Base field 6.6.703493.1
Generator \(w\), with minimal polynomial \(x^{6} - 2x^{5} - 5x^{4} + 11x^{3} + 2x^{2} - 9x + 1\); narrow class number \(1\) and class number \(1\).
Form
| Weight: | $[2, 2, 2, 2, 2, 2]$ |
| Level: | $[41,41,4w^{5} - 3w^{4} - 23w^{3} + 14w^{2} + 22w - 4]$ |
| Dimension: | $8$ |
| CM: | no |
| Base change: | no |
| Newspace dimension: | $11$ |
Hecke eigenvalues ($q$-expansion)
The Hecke eigenvalue field is $\Q(e)$ where $e$ is a root of the defining polynomial:
| \(x^{8} - 7x^{7} - 32x^{6} + 263x^{5} + 294x^{4} - 2952x^{3} - 1792x^{2} + 11040x + 8896\) |
Show full eigenvalues Hide large eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| 13 | $[13, 13, 3w^{5} - 2w^{4} - 17w^{3} + 10w^{2} + 16w - 3]$ | $-\frac{1049}{123248}e^{7} + \frac{2069}{123248}e^{6} + \frac{24893}{61624}e^{5} - \frac{50195}{123248}e^{4} - \frac{197469}{30812}e^{3} - \frac{943}{15406}e^{2} + \frac{271564}{7703}e + \frac{224197}{7703}$ |
| 13 | $[13, 13, w^{5} - w^{4} - 5w^{3} + 5w^{2} + 3w - 3]$ | $\phantom{-}\frac{399}{123248}e^{7} + \frac{3663}{123248}e^{6} - \frac{5655}{15406}e^{5} - \frac{108591}{123248}e^{4} + \frac{537477}{61624}e^{3} + \frac{108935}{15406}e^{2} - \frac{778803}{15406}e - \frac{292207}{7703}$ |
| 13 | $[13, 13, 2w^{5} - w^{4} - 12w^{3} + 4w^{2} + 13w]$ | $-\frac{1735}{61624}e^{7} + \frac{7123}{61624}e^{6} + \frac{30539}{30812}e^{5} - \frac{200805}{61624}e^{4} - \frac{162059}{15406}e^{3} + \frac{125426}{7703}e^{2} + \frac{352469}{7703}e + \frac{203398}{7703}$ |
| 13 | $[13, 13, -w^{2} + 3]$ | $\phantom{-}e$ |
| 41 | $[41, 41, -2w^{5} + w^{4} + 11w^{3} - 4w^{2} - 10w - 1]$ | $-\frac{139}{15406}e^{7} + \frac{5035}{30812}e^{6} - \frac{4337}{15406}e^{5} - \frac{153403}{30812}e^{4} + \frac{424381}{30812}e^{3} + \frac{1113893}{30812}e^{2} - \frac{628813}{7703}e - \frac{656863}{7703}$ |
| 41 | $[41, 41, -4w^{5} + 3w^{4} + 23w^{3} - 14w^{2} - 22w + 4]$ | $-1$ |
| 41 | $[41, 41, w^{5} - 6w^{3} + w^{2} + 7w - 2]$ | $-\frac{555}{15406}e^{7} + \frac{551}{7703}e^{6} + \frac{25687}{15406}e^{5} - \frac{30581}{15406}e^{4} - \frac{384441}{15406}e^{3} + \frac{49693}{7703}e^{2} + \frac{955511}{7703}e + \frac{623340}{7703}$ |
| 41 | $[41, 41, -3w^{5} + w^{4} + 18w^{3} - 4w^{2} - 20w - 1]$ | $-\frac{1287}{61624}e^{7} + \frac{1361}{30812}e^{6} + \frac{60149}{61624}e^{5} - \frac{69749}{61624}e^{4} - \frac{918219}{61624}e^{3} + \frac{24297}{30812}e^{2} + \frac{600229}{7703}e + \frac{525630}{7703}$ |
| 43 | $[43, 43, 3w^{5} - 2w^{4} - 17w^{3} + 11w^{2} + 16w - 6]$ | $-\frac{3867}{123248}e^{7} + \frac{23459}{123248}e^{6} + \frac{49703}{61624}e^{5} - \frac{695741}{123248}e^{4} - \frac{21205}{7703}e^{3} + \frac{1106869}{30812}e^{2} - \frac{50801}{15406}e - \frac{263109}{7703}$ |
| 43 | $[43, 43, -2w^{5} + w^{4} + 12w^{3} - 6w^{2} - 14w + 5]$ | $\phantom{-}\frac{187}{15406}e^{7} - \frac{3227}{30812}e^{6} - \frac{1045}{7703}e^{5} + \frac{95653}{30812}e^{4} - \frac{124045}{30812}e^{3} - \frac{648335}{30812}e^{2} + \frac{232100}{7703}e + \frac{370197}{7703}$ |
