Base field \(\Q(\sqrt{79}) \)
Generator \(w\), with minimal polynomial \(x^{2} - 79\); narrow class number \(6\) and class number \(3\).
Form
| Weight: | $[2, 2]$ |
| Level: | $[4, 2, 2]$ |
| Dimension: | $16$ |
| CM: | no |
| Base change: | no |
| Newspace dimension: | $42$ |
Hecke eigenvalues ($q$-expansion)
The Hecke eigenvalue field is $\Q(e)$ where $e$ is a root of the defining polynomial:
| \(x^{16} + 16x^{14} + 202x^{12} + 832x^{10} + 2659x^{8} + 832x^{6} + 202x^{4} + 16x^{2} + 1\) |
Show full eigenvalues Hide large eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| 2 | $[2, 2, -w + 9]$ | $\phantom{-}0$ |
| 3 | $[3, 3, w + 1]$ | $-e^{15} - 16e^{13} - 202e^{11} - 832e^{9} - 2659e^{7} - 832e^{5} - 202e^{3} - 16e$ |
| 3 | $[3, 3, w + 2]$ | $\phantom{-}e$ |
| 5 | $[5, 5, w + 2]$ | $\phantom{-}\frac{747}{3404}e^{14} + \frac{404061}{114034}e^{12} + \frac{457843}{10212}e^{10} + \frac{1929331}{10212}e^{8} + \frac{416245465}{684204}e^{6} + \frac{2656679}{10212}e^{4} + \frac{236125}{5106}e^{2} + \frac{2506307}{684204}$ |
| 5 | $[5, 5, w + 3]$ | $\phantom{-}\frac{92919}{228068}e^{14} + \frac{4444055}{684204}e^{12} + \frac{209132}{2553}e^{10} + \frac{228609257}{684204}e^{8} + \frac{181735708}{171051}e^{6} + \frac{2770999}{10212}e^{4} + \frac{32248075}{684204}e^{2} + \frac{156849}{114034}$ |
| 7 | $[7, 7, w + 3]$ | $\phantom{-}\frac{68025}{114034}e^{15} + \frac{6409481}{684204}e^{13} + \frac{1201775}{10212}e^{11} + \frac{78818300}{171051}e^{9} + \frac{986159605}{684204}e^{7} + \frac{90700}{2553}e^{5} + \frac{1927375}{684204}e^{3} - \frac{3621263}{228068}e$ |
| 7 | $[7, 7, w + 4]$ | $-\frac{216077}{684204}e^{15} - \frac{1748875}{342102}e^{13} - \frac{660953}{10212}e^{11} - \frac{187769365}{684204}e^{9} - \frac{605846647}{684204}e^{7} - \frac{4144207}{10212}e^{5} - \frac{7676771}{114034}e^{3} - \frac{1216189}{228068}e$ |
| 13 | $[13, 13, w + 1]$ | $-\frac{4295}{228068}e^{14} - \frac{28975}{114034}e^{12} - \frac{31273}{10212}e^{10} - \frac{4346275}{684204}e^{8} - \frac{9303535}{684204}e^{6} + \frac{1002655}{10212}e^{4} - \frac{339265}{342102}e^{2} - \frac{53585}{684204}$ |
| 13 | $[13, 13, w + 12]$ | $-\frac{142077}{228068}e^{14} - \frac{6791585}{684204}e^{12} - \frac{319604}{2553}e^{10} - \frac{349024667}{684204}e^{8} - \frac{277735876}{171051}e^{6} - \frac{4234753}{10212}e^{4} - \frac{73825669}{684204}e^{2} - \frac{239703}{114034}$ |
| 43 | $[43, 43, -w - 6]$ | $-\frac{68882}{57017}e^{15} - \frac{2196145}{114034}e^{13} - \frac{206696}{851}e^{11} - \frac{112959321}{114034}e^{9} - \frac{179618824}{57017}e^{7} - \frac{1369361}{1702}e^{5} - \frac{8447031}{57017}e^{3} - \frac{232533}{57017}e$ |
