Base field \(\Q(\sqrt{67}) \)
Generator \(w\), with minimal polynomial \(x^{2} - 67\); narrow class number \(2\) and class number \(1\).
Form
| Weight: | $[2, 2]$ |
| Level: | $[14, 14, -w + 9]$ |
| Dimension: | $13$ |
| CM: | no |
| Base change: | no |
| Newspace dimension: | $68$ |
Hecke eigenvalues ($q$-expansion)
The Hecke eigenvalue field is $\Q(e)$ where $e$ is a root of the defining polynomial:
| \(x^{13} - 25x^{11} + 6x^{10} + 235x^{9} - 113x^{8} - 999x^{7} + 712x^{6} + 1729x^{5} - 1601x^{4} - 564x^{3} + 618x^{2} + 47x - 64\) |
Show full eigenvalues Hide large eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| 2 | $[2, 2, -27w + 221]$ | $\phantom{-}1$ |
| 3 | $[3, 3, -w + 8]$ | $\phantom{-}e$ |
| 3 | $[3, 3, -w - 8]$ | $\phantom{-}\frac{3215}{53361}e^{12} + \frac{386}{53361}e^{11} - \frac{8858}{5929}e^{10} + \frac{310}{2541}e^{9} + \frac{739787}{53361}e^{8} - \frac{30425}{7623}e^{7} - \frac{3077843}{53361}e^{6} + \frac{167098}{5929}e^{5} + \frac{5077388}{53361}e^{4} - \frac{3353435}{53361}e^{3} - \frac{165218}{7623}e^{2} + \frac{565198}{53361}e + \frac{4415}{53361}$ |
| 7 | $[7, 7, -11w + 90]$ | $\phantom{-}\frac{2126}{53361}e^{12} + \frac{3653}{53361}e^{11} - \frac{5954}{5929}e^{10} - \frac{3683}{2541}e^{9} + \frac{523076}{53361}e^{8} + \frac{83920}{7623}e^{7} - \frac{2472359}{53361}e^{6} - \frac{209091}{5929}e^{5} + \frac{5713364}{53361}e^{4} + \frac{2139481}{53361}e^{3} - \frac{788126}{7623}e^{2} - \frac{115427}{53361}e + \frac{1065101}{53361}$ |
| 7 | $[7, 7, -11w - 90]$ | $-1$ |
| 11 | $[11, 11, 6w - 49]$ | $-\frac{50}{693}e^{12} + \frac{184}{4851}e^{11} + \frac{988}{539}e^{10} - \frac{1747}{1617}e^{9} - \frac{83810}{4851}e^{8} + \frac{55562}{4851}e^{7} + \frac{350138}{4851}e^{6} - \frac{4163}{77}e^{5} - \frac{571562}{4851}e^{4} + \frac{482099}{4851}e^{3} + \frac{17363}{693}e^{2} - \frac{13819}{693}e - \frac{5204}{4851}$ |
| 11 | $[11, 11, 6w + 49]$ | $-\frac{15310}{53361}e^{12} - \frac{10750}{53361}e^{11} + \frac{41798}{5929}e^{10} + \frac{56674}{17787}e^{9} - \frac{500662}{7623}e^{8} - \frac{679964}{53361}e^{7} + \frac{15081910}{53361}e^{6} - \frac{83745}{5929}e^{5} - \frac{28201087}{53361}e^{4} + \frac{6515995}{53361}e^{3} + \frac{2188015}{7623}e^{2} - \frac{1703690}{53361}e - \frac{2321089}{53361}$ |
| 17 | $[17, 17, 4w + 33]$ | $-\frac{10079}{53361}e^{12} - \frac{2795}{53361}e^{11} + \frac{27943}{5929}e^{10} + \frac{2945}{17787}e^{9} - \frac{338084}{7623}e^{8} + \frac{491594}{53361}e^{7} + \frac{10147763}{53361}e^{6} - \frac{486692}{5929}e^{5} - \frac{18142154}{53361}e^{4} + \frac{10712930}{53361}e^{3} + \frac{1054469}{7623}e^{2} - \frac{2578924}{53361}e - \frac{914654}{53361}$ |
