Base field 4.4.19225.1
Generator \(w\), with minimal polynomial \(x^{4} - x^{3} - 15x^{2} + 2x + 44\); narrow class number \(1\) and class number \(1\).
Form
| Weight: | $[2, 2, 2, 2]$ |
| Level: | $[16,4,\frac{1}{2}w^{3} - \frac{3}{2}w^{2} - \frac{5}{2}w + 8]$ |
| Dimension: | $12$ |
| CM: | no |
| Base change: | no |
| Newspace dimension: | $26$ |
Hecke eigenvalues ($q$-expansion)
The Hecke eigenvalue field is $\Q(e)$ where $e$ is a root of the defining polynomial:
| \(x^{12} + 4x^{11} - 23x^{10} - 109x^{9} + 126x^{8} + 940x^{7} + 140x^{6} - 3015x^{5} - 1835x^{4} + 3250x^{3} + 1835x^{2} - 1350x - 220\) |
Show full eigenvalues Hide large eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| 4 | $[4, 2, w + 2]$ | $\phantom{-}e$ |
| 4 | $[4, 2, -\frac{1}{2}w^{3} + \frac{3}{2}w^{2} + \frac{9}{2}w - 10]$ | $\phantom{-}0$ |
| 9 | $[9, 3, \frac{1}{2}w^{3} - \frac{5}{2}w^{2} - \frac{5}{2}w + 17]$ | $\phantom{-}\frac{10074419}{122946112}e^{11} + \frac{22294333}{122946112}e^{10} - \frac{135391157}{61473056}e^{9} - \frac{32312799}{6470848}e^{8} + \frac{2347002283}{122946112}e^{7} + \frac{5271418953}{122946112}e^{6} - \frac{7843515241}{122946112}e^{5} - \frac{4094641757}{30736528}e^{4} + \frac{532290153}{6470848}e^{3} + \frac{14906832519}{122946112}e^{2} - \frac{179289133}{3235424}e - \frac{403475125}{30736528}$ |
| 9 | $[9, 3, \frac{3}{2}w^{3} - \frac{11}{2}w^{2} - \frac{19}{2}w + 28]$ | $-\frac{2932849}{122946112}e^{11} - \frac{5028135}{122946112}e^{10} + \frac{42449823}{61473056}e^{9} + \frac{7478197}{6470848}e^{8} - \frac{841906049}{122946112}e^{7} - \frac{1277542819}{122946112}e^{6} + \frac{3559516939}{122946112}e^{5} + \frac{1096030283}{30736528}e^{4} - \frac{339256979}{6470848}e^{3} - \frac{5059755109}{122946112}e^{2} + \frac{110986983}{3235424}e + \frac{111276015}{30736528}$ |
| 11 | $[11, 11, \frac{1}{2}w^{3} - \frac{3}{2}w^{2} - \frac{9}{2}w + 11]$ | $-e - 1$ |
| 11 | $[11, 11, -w - 3]$ | $\phantom{-}\frac{7090775}{122946112}e^{11} + \frac{12801159}{122946112}e^{10} - \frac{100996771}{61473056}e^{9} - \frac{19043907}{6470848}e^{8} + \frac{1954376237}{122946112}e^{7} + \frac{3242769517}{122946112}e^{6} - \frac{8000693859}{122946112}e^{5} - \frac{1369991299}{15368264}e^{4} + \frac{745277481}{6470848}e^{3} + \frac{12020159405}{122946112}e^{2} - \frac{252692935}{3235424}e - \frac{305968123}{30736528}$ |
| 25 | $[25, 5, w^{3} - 3w^{2} - 7w + 15]$ | $\phantom{-}\frac{3440467}{30736528}e^{11} + \frac{429023}{1921033}e^{10} - \frac{23766381}{7684132}e^{9} - \frac{10072871}{1617712}e^{8} + \frac{434529283}{15368264}e^{7} + \frac{1675126053}{30736528}e^{6} - \frac{803570219}{7684132}e^{5} - \frac{2697685115}{15368264}e^{4} + \frac{257775547}{1617712}e^{3} + \frac{1329256341}{7684132}e^{2} - \frac{10639741}{101107}e - \frac{67662449}{3842066}$ |
