Properties

Base field 3.3.321.1
Weight [2, 2, 2]
Level norm 41
Level $[41, 41, -2w^{2} + 3w + 6]$
Label 3.3.321.1-41.1-b
Dimension 8
CM no
Base change no

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Base field 3.3.321.1

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

Form

Weight [2, 2, 2]
Level $[41, 41, -2w^{2} + 3w + 6]$
Label 3.3.321.1-41.1-b
Dimension 8
Is CM no
Is base change no
Parent newspace dimension 11

Hecke eigenvalues ($q$-expansion)

The Hecke eigenvalue field is $\Q(e)$ where $e$ is a root of the defining polynomial:
\(x^{8} - 5x^{7} - 3x^{6} + 39x^{5} - 13x^{4} - 89x^{3} + 45x^{2} + 53x - 22\)

  Show full eigenvalues   Hide large eigenvalues

Norm Prime Eigenvalue
3 $[3, 3, w + 1]$ $\phantom{-}e$
3 $[3, 3, w - 1]$ $-\frac{1}{23}e^{7} + \frac{8}{23}e^{6} + \frac{2}{23}e^{5} - \frac{68}{23}e^{4} - \frac{13}{23}e^{3} + \frac{174}{23}e^{2} + \frac{54}{23}e - \frac{100}{23}$
7 $[7, 7, w^{2} - 2]$ $\phantom{-}\frac{1}{23}e^{7} - \frac{8}{23}e^{6} - \frac{2}{23}e^{5} + \frac{68}{23}e^{4} - \frac{10}{23}e^{3} - \frac{128}{23}e^{2} + \frac{38}{23}e + \frac{54}{23}$
8 $[8, 2, 2]$ $\phantom{-}\frac{4}{23}e^{7} - \frac{9}{23}e^{6} - \frac{54}{23}e^{5} + \frac{111}{23}e^{4} + \frac{190}{23}e^{3} - \frac{328}{23}e^{2} - \frac{170}{23}e + \frac{193}{23}$
11 $[11, 11, -w^{2} + w + 1]$ $\phantom{-}\frac{6}{23}e^{7} - \frac{25}{23}e^{6} - \frac{12}{23}e^{5} + \frac{132}{23}e^{4} - \frac{60}{23}e^{3} - \frac{170}{23}e^{2} + \frac{90}{23}e + \frac{48}{23}$
23 $[23, 23, -w - 3]$ $-\frac{9}{23}e^{7} + \frac{26}{23}e^{6} + \frac{87}{23}e^{5} - \frac{198}{23}e^{4} - \frac{324}{23}e^{3} + \frac{370}{23}e^{2} + \frac{394}{23}e - \frac{118}{23}$
29 $[29, 29, -w^{2} + 2w + 4]$ $\phantom{-}\frac{2}{23}e^{7} - \frac{16}{23}e^{6} - \frac{4}{23}e^{5} + \frac{136}{23}e^{4} + \frac{3}{23}e^{3} - \frac{302}{23}e^{2} - \frac{39}{23}e + \frac{154}{23}$
31 $[31, 31, 2w - 3]$ $\phantom{-}\frac{8}{23}e^{7} - \frac{41}{23}e^{6} - \frac{16}{23}e^{5} + \frac{268}{23}e^{4} - \frac{80}{23}e^{3} - \frac{403}{23}e^{2} + \frac{120}{23}e + \frac{64}{23}$
41 $[41, 41, -2w^{2} + 3w + 6]$ $\phantom{-}1$
43 $[43, 43, w^{2} - 3w + 3]$ $\phantom{-}\frac{6}{23}e^{7} - \frac{25}{23}e^{6} - \frac{58}{23}e^{5} + \frac{224}{23}e^{4} + \frac{216}{23}e^{3} - \frac{469}{23}e^{2} - \frac{186}{23}e + \frac{140}{23}$
47 $[47, 47, w^{2} + w - 4]$ $\phantom{-}\frac{13}{23}e^{7} - \frac{58}{23}e^{6} - \frac{49}{23}e^{5} + \frac{401}{23}e^{4} - \frac{15}{23}e^{3} - \frac{790}{23}e^{2} + \frac{149}{23}e + \frac{334}{23}$
49 $[49, 7, 2w^{2} - 3w - 3]$ $-\frac{18}{23}e^{7} + \frac{75}{23}e^{6} + \frac{82}{23}e^{5} - \frac{511}{23}e^{4} - \frac{50}{23}e^{3} + \frac{947}{23}e^{2} - \frac{132}{23}e - \frac{374}{23}$
53 $[53, 53, w^{2} - 3w - 2]$ $\phantom{-}\frac{3}{23}e^{7} - \frac{24}{23}e^{6} + \frac{17}{23}e^{5} + \frac{158}{23}e^{4} - \frac{122}{23}e^{3} - \frac{292}{23}e^{2} - \frac{1}{23}e + \frac{162}{23}$
59 $[59, 59, 2w^{2} - w - 5]$ $-\frac{11}{23}e^{7} + \frac{42}{23}e^{6} + \frac{68}{23}e^{5} - \frac{334}{23}e^{4} - \frac{97}{23}e^{3} + \frac{764}{23}e^{2} + \frac{65}{23}e - \frac{272}{23}$
59 $[59, 59, w^{2} - w - 7]$ $\phantom{-}\frac{18}{23}e^{7} - \frac{75}{23}e^{6} - \frac{128}{23}e^{5} + \frac{626}{23}e^{4} + \frac{280}{23}e^{3} - \frac{1315}{23}e^{2} - \frac{144}{23}e + \frac{420}{23}$
59 $[59, 59, -w^{2} - w + 7]$ $\phantom{-}\frac{5}{23}e^{7} + \frac{6}{23}e^{6} - \frac{125}{23}e^{5} + \frac{18}{23}e^{4} + \frac{640}{23}e^{3} - \frac{180}{23}e^{2} - \frac{799}{23}e + \frac{132}{23}$
67 $[67, 67, 2w^{2} - 3w - 7]$ $-\frac{12}{23}e^{7} + \frac{50}{23}e^{6} + \frac{70}{23}e^{5} - \frac{379}{23}e^{4} - \frac{110}{23}e^{3} + \frac{800}{23}e^{2} - \frac{88}{23}e - \frac{372}{23}$
73 $[73, 73, -w^{2} + 4w - 5]$ $-\frac{9}{23}e^{7} + \frac{26}{23}e^{6} + \frac{87}{23}e^{5} - \frac{198}{23}e^{4} - \frac{278}{23}e^{3} + \frac{278}{23}e^{2} + \frac{256}{23}e + \frac{112}{23}$
79 $[79, 79, w^{2} - 8]$ $\phantom{-}\frac{6}{23}e^{7} - \frac{2}{23}e^{6} - \frac{104}{23}e^{5} + \frac{40}{23}e^{4} + \frac{492}{23}e^{3} - \frac{78}{23}e^{2} - \frac{738}{23}e - \frac{44}{23}$
79 $[79, 79, w^{2} - 5w + 5]$ $\phantom{-}\frac{8}{23}e^{7} - \frac{18}{23}e^{6} - \frac{62}{23}e^{5} + \frac{38}{23}e^{4} + \frac{288}{23}e^{3} + \frac{149}{23}e^{2} - \frac{570}{23}e - \frac{120}{23}$
Display number of eigenvalues

Atkin-Lehner eigenvalues

Norm Prime Eigenvalue
41 $[41, 41, -2w^{2} + 3w + 6]$ $-1$