^{2} + \frac{1669}{202}e + \frac{185}{101}$ |
| 19 | $[19, 19, w + 1]$ | $-1$ |
| 19 | $[19, 19, -w^{2} + 6]$ | $\phantom{-}\frac{115}{3232}e^{6} + \frac{507}{3232}e^{5} - \frac{915}{1616}e^{4} - \frac{6975}{3232}e^{3} + \frac{3773}{1616}e^{2} + \frac{2799}{404}e + \frac{543}{101}$ |
| 19 | $[19, 19, -w^{2} + 2w + 5]$ | $-\frac{11}{404}e^{6} - \frac{133}{808}e^{5} + \frac{293}{808}e^{4} + \frac{529}{202}e^{3} - \frac{849}{808}e^{2} - \frac{3497}{404}e - \frac{412}{101}$ |
| 19 | $[19, 19, -w + 2]$ | $-\frac{1}{16}e^{6} - \frac{5}{16}e^{5} + \frac{7}{8}e^{4} + \frac{69}{16}e^{3} - \frac{33}{8}e^{2} - \frac{25}{2}e + 6$ |
| 25 | $[25, 5, 2w^{2} - 2w - 13]$ | $\phantom{-}\frac{29}{1616}e^{6} + \frac{33}{1616}e^{5} - \frac{503}{808}e^{4} - \frac{1513}{1616}e^{3} + \frac{2917}{808}e^{2} + \frac{493}{101}e + \frac{244}{101}$ |
| 29 | $[29, 29, -w^{2} + 9]$ | $-\frac{47}{808}e^{6} - \frac{63}{202}e^{5} + \frac{401}{808}e^{4} + \frac{2877}{808}e^{3} + \frac{119}{808}e^{2} - \frac{2655}{404}e - \frac{366}{101}$ |
| 29 | $[29, 29, -w^{2} + 2w + 6]$ | $\phantom{-}\frac{47}{1616}e^{6} + \frac{353}{1616}e^{5} - \frac{75}{404}e^{4} - \frac{4695}{1616}e^{3} + \frac{62}{101}e^{2} + \frac{3903}{404}e - \frac{120}{101}$ |
| 29 | $[29, 29, w^{2} - 7]$ | $\phantom{-}\frac{77}{3232}e^{6} + \frac{617}{3232}e^{5} - \frac{235}{1616}e^{4} - \frac{10537}{3232}e^{3} - \frac{2327}{1616}e^{2} + \frac{1311}{101}e + \frac{714}{101}$ |
| 29 | $[29, 29, -w^{2} + 2w + 8]$ | $-\frac{29}{404}e^{6} - \frac{67}{202}e^{5} + \frac{501}{404}e^{4} + \frac{2119}{404}e^{3} - \frac{2501}{404}e^{2} - \frac{3641}{202}e + \frac{438}{101}$ |
| 31 | $[31, 31, -2w^{2} + w + 12]$ | $-\frac{63}{1616}e^{6} - \frac{413}{1616}e^{5} + \frac{16}{101}e^{4} + \frac{5279}{1616}e^{3} + \frac{1131}{404}e^{2} - \frac{3981}{404}e - \frac{847}{101}$ |
| 31 | $[31, 31, 2w^{2} - 3w - 11]$ | $\phantom{-}\frac{69}{808}e^{6} + \frac{385}{808}e^{5} - \frac{347}{404}e^{4} - \frac{4993}{808}e^{3} + \frac{769}{404}e^{2} + \frac{1740}{101}e + \frac{172}{101}$ |
| 41 | $[41, 41, -w]$ | $\phantom{-}\frac{57}{3232}e^{6} + \frac{441}{3232}e^{5} + \frac{91}{1616}e^{4} - \frac{3949}{3232}e^{3} - \frac{3677}{1616}e^{2} + \frac{19}{404}e + \frac{1107}{101}$ |
| 41 | $[41, 41, -w + 1]$ | $-\frac{151}{1616}e^{6} - \frac{743}{1616}e^{5} + \frac{815}{808}e^{4} + \frac{8491}{1616}e^{3} - \frac{2365}{808}e^{2} - \frac{1254}{101}e - \frac{156}{101}$ |
| 49 | $[49, 7, w^{3} + 2w^{2} - 10w - 20]$ | $-\frac{309}{3232}e^{6} - \frac{1285}{3232}e^{5} + \frac{2441}{1616}e^{4} + \frac{16985}{3232}e^{3} - \frac{13535}{1616}e^{2} - \frac{6567}{404}e + \frac{1037}{101}$ |
| 49 | $[49, 7, w^{3} - 5w^{2} - 3w + 27]$ | $\phantom{-}\frac{297}{3232}e^{6} + \frac{1341}{3232}e^{5} - \frac{2811}{1616}e^{4} - \frac{22405}{3232}e^{3} + \frac{15553}{1616}e^{2} + \frac{5049}{202}e - \frac{781}{101}$ |
| 61 | $[61, 61, 2w^{2} - 3w - 14]$ | $-\frac{37}{1616}e^{6} - \frac{63}{1616}e^{5} + \frac{123}{202}e^{4} + \frac{1805}{1616}e^{3} - \frac{971}{404}e^{2} - \frac{2213}{404}e - \frac{71}{101}$ |
| 61 | $[61, 61, 2w^{2} - w - 15]$ | $-\frac{113}{3232}e^{6} - \frac{853}{3232}e^{5} + \frac{539}{1616}e^{4} + \frac{14477}{3232}e^{3} - \frac{305}{1616}e^{2} - \frac{4143}{202}e - \frac{653}{101}$ |
Atkin-Lehner eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| $19$ | $[19, 19, w + 1]$ | $1$ |