Base field \(\Q(\sqrt{401}) \)
Generator \(w\), with minimal polynomial \(x^{2} - x - 100\); narrow class number \(5\) and class number \(5\).
Form
| Weight: | $[2, 2]$ |
| Level: | $[1, 1, 1]$ |
| Dimension: | $16$ |
| CM: | no |
| Base change: | yes |
| Newspace dimension: | $120$ |
Hecke eigenvalues ($q$-expansion)
The Hecke eigenvalue field is $\Q(e)$ where $e$ is a root of the defining polynomial:
| \(x^{16} - x^{15} - 22x^{14} + 20x^{13} + 192x^{12} - 154x^{11} - 854x^{10} + 585x^{9} + 2064x^{8} - 1169x^{7} - 2653x^{6} + 1197x^{5} + 1623x^{4} - 532x^{3} - 351x^{2} + 38x + 12\) |
Show full eigenvalues Hide large eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| 2 | $[2, 2, w]$ | $\phantom{-}e$ |
| 2 | $[2, 2, w + 1]$ | $\phantom{-}e$ |
| 5 | $[5, 5, w]$ | $-\frac{286}{331}e^{15} + \frac{59}{331}e^{14} + \frac{12511}{662}e^{13} - \frac{1275}{662}e^{12} - \frac{107585}{662}e^{11} - \frac{1489}{662}e^{10} + \frac{463061}{662}e^{9} + \frac{59965}{662}e^{8} - \frac{1044707}{662}e^{7} - \frac{119424}{331}e^{6} + \frac{1161221}{662}e^{5} + \frac{339253}{662}e^{4} - \frac{251672}{331}e^{3} - \frac{73459}{331}e^{2} + \frac{27221}{662}e + \frac{2478}{331}$ |
| 5 | $[5, 5, w + 4]$ | $-\frac{286}{331}e^{15} + \frac{59}{331}e^{14} + \frac{12511}{662}e^{13} - \frac{1275}{662}e^{12} - \frac{107585}{662}e^{11} - \frac{1489}{662}e^{10} + \frac{463061}{662}e^{9} + \frac{59965}{662}e^{8} - \frac{1044707}{662}e^{7} - \frac{119424}{331}e^{6} + \frac{1161221}{662}e^{5} + \frac{339253}{662}e^{4} - \frac{251672}{331}e^{3} - \frac{73459}{331}e^{2} + \frac{27221}{662}e + \frac{2478}{331}$ |
| 7 | $[7, 7, w + 1]$ | $\phantom{-}\frac{2960}{331}e^{15} - \frac{275}{331}e^{14} - \frac{65351}{331}e^{13} - \frac{117}{331}e^{12} + \frac{567851}{331}e^{11} + \frac{60001}{331}e^{10} - \frac{2470700}{331}e^{9} - \frac{515246}{331}e^{8} + \frac{5632425}{331}e^{7} + \frac{1669852}{331}e^{6} - \frac{6322200}{331}e^{5} - \frac{2229935}{331}e^{4} + \frac{2773180}{331}e^{3} + \frac{970497}{331}e^{2} - \frac{162203}{331}e - \frac{39685}{331}$ |
| 7 | $[7, 7, w + 5]$ | $\phantom{-}\frac{2960}{331}e^{15} - \frac{275}{331}e^{14} - \frac{65351}{331}e^{13} - \frac{117}{331}e^{12} + \frac{567851}{331}e^{11} + \frac{60001}{331}e^{10} - \frac{2470700}{331}e^{9} - \frac{515246}{331}e^{8} + \frac{5632425}{331}e^{7} + \frac{1669852}{331}e^{6} - \frac{6322200}{331}e^{5} - \frac{2229935}{331}e^{4} + \frac{2773180}{331}e^{3} + \frac{970497}{331}e^{2} - \frac{162203}{331}e - \frac{39685}{331}$ |
| 9 | $[9, 3, 3]$ | $\phantom{-}\frac{5569}{331}e^{15} - \frac{895}{662}e^{14} - \frac{246185}{662}e^{13} - \frac{3101}{662}e^{12} + \frac{2142061}{662}e^{11} + \frac{244601}{662}e^{10} - \frac{9335065}{662}e^{9} - \frac{1999773}{662}e^{8} + \frac{10662078}{331}e^{7} + \frac{6369937}{662}e^{6} - \frac{24007605}{662}e^{5} - \frac{4210475}{331}e^{4} + \frac{5297792}{331}e^{3} + \frac{3629341}{662}e^{2} - \frac{321040}{331}e - \frac{73079}{331}$ |
