Base field \(\Q(\sqrt{449}) \)
Generator \(w\), with minimal polynomial \(x^{2} - x - 112\); narrow class number \(1\) and class number \(1\).
Form
| Weight: | $[2, 2]$ |
| Level: | $[2,2,-w + 11]$ |
| Dimension: | $13$ |
| CM: | no |
| Base change: | no |
| Newspace dimension: | $22$ |
Hecke eigenvalues ($q$-expansion)
The Hecke eigenvalue field is $\Q(e)$ where $e$ is a root of the defining polynomial:
| \(x^{13} - 4x^{12} - 10x^{11} + 53x^{10} + 12x^{9} - 212x^{8} + 61x^{7} + 353x^{6} - 163x^{5} - 245x^{4} + 126x^{3} + 56x^{2} - 22x - 5\) |
Show full eigenvalues Hide large eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| 2 | $[2, 2, w + 10]$ | $\phantom{-}e$ |
| 2 | $[2, 2, w - 11]$ | $-1$ |
| 5 | $[5, 5, -116w - 1171]$ | $-\frac{40}{9}e^{12} + \frac{197}{9}e^{11} + \frac{200}{9}e^{10} - \frac{2224}{9}e^{9} + \frac{1696}{9}e^{8} + \frac{5968}{9}e^{7} - \frac{7559}{9}e^{6} - \frac{1448}{3}e^{5} + \frac{8254}{9}e^{4} - \frac{557}{9}e^{3} - \frac{2005}{9}e^{2} + \frac{230}{9}e + \frac{134}{9}$ |
| 5 | $[5, 5, 116w - 1287]$ | $\phantom{-}\frac{43}{9}e^{12} - \frac{194}{9}e^{11} - \frac{296}{9}e^{10} + \frac{2272}{9}e^{9} - \frac{880}{9}e^{8} - \frac{6817}{9}e^{7} + \frac{5342}{9}e^{6} + \frac{2318}{3}e^{5} - \frac{6142}{9}e^{4} - \frac{1912}{9}e^{3} + \frac{1468}{9}e^{2} + \frac{250}{9}e - \frac{14}{9}$ |
| 7 | $[7, 7, -1002w - 10115]$ | $-6e^{12} + 31e^{11} + 23e^{10} - 342e^{9} + 336e^{8} + 846e^{7} - 1376e^{6} - 406e^{5} + 1482e^{4} - 364e^{3} - 370e^{2} + 103e + 33$ |
| 7 | $[7, 7, 1002w - 11117]$ | $\phantom{-}\frac{50}{9}e^{12} - \frac{181}{9}e^{11} - \frac{538}{9}e^{10} + \frac{2312}{9}e^{9} + \frac{1246}{9}e^{8} - \frac{8549}{9}e^{7} - \frac{584}{9}e^{6} + \frac{4198}{3}e^{5} - \frac{296}{9}e^{4} - \frac{7262}{9}e^{3} - \frac{25}{9}e^{2} + \frac{1283}{9}e + \frac{170}{9}$ |
| 9 | $[9, 3, 3]$ | $\phantom{-}\frac{14}{9}e^{12} - \frac{46}{9}e^{11} - \frac{169}{9}e^{10} + \frac{602}{9}e^{9} + \frac{580}{9}e^{8} - \frac{2339}{9}e^{7} - \frac{989}{9}e^{6} + \frac{1228}{3}e^{5} + \frac{1054}{9}e^{4} - \frac{2330}{9}e^{3} - \frac{565}{9}e^{2} + \frac{491}{9}e + \frac{107}{9}$ |
| 11 | $[11, 11, 10w - 111]$ | $\phantom{-}\frac{16}{9}e^{12} - \frac{113}{9}e^{11} + \frac{73}{9}e^{10} + \frac{1129}{9}e^{9} - \frac{2473}{9}e^{8} - \frac{1732}{9}e^{7} + \frac{8429}{9}e^{6} - \frac{796}{3}e^{5} - \frac{8851}{9}e^{4} + \frac{4973}{9}e^{3} + \frac{2350}{9}e^{2} - \frac{1172}{9}e - \frac{257}{9}$ |
| 11 | $[11, 11, -10w - 101]$ | $\phantom{-}4e^{12} - 12e^{11} - 54e^{10} + 167e^{9} + 229e^{8} - 725e^{7} - 442e^{6} + 1292e^{5} + 392e^{4} - 919e^{3} - 125e^{2} + 185e + 29$ |
| 23 | $[23, 23, -32w + 355]$ | $-\frac{77}{9}e^{12} + \frac{316}{9}e^{11} + \frac{655}{9}e^{10} - \frac{3788}{9}e^{9} + \frac{68}{9}e^{8} + \frac{12086}{9}e^{7} - \frac{4690}{9}e^{6} - \frac{4663}{3}e^{5} + \frac{5327}{9}e^{4} + \frac{5408}{9}e^{3} - \frac{623}{9}e^{2} - \frac{869}{9}e - \frac{125}{9}$ |
