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: | $[36, 6, 6]$ |
| Dimension: | $9$ |
| CM: | no |
| Base change: | yes |
| Newspace dimension: | $97$ |
Hecke eigenvalues ($q$-expansion)
The Hecke eigenvalue field is $\Q(e)$ where $e$ is a root of the defining polynomial:
| \(x^{9} - 3x^{8} - 33x^{7} + 68x^{6} + 370x^{5} - 266x^{4} - 1572x^{3} - 988x^{2} + 324x + 288\) |
Show full eigenvalues Hide large eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| 4 | $[4, 2, 2]$ | $-1$ |
| 7 | $[7, 7, w - 7]$ | $\phantom{-}e$ |
| 7 | $[7, 7, w + 6]$ | $\phantom{-}e$ |
| 9 | $[9, 3, 3]$ | $-1$ |
| 19 | $[19, 19, w + 5]$ | $\phantom{-}\frac{18821}{721068}e^{8} - \frac{75901}{721068}e^{7} - \frac{527909}{721068}e^{6} + \frac{307043}{120178}e^{5} + \frac{2278403}{360534}e^{4} - \frac{2624893}{180267}e^{3} - \frac{3680530}{180267}e^{2} + \frac{1086764}{180267}e + \frac{247856}{60089}$ |
| 19 | $[19, 19, w - 6]$ | $\phantom{-}\frac{18821}{721068}e^{8} - \frac{75901}{721068}e^{7} - \frac{527909}{721068}e^{6} + \frac{307043}{120178}e^{5} + \frac{2278403}{360534}e^{4} - \frac{2624893}{180267}e^{3} - \frac{3680530}{180267}e^{2} + \frac{1086764}{180267}e + \frac{247856}{60089}$ |
| 23 | $[23, 23, w + 8]$ | $-\frac{29243}{360534}e^{8} + \frac{108391}{360534}e^{7} + \frac{447553}{180267}e^{6} - \frac{872269}{120178}e^{5} - \frac{9152329}{360534}e^{4} + \frac{7025948}{180267}e^{3} + \frac{18480923}{180267}e^{2} + \frac{2224103}{180267}e - \frac{1757864}{60089}$ |
| 23 | $[23, 23, -w + 9]$ | $-\frac{29243}{360534}e^{8} + \frac{108391}{360534}e^{7} + \frac{447553}{180267}e^{6} - \frac{872269}{120178}e^{5} - \frac{9152329}{360534}e^{4} + \frac{7025948}{180267}e^{3} + \frac{18480923}{180267}e^{2} + \frac{2224103}{180267}e - \frac{1757864}{60089}$ |
| 25 | $[25, 5, 5]$ | $-\frac{12865}{721068}e^{8} + \frac{15721}{240356}e^{7} + \frac{135725}{240356}e^{6} - \frac{288881}{180267}e^{5} - \frac{2173727}{360534}e^{4} + \frac{1589714}{180267}e^{3} + \frac{1466130}{60089}e^{2} + \frac{639859}{180267}e + \frac{29054}{60089}$ |
| 29 | $[29, 29, -w - 4]$ | $\phantom{-}\frac{15205}{360534}e^{8} - \frac{10166}{60089}e^{7} - \frac{144111}{120178}e^{6} + \frac{743173}{180267}e^{5} + \frac{3877039}{360534}e^{4} - \frac{4256635}{180267}e^{3} - \frac{2249603}{60089}e^{2} + \frac{1516954}{180267}e + \frac{615492}{60089}$ |
| 29 | $[29, 29, w - 5]$ | $\phantom{-}\frac{15205}{360534}e^{8} - \frac{10166}{60089}e^{7} - \frac{144111}{120178}e^{6} + \frac{743173}{180267}e^{5} + \frac{3877039}{360534}e^{4} - \frac{4256635}{180267}e^{3} - \frac{2249603}{60089}e^{2} + \frac{1516954}{180267}e + \frac{615492}{60089}$ |
