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: | $[16, 4, w^{2} - 8]$ |
| Dimension: | $14$ |
| CM: | no |
| Base change: | no |
| Newspace dimension: | $28$ |
Hecke eigenvalues ($q$-expansion)
The Hecke eigenvalue field is $\Q(e)$ where $e$ is a root of the defining polynomial:
| \(x^{14} + 6x^{13} - 14x^{12} - 141x^{11} - 60x^{10} + 1043x^{9} + 1491x^{8} - 2425x^{7} - 5740x^{6} - 360x^{5} + 4765x^{4} + 1475x^{3} - 1045x^{2} - 150x + 25\) |
Show full eigenvalues Hide large eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| 4 | $[4, 2, w^{2} - 2w - 6]$ | $\phantom{-}0$ |
| 4 | $[4, 2, -w^{2} + 7]$ | $\phantom{-}e$ |
| 5 | $[5, 5, -\frac{1}{2}w^{3} + 2w^{2} + \frac{7}{2}w - 14]$ | $\phantom{-}\frac{551376527}{31839983365}e^{13} + \frac{499494351}{6367996673}e^{12} - \frac{23508344819}{63679966730}e^{11} - \frac{62060676689}{31839983365}e^{10} + \frac{26266695785}{12735993346}e^{9} + \frac{103567309889}{6367996673}e^{8} + \frac{49710069379}{63679966730}e^{7} - \frac{1693709425899}{31839983365}e^{6} - \frac{2603610220}{107932147}e^{5} + \frac{382193340379}{6367996673}e^{4} + \frac{125071937444}{6367996673}e^{3} - \frac{191510674515}{6367996673}e^{2} + \frac{20759642601}{12735993346}e + \frac{33295126041}{12735993346}$ |
| 5 | $[5, 5, \frac{1}{2}w^{3} + \frac{1}{2}w^{2} - 6w - 9]$ | $\phantom{-}\frac{3419944329}{63679966730}e^{13} + \frac{8726636571}{31839983365}e^{12} - \frac{62296157449}{63679966730}e^{11} - \frac{210335578703}{31839983365}e^{10} + \frac{146098326201}{63679966730}e^{9} + \frac{1647411876037}{31839983365}e^{8} + \frac{1140129205508}{31839983365}e^{7} - \frac{4584010646554}{31839983365}e^{6} - \frac{19046226860}{107932147}e^{5} + \frac{513473688137}{6367996673}e^{4} + \frac{1902129368221}{12735993346}e^{3} + \frac{23061943157}{12735993346}e^{2} - \frac{168203423230}{6367996673}e - \frac{9340187182}{6367996673}$ |
| 11 | $[11, 11, \frac{1}{2}w^{3} - 2w^{2} - \frac{5}{2}w + 11]$ | $-\frac{234975063}{6367996673}e^{13} - \frac{3939380403}{31839983365}e^{12} + \frac{30574549084}{31839983365}e^{11} + \frac{102467283612}{31839983365}e^{10} - \frac{54487135941}{6367996673}e^{9} - \frac{933065558484}{31839983365}e^{8} + \frac{196711541906}{6367996673}e^{7} + \frac{3631760930679}{31839983365}e^{6} - \frac{4411216559}{107932147}e^{5} - \frac{1163512414239}{6367996673}e^{4} + \frac{116000828999}{6367996673}e^{3} + \frac{644527487502}{6367996673}e^{2} - \frac{48882828771}{6367996673}e - \frac{42030129531}{6367996673}$ |
| 11 | $[11, 11, \frac{1}{2}w^{3} + \frac{1}{2}w^{2} - 5w - 7]$ | $-\frac{551376527}{31839983365}e^{13} - \frac{499494351}{6367996673}e^{12} + \frac{23508344819}{63679966730}e^{11} + \frac{62060676689}{31839983365}e^{10} - \frac{26266695785}{12735993346}e^{9} - \frac{103567309889}{6367996673}e^{8} - \frac{49710069379}{63679966730}e^{7} + \frac{1693709425899}{31839983365}e^{6} + \frac{2603610220}{107932147}e^{5} - \frac{382193340379}{6367996673}e^{4} - \frac{125071937444}{6367996673}e^{3} + \frac{191510674515}{6367996673}e^{2} - \frac{20759642601}{12735993346}e - \frac{46031119387}{12735993346}$ |
