Base field 4.4.19821.1
Generator \(w\), with minimal polynomial \(x^{4} - x^{3} - 8x^{2} + 6x + 3\); narrow class number \(1\) and class number \(1\).
Form
| Weight: | $[2, 2, 2, 2]$ |
| Level: | $[21, 21, \frac{2}{3}w^{3} - \frac{1}{3}w^{2} - 5w - 1]$ |
| Dimension: | $9$ |
| CM: | no |
| Base change: | no |
| Newspace dimension: | $37$ |
Hecke eigenvalues ($q$-expansion)
The Hecke eigenvalue field is $\Q(e)$ where $e$ is a root of the defining polynomial:
| \(x^{9} + x^{8} - 42x^{7} - 31x^{6} + 539x^{5} + 196x^{4} - 2176x^{3} - 92x^{2} + 376x - 56\) |
Show full eigenvalues Hide large eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| 3 | $[3, 3, w]$ | $\phantom{-}1$ |
| 7 | $[7, 7, \frac{1}{3}w^{3} - \frac{2}{3}w^{2} - 2w + 2]$ | $\phantom{-}1$ |
| 9 | $[9, 3, w + 1]$ | $\phantom{-}e$ |
| 13 | $[13, 13, \frac{2}{3}w^{3} - \frac{1}{3}w^{2} - 5w]$ | $-\frac{749849}{1554452}e^{8} - \frac{917617}{1554452}e^{7} + \frac{15654677}{777226}e^{6} + \frac{30365471}{1554452}e^{5} - \frac{397989303}{1554452}e^{4} - \frac{59848882}{388613}e^{3} + \frac{395344644}{388613}e^{2} + \frac{110673737}{388613}e - \frac{44701380}{388613}$ |
| 16 | $[16, 2, 2]$ | $\phantom{-}\frac{470185}{1554452}e^{8} + \frac{585853}{1554452}e^{7} - \frac{9805199}{777226}e^{6} - \frac{19331117}{1554452}e^{5} + \frac{248898863}{1554452}e^{4} + \frac{37857019}{388613}e^{3} - \frac{494165597}{777226}e^{2} - \frac{68882781}{388613}e + \frac{29505633}{388613}$ |
| 17 | $[17, 17, \frac{1}{3}w^{3} - \frac{2}{3}w^{2} - 3w + 5]$ | $-\frac{110952}{388613}e^{8} - \frac{139270}{388613}e^{7} + \frac{9260029}{777226}e^{6} + \frac{9239783}{777226}e^{5} - \frac{58821481}{388613}e^{4} - \frac{73415997}{777226}e^{3} + \frac{467757443}{777226}e^{2} + \frac{69279518}{388613}e - \frac{26670730}{388613}$ |
| 19 | $[19, 19, -\frac{1}{3}w^{3} + \frac{2}{3}w^{2} + 2w - 5]$ | $-\frac{319358}{388613}e^{8} - \frac{391270}{388613}e^{7} + \frac{13322658}{388613}e^{6} + \frac{12916627}{388613}e^{5} - \frac{169147391}{388613}e^{4} - \frac{101260463}{388613}e^{3} + \frac{671405224}{388613}e^{2} + \frac{185046398}{388613}e - \frac{76587710}{388613}$ |
| 23 | $[23, 23, -\frac{1}{3}w^{3} + \frac{2}{3}w^{2} + 3w - 2]$ | $-\frac{194293}{1554452}e^{8} - \frac{244001}{1554452}e^{7} + \frac{2022005}{388613}e^{6} + \frac{7972569}{1554452}e^{5} - \frac{102414923}{1554452}e^{4} - \frac{30288693}{777226}e^{3} + \frac{203332461}{777226}e^{2} + \frac{24222600}{388613}e - \frac{12697098}{388613}$ |
| 25 | $[25, 5, \frac{1}{3}w^{3} + \frac{1}{3}w^{2} - 3w]$ | $\phantom{-}\frac{1300909}{1554452}e^{8} + \frac{1618093}{1554452}e^{7} - \frac{13554504}{388613}e^{6} - \frac{53433265}{1554452}e^{5} + \frac{687465023}{1554452}e^{4} + \frac{209438021}{777226}e^{3} - \frac{1362976747}{777226}e^{2} - \frac{190065044}{388613}e + \frac{78581854}{388613}$ |
| 25 | $[25, 5, -\frac{2}{3}w^{3} + \frac{1}{3}w^{2} + 5w - 3]$ | $-\frac{42323}{388613}e^{8} - \frac{93823}{777226}e^{7} + \frac{3570197}{777226}e^{6} + \frac{1548140}{388613}e^{5} - \frac{45991147}{777226}e^{4} - \frac{24259011}{777226}e^{3} + \frac{92310839}{388613}e^{2} + \frac{22494190}{388613}e - \frac{9045968}{388613}$ |
| 29 | $[29, 29, -\frac{2}{3}w^{3} + \frac{1}{3}w^{2} + 4w - 3]$ | $-\frac{131453}{1554452}e^{8} - \frac{179637}{1554452}e^{7} + \frac{2726041}{777226}e^{6} + \frac{6004287}{1554452}e^{5} - \frac{68557415}{1554452}e^{4} - \frac{12175894}{388613}e^{3} + \frac{67037564}{388613}e^{2} + \frac{23893651}{388613}e - \frac{5831580}{388613}$ |
