Base field \(\Q(\sqrt{313}) \)
Generator \(w\), with minimal polynomial \(x^{2} - x - 78\); narrow class number \(1\) and class number \(1\).
Form
| Weight: | $[2, 2]$ |
| Level: | $[1, 1, 1]$ |
| Dimension: | $12$ |
| CM: | no |
| Base change: | yes |
| Newspace dimension: | $18$ |
Hecke eigenvalues ($q$-expansion)
The Hecke eigenvalue field is $\Q(e)$ where $e$ is a root of the defining polynomial:
| \(x^{12} + x^{11} - 16x^{10} - 14x^{9} + 93x^{8} + 71x^{7} - 238x^{6} - 158x^{5} + 256x^{4} + 148x^{3} - 78x^{2} - 41x - 3\) |
Show full eigenvalues Hide large eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| 2 | $[2, 2, -3w + 28]$ | $\phantom{-}e$ |
| 2 | $[2, 2, -3w - 25]$ | $\phantom{-}e$ |
| 3 | $[3, 3, 26w - 243]$ | $-\frac{11}{59}e^{11} - \frac{17}{59}e^{10} + \frac{156}{59}e^{9} + \frac{223}{59}e^{8} - \frac{778}{59}e^{7} - \frac{1023}{59}e^{6} + \frac{1647}{59}e^{5} + \frac{1923}{59}e^{4} - \frac{1515}{59}e^{3} - \frac{1328}{59}e^{2} + \frac{552}{59}e + \frac{205}{59}$ |
| 3 | $[3, 3, -26w - 217]$ | $-\frac{11}{59}e^{11} - \frac{17}{59}e^{10} + \frac{156}{59}e^{9} + \frac{223}{59}e^{8} - \frac{778}{59}e^{7} - \frac{1023}{59}e^{6} + \frac{1647}{59}e^{5} + \frac{1923}{59}e^{4} - \frac{1515}{59}e^{3} - \frac{1328}{59}e^{2} + \frac{552}{59}e + \frac{205}{59}$ |
| 11 | $[11, 11, -2w - 17]$ | $\phantom{-}\frac{10}{59}e^{11} - \frac{6}{59}e^{10} - \frac{174}{59}e^{9} + \frac{103}{59}e^{8} + \frac{1131}{59}e^{7} - \frac{545}{59}e^{6} - \frac{3337}{59}e^{5} + \frac{998}{59}e^{4} + \frac{4220}{59}e^{3} - \frac{493}{59}e^{2} - \frac{1596}{59}e - \frac{63}{59}$ |
| 11 | $[11, 11, -2w + 19]$ | $\phantom{-}\frac{10}{59}e^{11} - \frac{6}{59}e^{10} - \frac{174}{59}e^{9} + \frac{103}{59}e^{8} + \frac{1131}{59}e^{7} - \frac{545}{59}e^{6} - \frac{3337}{59}e^{5} + \frac{998}{59}e^{4} + \frac{4220}{59}e^{3} - \frac{493}{59}e^{2} - \frac{1596}{59}e - \frac{63}{59}$ |
| 13 | $[13, 13, 2148w + 17927]$ | $\phantom{-}\frac{24}{59}e^{11} + \frac{21}{59}e^{10} - \frac{394}{59}e^{9} - \frac{272}{59}e^{8} + \frac{2384}{59}e^{7} + \frac{1229}{59}e^{6} - \frac{6463}{59}e^{5} - \frac{2313}{59}e^{4} + \frac{7473}{59}e^{3} + \frac{1755}{59}e^{2} - \frac{2497}{59}e - \frac{399}{59}$ |
| 13 | $[13, 13, 2148w - 20075]$ | $\phantom{-}\frac{24}{59}e^{11} + \frac{21}{59}e^{10} - \frac{394}{59}e^{9} - \frac{272}{59}e^{8} + \frac{2384}{59}e^{7} + \frac{1229}{59}e^{6} - \frac{6463}{59}e^{5} - \frac{2313}{59}e^{4} + \frac{7473}{59}e^{3} + \frac{1755}{59}e^{2} - \frac{2497}{59}e - \frac{399}{59}$ |
| 19 | $[19, 19, -292w + 2729]$ | $\phantom{-}e^{5} + e^{4} - 7e^{3} - 5e^{2} + 9e + 3$ |
| 19 | $[19, 19, 292w + 2437]$ | $\phantom{-}e^{5} + e^{4} - 7e^{3} - 5e^{2} + 9e + 3$ |
| 25 | $[25, 5, -5]$ | $\phantom{-}\frac{23}{59}e^{11} - \frac{2}{59}e^{10} - \frac{353}{59}e^{9} + \frac{54}{59}e^{8} + \frac{1852}{59}e^{7} - \frac{398}{59}e^{6} - \frac{3905}{59}e^{5} + \frac{1080}{59}e^{4} + \frac{3157}{59}e^{3} - \frac{951}{59}e^{2} - \frac{768}{59}e + \frac{333}{59}$ |
