Base field 5.5.160801.1
Generator \(w\), with minimal polynomial \(x^{5} - x^{4} - 5x^{3} + 4x^{2} + 3x - 1\); narrow class number \(1\) and class number \(1\).
Form
| Weight: | $[2, 2, 2, 2, 2]$ |
| Level: | $[27, 3, -w^{3} + w^{2} + 3w - 2]$ |
| Dimension: | $9$ |
| CM: | no |
| Base change: | no |
| Newspace dimension: | $25$ |
Hecke eigenvalues ($q$-expansion)
The Hecke eigenvalue field is $\Q(e)$ where $e$ is a root of the defining polynomial:
| \(x^{9} - 3x^{8} - 71x^{7} + 194x^{6} + 1730x^{5} - 4389x^{4} - 16430x^{3} + 39516x^{2} + 44872x - 103808\) |
Show full eigenvalues Hide large eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| 3 | $[3, 3, w^{4} - w^{3} - 5w^{2} + 3w + 3]$ | $-1$ |
| 9 | $[9, 3, -w^{4} + 5w^{2} - 3]$ | $\phantom{-}e$ |
| 9 | $[9, 3, -w^{4} + w^{3} + 5w^{2} - 3w - 2]$ | $\phantom{-}1$ |
| 13 | $[13, 13, -w^{4} + w^{3} + 4w^{2} - 3w - 1]$ | $\phantom{-}\frac{6896623}{4877833864}e^{8} - \frac{15795193}{4877833864}e^{7} - \frac{441686219}{4877833864}e^{6} + \frac{366912919}{2438916932}e^{5} + \frac{1170160041}{609729233}e^{4} - \frac{9361496893}{4877833864}e^{3} - \frac{9390696649}{609729233}e^{2} + \frac{6371631913}{1219458466}e + \frac{23947426614}{609729233}$ |
| 17 | $[17, 17, w^{4} - w^{3} - 5w^{2} + 3w + 1]$ | $-\frac{13843703}{9755667728}e^{8} + \frac{41067663}{9755667728}e^{7} + \frac{794433119}{9755667728}e^{6} - \frac{122933209}{609729233}e^{5} - \frac{7129203363}{4877833864}e^{4} + \frac{27400742327}{9755667728}e^{3} + \frac{10865077791}{1219458466}e^{2} - \frac{23279125491}{2438916932}e - \frac{8589679268}{609729233}$ |
| 19 | $[19, 19, -w^{3} + w^{2} + 4w - 2]$ | $-\frac{12421749}{9755667728}e^{8} + \frac{64196373}{9755667728}e^{7} + \frac{545162001}{9755667728}e^{6} - \frac{383133495}{1219458466}e^{5} - \frac{2170369555}{4877833864}e^{4} + \frac{41008677985}{9755667728}e^{3} - \frac{9916896813}{2438916932}e^{2} - \frac{31080571187}{2438916932}e + \frac{13365759444}{609729233}$ |
| 23 | $[23, 23, -w^{2} + 3]$ | $\phantom{-}\frac{4749455}{2438916932}e^{8} - \frac{2814211}{1219458466}e^{7} - \frac{316237917}{2438916932}e^{6} + \frac{234271557}{2438916932}e^{5} + \frac{7109990125}{2438916932}e^{4} - \frac{706910003}{609729233}e^{3} - \frac{15630733520}{609729233}e^{2} + \frac{2256521212}{609729233}e + \frac{42315035036}{609729233}$ |
| 31 | $[31, 31, w^{3} - 4w + 2]$ | $\phantom{-}\frac{3924433}{9755667728}e^{8} + \frac{51102519}{9755667728}e^{7} - \frac{612171265}{9755667728}e^{6} - \frac{166724416}{609729233}e^{5} + \frac{11745117473}{4877833864}e^{4} + \frac{38693706167}{9755667728}e^{3} - \frac{18456091081}{609729233}e^{2} - \frac{28742985191}{2438916932}e + \frac{54988837174}{609729233}$ |
| 32 | $[32, 2, 2]$ | $\phantom{-}\frac{1110785}{1219458466}e^{8} + \frac{1747405}{1219458466}e^{7} - \frac{172941495}{2438916932}e^{6} - \frac{60097101}{609729233}e^{5} + \frac{4611359983}{2438916932}e^{4} + \frac{4643107225}{2438916932}e^{3} - \frac{47499587669}{2438916932}e^{2} - \frac{11571776261}{1219458466}e + \frac{33910417397}{609729233}$ |
| 37 | $[37, 37, w^{3} - 3w - 1]$ | $-\frac{10061083}{4877833864}e^{8} + \frac{27015651}{4877833864}e^{7} + \frac{662125007}{4877833864}e^{6} - \frac{174920022}{609729233}e^{5} - \frac{7229337249}{2438916932}e^{4} + \frac{21205240407}{4877833864}e^{3} + \frac{29636518959}{1219458466}e^{2} - \frac{20414317239}{1219458466}e - \frac{34110686844}{609729233}$ |
| 53 | $[53, 53, -2w^{4} + w^{3} + 9w^{2} - 3w - 2]$ | $\phantom{-}\frac{18947363}{9755667728}e^{8} - \frac{13036411}{9755667728}e^{7} - \frac{1353890987}{9755667728}e^{6} + \frac{27362910}{609729233}e^{5} + \frac{16452057215}{4877833864}e^{4} - \frac{2632811507}{9755667728}e^{3} - \frac{39177782547}{1219458466}e^{2} - \frac{3948693749}{2438916932}e + \frac{57672782382}{609729233}$ |