| 29 | $[29, 29, -\frac{1}{3}w^{3} + \frac{2}{3}w^{2} + 2w]$ | $-\frac{441413}{777226}e^{8} - \frac{275149}{388613}e^{7} + \frac{18390839}{777226}e^{6} + \frac{18191505}{777226}e^{5} - \frac{116525972}{388613}e^{4} - \frac{143162139}{777226}e^{3} + \frac{461654450}{388613}e^{2} + \frac{131665765}{388613}e - \frac{52967172}{388613}$ |
| 37 | $[37, 37, \frac{1}{3}w^{3} + \frac{1}{3}w^{2} - 2w - 3]$ | $\phantom{-}\frac{441413}{777226}e^{8} + \frac{275149}{388613}e^{7} - \frac{18390839}{777226}e^{6} - \frac{18191505}{777226}e^{5} + \frac{116525972}{388613}e^{4} + \frac{143162139}{777226}e^{3} - \frac{461654450}{388613}e^{2} - \frac{131665765}{388613}e + \frac{53744398}{388613}$ |
| 41 | $[41, 41, -\frac{2}{3}w^{3} + \frac{1}{3}w^{2} + 5w - 4]$ | $\phantom{-}\frac{2353027}{1554452}e^{8} + \frac{2906335}{1554452}e^{7} - \frac{24535022}{388613}e^{6} - \frac{96008835}{1554452}e^{5} + \frac{1245551205}{1554452}e^{4} + \frac{376859171}{777226}e^{3} - \frac{2470907325}{777226}e^{2} - \frac{344224999}{388613}e + \frac{142926672}{388613}$ |
| 43 | $[43, 43, \frac{2}{3}w^{3} - \frac{1}{3}w^{2} - 6w]$ | $\phantom{-}\frac{832247}{777226}e^{8} + \frac{512435}{388613}e^{7} - \frac{34723215}{777226}e^{6} - \frac{33799197}{777226}e^{5} + \frac{220465560}{388613}e^{4} + \frac{263985725}{777226}e^{3} - \frac{875465443}{388613}e^{2} - \frac{236781834}{388613}e + \frac{101102146}{388613}$ |
| 47 | $[47, 47, \frac{2}{3}w^{3} - \frac{1}{3}w^{2} - 4w]$ | $-\frac{628015}{1554452}e^{8} - \frac{794263}{1554452}e^{7} + \frac{6538205}{388613}e^{6} + \frac{26186955}{1554452}e^{5} - \frac{331069217}{1554452}e^{4} - \frac{102363867}{777226}e^{3} + \frac{654648879}{777226}e^{2} + \frac{92379695}{388613}e - \frac{36617664}{388613}$ |
| 59 | $[59, 59, \frac{1}{3}w^{3} + \frac{1}{3}w^{2} - 2w - 4]$ | $\phantom{-}\frac{2415867}{1554452}e^{8} + \frac{2970699}{1554452}e^{7} - \frac{50388013}{777226}e^{6} - \frac{97977117}{1554452}e^{5} + \frac{1279408713}{1554452}e^{4} + \frac{191398038}{388613}e^{3} - \frac{1269693716}{388613}e^{2} - \frac{344942561}{388613}e + \frac{145128834}{388613}$ |
| 59 | $[59, 59, \frac{4}{3}w^{3} - \frac{5}{3}w^{2} - 10w + 9]$ | $\phantom{-}\frac{500389}{388613}e^{8} + \frac{1221557}{777226}e^{7} - \frac{41758133}{777226}e^{6} - \frac{20150667}{388613}e^{5} + \frac{530327039}{777226}e^{4} + \frac{315078575}{777226}e^{3} - \frac{1052705070}{388613}e^{2} - \frac{284062670}{388613}e + \frac{120233042}{388613}$ |
| 67 | $[67, 67, -\frac{1}{3}w^{3} + \frac{2}{3}w^{2} + 3w]$ | $-\frac{26377}{777226}e^{8} - \frac{28773}{777226}e^{7} + \frac{545170}{388613}e^{6} + \frac{851551}{777226}e^{5} - \frac{13612939}{777226}e^{4} - \frac{2298041}{388613}e^{3} + \frac{26408154}{388613}e^{2} - \frac{793474}{388613}e - \frac{3338128}{388613}$ |
| 71 | $[71, 71, \frac{2}{3}w^{3} - \frac{1}{3}w^{2} - 4w + 1]$ | $-\frac{275493}{388613}e^{8} - \frac{342507}{388613}e^{7} + \frac{11484949}{388613}e^{6} + \frac{22596939}{777226}e^{5} - \frac{145643645}{388613}e^{4} - \frac{88318143}{388613}e^{3} + \frac{1153727805}{777226}e^{2} + \frac{158393251}{388613}e - \frac{63344670}{388613}$ |
Atkin-Lehner eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| $3$ | $[3, 3, w]$ | $-1$ |
| $7$ | $[7, 7, \frac{1}{3}w^{3} - \frac{2}{3}w^{2} - 2w + 2]$ | $-1$ |