Properties

Label 2.2.205.1-9.1-d
Base field \(\Q(\sqrt{205}) \)
Weight $[2, 2]$
Level norm $9$
Level $[9, 3, 3]$
Dimension $10$
CM no
Base change yes

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Base field \(\Q(\sqrt{205}) \)

Generator \(w\), with minimal polynomial \(x^{2} - x - 51\); narrow class number \(4\) and class number \(2\).

Form

Weight: $[2, 2]$
Level: $[9, 3, 3]$
Dimension: $10$
CM: no
Base change: yes
Newspace dimension: $56$

Hecke eigenvalues ($q$-expansion)

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

\(x^{10} - 11x^{9} + 24x^{8} + 128x^{7} - 584x^{6} + 185x^{5} + 1909x^{4} - 1560x^{3} - 1392x^{2} + 224x + 164\)

  Show full eigenvalues   Hide large eigenvalues

Norm Prime Eigenvalue
3 $[3, 3, w]$ $\phantom{-}1$
3 $[3, 3, w + 2]$ $\phantom{-}1$
4 $[4, 2, 2]$ $\phantom{-}e$
5 $[5, 5, -w + 8]$ $\phantom{-}\frac{23}{34}e^{9} - \frac{171}{34}e^{8} - \frac{43}{34}e^{7} + \frac{1340}{17}e^{6} - \frac{1942}{17}e^{5} - \frac{7919}{34}e^{4} + \frac{12891}{34}e^{3} + \frac{5959}{34}e^{2} - \frac{927}{17}e - \frac{329}{17}$
7 $[7, 7, w + 1]$ $\phantom{-}\frac{4}{17}e^{9} - \frac{29}{17}e^{8} - \frac{29}{34}e^{7} + \frac{472}{17}e^{6} - \frac{572}{17}e^{5} - \frac{1613}{17}e^{4} + \frac{2015}{17}e^{3} + \frac{3463}{34}e^{2} - \frac{301}{17}e - \frac{195}{17}$
7 $[7, 7, w + 5]$ $\phantom{-}\frac{4}{17}e^{9} - \frac{29}{17}e^{8} - \frac{29}{34}e^{7} + \frac{472}{17}e^{6} - \frac{572}{17}e^{5} - \frac{1613}{17}e^{4} + \frac{2015}{17}e^{3} + \frac{3463}{34}e^{2} - \frac{301}{17}e - \frac{195}{17}$
13 $[13, 13, w + 3]$ $-\frac{27}{68}e^{9} + \frac{95}{34}e^{8} + 2e^{7} - \frac{93}{2}e^{6} + \frac{1641}{34}e^{5} + \frac{11663}{68}e^{4} - \frac{3041}{17}e^{3} - \frac{7241}{34}e^{2} + \frac{484}{17}e + \frac{512}{17}$
13 $[13, 13, w + 9]$ $-\frac{27}{68}e^{9} + \frac{95}{34}e^{8} + 2e^{7} - \frac{93}{2}e^{6} + \frac{1641}{34}e^{5} + \frac{11663}{68}e^{4} - \frac{3041}{17}e^{3} - \frac{7241}{34}e^{2} + \frac{484}{17}e + \frac{512}{17}$
17 $[17, 17, w]$ $-\frac{79}{68}e^{9} + \frac{142}{17}e^{8} + \frac{73}{17}e^{7} - \frac{4589}{34}e^{6} + \frac{5569}{34}e^{5} + \frac{30571}{68}e^{4} - \frac{19303}{34}e^{3} - \frac{15661}{34}e^{2} + \frac{1214}{17}e + \frac{1033}{17}$
17 $[17, 17, w + 16]$ $-\frac{79}{68}e^{9} + \frac{142}{17}e^{8} + \frac{73}{17}e^{7} - \frac{4589}{34}e^{6} + \frac{5569}{34}e^{5} + \frac{30571}{68}e^{4} - \frac{19303}{34}e^{3} - \frac{15661}{34}e^{2} + \frac{1214}{17}e + \frac{1033}{17}$