| 17 | $[17, 17, -4w + 33]$ | $\phantom{-}\frac{12470}{53361}e^{12} + \frac{3350}{53361}e^{11} - \frac{34228}{5929}e^{10} - \frac{1181}{17787}e^{9} + \frac{2868164}{53361}e^{8} - \frac{703001}{53361}e^{7} - \frac{12122720}{53361}e^{6} + \frac{647981}{5929}e^{5} + \frac{21013304}{53361}e^{4} - \frac{13758716}{53361}e^{3} - \frac{1038608}{7623}e^{2} + \frac{2944192}{53361}e + \frac{848465}{53361}$ |
| 25 | $[25, 5, -5]$ | $\phantom{-}\frac{11338}{53361}e^{12} + \frac{949}{7623}e^{11} - \frac{4409}{847}e^{10} - \frac{27547}{17787}e^{9} + \frac{2581465}{53361}e^{8} + \frac{6263}{53361}e^{7} - \frac{11029657}{53361}e^{6} + \frac{332478}{5929}e^{5} + \frac{2845900}{7623}e^{4} - \frac{9337375}{53361}e^{3} - \frac{1239691}{7623}e^{2} + \frac{2251154}{53361}e + \frac{1246648}{53361}$ |
| 29 | $[29, 29, -70w + 573]$ | $\phantom{-}\frac{943}{4851}e^{12} + \frac{604}{4851}e^{11} - \frac{2582}{539}e^{10} - \frac{3226}{1617}e^{9} + \frac{217048}{4851}e^{8} + \frac{39824}{4851}e^{7} - \frac{936310}{4851}e^{6} + \frac{3996}{539}e^{5} + \frac{1756582}{4851}e^{4} - \frac{349660}{4851}e^{3} - \frac{137524}{693}e^{2} + \frac{48107}{4851}e + \frac{134779}{4851}$ |
| 29 | $[29, 29, 151w - 1236]$ | $-\frac{14323}{53361}e^{12} - \frac{5218}{53361}e^{11} + \frac{39678}{5929}e^{10} + \frac{11758}{17787}e^{9} - \frac{3347371}{53361}e^{8} + \frac{576196}{53361}e^{7} + \frac{14171824}{53361}e^{6} - \frac{681328}{5929}e^{5} - \frac{24270718}{53361}e^{4} + \frac{15791137}{53361}e^{3} + \frac{149107}{1089}e^{2} - \frac{3787394}{53361}e - \frac{722782}{53361}$ |
| 31 | $[31, 31, -w - 6]$ | $\phantom{-}\frac{442}{5929}e^{12} + \frac{107}{5929}e^{11} - \frac{10974}{5929}e^{10} + \frac{446}{5929}e^{9} + \frac{101397}{5929}e^{8} - \frac{33813}{5929}e^{7} - \frac{412294}{5929}e^{6} + \frac{261892}{5929}e^{5} + \frac{616009}{5929}e^{4} - \frac{602776}{5929}e^{3} + \frac{4020}{847}e^{2} + \frac{76715}{5929}e - \frac{14719}{5929}$ |
| 31 | $[31, 31, w - 6]$ | $\phantom{-}\frac{9472}{53361}e^{12} + \frac{7594}{53361}e^{11} - \frac{25631}{5929}e^{10} - \frac{42382}{17787}e^{9} + \frac{2137195}{53361}e^{8} + \frac{616556}{53361}e^{7} - \frac{9212320}{53361}e^{6} - \frac{36312}{5929}e^{5} + \frac{17528125}{53361}e^{4} - \frac{2830354}{53361}e^{3} - \frac{1453906}{7623}e^{2} + \frac{1059746}{53361}e + \frac{1329058}{53361}$ |
