Base field 4.4.19025.1
Generator \(w\), with minimal polynomial \(x^{4} - 2x^{3} - 13x^{2} + 14x + 44\); narrow class number \(1\) and class number \(1\).
Form
| Weight: | $[2, 2, 2, 2]$ |
| Level: | $[20, 10, w + 1]$ |
| Dimension: | $12$ |
| CM: | no |
| Base change: | no |
| Newspace dimension: | $35$ |
Hecke eigenvalues ($q$-expansion)
The Hecke eigenvalue field is $\Q(e)$ where $e$ is a root of the defining polynomial:
| \(x^{12} - 40x^{10} - 2x^{9} + 557x^{8} + 44x^{7} - 3076x^{6} - 288x^{5} + 5367x^{4} + 964x^{3} - 2092x^{2} + 50x + 115\) |
Show full eigenvalues Hide large eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| 4 | $[4, 2, w^{2} - 2w - 6]$ | $\phantom{-}e$ |
| 4 | $[4, 2, -w^{2} + 7]$ | $\phantom{-}1$ |
| 5 | $[5, 5, -\frac{1}{2}w^{3} + 2w^{2} + \frac{7}{2}w - 14]$ | $\phantom{-}1$ |
| 5 | $[5, 5, \frac{1}{2}w^{3} + \frac{1}{2}w^{2} - 6w - 9]$ | $-\frac{294555}{24596968}e^{11} + \frac{31439}{12298484}e^{10} + \frac{5790205}{12298484}e^{9} - \frac{1619631}{24596968}e^{8} - \frac{78762137}{12298484}e^{7} + \frac{6516651}{12298484}e^{6} + \frac{418123553}{12298484}e^{5} - \frac{13573379}{12298484}e^{4} - \frac{119425033}{2236088}e^{3} - \frac{21527029}{3074621}e^{2} + \frac{95459047}{6149242}e + \frac{17288793}{24596968}$ |
| 11 | $[11, 11, \frac{1}{2}w^{3} - 2w^{2} - \frac{5}{2}w + 11]$ | $-\frac{9145}{3074621}e^{11} - \frac{5751}{6149242}e^{10} + \frac{2955141}{24596968}e^{9} + \frac{987935}{24596968}e^{8} - \frac{20816929}{12298484}e^{7} - \frac{7181919}{12298484}e^{6} + \frac{116985829}{12298484}e^{5} + \frac{42197417}{12298484}e^{4} - \frac{19439931}{1118044}e^{3} - \frac{100117591}{12298484}e^{2} + \frac{229833405}{24596968}e + \frac{39133279}{24596968}$ |
| 11 | $[11, 11, \frac{1}{2}w^{3} + \frac{1}{2}w^{2} - 5w - 7]$ | $-\frac{42285}{12298484}e^{11} - \frac{167799}{24596968}e^{10} + \frac{3450905}{24596968}e^{9} + \frac{1735925}{6149242}e^{8} - \frac{24557333}{12298484}e^{7} - \frac{48825563}{12298484}e^{6} + \frac{140487075}{12298484}e^{5} + \frac{259324905}{12298484}e^{4} - \frac{12180077}{559022}e^{3} - \frac{744775023}{24596968}e^{2} + \frac{117112481}{24596968}e + \frac{41013227}{12298484}$ |
| 31 | $[31, 31, \frac{1}{2}w^{2} + \frac{1}{2}w - 4]$ | $-\frac{271757}{24596968}e^{11} - \frac{5491}{3074621}e^{10} + \frac{10671217}{24596968}e^{9} + \frac{423337}{3074621}e^{8} - \frac{73425955}{12298484}e^{7} - \frac{33215925}{12298484}e^{6} + \frac{410359843}{12298484}e^{5} + \frac{223866559}{12298484}e^{4} - \frac{144646751}{2236088}e^{3} - \frac{448680415}{12298484}e^{2} + \frac{697466409}{24596968}e + \frac{91830477}{12298484}$ |
| 31 | $[31, 31, -\frac{1}{2}w^{2} + \frac{3}{2}w + 3]$ | $\phantom{-}\frac{435231}{24596968}e^{11} + \frac{147641}{6149242}e^{10} - \frac{8805429}{12298484}e^{9} - \frac{23882657}{24596968}e^{8} + \frac{124192745}{12298484}e^{7} + \frac{163214407}{12298484}e^{6} - \frac{698272027}{12298484}e^{5} - \frac{844439441}{12298484}e^{4} + \frac{228340109}{2236088}e^{3} + \frac{1268037541}{12298484}e^{2} - \frac{72268506}{3074621}e - \frac{205029681}{24596968}$ |
