Properties

Label 4.4.6809.1-31.1-b
Base field 4.4.6809.1
Weight $[2, 2, 2, 2]$
Level norm $31$
Level $[31, 31, -w^{3} + 6w + 2]$
Dimension $14$
CM no
Base change no

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Base field 4.4.6809.1

Generator \(w\), with minimal polynomial \(x^{4} - 5x^{2} - x + 1\); narrow class number \(1\) and class number \(1\).

Form

Weight: $[2, 2, 2, 2]$
Level: $[31, 31, -w^{3} + 6w + 2]$
Dimension: $14$
CM: no
Base change: no
Newspace dimension: $19$

Hecke eigenvalues ($q$-expansion)

The Hecke eigenvalue field is $\Q(e)$ where $e$ is a root of the defining polynomial:

\(x^{14} - 5x^{13} - 10x^{12} + 80x^{11} - 477x^{9} + 305x^{8} + 1313x^{7} - 1265x^{6} - 1603x^{5} + 1976x^{4} + 558x^{3} - 1101x^{2} + 143x + 83\)

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Norm Prime Eigenvalue
2 $[2, 2, w + 1]$ $\phantom{-}e$
5 $[5, 5, w^{3} - 4w]$ $-\frac{57}{391}e^{13} + \frac{13}{23}e^{12} + \frac{825}{391}e^{11} - \frac{3716}{391}e^{10} - \frac{3994}{391}e^{9} + \frac{23658}{391}e^{8} + \frac{5824}{391}e^{7} - \frac{71272}{391}e^{6} + \frac{9250}{391}e^{5} + \frac{101462}{391}e^{4} - \frac{31835}{391}e^{3} - \frac{57021}{391}e^{2} + \frac{19477}{391}e + \frac{6920}{391}$
8 $[8, 2, w^{3} - w^{2} - 4w + 3]$ $\phantom{-}\frac{361}{391}e^{13} - \frac{90}{23}e^{12} - \frac{4834}{391}e^{11} + \frac{25229}{391}e^{10} + \frac{20473}{391}e^{9} - \frac{157263}{391}e^{8} - \frac{19551}{391}e^{7} + \frac{463771}{391}e^{6} - \frac{68228}{391}e^{5} - \frac{644287}{391}e^{4} + \frac{165780}{391}e^{3} + \frac{344711}{391}e^{2} - \frac{100155}{391}e - \frac{32618}{391}$
11 $[11, 11, w^{3} - 5w + 1]$ $-\frac{6}{391}e^{13} + \frac{5}{23}e^{12} - \frac{263}{391}e^{11} - \frac{741}{391}e^{10} + \frac{4251}{391}e^{9} + \frac{62}{391}e^{8} - \frac{20645}{391}e^{7} + \frac{12521}{391}e^{6} + \frac{39765}{391}e^{5} - \frac{30169}{391}e^{4} - \frac{26667}{391}e^{3} + \frac{16079}{391}e^{2} + \frac{3038}{391}e + \frac{1140}{391}$
17 $[17, 17, -w^{3} - w^{2} + 5w + 4]$ $\phantom{-}\frac{135}{391}e^{13} - \frac{32}{23}e^{12} - \frac{1707}{391}e^{11} + \frac{8266}{391}e^{10} + \frac{6990}{391}e^{9} - \frac{46360}{391}e^{8} - \frac{9966}{391}e^{7} + \frac{120421}{391}e^{6} + \frac{1264}{391}e^{5} - \frac{144839}{391}e^{4} + \frac{5101}{391}e^{3} + \frac{66172}{391}e^{2} - \frac{5795}{391}e - \frac{4536}{391}$
23 $[23, 23, -w^{2} + 2]$ $\phantom{-}\frac{753}{391}e^{13} - \frac{179}{23}e^{12} - \frac{10590}{391}e^{11} + \frac{50963}{391}e^{10} + \frac{49285}{391}e^{9} - \frac{323318}{391}e^{8} - \frac{68830}{391}e^{7} + \frac{971092}{391}e^{6} - \frac{99684}{391}e^{5} - \frac{1374404}{391}e^{4} + \frac{337377}{391}e^{3} + \frac{750561}{391}e^{2} - \frac{221741}{391}e - \frac{73081}{391}$