| 31 | $[31, 31, \frac{1}{2}w^{2} + \frac{1}{2}w - 4]$ | $-\frac{2280099241}{63679966730}e^{13} - \frac{4737298869}{31839983365}e^{12} + \frac{53752547597}{63679966730}e^{11} + \frac{121888228302}{31839983365}e^{10} - \frac{394742841063}{63679966730}e^{9} - \frac{1082738135889}{31839983365}e^{8} + \frac{422875957014}{31839983365}e^{7} + \frac{3992622054474}{31839983365}e^{6} + \frac{1132733094}{107932147}e^{5} - \frac{1144318042215}{6367996673}e^{4} - \frac{315109182537}{12735993346}e^{3} + \frac{1112423384359}{12735993346}e^{2} - \frac{25270658137}{6367996673}e - \frac{25784909551}{6367996673}$ |
| 31 | $[31, 31, -\frac{1}{2}w^{2} + \frac{3}{2}w + 3]$ | $\phantom{-}\frac{1946650657}{31839983365}e^{13} + \frac{16532757437}{63679966730}e^{12} - \frac{41915306654}{31839983365}e^{11} - \frac{402250327091}{63679966730}e^{10} + \frac{242414704631}{31839983365}e^{9} + \frac{3224658008117}{63679966730}e^{8} - \frac{16599505276}{31839983365}e^{7} - \frac{4803672637966}{31839983365}e^{6} - \frac{7410571766}{107932147}e^{5} + \frac{774945408571}{6367996673}e^{4} + \frac{205222086388}{6367996673}e^{3} - \frac{365953028645}{12735993346}e^{2} + \frac{263105388127}{12735993346}e + \frac{7082530439}{6367996673}$ |
| 41 | $[41, 41, \frac{1}{2}w^{2} + \frac{1}{2}w - 6]$ | $\phantom{-}\frac{4038659942}{31839983365}e^{13} + \frac{32164817797}{63679966730}e^{12} - \frac{94539435814}{31839983365}e^{11} - \frac{810462083451}{63679966730}e^{10} + \frac{684384530471}{31839983365}e^{9} + \frac{6954971993057}{63679966730}e^{8} - \frac{1428115481846}{31839983365}e^{7} - \frac{12081856353666}{31839983365}e^{6} - \frac{3072202037}{107932147}e^{5} + \frac{3125011893313}{6367996673}e^{4} + \frac{200826225686}{6367996673}e^{3} - \frac{3112702689253}{12735993346}e^{2} + \frac{292398025555}{12735993346}e + \frac{79964499634}{6367996673}$ |
| 41 | $[41, 41, 2w^{3} - \frac{15}{2}w^{2} - \frac{25}{2}w + 50]$ | $-\frac{782143801}{31839983365}e^{13} - \frac{9599643939}{63679966730}e^{12} + \frac{10266352173}{31839983365}e^{11} + \frac{44814645405}{12735993346}e^{10} + \frac{63701820576}{31839983365}e^{9} - \frac{1632546256291}{63679966730}e^{8} - \frac{1297921123354}{31839983365}e^{7} + \frac{1804380763144}{31839983365}e^{6} + \frac{16344072919}{107932147}e^{5} + \frac{120156413816}{6367996673}e^{4} - \frac{731973522069}{6367996673}e^{3} - \frac{452372736793}{12735993346}e^{2} + \frac{160284930335}{12735993346}e + \frac{4467472579}{6367996673}$ |
| 41 | $[41, 41, \frac{5}{2}w^{2} - \frac{1}{2}w - 17]$ | $\phantom{-}\frac{1334952694}{31839983365}e^{13} + \frac{6613490737}{31839983365}e^{12} - \frac{25134676362}{31839983365}e^{11} - \frac{161676527801}{31839983365}e^{10} + \frac{73328223053}{31839983365}e^{9} + \frac{261668017676}{6367996673}e^{8} + \frac{162586584004}{6367996673}e^{7} - \frac{3997496719866}{31839983365}e^{6} - \frac{15323706449}{107932147}e^{5} + \frac{728494527152}{6367996673}e^{4} + \frac{987935318445}{6367996673}e^{3} - \frac{286506350454}{6367996673}e^{2} - \frac{273722943975}{6367996673}e + \frac{39709721879}{6367996673}$ |