| 29 | $[29, 29, 20w + 167]$ | $-\frac{10}{59}e^{11} + \frac{6}{59}e^{10} + \frac{174}{59}e^{9} - \frac{103}{59}e^{8} - \frac{1131}{59}e^{7} + \frac{486}{59}e^{6} + \frac{3278}{59}e^{5} - \frac{467}{59}e^{4} - \frac{3807}{59}e^{3} - \frac{687}{59}e^{2} + \frac{1006}{59}e + \frac{417}{59}$ |
| 29 | $[29, 29, 20w - 187]$ | $-\frac{10}{59}e^{11} + \frac{6}{59}e^{10} + \frac{174}{59}e^{9} - \frac{103}{59}e^{8} - \frac{1131}{59}e^{7} + \frac{486}{59}e^{6} + \frac{3278}{59}e^{5} - \frac{467}{59}e^{4} - \frac{3807}{59}e^{3} - \frac{687}{59}e^{2} + \frac{1006}{59}e + \frac{417}{59}$ |
| 49 | $[49, 7, -7]$ | $-\frac{17}{59}e^{11} - \frac{37}{59}e^{10} + \frac{225}{59}e^{9} + \frac{468}{59}e^{8} - \frac{961}{59}e^{7} - \frac{1817}{59}e^{6} + \frac{1419}{59}e^{5} + \frac{1867}{59}e^{4} - \frac{625}{59}e^{3} + \frac{726}{59}e^{2} + \frac{247}{59}e - \frac{5}{59}$ |
| 71 | $[71, 71, -240w + 2243]$ | $\phantom{-}\frac{10}{59}e^{11} - \frac{6}{59}e^{10} - \frac{174}{59}e^{9} + \frac{103}{59}e^{8} + \frac{1072}{59}e^{7} - \frac{722}{59}e^{6} - \frac{2924}{59}e^{5} + \frac{2355}{59}e^{4} + \frac{3571}{59}e^{3} - \frac{2853}{59}e^{2} - \frac{1242}{59}e + \frac{468}{59}$ |
| 71 | $[71, 71, 240w + 2003]$ | $\phantom{-}\frac{10}{59}e^{11} - \frac{6}{59}e^{10} - \frac{174}{59}e^{9} + \frac{103}{59}e^{8} + \frac{1072}{59}e^{7} - \frac{722}{59}e^{6} - \frac{2924}{59}e^{5} + \frac{2355}{59}e^{4} + \frac{3571}{59}e^{3} - \frac{2853}{59}e^{2} - \frac{1242}{59}e + \frac{468}{59}$ |
| 79 | $[79, 79, -3368w + 31477]$ | $\phantom{-}\frac{71}{59}e^{11} + \frac{40}{59}e^{10} - \frac{1082}{59}e^{9} - \frac{431}{59}e^{8} + \frac{5912}{59}e^{7} + \frac{1588}{59}e^{6} - \frac{13940}{59}e^{5} - \frac{2602}{59}e^{4} + \frac{13501}{59}e^{3} + \frac{2323}{59}e^{2} - \frac{3697}{59}e - \frac{937}{59}$ |
| 79 | $[79, 79, -3368w - 28109]$ | $\phantom{-}\frac{71}{59}e^{11} + \frac{40}{59}e^{10} - \frac{1082}{59}e^{9} - \frac{431}{59}e^{8} + \frac{5912}{59}e^{7} + \frac{1588}{59}e^{6} - \frac{13940}{59}e^{5} - \frac{2602}{59}e^{4} + \frac{13501}{59}e^{3} + \frac{2323}{59}e^{2} - \frac{3697}{59}e - \frac{937}{59}$ |
| 83 | $[83, 83, -84w - 701]$ | $\phantom{-}\frac{69}{59}e^{11} - \frac{6}{59}e^{10} - \frac{1177}{59}e^{9} + \frac{103}{59}e^{8} + \frac{7267}{59}e^{7} - \frac{368}{59}e^{6} - \frac{19739}{59}e^{5} - \frac{477}{59}e^{4} + \frac{22628}{59}e^{3} + \frac{2752}{59}e^{2} - \frac{7791}{59}e - \frac{1479}{59}$ |
| 83 | $[83, 83, 84w - 785]$ | $\phantom{-}\frac{69}{59}e^{11} - \frac{6}{59}e^{10} - \frac{1177}{59}e^{9} + \frac{103}{59}e^{8} + \frac{7267}{59}e^{7} - \frac{368}{59}e^{6} - \frac{19739}{59}e^{5} - \frac{477}{59}e^{4} + \frac{22628}{59}e^{3} + \frac{2752}{59}e^{2} - \frac{7791}{59}e - \frac{1479}{59}$ |
Atkin-Lehner eigenvalues
This form has no Atkin-Lehner eigenvalues since the level is \((1)\).