Properties

Label 4.4.14197.1-13.1-b
Base field 4.4.14197.1
Weight $[2, 2, 2, 2]$
Level norm $13$
Level $[13, 13, -w + 2]$
Dimension $10$
CM no
Base change no

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Base field 4.4.14197.1

Generator \(w\), with minimal polynomial \(x^{4} - 6x^{2} - 3x + 1\); narrow class number \(2\) and class number \(1\).

Form

Weight: $[2, 2, 2, 2]$
Level: $[13, 13, -w + 2]$
Dimension: $10$
CM: no
Base change: no
Newspace dimension: $20$

Hecke eigenvalues ($q$-expansion)

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

\(x^{10} + 10x^{9} + 7x^{8} - 208x^{7} - 598x^{6} + 530x^{5} + 3318x^{4} + 1548x^{3} - 4023x^{2} - 3402x - 486\)

  Show full eigenvalues   Hide large eigenvalues

Norm Prime Eigenvalue
7 $[7, 7, w - 1]$ $\phantom{-}e$
9 $[9, 3, w^{3} - 5w - 2]$ $-\frac{52909}{2374677}e^{9} - \frac{430789}{2374677}e^{8} + \frac{418202}{2374677}e^{7} + \frac{10126540}{2374677}e^{6} + \frac{674035}{124983}e^{5} - \frac{49983008}{2374677}e^{4} - \frac{8694526}{263853}e^{3} + \frac{771501}{29317}e^{2} + \frac{3243322}{87951}e + \frac{31322}{29317}$
9 $[9, 3, w^{3} - w^{2} - 4w]$ $\phantom{-}\frac{34777}{2374677}e^{9} + \frac{264313}{2374677}e^{8} - \frac{410177}{2374677}e^{7} - \frac{6358627}{2374677}e^{6} - \frac{267043}{124983}e^{5} + \frac{33622253}{2374677}e^{4} + \frac{10738511}{791559}e^{3} - \frac{5169191}{263853}e^{2} - \frac{930455}{87951}e - \frac{22668}{29317}$
13 $[13, 13, -w + 2]$ $\phantom{-}1$
16 $[16, 2, 2]$ $\phantom{-}\frac{5467}{263853}e^{9} + \frac{136691}{791559}e^{8} - \frac{120205}{791559}e^{7} - \frac{3254386}{791559}e^{6} - \frac{225899}{41661}e^{5} + \frac{16656817}{791559}e^{4} + \frac{27418999}{791559}e^{3} - \frac{7095860}{263853}e^{2} - \frac{3851408}{87951}e - \frac{196155}{29317}$
17 $[17, 17, w^{3} - w^{2} - 5w]$ $-\frac{50264}{2374677}e^{9} - \frac{380738}{2374677}e^{8} + \frac{619762}{2374677}e^{7} + \frac{9340862}{2374677}e^{6} + \frac{371924}{124983}e^{5} - \frac{52239247}{2374677}e^{4} - \frac{5676878}{263853}e^{3} + \frac{2861981}{87951}e^{2} + \frac{691912}{29317}e + \frac{147596}{29317}$
19 $[19, 19, -w^{3} + w^{2} + 6w]$ $-\frac{16090}{2374677}e^{9} - \frac{169633}{2374677}e^{8} - \frac{98224}{2374677}e^{7} + \frac{3850861}{2374677}e^{6} + \frac{498622}{124983}e^{5} - \frac{16414163}{2374677}e^{4} - \frac{18491236}{791559}e^{3} + \frac{314269}{87951}e^{2} + \frac{2552566}{87951}e + \frac{246568}{29317}$
23 $[23, 23, -w^{2} + w + 3]$ $-\frac{3206}{2374677}e^{9} - \frac{37856}{2374677}e^{8} - \frac{77054}{2374677}e^{7} + \frac{638156}{2374677}e^{6} + \frac{162899}{124983}e^{5} + \frac{1504334}{2374677}e^{4} - \frac{4919491}{791559}e^{3} - \frac{2448314}{263853}e^{2} + \frac{527927}{87951}e + \frac{186506}{29317}$
29 $[29, 29, w^{3} - 5w]$ $\phantom{-}\frac{23326}{791559}e^{9} + \frac{192472}{791559}e^{8} - \frac{165863}{791559}e^{7} - \frac{4511812}{791559}e^{6} - \frac{326032}{41661}e^{5} + \frac{21906404}{791559}e^{4} + \frac{13077166}{263853}e^{3} - \frac{7903288}{263853}e^{2} - \frac{5666626}{87951}e - \frac{428024}{29317}$
31 $[31, 31, w^{3} - 6w - 1]$ $\phantom{-}\frac{18172}{791559}e^{9} + \frac{17042}{87951}e^{8} - \frac{12382}{87951}e^{7} - \frac{3592310}{791559}e^{6} - \frac{10189}{1543}e^{5} + \frac{646454}{29317}e^{4} + \frac{32151694}{791559}e^{3} - \frac{2159696}{87951}e^{2} - \frac{4448420}{87951}e - \frac{413058}{29317}$
