Base field 4.4.12725.1
Generator \(w\), with minimal polynomial \(x^{4} - 2x^{3} - 10x^{2} + 11x + 29\); narrow class number \(1\) and class number \(1\).
Form
| Weight: | $[2, 2, 2, 2]$ |
| Level: | $[16, 2, 2]$ |
| Dimension: | $12$ |
| CM: | no |
| Base change: | yes |
| Newspace dimension: | $19$ |
Hecke eigenvalues ($q$-expansion)
The Hecke eigenvalue field is $\Q(e)$ where $e$ is a root of the defining polynomial:
| \(x^{12} - 7x^{11} - 45x^{10} + 350x^{9} + 392x^{8} - 4679x^{7} + 1153x^{6} + 20492x^{5} - 13640x^{4} - 28280x^{3} + 20368x^{2} + 6776x - 3632\) |
Show full eigenvalues Hide large eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| 11 | $[11, 11, -w - 1]$ | $-\frac{21847101049967163}{323384394903557595496}e^{11} + \frac{93247121741079715}{323384394903557595496}e^{10} + \frac{1587834186538511809}{323384394903557595496}e^{9} - \frac{725601522704032223}{40423049362944699437}e^{8} - \frac{9846676654608574891}{80846098725889398874}e^{7} + \frac{115460520085995607137}{323384394903557595496}e^{6} + \frac{393503039567662632459}{323384394903557595496}e^{5} - \frac{421683076796184194161}{161692197451778797748}e^{4} - \frac{24130649067744078991}{5774721337563528491}e^{3} + \frac{61139369652042280773}{11549442675127056982}e^{2} + \frac{90229053789152545249}{40423049362944699437}e + \frac{14291690998435591945}{40423049362944699437}$ |
| 11 | $[11, 11, w^{2} - 5]$ | $\phantom{-}e$ |
| 11 | $[11, 11, -w^{2} + 2w + 4]$ | $\phantom{-}e$ |
| 11 | $[11, 11, w - 2]$ | $-\frac{21847101049967163}{323384394903557595496}e^{11} + \frac{93247121741079715}{323384394903557595496}e^{10} + \frac{1587834186538511809}{323384394903557595496}e^{9} - \frac{725601522704032223}{40423049362944699437}e^{8} - \frac{9846676654608574891}{80846098725889398874}e^{7} + \frac{115460520085995607137}{323384394903557595496}e^{6} + \frac{393503039567662632459}{323384394903557595496}e^{5} - \frac{421683076796184194161}{161692197451778797748}e^{4} - \frac{24130649067744078991}{5774721337563528491}e^{3} + \frac{61139369652042280773}{11549442675127056982}e^{2} + \frac{90229053789152545249}{40423049362944699437}e + \frac{14291690998435591945}{40423049362944699437}$ |
| 16 | $[16, 2, 2]$ | $-1$ |
| 19 | $[19, 19, w^{2} - 2w - 5]$ | $-\frac{159004110069899763}{323384394903557595496}e^{11} + \frac{1100362658285904987}{323384394903557595496}e^{10} + \frac{7456649656901597841}{323384394903557595496}e^{9} - \frac{14031056898767119027}{80846098725889398874}e^{8} - \frac{9826340632268944555}{40423049362944699437}e^{7} + \frac{788116081982594543565}{323384394903557595496}e^{6} + \frac{70186973408556936187}{323384394903557595496}e^{5} - \frac{1915455566037182150481}{161692197451778797748}e^{4} + \frac{27120041070719265717}{11549442675127056982}e^{3} + \frac{109008708761084415377}{5774721337563528491}e^{2} - \frac{41190881775707565929}{40423049362944699437}e - \frac{53515687508385896327}{40423049362944699437}$ |
| 19 | $[19, 19, -w^{2} + 6]$ | $-\frac{159004110069899763}{323384394903557595496}e^{11} + \frac{1100362658285904987}{323384394903557595496}e^{10} + \frac{7456649656901597841}{323384394903557595496}e^{9} - \frac{14031056898767119027}{80846098725889398874}e^{8} - \frac{9826340632268944555}{40423049362944699437}e^{7} + \frac{788116081982594543565}{323384394903557595496}e^{6} + \frac{70186973408556936187}{323384394903557595496}e^{5} - \frac{1915455566037182150481}{161692197451778797748}e^{4} + \frac{27120041070719265717}{11549442675127056982}e^{3} + \frac{109008708761084415377}{5774721337563528491}e^{2} - \frac{41190881775707565929}{40423049362944699437}e - \frac{53515687508385896327}{40423049362944699437}$ |
