Base field \(\Q(\sqrt{82}) \)
Generator \(w\), with minimal polynomial \(x^{2} - 82\); narrow class number \(4\) and class number \(4\).
Form
| Weight: | $[2, 2]$ |
| Level: | $[4, 2, 2]$ |
| Dimension: | $24$ |
| CM: | no |
| Base change: | no |
| Newspace dimension: | $60$ |
Hecke eigenvalues ($q$-expansion)
The Hecke eigenvalue field is $\Q(e)$ where $e$ is a root of the defining polynomial:
| \(x^{24} + 208x^{20} + 12768x^{16} + 232204x^{12} + 359332x^{8} + 119520x^{4} + 5184\) |
Show full eigenvalues Hide large eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| 2 | $[2, 2, w]$ | $\phantom{-}0$ |
| 3 | $[3, 3, w + 1]$ | $\phantom{-}e$ |
| 3 | $[3, 3, w + 2]$ | $-\frac{48427643}{1618751245248}e^{23} - \frac{2519166179}{404687811312}e^{19} - \frac{6448953905}{16861992138}e^{15} - \frac{2823275796473}{404687811312}e^{11} - \frac{4575511502159}{404687811312}e^{7} - \frac{18696404571}{3747109364}e^{3}$ |
| 11 | $[11, 11, w + 4]$ | $-\frac{767180819}{10521883094112}e^{21} - \frac{39843145259}{2630470773528}e^{17} - \frac{101593924517}{109602948897}e^{13} - \frac{43870608081101}{2630470773528}e^{9} - \frac{56234187718391}{2630470773528}e^{5} + \frac{80287782391}{73068632598}e$ |
| 11 | $[11, 11, w + 7]$ | $\phantom{-}\frac{1852662733}{31565649282336}e^{23} + \frac{47988443423}{3945706160292}e^{19} + \frac{973075135213}{1315235386764}e^{15} + \frac{102925920116671}{7891412320584}e^{11} + \frac{82293565494169}{7891412320584}e^{7} - \frac{986996645083}{109602948897}e^{3}$ |
| 13 | $[13, 13, w + 2]$ | $\phantom{-}\frac{68405999185}{252525194258688}e^{23} + \frac{3558816229075}{63131298564672}e^{19} + \frac{18225324522641}{5260941547056}e^{15} + \frac{3992266701255331}{63131298564672}e^{11} + \frac{6508026613762285}{63131298564672}e^{7} + \frac{57017245933561}{1753647182352}e^{3}$ |
| 13 | $[13, 13, w + 11]$ | $-\frac{1125427507}{42087532376448}e^{21} - \frac{58932099145}{10521883094112}e^{17} - \frac{306442043243}{876823591176}e^{13} - \frac{70499946266953}{10521883094112}e^{9} - \frac{191808487363687}{10521883094112}e^{5} - \frac{2075025367723}{292274530392}e$ |
| 19 | $[19, 19, w + 5]$ | $-\frac{1320414671}{5260941547056}e^{21} - \frac{68621838041}{1315235386764}e^{17} - \frac{701082859559}{219205897794}e^{13} - \frac{76142803415645}{1315235386764}e^{9} - \frac{109312307060243}{1315235386764}e^{5} - \frac{728524730366}{36534316299}e$ |
| 19 | $[19, 19, w + 14]$ | $\phantom{-}\frac{58995626539}{63131298564672}e^{23} + \frac{3066511309219}{15782824641168}e^{19} + \frac{3917747090945}{328808846691}e^{15} + \frac{3408685355747857}{15782824641168}e^{11} + \frac{5009060414687119}{15782824641168}e^{7} + \frac{37653893624365}{438411795588}e^{3}$ |
| 23 | $[23, 23, w + 6]$ | $-\frac{1467013807}{42087532376448}e^{22} - \frac{75116407453}{10521883094112}e^{18} - \frac{370044614231}{876823591176}e^{14} - \frac{70415069508253}{10521883094112}e^{10} + \frac{129326339349581}{10521883094112}e^{6} + \frac{1854027625459}{97424843464}e^{2}$ |
