Base field 4.4.16448.1
Generator \(w\), with minimal polynomial \(x^{4} - 2x^{3} - 6x^{2} + 2\); narrow class number \(2\) and class number \(1\).
Form
| Weight: | $[2, 2, 2, 2]$ |
| Level: | $[13, 13, -w^{3} + 3w^{2} + 3w - 1]$ |
| Dimension: | $13$ |
| CM: | no |
| Base change: | no |
| Newspace dimension: | $30$ |
Hecke eigenvalues ($q$-expansion)
The Hecke eigenvalue field is $\Q(e)$ where $e$ is a root of the defining polynomial:
| \(x^{13} + 8x^{12} + 11x^{11} - 66x^{10} - 202x^{9} + 64x^{8} + 804x^{7} + 565x^{6} - 985x^{5} - 1282x^{4} + 21x^{3} + 452x^{2} + 83x - 1\) |
Show full eigenvalues Hide large eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| 2 | $[2, 2, w]$ | $\phantom{-}e$ |
| 5 | $[5, 5, -w^{3} + 3w^{2} + 2w - 1]$ | $\phantom{-}\frac{336}{97}e^{12} + \frac{2218}{97}e^{11} + \frac{620}{97}e^{10} - \frac{22907}{97}e^{9} - \frac{35964}{97}e^{8} + \frac{70238}{97}e^{7} + \frac{171078}{97}e^{6} - \frac{43455}{97}e^{5} - \frac{264062}{97}e^{4} - \frac{69501}{97}e^{3} + \frac{93499}{97}e^{2} + \frac{22665}{97}e + \frac{38}{97}$ |
| 5 | $[5, 5, w - 1]$ | $-\frac{92}{97}e^{12} - \frac{605}{97}e^{11} - \frac{179}{97}e^{10} + \frac{6174}{97}e^{9} + \frac{9875}{97}e^{8} - \frac{18375}{97}e^{7} - \frac{46531}{97}e^{6} + \frac{9172}{97}e^{5} + \frac{70947}{97}e^{4} + \frac{22283}{97}e^{3} - \frac{24251}{97}e^{2} - \frac{7214}{97}e - \frac{179}{97}$ |
| 11 | $[11, 11, w^{3} - 2w^{2} - 5w - 1]$ | $\phantom{-}\frac{312}{97}e^{12} + \frac{2115}{97}e^{11} + \frac{839}{97}e^{10} - \frac{21638}{97}e^{9} - \frac{36416}{97}e^{8} + \frac{64833}{97}e^{7} + \frac{170609}{97}e^{6} - \frac{34344}{97}e^{5} - \frac{262993}{97}e^{4} - \frac{74105}{97}e^{3} + \frac{94823}{97}e^{2} + \frac{22702}{97}e - \frac{325}{97}$ |
| 13 | $[13, 13, -w^{3} + 3w^{2} + 3w - 1]$ | $\phantom{-}1$ |
| 17 | $[17, 17, 2w^{3} - 5w^{2} - 9w + 5]$ | $\phantom{-}\frac{1102}{97}e^{12} + \frac{7405}{97}e^{11} + \frac{2627}{97}e^{10} - \frac{76189}{97}e^{9} - \frac{125069}{97}e^{8} + \frac{231633}{97}e^{7} + \frac{590642}{97}e^{6} - \frac{136780}{97}e^{5} - \frac{915594}{97}e^{4} - \frac{237063}{97}e^{3} + \frac{333770}{97}e^{2} + \frac{70364}{97}e - \frac{1059}{97}$ |
| 23 | $[23, 23, -w^{3} + 3w^{2} + 4w - 5]$ | $\phantom{-}\frac{1226}{97}e^{12} + \frac{8212}{97}e^{11} + \frac{2805}{97}e^{10} - \frac{84540}{97}e^{9} - \frac{137704}{97}e^{8} + \frac{257441}{97}e^{7} + \frac{651055}{97}e^{6} - \frac{153929}{97}e^{5} - \frac{1008595}{97}e^{4} - \frac{259674}{97}e^{3} + \frac{366182}{97}e^{2} + \frac{77173}{97}e - \frac{881}{97}$ |