| 11 | $[11, 11, w + 3]$ | $-\frac{2631}{662}e^{15} + \frac{37}{662}e^{14} + \frac{29175}{331}e^{13} + \frac{2169}{331}e^{12} - \frac{254779}{331}e^{11} - \frac{43349}{331}e^{10} + \frac{1114285}{331}e^{9} + \frac{586807}{662}e^{8} - \frac{2552721}{331}e^{7} - \frac{1737607}{662}e^{6} + \frac{5753967}{662}e^{5} + \frac{2214327}{662}e^{4} - \frac{2526021}{662}e^{3} - \frac{467261}{331}e^{2} + \frac{139833}{662}e + \frac{18651}{331}$ |
| 11 | $[11, 11, w + 7]$ | $-\frac{2631}{662}e^{15} + \frac{37}{662}e^{14} + \frac{29175}{331}e^{13} + \frac{2169}{331}e^{12} - \frac{254779}{331}e^{11} - \frac{43349}{331}e^{10} + \frac{1114285}{331}e^{9} + \frac{586807}{662}e^{8} - \frac{2552721}{331}e^{7} - \frac{1737607}{662}e^{6} + \frac{5753967}{662}e^{5} + \frac{2214327}{662}e^{4} - \frac{2526021}{662}e^{3} - \frac{467261}{331}e^{2} + \frac{139833}{662}e + \frac{18651}{331}$ |
| 29 | $[29, 29, w + 6]$ | $\phantom{-}\frac{4697}{662}e^{15} - \frac{447}{662}e^{14} - \frac{103683}{662}e^{13} - \frac{65}{662}e^{12} + \frac{900975}{662}e^{11} + \frac{95047}{662}e^{10} - \frac{3922157}{662}e^{9} - \frac{409726}{331}e^{8} + \frac{8954063}{662}e^{7} + \frac{2655111}{662}e^{6} - \frac{5041452}{331}e^{5} - \frac{1770626}{331}e^{4} + \frac{4457023}{662}e^{3} + \frac{770882}{331}e^{2} - \frac{136394}{331}e - \frac{33219}{331}$ |
| 29 | $[29, 29, w + 22]$ | $\phantom{-}\frac{4697}{662}e^{15} - \frac{447}{662}e^{14} - \frac{103683}{662}e^{13} - \frac{65}{662}e^{12} + \frac{900975}{662}e^{11} + \frac{95047}{662}e^{10} - \frac{3922157}{662}e^{9} - \frac{409726}{331}e^{8} + \frac{8954063}{662}e^{7} + \frac{2655111}{662}e^{6} - \frac{5041452}{331}e^{5} - \frac{1770626}{331}e^{4} + \frac{4457023}{662}e^{3} + \frac{770882}{331}e^{2} - \frac{136394}{331}e - \frac{33219}{331}$ |
| 41 | $[41, 41, w + 13]$ | $-\frac{16729}{662}e^{15} + \frac{769}{331}e^{14} + \frac{369555}{662}e^{13} + \frac{605}{662}e^{12} - \frac{3213181}{662}e^{11} - \frac{334699}{662}e^{10} + \frac{13989981}{662}e^{9} + \frac{1437711}{331}e^{8} - \frac{15959019}{331}e^{7} - \frac{4657660}{331}e^{6} + \frac{17934973}{331}e^{5} + \frac{12420017}{662}e^{4} - \frac{15777611}{662}e^{3} - \frac{5377797}{662}e^{2} + \frac{939151}{662}e + \frac{107766}{331}$ |
| 41 | $[41, 41, w + 27]$ | $-\frac{16729}{662}e^{15} + \frac{769}{331}e^{14} + \frac{369555}{662}e^{13} + \frac{605}{662}e^{12} - \frac{3213181}{662}e^{11} - \frac{334699}{662}e^{10} + \frac{13989981}{662}e^{9} + \frac{1437711}{331}e^{8} - \frac{15959019}{331}e^{7} - \frac{4657660}{331}e^{6} + \frac{17934973}{331}e^{5} + \frac{12420017}{662}e^{4} - \frac{15777611}{662}e^{3} - \frac{5377797}{662}e^{2} + \frac{939151}{662}e + \frac{107766}{331}$ |
| 43 | $[43, 43, w + 16]$ | $\phantom{-}\frac{8311}{662}e^{15} - \frac{681}{662}e^{14} - \frac{91899}{331}e^{13} - \frac{917}{331}e^{12} + \frac{800025}{331}e^{11} + \frac{88344}{331}e^{10} - \frac{3487881}{331}e^{9} - \frac{1460549}{662}e^{8} + \frac{7968877}{331}e^{7} + \frac{4671589}{662}e^{6} - \frac{17943547}{662}e^{5} - \frac{6189775}{662}e^{4} + \frac{7923587}{662}e^{3} + \frac{1335859}{331}e^{2} - \frac{486075}{662}e - \frac{54352}{331}$ |
