Base field 4.4.16609.1
Generator \(w\), with minimal polynomial \(x^{4} - 7x^{2} - x + 9\); narrow class number \(1\) and class number \(1\).
Form
| Weight: | $[2, 2, 2, 2]$ |
| Level: | $[8, 2, w^{3} - w^{2} - 4w + 5]$ |
| Dimension: | $12$ |
| CM: | no |
| Base change: | no |
| Newspace dimension: | $18$ |
Hecke eigenvalues ($q$-expansion)
The Hecke eigenvalue field is $\Q(e)$ where $e$ is a root of the defining polynomial:
| \(x^{12} - 4x^{11} - 9x^{10} + 49x^{9} + 14x^{8} - 208x^{7} + 55x^{6} + 374x^{5} - 161x^{4} - 270x^{3} + 96x^{2} + 61x + 3\) |
Show full eigenvalues Hide large eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| 2 | $[2, 2, -w^{2} + w + 4]$ | $\phantom{-}e$ |
| 3 | $[3, 3, -w^{2} + w + 3]$ | $-\frac{7}{43}e^{11} + \frac{11}{43}e^{10} + \frac{102}{43}e^{9} - \frac{126}{43}e^{8} - \frac{576}{43}e^{7} + \frac{481}{43}e^{6} + \frac{1637}{43}e^{5} - 16e^{4} - \frac{2356}{43}e^{3} + \frac{155}{43}e^{2} + \frac{1277}{43}e + \frac{211}{43}$ |
| 5 | $[5, 5, w^{2} - 4]$ | $\phantom{-}\frac{15}{43}e^{11} - \frac{42}{43}e^{10} - \frac{194}{43}e^{9} + \frac{528}{43}e^{8} + \frac{921}{43}e^{7} - \frac{2290}{43}e^{6} - \frac{2052}{43}e^{5} + 94e^{4} + \frac{2143}{43}e^{3} - \frac{2390}{43}e^{2} - \frac{826}{43}e + \frac{27}{43}$ |
| 8 | $[8, 2, w^{3} - w^{2} - 4w + 5]$ | $-1$ |
| 17 | $[17, 17, -w^{2} + w + 5]$ | $\phantom{-}\frac{28}{43}e^{11} - \frac{44}{43}e^{10} - \frac{408}{43}e^{9} + \frac{547}{43}e^{8} + \frac{2175}{43}e^{7} - \frac{2354}{43}e^{6} - \frac{5258}{43}e^{5} + 96e^{4} + \frac{5726}{43}e^{3} - \frac{2426}{43}e^{2} - \frac{2313}{43}e - \frac{27}{43}$ |
| 27 | $[27, 3, -w^{3} + 4w - 2]$ | $\phantom{-}\frac{52}{43}e^{11} - \frac{137}{43}e^{10} - \frac{684}{43}e^{9} + \frac{1710}{43}e^{8} + \frac{3339}{43}e^{7} - \frac{7308}{43}e^{6} - \frac{7879}{43}e^{5} + 290e^{4} + \frac{9387}{43}e^{3} - \frac{6637}{43}e^{2} - \frac{4658}{43}e - \frac{431}{43}$ |
| 29 | $[29, 29, -w^{2} + w + 1]$ | $\phantom{-}\frac{27}{43}e^{11} - \frac{67}{43}e^{10} - \frac{332}{43}e^{9} + \frac{787}{43}e^{8} + \frac{1460}{43}e^{7} - \frac{3133}{43}e^{6} - \frac{2997}{43}e^{5} + 118e^{4} + \frac{3092}{43}e^{3} - \frac{2969}{43}e^{2} - \frac{1504}{43}e + \frac{255}{43}$ |
| 29 | $[29, 29, -w^{3} + 4w + 2]$ | $\phantom{-}\frac{3}{43}e^{11} - \frac{17}{43}e^{10} - \frac{13}{43}e^{9} + \frac{183}{43}e^{8} - \frac{91}{43}e^{7} - \frac{630}{43}e^{6} + \frac{570}{43}e^{5} + 20e^{4} - \frac{1042}{43}e^{3} - \frac{521}{43}e^{2} + \frac{583}{43}e + \frac{57}{43}$ |
| 37 | $[37, 37, -2w^{3} + 3w^{2} + 9w - 13]$ | $\phantom{-}\frac{19}{43}e^{11} - \frac{79}{43}e^{10} - \frac{197}{43}e^{9} + \frac{987}{43}e^{8} + \frac{642}{43}e^{7} - \frac{4205}{43}e^{6} - \frac{862}{43}e^{5} + 165e^{4} + \frac{897}{43}e^{3} - \frac{3615}{43}e^{2} - \frac{794}{43}e - \frac{112}{43}$ |
| 41 | $[41, 41, w^{3} - w^{2} - 5w + 2]$ | $-\frac{5}{43}e^{11} + \frac{14}{43}e^{10} + \frac{79}{43}e^{9} - \frac{176}{43}e^{8} - \frac{565}{43}e^{7} + \frac{792}{43}e^{6} + \frac{2232}{43}e^{5} - 37e^{4} - \frac{4226}{43}e^{3} + \frac{1155}{43}e^{2} + \frac{2626}{43}e + \frac{249}{43}$ |