| 49 | $[49, 7, 2w^{5} - w^{4} - 12w^{3} + 5w^{2} + 13w - 4]$ | $-\frac{1963}{61624}e^{7} + \frac{18235}{61624}e^{6} + \frac{8251}{30812}e^{5} - \frac{539309}{61624}e^{4} + \frac{101339}{7703}e^{3} + \frac{902157}{15406}e^{2} - \frac{746402}{7703}e - \frac{913016}{7703}$ |
| 64 | $[64, 2, -2]$ | $\phantom{-}\frac{2983}{123248}e^{7} - \frac{14431}{123248}e^{6} - \frac{44763}{61624}e^{5} + \frac{420633}{123248}e^{4} + \frac{79935}{15406}e^{3} - \frac{644281}{30812}e^{2} - \frac{115601}{7703}e + \frac{118235}{7703}$ |
| 71 | $[71, 71, -2w^{5} + w^{4} + 12w^{3} - 4w^{2} - 12w - 1]$ | $-\frac{1295}{123248}e^{7} + \frac{5139}{123248}e^{6} + \frac{23549}{61624}e^{5} - \frac{153521}{123248}e^{4} - \frac{133747}{30812}e^{3} + \frac{228725}{30812}e^{2} + \frac{314095}{15406}e + \frac{62411}{7703}$ |
| 71 | $[71, 71, -3w^{5} + 2w^{4} + 17w^{3} - 9w^{2} - 15w]$ | $-\frac{2325}{61624}e^{7} + \frac{2711}{7703}e^{6} + \frac{21001}{61624}e^{5} - \frac{640255}{61624}e^{4} + \frac{895269}{61624}e^{3} + \frac{2087289}{30812}e^{2} - \frac{795292}{7703}e - \frac{885904}{7703}$ |
| 83 | $[83, 83, 4w^{5} - 2w^{4} - 24w^{3} + 9w^{2} + 26w - 1]$ | $\phantom{-}\frac{1735}{61624}e^{7} - \frac{7123}{61624}e^{6} - \frac{30539}{30812}e^{5} + \frac{200805}{61624}e^{4} + \frac{162059}{15406}e^{3} - \frac{125426}{7703}e^{2} - \frac{360172}{7703}e - \frac{172586}{7703}$ |
| 83 | $[83, 83, -4w^{5} + 2w^{4} + 23w^{3} - 9w^{2} - 23w]$ | $-\frac{105}{61624}e^{7} + \frac{937}{30812}e^{6} - \frac{4717}{61624}e^{5} - \frac{55751}{61624}e^{4} + \frac{229975}{61624}e^{3} + \frac{224669}{30812}e^{2} - \frac{217906}{7703}e - \frac{245952}{7703}$ |
| 97 | $[97, 97, -w^{5} + 7w^{3} + w^{2} - 11w - 3]$ | $\phantom{-}\frac{2965}{61624}e^{7} - \frac{22473}{61624}e^{6} - \frac{23819}{30812}e^{5} + \frac{655811}{61624}e^{4} - \frac{141145}{15406}e^{3} - \frac{1034163}{15406}e^{2} + \frac{638636}{7703}e + \frac{782816}{7703}$ |
| 97 | $[97, 97, -2w^{5} + w^{4} + 11w^{3} - 6w^{2} - 9w + 3]$ | $-\frac{2787}{123248}e^{7} - \frac{2419}{123248}e^{6} + \frac{9933}{7703}e^{5} + \frac{106587}{123248}e^{4} - \frac{1384281}{61624}e^{3} - \frac{104854}{7703}e^{2} + \frac{1741543}{15406}e + \frac{786103}{7703}$ |
| 113 | $[113, 113, -4w^{5} + 2w^{4} + 24w^{3} - 9w^{2} - 27w + 2]$ | $\phantom{-}\frac{5315}{123248}e^{7} - \frac{6459}{123248}e^{6} - \frac{135731}{61624}e^{5} + \frac{175165}{123248}e^{4} + \frac{275448}{7703}e^{3} - \frac{8971}{30812}e^{2} - \frac{2804027}{15406}e - \frac{1039001}{7703}$ |
| 113 | $[113, 113, 3w^{5} - 2w^{4} - 18w^{3} + 10w^{2} + 20w - 6]$ | $-\frac{2421}{61624}e^{7} + \frac{12177}{61624}e^{6} + \frac{36185}{30812}e^{5} - \frac{351415}{61624}e^{4} - \frac{126649}{15406}e^{3} + \frac{244092}{7703}e^{2} + \frac{177216}{7703}e + \frac{50838}{7703}$ |
Atkin-Lehner eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| $41$ | $[41,41,4w^{5} - 3w^{4} - 23w^{3} + 14w^{2} + 22w - 4]$ | $1$ |