| 41 | $[41, 41, \frac{1}{2}w^{2} - \frac{3}{2}w - 5]$ | $-\frac{1628541037}{12735993346}e^{13} - \frac{19722320253}{31839983365}e^{12} + \frac{77366284509}{31839983365}e^{11} + \frac{474219124747}{31839983365}e^{10} - \frac{50300202154}{6367996673}e^{9} - \frac{3700158004054}{31839983365}e^{8} - \frac{837799702085}{12735993346}e^{7} + \frac{10211262016994}{31839983365}e^{6} + \frac{39079214852}{107932147}e^{5} - \frac{1079598203443}{6367996673}e^{4} - \frac{3953838308885}{12735993346}e^{3} - \frac{257490505861}{12735993346}e^{2} + \frac{803063645175}{12735993346}e + \frac{50127965573}{12735993346}$ |
| 61 | $[61, 61, -\frac{1}{2}w^{3} + w^{2} + \frac{7}{2}w - 1]$ | $-\frac{2803564178}{31839983365}e^{13} - \frac{15879190614}{31839983365}e^{12} + \frac{43596537914}{31839983365}e^{11} + \frac{374705571432}{31839983365}e^{10} + \frac{57209733389}{31839983365}e^{9} - \frac{560226473268}{6367996673}e^{8} - \frac{641749657574}{6367996673}e^{7} + \frac{6782783385997}{31839983365}e^{6} + \frac{42788848627}{107932147}e^{5} - \frac{66205821518}{6367996673}e^{4} - \frac{1743744450596}{6367996673}e^{3} - \frac{490369513723}{6367996673}e^{2} + \frac{134753336759}{6367996673}e + \frac{41965856846}{6367996673}$ |
| 61 | $[61, 61, -\frac{1}{2}w^{3} + \frac{1}{2}w^{2} + 4w - 3]$ | $\phantom{-}\frac{1952849251}{31839983365}e^{13} + \frac{11035081127}{31839983365}e^{12} - \frac{32448281847}{31839983365}e^{11} - \frac{53528491078}{6367996673}e^{10} + \frac{1618667669}{6367996673}e^{9} + \frac{2119097985251}{31839983365}e^{8} + \frac{1900729711801}{31839983365}e^{7} - \frac{6039217541983}{31839983365}e^{6} - \frac{27735047378}{107932147}e^{5} + \frac{758133852761}{6367996673}e^{4} + \frac{1336468979478}{6367996673}e^{3} + \frac{489327698}{6367996673}e^{2} - \frac{103918338830}{6367996673}e - \frac{73050402984}{6367996673}$ |
| 71 | $[71, 71, \frac{1}{2}w^{3} + w^{2} - \frac{11}{2}w - 13]$ | $\phantom{-}\frac{4500655351}{63679966730}e^{13} + \frac{25531644013}{63679966730}e^{12} - \frac{6688847668}{6367996673}e^{11} - \frac{592474258239}{63679966730}e^{10} - \frac{16460415681}{6367996673}e^{9} + \frac{4265841568289}{63679966730}e^{8} + \frac{5682309310571}{63679966730}e^{7} - \frac{4526748215943}{31839983365}e^{6} - \frac{36325190975}{107932147}e^{5} - \frac{476392220252}{6367996673}e^{4} + \frac{2747462159451}{12735993346}e^{3} + \frac{775338322287}{6367996673}e^{2} - \frac{123059493218}{6367996673}e - \frac{177822773393}{12735993346}$ |