| 37 | $[37, 37, -w - 3]$ | $-\frac{27901}{180267}e^{8} + \frac{69437}{120178}e^{7} + \frac{279898}{60089}e^{6} - \frac{2515826}{180267}e^{5} - \frac{16687187}{360534}e^{4} + \frac{13781144}{180267}e^{3} + \frac{10983771}{60089}e^{2} + \frac{574447}{180267}e - \frac{3754858}{60089}$ |
| 37 | $[37, 37, w - 4]$ | $-\frac{27901}{180267}e^{8} + \frac{69437}{120178}e^{7} + \frac{279898}{60089}e^{6} - \frac{2515826}{180267}e^{5} - \frac{16687187}{360534}e^{4} + \frac{13781144}{180267}e^{3} + \frac{10983771}{60089}e^{2} + \frac{574447}{180267}e - \frac{3754858}{60089}$ |
| 41 | $[41, 41, -w - 9]$ | $\phantom{-}\frac{13509}{120178}e^{8} - \frac{163079}{360534}e^{7} - \frac{1164643}{360534}e^{6} + \frac{3923677}{360534}e^{5} + \frac{10869787}{360534}e^{4} - \frac{7106779}{120178}e^{3} - \frac{20727106}{180267}e^{2} - \frac{703652}{180267}e + \frac{1894838}{60089}$ |
| 41 | $[41, 41, w - 10]$ | $\phantom{-}\frac{13509}{120178}e^{8} - \frac{163079}{360534}e^{7} - \frac{1164643}{360534}e^{6} + \frac{3923677}{360534}e^{5} + \frac{10869787}{360534}e^{4} - \frac{7106779}{120178}e^{3} - \frac{20727106}{180267}e^{2} - \frac{703652}{180267}e + \frac{1894838}{60089}$ |
| 43 | $[43, 43, -w - 2]$ | $\phantom{-}\frac{14927}{120178}e^{8} - \frac{89260}{180267}e^{7} - \frac{647714}{180267}e^{6} + \frac{2146067}{180267}e^{5} + \frac{12124645}{360534}e^{4} - \frac{7740861}{120178}e^{3} - \frac{22669468}{180267}e^{2} - \frac{756593}{180267}e + \frac{1859080}{60089}$ |
| 43 | $[43, 43, w - 3]$ | $\phantom{-}\frac{14927}{120178}e^{8} - \frac{89260}{180267}e^{7} - \frac{647714}{180267}e^{6} + \frac{2146067}{180267}e^{5} + \frac{12124645}{360534}e^{4} - \frac{7740861}{120178}e^{3} - \frac{22669468}{180267}e^{2} - \frac{756593}{180267}e + \frac{1859080}{60089}$ |
| 47 | $[47, 47, -w - 1]$ | $\phantom{-}\frac{10283}{360534}e^{8} - \frac{51775}{360534}e^{7} - \frac{128386}{180267}e^{6} + \frac{422501}{120178}e^{5} + \frac{1833565}{360534}e^{4} - \frac{3641108}{180267}e^{3} - \frac{2520368}{180267}e^{2} + \frac{1457776}{180267}e - \frac{378180}{60089}$ |
| 47 | $[47, 47, w - 2]$ | $\phantom{-}\frac{10283}{360534}e^{8} - \frac{51775}{360534}e^{7} - \frac{128386}{180267}e^{6} + \frac{422501}{120178}e^{5} + \frac{1833565}{360534}e^{4} - \frac{3641108}{180267}e^{3} - \frac{2520368}{180267}e^{2} + \frac{1457776}{180267}e - \frac{378180}{60089}$ |
| 53 | $[53, 53, 2w - 13]$ | $\phantom{-}\frac{25943}{180267}e^{8} - \frac{66959}{120178}e^{7} - \frac{516077}{120178}e^{6} + \frac{4915283}{360534}e^{5} + \frac{7597064}{180267}e^{4} - \frac{13921003}{180267}e^{3} - \frac{9896539}{60089}e^{2} + \frac{1609015}{180267}e + \frac{3433908}{60089}$ |
Atkin-Lehner eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| $4$ | $[4, 2, 2]$ | $1$ |
| $9$ | $[9, 3, 3]$ | $1$ |