| 43 | $[43, 43, w - 6]$ | $\phantom{-}\frac{354447}{114034}e^{15} + \frac{2825315}{57017}e^{13} + \frac{531824}{851}e^{11} + \frac{145333824}{57017}e^{9} + \frac{462155056}{57017}e^{7} + \frac{1761667}{851}e^{5} + \frac{41215067}{114034}e^{3} + \frac{598302}{57017}e$ |
| 47 | $[47, 47, w + 19]$ | $\phantom{-}\frac{176523}{114034}e^{15} + \frac{16632361}{684204}e^{13} + \frac{3118573}{10212}e^{11} + \frac{204531316}{171051}e^{9} + \frac{2559152531}{684204}e^{7} + \frac{235364}{2553}e^{5} + \frac{5001485}{684204}e^{3} - \frac{9567283}{228068}e$ |
| 47 | $[47, 47, w + 28]$ | $\phantom{-}\frac{1723103}{684204}e^{15} + \frac{13898611}{342102}e^{13} + \frac{5248367}{10212}e^{11} + \frac{1478516989}{684204}e^{9} + \frac{4757519149}{684204}e^{7} + \frac{29603395}{10212}e^{5} + \frac{60270017}{114034}e^{3} + \frac{9548101}{228068}e$ |
| 59 | $[59, 59, w + 16]$ | $\phantom{-}\frac{10155}{57017}e^{15} + \frac{956759}{342102}e^{13} + \frac{179405}{5106}e^{11} + \frac{23532520}{171051}e^{9} + \frac{147324865}{342102}e^{7} + \frac{27080}{2553}e^{5} + \frac{287725}{342102}e^{3} - \frac{3049617}{114034}e$ |
| 59 | $[59, 59, w + 43]$ | $\phantom{-}\frac{355117}{342102}e^{15} + \frac{2856425}{171051}e^{13} + \frac{1077913}{5106}e^{11} + \frac{301568615}{342102}e^{9} + \frac{968184887}{342102}e^{7} + \frac{5525387}{5106}e^{5} + \frac{12263091}{57017}e^{3} + \frac{1942719}{114034}e$ |
| 71 | $[71, 71, w + 24]$ | $-\frac{1610515}{684204}e^{15} - \frac{12935357}{342102}e^{13} - \frac{4879603}{10212}e^{11} - \frac{1360164323}{684204}e^{9} - \frac{4361514857}{684204}e^{7} - \frac{23691365}{10212}e^{5} - \frac{55237821}{114034}e^{3} - \frac{8750715}{228068}e$ |
| 71 | $[71, 71, w + 47]$ | $\phantom{-}\frac{47715}{114034}e^{15} + \frac{4495963}{684204}e^{13} + \frac{842965}{10212}e^{11} + \frac{55285780}{171051}e^{9} + \frac{691509875}{684204}e^{7} + \frac{63620}{2553}e^{5} + \frac{1351925}{684204}e^{3} + \frac{2706039}{228068}e$ |
| 73 | $[73, 73, 3w - 28]$ | $\phantom{-}\frac{29412}{57017}e^{14} + \frac{923763}{114034}e^{12} + \frac{86602}{851}e^{10} + \frac{22719136}{57017}e^{8} + \frac{71060006}{57017}e^{6} + \frac{26144}{851}e^{4} + \frac{138890}{57017}e^{2} - \frac{956123}{114034}$ |
| 73 | $[73, 73, 12w - 107]$ | $-\frac{29412}{57017}e^{14} - \frac{923763}{114034}e^{12} - \frac{86602}{851}e^{10} - \frac{22719136}{57017}e^{8} - \frac{71060006}{57017}e^{6} - \frac{26144}{851}e^{4} - \frac{138890}{57017}e^{2} + \frac{1412259}{114034}$ |
| 79 | $[79, 79, -w]$ | $-\frac{29583}{4958}e^{15} - \frac{471495}{4958}e^{13} - \frac{44376}{37}e^{11} - \frac{24241943}{4958}e^{9} - \frac{38562744}{2479}e^{7} - \frac{293991}{74}e^{5} - \frac{4320531}{4958}e^{3} - \frac{49923}{2479}e$ |
Atkin-Lehner eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| $2$ | $[2, 2, -w + 9]$ | $-1$ |