| 25 | $[25, 5, -w^{2} + 3w + 1]$ | $-\frac{466}{97}e^{12} - \frac{3172}{97}e^{11} - \frac{1301}{97}e^{10} + \frac{32521}{97}e^{9} + \frac{55179}{97}e^{8} - \frac{97858}{97}e^{7} - \frac{259132}{97}e^{6} + \frac{53303}{97}e^{5} + \frac{401635}{97}e^{4} + \frac{108615}{97}e^{3} - \frac{146532}{97}e^{2} - \frac{31526}{97}e - \frac{137}{97}$ |
| 29 | $[29, 29, -2w^{3} + 6w^{2} + 5w - 1]$ | $-\frac{729}{97}e^{12} - \frac{4911}{97}e^{11} - \frac{1799}{97}e^{10} + \frac{50392}{97}e^{9} + \frac{83222}{97}e^{8} - \frac{152064}{97}e^{7} - \frac{391103}{97}e^{6} + \frac{84723}{97}e^{5} + \frac{601776}{97}e^{4} + \frac{166140}{97}e^{3} - \frac{213875}{97}e^{2} - \frac{50468}{97}e - \frac{247}{97}$ |
| 31 | $[31, 31, -w^{2} + 2w + 1]$ | $-\frac{461}{97}e^{12} - \frac{3001}{97}e^{11} - \frac{607}{97}e^{10} + \frac{31323}{97}e^{9} + \frac{46624}{97}e^{8} - \frac{98870}{97}e^{7} - \frac{224959}{97}e^{6} + \frac{73727}{97}e^{5} + \frac{351041}{97}e^{4} + \frac{72795}{97}e^{3} - \frac{128976}{97}e^{2} - \frac{24093}{97}e + \frac{945}{97}$ |
| 43 | $[43, 43, -w^{3} + 3w^{2} + 2w - 3]$ | $-\frac{1148}{97}e^{12} - \frac{7756}{97}e^{11} - \frac{2959}{97}e^{10} + \frac{79567}{97}e^{9} + \frac{132868}{97}e^{8} - \frac{239899}{97}e^{7} - \frac{625402}{97}e^{6} + \frac{132636}{97}e^{5} + \frac{968964}{97}e^{4} + \frac{263967}{97}e^{3} - \frac{353995}{97}e^{2} - \frac{79694}{97}e + \frac{1309}{97}$ |
| 43 | $[43, 43, -w^{3} + 2w^{2} + 6w - 3]$ | $-\frac{90}{97}e^{12} - \frac{556}{97}e^{11} + \frac{21}{97}e^{10} + \frac{5947}{97}e^{9} + \frac{7617}{97}e^{8} - \frac{19808}{97}e^{7} - \frac{38837}{97}e^{6} + \frac{18719}{97}e^{5} + \frac{62524}{97}e^{4} + \frac{6791}{97}e^{3} - \frac{24232}{97}e^{2} - \frac{1389}{97}e + \frac{700}{97}$ |
| 59 | $[59, 59, 2w^{3} - 6w^{2} - 7w + 7]$ | $-\frac{1262}{97}e^{12} - \frac{8415}{97}e^{11} - \frac{2719}{97}e^{10} + \frac{86686}{97}e^{9} + \frac{139742}{97}e^{8} - \frac{264142}{97}e^{7} - \frac{661701}{97}e^{6} + \frac{157459}{97}e^{5} + \frac{1022857}{97}e^{4} + \frac{269258}{97}e^{3} - \frac{366524}{97}e^{2} - \frac{81628}{97}e + \frac{191}{97}$ |
