Base field \(\Q(\sqrt{229}) \)
Generator \(w\), with minimal polynomial \(x^{2} - x - 57\); narrow class number \(3\) and class number \(3\).
Form
| Weight: | $[2, 2]$ |
| Level: | $[1, 1, 1]$ |
| Dimension: | $8$ |
| CM: | no |
| Base change: | yes |
| Newspace dimension: | $27$ |
Hecke eigenvalues ($q$-expansion)
The Hecke eigenvalue field is $\Q(e)$ where $e$ is a root of the defining polynomial:
| \(x^{8} - 17x^{6} - 5x^{5} + 84x^{4} + 44x^{3} - 99x^{2} - 60x + 4\) |
Show full eigenvalues Hide large eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| 3 | $[3, 3, w]$ | $\phantom{-}e$ |
| 3 | $[3, 3, w + 2]$ | $\phantom{-}e$ |
| 4 | $[4, 2, 2]$ | $-\frac{45}{284}e^{7} + \frac{17}{142}e^{6} + \frac{733}{284}e^{5} - \frac{373}{284}e^{4} - \frac{1727}{142}e^{3} + \frac{230}{71}e^{2} + \frac{3823}{284}e + \frac{215}{142}$ |
| 5 | $[5, 5, w + 1]$ | $\phantom{-}\frac{11}{568}e^{7} - \frac{1}{284}e^{6} - \frac{135}{568}e^{5} - \frac{237}{568}e^{4} + \frac{277}{284}e^{3} + \frac{248}{71}e^{2} - \frac{985}{568}e - \frac{1015}{284}$ |
| 5 | $[5, 5, w + 3]$ | $\phantom{-}\frac{11}{568}e^{7} - \frac{1}{284}e^{6} - \frac{135}{568}e^{5} - \frac{237}{568}e^{4} + \frac{277}{284}e^{3} + \frac{248}{71}e^{2} - \frac{985}{568}e - \frac{1015}{284}$ |
| 11 | $[11, 11, w + 1]$ | $-\frac{20}{71}e^{7} + \frac{23}{71}e^{6} + \frac{310}{71}e^{5} - \frac{221}{71}e^{4} - \frac{1401}{71}e^{3} + \frac{401}{71}e^{2} + \frac{1423}{71}e - \frac{14}{71}$ |
| 11 | $[11, 11, w + 9]$ | $-\frac{20}{71}e^{7} + \frac{23}{71}e^{6} + \frac{310}{71}e^{5} - \frac{221}{71}e^{4} - \frac{1401}{71}e^{3} + \frac{401}{71}e^{2} + \frac{1423}{71}e - \frac{14}{71}$ |
| 17 | $[17, 17, w + 2]$ | $\phantom{-}\frac{123}{568}e^{7} - \frac{37}{284}e^{6} - \frac{1871}{568}e^{5} + \frac{603}{568}e^{4} + \frac{4001}{284}e^{3} - \frac{125}{71}e^{2} - \frac{7193}{568}e + \frac{501}{284}$ |
| 17 | $[17, 17, w + 14]$ | $\phantom{-}\frac{123}{568}e^{7} - \frac{37}{284}e^{6} - \frac{1871}{568}e^{5} + \frac{603}{568}e^{4} + \frac{4001}{284}e^{3} - \frac{125}{71}e^{2} - \frac{7193}{568}e + \frac{501}{284}$ |
| 19 | $[19, 19, w]$ | $-\frac{33}{142}e^{7} + \frac{3}{71}e^{6} + \frac{547}{142}e^{5} + \frac{1}{142}e^{4} - \frac{1257}{71}e^{3} - \frac{207}{71}e^{2} + \frac{2387}{142}e + \frac{347}{71}$ |
| 19 | $[19, 19, w + 18]$ | $-\frac{33}{142}e^{7} + \frac{3}{71}e^{6} + \frac{547}{142}e^{5} + \frac{1}{142}e^{4} - \frac{1257}{71}e^{3} - \frac{207}{71}e^{2} + \frac{2387}{142}e + \frac{347}{71}$ |
| 37 | $[37, 37, -w - 4]$ | $\phantom{-}\frac{13}{568}e^{7} - \frac{27}{284}e^{6} + \frac{47}{568}e^{5} + \frac{133}{568}e^{4} - \frac{757}{284}e^{3} + \frac{306}{71}e^{2} + \frac{4361}{568}e - \frac{2413}{284}$ |
| 37 | $[37, 37, w - 5]$ | $\phantom{-}\frac{13}{568}e^{7} - \frac{27}{284}e^{6} + \frac{47}{568}e^{5} + \frac{133}{568}e^{4} - \frac{757}{284}e^{3} + \frac{306}{71}e^{2} + \frac{4361}{568}e - \frac{2413}{284}$ |
| 43 | $[43, 43, w + 16]$ | $\phantom{-}\frac{15}{284}e^{7} - \frac{53}{142}e^{6} - \frac{55}{284}e^{5} + \frac{1355}{284}e^{4} - \frac{371}{142}e^{3} - \frac{1118}{71}e^{2} + \frac{2607}{284}e + \frac{1301}{142}$ |
| 43 | $[43, 43, w + 26]$ | $\phantom{-}\frac{15}{284}e^{7} - \frac{53}{142}e^{6} - \frac{55}{284}e^{5} + \frac{1355}{284}e^{4} - \frac{371}{142}e^{3} - \frac{1118}{71}e^{2} + \frac{2607}{284}e + \frac{1301}{142}$ |
| 49 | $[49, 7, -7]$ | $\phantom{-}\frac{171}{568}e^{7} - \frac{93}{284}e^{6} - \frac{2615}{568}e^{5} + \frac{2099}{568}e^{4} + \frac{5313}{284}e^{3} - \frac{721}{71}e^{2} - \frac{6121}{568}e + \frac{2449}{284}$ |
| 53 | $[53, 53, -w - 10]$ | $\phantom{-}\frac{85}{568}e^{7} - \frac{111}{284}e^{6} - \frac{1353}{568}e^{5} + \frac{2661}{568}e^{4} + \frac{3483}{284}e^{3} - \frac{1014}{71}e^{2} - \frac{9367}{568}e + \frac{1503}{284}$ |
| 53 | $[53, 53, w - 11]$ | $\phantom{-}\frac{85}{568}e^{7} - \frac{111}{284}e^{6} - \frac{1353}{568}e^{5} + \frac{2661}{568}e^{4} + \frac{3483}{284}e^{3} - \frac{1014}{71}e^{2} - \frac{9367}{568}e + \frac{1503}{284}$ |
| 61 | $[61, 61, w + 15]$ | $-\frac{1}{8}e^{7} - \frac{1}{4}e^{6} + \frac{21}{8}e^{5} + \frac{23}{8}e^{4} - \frac{55}{4}e^{3} - 8e^{2} + \frac{91}{8}e + \frac{17}{4}$ |
| 61 | $[61, 61, w + 45]$ | $-\frac{1}{8}e^{7} - \frac{1}{4}e^{6} + \frac{21}{8}e^{5} + \frac{23}{8}e^{4} - \frac{55}{4}e^{3} - 8e^{2} + \frac{91}{8}e + \frac{17}{4}$ |
Atkin-Lehner eigenvalues
This form has no Atkin-Lehner eigenvalues since the level is \((1)\).