Base field 4.4.16317.1
Generator \(w\), with minimal polynomial \(x^4 - 2 x^3 - 4 x^2 + 5 x + 1\); 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:
| \(x^{12} - 39 x^{10} + 540 x^8 - 3217 x^6 + 8303 x^4 - 7948 x^2 + 2476\) |
Show full eigenvalues Hide large eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| 5 | $[5, 5, w + 1]$ | $\phantom{-}e$ |
| 5 | $[5, 5, -w + 2]$ | $\phantom{-}e$ |
| 7 | $[7, 7, -w^2 + 2 w + 1]$ | $-\frac{4679}{189463} e^{10} + \frac{180218}{189463} e^8 - \frac{2436906}{189463} e^6 + \frac{13741647}{189463} e^4 - \frac{30428771}{189463} e^2 + \frac{16152958}{189463}$ |
| 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 | $[9, 3, w^3 - w^2 - 4 w]$ | $\phantom{-}\frac{1585}{189463} e^{10} - \frac{60603}{189463} e^8 + \frac{820556}{189463} e^6 - \frac{4714028}{189463} e^4 + \frac{10852779}{189463} e^2 - \frac{5480522}{189463}$ |
| 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 | $[17, 17, -w^3 + 2 w^2 + 3 w - 2]$ | $\phantom{-}\frac{301}{189463} e^{11} - \frac{13780}{189463} e^9 + \frac{232689}{189463} e^7 - \frac{1747982}{189463} e^5 + \frac{5447667}{189463} e^3 - \frac{4802667}{189463} e$ |
| 17 | $[17, 17, -w^3 + w^2 + 4 w - 2]$ | $\phantom{-}\frac{301}{189463} e^{11} - \frac{13780}{189463} e^9 + \frac{232689}{189463} e^7 - \frac{1747982}{189463} e^5 + \frac{5447667}{189463} e^3 - \frac{4802667}{189463} e$ |
| 25 | $[25, 5, w^2 - w - 3]$ | $-1$ |
| 37 | $[37, 37, 2 w - 1]$ | $\phantom{-}\frac{4077}{189463} e^{10} - \frac{152658}{189463} e^8 + \frac{1971528}{189463} e^6 - \frac{10245683}{189463} e^4 + \frac{19722900}{189463} e^2 - \frac{8442254}{189463}$ |
| 43 | $[43, 43, -w^3 + 2 w^2 + 2 w - 2]$ | $-\frac{3175}{378926} e^{10} + \frac{54423}{189463} e^8 - \frac{614457}{189463} e^6 + \frac{5079301}{378926} e^4 - \frac{3401648}{189463} e^2 + \frac{1461434}{189463}$ |
| 43 | $[43, 43, w^3 - w^2 - 3 w + 1]$ | $-\frac{3175}{378926} e^{10} + \frac{54423}{189463} e^8 - \frac{614457}{189463} e^6 + \frac{5079301}{378926} e^4 - \frac{3401648}{189463} e^2 + \frac{1461434}{189463}$ |
| 59 | $[59, 59, 2 w - 5]$ | $-\frac{7397}{378926} e^{11} + \frac{140995}{189463} e^9 - \frac{1875629}{189463} e^7 + \frac{20698411}{378926} e^5 - \frac{22830732}{189463} e^3 + \frac{13606200}{189463} e$ |
| 59 | $[59, 59, -2 w - 3]$ | $-\frac{7397}{378926} e^{11} + \frac{140995}{189463} e^9 - \frac{1875629}{189463} e^7 + \frac{20698411}{378926} e^5 - \frac{22830732}{189463} e^3 + \frac{13606200}{189463} e$ |
| 79 | $[79, 79, -w^3 + 2 w^2 + 5 w - 4]$ | $-\frac{11921}{378926} e^{10} + \frac{233221}{189463} e^8 - \frac{3230872}{189463} e^6 + \frac{37764141}{378926} e^4 - \frac{43474477}{189463} e^2 + \frac{23312920}{189463}$ |
| 79 | $[79, 79, -w^3 + w^2 + 6 w - 2]$ | $-\frac{11921}{378926} e^{10} + \frac{233221}{189463} e^8 - \frac{3230872}{189463} e^6 + \frac{37764141}{378926} e^4 - \frac{43474477}{189463} e^2 + \frac{23312920}{189463}$ |
| 83 | $[83, 83, -w^3 + 7 w]$ | $-\frac{32301}{378926} e^{11} + \frac{621046}{189463} e^9 - \frac{8365791}{189463} e^7 + \frac{93720237}{378926} e^5 - \frac{103074803}{189463} e^3 + \frac{53932070}{189463} e$ |
| 83 | $[83, 83, -w^3 + 2 w^2 + 4 w - 2]$ | $\phantom{-}\frac{6038}{189463} e^{11} - \frac{231104}{189463} e^9 + \frac{3094082}{189463} e^7 - \frac{17220029}{189463} e^5 + \frac{38139849}{189463} e^3 - \frac{22251068}{189463} e$ |
| 83 | $[83, 83, -w^3 + w^2 + 5 w - 3]$ | $\phantom{-}\frac{6038}{189463} e^{11} - \frac{231104}{189463} e^9 + \frac{3094082}{189463} e^7 - \frac{17220029}{189463} e^5 + \frac{38139849}{189463} e^3 - \frac{22251068}{189463} e$ |
| 83 | $[83, 83, w^2 - 2 w - 6]$ | $-\frac{32301}{378926} e^{11} + \frac{621046}{189463} e^9 - \frac{8365791}{189463} e^7 + \frac{93720237}{378926} e^5 - \frac{103074803}{189463} e^3 + \frac{53932070}{189463} e$ |
Atkin-Lehner eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| $25$ | $[25, 5, w^2 - w - 3]$ | $1$ |