| 41 | $[41, 41, \frac{1}{2}w^{2} + \frac{1}{2}w - 6]$ | $\phantom{-}\frac{186981}{24596968}e^{11} - \frac{25242}{3074621}e^{10} - \frac{3643945}{12298484}e^{9} + \frac{7679593}{24596968}e^{8} + \frac{48632405}{12298484}e^{7} - \frac{50876465}{12298484}e^{6} - \frac{245974101}{12298484}e^{5} + \frac{255383903}{12298484}e^{4} + \frac{58913003}{2236088}e^{3} - \frac{260186537}{12298484}e^{2} - \frac{55483107}{6149242}e + \frac{97748029}{24596968}$ |
| 41 | $[41, 41, 2w^{3} - \frac{15}{2}w^{2} - \frac{25}{2}w + 50]$ | $\phantom{-}\frac{180959}{3074621}e^{11} + \frac{23447}{3074621}e^{10} - \frac{28725251}{12298484}e^{9} - \frac{5576111}{12298484}e^{8} + \frac{197326137}{6149242}e^{7} + \frac{47439929}{6149242}e^{6} - \frac{529491506}{3074621}e^{5} - \frac{147482283}{3074621}e^{4} + \frac{152298059}{559022}e^{3} + \frac{713858887}{6149242}e^{2} - \frac{692723221}{12298484}e - \frac{128876193}{12298484}$ |
| 41 | $[41, 41, \frac{5}{2}w^{2} - \frac{1}{2}w - 17]$ | $-\frac{371863}{24596968}e^{11} - \frac{69353}{24596968}e^{10} + \frac{14965847}{24596968}e^{9} + \frac{3007453}{24596968}e^{8} - \frac{51839399}{6149242}e^{7} - \frac{10622217}{6149242}e^{6} + \frac{275479419}{6149242}e^{5} + \frac{32378460}{3074621}e^{4} - \frac{143920887}{2236088}e^{3} - \frac{926019491}{24596968}e^{2} + \frac{233488285}{24596968}e + \frac{279389379}{24596968}$ |
| 41 | $[41, 41, \frac{1}{2}w^{2} - \frac{3}{2}w - 5]$ | $\phantom{-}\frac{913511}{24596968}e^{11} + \frac{76941}{24596968}e^{10} - \frac{18296031}{12298484}e^{9} - \frac{1064557}{6149242}e^{8} + \frac{253943223}{12298484}e^{7} + \frac{32996077}{12298484}e^{6} - \frac{1378362119}{12298484}e^{5} - \frac{192028055}{12298484}e^{4} + \frac{403391269}{2236088}e^{3} + \frac{1214721873}{24596968}e^{2} - \frac{150311196}{3074621}e + \frac{12897957}{12298484}$ |
| 61 | $[61, 61, -\frac{1}{2}w^{3} + w^{2} + \frac{7}{2}w - 1]$ | $-\frac{320253}{24596968}e^{11} - \frac{46735}{12298484}e^{10} + \frac{6500473}{12298484}e^{9} + \frac{3858341}{24596968}e^{8} - \frac{92319217}{12298484}e^{7} - \frac{25450213}{12298484}e^{6} + \frac{526265557}{12298484}e^{5} + \frac{118285011}{12298484}e^{4} - \frac{178925211}{2236088}e^{3} - \frac{85329575}{6149242}e^{2} + \frac{198100383}{6149242}e - \frac{135758999}{24596968}$ |
| 61 | $[61, 61, -\frac{1}{2}w^{3} + \frac{1}{2}w^{2} + 4w - 3]$ | $-\frac{210333}{24596968}e^{11} - \frac{21232}{3074621}e^{10} + \frac{8313851}{24596968}e^{9} + \frac{1801749}{6149242}e^{8} - \frac{56544145}{12298484}e^{7} - \frac{52264515}{12298484}e^{6} + \frac{293740373}{12298484}e^{5} + \frac{302835009}{12298484}e^{4} - \frac{70160491}{2236088}e^{3} - \frac{633619125}{12298484}e^{2} - \frac{237845133}{24596968}e + \frac{159549073}{12298484}$ |