| 29 | $[29, 29, w + 1]$ | $-\frac{1924175}{30736528}e^{11} - \frac{6132361}{61473056}e^{10} + \frac{53111595}{30736528}e^{9} + \frac{4538751}{1617712}e^{8} - \frac{970601165}{61473056}e^{7} - \frac{750399123}{30736528}e^{6} + \frac{3544889905}{61473056}e^{5} + \frac{2352863601}{30736528}e^{4} - \frac{65885989}{808856}e^{3} - \frac{4274917839}{61473056}e^{2} + \frac{67533849}{1617712}e + \frac{64981495}{15368264}$ |
| 29 | $[29, 29, \frac{1}{2}w^{3} - \frac{3}{2}w^{2} - \frac{9}{2}w + 9]$ | $-\frac{2411827}{30736528}e^{11} - \frac{11503385}{61473056}e^{10} + \frac{64410851}{30736528}e^{9} + \frac{8235139}{1617712}e^{8} - \frac{1098013293}{61473056}e^{7} - \frac{1319792471}{30736528}e^{6} + \frac{3532513121}{61473056}e^{5} + \frac{3958723345}{30736528}e^{4} - \frac{56318879}{808856}e^{3} - \frac{6617701055}{61473056}e^{2} + \frac{85101633}{1617712}e + \frac{169493255}{15368264}$ |
| 31 | $[31, 31, -\frac{1}{2}w^{3} + \frac{5}{2}w^{2} + \frac{5}{2}w - 16]$ | $\phantom{-}\frac{1755}{1921033}e^{11} + \frac{197721}{15368264}e^{10} - \frac{219401}{7684132}e^{9} - \frac{34752}{101107}e^{8} + \frac{5046673}{15368264}e^{7} + \frac{5459758}{1921033}e^{6} - \frac{32964593}{15368264}e^{5} - \frac{55237721}{7684132}e^{4} + \frac{3431015}{404428}e^{3} - \frac{24978617}{15368264}e^{2} - \frac{5838183}{404428}e + \frac{29142699}{3842066}$ |
| 31 | $[31, 31, \frac{1}{2}w^{3} - \frac{3}{2}w^{2} - \frac{9}{2}w + 5]$ | $\phantom{-}\frac{517999}{61473056}e^{11} + \frac{5266653}{61473056}e^{10} - \frac{3165821}{30736528}e^{9} - \frac{7190699}{3235424}e^{8} - \frac{85420765}{61473056}e^{7} + \frac{1087757069}{61473056}e^{6} + \frac{1273216071}{61473056}e^{5} - \frac{720935459}{15368264}e^{4} - \frac{211092699}{3235424}e^{3} + \frac{1422227351}{61473056}e^{2} + \frac{58049475}{1617712}e + \frac{9919491}{15368264}$ |
| 31 | $[31, 31, -w + 3]$ | $-\frac{14573257}{122946112}e^{11} - \frac{35636729}{122946112}e^{10} + \frac{192634365}{61473056}e^{9} + \frac{51240285}{6470848}e^{8} - \frac{3216903059}{122946112}e^{7} - \frac{8268981683}{122946112}e^{6} + \frac{9853545373}{122946112}e^{5} + \frac{3134649285}{15368264}e^{4} - \frac{551298935}{6470848}e^{3} - \frac{21147477651}{122946112}e^{2} + \frac{202990713}{3235424}e + \frac{553850677}{30736528}$ |
| 31 | $[31, 31, \frac{3}{2}w^{3} - \frac{11}{2}w^{2} - \frac{19}{2}w + 29]$ | $-\frac{971721}{61473056}e^{11} - \frac{3152015}{61473056}e^{10} + \frac{14208415}{30736528}e^{9} + \frac{4755437}{3235424}e^{8} - \frac{285781081}{61473056}e^{7} - \frac{851058395}{61473056}e^{6} + \frac{1263301235}{61473056}e^{5} + \frac{780066623}{15368264}e^{4} - \frac{136806651}{3235424}e^{3} - \frac{3754575773}{61473056}e^{2} + \frac{59040775}{1617712}e + \frac{48890231}{15368264}$ |
| 59 | $[59, 59, 2w^{2} - w - 13]$ | $\phantom{-}\frac{2877473}{61473056}e^{11} + \frac{7853923}{61473056}e^{10} - \frac{37002451}{30736528}e^{9} - \frac{11074405}{3235424}e^{8} + \frac{577315197}{61473056}e^{7} + \frac{1728336643}{61473056}e^{6} - \frac{1456387175}{61473056}e^{5} - \frac{1209944581}{15368264}e^{4} + \frac{37719083}{3235424}e^{3} + \frac{3004443209}{61473056}e^{2} - \frac{25233475}{1617712}e - \frac{43566995}{15368264}$ |