31 $[31, 31, w^{2} - 2]$ $\phantom{-}\frac{4214}{263853}e^{9} + \frac{102547}{791559}e^{8} - \frac{99299}{791559}e^{7} - \frac{2400281}{791559}e^{6} - \frac{164416}{41661}e^{5} + \frac{11477240}{791559}e^{4} + \frac{19915370}{791559}e^{3} - \frac{1168057}{87951}e^{2} - \frac{2943916}{87951}e - \frac{312610}{29317}$
37 $[37, 37, -w - 3]$ $-\frac{22235}{791559}e^{9} - \frac{187655}{791559}e^{8} + \frac{114997}{791559}e^{7} + \frac{4296239}{791559}e^{6} + \frac{366188}{41661}e^{5} - \frac{18971395}{791559}e^{4} - \frac{14366744}{263853}e^{3} + \frac{4046447}{263853}e^{2} + \frac{6155014}{87951}e + \frac{605866}{29317}$
37 $[37, 37, w^{3} - 4w - 1]$ $\phantom{-}\frac{506}{13887}e^{9} + \frac{13201}{41661}e^{8} - \frac{6152}{41661}e^{7} - \frac{303977}{41661}e^{6} - \frac{515206}{41661}e^{5} + \frac{1378973}{41661}e^{4} + \frac{3129914}{41661}e^{3} - \frac{124444}{4629}e^{2} - \frac{143903}{1543}e - \frac{40434}{1543}$
37 $[37, 37, w^{3} - 7w - 4]$ $\phantom{-}\frac{52468}{2374677}e^{9} + \frac{403690}{2374677}e^{8} - \frac{603935}{2374677}e^{7} - \frac{9747253}{2374677}e^{6} - \frac{417580}{124983}e^{5} + \frac{52511612}{2374677}e^{4} + \frac{1814836}{87951}e^{3} - \frac{8905523}{263853}e^{2} - \frac{1392800}{87951}e + \frac{1630}{29317}$
37 $[37, 37, w^{2} - 3]$ $-\frac{9626}{791559}e^{9} - \frac{25256}{263853}e^{8} + \frac{29518}{263853}e^{7} + \frac{1756216}{791559}e^{6} + \frac{34061}{13887}e^{5} - \frac{2713990}{263853}e^{4} - \frac{10926950}{791559}e^{3} + \frac{741538}{87951}e^{2} + \frac{1104661}{87951}e + \frac{243216}{29317}$
43 $[43, 43, w^{2} + w - 3]$ $-\frac{5348}{124983}e^{9} - \frac{45158}{124983}e^{8} + \frac{33904}{124983}e^{7} + \frac{1061288}{124983}e^{6} + \frac{1518293}{124983}e^{5} - \frac{5180872}{124983}e^{4} - \frac{349459}{4629}e^{3} + \frac{618074}{13887}e^{2} + \frac{141313}{1543}e + \frac{27378}{1543}$
43 $[43, 43, w^{3} - w^{2} - 5w - 1]$ $-\frac{430}{41661}e^{9} - \frac{3467}{41661}e^{8} + \frac{3430}{41661}e^{7} + \frac{27431}{13887}e^{6} + \frac{113162}{41661}e^{5} - \frac{409864}{41661}e^{4} - \frac{845696}{41661}e^{3} + \frac{128416}{13887}e^{2} + \frac{156290}{4629}e + \frac{10612}{1543}$
47 $[47, 47, -w^{3} + w^{2} + 5w - 4]$ $\phantom{-}\frac{24749}{791559}e^{9} + \frac{7841}{29317}e^{8} - \frac{38797}{263853}e^{7} - \frac{4844203}{791559}e^{6} - \frac{138214}{13887}e^{5} + \frac{7204381}{263853}e^{4} + \frac{46348136}{791559}e^{3} - \frac{6269513}{263853}e^{2} - \frac{6084116}{87951}e - \frac{402384}{29317}$
53 $[53, 53, w^{3} - 6w]$ $-\frac{5248}{2374677}e^{9} - \frac{34699}{2374677}e^{8} + \frac{29195}{2374677}e^{7} + \frac{555208}{2374677}e^{6} + \frac{71269}{124983}e^{5} + \frac{1557742}{2374677}e^{4} - \frac{1773322}{791559}e^{3} - \frac{1616803}{263853}e^{2} + \frac{66759}{29317}e + \frac{7226}{29317}$
61 $[61, 61, w^{3} - w^{2} - 7w - 2]$ $-\frac{102095}{2374677}e^{9} - \frac{774602}{2374677}e^{8} + \frac{1257442}{2374677}e^{7} + \frac{19097279}{2374677}e^{6} + \frac{770873}{124983}e^{5} - \frac{108960310}{2374677}e^{4} - \frac{36382330}{791559}e^{3} + \frac{19810810}{263853}e^{2} + \frac{4828276}{87951}e - \frac{214346}{29317}$
Display number of eigenvalues

Atkin-Lehner eigenvalues

Norm Prime Eigenvalue
$13$ $[13, 13, -w + 2]$ $-1$