| 25 | $[25, 5, -2w^{2} + 2w + 11]$ | $\phantom{-}\frac{70266084504776355}{323384394903557595496}e^{11} - \frac{220446699194268629}{323384394903557595496}e^{10} - \frac{4498962361171508007}{323384394903557595496}e^{9} + \frac{4478641876833724377}{161692197451778797748}e^{8} + \frac{11883927094364839202}{40423049362944699437}e^{7} - \frac{57070943393202499029}{323384394903557595496}e^{6} - \frac{860025901969912848701}{323384394903557595496}e^{5} - \frac{69545181201578951263}{80846098725889398874}e^{4} + \frac{65728847330603063086}{5774721337563528491}e^{3} + \frac{30468210334243713547}{5774721337563528491}e^{2} - \frac{629845232122681525494}{40423049362944699437}e + \frac{23708881037097234849}{40423049362944699437}$ |
| 29 | $[29, 29, w]$ | $\phantom{-}\frac{198321533993759243}{161692197451778797748}e^{11} - \frac{1070412726352570483}{161692197451778797748}e^{10} - \frac{10491654330882917405}{161692197451778797748}e^{9} + \frac{26029691968366027945}{80846098725889398874}e^{8} + \frac{75893663408076381211}{80846098725889398874}e^{7} - \frac{658511012536058529115}{161692197451778797748}e^{6} - \frac{640565695447093436339}{161692197451778797748}e^{5} + \frac{1349690068150460441877}{80846098725889398874}e^{4} + \frac{20120134849113226347}{11549442675127056982}e^{3} - \frac{254928388533800035697}{11549442675127056982}e^{2} + \frac{449337141217892970439}{40423049362944699437}e + \frac{274705403298587598285}{40423049362944699437}$ |
| 29 | $[29, 29, 2w^{2} - w - 10]$ | $-\frac{73917586864187845}{80846098725889398874}e^{11} + \frac{765310741162600963}{161692197451778797748}e^{10} + \frac{7824094695310261287}{161692197451778797748}e^{9} - \frac{35920597950579886179}{161692197451778797748}e^{8} - \frac{28397774805288270981}{40423049362944699437}e^{7} + \frac{102293073197994245543}{40423049362944699437}e^{6} + \frac{494292841057548116343}{161692197451778797748}e^{5} - \frac{1197453204609488753359}{161692197451778797748}e^{4} - \frac{26102542429579211823}{11549442675127056982}e^{3} + \frac{4839759457870191193}{11549442675127056982}e^{2} - \frac{266271353166481505736}{40423049362944699437}e + \frac{236001256404172190347}{40423049362944699437}$ |
| 29 | $[29, 29, -2w^{2} + 3w + 9]$ | $-\frac{73917586864187845}{80846098725889398874}e^{11} + \frac{765310741162600963}{161692197451778797748}e^{10} + \frac{7824094695310261287}{161692197451778797748}e^{9} - \frac{35920597950579886179}{161692197451778797748}e^{8} - \frac{28397774805288270981}{40423049362944699437}e^{7} + \frac{102293073197994245543}{40423049362944699437}e^{6} + \frac{494292841057548116343}{161692197451778797748}e^{5} - \frac{1197453204609488753359}{161692197451778797748}e^{4} - \frac{26102542429579211823}{11549442675127056982}e^{3} + \frac{4839759457870191193}{11549442675127056982}e^{2} - \frac{266271353166481505736}{40423049362944699437}e + \frac{236001256404172190347}{40423049362944699437}$ |
