Base field 4.4.17725.1
Generator \(w\), with minimal polynomial \(x^{4} - 2x^{3} - 12x^{2} + 13x + 41\); narrow class number \(1\) and class number \(1\).
Form
| Weight: | $[2, 2, 2, 2]$ |
| Level: | $[9, 3, -w^{3} + 3w^{2} + 5w - 15]$ |
| Dimension: | $8$ |
| CM: | no |
| Base change: | no |
| Newspace dimension: | $13$ |
Hecke eigenvalues ($q$-expansion)
The Hecke eigenvalue field is $\Q(e)$ where $e$ is a root of the defining polynomial:
| \(x^{8} - 4x^{7} - 49x^{6} + 190x^{5} + 790x^{4} - 2809x^{3} - 4804x^{2} + 12608x + 10144\) |
Show full eigenvalues Hide large eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| 9 | $[9, 3, -w^{3} + 3w^{2} + 5w - 15]$ | $-1$ |
| 9 | $[9, 3, -w^{3} + 8w + 8]$ | $\phantom{-}e$ |
| 16 | $[16, 2, 2]$ | $-\frac{9958}{5959189}e^{7} + \frac{81083}{11918378}e^{6} + \frac{498967}{5959189}e^{5} - \frac{2843193}{11918378}e^{4} - \frac{7674421}{5959189}e^{3} + \frac{13620547}{5959189}e^{2} + \frac{69077153}{11918378}e - \frac{26609499}{5959189}$ |
| 19 | $[19, 19, w + 1]$ | $-\frac{30713}{47673512}e^{7} - \frac{62619}{11918378}e^{6} + \frac{1851921}{47673512}e^{5} + \frac{4075249}{23836756}e^{4} - \frac{14169715}{23836756}e^{3} - \frac{46607231}{47673512}e^{2} + \frac{24929119}{11918378}e - \frac{27690616}{5959189}$ |
| 19 | $[19, 19, -w^{2} + 6]$ | $-\frac{128221}{11918378}e^{7} - \frac{13521}{5959189}e^{6} + \frac{5552873}{11918378}e^{5} + \frac{391582}{5959189}e^{4} - \frac{38221220}{5959189}e^{3} - \frac{14351753}{11918378}e^{2} + \frac{168381325}{5959189}e + \frac{75690684}{5959189}$ |
| 19 | $[19, 19, -w^{2} + 2w + 5]$ | $\phantom{-}\frac{263797}{23836756}e^{7} + \frac{260569}{11918378}e^{6} - \frac{10957837}{23836756}e^{5} - \frac{4409595}{5959189}e^{4} + \frac{71549953}{11918378}e^{3} + \frac{164930719}{23836756}e^{2} - \frac{298791219}{11918378}e - \frac{90157606}{5959189}$ |
| 19 | $[19, 19, -w + 2]$ | $-\frac{1000275}{23836756}e^{7} + \frac{51250}{5959189}e^{6} + \frac{48785827}{23836756}e^{5} - \frac{440885}{11918378}e^{4} - \frac{379861173}{11918378}e^{3} - \frac{203635001}{23836756}e^{2} + \frac{936921521}{5959189}e + \frac{658908748}{5959189}$ |
| 25 | $[25, 5, 2w^{2} - 2w - 13]$ | $-\frac{462761}{23836756}e^{7} + \frac{47987}{5959189}e^{6} + \frac{23553277}{23836756}e^{5} - \frac{1086639}{11918378}e^{4} - \frac{193968787}{11918378}e^{3} - \frac{117527927}{23836756}e^{2} + \frac{518213016}{5959189}e + \frac{387230824}{5959189}$ |
| 29 | $[29, 29, -w^{2} + 9]$ | $\phantom{-}\frac{1124857}{23836756}e^{7} + \frac{105157}{5959189}e^{6} - \frac{53047465}{23836756}e^{5} - \frac{10409679}{11918378}e^{4} + \frac{399212801}{11918378}e^{3} + \frac{401931943}{23836756}e^{2} - \frac{956112619}{5959189}e - \frac{705381118}{5959189}$ |
| 29 | $[29, 29, -w^{2} + 2w + 6]$ | $-\frac{841045}{23836756}e^{7} - \frac{98698}{5959189}e^{6} + \frac{39494305}{23836756}e^{5} + \frac{8776819}{11918378}e^{4} - \frac{295628493}{11918378}e^{3} - \frac{309059679}{23836756}e^{2} + \frac{700704953}{5959189}e + \frac{517510848}{5959189}$ |