29 $[29, 29, -w^{2} - w + 3]$ $\phantom{-}\frac{921}{391}e^{13} - \frac{227}{23}e^{12} - \frac{12219}{391}e^{11} + \frac{63109}{391}e^{10} + \frac{50685}{391}e^{9} - \frac{389569}{391}e^{8} - \frac{42862}{391}e^{7} + \frac{1137797}{391}e^{6} - \frac{185165}{391}e^{5} - \frac{1570905}{391}e^{4} + \frac{421699}{391}e^{3} + \frac{845012}{391}e^{2} - \frac{244636}{391}e - \frac{85060}{391}$
31 $[31, 31, -w^{3} + 6w + 2]$ $-1$
43 $[43, 43, w^{3} - 6w]$ $-\frac{453}{391}e^{13} + \frac{113}{23}e^{12} + \frac{5754}{391}e^{11} - \frac{30726}{391}e^{10} - \frac{21761}{391}e^{9} + \frac{184541}{391}e^{8} + \frac{9017}{391}e^{7} - \frac{522283}{391}e^{6} + \frac{102797}{391}e^{5} + \frac{696773}{391}e^{4} - \frac{193058}{391}e^{3} - \frac{361570}{391}e^{2} + \frac{101903}{391}e + \frac{35631}{391}$
47 $[47, 47, 2w^{3} - 9w]$ $\phantom{-}\frac{1095}{391}e^{13} - \frac{257}{23}e^{12} - \frac{15540}{391}e^{11} + \frac{73259}{391}e^{10} + \frac{73640}{391}e^{9} - \frac{465266}{391}e^{8} - \frac{110030}{391}e^{7} + \frac{1398724}{391}e^{6} - \frac{122340}{391}e^{5} - \frac{1982394}{391}e^{4} + \frac{466609}{391}e^{3} + \frac{1086431}{391}e^{2} - \frac{310842}{391}e - \frac{108345}{391}$
47 $[47, 47, -2w^{3} + w^{2} + 8w - 2]$ $\phantom{-}\frac{839}{391}e^{13} - \frac{197}{23}e^{12} - \frac{12164}{391}e^{11} + \frac{56892}{391}e^{10} + \frac{59907}{391}e^{9} - \frac{366565}{391}e^{8} - \frac{100837}{391}e^{7} + \frac{1117588}{391}e^{6} - \frac{59689}{391}e^{5} - \frac{1602120}{391}e^{4} + \frac{327822}{391}e^{3} + \frac{882422}{391}e^{2} - \frac{230617}{391}e - \frac{87075}{391}$
53 $[53, 53, -4w^{3} + 2w^{2} + 18w - 5]$ $-\frac{658}{391}e^{13} + \frac{165}{23}e^{12} + \frac{8824}{391}e^{11} - \frac{46464}{391}e^{10} - \frac{37415}{391}e^{9} + \frac{291317}{391}e^{8} + \frac{35924}{391}e^{7} - \frac{865469}{391}e^{6} + \frac{122455}{391}e^{5} + \frac{1214815}{391}e^{4} - \frac{295788}{391}e^{3} - \frac{661782}{391}e^{2} + \frac{178201}{391}e + \frac{67543}{391}$
59 $[59, 59, w^{2} - w - 5]$ $\phantom{-}\frac{1177}{391}e^{13} - \frac{287}{23}e^{12} - \frac{15986}{391}e^{11} + \frac{80649}{391}e^{10} + \frac{69501}{391}e^{9} - \frac{504301}{391}e^{8} - \frac{73951}{391}e^{7} + \frac{1494396}{391}e^{6} - \frac{213017}{391}e^{5} - \frac{2097022}{391}e^{4} + \frac{551102}{391}e^{3} + \frac{1151854}{391}e^{2} - \frac{336591}{391}e - \frac{120406}{391}$