| 43 | $[43, 43, w + 26]$ | $\phantom{-}\frac{8311}{662}e^{15} - \frac{681}{662}e^{14} - \frac{91899}{331}e^{13} - \frac{917}{331}e^{12} + \frac{800025}{331}e^{11} + \frac{88344}{331}e^{10} - \frac{3487881}{331}e^{9} - \frac{1460549}{662}e^{8} + \frac{7968877}{331}e^{7} + \frac{4671589}{662}e^{6} - \frac{17943547}{662}e^{5} - \frac{6189775}{662}e^{4} + \frac{7923587}{662}e^{3} + \frac{1335859}{331}e^{2} - \frac{486075}{662}e - \frac{54352}{331}$ |
| 47 | $[47, 47, w + 2]$ | $-\frac{10284}{331}e^{15} + \frac{904}{331}e^{14} + \frac{227153}{331}e^{13} + \frac{1374}{331}e^{12} - \frac{1974817}{331}e^{11} - \frac{215163}{331}e^{10} + \frac{8597602}{331}e^{9} + \frac{1810821}{331}e^{8} - \frac{19615531}{331}e^{7} - \frac{5826767}{331}e^{6} + \frac{22047998}{331}e^{5} + \frac{7746303}{331}e^{4} - \frac{9703730}{331}e^{3} - \frac{3356543}{331}e^{2} + \frac{579513}{331}e + \frac{138261}{331}$ |
| 47 | $[47, 47, w + 44]$ | $-\frac{10284}{331}e^{15} + \frac{904}{331}e^{14} + \frac{227153}{331}e^{13} + \frac{1374}{331}e^{12} - \frac{1974817}{331}e^{11} - \frac{215163}{331}e^{10} + \frac{8597602}{331}e^{9} + \frac{1810821}{331}e^{8} - \frac{19615531}{331}e^{7} - \frac{5826767}{331}e^{6} + \frac{22047998}{331}e^{5} + \frac{7746303}{331}e^{4} - \frac{9703730}{331}e^{3} - \frac{3356543}{331}e^{2} + \frac{579513}{331}e + \frac{138261}{331}$ |
| 73 | $[73, 73, w + 33]$ | $-\frac{10493}{331}e^{15} + \frac{2047}{662}e^{14} + \frac{463385}{662}e^{13} - \frac{1481}{662}e^{12} - \frac{4027299}{662}e^{11} - \frac{403521}{662}e^{10} + \frac{17529279}{662}e^{9} + \frac{3547843}{662}e^{8} - \frac{19995424}{331}e^{7} - \frac{11569413}{662}e^{6} + \frac{44958699}{662}e^{5} + \frac{7729885}{331}e^{4} - \frac{9898007}{331}e^{3} - \frac{6693447}{662}e^{2} + \frac{589962}{331}e + \frac{134012}{331}$ |
| 73 | $[73, 73, w + 39]$ | $-\frac{10493}{331}e^{15} + \frac{2047}{662}e^{14} + \frac{463385}{662}e^{13} - \frac{1481}{662}e^{12} - \frac{4027299}{662}e^{11} - \frac{403521}{662}e^{10} + \frac{17529279}{662}e^{9} + \frac{3547843}{662}e^{8} - \frac{19995424}{331}e^{7} - \frac{11569413}{662}e^{6} + \frac{44958699}{662}e^{5} + \frac{7729885}{331}e^{4} - \frac{9898007}{331}e^{3} - \frac{6693447}{662}e^{2} + \frac{589962}{331}e + \frac{134012}{331}$ |
| 83 | $[83, 83, -4w - 37]$ | $\phantom{-}\frac{7493}{662}e^{15} - \frac{709}{662}e^{14} - \frac{82595}{331}e^{13} - \frac{143}{331}e^{12} + \frac{716401}{331}e^{11} + \frac{77233}{331}e^{10} - \frac{3110181}{331}e^{9} - \frac{1327713}{662}e^{8} + \frac{7070759}{331}e^{7} + \frac{4307345}{662}e^{6} - \frac{15813843}{662}e^{5} - \frac{5760143}{662}e^{4} + \frac{6891829}{662}e^{3} + \frac{1254680}{331}e^{2} - \frac{393223}{662}e - \frac{49644}{331}$ |
Atkin-Lehner eigenvalues
This form has no Atkin-Lehner eigenvalues since the level is \((1)\).