Base field \(\Q(\sqrt{205}) \)
Generator \(w\), with minimal polynomial \(x^{2} - x - 51\); narrow class number \(4\) and class number \(2\).
Form
| Weight: | $[2, 2]$ |
| Level: | $[15, 15, w + 2]$ |
| Dimension: | $14$ |
| CM: | no |
| Base change: | no |
| Newspace dimension: | $100$ |
Hecke eigenvalues ($q$-expansion)
The Hecke eigenvalue field is $\Q(e)$ where $e$ is a root of the defining polynomial:
| \(x^{14} - 9x^{13} + 14x^{12} + 87x^{11} - 275x^{10} - 151x^{9} + 1152x^{8} - 194x^{7} - 2004x^{6} + 576x^{5} + 1556x^{4} - 388x^{3} - 456x^{2} + 84x + 16\) |
Show full eigenvalues Hide large eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| 3 | $[3, 3, w]$ | $\phantom{-}e$ |
| 3 | $[3, 3, w + 2]$ | $\phantom{-}1$ |
| 4 | $[4, 2, 2]$ | $-\frac{11}{9}e^{13} + \frac{119}{18}e^{12} + \frac{77}{6}e^{11} - \frac{213}{2}e^{10} - \frac{152}{9}e^{9} + \frac{1209}{2}e^{8} - 135e^{7} - \frac{26665}{18}e^{6} + \frac{6233}{18}e^{5} + \frac{13312}{9}e^{4} - \frac{910}{3}e^{3} - \frac{4627}{9}e^{2} + \frac{917}{9}e + \frac{161}{9}$ |
| 5 | $[5, 5, -w + 8]$ | $\phantom{-}1$ |
| 7 | $[7, 7, w + 1]$ | $\phantom{-}\frac{56}{9}e^{13} - \frac{361}{9}e^{12} - \frac{52}{3}e^{11} + 512e^{10} - \frac{3709}{9}e^{9} - 2151e^{8} + 1960e^{7} + \frac{38141}{9}e^{6} - \frac{24415}{9}e^{5} - \frac{33823}{9}e^{4} + \frac{4204}{3}e^{3} + \frac{11053}{9}e^{2} - \frac{2222}{9}e - \frac{440}{9}$ |
| 7 | $[7, 7, w + 5]$ | $-\frac{28}{9}e^{13} + \frac{379}{18}e^{12} + \frac{7}{6}e^{11} - 251e^{10} + \frac{5113}{18}e^{9} + \frac{1853}{2}e^{8} - \frac{2361}{2}e^{7} - \frac{14179}{9}e^{6} + \frac{13589}{9}e^{5} + \frac{11012}{9}e^{4} - \frac{2084}{3}e^{3} - \frac{3191}{9}e^{2} + \frac{823}{9}e + \frac{148}{9}$ |
| 13 | $[13, 13, w + 3]$ | $...$ |
| 13 | $[13, 13, w + 9]$ | $\phantom{-}\frac{10}{9}e^{13} - \frac{68}{9}e^{12} - \frac{1}{6}e^{11} + \frac{179}{2}e^{10} - \frac{932}{9}e^{9} - \frac{655}{2}e^{8} + \frac{847}{2}e^{7} + \frac{10097}{18}e^{6} - \frac{4805}{9}e^{5} - \frac{4145}{9}e^{4} + \frac{719}{3}e^{3} + \frac{1400}{9}e^{2} - \frac{247}{9}e - \frac{94}{9}$ |
| 17 | $[17, 17, w]$ | $-\frac{29}{3}e^{13} + \frac{187}{3}e^{12} + \frac{53}{2}e^{11} - \frac{1587}{2}e^{10} + \frac{1927}{3}e^{9} + \frac{6635}{2}e^{8} - \frac{6061}{2}e^{7} - \frac{38983}{6}e^{6} + \frac{12448}{3}e^{5} + \frac{17110}{3}e^{4} - 2117e^{3} - \frac{5479}{3}e^{2} + \frac{1097}{3}e + \frac{194}{3}$ |
| 17 | $[17, 17, w + 16]$ | $-\frac{11}{6}e^{13} + \frac{35}{3}e^{12} + \frac{13}{2}e^{11} - \frac{307}{2}e^{10} + \frac{332}{3}e^{9} + 676e^{8} - \frac{1163}{2}e^{7} - \frac{4147}{3}e^{6} + \frac{2678}{3}e^{5} + \frac{3755}{3}e^{4} - 529e^{3} - \frac{1229}{3}e^{2} + \frac{328}{3}e + \frac{46}{3}$ |
| 31 | $[31, 31, -w - 4]$ | $\phantom{-}\frac{20}{9}e^{13} - \frac{191}{18}e^{12} - \frac{203}{6}e^{11} + 200e^{10} + \frac{2545}{18}e^{9} - \frac{2601}{2}e^{8} - \frac{109}{2}e^{7} + \frac{30824}{9}e^{6} - \frac{3040}{9}e^{5} - \frac{31978}{9}e^{4} + \frac{1348}{3}e^{3} + \frac{11449}{9}e^{2} - \frac{1610}{9}e - \frac{440}{9}$ |