59 $[59, 59, 3w^{3} - 15w - 5]$ $\phantom{-}\frac{574}{391}e^{13} - \frac{141}{23}e^{12} - \frac{7423}{391}e^{11} + \frac{38436}{391}e^{10} + \frac{29286}{391}e^{9} - \frac{231408}{391}e^{8} - \frac{18410}{391}e^{7} + \frac{656410}{391}e^{6} - \frac{122138}{391}e^{5} - \frac{879032}{391}e^{4} + \frac{250108}{391}e^{3} + \frac{461871}{391}e^{2} - \frac{136451}{391}e - \frac{47282}{391}$
71 $[71, 71, w^{3} - 3w - 3]$ $-\frac{235}{391}e^{13} + \frac{54}{23}e^{12} + \frac{3319}{391}e^{11} - \frac{15142}{391}e^{10} - \frac{15904}{391}e^{9} + \frac{94574}{391}e^{8} + \frac{26124}{391}e^{7} - \frac{280190}{391}e^{6} + \frac{16336}{391}e^{5} + \frac{391830}{391}e^{4} - \frac{87485}{391}e^{3} - \frac{210042}{391}e^{2} + \frac{61381}{391}e + \frac{18844}{391}$
81 $[81, 3, -3]$ $-\frac{2413}{391}e^{13} + \frac{581}{23}e^{12} + \frac{32970}{391}e^{11} - \frac{162915}{391}e^{10} - \frac{145880}{391}e^{9} + \frac{1015598}{391}e^{8} + \frac{173693}{391}e^{7} - \frac{2996328}{391}e^{6} + \frac{369948}{391}e^{5} + \frac{4176882}{391}e^{4} - \frac{1038140}{391}e^{3} - \frac{2267655}{391}e^{2} + \frac{646882}{391}e + \frac{229735}{391}$
83 $[83, 83, -2w^{3} + 8w + 3]$ $\phantom{-}\frac{1352}{391}e^{13} - \frac{337}{23}e^{12} - \frac{17634}{391}e^{11} + \frac{93073}{391}e^{10} + \frac{70438}{391}e^{9} - \frac{569712}{391}e^{8} - \frac{45728}{391}e^{7} + \frac{1647210}{391}e^{6} - \frac{299730}{391}e^{5} - \frac{2248616}{391}e^{4} + \frac{631150}{391}e^{3} + \frac{1196983}{391}e^{2} - \frac{358208}{391}e - \frac{120421}{391}$
89 $[89, 89, -2w^{3} + 11w - 2]$ $-\frac{689}{391}e^{13} + \frac{164}{23}e^{12} + \frac{9355}{391}e^{11} - \frac{45405}{391}e^{10} - \frac{41453}{391}e^{9} + \frac{278995}{391}e^{8} + \frac{53140}{391}e^{7} - \frac{811269}{391}e^{6} + \frac{83337}{391}e^{5} + \frac{1116093}{391}e^{4} - \frac{253121}{391}e^{3} - \frac{599365}{391}e^{2} + \frac{158968}{391}e + \frac{62094}{391}$
101 $[101, 101, 2w^{3} - w^{2} - 10w]$ $\phantom{-}\frac{30}{17}e^{13} - 7e^{12} - \frac{436}{17}e^{11} + \frac{2022}{17}e^{10} + \frac{2154}{17}e^{9} - \frac{13026}{17}e^{8} - \frac{3654}{17}e^{7} + \frac{39684}{17}e^{6} - \frac{2016}{17}e^{5} - \frac{56810}{17}e^{4} + \frac{11530}{17}e^{3} + \frac{31261}{17}e^{2} - \frac{8084}{17}e - \frac{3218}{17}$
101 $[101, 101, 2w^{3} - 2w^{2} - 10w + 3]$ $\phantom{-}\frac{1066}{391}e^{13} - \frac{252}{23}e^{12} - \frac{15182}{391}e^{11} + \frac{72219}{391}e^{10} + \frac{72095}{391}e^{9} - \frac{461708}{391}e^{8} - \frac{105743}{391}e^{7} + \frac{1398963}{391}e^{6} - \frac{136395}{391}e^{5} - \frac{2001201}{391}e^{4} + \frac{494705}{391}e^{3} + \frac{1109732}{391}e^{2} - \frac{326917}{391}e - \frac{111046}{391}$
Display number of eigenvalues

Atkin-Lehner eigenvalues

Norm Prime Eigenvalue
$31$ $[31, 31, -w^{3} + 6w + 2]$ $1$