Base field 4.4.16317.1
Generator \(w\), with minimal polynomial \(x^{4} - 2x^{3} - 4x^{2} + 5x + 1\); narrow class number \(2\) and class number \(1\).
Form
| Weight: | $[2, 2, 2, 2]$ |
| Level: | $[25, 5, w^{2} - w - 3]$ |
| Dimension: | $12$ |
| CM: | no |
| Base change: | yes |
| Newspace dimension: | $50$ |
Hecke eigenvalues ($q$-expansion)
The Hecke eigenvalue field is $\Q(e)$ where $e$ is a root of the defining polynomial:
| \(x^{12} - 39x^{10} + 540x^{8} - 3217x^{6} + 8303x^{4} - 7948x^{2} + 2476\) |
Show full eigenvalues Hide large eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| 5 | $[5, 5, w + 1]$ | $\phantom{-}e$ |
| 5 | $[5, 5, -w + 2]$ | $\phantom{-}e$ |
| 7 | $[7, 7, -w^{2} + 2w + 1]$ | $-\frac{4679}{189463}e^{10} + \frac{180218}{189463}e^{8} - \frac{2436906}{189463}e^{6} + \frac{13741647}{189463}e^{4} - \frac{30428771}{189463}e^{2} + \frac{16152958}{189463}$ |
| 7 | $[7, 7, -w^{2} + 2]$ | $-\frac{4679}{189463}e^{10} + \frac{180218}{189463}e^{8} - \frac{2436906}{189463}e^{6} + \frac{13741647}{189463}e^{4} - \frac{30428771}{189463}e^{2} + \frac{16152958}{189463}$ |
| 9 | $[9, 3, w^{3} - w^{2} - 4w]$ | $\phantom{-}\frac{1585}{189463}e^{10} - \frac{60603}{189463}e^{8} + \frac{820556}{189463}e^{6} - \frac{4714028}{189463}e^{4} + \frac{10852779}{189463}e^{2} - \frac{5480522}{189463}$ |
| 16 | $[16, 2, 2]$ | $-\frac{1509}{378926}e^{10} + \frac{29506}{189463}e^{8} - \frac{397897}{189463}e^{6} + \frac{4313591}{378926}e^{4} - \frac{4266875}{189463}e^{2} + \frac{2217031}{189463}$ |
| 17 | $[17, 17, -w^{3} + 2w^{2} + 3w - 2]$ | $\phantom{-}\frac{301}{189463}e^{11} - \frac{13780}{189463}e^{9} + \frac{232689}{189463}e^{7} - \frac{1747982}{189463}e^{5} + \frac{5447667}{189463}e^{3} - \frac{4802667}{189463}e$ |
| 17 | $[17, 17, -w^{3} + w^{2} + 4w - 2]$ | $\phantom{-}\frac{301}{189463}e^{11} - \frac{13780}{189463}e^{9} + \frac{232689}{189463}e^{7} - \frac{1747982}{189463}e^{5} + \frac{5447667}{189463}e^{3} - \frac{4802667}{189463}e$ |
| 25 | $[25, 5, w^{2} - w - 3]$ | $-1$ |
| 37 | $[37, 37, 2w - 1]$ | $\phantom{-}\frac{4077}{189463}e^{10} - \frac{152658}{189463}e^{8} + \frac{1971528}{189463}e^{6} - \frac{10245683}{189463}e^{4} + \frac{19722900}{189463}e^{2} - \frac{8442254}{189463}$ |
| 43 | $[43, 43, -w^{3} + 2w^{2} + 2w - 2]$ | $-\frac{3175}{378926}e^{10} + \frac{54423}{189463}e^{8} - \frac{614457}{189463}e^{6} + \frac{5079301}{378926}e^{4} - \frac{3401648}{189463}e^{2} + \frac{1461434}{189463}$ |
| 43 | $[43, 43, w^{3} - w^{2} - 3w + 1]$ | $-\frac{3175}{378926}e^{10} + \frac{54423}{189463}e^{8} - \frac{614457}{189463}e^{6} + \frac{5079301}{378926}e^{4} - \frac{3401648}{189463}e^{2} + \frac{1461434}{189463}$ |
| 59 | $[59, 59, 2w - 5]$ | $-\frac{7397}{378926}e^{11} + \frac{140995}{189463}e^{9} - \frac{1875629}{189463}e^{7} + \frac{20698411}{378926}e^{5} - \frac{22830732}{189463}e^{3} + \frac{13606200}{189463}e$ |
| 59 | $[59, 59, -2w - 3]$ | $-\frac{7397}{378926}e^{11} + \frac{140995}{189463}e^{9} - \frac{1875629}{189463}e^{7} + \frac{20698411}{378926}e^{5} - \frac{22830732}{189463}e^{3} + \frac{13606200}{189463}e$ |
| 79 | $[79, 79, -w^{3} + 2w^{2} + 5w - 4]$ | $-\frac{11921}{378926}e^{10} + \frac{233221}{189463}e^{8} - \frac{3230872}{189463}e^{6} + \frac{37764141}{378926}e^{4} - \frac{43474477}{189463}e^{2} + \frac{23312920}{189463}$ |
| 79 | $[79, 79, -w^{3} + w^{2} + 6w - 2]$ | $-\frac{11921}{378926}e^{10} + \frac{233221}{189463}e^{8} - \frac{3230872}{189463}e^{6} + \frac{37764141}{378926}e^{4} - \frac{43474477}{189463}e^{2} + \frac{23312920}{189463}$ |
| 83 | $[83, 83, -w^{3} + 7w]$ | $-\frac{32301}{378926}e^{11} + \frac{621046}{189463}e^{9} - \frac{8365791}{189463}e^{7} + \frac{93720237}{378926}e^{5} - \frac{103074803}{189463}e^{3} + \frac{53932070}{189463}e$ |
| 83 | $[83, 83, -w^{3} + 2w^{2} + 4w - 2]$ | $\phantom{-}\frac{6038}{189463}e^{11} - \frac{231104}{189463}e^{9} + \frac{3094082}{189463}e^{7} - \frac{17220029}{189463}e^{5} + \frac{38139849}{189463}e^{3} - \frac{22251068}{189463}e$ |
| 83 | $[83, 83, -w^{3} + w^{2} + 5w - 3]$ | $\phantom{-}\frac{6038}{189463}e^{11} - \frac{231104}{189463}e^{9} + \frac{3094082}{189463}e^{7} - \frac{17220029}{189463}e^{5} + \frac{38139849}{189463}e^{3} - \frac{22251068}{189463}e$ |
| 83 | $[83, 83, w^{2} - 2w - 6]$ | $-\frac{32301}{378926}e^{11} + \frac{621046}{189463}e^{9} - \frac{8365791}{189463}e^{7} + \frac{93720237}{378926}e^{5} - \frac{103074803}{189463}e^{3} + \frac{53932070}{189463}e$ |
Atkin-Lehner eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| $25$ | $[25, 5, w^{2} - w - 3]$ | $1$ |