| 31 | $[31, 31, w - 5]$ | $\phantom{-}\frac{95}{9}e^{13} - \frac{637}{9}e^{12} - \frac{28}{3}e^{11} + 858e^{10} - \frac{8224}{9}e^{9} - 3274e^{8} + 3924e^{7} + \frac{52112}{9}e^{6} - \frac{46570}{9}e^{5} - \frac{42100}{9}e^{4} + \frac{7393}{3}e^{3} + \frac{12634}{9}e^{2} - \frac{3062}{9}e - \frac{488}{9}$ |
| 41 | $[41, 41, 3w - 22]$ | $-\frac{17}{3}e^{13} + \frac{106}{3}e^{12} + 25e^{11} - 474e^{10} + \frac{850}{3}e^{9} + 2153e^{8} - 1580e^{7} - \frac{13595}{3}e^{6} + \frac{7195}{3}e^{5} + \frac{12523}{3}e^{4} - 1396e^{3} - \frac{4144}{3}e^{2} + \frac{878}{3}e + \frac{134}{3}$ |
| 47 | $[47, 47, w + 19]$ | $\phantom{-}\frac{38}{3}e^{13} - \frac{253}{3}e^{12} - 15e^{11} + 1028e^{10} - \frac{3151}{3}e^{9} - 3965e^{8} + 4532e^{7} + \frac{21275}{3}e^{6} - \frac{17701}{3}e^{5} - \frac{17212}{3}e^{4} + 2726e^{3} + \frac{5110}{3}e^{2} - \frac{1046}{3}e - \frac{200}{3}$ |
| 47 | $[47, 47, w + 27]$ | $-\frac{41}{3}e^{13} + \frac{262}{3}e^{12} + 44e^{11} - 1130e^{10} + \frac{2545}{3}e^{9} + 4851e^{8} - 4198e^{7} - \frac{29162}{3}e^{6} + \frac{17941}{3}e^{5} + \frac{25999}{3}e^{4} - 3219e^{3} - \frac{8458}{3}e^{2} + \frac{1805}{3}e + \frac{332}{3}$ |
| 53 | $[53, 53, w + 14]$ | $-\frac{19}{2}e^{13} + 61e^{12} + \frac{57}{2}e^{11} - \frac{1569}{2}e^{10} + 611e^{9} + 3336e^{8} - \frac{5939}{2}e^{7} - 6639e^{6} + 4183e^{5} + 5919e^{4} - 2199e^{3} - 1937e^{2} + 388e + 78$ |
| 53 | $[53, 53, w + 38]$ | $\phantom{-}\frac{25}{6}e^{13} - \frac{185}{6}e^{12} + \frac{35}{2}e^{11} + 328e^{10} - \frac{3521}{6}e^{9} - 907e^{8} + \frac{4367}{2}e^{7} + \frac{5317}{6}e^{6} - \frac{8095}{3}e^{5} - \frac{706}{3}e^{4} + 1172e^{3} - \frac{137}{3}e^{2} - \frac{311}{3}e - \frac{2}{3}$ |
| 59 | $[59, 59, -w - 10]$ | $-\frac{5}{3}e^{13} + \frac{59}{6}e^{12} + \frac{23}{2}e^{11} - 142e^{10} + \frac{245}{6}e^{9} + \frac{1421}{2}e^{8} - \frac{715}{2}e^{7} - \frac{4769}{3}e^{6} + \frac{1831}{3}e^{5} + \frac{4429}{3}e^{4} - 375e^{3} - \frac{1381}{3}e^{2} + \frac{236}{3}e + \frac{20}{3}$ |
| 59 | $[59, 59, w - 11]$ | $-\frac{17}{3}e^{13} + \frac{239}{6}e^{12} - \frac{19}{2}e^{11} - 446e^{10} + \frac{3779}{6}e^{9} + \frac{2863}{2}e^{8} - \frac{4739}{2}e^{7} - \frac{6014}{3}e^{6} + \frac{8185}{3}e^{5} + \frac{3847}{3}e^{4} - 1012e^{3} - \frac{943}{3}e^{2} + \frac{164}{3}e + \frac{68}{3}$ |
| 61 | $[61, 61, 2w - 13]$ | $\phantom{-}\frac{16}{9}e^{13} - \frac{116}{9}e^{12} + \frac{16}{3}e^{11} + 142e^{10} - \frac{2060}{9}e^{9} - 435e^{8} + 881e^{7} + \frac{4978}{9}e^{6} - \frac{10064}{9}e^{5} - \frac{3023}{9}e^{4} + \frac{1496}{3}e^{3} + \frac{1016}{9}e^{2} - \frac{406}{9}e - \frac{154}{9}$ |
Atkin-Lehner eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| $3$ | $[3, 3, w + 2]$ | $-1$ |
| $5$ | $[5, 5, -w + 8]$ | $-1$ |