Base field \(\Q(\sqrt{197}) \)
Generator \(w\), with minimal polynomial \(x^{2} - x - 49\); narrow class number \(1\) and class number \(1\).
Form
| Weight: | $[2, 2]$ |
| Level: | $[16, 4, 4]$ |
| Dimension: | $8$ |
| CM: | no |
| Base change: | yes |
| Newspace dimension: | $40$ |
Hecke eigenvalues ($q$-expansion)
The Hecke eigenvalue field is $\Q(e)$ where $e$ is a root of the defining polynomial:
| \(x^{8} + x^{7} - 31x^{6} - 20x^{5} + 282x^{4} + 148x^{3} - 733x^{2} - 468x - 36\) |
Show full eigenvalues Hide large eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| 4 | $[4, 2, 2]$ | $\phantom{-}0$ |
| 7 | $[7, 7, w - 7]$ | $\phantom{-}e$ |
| 7 | $[7, 7, w + 6]$ | $\phantom{-}e$ |
| 9 | $[9, 3, 3]$ | $\phantom{-}\frac{843}{153928}e^{7} + \frac{3345}{153928}e^{6} - \frac{21683}{153928}e^{5} - \frac{31021}{76964}e^{4} + \frac{89789}{76964}e^{3} + \frac{21414}{19241}e^{2} - \frac{363095}{153928}e + \frac{308539}{76964}$ |
| 19 | $[19, 19, w + 5]$ | $-\frac{1129}{115446}e^{7} - \frac{4069}{115446}e^{6} + \frac{34015}{115446}e^{5} + \frac{49648}{57723}e^{4} - \frac{51534}{19241}e^{3} - \frac{295691}{57723}e^{2} + \frac{747757}{115446}e + \frac{95999}{19241}$ |
| 19 | $[19, 19, w - 6]$ | $-\frac{1129}{115446}e^{7} - \frac{4069}{115446}e^{6} + \frac{34015}{115446}e^{5} + \frac{49648}{57723}e^{4} - \frac{51534}{19241}e^{3} - \frac{295691}{57723}e^{2} + \frac{747757}{115446}e + \frac{95999}{19241}$ |
| 23 | $[23, 23, w + 8]$ | $-\frac{1199}{57723}e^{7} + \frac{1405}{57723}e^{6} + \frac{41339}{57723}e^{5} - \frac{40670}{57723}e^{4} - \frac{135755}{19241}e^{3} + \frac{214840}{57723}e^{2} + \frac{1180439}{57723}e + \frac{70732}{19241}$ |
| 23 | $[23, 23, -w + 9]$ | $-\frac{1199}{57723}e^{7} + \frac{1405}{57723}e^{6} + \frac{41339}{57723}e^{5} - \frac{40670}{57723}e^{4} - \frac{135755}{19241}e^{3} + \frac{214840}{57723}e^{2} + \frac{1180439}{57723}e + \frac{70732}{19241}$ |
| 25 | $[25, 5, 5]$ | $-\frac{2333}{115446}e^{7} - \frac{2273}{115446}e^{6} + \frac{67631}{115446}e^{5} + \frac{11945}{57723}e^{4} - \frac{97902}{19241}e^{3} - \frac{23416}{57723}e^{2} + \frac{1563287}{115446}e + \frac{161393}{19241}$ |
| 29 | $[29, 29, -w - 4]$ | $-\frac{20129}{461784}e^{7} - \frac{19067}{461784}e^{6} + \frac{587129}{461784}e^{5} + \frac{144083}{230892}e^{4} - \frac{801601}{76964}e^{3} - \frac{165187}{57723}e^{2} + \frac{10066589}{461784}e + \frac{565053}{76964}$ |
| 29 | $[29, 29, w - 5]$ | $-\frac{20129}{461784}e^{7} - \frac{19067}{461784}e^{6} + \frac{587129}{461784}e^{5} + \frac{144083}{230892}e^{4} - \frac{801601}{76964}e^{3} - \frac{165187}{57723}e^{2} + \frac{10066589}{461784}e + \frac{565053}{76964}$ |
| 37 | $[37, 37, -w - 3]$ | $\phantom{-}\frac{2509}{461784}e^{7} + \frac{11599}{461784}e^{6} - \frac{95941}{461784}e^{5} - \frac{162535}{230892}e^{4} + \frac{214309}{76964}e^{3} + \frac{286820}{57723}e^{2} - \frac{5319625}{461784}e - \frac{494917}{76964}$ |
| 37 | $[37, 37, w - 4]$ | $\phantom{-}\frac{2509}{461784}e^{7} + \frac{11599}{461784}e^{6} - \frac{95941}{461784}e^{5} - \frac{162535}{230892}e^{4} + \frac{214309}{76964}e^{3} + \frac{286820}{57723}e^{2} - \frac{5319625}{461784}e - \frac{494917}{76964}$ |
| 41 | $[41, 41, -w - 9]$ | $-\frac{15593}{461784}e^{7} - \frac{4355}{461784}e^{6} + \frac{481961}{461784}e^{5} + \frac{14963}{230892}e^{4} - \frac{719985}{76964}e^{3} - \frac{92074}{57723}e^{2} + \frac{11305901}{461784}e + \frac{830585}{76964}$ |
| 41 | $[41, 41, w - 10]$ | $-\frac{15593}{461784}e^{7} - \frac{4355}{461784}e^{6} + \frac{481961}{461784}e^{5} + \frac{14963}{230892}e^{4} - \frac{719985}{76964}e^{3} - \frac{92074}{57723}e^{2} + \frac{11305901}{461784}e + \frac{830585}{76964}$ |
| 43 | $[43, 43, -w - 2]$ | $\phantom{-}\frac{6733}{115446}e^{7} + \frac{4531}{115446}e^{6} - \frac{198151}{115446}e^{5} - \frac{24700}{57723}e^{4} + \frac{275855}{19241}e^{3} + \frac{66638}{57723}e^{2} - \frac{3923059}{115446}e - \frac{110417}{19241}$ |
| 43 | $[43, 43, w - 3]$ | $\phantom{-}\frac{6733}{115446}e^{7} + \frac{4531}{115446}e^{6} - \frac{198151}{115446}e^{5} - \frac{24700}{57723}e^{4} + \frac{275855}{19241}e^{3} + \frac{66638}{57723}e^{2} - \frac{3923059}{115446}e - \frac{110417}{19241}$ |
| 47 | $[47, 47, -w - 1]$ | $-\frac{1199}{115446}e^{7} + \frac{1405}{115446}e^{6} + \frac{41339}{115446}e^{5} - \frac{20335}{57723}e^{4} - \frac{77498}{19241}e^{3} + \frac{49697}{57723}e^{2} + \frac{1873115}{115446}e + \frac{131571}{19241}$ |
| 47 | $[47, 47, w - 2]$ | $-\frac{1199}{115446}e^{7} + \frac{1405}{115446}e^{6} + \frac{41339}{115446}e^{5} - \frac{20335}{57723}e^{4} - \frac{77498}{19241}e^{3} + \frac{49697}{57723}e^{2} + \frac{1873115}{115446}e + \frac{131571}{19241}$ |
| 53 | $[53, 53, 2w - 13]$ | $-\frac{16115}{461784}e^{7} - \frac{9713}{461784}e^{6} + \frac{506891}{461784}e^{5} + \frac{71969}{230892}e^{4} - \frac{740983}{76964}e^{3} - \frac{89722}{57723}e^{2} + \frac{10105943}{461784}e + \frac{402075}{76964}$ |
Atkin-Lehner eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| $4$ | $[4, 2, 2]$ | $1$ |