| 29 | $[29, 29, w - 1]$ | $\phantom{-}\frac{198321533993759243}{161692197451778797748}e^{11} - \frac{1070412726352570483}{161692197451778797748}e^{10} - \frac{10491654330882917405}{161692197451778797748}e^{9} + \frac{26029691968366027945}{80846098725889398874}e^{8} + \frac{75893663408076381211}{80846098725889398874}e^{7} - \frac{658511012536058529115}{161692197451778797748}e^{6} - \frac{640565695447093436339}{161692197451778797748}e^{5} + \frac{1349690068150460441877}{80846098725889398874}e^{4} + \frac{20120134849113226347}{11549442675127056982}e^{3} - \frac{254928388533800035697}{11549442675127056982}e^{2} + \frac{449337141217892970439}{40423049362944699437}e + \frac{274705403298587598285}{40423049362944699437}$ |
| 31 | $[31, 31, w^{3} - 6w - 6]$ | $\phantom{-}\frac{77552408814454605}{80846098725889398874}e^{11} - \frac{603908907763942933}{161692197451778797748}e^{10} - \frac{9732004170180454625}{161692197451778797748}e^{9} + \frac{29652055076912779345}{161692197451778797748}e^{8} + \frac{48487951022598650814}{40423049362944699437}e^{7} - \frac{98186137124785720548}{40423049362944699437}e^{6} - \frac{1468558663197489240929}{161692197451778797748}e^{5} + \frac{1889152703064448123801}{161692197451778797748}e^{4} + \frac{288136386612647977801}{11549442675127056982}e^{3} - \frac{233596737084626863813}{11549442675127056982}e^{2} - \frac{698044387328259660146}{40423049362944699437}e + \frac{374019515723953510130}{40423049362944699437}$ |
| 31 | $[31, 31, -w^{3} + 3w^{2} + 3w - 11]$ | $\phantom{-}\frac{77552408814454605}{80846098725889398874}e^{11} - \frac{603908907763942933}{161692197451778797748}e^{10} - \frac{9732004170180454625}{161692197451778797748}e^{9} + \frac{29652055076912779345}{161692197451778797748}e^{8} + \frac{48487951022598650814}{40423049362944699437}e^{7} - \frac{98186137124785720548}{40423049362944699437}e^{6} - \frac{1468558663197489240929}{161692197451778797748}e^{5} + \frac{1889152703064448123801}{161692197451778797748}e^{4} + \frac{288136386612647977801}{11549442675127056982}e^{3} - \frac{233596737084626863813}{11549442675127056982}e^{2} - \frac{698044387328259660146}{40423049362944699437}e + \frac{374019515723953510130}{40423049362944699437}$ |
| 41 | $[41, 41, w^{3} - 4w^{2} - 2w + 16]$ | $\phantom{-}\frac{402486608189708411}{323384394903557595496}e^{11} - \frac{2987563929176857803}{323384394903557595496}e^{10} - \frac{17715617248321191369}{323384394903557595496}e^{9} + \frac{38012145675630483981}{80846098725889398874}e^{8} + \frac{17911417500041685752}{40423049362944699437}e^{7} - \frac{2112328417678572405101}{323384394903557595496}e^{6} + \frac{513625226348971688485}{323384394903557595496}e^{5} + \frac{4917901820179651996009}{161692197451778797748}e^{4} - \frac{174192628573947076769}{11549442675127056982}e^{3} - \frac{255360920875431605692}{5774721337563528491}e^{2} + \frac{731474456771715328459}{40423049362944699437}e + \frac{368349078824565582232}{40423049362944699437}$ |
| 41 | $[41, 41, w^{3} - 5w^{2} - 2w + 24]$ | $\phantom{-}\frac{402486608189708411}{323384394903557595496}e^{11} - \frac{2987563929176857803}{323384394903557595496}e^{10} - \frac{17715617248321191369}{323384394903557595496}e^{9} + \frac{38012145675630483981}{80846098725889398874}e^{8} + \frac{17911417500041685752}{40423049362944699437}e^{7} - \frac{2112328417678572405101}{323384394903557595496}e^{6} + \frac{513625226348971688485}{323384394903557595496}e^{5} + \frac{4917901820179651996009}{161692197451778797748}e^{4} - \frac{174192628573947076769}{11549442675127056982}e^{3} - \frac{255360920875431605692}{5774721337563528491}e^{2} + \frac{731474456771715328459}{40423049362944699437}e + \frac{368349078824565582232}{40423049362944699437}$ |
