Properties

Label 4.4.18496.1-9.2-c
Base field \(\Q(\sqrt{2}, \sqrt{17})\)
Weight $[2, 2, 2, 2]$
Level norm $9$
Level $[9,3,-\frac{1}{3}w^{3} + \frac{11}{3}w + \frac{7}{3}]$
Dimension $10$
CM no
Base change no

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Base field \(\Q(\sqrt{2}, \sqrt{17})\)

Generator \(w\), with minimal polynomial \(x^{4} - 2x^{3} - 11x^{2} + 12x + 2\); narrow class number \(2\) and class number \(1\).

Form

Weight: $[2, 2, 2, 2]$
Level: $[9,3,-\frac{1}{3}w^{3} + \frac{11}{3}w + \frac{7}{3}]$
Dimension: $10$
CM: no
Base change: no
Newspace dimension: $24$

Hecke eigenvalues ($q$-expansion)

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

\(x^{10} + 2x^{9} - 12x^{8} - 23x^{7} + 46x^{6} + 80x^{5} - 68x^{4} - 96x^{3} + 32x^{2} + 32x + 3\)

  Show full eigenvalues   Hide large eigenvalues

Norm Prime Eigenvalue
2 $[2, 2, \frac{4}{9}w^{3} - \frac{2}{3}w^{2} - \frac{47}{9}w + \frac{20}{9}]$ $\phantom{-}e$
2 $[2, 2, \frac{5}{9}w^{3} - \frac{4}{3}w^{2} - \frac{52}{9}w + \frac{79}{9}]$ $\phantom{-}\frac{3}{25}e^{9} + \frac{7}{25}e^{8} - \frac{42}{25}e^{7} - \frac{83}{25}e^{6} + \frac{202}{25}e^{5} + \frac{299}{25}e^{4} - \frac{396}{25}e^{3} - \frac{69}{5}e^{2} + \frac{231}{25}e + \frac{73}{25}$
9 $[9, 3, \frac{1}{3}w^{3} - \frac{11}{3}w - \frac{1}{3}]$ $\phantom{-}\frac{3}{25}e^{9} + \frac{2}{25}e^{8} - \frac{47}{25}e^{7} - \frac{23}{25}e^{6} + \frac{252}{25}e^{5} + \frac{79}{25}e^{4} - \frac{526}{25}e^{3} - \frac{19}{5}e^{2} + \frac{331}{25}e + \frac{13}{25}$
9 $[9, 3, w^{3} - w^{2} - 12w - 1]$ $\phantom{-}1$
17 $[17, 17, -\frac{1}{9}w^{3} - \frac{1}{3}w^{2} + \frac{14}{9}w + \frac{49}{9}]$ $-\frac{2}{5}e^{9} + \frac{26}{5}e^{7} + \frac{1}{5}e^{6} - \frac{113}{5}e^{5} - \frac{4}{5}e^{4} + \frac{187}{5}e^{3} - \frac{84}{5}e - \frac{6}{5}$
17 $[17, 17, -\frac{1}{9}w^{3} + \frac{2}{3}w^{2} + \frac{5}{9}w - \frac{59}{9}]$ $\phantom{-}\frac{16}{25}e^{9} + \frac{34}{25}e^{8} - \frac{169}{25}e^{7} - \frac{361}{25}e^{6} + \frac{494}{25}e^{5} + \frac{1048}{25}e^{4} - \frac{407}{25}e^{3} - \frac{153}{5}e^{2} + \frac{82}{25}e - \frac{59}{25}$
25 $[25, 5, -\frac{7}{9}w^{3} + \frac{2}{3}w^{2} + \frac{89}{9}w + \frac{19}{9}]$ $\phantom{-}\frac{4}{25}e^{9} - \frac{14}{25}e^{8} - \frac{71}{25}e^{7} + \frac{161}{25}e^{6} + \frac{411}{25}e^{5} - \frac{553}{25}e^{4} - \frac{843}{25}e^{3} + \frac{108}{5}e^{2} + \frac{458}{25}e + \frac{34}{25}$
25 $[25, 5, -\frac{5}{9}w^{3} + \frac{1}{3}w^{2} + \frac{52}{9}w + \frac{11}{9}]$ $-\frac{7}{25}e^{9} - \frac{13}{25}e^{8} + \frac{93}{25}e^{7} + \frac{162}{25}e^{6} - \frac{413}{25}e^{5} - \frac{626}{25}e^{4} + \frac{719}{25}e^{3} + \frac{161}{5}e^{2} - \frac{414}{25}e - \frac{222}{25}$
47 $[47, 47, -\frac{1}{9}w^{3} + \frac{2}{3}w^{2} + \frac{5}{9}w - \frac{41}{9}]$ $\phantom{-}\frac{23}{25}e^{9} + \frac{52}{25}e^{8} - \frac{257}{25}e^{7} - \frac{558}{25}e^{6} + \frac{832}{25}e^{5} + \frac{1669}{25}e^{4} - \frac{796}{25}e^{3} - \frac{274}{5}e^{2} + \frac{46}{25}e + \frac{148}{25}$