| 43 | $[43, 43, -w^{3} + w^{2} + 3w - 4]$ | $-\frac{76}{43}e^{11} + \frac{230}{43}e^{10} + \frac{874}{43}e^{9} - \frac{2787}{43}e^{8} - \frac{3385}{43}e^{7} + \frac{11531}{43}e^{6} + \frac{5512}{43}e^{5} - 449e^{4} - \frac{4061}{43}e^{3} + \frac{10848}{43}e^{2} + \frac{1542}{43}e - \frac{154}{43}$ |
| 61 | $[61, 61, w^{2} - 2w - 2]$ | $\phantom{-}\frac{81}{43}e^{11} - \frac{201}{43}e^{10} - \frac{996}{43}e^{9} + \frac{2404}{43}e^{8} + \frac{4251}{43}e^{7} - \frac{9786}{43}e^{6} - \frac{7744}{43}e^{5} + 375e^{4} + \frac{5621}{43}e^{3} - \frac{9122}{43}e^{2} - \frac{986}{43}e + \frac{335}{43}$ |
| 61 | $[61, 61, 2w^{2} - w - 8]$ | $-\frac{12}{43}e^{11} + \frac{25}{43}e^{10} + \frac{181}{43}e^{9} - \frac{302}{43}e^{8} - \frac{1098}{43}e^{7} + \frac{1230}{43}e^{6} + \frac{3396}{43}e^{5} - 46e^{4} - \frac{5077}{43}e^{3} + \frac{923}{43}e^{2} + \frac{2656}{43}e + \frac{374}{43}$ |
| 67 | $[67, 67, -4w^{3} + 5w^{2} + 18w - 22]$ | $-\frac{7}{43}e^{11} + \frac{97}{43}e^{10} - \frac{70}{43}e^{9} - \frac{1158}{43}e^{8} + \frac{1273}{43}e^{7} + \frac{4695}{43}e^{6} - \frac{4770}{43}e^{5} - 182e^{4} + \frac{6072}{43}e^{3} + \frac{4756}{43}e^{2} - \frac{2249}{43}e - \frac{520}{43}$ |
| 73 | $[73, 73, -w^{2} - 2w + 2]$ | $\phantom{-}\frac{22}{43}e^{11} - \frac{53}{43}e^{10} - \frac{253}{43}e^{9} + \frac{568}{43}e^{8} + \frac{981}{43}e^{7} - \frac{1911}{43}e^{6} - \frac{1539}{43}e^{5} + 53e^{4} + \frac{672}{43}e^{3} - \frac{954}{43}e^{2} + \frac{391}{43}e + \frac{332}{43}$ |
| 79 | $[79, 79, w^{2} - 2w - 4]$ | $\phantom{-}e^{11} - 2e^{10} - 12e^{9} + 22e^{8} + 48e^{7} - 80e^{6} - 74e^{5} + 117e^{4} + 32e^{3} - 66e^{2} + 6e + 8$ |
| 89 | $[89, 89, w^{2} + w - 5]$ | $-\frac{10}{43}e^{11} + \frac{28}{43}e^{10} + \frac{158}{43}e^{9} - \frac{395}{43}e^{8} - \frac{1001}{43}e^{7} + \frac{1971}{43}e^{6} + \frac{3174}{43}e^{5} - 95e^{4} - \frac{4711}{43}e^{3} + \frac{2912}{43}e^{2} + \frac{2242}{43}e - \frac{18}{43}$ |
| 89 | $[89, 89, 2w^{3} - 2w^{2} - 9w + 8]$ | $-\frac{24}{43}e^{11} + \frac{50}{43}e^{10} + \frac{276}{43}e^{9} - \frac{604}{43}e^{8} - \frac{863}{43}e^{7} + \frac{2503}{43}e^{6} - \frac{88}{43}e^{5} - 98e^{4} + \frac{3090}{43}e^{3} + \frac{2620}{43}e^{2} - \frac{1955}{43}e - \frac{585}{43}$ |
| 89 | $[89, 89, w^{3} - 5w - 5]$ | $\phantom{-}\frac{95}{43}e^{11} - \frac{266}{43}e^{10} - \frac{1071}{43}e^{9} + \frac{3086}{43}e^{8} + \frac{3984}{43}e^{7} - \frac{11995}{43}e^{6} - \frac{5901}{43}e^{5} + 430e^{4} + \frac{3195}{43}e^{3} - \frac{9346}{43}e^{2} - \frac{573}{43}e + \frac{171}{43}$ |
| 89 | $[89, 89, -w^{3} + 2w^{2} + 4w - 4]$ | $\phantom{-}\frac{21}{43}e^{11} - \frac{33}{43}e^{10} - \frac{263}{43}e^{9} + \frac{292}{43}e^{8} + \frac{1212}{43}e^{7} - \frac{583}{43}e^{6} - \frac{2761}{43}e^{5} - 10e^{4} + \frac{3284}{43}e^{3} + \frac{1384}{43}e^{2} - \frac{1681}{43}e - \frac{117}{43}$ |
Atkin-Lehner eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| $8$ | $[8, 2, w^{3} - w^{2} - 4w + 5]$ | $1$ |