| 71 | $[71, 71, -\frac{1}{2}w^{3} + \frac{5}{2}w^{2} + 2w - 17]$ | $-\frac{1013993867}{63679966730}e^{13} - \frac{4818919503}{31839983365}e^{12} + \frac{1329924291}{31839983365}e^{11} + \frac{114179134399}{31839983365}e^{10} + \frac{168764971301}{31839983365}e^{9} - \frac{852302734986}{31839983365}e^{8} - \frac{3691320968903}{63679966730}e^{7} + \frac{2002523431137}{31839983365}e^{6} + \frac{20626980742}{107932147}e^{5} + \frac{40536972130}{6367996673}e^{4} - \frac{1913188280355}{12735993346}e^{3} - \frac{319307561589}{12735993346}e^{2} + \frac{296049318235}{12735993346}e - \frac{79305722503}{12735993346}$ |
| 81 | $[81, 3, -3]$ | $\phantom{-}\frac{669501953}{31839983365}e^{13} + \frac{2104310844}{31839983365}e^{12} - \frac{38156884251}{63679966730}e^{11} - \frac{57683503687}{31839983365}e^{10} + \frac{394660479379}{63679966730}e^{9} + \frac{566717325253}{31839983365}e^{8} - \frac{1833471705191}{63679966730}e^{7} - \frac{2445093516471}{31839983365}e^{6} + \frac{6379270487}{107932147}e^{5} + \frac{877500266715}{6367996673}e^{4} - \frac{291094864398}{6367996673}e^{3} - \frac{433424511953}{6367996673}e^{2} + \frac{267898306065}{12735993346}e - \frac{20382827387}{12735993346}$ |
| 89 | $[89, 89, -w^{3} + \frac{3}{2}w^{2} + \frac{13}{2}w - 9]$ | $\phantom{-}\frac{4023051926}{31839983365}e^{13} + \frac{6195034107}{12735993346}e^{12} - \frac{94703887884}{31839983365}e^{11} - \frac{770988904599}{63679966730}e^{10} + \frac{697531849037}{31839983365}e^{9} + \frac{6469933543457}{63679966730}e^{8} - \frac{320966314769}{6367996673}e^{7} - \frac{2147373747026}{6367996673}e^{6} + \frac{316223792}{107932147}e^{5} + \frac{2489666673031}{6367996673}e^{4} - \frac{253045569724}{6367996673}e^{3} - \frac{2274689050433}{12735993346}e^{2} + \frac{664875634717}{12735993346}e + \frac{44335293075}{6367996673}$ |
| 89 | $[89, 89, w^{3} - \frac{7}{2}w^{2} - \frac{11}{2}w + 20]$ | $-\frac{3969201467}{31839983365}e^{13} - \frac{35103043177}{63679966730}e^{12} + \frac{165883808091}{63679966730}e^{11} + \frac{852663850461}{63679966730}e^{10} - \frac{867239566169}{63679966730}e^{9} - \frac{6809319657213}{63679966730}e^{8} - \frac{883866904127}{63679966730}e^{7} + \frac{2010519675931}{6367996673}e^{6} + \frac{19693278175}{107932147}e^{5} - \frac{1591161689401}{6367996673}e^{4} - \frac{611102306065}{6367996673}e^{3} + \frac{947177541955}{12735993346}e^{2} - \frac{166181582641}{6367996673}e - \frac{28580876985}{12735993346}$ |
| 89 | $[89, 89, -w^{3} - \frac{1}{2}w^{2} + \frac{19}{2}w + 12]$ | $\phantom{-}\frac{7447351562}{31839983365}e^{13} + \frac{72512312911}{63679966730}e^{12} - \frac{287698247461}{63679966730}e^{11} - \frac{1760662289977}{63679966730}e^{10} + \frac{1045569945621}{63679966730}e^{9} + \frac{14024805146011}{63679966730}e^{8} + \frac{6482426557541}{63679966730}e^{7} - \frac{4101129128771}{6367996673}e^{6} - \frac{63484153179}{107932147}e^{5} + \frac{3097266224796}{6367996673}e^{4} + \frac{2813627426861}{6367996673}e^{3} - \frac{1727483510107}{12735993346}e^{2} - \frac{151125577275}{6367996673}e + \frac{125820700335}{12735993346}$ |
Atkin-Lehner eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| $4$ | $[4, 2, w^{2} - 2w - 6]$ | $-1$ |