Base field \(\Q(\sqrt{493}) \)
Generator \(w\), with minimal polynomial \(x^{2} - x - 123\); narrow class number \(2\) and class number \(2\).
Form
| Weight: | $[2, 2]$ |
| Level: | $[4, 2, 2]$ |
| Dimension: | $12$ |
| CM: | no |
| Base change: | no |
| Newspace dimension: | $118$ |
Hecke eigenvalues ($q$-expansion)
The Hecke eigenvalue field is $\Q(e)$ where $e$ is a root of the defining polynomial:
| \(x^{12} + 23x^{10} + 186x^{8} + 638x^{6} + 925x^{4} + 483x^{2} + 16\) |
Show full eigenvalues Hide large eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| 3 | $[3, 3, w]$ | $\phantom{-}e$ |
| 3 | $[3, 3, w + 2]$ | $\phantom{-}\frac{21}{88}e^{11} + \frac{229}{44}e^{9} + \frac{153}{4}e^{7} + 108e^{5} + \frac{8909}{88}e^{3} + \frac{791}{44}e$ |
| 4 | $[4, 2, 2]$ | $\phantom{-}1$ |
| 11 | $[11, 11, w + 1]$ | $\phantom{-}\frac{17}{88}e^{11} + \frac{47}{11}e^{9} + \frac{129}{4}e^{7} + \frac{193}{2}e^{5} + \frac{9257}{88}e^{3} + \frac{751}{22}e$ |
| 11 | $[11, 11, w + 9]$ | $\phantom{-}\frac{17}{88}e^{11} + \frac{47}{11}e^{9} + \frac{129}{4}e^{7} + \frac{193}{2}e^{5} + \frac{9169}{88}e^{3} + \frac{597}{22}e$ |
| 13 | $[13, 13, w - 11]$ | $\phantom{-}\frac{127}{440}e^{10} + \frac{344}{55}e^{8} + \frac{909}{20}e^{6} + 125e^{4} + \frac{9327}{88}e^{2} + \frac{282}{55}$ |
| 13 | $[13, 13, w + 10]$ | $\phantom{-}\frac{73}{440}e^{10} + \frac{196}{55}e^{8} + \frac{511}{20}e^{6} + 69e^{4} + \frac{5233}{88}e^{2} + \frac{428}{55}$ |
| 17 | $[17, 17, w + 8]$ | $\phantom{-}\frac{1}{88}e^{11} + \frac{13}{44}e^{9} + \frac{11}{4}e^{7} + \frac{21}{2}e^{5} + \frac{1101}{88}e^{3} - \frac{87}{44}e$ |
| 25 | $[25, 5, -5]$ | $\phantom{-}\frac{25}{44}e^{10} + \frac{135}{11}e^{8} + \frac{177}{2}e^{6} + 239e^{4} + \frac{8561}{44}e^{2} + \frac{51}{11}$ |
| 29 | $[29, 29, w + 14]$ | $\phantom{-}\frac{13}{88}e^{11} + \frac{147}{44}e^{9} + \frac{105}{4}e^{7} + 85e^{5} + \frac{9517}{88}e^{3} + \frac{1861}{44}e$ |
| 31 | $[31, 31, w + 5]$ | $\phantom{-}\frac{9}{11}e^{11} + \frac{391}{22}e^{9} + \frac{259}{2}e^{7} + \frac{715}{2}e^{5} + \frac{6805}{22}e^{3} + \frac{315}{11}e$ |
| 31 | $[31, 31, w + 25]$ | $\phantom{-}\frac{9}{88}e^{11} + \frac{95}{44}e^{9} + \frac{59}{4}e^{7} + \frac{67}{2}e^{5} + \frac{493}{88}e^{3} - \frac{981}{44}e$ |
| 37 | $[37, 37, w + 3]$ | $-\frac{2}{55}e^{11} - \frac{52}{55}e^{9} - \frac{91}{10}e^{7} - \frac{79}{2}e^{5} - \frac{1631}{22}e^{3} - \frac{5167}{110}e$ |
| 37 | $[37, 37, w + 33]$ | $-\frac{57}{220}e^{11} - \frac{631}{110}e^{9} - \frac{217}{5}e^{7} - \frac{261}{2}e^{5} - \frac{6255}{44}e^{3} - \frac{2069}{55}e$ |
| 41 | $[41, 41, w]$ | $-\frac{24}{55}e^{11} - \frac{514}{55}e^{9} - \frac{331}{5}e^{7} - 170e^{5} - \frac{1195}{11}e^{3} + \frac{1668}{55}e$ |
| 41 | $[41, 41, w + 40]$ | $-\frac{27}{110}e^{11} - \frac{296}{55}e^{9} - \frac{199}{5}e^{7} - 113e^{5} - \frac{2311}{22}e^{3} - \frac{973}{55}e$ |
| 49 | $[49, 7, -7]$ | $\phantom{-}\frac{5}{11}e^{10} + \frac{108}{11}e^{8} + 71e^{6} + 194e^{4} + \frac{1820}{11}e^{2} + \frac{54}{11}$ |
| 53 | $[53, 53, -4w - 43]$ | $\phantom{-}\frac{51}{220}e^{10} + \frac{553}{110}e^{8} + \frac{181}{5}e^{6} + \frac{191}{2}e^{4} + \frac{2989}{44}e^{2} - \frac{178}{55}$ |
| 53 | $[53, 53, 4w - 47]$ | $-\frac{7}{40}e^{10} - \frac{19}{5}e^{8} - \frac{559}{20}e^{6} - 80e^{4} - \frac{623}{8}e^{2} - \frac{57}{5}$ |
| 59 | $[59, 59, -w - 13]$ | $-\frac{67}{88}e^{10} - \frac{182}{11}e^{8} - \frac{481}{4}e^{6} - 327e^{4} - \frac{22903}{88}e^{2} + \frac{8}{11}$ |
Atkin-Lehner eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| $4$ | $[4, 2, 2]$ | $-1$ |