| 23 | $[23, 23, 32w + 323]$ | $\phantom{-}13e^{12} - 64e^{11} - 65e^{10} + 722e^{9} - 550e^{8} - 1933e^{7} + 2442e^{6} + 1395e^{5} - 2633e^{4} + 198e^{3} + 590e^{2} - 83e - 24$ |
| 41 | $[41, 41, -8w - 81]$ | $\phantom{-}\frac{169}{9}e^{12} - \frac{743}{9}e^{11} - \frac{1223}{9}e^{10} + \frac{8689}{9}e^{9} - \frac{2689}{9}e^{8} - \frac{25924}{9}e^{7} + \frac{18113}{9}e^{6} + \frac{8606}{3}e^{5} - \frac{19984}{9}e^{4} - \frac{6187}{9}e^{3} + \frac{3745}{9}e^{2} + \frac{655}{9}e + \frac{49}{9}$ |
| 41 | $[41, 41, -8w + 89]$ | $\phantom{-}\frac{41}{9}e^{12} - \frac{172}{9}e^{11} - \frac{340}{9}e^{10} + \frac{2078}{9}e^{9} - \frac{176}{9}e^{8} - \frac{6767}{9}e^{7} + \frac{3232}{9}e^{6} + \frac{2719}{3}e^{5} - \frac{4364}{9}e^{4} - \frac{3347}{9}e^{3} + \frac{1505}{9}e^{2} + \frac{356}{9}e - \frac{109}{9}$ |
| 53 | $[53, 53, 4w - 45]$ | $\phantom{-}\frac{131}{9}e^{12} - \frac{532}{9}e^{11} - \frac{1150}{9}e^{10} + \frac{6452}{9}e^{9} + \frac{256}{9}e^{8} - \frac{21203}{9}e^{7} + \frac{7210}{9}e^{6} + \frac{8659}{3}e^{5} - \frac{8846}{9}e^{4} - \frac{11357}{9}e^{3} + \frac{1217}{9}e^{2} + \frac{1886}{9}e + \frac{350}{9}$ |
| 53 | $[53, 53, 4w + 41]$ | $-\frac{23}{9}e^{12} + \frac{64}{9}e^{11} + \frac{322}{9}e^{10} - \frac{881}{9}e^{9} - \frac{1489}{9}e^{8} + \frac{3743}{9}e^{7} + \frac{3374}{9}e^{6} - \frac{2152}{3}e^{5} - \frac{3862}{9}e^{4} + \frac{4454}{9}e^{3} + \frac{1879}{9}e^{2} - \frac{995}{9}e - \frac{323}{9}$ |
| 59 | $[59, 59, -3892w + 43181]$ | $-6e^{12} + 42e^{11} - 25e^{10} - 423e^{9} + 898e^{8} + 688e^{7} - 3059e^{6} + 718e^{5} + 3178e^{4} - 1611e^{3} - 805e^{2} + 342e + 86$ |
| 59 | $[59, 59, 3892w + 39289]$ | $-\frac{13}{9}e^{12} + \frac{125}{9}e^{11} - \frac{223}{9}e^{10} - \frac{1099}{9}e^{9} + \frac{3910}{9}e^{8} + \frac{163}{9}e^{7} - \frac{12428}{9}e^{6} + \frac{2623}{3}e^{5} + \frac{12673}{9}e^{4} - \frac{10673}{9}e^{3} - \frac{3319}{9}e^{2} + \frac{2489}{9}e + \frac{449}{9}$ |
| 61 | $[61, 61, 770w - 8543]$ | $-\frac{19}{9}e^{12} + \frac{164}{9}e^{11} - \frac{211}{9}e^{10} - \frac{1582}{9}e^{9} + \frac{4402}{9}e^{8} + \frac{1897}{9}e^{7} - \frac{14474}{9}e^{6} + \frac{1660}{3}e^{5} + \frac{15145}{9}e^{4} - \frac{8246}{9}e^{3} - \frac{3982}{9}e^{2} + \frac{1817}{9}e + \frac{335}{9}$ |
| 61 | $[61, 61, -770w - 7773]$ | $\phantom{-}\frac{31}{3}e^{12} - \frac{179}{3}e^{11} - \frac{35}{3}e^{10} + \frac{1897}{3}e^{9} - \frac{2704}{3}e^{8} - \frac{4006}{3}e^{7} + \frac{9893}{3}e^{6} - 17e^{5} - \frac{10171}{3}e^{4} + \frac{4112}{3}e^{3} + \frac{2233}{3}e^{2} - \frac{866}{3}e - \frac{176}{3}$ |
| 67 | $[67, 67, 158w + 1595]$ | $\phantom{-}\frac{53}{9}e^{12} - \frac{331}{9}e^{11} + \frac{59}{9}e^{10} + \frac{3395}{9}e^{9} - \frac{6011}{9}e^{8} - \frac{6095}{9}e^{7} + \frac{21067}{9}e^{6} - \frac{1328}{3}e^{5} - \frac{21728}{9}e^{4} + \frac{11590}{9}e^{3} + \frac{5243}{9}e^{2} - \frac{2755}{9}e - \frac{556}{9}$ |
Atkin-Lehner eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| $2$ | $[2,2,-w + 11]$ | $1$ |