47 $[47, 47, \frac{1}{9}w^{3} + \frac{1}{3}w^{2} - \frac{14}{9}w - \frac{13}{9}]$ $\phantom{-}\frac{4}{25}e^{9} - \frac{24}{25}e^{8} - \frac{81}{25}e^{7} + \frac{256}{25}e^{6} + \frac{536}{25}e^{5} - \frac{768}{25}e^{4} - \frac{1253}{25}e^{3} + \frac{118}{5}e^{2} + \frac{758}{25}e + \frac{39}{25}$
47 $[47, 47, -\frac{1}{9}w^{3} + \frac{2}{3}w^{2} + \frac{5}{9}w - \frac{23}{9}]$ $-\frac{19}{25}e^{9} - \frac{46}{25}e^{8} + \frac{206}{25}e^{7} + \frac{504}{25}e^{6} - \frac{646}{25}e^{5} - \frac{1592}{25}e^{4} + \frac{698}{25}e^{3} + \frac{302}{5}e^{2} - \frac{363}{25}e - \frac{174}{25}$
47 $[47, 47, \frac{1}{9}w^{3} + \frac{1}{3}w^{2} - \frac{14}{9}w - \frac{31}{9}]$ $\phantom{-}\frac{1}{5}e^{9} + \frac{1}{5}e^{8} - \frac{17}{5}e^{7} - 3e^{6} + \frac{99}{5}e^{5} + \frac{76}{5}e^{4} - 47e^{3} - 28e^{2} + \frac{207}{5}e + 11$
49 $[49, 7, -\frac{4}{9}w^{3} + \frac{2}{3}w^{2} + \frac{38}{9}w - \frac{29}{9}]$ $-\frac{22}{25}e^{9} - \frac{28}{25}e^{8} + \frac{248}{25}e^{7} + \frac{287}{25}e^{6} - \frac{848}{25}e^{5} - \frac{791}{25}e^{4} + \frac{1044}{25}e^{3} + \frac{126}{5}e^{2} - \frac{494}{25}e - \frac{147}{25}$
49 $[49, 7, \frac{4}{9}w^{3} - \frac{2}{3}w^{2} - \frac{38}{9}w + \frac{11}{9}]$ $\phantom{-}\frac{4}{5}e^{9} + \frac{4}{5}e^{8} - \frac{48}{5}e^{7} - 9e^{6} + \frac{186}{5}e^{5} + \frac{149}{5}e^{4} - 56e^{3} - 34e^{2} + \frac{148}{5}e + 8$
89 $[89, 89, \frac{20}{9}w^{3} - \frac{16}{3}w^{2} - \frac{208}{9}w + \frac{325}{9}]$ $-\frac{12}{25}e^{9} - \frac{48}{25}e^{8} + \frac{123}{25}e^{7} + \frac{497}{25}e^{6} - \frac{358}{25}e^{5} - \frac{1401}{25}e^{4} + \frac{439}{25}e^{3} + \frac{201}{5}e^{2} - \frac{349}{25}e + \frac{68}{25}$
89 $[89, 89, \frac{2}{3}w^{3} - w^{2} - \frac{25}{3}w + \frac{7}{3}]$ $\phantom{-}\frac{17}{25}e^{9} + \frac{43}{25}e^{8} - \frac{193}{25}e^{7} - \frac{477}{25}e^{6} + \frac{678}{25}e^{5} + \frac{1516}{25}e^{4} - \frac{899}{25}e^{3} - \frac{266}{5}e^{2} + \frac{409}{25}e - \frac{38}{25}$
89 $[89, 89, -\frac{2}{3}w^{3} + w^{2} + \frac{25}{3}w - \frac{19}{3}]$ $-\frac{16}{25}e^{9} + \frac{6}{25}e^{8} + \frac{184}{25}e^{7} - \frac{94}{25}e^{6} - \frac{619}{25}e^{5} + \frac{487}{25}e^{4} + \frac{547}{25}e^{3} - \frac{152}{5}e^{2} + \frac{93}{25}e + \frac{89}{25}$
89 $[89, 89, -2w^{3} + 3w^{2} + 23w - 11]$ $\phantom{-}\frac{6}{25}e^{9} - \frac{11}{25}e^{8} - \frac{59}{25}e^{7} + \frac{109}{25}e^{6} + \frac{129}{25}e^{5} - \frac{277}{25}e^{4} + \frac{133}{25}e^{3} + \frac{2}{5}e^{2} - \frac{338}{25}e + \frac{321}{25}$
103 $[103, 103, \frac{8}{9}w^{3} - \frac{4}{3}w^{2} - \frac{94}{9}w + \frac{67}{9}]$ $-\frac{23}{25}e^{9} - \frac{22}{25}e^{8} + \frac{312}{25}e^{7} + \frac{248}{25}e^{6} - \frac{1432}{25}e^{5} - \frac{874}{25}e^{4} + \frac{2676}{25}e^{3} + \frac{239}{5}e^{2} - \frac{1771}{25}e - \frac{463}{25}$
103 $[103, 103, -\frac{38}{9}w^{3} + \frac{28}{3}w^{2} + \frac{406}{9}w - \frac{541}{9}]$ $-\frac{9}{5}e^{9} - \frac{14}{5}e^{8} + \frac{108}{5}e^{7} + 30e^{6} - \frac{426}{5}e^{5} - \frac{449}{5}e^{4} + 141e^{3} + 81e^{2} - \frac{433}{5}e - 22$
Display number of eigenvalues

Atkin-Lehner eigenvalues

Norm Prime Eigenvalue
$9$ $[9,3,-\frac{1}{3}w^{3} + \frac{11}{3}w + \frac{7}{3}]$ $-1$