| 71 | $[71, 71, \frac{1}{2}w^{3} + w^{2} - \frac{11}{2}w - 13]$ | $-\frac{1465535}{24596968}e^{11} - \frac{23219}{12298484}e^{10} + \frac{57918917}{24596968}e^{9} + \frac{3251543}{12298484}e^{8} - \frac{396314639}{12298484}e^{7} - \frac{69372131}{12298484}e^{6} + \frac{2122204757}{12298484}e^{5} + \frac{481441091}{12298484}e^{4} - \frac{615567233}{2236088}e^{3} - \frac{621889483}{6149242}e^{2} + \frac{1512536217}{24596968}e + \frac{28206659}{6149242}$ |
| 71 | $[71, 71, -\frac{1}{2}w^{3} + \frac{5}{2}w^{2} + 2w - 17]$ | $-\frac{20083}{1118044}e^{11} - \frac{57009}{2236088}e^{10} + \frac{1621403}{2236088}e^{9} + \frac{1132977}{1118044}e^{8} - \frac{5646265}{559022}e^{7} - \frac{7654679}{559022}e^{6} + \frac{30509761}{559022}e^{5} + \frac{19877992}{279511}e^{4} - \frac{93658493}{1118044}e^{3} - \frac{255917523}{2236088}e^{2} - \frac{3911963}{2236088}e + \frac{19228943}{1118044}$ |
| 81 | $[81, 3, -3]$ | $\phantom{-}\frac{188339}{12298484}e^{11} + \frac{177613}{24596968}e^{10} - \frac{3809259}{6149242}e^{9} - \frac{7920381}{24596968}e^{8} + \frac{107394117}{12298484}e^{7} + \frac{57699547}{12298484}e^{6} - \frac{601517369}{12298484}e^{5} - \frac{310299879}{12298484}e^{4} + \frac{24068546}{279511}e^{3} + \frac{1019362885}{24596968}e^{2} - \frac{260287145}{12298484}e - \frac{18186101}{24596968}$ |
| 89 | $[89, 89, -w^{3} + \frac{3}{2}w^{2} + \frac{13}{2}w - 9]$ | $-\frac{83735}{3074621}e^{11} - \frac{333349}{24596968}e^{10} + \frac{26365775}{24596968}e^{9} + \frac{1979452}{3074621}e^{8} - \frac{45080304}{3074621}e^{7} - \frac{30704956}{3074621}e^{6} + \frac{244247155}{3074621}e^{5} + \frac{178292720}{3074621}e^{4} - \frac{75352537}{559022}e^{3} - \frac{2523018259}{24596968}e^{2} + \frac{682606421}{24596968}e + \frac{26997515}{3074621}$ |
| 89 | $[89, 89, w^{3} - \frac{7}{2}w^{2} - \frac{11}{2}w + 20]$ | $\phantom{-}\frac{2090973}{24596968}e^{11} + \frac{702357}{24596968}e^{10} - \frac{83431019}{24596968}e^{9} - \frac{32042919}{24596968}e^{8} + \frac{288777701}{6149242}e^{7} + \frac{118998189}{6149242}e^{6} - \frac{785923573}{3074621}e^{5} - \frac{668842047}{6149242}e^{4} + \frac{942095397}{2236088}e^{3} + \frac{5275497559}{24596968}e^{2} - \frac{2396861109}{24596968}e - \frac{402005405}{24596968}$ |
| 89 | $[89, 89, -w^{3} - \frac{1}{2}w^{2} + \frac{19}{2}w + 12]$ | $\phantom{-}\frac{12849}{6149242}e^{11} + \frac{39087}{3074621}e^{10} - \frac{355134}{3074621}e^{9} - \frac{1369493}{3074621}e^{8} + \frac{6778540}{3074621}e^{7} + \frac{15983432}{3074621}e^{6} - \frac{54071002}{3074621}e^{5} - \frac{69003816}{3074621}e^{4} + \frac{30309111}{559022}e^{3} + \frac{85320211}{3074621}e^{2} - \frac{130312925}{3074621}e - \frac{1708125}{3074621}$ |
Atkin-Lehner eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| $4$ | $[4, 2, -w^{2} + 7]$ | $-1$ |
| $5$ | $[5, 5, -\frac{1}{2}w^{3} + 2w^{2} + \frac{7}{2}w - 14]$ | $-1$ |