| 59 | $[59, 59, \frac{9}{2}w^{3} - \frac{31}{2}w^{2} - \frac{61}{2}w + 85]$ | $\phantom{-}\frac{2892139}{122946112}e^{11} + \frac{9959327}{122946112}e^{10} - \frac{32475299}{61473056}e^{9} - \frac{13832519}{6470848}e^{8} + \frac{323818541}{122946112}e^{7} + \frac{2085239577}{122946112}e^{6} + \frac{766850037}{122946112}e^{5} - \frac{330042617}{7684132}e^{4} - \frac{341493907}{6470848}e^{3} + \frac{1886995333}{122946112}e^{2} + \frac{158582369}{3235424}e - \frac{35235675}{30736528}$ |
| 61 | $[61, 61, 2w^{3} - 6w^{2} - 15w + 31]$ | $\phantom{-}\frac{20776505}{122946112}e^{11} + \frac{43907711}{122946112}e^{10} - \frac{281830551}{61473056}e^{9} - \frac{63445821}{6470848}e^{8} + \frac{4968538953}{122946112}e^{7} + \frac{10273917531}{122946112}e^{6} - \frac{17114367555}{122946112}e^{5} - \frac{7831699667}{30736528}e^{4} + \frac{1217205323}{6470848}e^{3} + \frac{26743793677}{122946112}e^{2} - \frac{401507791}{3235424}e - \frac{424868183}{30736528}$ |
| 61 | $[61, 61, -\frac{3}{2}w^{3} + \frac{11}{2}w^{2} + \frac{21}{2}w - 34]$ | $-\frac{218049}{7684132}e^{11} - \frac{6123819}{61473056}e^{10} + \frac{21197429}{30736528}e^{9} + \frac{534419}{202214}e^{8} - \frac{291651411}{61473056}e^{7} - \frac{329278061}{15368264}e^{6} + \frac{524424507}{61473056}e^{5} + \frac{1787952289}{30736528}e^{4} - \frac{1275661}{1617712}e^{3} - \frac{1796631205}{61473056}e^{2} + \frac{34477451}{1617712}e - \frac{157067879}{15368264}$ |
| 71 | $[71, 71, \frac{3}{2}w^{3} - \frac{11}{2}w^{2} - \frac{19}{2}w + 32]$ | $-\frac{342963}{3842066}e^{11} - \frac{4710341}{30736528}e^{10} + \frac{38008839}{15368264}e^{9} + \frac{429303}{101107}e^{8} - \frac{699625005}{30736528}e^{7} - \frac{278780477}{7684132}e^{6} + \frac{2610807077}{30736528}e^{5} + \frac{1692295947}{15368264}e^{4} - \frac{103845603}{808856}e^{3} - \frac{2803467755}{30736528}e^{2} + \frac{61846369}{808856}e - \frac{2262437}{7684132}$ |
| 71 | $[71, 71, -\frac{1}{2}w^{3} + \frac{5}{2}w^{2} + \frac{5}{2}w - 13]$ | $\phantom{-}\frac{223637}{15368264}e^{11} + \frac{484003}{30736528}e^{10} - \frac{6089445}{15368264}e^{9} - \frac{327253}{808856}e^{8} + \frac{108719327}{30736528}e^{7} + \frac{42066317}{15368264}e^{6} - \frac{382300651}{30736528}e^{5} - \frac{51328431}{15368264}e^{4} + \frac{7175599}{404428}e^{3} - \frac{272384955}{30736528}e^{2} - \frac{11461863}{808856}e + \frac{51051163}{7684132}$ |
| 79 | $[79, 79, -3w^{3} + 10w^{2} + 19w - 51]$ | $-\frac{2108127}{15368264}e^{11} - \frac{2288993}{7684132}e^{10} + \frac{14441269}{3842066}e^{9} + \frac{6708091}{808856}e^{8} - \frac{65049912}{1921033}e^{7} - \frac{1117800277}{15368264}e^{6} + \frac{943279647}{7684132}e^{5} + \frac{1815246995}{7684132}e^{4} - \frac{150344819}{808856}e^{3} - \frac{1851684169}{7684132}e^{2} + \frac{13038143}{101107}e + \frac{58164940}{1921033}$ |
Atkin-Lehner eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| $4$ | $[4,2,\frac{1}{2}w^{3} - \frac{3}{2}w^{2} - \frac{9}{2}w + 10]$ | $-1$ |