| 59 | $[59, 59, w^{3} - w^{2} - 5w - 2]$ | $\phantom{-}\frac{271147792955171011}{161692197451778797748}e^{11} - \frac{1379904749226493773}{161692197451778797748}e^{10} - \frac{14883641075259301893}{161692197451778797748}e^{9} + \frac{33359765078926764165}{80846098725889398874}e^{8} + \frac{59171875356705075270}{40423049362944699437}e^{7} - \frac{832917718830637164675}{161692197451778797748}e^{6} - \frac{1335549090717465502821}{161692197451778797748}e^{5} + \frac{1667455434878709863067}{80846098725889398874}e^{4} + \frac{106847810345796455010}{5774721337563528491}e^{3} - \frac{299456321500043542299}{11549442675127056982}e^{2} - \frac{559321698129701311965}{40423049362944699437}e + \frac{198052467551441838270}{40423049362944699437}$ |
| 59 | $[59, 59, 2w^{2} - w - 13]$ | $\phantom{-}\frac{271147792955171011}{161692197451778797748}e^{11} - \frac{1379904749226493773}{161692197451778797748}e^{10} - \frac{14883641075259301893}{161692197451778797748}e^{9} + \frac{33359765078926764165}{80846098725889398874}e^{8} + \frac{59171875356705075270}{40423049362944699437}e^{7} - \frac{832917718830637164675}{161692197451778797748}e^{6} - \frac{1335549090717465502821}{161692197451778797748}e^{5} + \frac{1667455434878709863067}{80846098725889398874}e^{4} + \frac{106847810345796455010}{5774721337563528491}e^{3} - \frac{299456321500043542299}{11549442675127056982}e^{2} - \frac{559321698129701311965}{40423049362944699437}e + \frac{198052467551441838270}{40423049362944699437}$ |
| 61 | $[61, 61, w^{3} - w^{2} - 6w + 3]$ | $-\frac{739838898242753451}{323384394903557595496}e^{11} + \frac{4067457302872091167}{323384394903557595496}e^{10} + \frac{39203365801281714961}{323384394903557595496}e^{9} - \frac{24928239804086268590}{40423049362944699437}e^{8} - \frac{72182386580770848782}{40423049362944699437}e^{7} + \frac{2570635791136129973369}{323384394903557595496}e^{6} + \frac{2775925330191622996991}{323384394903557595496}e^{5} - \frac{5431680506616891742717}{161692197451778797748}e^{4} - \frac{161109777081674850125}{11549442675127056982}e^{3} + \frac{504539926670963014479}{11549442675127056982}e^{2} + \frac{203312165753898974280}{40423049362944699437}e - \frac{174210572670420021548}{40423049362944699437}$ |
| 61 | $[61, 61, -w^{3} + 2w^{2} + 5w - 3]$ | $-\frac{739838898242753451}{323384394903557595496}e^{11} + \frac{4067457302872091167}{323384394903557595496}e^{10} + \frac{39203365801281714961}{323384394903557595496}e^{9} - \frac{24928239804086268590}{40423049362944699437}e^{8} - \frac{72182386580770848782}{40423049362944699437}e^{7} + \frac{2570635791136129973369}{323384394903557595496}e^{6} + \frac{2775925330191622996991}{323384394903557595496}e^{5} - \frac{5431680506616891742717}{161692197451778797748}e^{4} - \frac{161109777081674850125}{11549442675127056982}e^{3} + \frac{504539926670963014479}{11549442675127056982}e^{2} + \frac{203312165753898974280}{40423049362944699437}e - \frac{174210572670420021548}{40423049362944699437}$ |
Atkin-Lehner eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| $16$ | $[16, 2, 2]$ | $1$ |