| 37 | $[37, 37, -21w - 172]$ | $\phantom{-}\frac{19}{53361}e^{12} - \frac{4874}{53361}e^{11} - \frac{547}{5929}e^{10} + \frac{35912}{17787}e^{9} + \frac{57922}{53361}e^{8} - \frac{901633}{53361}e^{7} - \frac{7543}{7623}e^{6} + \frac{388688}{5929}e^{5} - \frac{1365317}{53361}e^{4} - \frac{822868}{7623}e^{3} + \frac{584219}{7623}e^{2} + \frac{1830041}{53361}e - \frac{127808}{7623}$ |
| 37 | $[37, 37, -21w + 172]$ | $-\frac{11026}{53361}e^{12} - \frac{1591}{7623}e^{11} + \frac{4223}{847}e^{10} + \frac{66436}{17787}e^{9} - \frac{2435497}{53361}e^{8} - \frac{1143815}{53361}e^{7} + \frac{10316158}{53361}e^{6} + \frac{222204}{5929}e^{5} - \frac{388924}{1089}e^{4} + \frac{560281}{53361}e^{3} + \frac{1483945}{7623}e^{2} + \frac{190342}{53361}e - \frac{1362436}{53361}$ |
| 43 | $[43, 43, 2w - 15]$ | $\phantom{-}\frac{3578}{17787}e^{12} - \frac{703}{17787}e^{11} - \frac{29478}{5929}e^{10} + \frac{1641}{847}e^{9} + \frac{812024}{17787}e^{8} - \frac{68540}{2541}e^{7} - \frac{3297458}{17787}e^{6} + \frac{877483}{5929}e^{5} + \frac{5114414}{17787}e^{4} - \frac{5202194}{17787}e^{3} - \frac{82091}{2541}e^{2} + \frac{934369}{17787}e - \frac{64555}{17787}$ |
| 43 | $[43, 43, 2w + 15]$ | $\phantom{-}\frac{494}{1089}e^{12} + \frac{4358}{53361}e^{11} - \frac{66589}{5929}e^{10} + \frac{9169}{17787}e^{9} + \frac{5578757}{53361}e^{8} - \frac{1504973}{53361}e^{7} - \frac{23521166}{53361}e^{6} + \frac{177525}{847}e^{5} + \frac{40672076}{53361}e^{4} - \frac{25485113}{53361}e^{3} - \frac{2044610}{7623}e^{2} + \frac{792871}{7623}e + \frac{1594601}{53361}$ |
| 67 | $[67, 67, -w]$ | $-\frac{139}{5929}e^{12} + \frac{115}{5929}e^{11} + \frac{2747}{5929}e^{10} - \frac{6049}{5929}e^{9} - \frac{16612}{5929}e^{8} + \frac{84625}{5929}e^{7} + \frac{1622}{5929}e^{6} - \frac{458604}{5929}e^{5} + \frac{313925}{5929}e^{4} + \frac{892775}{5929}e^{3} - \frac{17289}{121}e^{2} - \frac{176867}{5929}e + \frac{173688}{5929}$ |
| 73 | $[73, 73, -3w - 26]$ | $-\frac{25912}{53361}e^{12} - \frac{1159}{7623}e^{11} + \frac{10198}{847}e^{10} + \frac{19441}{17787}e^{9} - \frac{6009379}{53361}e^{8} + \frac{783400}{53361}e^{7} + \frac{25658341}{53361}e^{6} - \frac{945993}{5929}e^{5} - \frac{6576844}{7623}e^{4} + \frac{21528841}{53361}e^{3} + \frac{2876389}{7623}e^{2} - \frac{4189832}{53361}e - \frac{2873455}{53361}$ |
Atkin-Lehner eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| $2$ | $[2, 2, -27w + 221]$ | $-1$ |
| $7$ | $[7, 7, -11w - 90]$ | $1$ |