| 29 | $[29, 29, w^{2} - 7]$ | $\phantom{-}\frac{304999}{23836756}e^{7} + \frac{202729}{11918378}e^{6} - \frac{12666883}{23836756}e^{5} - \frac{3430566}{5959189}e^{4} + \frac{78006787}{11918378}e^{3} + \frac{140101257}{23836756}e^{2} - \frac{259724267}{11918378}e - \frac{95244756}{5959189}$ |
| 29 | $[29, 29, -w^{2} + 2w + 8]$ | $-\frac{4374409}{47673512}e^{7} + \frac{46145}{11918378}e^{6} + \frac{213517825}{47673512}e^{5} + \frac{8169729}{23836756}e^{4} - \frac{1672403227}{23836756}e^{3} - \frac{994608215}{47673512}e^{2} + \frac{4177350785}{11918378}e + \frac{1387596990}{5959189}$ |
| 31 | $[31, 31, -2w^{2} + w + 12]$ | $\phantom{-}\frac{570967}{23836756}e^{7} + \frac{82599}{5959189}e^{6} - \frac{26510827}{23836756}e^{5} - \frac{6491273}{11918378}e^{4} + \frac{194196463}{11918378}e^{3} + \frac{192019757}{23836756}e^{2} - \frac{442758921}{5959189}e - \frac{267771612}{5959189}$ |
| 31 | $[31, 31, 2w^{2} - 3w - 11]$ | $\phantom{-}\frac{514957}{23836756}e^{7} - \frac{109790}{5959189}e^{6} - \frac{26405649}{23836756}e^{5} + \frac{5467653}{11918378}e^{4} + \frac{220361915}{11918378}e^{3} + \frac{56898867}{23836756}e^{2} - \frac{592366901}{5959189}e - \frac{445831380}{5959189}$ |
| 41 | $[41, 41, -w]$ | $\phantom{-}\frac{564413}{23836756}e^{7} - \frac{194836}{5959189}e^{6} - \frac{29831665}{23836756}e^{5} + \frac{11618937}{11918378}e^{4} + \frac{252620681}{11918378}e^{3} - \frac{63035645}{23836756}e^{2} - \frac{685277056}{5959189}e - \frac{393365126}{5959189}$ |
| 41 | $[41, 41, -w + 1]$ | $\phantom{-}\frac{1959819}{47673512}e^{7} + \frac{110931}{11918378}e^{6} - \frac{93710907}{47673512}e^{5} - \frac{14190675}{23836756}e^{4} + \frac{713679085}{23836756}e^{3} + \frac{645410941}{47673512}e^{2} - \frac{1719793497}{11918378}e - \frac{597978664}{5959189}$ |
| 49 | $[49, 7, w^{3} + 2w^{2} - 10w - 20]$ | $-\frac{819057}{23836756}e^{7} - \frac{164464}{5959189}e^{6} + \frac{37207653}{23836756}e^{5} + \frac{13622731}{11918378}e^{4} - \frac{267674283}{11918378}e^{3} - \frac{380658875}{23836756}e^{2} + \frac{602718066}{5959189}e + \frac{470075926}{5959189}$ |
| 49 | $[49, 7, w^{3} - 5w^{2} - 3w + 27]$ | $\phantom{-}\frac{666673}{5959189}e^{7} - \frac{56257}{11918378}e^{6} - \frac{32737832}{5959189}e^{5} - \frac{5920985}{11918378}e^{4} + \frac{514181625}{5959189}e^{3} + \frac{166947094}{5959189}e^{2} - \frac{5130561831}{11918378}e - \frac{1784793036}{5959189}$ |
| 61 | $[61, 61, 2w^{2} - 3w - 14]$ | $\phantom{-}\frac{2378201}{47673512}e^{7} - \frac{241745}{11918378}e^{6} - \frac{121718385}{47673512}e^{5} + \frac{9326599}{23836756}e^{4} + \frac{1001960327}{23836756}e^{3} + \frac{408987287}{47673512}e^{2} - \frac{2630510099}{11918378}e - \frac{870649382}{5959189}$ |
| 61 | $[61, 61, 2w^{2} - w - 15]$ | $-\frac{3414929}{47673512}e^{7} + \frac{94689}{11918378}e^{6} + \frac{163846793}{47673512}e^{5} + \frac{887137}{23836756}e^{4} - \frac{1260430779}{23836756}e^{3} - \frac{618002583}{47673512}e^{2} + \frac{3097908161}{11918378}e + \frac{958123298}{5959189}$ |
Atkin-Lehner eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| $9$ | $[9, 3, -w^{3} + 3w^{2} + 5w - 15]$ | $1$ |