| 23 | $[23, 23, w + 17]$ | $\phantom{-}\frac{1467013807}{42087532376448}e^{22} + \frac{75116407453}{10521883094112}e^{18} + \frac{370044614231}{876823591176}e^{14} + \frac{70415069508253}{10521883094112}e^{10} - \frac{129326339349581}{10521883094112}e^{6} - \frac{1854027625459}{97424843464}e^{2}$ |
| 25 | $[25, 5, -5]$ | $-\frac{302050979}{1753647182352}e^{20} - \frac{15656583485}{438411795588}e^{16} - \frac{79483756883}{36534316299}e^{12} - \frac{16909215388385}{438411795588}e^{8} - \frac{16281574088363}{438411795588}e^{4} - \frac{492068989}{12178105433}$ |
| 29 | $[29, 29, w + 13]$ | $-\frac{73144911409}{84175064752896}e^{23} - \frac{3803201850115}{21043766188224}e^{19} - \frac{19450778849345}{1753647182352}e^{15} - \frac{4241915507253571}{21043766188224}e^{11} - \frac{6496026028494541}{21043766188224}e^{7} - \frac{19556158609827}{194849686928}e^{3}$ |
| 29 | $[29, 29, w + 16]$ | $\phantom{-}\frac{2130937915}{14029177458816}e^{21} + \frac{12311308761}{389699373856}e^{17} + \frac{566706439079}{292274530392}e^{13} + \frac{123616406656609}{3507294364704}e^{9} + \frac{21159716998519}{389699373856}e^{5} + \frac{2132681102723}{97424843464}e$ |
| 31 | $[31, 31, w + 12]$ | $-\frac{2316299215}{3237502490496}e^{22} - \frac{120338711677}{809375622624}e^{18} - \frac{614251450091}{67447968552}e^{14} - \frac{133080699261181}{809375622624}e^{10} - \frac{183291647687539}{809375622624}e^{6} - \frac{373522602189}{7494218728}e^{2}$ |
| 31 | $[31, 31, w + 19]$ | $\phantom{-}\frac{2316299215}{3237502490496}e^{22} + \frac{120338711677}{809375622624}e^{18} + \frac{614251450091}{67447968552}e^{14} + \frac{133080699261181}{809375622624}e^{10} + \frac{183291647687539}{809375622624}e^{6} + \frac{373522602189}{7494218728}e^{2}$ |
| 41 | $[41, 41, w]$ | $\phantom{-}0$ |
| 49 | $[49, 7, -7]$ | $\phantom{-}\frac{614529805}{3507294364704}e^{20} + \frac{31902210169}{876823591176}e^{16} + \frac{162549407465}{73068632598}e^{12} + \frac{35009828041891}{876823591176}e^{8} + \frac{43427127345961}{876823591176}e^{4} + \frac{273867373513}{24356210866}$ |
| 53 | $[53, 53, w + 20]$ | $\phantom{-}\frac{8453215447}{9352784972544}e^{23} + \frac{1317811555007}{7014588729408}e^{19} + \frac{2243429786423}{194849686928}e^{15} + \frac{486971738474453}{2338196243136}e^{11} + \frac{2079124449931793}{7014588729408}e^{7} + \frac{15848103358909}{194849686928}e^{3}$ |
| 53 | $[53, 53, w + 33]$ | $-\frac{3846654517}{14029177458816}e^{21} - \frac{66566996965}{1169098121568}e^{17} - \frac{1017603146237}{292274530392}e^{13} - \frac{219218811724879}{3507294364704}e^{9} - \frac{91375561455499}{1169098121568}e^{5} - \frac{1839974838381}{97424843464}e$ |
Atkin-Lehner eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| $2$ | $[2, 2, w]$ | $1$ |