| 59 | $[59, 59, -w^{4} + 5w^{2} + w - 4]$ | $\phantom{-}\frac{2183213}{9755667728}e^{8} + \frac{39942967}{9755667728}e^{7} - \frac{285880033}{9755667728}e^{6} - \frac{510294025}{2438916932}e^{5} + \frac{4798643937}{4877833864}e^{4} + \frac{26518184067}{9755667728}e^{3} - \frac{27156408899}{2438916932}e^{2} - \frac{10773568437}{2438916932}e + \frac{18349204666}{609729233}$ |
| 61 | $[61, 61, -w^{4} + w^{3} + 5w^{2} - 4w]$ | $-\frac{1328759}{1219458466}e^{8} + \frac{8234831}{1219458466}e^{7} + \frac{38476261}{1219458466}e^{6} - \frac{199512678}{609729233}e^{5} + \frac{318003377}{609729233}e^{4} + \frac{5563306443}{1219458466}e^{3} - \frac{10797784539}{609729233}e^{2} - \frac{9018802336}{609729233}e + \frac{44403312334}{609729233}$ |
| 67 | $[67, 67, -w^{4} + 6w^{2} + 2w - 4]$ | $-\frac{13550519}{9755667728}e^{8} + \frac{77790363}{9755667728}e^{7} + \frac{581387923}{9755667728}e^{6} - \frac{926063541}{2438916932}e^{5} - \frac{1980182899}{4877833864}e^{4} + \frac{46256704791}{9755667728}e^{3} - \frac{13722137543}{2438916932}e^{2} - \frac{22059875473}{2438916932}e + \frac{16108330818}{609729233}$ |
| 71 | $[71, 71, 2w^{4} - w^{3} - 9w^{2} + 4w + 5]$ | $\phantom{-}\frac{10063295}{4877833864}e^{8} - \frac{18053839}{4877833864}e^{7} - \frac{650588795}{4877833864}e^{6} + \frac{98517027}{609729233}e^{5} + \frac{7014896797}{2438916932}e^{4} - \frac{8279293427}{4877833864}e^{3} - \frac{29358846675}{1219458466}e^{2} + \frac{2694567}{1219458466}e + \frac{39572463662}{609729233}$ |
| 79 | $[79, 79, 2w^{4} - w^{3} - 10w^{2} + 2w + 7]$ | $\phantom{-}\frac{2322205}{1219458466}e^{8} + \frac{1051253}{1219458466}e^{7} - \frac{85363137}{609729233}e^{6} - \frac{33324698}{609729233}e^{5} + \frac{4174123613}{1219458466}e^{4} + \frac{467334412}{609729233}e^{3} - \frac{38364188073}{1219458466}e^{2} + \frac{140312377}{609729233}e + \frac{46151751272}{609729233}$ |
| 83 | $[83, 83, -w^{4} + 2w^{3} + 5w^{2} - 7w - 2]$ | $-\frac{1213625}{609729233}e^{8} + \frac{6679675}{1219458466}e^{7} + \frac{145511643}{1219458466}e^{6} - \frac{300920953}{1219458466}e^{5} - \frac{1467933256}{609729233}e^{4} + \frac{1881154418}{609729233}e^{3} + \frac{24158746007}{1219458466}e^{2} - \frac{4982459280}{609729233}e - \frac{34819943402}{609729233}$ |
| 83 | $[83, 83, -w^{4} + w^{3} + 4w^{2} - 3w + 3]$ | $-\frac{498674}{609729233}e^{8} + \frac{21478293}{2438916932}e^{7} + \frac{12397896}{609729233}e^{6} - \frac{1109773903}{2438916932}e^{5} + \frac{1016730309}{2438916932}e^{4} + \frac{16269730389}{2438916932}e^{3} - \frac{12763388285}{1219458466}e^{2} - \frac{13979496101}{609729233}e + \frac{13717478946}{609729233}$ |
| 83 | $[83, 83, w^{4} - w^{3} - 5w^{2} + 4w - 1]$ | $-\frac{7700417}{9755667728}e^{8} - \frac{10165071}{9755667728}e^{7} + \frac{779088013}{9755667728}e^{6} + \frac{33293451}{1219458466}e^{5} - \frac{12288304943}{4877833864}e^{4} + \frac{1401802829}{9755667728}e^{3} + \frac{69189934731}{2438916932}e^{2} - \frac{9748063291}{2438916932}e - \frac{51134821686}{609729233}$ |
| 83 | $[83, 83, -w^{4} + w^{3} + 4w^{2} - 4w - 2]$ | $-\frac{6281781}{4877833864}e^{8} - \frac{479079}{4877833864}e^{7} + \frac{512512577}{4877833864}e^{6} + \frac{21261733}{1219458466}e^{5} - \frac{7007810457}{2438916932}e^{4} - \frac{1940265051}{4877833864}e^{3} + \frac{35297186211}{1219458466}e^{2} + \frac{873340361}{1219458466}e - \frac{45210908020}{609729233}$ |
Atkin-Lehner eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| $3$ | $[3, 3, w^{4} - w^{3} - 5w^{2} + 3w + 3]$ | $1$ |
| $9$ | $[9, 3, -w^{4} + w^{3} + 5w^{2} - 3w - 2]$ | $-1$ |