31 $[31, 31, -w - 4]$ $-\frac{9}{17}e^{9} + \frac{67}{17}e^{8} + \frac{14}{17}e^{7} - \frac{1039}{17}e^{6} + \frac{1563}{17}e^{5} + \frac{2921}{17}e^{4} - \frac{5170}{17}e^{3} - \frac{1739}{17}e^{2} + \frac{903}{17}e + \frac{117}{17}$
31 $[31, 31, w - 5]$ $-\frac{9}{17}e^{9} + \frac{67}{17}e^{8} + \frac{14}{17}e^{7} - \frac{1039}{17}e^{6} + \frac{1563}{17}e^{5} + \frac{2921}{17}e^{4} - \frac{5170}{17}e^{3} - \frac{1739}{17}e^{2} + \frac{903}{17}e + \frac{117}{17}$
41 $[41, 41, 3w - 22]$ $\phantom{-}\frac{63}{34}e^{9} - \frac{229}{17}e^{8} - \frac{96}{17}e^{7} + \frac{3659}{17}e^{6} - \frac{4767}{17}e^{5} - \frac{23347}{34}e^{4} + 966e^{3} + \frac{10719}{17}e^{2} - \frac{2774}{17}e - 76$
47 $[47, 47, w + 19]$ $\phantom{-}\frac{1}{34}e^{9} - \frac{13}{34}e^{8} + \frac{43}{34}e^{7} + \frac{54}{17}e^{6} - \frac{429}{17}e^{5} + \frac{899}{34}e^{4} + \frac{2417}{34}e^{3} - \frac{3747}{34}e^{2} - \frac{519}{17}e + \frac{248}{17}$
47 $[47, 47, w + 27]$ $\phantom{-}\frac{1}{34}e^{9} - \frac{13}{34}e^{8} + \frac{43}{34}e^{7} + \frac{54}{17}e^{6} - \frac{429}{17}e^{5} + \frac{899}{34}e^{4} + \frac{2417}{34}e^{3} - \frac{3747}{34}e^{2} - \frac{519}{17}e + \frac{248}{17}$
53 $[53, 53, w + 14]$ $-\frac{111}{68}e^{9} + \frac{205}{17}e^{8} + \frac{117}{34}e^{7} - \frac{6433}{34}e^{6} + \frac{9167}{34}e^{5} + \frac{38159}{68}e^{4} - \frac{30615}{34}e^{3} - \frac{7188}{17}e^{2} + \frac{2529}{17}e + \frac{851}{17}$
53 $[53, 53, w + 38]$ $-\frac{111}{68}e^{9} + \frac{205}{17}e^{8} + \frac{117}{34}e^{7} - \frac{6433}{34}e^{6} + \frac{9167}{34}e^{5} + \frac{38159}{68}e^{4} - \frac{30615}{34}e^{3} - \frac{7188}{17}e^{2} + \frac{2529}{17}e + \frac{851}{17}$
59 $[59, 59, -w - 10]$ $\phantom{-}\frac{43}{34}e^{9} - \frac{315}{34}e^{8} - \frac{57}{17}e^{7} + \frac{2495}{17}e^{6} - 198e^{5} - \frac{15393}{34}e^{4} + \frac{22549}{34}e^{3} + \frac{6505}{17}e^{2} - 77e - \frac{702}{17}$
59 $[59, 59, w - 11]$ $\phantom{-}\frac{43}{34}e^{9} - \frac{315}{34}e^{8} - \frac{57}{17}e^{7} + \frac{2495}{17}e^{6} - 198e^{5} - \frac{15393}{34}e^{4} + \frac{22549}{34}e^{3} + \frac{6505}{17}e^{2} - 77e - \frac{702}{17}$
61 $[61, 61, 2w - 13]$ $-\frac{23}{34}e^{9} + \frac{88}{17}e^{8} + \frac{4}{17}e^{7} - \frac{1346}{17}e^{6} + 130e^{5} + \frac{7117}{34}e^{4} - \frac{7165}{17}e^{3} - \frac{1645}{17}e^{2} + 74e + \frac{145}{17}$
Display number of eigenvalues

Atkin-Lehner eigenvalues

Norm Prime Eigenvalue
$3$ $[3, 3, w]$ $-1$
$3$ $[3, 3, w + 2]$ $-1$