| 73 | $[73, 73, 3w^{3} - 6w^{2} - 16w - 5]$ | $-\frac{193}{97}e^{12} - \frac{1285}{97}e^{11} - \frac{385}{97}e^{10} + \frac{13321}{97}e^{9} + \frac{21084}{97}e^{8} - \frac{41117}{97}e^{7} - \frac{100331}{97}e^{6} + \frac{26065}{97}e^{5} + \frac{155208}{97}e^{4} + \frac{39978}{97}e^{3} - \frac{55135}{97}e^{2} - \frac{12559}{97}e + \frac{197}{97}$ |
| 79 | $[79, 79, -5w^{3} + 14w^{2} + 20w - 17]$ | $\phantom{-}\frac{1586}{97}e^{12} + \frac{10630}{97}e^{11} + \frac{3594}{97}e^{10} - \frac{109686}{97}e^{9} - \frac{177969}{97}e^{8} + \frac{336091}{97}e^{7} + \frac{843069}{97}e^{6} - \frac{209696}{97}e^{5} - \frac{1310392}{97}e^{4} - \frac{322437}{97}e^{3} + \frac{481734}{97}e^{2} + \frac{95824}{97}e - \frac{2517}{97}$ |
| 81 | $[81, 3, -3]$ | $-\frac{309}{97}e^{12} - \frac{1993}{97}e^{11} - \frac{248}{97}e^{10} + \frac{21152}{97}e^{9} + \frac{29634}{97}e^{8} - \frac{69553}{97}e^{7} - \frac{145876}{97}e^{6} + \frac{63263}{97}e^{5} + \frac{231880}{97}e^{4} + \frac{28751}{97}e^{3} - \frac{89508}{97}e^{2} - \frac{8969}{97}e + \frac{528}{97}$ |
| 83 | $[83, 83, -w - 3]$ | $\phantom{-}\frac{510}{97}e^{12} + \frac{3280}{97}e^{11} + \frac{560}{97}e^{10} - \frac{34023}{97}e^{9} - \frac{49953}{97}e^{8} + \frac{105520}{97}e^{7} + \frac{240317}{97}e^{6} - \frac{70249}{97}e^{5} - \frac{369176}{97}e^{4} - \frac{95098}{97}e^{3} + \frac{126677}{97}e^{2} + \frac{31248}{97}e + \frac{560}{97}$ |
| 83 | $[83, 83, w^{2} - 3w - 7]$ | $-\frac{995}{97}e^{12} - \frac{6772}{97}e^{11} - \frac{2791}{97}e^{10} + \frac{69234}{97}e^{9} + \frac{117465}{97}e^{8} - \frac{207176}{97}e^{7} - \frac{549262}{97}e^{6} + \frac{109243}{97}e^{5} + \frac{846222}{97}e^{4} + \frac{236136}{97}e^{3} - \frac{304478}{97}e^{2} - \frac{69854}{97}e - \frac{75}{97}$ |
| 89 | $[89, 89, w^{2} - 2w - 7]$ | $-\frac{916}{97}e^{12} - \frac{6146}{97}e^{11} - \frac{2166}{97}e^{10} + \frac{63129}{97}e^{9} + \frac{103546}{97}e^{8} - \frac{191078}{97}e^{7} - \frac{487946}{97}e^{6} + \frac{109165}{97}e^{5} + \frac{752667}{97}e^{4} + \frac{203001}{97}e^{3} - \frac{269826}{97}e^{2} - \frac{60684}{97}e + \frac{841}{97}$ |
| 101 | $[101, 101, -2w^{3} + 5w^{2} + 8w - 3]$ | $\phantom{-}\frac{287}{97}e^{12} + \frac{1842}{97}e^{11} + \frac{279}{97}e^{10} - \frac{19140}{97}e^{9} - \frac{27591}{97}e^{8} + \frac{59708}{97}e^{7} + \frac{132537}{97}e^{6} - \frac{41598}{97}e^{5} - \frac{201889}{97}e^{4} - \frac{49623}{97}e^{3} + \frac{66989}{97}e^{2} + \frac{16577}{97}e + \frac{85}{97}$ |
Atkin-Lehner eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| $13$ | $[13, 13, -w^{3} + 3w^{2} + 3w - 1]$ | $-1$ |